Thomas Dittrich, Peter Hanggi, Gert-Ludwig Ingold,
Bernhard Kramer, Gerd Schon and Wilhelm Zwerger
Quantum Transport
and Dissipation
№ Cote!
Xnetitut Henri Poincnrtf
* BIBUOTHfeQUE
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№ Inventaire :
® WILEY-VCH
Weinheim • Berlin • New York • Chichester
Brisbane • Singapore • Toronto
Priv.-Doz. Dr. Thomas Dittrich
Max-Planck-Institut
fur Physik komplexer Systeme
Bayreuther StraBe 40, Haus 16
D-01187 Dresden
Prof. Dr. Gert-Ludwig Ingold
Institut fur Physik
Universitat Augsburg
Memminger StraBe 6
D-86135 Augsburg
Prof. Dr. Gerd Schon
Institut fiir Theoretische Festkbrperphysik
Universitat Karlsruhe
EngesserstraBe 7
D-76128 Karlsruhe
Prof. Dr. Peter Hanggi
Institut fiir Physik
Universitat Augsburg
Memminger StraBe 6
D-86135 Augsburg
Prof. Dr. Bernhard Kramer
1. Institut fiir Theoretische Physik
Universitat Hamburg
JungiusstraBe 9
D-20355 Hamburg
Prof. Dr. Wilhelm Zwcrgcr
Sektion Physik
Universitat Munchen
TheresienstraBe 37
D-80333 Munchen
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V
Preface
In recent years, systems intermediate between macroscopic and microscopic scales,
where q lantum coherence becomes important, have developed into a main area nf
research in modern condensed-matter physics. The notion of “mesoscopic system:,
has beer coined to characterize this field. One of its fascinating aspects is th'1'
connects quite different subjects ranging from quantum transport theory to qua:J uni
chaos. Here, technical applications, for instance single-electron transistors, are close;
tied to fundamental questions in quantum and statistical mechanics. As a result o!
the rapid growth of the literature, it has become very difficult for young scientists to
obtain ar overview of the foundations of this interdisciplinary field.
The aim of this book is to provide an introduction into some of the theoretical
aspects. At the same time, we intend to guide towards the forefront of ongoing research
in an im nensely vivid subject. This goal is almost impossible to achieve1. We can only
hope that the compromise attempted here will prove successful.
In order to cover the wide range of topics, the book has been written by several
authors. Each of the six chapters is aimed at a reasonably self-contained introduction
into t he principal issues of a particular subfield, which we believe to be representative.
The first chapter deals with the basic notions in quantum transport theory, including
weak localization, universal conductance fluctuations and the Landauer-Biittiker scat-
tering formalism. It also provides some background for diagrammatic techniques in
weakly disordered systems. The1 second chapter introduces the theory of the quantum
Hall effect, both integer and fractional. The concept of localization as well as Laugh-
lin’s theory of the effect of interactions in tin1 fractional case1 are treated here on an
elementary level. Quantization of conductance in quantum wires, including the effects
of impurity scattering and interactions, is outlined here. The subject of the following
chapter ire single-electron tunneling and Coulomb-blockade phenomena for metallic
and supc rconducting junction devices. Beyond the standard rate equation, it contains
a discussion of the effect of the electromagnetic environment and of nonperturbative
effects.
In ch ipter four, concepts for the1 description of dissipation in quantum systems, in
particular the role of the coupling to a heat bath, are explained. This chapter als11
includes the Fevnman path-integral formulation via influence functionals and applies
this formalism to dissipative tunneling. The fifth chapter is devoted to the theory
strongly driven quantum systems. On basis of the general Floquet theory, archetypal
problenu like driven tunneling together with its dynamical destruction in a dri-
double well and the control of a quantum dynamics via a sequence of coherent laser
pulses are discussed there. Quantum chaos and its relation to transport phenomena a
treated i i the final chapter. The main topics are semiclassical methods, the interplay
chaos wi h tunneling and localization, and quantum chaos in scattering and dissipative
systems.
The present volume introduces many of the basic ideas and techniques in this field
which me necessary to master the more specialized original literature1. As such it is
mainlv ; ddressed to those who intend to start research in this very active area. It
VI
should also be useful for physicists in other fields who would like to learn about this
subject or to experimentalists for a better understanding of the crucial theoretical
concepts. It is obvious that many more interesting topics like persistent currents and
the physics of semiconductor quantum dots could not be included or could only be
touched very briefly. Nevertheless, we are confident that the present volume reflects at
least the cornerstones of the physics of mesoscopic systems and conveys the excitement
in this area of research.
We would like to thank many colleagues and friends who contributed by numer-
ous constructive criticisms to the book. In particular we mention Christoph Bruder,
Giuseppe Falci, Rosario Fazio, Andrea Fechner, Milena Grifoni, Frank Grofimann,
Gregor Hackenbroich, Frank Hekking, Sigmund Kohler, Jiirgen Konig, Stefan Linz,
Peter Reimann, Maura Sassetti, Herbert Schoeller, Jens Siewert, Lydia Sohn, Sigmund
Stintzing, Ralf Utermann, David Wharam, and Andrei Zaikin for many stimulating
discussions and for critically reading parts of the manuscript. Sigmund Kohler was
also of great help in the preparation of the camera-ready version of the book.
The idea for this book has grown out of a summer school held at the Physikalisch-
Technische Bundesanstalt in Braunschweig several years ago. Without the generous
support and hospitality of this institution, the project would never have been started.
Support by the European Union within SCIENCE, HCM, and TMR networks and
by the Deutsche Forschungsgemeinschaft is gratefully acknowledged by several of the
authors. The discussions within these networks, especially with young European col-
leagues active in the field, helped considerably to motivate the completion of this
volume. One of the authors (G.S.) also acknowledges the support by an Alexander-
von-Humboldt award of the Academy of Finland and the hospitality of the Helsinki
University of Technology, where chapter three could be finished.
May 1997
Thomas Dittrich
Gert-Ludwig Ingold
Peter Hanggi
Bernhard Kramer
Gerd Schon
Wilhelm Zwerger
vji
Contents
1	Theory of Coherent Transport	1
1.1	Introduction......................................................... 1
1.2	Scattering and Linear	Response....................................... 7
1.2.1	Disorder, density of states and scattering.................... 8
1.2.2	Green functions.............................................. 13
1.2.3	Kubo-Greenwood formula...................................... 2 ,
1.3	Weak Localization .................................................. 24
1.3.1	Coherent Backscattering...................................... 21
1.3.2	Aharonov-Bohm interference effects.......................... 3 1
1.3.3	Dephasing................................................... I1'
1.4	Universal Conductance	Fluctuations ................................
1.4.1 Self-averaging................................................ 45
1.4.2 Diagrammatic theory.......................................... 1 ~
1.4.3	Random matrix theory........................................
1.5	The Landauer-Biittiker formalism.................................... 60
1.5.1 Landauer formula.............................................. 61
1.5.2	Scattering theory of multiprobe conductances ................ 64
1.5.3 Symmetry of electrical conduction............................. 72
References............................................................... 74
2	Quantization of Transport	79
2.1	Moninteracting electrons in reduced dimensions ..................... 79
2.1.1 The Landau model.............................................. 79
2.1.2 Confinement and magnetic held................................. 82
2.1.3 Influence of impurities....................................... 88
'2.1.4 Percolation limit and localization........................... 94
2.2	The integer quantum Hall effect.................................... 100
2.2.1	Experiment.................................................. 100
2.2.2	Electrical conductivity..................................... 102
2.2.3	Understanding the integer quantum Hall	effect............... 105
2.3	Coulomb interaction and magnetic held.............................. 110
2.3.1	The fractional quantum Hall effect ......................... 110
2.3.2	Few electrons with Coulomb interact ion	.................... HI
2.3.3	Fractional quantum Hall states ............................. 11
2.4	Conductance quantization........................................... 12o
2.4.1	Experimental results........................................ 125
2.4.2	Conductance quantization in ideal quantum	wires............. 12
viii
2.4.3	Adiabatic constrictions......................................... 128
2.4.4	Disorder......................................................... 132
2.4.5	Electron-phonon interaction..................................... 134
2.4.6	Electron-electron interaction................................... 135
2.5	Conclusion............................................................. 143
References.................................................................. 143
3	Single-Electron Tunneling	149
3.1	Introduction............................................................149
3.2	Charging energy and single-electron devices ........................... 151
3.2.1 The energy scale ................................................ 152
3.2.2	Single-electron box............................................. 152
3.2.3	Single-electron transistor...................................... 154
3.3	Tunneling rates and I-V characteristics................................ 156
3.3.1	The single-electron tunneling rate............................... 156
3.3.2	Master equation for sequential tunneling..........................158
3.3.3	Cotunneling processes ............................................161
3.3.4	Broadening of the steps ..........................................162
3.4	Influence of the electromagnetic environment........................... 163
3.4.1	The model Hamiltonian............................................ 163
3.4.2	The single-electron tunneling rate................................166
3.4.3	General properties .............................................. 167
3.4.4	The effect of an Ohmic resistor ................................. 169
3.4.5	Other environments............................................... 171
3.5	Charging effects and superconductivity..................................172
3.5.1	Charging effects on quasiparticle tunneling.......................173
3.5.2	Two-electron tunneling, Andreev reflection....................... 174
3.5.3	Parity effects in small superconductors.......................... 176
3.5.4	I-V characteristics of NSN transistors........................... 180
3.5.5	Coherent Cooper-pair tunneling................................... 183
3.5.6	Andreev spectroscopy of Josephson tunneling ......................184
3.5.7	Incoherent Cooper-pair tunneling................................. 188
3.6	Effective-action description........................................... 189
3.6.1	The effective action in imaginary times.......................... 189
3.6.2	Single-particle and Cooper-pair tunneling........................ 190
3.6.3	Higher-order processes........................................... 192
3.6.4	Josephson current through SNS transistors.........................194
3.6.5	Proximity effect..................................................196
3.7	Real-time evolution of the density matrix.............................. 198
3.7.1	Phase representation............................................. 199
3.7.2	Charge representation ............................................200
3.7.3	Diagrams and rules................................................201
3.7.4	Simple examples, SET and cotunneling.............................2
3.7.5	Resonant tunneling................................................2:!
3.7.6	The current.......................................................20C
3.7.7	Charge fluctuations in the single-electron	box ...................207
3.7.8	Conductance oscillations in the SET transistor....................208
3.8	Outlook..................................................................209
References.....................................................................210
4	Dissipative Quantum Systems	213
4.1	Introduction...........................................................213
4.2	Description of dissipation	in quantum mechanics........................214
4.2.1	Hamiltonian for system and heat	bath ..........................214
4.2.2	Elimination of the heat bath....................................215
4.2.3	Spectral density of bath modes	..............................216
4.2.4	Rubin model ....................................................217
4.3	Density matrices.......................................................219
4.4	Linear damped systems..................................................221
4.4.1	Response function...............................................221
4.4.2	Fluctuation-dissipation theorem................................22 -
4.4.3	Correlation functions of the damped	harmonic oscillator .... 22~
4.4.4	Free damped particle...........................................2'28
4.5	Short introduction to path integrals...................................2
4.6	Dissipation within the path-integral formalism.............................
4.6.1	Influence functional............................................2'2
4.6.2	Elimination of the heat bath....................................222
4.6.3	Evaluation of the fluctuations..................................2 2
4.6.4	Effective action................................................2.5'1
4.7	Decay of a metastable state ...........................................2•!'>
4.7.1	Crossover temperature...........................................21!
4.7.2	Imaginary part of the free energy...............................2 13
4.7.3	Above crossover.................................................2 >
4.7.4	Zero temperature................................................2 15
References..................................................................217
5	Driven Quantum Systems
249
5.1	Introduction..........................................................249
5.2	Time-dependent interactions...........................................256
5.2.1 Laser interactions.............................................T-
5.2.2 Spin magnetic resonance........................................251
5.3	Floquet and generalized Floquet theory................................2 ’
X
5.3.1	Floquet theory ................................................252
5.3.2	General properties of Floquet theory...........................254
5.3.3	Time-evolution operators for Floquet Hamiltonians..............256
5.3.4	Generalized Floquet theory ....................................258
5.3.5	The (t,	 259
5.4	Exactly solvable driven quantum systems..............................261
5.4.1	Driven quantum oscillators.....................................261
5.4.2	Periodically driven two-level systems..........................263
5.4.3 Quantum systems driven by circularly polarized fields...........268
5.5	Numerical approaches to periodically driven quantum systems.........270
5.5.1	Method of Floquet matrix.......................................270
5.5.2	Matrix-continued-fraction method...............................271
5.6	Coherent tunneling in driven bistable systems........................273
5.6.1 Limits of slow and fast driving.................................275
5.6.2 Driven tunneling near a resonance...............................276
5.6.3	Coherent destruction of tunneling..............................277
5.6.4	Two-state approximation to driven tunneling....................280
5.7	Laser control of quantum dynamics..................................  281
5.8	Conclusions and outlook..............................................283
References.................................................................284
6	Chaos, Coherence, and Dissipation	287
6.1	Introduction.........................................................287
6.2	Quasiclassical chaos.................................................290
6.2.1	Time-domain aspects............................................290
6.2.2	Energy-domain aspects..........................................303
6.3	Chaos and quantum coherence..........................................317
6.3.1	Chaotic tunneling..............................................317
6.3.2	Quantum chaos in extended systems .............................323
6.4	Quantum chaos in open systems........................................333
6.4.1	Chaotic scattering ............................................333
6.4.2	Quantum chaos with dissipation.................................341
6.5	Conclusion...........................................................350
References.................................................................352
Author Index	359
Subject Index	367
Wilhelm Zwerger
Theory of Coherent Transport
1.1 Introduction
Traditionally the theory of conduction in degenerate electron systems like metals is
based or the Drude-Sommerfeld model and the semiclassical Boltzmann equation [1].
In its most simple version this theory states that conduction electrons with density n
and effective mass M respond to a de electric field E(x) with a current density ji(x)
which is local and linear in the field to lowest order
j(x) = aE(x)	(1.1)
provided E(x) varies slowly on lengths comparable to the mean free; path. The ass*
ciated conductivity tensor cr is an example of a linear kinetic coefficient which obeys
the general Onsager-Casimir symmetry relations [2]
<7y(B) = <7Jt(-B)	(l.'.i)
of noneq lilibrium thermodynamics in the presence of a magnetic field B. In the sim
plest case of an isotropic material and В = 0 the conductivity tensor is proportion;* 1
to the in it tensor cr = al with
Here rlr is a transport collision time which has to be calculated microscopically by
solving the Boltzmann equation with an appropriate collision term [1]. The basic
assumptions which enter into the derivation of these standard results are the following:
(i) between collisions the electrons move like classical particles being affected only
by he external fields but not by electron-electron interactions.
(ii) collisions are instantaneous events, ocurring statistically with probability 1/r,,
per unit time. In particular successive collisions an1 assumed to be independent
1'21.
2 Theory of Coherent Transport
Now in reality electrons are neither classical nor noninteracting and — as we will see
— there is no reason that successive collisions are independent, provided they are
caused by a static scattering potential. It is therefore very surprising that in many
cases these crude assumptions are nevertheless sufficient to describe conduction in real
systems. Let us therefore examine the arguments which are usually given to justify
the assumptions above. Since the electrons which are relevant for conduction are those
close to the Fermi level [1] they have momenta of the order hkp with kp the Fermi
wave vector. According to Ehrenfest’s theorem wave packets move classically provided
that the uncertainty Д/с in wave vector is much smaller than kp itself and the effective
potential varies slowly on the scale of the uncertainty Д:г in position. Taking the mean
free path I = VpTtI as the typical value of Дт, we find that the condition Д/с C kp and
the uncertainty relation ДхД/с >1/2 require that
kFl-»l.	(1.4)
Thus a classical description of electronic motion is expected to be valid provided the
Fermi wavelength Ар = 2тг//ср is much smaller than the average distance between two
scattering events. This remains valid irrespective of whether the size of the scattering
centers is also large compared to Ap or not. In the latter case it is only that the
transition probabilities W(fc -> k'} in the Boltzmann equation [1,2]
/	j
dtf + v\7xf + F- Vnkf = ~ [W(k -+ k')f(k) - W(k' -> k)f(k')] (1.5)
J (2ТГ)'5
for the phase space distribution /(x, k, i) have to be calculated quantum mechanically
(note that the collision term contains no factors 1-/ which suppress scattering into
occupied states [1]. It is therefore linear in the distribution function as it must be
for noriinteracting electrons, irrespective of whether the symmetry IT(fc —> fcz) =
II’(fc' fc) is obeyed or not). The assignment of a classical distribution function to
quantum mechanical particles for which position and momentum cannot have sharp
values simultaneously is possible since in the usual “semiclassical” description of Bloch
electrons [1] one considers wave packets with a spread ДА: kp ~ l/« (a is the
lattice constant). These wave packets have thus both a well defined wave vector and
a well defined position, i.e. Дх is much smaller than the scale over which the external
fields are allowed to vary. As a result the phase space is divided into volume elements
d3xd3p A> Zr3 which contain many microstates and therefore a classical description
is possible [2]. To avoid confusion with our later notation, it should be pointed out
that the notion “semiclassical” for the Boltzmann equation refers to the fact that the
rapidly varying periodic potential of the average ion arrangement is taken into account
in a fully quantum mechanical way. Thus for instance the deviation of .1/ from the
bare electron mass due to bandstructure effects is clearly a quantum effect. However
the applied fields and in particular the motion of the electron in response to them are
treated in a classical manner. This is in marked contrast to the semiclassical theory in
section 1.3 which really includes the phase of the electronic wave function.
The above qualitative arguments show that the assumption of classical electron
motion is justified as long as kyl >> 1. Quantum effects in transport are therefore
1.1 Introduction 3
expected to appear in systems with strong disorder or low densities. Indeed we will see
in section 1.3 that the classical theory is only the leading result in the small parameter
1/kpl < 1. Already the following term in the expansion leads to a decrease of the
conductivity compared with its classical value which is due to quantum interference ef-
fects. Eventually these effects lead to a localization of the electronic states at the Fermi
energy. la the three dimensional case this occurs if kFl is of order one while localization
happens even for arbitrary large values of kPl in one and two dimensions. As a result
the de cc nductivity at zero temperature will vanish identically, a phenomenon which
is called Anderson localization [3]. In the following we will not deal with this very
intricate problem but will consistently remain in the so-called metallic regime, which is
defined b.- the property that the conductance G of a system with size L scales like Ld 2
as expected for an Ohmic conductor (see (1.38), in d = 1,2 a metallic regime at zero
temperature is possible only in systems which are smaller than the localization lengt’
£). A crucial property of transport in the metallic limit is the fact that the motion of
the elections is diffusive. Indeed there is an exact relation between the conductivity
and the ciffusion tensor D which is the well known Einstein relation
This relation is very general and holds both in a fully quantum mechanical treatment
as well as for the interacting electron system. In our discussion here for noninteracting
electrons ,he thermodynamic derivative дп/др. which is generally related to the isother-
mal compressibility kt by dn/d^i = п?к,г is (at T = 0) equal to the density of states
2p(fp) at ' he Fermi level (the factor 2 is due to the spin degeneracy, i.e. p{ep) is the den-
sity of sta ,es per spin, note also that for an ideal classical gas one has дп/др = п/квТ\
The relation (1.6) may be proven by a rather general thermodynamic argument [2]: by
definition of D the particle current density in response to a small gradient in density
is — DVn, while <rE is the electrical current density to linear order in the electric ,
E. Thus f both Vzi and E are nonvanishing, the total electric current density is Uir
sum of the conduction and diffusion current
jtot = <rE + cDW.	(1.7
Now for a charged system in the presence of a field E = — V<A thermodynamic equi-
librium re juires that the electrochemical potential /zcl = /z - < (!) is constant in space.
As a result Jtot has to vanish if V/z = —eE or (др/dn}\hi = —cE provided there is
local thermodynamic equilibrium. Evidently this condition immediately leads to the
Einstein relation (1.6).
A seco id and indeed very subtle assumption usually made in classical transport
theory is to neglect electron-electron interactions. Now for a single band and in the
absence of umklapp processes one expects that collisions between the electrons them-
selves should not affect the conductivity since the total momentum of the electron
system is lonserved. However this argument is only applicable1 in a translation invari-
ant system. In the presenci1 of a finite disorder, however, the effective electron-electron
4 Theory of Coherent Transport
interaction is different from the standard screened interaction of a pure electron gas
usually discussed in the textbooks [1,4]- As a result of this disorder-dependent effec-
tive interaction there are quantum corrections to the classical conductivity which are of
the same order as those arising from quantum interference of noninteracting electrons
[3,5]. Again we will not deal with this rather difficult question here, which is related
to the general problem of determining the range of validity of the Fermi liquid concept
in disordered systems [6]. Instead we will assume throughout that the electrons may
be described in terms of a Fermi liquid. Thus the interacting system is equivalent
to a set of Quasiparticles which behave like free Fermions for low temperatures and
small perturbations [4]. At T — 0 the quasiparticles are filled up to momentum ky
which is related to the total density in precisely the same manner as for noninteracting
electrons. The independent electron approximation is then applicable provided cer-
tain physical properties are renormalized. The standard example is the effective mass,
which is different from the result obtained in a bandstructure calculation with non-
interacting electrons. In principle it is this mass which enters into the simple Drude
expression (1.1) although it is in general very difficult to separate band structure from
many body effects through a measurement of the frequency and wave vector dependent
conductivity [4]. A further effect of the interactions is that the transition probabilities
W(fc —> fc') between momenta к and k’ which determine the collision rate l/rtr are
those1 for quasiparticles rather than bare electrons. As a result it is e.g. the screened
interaction of a charged impurity which enters into the calculation of II’(fc k').
Finally the electric field E occuring in the Boltzmann equation is the effective i.e.
internal electric field acting on the particles which includes the effect of screening of
an externally applied field [4]. Apart from these conceptually very important points,
however, interactions do not affect the basic validity of classical transport theory.
A similar situation appeared to be valid with respect to quantum effects. For a
long time it therefore seemed that the complicated rigorous formulations of transport
like the Kubo formula (see below) merely served to justify the simple minded Boltz-
mann or Drude-Sommerfeld approach. This situation changed drastically in 1979 when
Abrahams, Anderson, Licciardello and Ramakrishnan [7] in their proposal of a scaling
theory of localization suggested the way how to calculate systematically the leading
quantum corrections to the classical conductivity. They were first calculated explicitely
by Gorkov, Larkin and Khrnelnitskii [8] whose work started a new held in the study of
electronic transport which is called weak localization. In its simplest form it shows t hat
the leading correction in order \/kpl to the classical conductivity is due to coherent
backscattering of electrons from momenta к to — к which leads to a decrease of the
conductivity compared to its classical value. Crucial for this idea was the fundamental
recognition that elastic scattering at impurities does not destroy the phase memory
of an electron [9]. The ability to exhibit quantum mechanical interference is therefore
retained for arbitrary many scatterings provided there are no processes which break
the time reversal invariance. At low temperatures the rate l/rv for collisions destroy-
ing phase coherence is generally much smaller than the elastic scattering rate l/~tr.
Therefore the elastic mean free path I becomes much shorter than the phase1 coherence
length L,fi which is defined as the typical distance an electron moves before it looses
1.1 Introduction
•j
phase memory. In such a situation the assumptions (i) and (ii) made in the classical
description of transport are no longer valid: successive collisions are not independent
and the Boltzmann equation which is based on uncorrelated collisions — and thus ei-
fectively assumes that tv and rtr are identical — no longer applies. Thus even for free
and independent electrons with instantaneous collisions and well defined trajectories
there are deviations from the classical picture, which are due to the quantum inter-
ference between successive elastic scattering events. These quantum effects, however,
are observable only at low temperatures T because with increasing T collisions which
destroy the phase memory become more frequent. Eventually, when r,f becomes of
order rtr one recovers the classical behaviour.
While the phenomenon of weak localization displays quantum interference effects
in macroscopic samples another new field where quantum interference shows up in
transport properties was discovered only a few years later in the so-called mesoscopic
systems, in brief a mesoscopic system is one whose typical size L is small enough tha1
electrons can move through the whole sample without loosing phase coherence, i.e. tin
requirement is that [10]
L<LV.	(l.H
Clearly since Lv is in general increasing with decreasing temperature, any system will
eventually become mesoscopic. In practice, however, typical values of in the regime
below one Kelvin or so are of order 1pm. To study mesoscopic phenomena one therefore
needs devices in the micron or submicron range at very low temperatures. One of the
central phenomena discovered in these samples are the so-called universal conductance
fluctuations (UCF). They are apparently random but static (i.e. reproducible 1), sample
specific fluctations of the linear conductance G of a mesoscopic system as a function of
the Fermi energy or a magnetic field. These fluctuations are called universal because
their variance at T = 0
Zc2A2
VarG = (G2)- (G)2 = с I- j	(1.9)
has a universal magnitude with a numerical constant c which only weakly depends on
dimension and sample shape but not on its size or any material parameters. An expla-
nation for this striking behaviour was given by Altshuler, Khmelnitskii, Lee and Stone
[11, 12] wl о showed that the fluctuations arise, from the random quantum interference
between t ie many electron paths which contribute to the conductance in a diffusive
system. Their sample specific nature enables one to use the precise dependence of the
conductance e.g. on the magnetic field as a “fingerprint’’ of a particular sample. More-
over it she ws that the conductance of mesoscopic systems is not self-averaging. What
is meant by this is that the usual assumption that the behaviour of one particular
realization (one hardly performs measurements on an ensemble 1) is identical with the
ensemble averaged behaviour for large system sizes does not hold. Indeed for trans-
port through a static random potential one would naively expect that self-averaging
occurs on scales L I. however this is not the case. Instead, quantum mechanically
6 Theory of Coherent Transport
the electrons “remember” the specific impurity configuration for system sizes up to
and thus self-averaging only sets in beyond the mesoscopic regime, i.e. for L > Lv. A
striking consequence of this phase memory is the observation [13,14] that the motion
of a single scatterer by an amount of order Ap for instance, in a two-dimensional system
is sufficient to change the conductance by an amount
which is independent of the size of the system (n,- is the two-dimensional impurity
concentration).
The realization that transport in mesoscopic systems is phase coherent over the
whole sample shows that a fully quantum mechanical formulation of conduction is re-
quired. In particular, if the wave-like nature of the electrons plays a role, the relation
between the current and the electric field will not be local on scales of order I. It turns
out that the generalization of Ohm's law (1.1) in a rigorous quantum mechanical re-
sponse calculation based on the Kubo formula (or Kubo-Greenwood for noninteracting
electrons) has the form
j(x) = У d3i' cr(x, x')E(x').	(1-11)
Thus although the local current density is linear in the field, the relation is nonlocal on
a scale of order Llf, 3> 1. Moreover it cannot be made local by a Fodrier transformation
as is usually assumed. Indeed the conductivity tensor is a function of both x and x'
and there is no translation invariance for a specific sample unless thermal averaging or
dephasing effectively allow to perform an average over the impurity configuration. Now
in practice one is often not interested in the spatial dependent local conductivity tensor
<t(x, x1) but only in the global conductance G of a specific sample, which is the quantity
actually measured in experiments. Typically such measurements are performed in a so-
called multiprobe geometry. An idealized description of such an experiment consists
of a finite region containing an arbitrary scattering potential which is connected to
n = I...., perfect, semi-infinite leads (see Fig. 1.12 below). The leads are similar
to waveguides for electrons and have discrete modes (or channels) whose energy depends
on the nature of the confining potential. If the voltages V’„ in the asymptotic region of
lead n are not all identical there will be net currents Im flowing out of lead in and the'
general global form of Ohm’s law is given by
/v(
=	(1.12)
71— 1
With the assumption that the transport through the whole structure is completely
coherent, the associated conductance coefficients Gmn can be expressed simply by the
quantum mechanical transmission probabilities through the sample. Such a relation
between conductance and a scattering matrix was originally derived bv Landauer [15].
The final and presently used form of this relation was given by Biittiker [16], incor-
porating the essential point that current and voltage probes have to be treated on
1-2 Scattering and Linear Response
an equal footing. This formulation of transport in mesoscopic structures has lead t-
an understanding of very many phenomena, ranging from symmetries and nor.' .
aspects to a very powerful explanation of the quantum Hall effect.
A particularly nice and simple application of the Landau er formula has been di-
covered in the so-called quantum point contacts (QPC) [17,18]. These are two prob'
structures in which a (usually) two-dimensional electron gas is forced to flow throng!
an electrostatically defined small hole whose width is only of the order of a few Ferm
wavelmgths. Moreover the motion is ballistic rather than diffusive, i.e. the mean fr<<
path ,s much longer than the constriction size and the confining potential is smooi I
enough to eliminate backscattering. In this situation all modes whose transverse eneir
at the minimum width of the contact is below the Fermi energy will be transmitted
probability one while all others cannot get through. The simple two probe Landaue-
formula
with ja as the transmission probability in mode a then leads to the striking pi ’!  '
that the conductance of a quantum point contact
Gqpc =	(- - '
is qua itized in units of the fundamental value 2e2/h = (12.9 kQ) 1 with N the integei
number of open channels. Thus by continuously changing the constriction width via tin
electrostatic confinement, the conductance changes in a step-like fashion with plateau
values as given by (1.14).
The present chapter tries to give a brief introduction to the theoretical description
of the basic phenomena in quantum transport. In general our treatment is restricted
to the most simple cases. Information on more specialized topics can therefore only be
obtained from the original literature (see also the book edited by Altshuler. Lee and
Webb [19] and the recent Les Houches Lectures 1994 [20]). Moreover the important
experi nental aspect is not treated here at all and we refer to the review by Beenakkei
and ven Houten [21] for further information. Nevertheless we hope that this introduc-
tion still captures the essential physical ideas and enough of the formalism needed to
make 1 he original theoretical literature more accessible.
1.2 Scattering and Linear Response
In this section we will introduce the basic model and the formalism to calculate th-
conductivity of noninteracting electrons in a fully quantum mechanical way. For sim
plicity we assume’ that only elastic scattering due to impurities or boundaries is pres-- .
i.e. we are dealing with the1 residual resistance’ at low temperatures. Inelastic scatter11-
e.g. di (’ to phonons is neglected and will later be introduced only in the sense hr
8 Theory of Coherent Transport
it provides a cutoff for coherent propagation of the electrons. Most of our discussion
will be based on the standard Kubo-Greenwood formula for the conductivity in linear
response. However we will also present an elementary application of the alternative
approach to transport proposed by Landauer [15,22]. This approach views the local
electric field as the response to an incoming current rather than calculating the currents
in response to a given electric field as is done in the more familiar Kubo-Greenwood
formulation.
1.2.1 Disorder, density of states and scattering
We consider a system of noninteracting electrons with charge — e (e > 0) which move
in a static potential U(x) and — possibly — a magnetic field В = V x A. The exact
many particle eigenstates may then be obtained by antisymmetrization from the single
particle eigenstates of the Hamiltonian
~(р + еА)2 + Щх).	(1.15)
Z l\i
The bandstructure and many body effects are incorporated here in the effective mass
M. Moreover spin is not explicitely taken into account since in the absence of spin-
dependent interactions it just leads to a degeneracy factor 2 for each state. In principle
the scattering potential U(x) consists of both the confining potential and the disorder
contribution which describes the deviations from perfect periodicity. For the present
discussion let us first assume that the electrons move in an unbounded volume with no
confining potential. An idealized model for the scattering potential describing struc-
tural disorder due to randomly distributed impurities is then given by
tv,	t
U(x) = ^ u(x — Xj).	(1-16)
j=i
This describes a collection of identical impurities with individual scattering potential
u(x) which are located at random positions x3. With пг = Nt/V as the impurity
concentration (which is assumed to be finite in the thermodynamic limit V —> oo)
the average distance f between the scattering centers is equal to ntT1^3. Provided the
range of u(x) is small compared to f, different impurity potentials do not overlap and
therefore the electronic motion separates into successive scatterings at single impurities
to lowest order. In principle, for a given distribution of the x, the diagonalization of
H is a simple one particle problem which leads to a complete and orthonormal set of
eigenstates ipn(x) with eigenvalues ea. In the limit of an infinite volume the spectrum
will be continuous (for simplicity we neglect the possibility of isolated eigenvalues) and
is characterized by a smooth density of states
p(e) = Jim 5Z<5(s - tQ).	(1.17)
-+O° I n
This quantity determines the complete thermodynamic behaviour of the electron sys-
tem from its grand canonical potential Q via
Jim —~ = У <кр(г) ln(l + (AT-P) .	(1.18)
1.2 Scattering and Linear Response П
Knowledge of (1.18) thus allows to calculate the specific heat or other equilibrium
properties of the electron gas in the presence of disorder [23]. In the trivial case of a
vanishing scattering potential the eigenstates are plane waves т/^0>(а:) = exp(ifc®) with
energy £k = Fi2k2/2M. The corresponding free density of states for a d-dime”4;
system is then (using limlz_+oo(l/Vr) — (27r)_rf/drfA:)
~ (1.1;))
J {£71}	fl
with K,i = (2тг2)-1, (27г)"1 or ~ 1 in d — 3, 2,1 respectively and ke = (2Ж)1/2/h the
wavenu.nber for given energy e.
As tn introduction to the problem of what happens at finite disorder, let us first
study the elementary quantum mechanics of scattering from a single impurity. Turning
on a sp.ierically symmetric short range potential u(x) at the origin we know that an
undistorted plane wave ~ exp(ifcx) is transformed into a scattering state ipk(x)
at the same energy. In the three-dimensional case the scattering state behaves asymp-
totical!} like an outgoing spherical wave exp(ib’)/r (r = |x|). Here we will discuss the
more interesting case of two dimensions, where the asymptotic behaviour is a cylindrical
wave [2^]
,	( i \ 1/2
1M®) e'kx + fk(6) -) e'kr .	(1.20)
Here в is the angle between the outgoing wave fc' = kx and the incoming direction
fc while ft is the corresponding dimensionless scattering amplitude. It determines the
differential cross section by the relation a(0) = \fk(d)\2/k [24]. The deviation
from a plane wave which is caused by the scattering gives rise to a spatial dependence
of the probability density |i[>k (x) |2. As a result the total electron density n(x) which
is obtained by adding up the different wave vectors к with a Fermi distribution
/Ы = ------------------7—7	(1.21 I
ехр(Ь(е*. - д)) + 1
via
"(a) = [ 7TTz/[-’*)|tA(a:)i2	(1.2:
./ ( 27Г)
is also nonuniform in the vicinity of the scatterer. It reaches the constant free particle
value' = А,р/4тг (no spin included) only far from the impurity. Indeed, as is wi
known, the sharpness of the Fermi distribution at temperatures small compared to
the Fermi temperature Tp leads to Friedel oscillations in the density with wave vectoi
2k].' [25]. Quite remarkably, recent experiments using a scanning t unneling microscope
(STM) have been able to make, these oscillations visible in real space [26]. As an
example Fig. 1.1 shows an STM image of a two dimensional electron gas due to a
band of surface states in copper. One clearly sees standing waves which are formed
around two impurities at which the electrons are scattered. To understand this pictlire
10 Theory of Coherent Transport
Fig. 1.1: An STM image of standing electron waves at the surface of a copper crystal with two local
defects. Reprinted with permission by D. Eigler.
quantitatively one uses the fact [27] that the local tunneling current between an STM
tip and a conducting sample at T — 0 is directly proportional to the local density of
states
p(x,s) = 52 W1)I2 5(£ - -<»)	(1/23)
* (I
at the Fermi energy Ep. For a single impurity (1.23) may be evaluated in terms of the
exact scattering states 'фк(х). Using <5(£р — Ek) =	— kp)M/h2kp and
(2тг)~2 kdk [d^k
V V Jo J
we find that the local density of states at the Fermi energy
W	p
p(.x,Ep) =	~ / d^Kr(®)|2	(1.24)
(2тг/г)2 J
is just the angular average of the square of the corresponding wave function. Using the
asymptotic form (1.20) and elementary scattering theory in d = 2. the square of the
wave function is asymptotically given by [28]
lim Ь(х)]2 = 1 - — <5(^ - QJ +	+ R.e[AWe2ih’M(^ + «J- (1-25)
kr -w>	Г	Г КГ
1.2 Scattering and Linear Response 11
Here £lk and Slx denote the directions of the unit vectors к and x respectively, while <7tot
is the total scattering cross section. Evidently the second and third term in (1.25) cancel
each otter upon integration over The last term, however, which is proportional to
the bad scattering amplitude Д(тг) gives rise to an oscillatory behaviour of the local
density of states with wave vector 2kp
p(x,£F) = pW
1 + /lspRe(A'w'2i‘")
(1.2F 
which is observed by the STM. We see that the Friedel oscillations in p(x,e)  '
only rattier slowly like 1/r in d = 2. In practice the smearing of the Fermi dislrilat-
tion leads to an eventually exponential decay like exp(-const (T/Tp)kPr) at any finite
temperature T, however for T Tp this effect is negligible on the scale of interest.
As a further interesting application of these ideas we briefly discuss how the solution
of the scattering problem at single impurities may be used to determine' the residua!
resistivity, i.e. a transport property at least in the Boltzmann-Drude limit. This ap-
proach is based on a suggestion by Landauer many years ago [15], where conduction
is treated in a way in which a given incident current gives rise to a resulting electric
field due to scattering. Since the local electron density varies strongly in the vicinity of
the impurity this field is far from uniform. The potential drop associated with a finite
transport current is therefore localized around the scattering centers. Qualitatively
one expects that with carriers impinging on one side of the scatterer the local density
will be enhanced before and depleted after the barrier. As a result Landauer argued
[22] that a dipolar density- and potential distribution should be present around each
impurity. A scattering theoretic derivation of the associated so-called Landauer resis-
tivity dipole may be given in a rather simple manner [28]: the situation with a finite
transport current with density j = — nev is described by assuming that the incoming
wave numbers к are distributed according to a shifted Fermi distribution f (Ck-Mv/h)-
The resulting stationary density around an impurity is then given by
f d^A*
=	(1.27)
Expanding this to linear order in v and using
(1.28>
<JEk
for the derivative of the Fermi distribution as T->() we find that at zero temperatuie
the cumnt-induced density change А/г.(ж) = n(x) — n(x, v = 0) is determined by
дп(х) =	[ tKhtu • к |(a:)|2 + . . .	(L2:i'
n (2tt)“ J
To lowes'. order in the transport velocity v the full density thus acquires a contribution
which contains an angular average of the square of the wave function at the Fern '
12 Theory of Coherent Transport
energy, similar to the local density of states (1.24). Since v determines a preferred
direction, however, there is an additional factor v  k. The precise form of <5п(ж) clearly
depends on the particular form of the scattering potential. For large distances r = |ж|,
however, the induced density is quite generally given by a dipolar distribution as pre-
dicted by Landauer though with additional Friedel oscillations [28]. To see this, we
consider the two-dimensional case and use the asymptotic form (1.25) for the proba-
bility density of the corresponding scattering states. With //0)(sF) = Л//2тг/?2 as the
constant density of states in the ideal case, we find that <5n(x) can indeed be written
in the form of a dipole
<5n(x) — -ep(0) (sF) —
The corresponding dipole moment
p{r) = ^- o-tr +	— Re (лр.(7г)ей'ггг)
2тгв	kp '	'
(1.30)
(1-31)
however is oscillating due the finite backscattering amplitude АР(тг) which also gives
rise to the static Friedel oscillations (1.26) in the absence of a transport current. In
contrast to the oscillations in the local density of states at v = 0 which decay like
l/r^"1 for all distances, the dynamic Friedel oscillations in the total density which are
proportional to v exhibit a l/r^"1 decay only as long as г C h/Mv [28]. For larger
distances the oscillations decay like l/rd as in the static case. The constant term in
p(r) is proportional to the transport cross section (d = 2)
<?tr
dd (1 - cos d)cr(d)
(1-32)
of the impurity which gives a dominant weight to backscattering в — тг (the energy at
which <7tr has to be evaluated is again the Fermi eneYgy). An example for the spatial
dependence of the full density around a scattering center in the presence of a finite
transport current is shown in Fig. 1.2. It is based on a numerical evaluation of (1.27)
for the particular case of a hard disk potential of radius a, for which the exact scattering
states are known [28]. Due to the large value of the transport velocity v — 0.2 tv there
is a strong asymmetry in the direction of the incoming current and the associated
dipolar contribution is superimposed by rather pronounced Friedel oscillations.
Using the results for the current-induced density change 6n(x) at a single impurity,
it is straightforward to see how the classical Boltzmann-Drude expression p = M/пе2т\г
for the resistivity may be derived from a fully quantum mechanical description. Since
there is charge neutrality on scales larger than the screening length, a change <5п(ж) in
the local electron density gives rise to an electrostatic potential
e<f0(a;) = ^ri/i(x).	(1.33)
an
With dp/dn as the inverse density of states at the Fermi energy, we see that the
definition of p(r) in (1.30) was chosen just in such a way that it coincides with the
1.2 Scattering and Linear Response
Fig. l.'S: The normalized local electron density around a hard disk with k^a = 1. The incoming
current flows in positive ^-direction and the coordinates are given in units of the disk radius a.
actual effective dipole moment of the impurity which is induced by the externally giv<>-
current. Now in order to obtain a finite resistance, we need a finite concentration nt 01
impurities and thus in principle one should calculate the scattering states for a problem
with infinitely many impurities. To lowest order in m, however, it is plausible to assume
that the induced dipole moment at each impurity is not affected by the presence of
others. In addition we neglect the oscillatory contribution to p. The spatially averaged
electric field associated with a concentration n, of identical dipole moments (p) is tlm ,
given by
(E) = —2тгпг(р)	(l..m
(the factor 2тг is characteristic for a two-dimensional system, however the final resuh
is clear!y independent of dimension). Inserting the value {p} = hkvaXrv/2ire obtaim
from (1.31) and equating (2?) with pj by Ohm’s law, we find that the resistivity p i;
indeed given by the Boltzmann-Drude result with a scattering rate rj1 = 7qt’F<Ttr as
expected. This simple derivation should be compared with the corresponding much
more complicated one based on an approximate evaluation of the exact Kubo formula
[23]. In particular the incorporation of the factor 1 — cost* in the appropriate cross
section there requires consideration of so-called vertex corrections, while in the present
formulation it is just a simple consequence of the asymptotic form of the scatter;-
wave function. From a conceptual point of view, it is clear from the assumptions mad-
above t, lat the Boltzmann-Drude result is based on neglecting correlations between t
scattering at different impurities and thus it will break down whenever such correlation
are prevent.
Coming back to the general scattering potential (1-16) with many identical iinpm:
14 Theory of Coherent Transport
ties at random positions xv it is clear that the exact one-particle wave functions ifia(x)
and the corresponding energies ea will depend on the specific realization of U(x) (for
notational simplicity we suppress the vector character of x in most of the following).
For macroscopically large systems, however, it is a very good approximation to assume
that there is spatial homogeneity and thus translation invariance on average. This
means that the average () over all possible realizations of U(x) fulfills
(U(x})... U(xn)) ~ {U^Xi + a).. .U(xn + a))	(1.35)
for any fixed vector a and all averages of the above type. In addition to that we assume
that the correlation between the random potential at different points decays to zero at
large separation, i.e. we have
lim (U(x)U(y)) = (U(x))(U(y))	(1.36)
|z—1/|—>oo
and similarly for more complicated averages. With these two assumptions it is now
possible to prove [29] that the density of states and thus all thermodynamic quantities
are self-averaged. Formally this property can be summarized in a generalized form by
the statement that for any function F the quantity
limitrF(H)	(1.37)
V-4-OO у
tends to a nonrandom limit with probability one, independent of the specific realization
of U(x). In particular for the density of states p(e) which is obtained from (1-37) by
choosing F(H) =	self-averaging means that even though the individual £n and
w„(.r) depend on the precise impurity locations, the density of states in a macroscopic
sample is independent of that and thus is not a statistical variable. This is a rather
nontrivial result and indeed the discovery of the universal conductance fluctuations
to be discussed in section 1.4 showed that the simple self-averaging valid for single
particle properties does not hold for the conductance or other observables depending
on the correlation between different eigenstates. Using the Einstein relation (1.6) and
assuming that the diffusion constant D is fixed, the fluctuations of the conductance
of a d-dimensional box of size L have a contribution from density of state fluctuations
VarpG Varp
(G)2 W2 	}
Naively one would expect that Var p scales like the inverse volume L~J and thus the
associated fluctuations in G would vanish as L —> oc for any d < 4. It turns out,
however, that for metallic systems the variance at T = (1 decays much more slowly
than L d. behaving like
Varp ~ L-'2(rf-2>	(1.40)
1.2 Scattering and Linear Response 15
for all d < 4. In particular in d = 2, Var p is independent of the size of the system
as long as L remains smaller than the localization length while in d = 1 it ev<
increases like L2 as long as L < £. For metallic systems the result (1.40) thus implies
that Varp G is of order one, independent of the system size. As was shown by Altshuk г
and Shklovskii [30] this contribution accounts for half of the UCF at T — 0 (see
section 1.4 below).
An example for a random potential which is translation invariant on average a.iri
whose correlations decay to zero asymptotically is provided by the impurity poteiV i. !
(1.16) with x3 being mutually independent and distributed uniformly over the who!
volume V. The generating functional
Ф[9] = ^exp (У ddxg(x)U(x)^	(1.1: -
which a lows to calculate arbitrary averages by functional differentiation is then given
by [29]
Ф[р] = exp {п, У d'h' |exp (J ddx g(x)u(x — .t')^ — ij J .	(1-42)
A particularly simple model which is commonly used in discussions of transport in
disordered systems is based on assuming that U(x) is a Gaussian random process [31].
This so-called Edwards model can be obtained from the above model of structur
disorder by taking the limit of a high density of very weak scatterers. Indeed let
us take .he strength Uo of u(x) to zero but let nt diverge such that the product /гр.-.;
remains constant. Then after subtracting the trivial average U = (U(.;;)), the expansion
of (1-42) to second order in u(.r) gives a characteristic functional of the form
$[.?] = exp | i I d'hj I ddx2 B(.t, - :r2)t;(.ri )c;(.r2) j .	(1. U!
This describes a Gaussian random process with zero mean (U(x)) ~ 0 and correlation
function
([/(.ri)[/(.r2)) = B{xt - .r2) = ?h У d'hu(a;i - x)u(j-2 - :r).	(1-44)
Provided the particle’s energy is not far from the average potential I' the associated
wave fun ’tions vary on a scale much larger than the characteristic fluctuation of the
random potential. In this case U(x) may be replaced by Gaussian white noise
([/(.7,-i)G(.r2)) = 7i,Uo<5(xi - .;;2).	(1.45)
I liis is the modi1! which we will use below to describe scattering in a disordered sample.
1.2.2 Green functions
A conven ent wav to describe the quantum mechanical propagation of single particles in
a random medium is via so-called Green functions (GF) [23]. In our present treatment
16 Theory of Coherent Transport
where no interactions are taken into account, they are simply related to the matrix
elements of the so-called resolvent operator
G(z) = (z - H)-1	(1.46)
with z a complex number. Obviously G(z} exists only for those values of z which do
not belong to the spectrum of H. A continuous spectrum therefore leads to a branch
cut of G(z) along the real axis. The corresponding density of states is related to the
jump of G(z') across the cut. Indeed using the formal relation
(e - H ± iO)"1 =	i^(e - H)
£ - H
we can write
p(f) = 77 tr<5(e - H) — т~ту Im trG(c ±iO).	(1-47)
V	7Г V
The trace can be evaluated in any basis and thus for instance (1.47) may be written as
p(e) = T J- hn [ <idx(x\G(E ± iO) Im) = =p- Im(.r; = 0|G(e ± iO) к = 0)	(1.48)
7rV J	7Г
using translational invariance on average. The matrix elements of G
G|G(e ± iO)K) =. £ (MzME) = gr,a(a	(1.49)
a £	Itl
are called retarded (+i0) or advanced (~i0) Green functions G[<,A. They have a simple
physical interpretation in terms of the quantum mechanical propagation of states at
fixed energy £. Indeed the amplitude I\(x' —> x, t) for a particle to propagate from x1
to ,c in a given time t > 0 may be written as the matrix element
K(r' -> ,r, t) = (,i,j exp(-iHt/ft)|.r')	(1.50)
of the general time evolution operator. Instead of time t it is often better to fix
the particle’s energy £ (e.g. at the Fermi energy). This leads to the corresponding
propagator at fixed energy [32]
K(x’ .r,£) = /'0°dfoi(f+il”'/f'A'(.r' x.t)	(1.51)
Ju
which is related to the retarded GF by a trivial factor
G^G.-F) = ~J<(x’ -o.r.s).	(1.52)
In order to obtain a corresponding interpretation for the advanced GF we note that
G(s - iO) = (G(e 4- iO)) . As a result the advanced function GA(.i',.r') is related t.o the
1.2 Scattering and Linear Response 17
О
----- +
к
q
+ —
к
к
к
-q
— 4-
к k+q к
Fig. 1.3: Free propagation with momentum к and lowest order scattering processes with momentum
transfer 0 or ±q.
amplituc e to go from x to x' (i.e. initial and final positions are reversed) at energy
via
G^(;r,x') = ^A'*(.r-> .-c'.e).	(l.t
Apart from its conceptual importance the interpretation of the GF in terms of Feyn-
man amplitudes has lead to a better intuitive understanding of interference effects in
quantum transport. Indeed in the semiclassical limit the fixed energy propagator ma’-’
be written in terms of all possible classical paths a from x' to x [32]
K{x' -> x,e) ~ 52 A„exp(iS(,//l).	(1.54;
Here Д, is a prefactor determined by the Gaussian fluctuations around the trajectory a
while S,-, is the corresponding classical action at fixed s. Using the semiclassical result
(1.54) foi the GF it is possible to give an interpretation of both weak localization [33]
and UCF [34] in terms of quantum interference between different classical trajectories.
Evide itly for a given potential U(x) the calculation of the GF (1.49) requires a
diagonalization of the full many impurity scattering problem which is impossible in
practice. In the case of a random potential like (1.45) the GF are themselves random
variables and a statistical description is required. In the limit of weak disorder kpl ~X-
1 there is a simple approximation at least for the average GF (вГ1,л). Indeed in
perturbation theory the propagation of an electron may be viewed as a sequence’ of
free propagations which are interrupted by successive scatterings at the full UlxY
Pictoriall./ this is shown in Fig. 1.3 by a series of lines representing the zeroth ordc’
propagate r
(k\G^(z)\k') = —1—(1.55)
z — £ к
which is < ressed by a sequence of dotted lines representing the potential U(x). Rea<.
from left, to right at each vertex the momentum </ carried by the dotted line with
weight U{q)/V is transferred to the next G(°l. After n scatterings an incoming wave
vector к is therefore changed to к' = к + yj + .. . + q„. For a given к and A:' one
has to su n over all possible </,. In the case of the Gaussian potential (1.45) with
(U('/i )U(к?)) = щи'о\''6qi _q2 this average' is performed by summing over all possible
pairwise contractions as indicated in Fig. 1.4 up to fourth order (since (U) ~ 0, only
diagrams with an even number of impurity lines contribute). In order to obtain a finite
energy width of an average plane wave state it, is necessary to sum the perturbation
18 Theory of Coherent Transport
A
< к | <G> | к > = G”’ e q/ \-q
к k+q к
Fig. 1.4: Perturbation expansion of the disorder averaged Green function up to fourth order for a
Gaussian random potential.
series to infinite order, at least approximately [23]. This may be done by using the
concept of irreducible diagrams [35] which are defined by the property that they cannot
be divided into separate pieces by cutting only a single G^-line. Thus e.g. in Fig. 1.4
all four diagrams except the second one are irreducible. It is then convenient to define
the irreducible self energy E by the sum of all possible irreducible diagrams with their
two free G(0,-ends omitted. Then since any number of E’s may be joined together by
G(l,)’s it is easy to see by iteration that the exact average GF fulfills the Dyson equation
(all indices omitted)
{G} = G(0)+ GW^(G}	(1.56)
with the formal solution
<G> = (L57)
Since (1-57) already sums up an infinite series, we calculate E to lowest order in пгь‘д
which is just, the triangle in the first diagram of Fig. 1.4 and corresponds to the standard
Born approximation in quantum mechanical scattering. The result is
1  	1	Г	nW f £)
Eu(fc, г) = muo TV E------- = «X / <1-	(1-58)
Here the (/-integration has been transformed to an integral over the zeroth order density
of states	Taking z — e + id the associated retarded self energy has both a real
and an imaginary part. The real part (which is naively divergent at large energies and
thus needs to be calenlated in a more realistic model than a parabolic band [23]) can
be absorbed into a renormalization of the bare energies Ek and is therefore omitted in
the following. Physically more important is the imaginary part
Im£B(e) = -япХ/'Е)	(1.59)
which may be replaced by a finite constant 7 = -IihEb^t) и* bhc vicinity of the
Fermi energy. The broadening of the corresponding averaged GF
(kKG^in =-------(1.60)
e - £k + n
1.2 Scattering and Linear Response 19
describes the fact that in a random potential plane wave states |fc) are not eigenstates
of H but have a finite lifetime. Since 7 is the associated energy uncertainty the corre-
sponding scattering time r is simply
7 = ^/2т = hvv/2l	(1.61)
with Z tie mean free path. In section 1.2.3 below we will show that within our simple
model for U(x), the scattering time т which characterizes the broadening of plane wave
states is identical with that appearing in the Boltzmann-Drude conductivity. Indeed
the equivalence between r and Ttr is valid here since U(x) only leads to s-wave scattering
for which the cos ^-contribution in (1.32) vanishes because the differential cross-section
is independent of (?. It should be emphasized that the appearance of a. finite broadening
7 yt fl does not imply any sort of dissipation since energy is still conserved. Moreover
the cond tion kpl 1 for a weakly disordered system is equivalent to 7 c eK, i.e. th'1
broadeni ig is much smaller than the energy of the states at ej.. The averaged GF in
real spac >
(27r)3 ,F^. + i7	(l.P
can be. calculated exactly using the residue theorem (terms of order (/pd)'2 are in
glected) ; nd is given by (r =	— x'|)
\1 eiAVr
The outgoing spherical wave G'(0),u ~ e’fcr/7- js fj1UK multiplied by an exponentially
decaying actor, i.e. a wave on average propagates only a distance I without scattering.
Flic finite width of the states \k} also leads to a change in the density of states compart'd
to the res lit (1.19) in the absence of disorder. Taking 7 to be energy independent, for
instance 1 tads to
,	м П 1	,
Aw(f) = 7-77 7 + ~ arctan -	(1.64)
2tt/i л 7 /
for the two-dimensional case. i.e. the step function in the fret' density of states is
smeared < ut by 7. In particular the density of states is now nonzero for all ener-
gies, howtver the approximation (1.64) is only valid for f 7. Indeed in the limb
—> —-00 one is in the regime of strongly localized states which are bound in po-
tential we Is formed by large negative fluctuations of . In this limit there i.. .in
asymptoti-ally exact theory for p(e) due to Lifschitz which shows that p(e -> -> )
vanishes exponentially like exp(-|e|2''^2) (d < 4) in the white noise mode! (1.45'
while1 p(t* -> —00) ~ exp(-e2) in the more general case (1-44) with B(0) < 00 [29].
the weak disorder limit, however, which is of interest here the density of states differs
from its fr'e value1 only by terms of order l/ki.-l. Incidentally this also shows that the
inclusion < f t he diagrams of the last type in Fig. 1.4 which nitty be taken into account
20 Theory of Coherent Transport
by replacing p^ by the exact p in (1.58) (this is the so-called self consistent Born ap-
proximation [35]) will not change (G) to lowest order in q/ep = l/fcFf. As regards the
diagrams with intersecting impurity lines, they contain additional momentum space
restrictions compared to the geometric series of the nonintersecting contractions which
are summed in the Born approximation. As a result it is not difficult to show that they
are smaller by a factor of l/fcpl and thus are negligible in (G). As will be shown in sec-
tion 1.3.1 however, diagrams with intersecting impurity lines are crucial for obtaining
the leading quantum corrections to the Boltzmann-Drude conductivity.
1.2.3 Kubo-Greenwood formula
The basic quantity of interest in electrical transport is the nonlocal conductivity tensor
cr(.;:,.r') relating the applied field Elx') and the induced current density /(.r) by the
general relation (1.11). For all transport coefficients in linear response there is an exact
formalism which allows to calculate them from the knowledge of time-dependent corre-
lation functions in the absence of the perturbation [4, 23]. Here we will not develop the
theory in its most general form but restrict ourselves to calculating the dc-conductivity
of noninteracting electrons, following a presentation of Baranger and Stone [36]. The
effect of a time-dependent electric field E(x,t) = — W(x,t) leads to a Hamiltonian
H(t) = Я- eV(x,t).	(1.65)
The time dependence is chosen as
V(x,t) = V'(x) cos(wt) exp(—<5|t|)	(1.66)
where the rate1 6 at which the perturbation is turned on and off is taken to zero at
the end of the calculation, however before the static limit 0. The sample is made
infinitely large by attaching a number of semi-infinite ideal leads. In each of them the
scalar potential V(.r) is required to reach a constant value asymptotically such that
the electric field is nonzero only in the region where some scattering occurs. Since
the infinite system has a continuous spectrum, the eigenstates of H are indexed by a
continuous variable a. The complete orthonormal set of its exact eigenstates
fulfill
I ddx ф^х^х^х) = 6(a - (5) I do ф*п{х)фа(х') = <5(;r - x').	(1.67)
In order to calculate the expectation value of the current density we need the time-
dependent density matrix p(f) which obeys the von Neumann equation ih()ip = [/7(f). £?]
To lowest order in the perturbation we may linearize this around the equilibrium den-
sity matrix p(0) = Jda/(£„)|rt)(o| with /(e) the usual Fermi function. The first order
deviation pH) = 5— then obeys
= [H, p01] + [Я1, p(u)],	(1.68)
Introducing /j„ -- /(s.d — /(t<>) and Vltlj =	V(.c)|/) this leads to
= “Gwia - t’/jJ’^rt'osM) e lV/
(1.69)
1.2 Scattering and Linear Response 21
for the matrix elements. This equation is easily solved with the initial condition pW --
0 as t — ► — oo giving for t < 0
6a/3^ < 0) = — -ffjaVafje 1 * * * [	-	+ (w —> —w) j .	(1.7(1'
2	\s0a - hu + ihd	J	'
We can now calculate the current density to first order in the field from
(1.7''
(For a finite magnetic field В ф 0 there are also equilibrium, i.e. persistent magne
tization currents due to however these do not contribute to the transport, [36]).
Here
\cti
ЗрЬ'Ь =	(1.72)
&1V1
are the matrix elements of the current density. The operator D is defined by
fDg = f(x)Dg(x) -ц(х)О'/(т) = -fjD'f	(1.73)
with D - V + (ie/h)A the covariant derivative. To proceed further we take matrix
elements of the continuity' equation	+ Vj = 0 which leads to
= 'VjaAb	(1-74)
as a consequence of number conservation. By a partial integration the matrix elements
Г„о of th1 scalar potential may be expressed in terms of the local electric field via
i/^ f
flWt = — / drfr  E(x).	(1.75)
£,)n J
I sing this and collecting the in-phase contributions J(.r) cos(u?/) to the current we
obt ain
M = ~ [ dev [ dd^j,,„(.,:) [ d-n.aCH ' E(x') f---------------2—-— + -mA .
(1.7G)
l inally we take* the limit m —> 0 which leads to
<т(.г..г') = тгЛ / da [ <1*4 (“/'(;7,)йЪ“^>) - 7^“^)	(1-77,
lor tin* no ilocal conductivity tensor defined in (1.11) (here the derivative of the Fermi
function vith respect to the energy is indicated by a prime while й denotes the tensor
product of two vectors). This is tin1 most general form of the Kubo-Greenwood formula
which allows to determine the linear conductivitv from the current matrix elements of
22 Theory of Coherent Transport
the unperturbed system. In particular at zero magnetic field В = 0 the principal
value term vanishes (see section 1.5.2 below) and thus as T —> 0 the conductivity
depends only on the states at the Fermi energy. A more detailed discussion of the
nonlocal conductivity cr(x,x') will be given in section 1.5.2. At present we will only
be concerned with the macroscopic conductivity tensor <r which relates the spatially
averaged current j — V~l f Л j(x) to the spatially averaged electric field (here and in
the following the limit V —> oo is always understood). As we have seen in section 1.2.1,
the local electric field E(x) in a transport situation exhibits strong spatial variations.
Nevertheless it turns out (see section 1.5.2) that for the macroscopic dc-conductivity
the electric field can effectively be considered as constant such that
cr = V-1 У d^y d^'<r(.r, i').	(1-78)
Using f d'h: j(x) — — ev, the macroscopic conductivity tensor a may be expressed in
terms of the matrix elements vltp of the velocity operator exactly as in (1.77). For
analytical calculations it is convenient to replace the exact eigenstate summation by a
general trace. After some partial integrations [36] the zero temperature longitudinal
dc-conductivity can then be written in the standard Kubo form (no spin degeneracy
included)
Тех = -у- и [ut<5(ef - H)vx3(£p ~ Я)]	(1.79)
or — more generally —
Re ст„(Г,~	+ k-- tr \vx6(ef - H)vx6(£l.- + Гш - H)]
(1.80)
at finite temperature and frequency. These results show that the principal value terms
in (1.77) disappear if one considers the macroscopic longitudinal conductivity. For
weak disorder and zero magnetic field the appropriate basis to evaluate the trace are
plane wave states |/c) which are eigenstates of the operator vx with eigenvalue hkt/M.
Using <5(t - H) = i(G^ - Ga)/2tt and including a factor of two for the spin degree of
freedom, the longitudinal dc-conductivity at T = 0 is given by
fE^'i(A:](Gr-Ga)|U)(U](Ga-G1<)|A-)	(1.81)
27r \M) 1 fr
where tin1 indices on G and a have been omitted for simplicity. As was noted above,
in a system with random disorder the GF and therefore also the conductivity are
random variables. Thus (1.81) may in principle be used to calculate the full probability
distribution of the conductivity. In the metallic limit it may be shown [37] that the
corresponding probability distribution is Gaussian and thus completely determined by
its average and variance. The latter will be discussed in section 1.4 where the variance
of the conductance turns out to have a universal value of order (e2/h}2. Here we will
concentrate on the average conductivity which requires us to evaluate the disorder
1.2 Scattering and Linear Response
23
average of products of GF of the type (GRGA), (GRGR) and (GAGA). This is easilv
done in the case of the GRGR- and GAGA-terms which may be reduced to single GF
via a Ward identity [35]
ft2
- £ k'(k\GR\k’)(k'\GR\k) = V* (k\GR\k)	(1.82)
and analogously for GAGA. These contributions are therefore directly related to tin'
density of states and thus are self-averaging. They do not contribute to the weak
localization interference effects in macroscopic samples or the mesoscopic Hint nation
of the co iductance, at least in the limit kpl > 1. In this limit a may thus be obtained
from the approximate form
fi p	\ 2
(1.83)
which wi 1 be used in the following to determine both the average conductivity and its
fluctuations.
From (1.83) it is straightforward to show that the classical Boltzmann-Drude re-
sult a = пе2т/М for the averaged conductivity arises by making the factorization
approximation
(GrGa) » (GR) (GA)	(1.8 1;
in which he GF are diagonal ~ 5kk,. Indeed, using (1.60). the right hand side of (1.3
is a strongly peaked Lorentzian at Ek = £p. In the limit (fcF/)*' = 7/£p -C 1 we <;>
thus replace k* by A.p/3 and evaluate the remaining integral as
[ ^{Gr)(Ga) « /%.-) Г 7Г—(1.85;
./ (2tt)3	Jo (Ek ~ £p)- + у2	7
I'sing p(e .) = 3n/4sp and 7 = Л./2т, we obtain precisely the Boltzmann-Drude expres-
sion for 0 Moreover from the Einstein relation (1-6), the associated diffusion constant
D = i'i.Z/3 is identical with that obtained from elementary kinetic theory. A physically
intuitive form of the result (1.83) for the conductivity is obtained by noting that in
analogy to the real space amplitude (1.52) the quantity
ih(k\GR\k'} = A(k' -a k)	(1.86)
may be interpreted as the amplitude that an electron with energy e is scattered from
momentum fc' to fc. Reversing tht* role of к and к' and using кгк'г —> к  к'/3 due to
overall isotropy we mav write (1.83) in the suggestive form
,2
(й) 5Fp
тгЛ
(1-87)
24 Theory of Coherent Transport
The longitudinal conductivity is thus an average of the product к к' over all scat-
tering processes from initial momentum к to final momentum к' with probability
\A(k -> fc')|2. The factorization approximation (1.84) which leads to the Boltzmann-
Drude result replaces (|A|2) by |(A)|2. Since (A) ~ dkk> only those scattering processes
contribute to the classical conductivity in which final and initial momenta coincide.
Clearly this is only a small subset of all possible processes and it is evident from (1.87)
that in general g will be smaller than its classical value because к  к' is largest if
к = к1. In particular one recognizes that the potentially strongest reduction is due
to exact backscattering processes in which к' = —к. Indeed this is the basic physical
mechanism behind the weak localization correction to the Boltzmann-Drude theory
which will be discussed in the following.
1.3 Weak Localization
In this section we give an introduction to the calculation of the weak localization cor-
rections to the Boltzmann-Drude result which arise from quantum interference effects.
It is shown that the basic mechanism of 2A:F-backscattering can be described in terms of
the so-called Cooperon diagrams. The quantum correction to the classical conductivity
is thus expressed by the eigenvalues of a diffusion equation. Weak localization leads
to a negative magnetoresistancc and allows to observe Aharonov-Bohm interference
effects in non-simply connected conductors. It is destroyed by processes which break
time reversal invariance leading to “dephasing”, a concept which is discussed in section
1.3.3.
1.3.1 Coherent Backscattering
We have seen above that the classical Boltzmann-Drude result for the average con-
ductivity is obtained from the exact Kubo formula by assuming that the average of
the probability to be scattered from к to к1 factorizes into the square of the aver-
age amplitude. In order to understand why the Boltzmann-Drude result, gives the
dominant contribution to g and what are the leading corrections to it. we consider
a general process in which an initial momentum к is scattered into к'. Such a pro-
cess may be caused by an arbitrary number n = 1,2,... of individual scatterings
к —> fc, —> к-i —> • • • —> ktl = к'. Obviously there are zd possibilities i for the order
in which these scatterings may occur which all add up coherently in principle. The
associated total probability is thus
|A(fc -у л')!2 = I E Цк fc')|2 = E A^k k'^A^k k'Y (1.88)
i	у
Now different A,’s have widely different phases and therefore terms with i -£ j are
cancelled out in (|.4|2). However there are two exceptions: One is the diagonal terms
i = j which give a real and positive contribution. Keeping these terms only we have
(Lil2) « (,4)j2 which is equivalent to the classical limit. However then* is another
1.3 Weak Localization 25
Fig. I.E: A closed electron trajectory i and its time reversed version j — i scattered at several
impuritks (x).
class of terms which also do not cancel on averaging. These are processes in which
the se<i icnce of scatterings in j is exactly the time reverse i of that in г. Since ;
starts with fc and ends with fc' while i is path with initial momentum —fc' and fin d
moment ши —fc this is possible only if fc' = -fc. Thus the contribution j = 1, describe s
backsca.tering processes with total momentum transfer 2fcK which — as we have seen
in (1.87) — are. the most effective processes in decreasing о below its classical value.
For a ii.ore intuitive understanding it is convenient to interpret the two terms i a . .
j as an amplitude for a particle (A,) and a hole (A*) to propagate from fc to k'.
This interpretation should, however, not be taken literally; it is just a useful wav
of thinking of the two terms in the square. Using this picture the diagonal tern,
correspond to trajectories in which the particle and the hole are scattered in identical
order, i. . they visit the same impurities retracing each others path. By contrast, in
the cont -ibution j — i the hole exactly retraces the particle’s path in the opposite sense
as iridic; ted in Fig. 1.5. In a real space picture one may use the semiclassical Feynman
path representation (1.54) of the propagator К to go from one side of the sample to
the othe' side at fixed energy 6|.-. In an expansion of
|ЛТ = E 1<п1<а = E ДЛА'ХР - 5ri))	(1.891
aft	Ck/J
as a sum over different classical paths a the phase's Sn — cancel only for the diagonal
terms J = rt and the time reversed contributions /3 = <1. The possibility 3 = Л requires
a clost’tl electron trajectory and the interference occurs between two partial wa rns
traversing these trajectories in opposite direction. Due to Л‘л = I\n the probability
for going around such a loop is \1<л + A'(>,2 = 4|A'n|2 instead of 2|I\„|'2 in the diagonal
approxin ation. The quantum probability to return to an arbitrary point is thus twice
as large as classically. As a consequence the probability to propagate through til-
whole sample and thus the conductivity is reduced compared to its classical limit. 1
a quantitative calculation it is convenient to go back to the momentum space picture
38]. Sinoe the nimijc of the nonlocal conductivity <r(.r,.r') depends only on ,r — .r'
26 Theory of Coherent Transport
Fig. 1.6: A graphical representation of the disorder average of the product GRGA in (1-91), (1-92)
defining the vertex function Г. The thick lines denote averaged Green functions.
we may consider the general wave vector- and frequency-dependent conductivity
lie2 t h \ 2 1
cr(<7,w) = - 1 — 1 — £ к  /с'ФН'(</,и).	(1-90)
7Г \M 3v 77?
\	/ kb'
Here we have introduced the function [39]
$kk’(q,iG) = (GR(k+,k'+,EF + hu)GA(kL,k_,El.-)')	(1.91)
with k± = fc±<?/2 (similarly for к') as a generalization of the disorder average {GRGA)
to finite q and ш. Obviously (1.90) reduces to our earlier result (1.83) for q =	— 0.
To calculate Ф we use a graphical summation in terms of diagrams as in the average
density of states. It is evident that the disorder average in Ф factorizes if we neglect all
impurity lines which connect the particle line GR with the hole line GA. The nontrivial
contributions are described by a (reducible) vertex function ПЯм) such that
= GR+G^6kk. + GR G^J^q^G^G*	(1.92)
(see Fig. 1.6 where GR is indicated by a right- and GA by a left-going line). Here and
in all of the following formulas the GF G^’A are already the averaged functions which
wo take to be of the simple form used in (1.60). This incorporates all diagrams in
which the impurity lines stay within the particle or hole contribution seperately. The
quantum corrections to the Boltzmann-Drude result, are now contained in the function
Г which - - unfortunately cannot be calculated exactly. One therefore tries to sum
an infinite class of terms which are expected to give the dominant contribution at least
in the limit kvl ,2> 1. These are the so called “diffuson” and “Cooperon" diagrams.
The diffuson contribution is closely connected with the conservation of particle number,
which apart from energy is the only conserved quantity for electrons moving in
a static random potential. Specifically particle conservation implies that the density
relaxation function Ф(г/,ш) = Фкк'(д^) is proportional to the diffusion propagator
( —icj + Dq2)~l in the limit q,ai —> 0 [40]. To verify this behaviour we use a ladder
approximation to calculate Г. Summing up the diagrams in Fig. 1.7 amounts to solving
the Dyson type equation f(u) = u2/V + <F(0) i.e.
1.3 Weak Localization 27
Fig. l.T: Perturbation expansion and Dyson equation for the vertex function Г in terms of ladder
diagrams which lead to the diffuson.
where
u2
= 7/52 (cf + ^)СА„.д(£р)	(1.94 j
1 k
and u'2 = For q,ai -> 0 the function <(r/,w) may be calculated by expand-
ing £k-4 « Ek - hvk  q Here a contribution eq = h2q2/2M is neglected although
the final result contains a term Dq'2. This is justified because hDq2 in th.
limit kyl 1. Since G1!GA is strongly peaked near we replace / <13/.-(2тг) ;! '
р(Еу) da; J df4/47r with x ~ ek~ eF. Evaluating the integral over x by the rose...
theorem we obtain
= / -7— t----:---—--------- ~ 1 + ^7 - Dxq2	(1.9o;
J 4tt 1 — iiur + irvk  q
where we have used that fdiikvk q — 0 and J' dftk(yk  q)'2 = (Ттг/З)v^q2. This form is
valid in the limit air << 1 and ql <C 1. Using (1.95) the diffuson or particle-hole ladde.'
(1.93) gives
with D ~ i'i.l/3 the classical diffusion constant. The associated density relaxation func-
tion Ф((пш) thus has the required diffusion pole. Indeed this should be valid through
out the conducting regime however with a renormalized diffusion constant which
smaller han its classical value. By contrast, in the localized phase one expects thai
Ф(</.и;	0) ~ (—icv(l + (<]()'))“1 with £, the localization length [39]. Considering
the diffuson contribution to the conductivity it is clear from (1.90) that Г^, gives
contribution to cr(q = 0, u) since it. is independent of к and к1 while G[i,A are even in k
and therefore "£.kk, к  k'.. . vanishes due to antisymmetry. This is no longer true if ! !
scattering is not, point like as in our model (1.45). however the ladder approxima
for Г only guarantees that the transport relaxation time in <7 contains the correct fact (
1 - cosh as in (1.32) [23] but does not give a localization correction to flu* conductive
[39].
We row turn to the Cooperon which is the contribution where particle1 and hole
are scat) ‘red in reversed order. Consider for example a process with three momentum
transfers к -о kt —> к/ —> кл = —к for the particle (GK) and the corresponding
28 Theory of Coherent Transport
к к, 1^ -к
к -к2	-к, -к
Fig. 1.8: An example of a maximally crossed diagram fors (GRGA) with three scattering processes.
ones k —> — k'2 —> — k\ —> —к for the hole (GA). It is evident that the associated
diagram (see Fig. 1.8, note that also the lower line is read from left to right) consists
of maximally crossed impurity lines. The sum of all diagrams of this type can be
calculated without doing any new work by exploiting time reversal invariance. Since
к —> —к under this operation we have
(fc'„|GA|fc_) = (-fc_|GA| - V ) .	(1.97)
As a result the function w) in (1-91) must remain invariant under the exchange
fc_ о — fc'_ which directly implies that
Г**:'(<7,и>) = r(k-k'+q}/2(k'-k+q)/2(k + k',	(1.98)
quite generally. Since the particle-hole ladder is obtained from a time reversal operation
by “detwisting” (i.e. turning around the lower line in Fig. 1.8) the maximally crossed
diagrams, the Cooperon contribution is obtained from the diffuson (1.96) simply by
replacing q by к -f- к' = Q. In the limit Q —> 0 the maximally crossed diagrams thus
sum up to
„2	1
r(c)(Q,eo)^—-----	(1.99)
tv —iw + DQ1
in analogy to (1.96). The singularity now appears near Q and w equal to zero, i.e. for
vanishing total momentum к + к' ~ 0 which leads to a singular contribution due to
backscattering. It is called a Cooperon because for electrons with some weak attractive
interaction the formation of Cooper pairs is described diagrammaficallv by the same
type of particle-particle ladder diagrams [23]. Inserting the contribution (1.99) into
the general expression (Г.90) and replacing к • к' by -fc2- for backscattering near the
Fermi surface, the associated contribution to the conductivity at </ = 0 is
A<7(w) = -—----------------------------------------Ц—.	(1.100)
k 1 Зтг p r J (2тг)3 A к к> V -iu + DQ2	v	'
Here we have set w = 0 and fc' = —fc in the averaged GF and have used that GR)A =
G»-a. Since GkGa is strongly peaked near the fc-integral may be calculated as
1.3 Weak Localization
(X = Ek - £p)
which finally gives
...	2e2 1 „	1
(1.1b.
(1.102,
Now ( -iw + DQ2) 1 = /Q°° dtelw(P(Q,t) is directly related to the spatial Fourier trans
form P(Q,t) = exp( — DQ2t) of the standard Gaussian solution
Р^°\х,0 = (^Dt')~d^
( x2 \
exp----—
\ 4DtJ
(1.1П2!
of the diffusion equation dtP{x,t) = DV2P{xp) with initial condition P{x,t = 0;
<5(x) ii an unbounded d-dimensional system. Since (1/V) Eq P(Q, t) - > P(x = 0.
the weik localization correction to the conductivity at w = 0 can also be written a<
2f>^ r<x> t 4
Д<7 = —-D / dtPM(t)
7ГЛ JO
(1.ИЧ)
where P^(t) is the probability density at the origin of a diffusion process stai
there ft t — 0. The index (tp) is introduced here for later purposes to indicate th'1
influen :e of phase breaking processes which suppress the coherent backscattering for
closed trajectories lasting longer than a characteristic time r^. Thus P^^t) differs
from the naive result P^X3\t) — (4тг£Э£) “d/<2 obtained from (1.103) by a cutoff at long
times < tie to dephasing and also by the fact that for a finite' system with boundaries,
the sin i over the momenta Q in (1.102) must be replaced by a an appropriate eigen-
value expansion. Before discussing this in more detail we would like to point out that
f (It P( )d'h; is just the average time a diffusing particle spends in a volume element dfh
near its starting point. The integral is also directly related to the return probability
П(0) which, however, can only be defined properly for a diffusion process on a lattice.
Indeed with discrete time steps N = 0,1,2,..., the return probability 11(0) is related
to the probability P{N) to be at the starting point at step N by [11]
E пю =
N=0
1
П- П(0) 
(1.105)
A return to the origin at. some time is therefore certain (П(0) = 1) if P(N) diverges
which ii true for any dimension d < 2 since P{N —> oo) ~ N~lld2.
In < rder to evaluate the quantum correction Au to the Boltzmann-Drude resul.
quantitatively we must take into account both the finite sample size as well as th<
finite p iase coherence time (“dephasing”). As is discussed in more detail in section
1.3.3 below, the* effect of dephasing leads to a cutoff rv in time beyond which two
30 Theory of Coherent Transport
partial waves can no longer interfere constructively. Assuming an exponential decay of
the ability for constructive interference, this leads to the approximation
PM(t) « P(t)e-!/T”	(1.106)
in (1.104) which is equivalent to the replacement —iw —> —iw + 1/r^, in (1.102). For
a finite sample the solution of the diffusion equation is determined by the appropriate
boundary conditions [42]. At a boundary |„ between the conductor and vacuum (or an
insulating region) we require that the normal derivative
dnP(x, t) |„= 0	(1.107)
vanishes. Physically this corresponds to demanding that there is no current normal
to the boundaries. At the interface |/ to an ideal lead where diffusion is replaced by
ballistic propagation, we require a vanishing excess density
P(x,t)\l=0.	(1.108)
The general solution of the diffusion equation with these boundary conditions can then
be obtained from an appropriate eigenfunction expansion
P(x, t) = 52 Pn(x) exp(—A„t)
with eigenvalues An > 0 and eigenfunctions Pn(x) obeying -DV2P„ = XnPn (the
initial point is always taken at x() = 0 which allows to suppress the corresponding
eigenfunctions Pn(xQ in the expansion of the transition probability density F(a;,t) =
F(0 —> x,t)). The sum over momenta Q in (1.102) is then replaced by a sum over
the appropriate eigenmodes n with Xn playing the role of DQ2. The weak localization
contribution to the conductivity can thus generally be expressed as
2f>2	1	1
ACT(W) =------D-У--------------------.	(1.109)
тгЛ V „ -iw + \/tv + An
The prime on the sum over n indicates that the summation must be cutoff at large
values Xn > 1/r because the diffusion approximation breaks down for short times of the
order of the average collision time r. Similarly, the integral /q00 di in (1.104) has a short
time cutoff т which eliminates the divergence which would arise from the behaviour
P(t —> 0) ~ D‘4'2 (for short times the boundary is not felt and the solution of the
diffusion equation behaves like that in an unbounded domain, in reality the dynamics
at short times t < т is ballistic rather than diffusive).
For a comparison between the magnitude of the weak localization correction and the
zeroth order Boltzmann-Drude result <r0 it is convenient to express the total averaged
conductivity <7 = <70 + Ac as
op) = C7O ( 1-----f
\ тгпр(сг-) JQ
1
— ice + DQ'2 + 1/ту
(1.110)
1.3 Weak Localization 31
Here we have assumed that шт 1 such that the Boltzmann-Drude conductivity
is frequency independent while Jq is an abbreviation for p 52 ^ —> J d%>/(27t)'1 or the
more general eigenvalue sum in (1.109). Note that the Einstein relation <j = 2e2p(cF)D
for non interacting electrons at T = 0 gives simply cr0 here because our D is still the
diffusio r coefficient D — vvl/3 of kinetic theory, i.e. to zeroth order in . In a
bulk th ee-dimensional system with 1/t^ = 0, the evaluation of the integral in (1.110)
with an upper cutoff Qmax « l/l gives
.	const. \
<73d(w = 0) = <70 1 — T-.—-ГТ I .	(1.111)
\	W,]2)
The weak localization correction is thus of second order in the small expansion param-
eter (/cF!)-1. Now in fact the expression (1.110) only includes the 2/cF-backscatteriug
correction to о but is not the leading term in (A’fO-1 here. Indeed it turns out th.<!
there are other nonsingular contributions to the conductivity which are of order (fcFl) 1
relative to <7q [43]. The weak localization correction is therefore not very interesting f<c
bulk three-dimensional samples, basically because in d = 3 the integral Q~2 is finite
at small Q l/l without a cutoff. Weak localization, however, becomes the de
inant and — in particular nontrivially temperature and magnetic field-dependent
correcticn to <t0 in systems with reduced dimensionality. To see this we consider a
metallic film with thickness Lz I. Assuming it is infinite in the x, ^/-direction, th
exact so ution of the diffusion equation with boundary conditions dzP = 0 at z = и
and z = L, and initial condition P(x, t = 0) — <5(ж) is
P(x, t) = 7г,(оо)(гсц, if) -i- 52 cos ) exp(-Ant) (1.112)
n=-oo \ b2 /
with eigenvalues A„ = D (mr/Lz)2. Due to the cutoff at only those eigenmodes
contribute to Au for which An < 1/tv, i.e. |n| < Lz/tLp where we have defined
L^^Dt^'1'2	(1.11
as the phase coherence length for diffusive motion. Now provided that this length is
larger than the film thickness we see that only the n = 0 term contributes to (1.112) in; i
thus P(t‘. ss Р?0^(t)/L,. In this limit the film behaves like an ideal two-dimensi nal
system, indeed quite generally for weak localization it is the phase coherence le-
whicl. has to be larger than the relevant system size to see the effects of red......
dimensio tality. In our example of a thin film with Lz < Lp, the weak localization
correction to the sheet conductance = crL, is now simply given by
A<72d = Г AtP^Xt} = - In	(1.114)
Til Jt	Til \T /
where we have used sharp upper and lower cutoffs т,р and t. Evidently д\<72<1 is universal
up to the logarithmic factor which diverges in the limit tp oc. In the absence of
32 Theory of Coherent Transport
any phase breaking processes we therefore find that in two dimensions already the
leading correction to the Boltzmann-Drude result is divergent. This is an indication —
but certainly not a proof — that at T = 0 all states in a two-dimensional system are
localized even if kpl 13> 1. Moreover, writing the Boltzmann-Drude result for the two-
dimensional sheet conductance in the form cr0,2d = {e^/h^kpl (i.e. the universal quantum
conductance e2/h times the large factor kpl 1), we see that Acr2<i is of relative order
(kpl). For a quantitative estimate of the weak localization correction we need the
phase breaking time tv. As will be shown in section 1.3.3, the corresponding rate r~l
typically vanishes with a power law т~1 (T —> 0) ~ Tp at low temperatures (for instance
we have p = 1 from electron-electron interactions in a two-dimensional situation). As a
consequence we obtain a logarithmic decrease of the conductivity at low temperatures
Aa2d(T->0) = -p— In M	(1.115)
7ГП \ 1 /
which was first observed by Dolan and Osheroff in 1979 [44] (a similar logarithmic de-
crease of the conductivity in two dimensions is also obtained from Coulomb interaction
effects [5], it may be distinguished from the weak localization contribution discussed
here by the fact that only the latter is sensitive to weak magnetic fields, see section
1.3.2). It should be noted that a decrease of the conductivity with temperature is
fundamentally different from the classical behaviour where a increases when the tem-
perature is lowered due to the freezing out of inelastic collisions. Such effects are not
included in our calculations. Indeed they are irrelevant at the temperatures of interest
here, where the Boltzmann-Drude conductivity is just a constant.
Weak localization effects which are larger than of relative order l/kyl are obtained
by considering a quasi one-dimensional situation. This is most easily realized by cre-
ating a narrow channel of width Ly C Lv in a genuine two-dimensional electron gas of
a semiconductor heterostructure [21]. Solving the corresponding diffusion equation in
d = ‘2 with boundary conditions dyP = 0 at у = 0 and у = Ly, the probability density
at the origin at time t reduces to a one-dimensional behaviour P(t.) « P[°°\t)/Ly for
Ly, A> Ly. The conductance G = cr^Ly/Lx of a wire with length L.r then has a weak
localization correction
AG = J±DL Г	(1.116)
Trh LT Jo (AttDI)1/'2	тек Lr
which is independent of the channel width (still keeping Ly C < £, . note that in
(1 = 1 there is no need for a lower cutoff r in the integral over t since G 1/2 is integrable
near zero). Moreover AG/Gq is of order (Lv/Ly}(kviyx. i.e. much larger than in the
two-dimensional case. We will not discuss this further except for a few final remarks:
(i) In spite of its name, weak localization should not be confused with the much
harder problem of true (or “strong”) localization, i.e. the effect that for sufficiently
small values of k?l the dc-conductivity vanishes at T = 0 due to the localization
of the corresponding eigenstates. In fact in d — 1 it can be shown [29] that all
1.3 Weak Localization 3c
states are localized even if kFl 3> 1 with a localization length which is of ordei
145]
(1.117!
where N is the number of discrete transverse eigenmodes (“channels”) in tl
wire. Qualitatively the result (1.117) may already be obtained by comparing
the weak localization correction AG in (1.116) with the Boltzmann-Drude result
Go = (e2/h.)(Lv/Lx)kpl. Indeed, while AG should clearly remain much sma”
than Go for perturbation theory to be valid, we may naively determine a vahm
l,y, ~ kFLyl such that Go + AG = 0. Since localization effects can only 1-
observed if Lv is larger than the localization length £ [45], this leads precise!;, io
the result (1.117) with N w kPLy the number of channels in a wire of width L, Л
s.milar consideration can be applied to the two-dimensional case using the row.''
(1.114) and In]^/?-) = 21n(L¥,//). From the naive extrapolation a0 + Ac — I,1 .
obtain a value Lv which gives a corresponding localization length
/ 7Г \
6>d ~ I -kvl ] .	(1.1J«!
Since £2d remains finite even for very large values of kpl, this indicates that als<-
iii two dimensions all eigenstates are localized for arbitrarily weak disorder as in
d — 1. For weak disorder, however, the associated localization length is obviously
extremely large and thus localization in <7 = 2 is not observed in practice in
systems with kpl 1.
(ii) In our present treatment we have completely neglected the effect of spin which
only appears through a trivial degeneracy factor two. In reality spin does p’
a role in systems in which there is either an appreciable spin-orbit interaction
oi when magnetic impurities are present. The corresponding extension of t’
t heory is straightforward in principle [5] and leads to a variety of novel effects, hi
particular it turns out that spin-orbit scattering may reverse the sign of the w< •’
localization correction. The resulting weak-antilocalization may be trace,! Li'i.
tc the elementary quantum-mechanical effect that rotating a spin 1 /2 around ;
pioduces a phase factor -1. For a discussion of these issues see e.g. Ref. [33.3S].
(iii) As was explained above, weak localization is due to constructive interference lei
backscattering in a random medium. For electrons, whose wave nature disappears
in a purely classical description it is thus a true quantum effect. Obviously,
hi wever, the same effect may occur for arbitrary classical waves in a disordered
ni-'dium, e.g. for acoustic or electromagnetic waves. In particular then' is a
coherent backscattering contribution for the propagation of light in a sample
with random scattering. The corresponding theory has been reviewd e.g. by
M Stephen in Ref. [19].
34 Theory of Coherent Transport
1.3.2 Aharonov-Bohm interference effects
One of the fundamental interference effects in quantum mechanics is the Aharonov-
Bohm (AB) effect. Formally it results from the fact that it is the vector potential A
rather than the associated magnetic field В = V x A which enters into the Hamiltonian.
As a striking consequence of this there is a flux-dependent interference of electrons
moving outside of an ideal solenoid which encloses a finite magnetic flux ф, although
the magnetic field is zero at any point where the electrons actually are, i.e. where the
wavefunction is nonvanishing. Historically the AB effect was verified with electrons in
vacuum but it seemed impossible to see a similar effect for electrons in a conductor
since the unavoidable scattering processes were expected to destroy any interference
phenomena. This naive picture, however, failed to recognize the fact that collisions with
static impurities do not destroy phase memory and thus at sufficiently low temperatures
where Lv I AB interference effects should be observable. In the following we will
discuss the effects of a magnetic field in the context of weak localization, for a general
review of AB effects in mesoscopic systems see the references [46,47].
Classically a uniform magnetic field В = Bez leads to a bending of the electron
trajectories into cyclotron orbits with radius rc = where wr = eB/M is the
associated cyclotron frequency. In a quantum mechanical treatment the cyclotron
motion is quantized into Landau levels leading to a completely discrete spectrum En =
hwc(n + 1/2), n = 0,1,... in d = 2 where the motion along the field is quenched. Now
in the weak localization context the Landau quantization is in fact negligible since the
relevant fields are small enough that rc U> I or equivalently wcr < 1. Thus an electron
is scattered long before a cyclotron orbit is closed and to lowest order it therefore still
moves on essentially straight lines between two scattering events. Up to corrections
of order (wcr)2. the longitudinal component of the two-dimensional Boltzmann-Drude
conductivity tensor in a finite magnetic field [21]
_ cr0 / 1	-wc7- \
°’0 ~ 1 + (wcr)2	1	/
thus coincides with its value at В = 0. For a quantum mechanical particle, however, the
magnetic field is relevant even if the bending of the classical trajectories is negligible.
Indeed, quantum mechanically, an electron moving between two points 1 and 2 acquires
a phase factor exp(-(ie/h) jf da: • A). Since V x A = В is nonzero, this phase factor
is “nonintegrable”, i.e. it depends on the specific path taken between two given points
1 and 2. Now in the limit kyl » I a quasiclassical description is possible in which
the electron trajectories are approximated by straight lines. The resulting disorder
averaged GF Ga in the presence of a magnetic field are then obtained from their zero
field expressions by a simple multiplication with a phase factor
(i||2-') и (.т/С^х') exp (-уА(ж)  (x — x')^ .	(1.120)
This result is obtained by considering the representation (1.49) of the GF in terms of
exact eigenstates and approximating the effect of a magnetic field simply by a phase
1.3 Weak Localization 35
factor i/’a,A(x) ~ 'фа(^) exp(-(ie//i) A(x) • x). Using the fact that for weak fields with
1 the vector potential changes appreciably only over distances much larger
than I which is the scale on which the averaged GF decays (see (1.63)) this leads
immeciately to (1.120) (originally this argument was first used in the context of the*
theory of superconductivity with the zero temperature coherence length £0 playing the
role of I [35]). Let us now examine how the additional phase factor in (1.120) affects the
calculi' tion of the quantum corrections to the Boltzrnann-Drudc conductivity. As was
explained above, these corrections may formally be expressed in terms of the vertex
funetkn Г, which in the ladder approximation is completely determined by the product
C of two averaged GF as given in (1.94). In position space we have to evaluate
G(rr - z') = u2(x|G>'>^'|G^)	(1.121)
in a w >ak magnetic field В 0. Using (1.120) it is obvious that the phase factm.
cancel each other and thus to the order we are considering, the diffuson is unaffeco d
by the field. In momentum space this result may be obtained more directly by real’
that the approximation (1.120) is equivalent to the so called Peierls substitution fe; ' >
k' + eA/h. Because the diffuson only depends on q = k — к' the vector potcuiial
cancels. Physically this is a consequence of the fact, that for the diffuson the particle  -:
hole fo low identical paths and thus there is no enclosed magnetic flux. The situation
is quite different however, for the Cooperon contribution where particle and hole
scattered in reversed order. The relevant function £ is therefore given by
^(.z- - .;;') = u.2(z|G>-')(z|G*|;r') « (c(.z - ?) exp (—A(i)  (x - x')) (Ы22.
thus acquiring twice the phase factor from (1.120). In momentum space this follows
again f'om the Peierls substitution since the Cooperon momentum is changed into
Q k — к1 —>Q + 2eA/h. Physically the appearance of a phase factor for charge
—2c is due to the fact that the two interfering time reversed paths both enclose the
full flux ф (see also the discussion of the Sharvin-Sharvin experiment below). The
additio nil phase factor for the Cooperon in the presence of a weak magnetic field may
now be easily incorporated into the calculation of the corresponding weak localiza* is 
correction. In fact it is sufficient to change the diffusive eigenvalue equation into
-D^V + iyAj Pn(x) = Xt,Pn(x).	(l.U U
Moreover the boundary condition (1.107) of vanishing outgoing current must i
generalized by using the covariant derivative dn + 2i(e/7i) A|u similar to the Ginz. ...
Landau equations in superconductivity [48]. In the following this is applied to calculau
tin1 weak localization correction to the conductance of a thin film with thickness L
in a perpendicular magnetic field В = Be:. Up to a factor h the eigenvalues A„ ot
(1.123) ire formally identical with the energy levels of the Schrodinger equation for
particle with mass Л/ = h/2D » M/k^l and charge —2c in the presence of a magnetic
field. T icy are therefore given by the Landau result Xn = wc(n + 1/2) with an effective
36 Theory of Coherent Transport
cyclotron frequency wc = 2eB/M = 2D//^ where /д = (h/2eB)1/Z2 is the magnetic
length for a particle with charge 2e (note that Tc is larger than by a factor kpl _2> 1
and thus the Landau quantization is relevant for the Cooperon equation (1.123) while
it is negligible for individual electrons). For a film with total area S each of these levels
is S/2Trlg-fold degenerate and thus, using (1.109), the weak localization correction to
the dc-sheet conductance cr2 = is given by
л ,n,	2c2 1 S v-,/1	2D/	1\\
До2(В)- 7rhDS27T^ E + Z2 (n +
(1.124)
The sum over n has to be cutoff at values such that Xnr « 1 which gives 7imax ~
(/H/Z)2 W 1. Defining the phase coherence length t.tJ = (Drv)}/2 as in (1.113) with
D = l2/3т the classical diffusion constant, evaluation of (1.124) in the limit I <' Lv
then gives
4’2<B> = ~2^
(1.125)
where Ф is the standard Digamma function. There are now two different magnetic
field regimes, depending on whether /д 2> L^, or C L^. They are seperated by a
characteristic field
Bv
Фо
2L2
-В
(1.126)
(7>o = h/c is the flux quantum for a .single charge e) which is about 10 Gauss for
Lip = 1pm. For fields much smaller than Bv. the area L2 which an electron sweeps
out in its diffusive motion before it looses phase coherence contains a magnetic flux
much less than the fundamental unit ф0. In this limit we use the asymptotic expansion
Ф(1/2 + z) = lnc + l/24z2 for c > x, to obtain
Acr2(B < В J = In
U J 3/i \bJ
(1.127)
While the В = 0 term obviously reproduces the previous result (1.114), we find that the
magnetic field contribution is quadratic in В and reduces the zero field weak localization
correction. It leads to a negative niagnetoresistance ~ B2 at weak fields which is indeed
observed experimentally. The physical origin of this effect is the destruction of time
reversal invariance by the magnetic field. The coherent backscat tering which strongly
increases the sheet resistance if 3> I is therefore suppressed, leading to an increase
in the conductance with increasing field. In the opposite limit, В <>• B.„ where 1ц Lv
the Diganima function in (1.125) is negligible compared to the first term and thus
Ai72(B»BJ = --^—In
Фо
An BP’
(1.128)
1.3 Weak Localization	3~
Fig. 1.9: Schematic view of the such
cylinder with length Lz and circumft i
ence L used in the Sharvin-Sharvin ex-
periment with attached current leads J
The magnitude of the weak localization correction thus becomes logarithmically small- '
with increasing В and vanishes for fields of order фо/l2 where the backscattering 1
rection is completely destroyed. Apart from its intrinsic interest the study of the lit id
dependence of weak localization is a very important sub ject in quantum transpoit m
general. Indeed the relations (1.126), (1.127) or their generalizations to different ge-
ometries allow to measure directly the phase coherence length L^, which is crucial for
determining the range over which coherent propagation of electrons occurs in practice.
For example, extending the above considerations to quasi one-dimensional systems, ZF
turns cut to be larger than the value given in (1.126) by a factor Ly 1. A further
enhancement occurs if the motion across the win* is ballistic (L;/	/) due to the so
called dux cancellation effect [21].
In .he remaining part of this section we want to discuss how the AB effect can
directly be observed in a transport experiment even if the motion of the electrons is
diffusive. The basic experiment was performed by Sharvin and Sharvin in 1981 (see e.g.
[-17]). Imagine that the. above film is wrapped around to form a long hollow cylinder
with circumference L = Ly, length Lz Lv and thickness L, <d Lv (.r and z are
interch rnged compared to the notation above). Now insert a long solenoid along the
cylinder axis which carries a finite magnetic flux ф but ideally has vanishing magnetic
field oi tside, i.e. В = ф62(х)ег (see Fig. 1.9, in practice this cannot be done due to
the sm dlness of L ss Lv and a uniform field is applied, giving ф — BL2/^). Since
В = 0 at any point where the electrons move, the classical conductivity is unaffected
by the flux. Quantum mechanically, however, electrons which encircle the solenoid
on different sides exhibit an AB interference provided their motion is coherent aloiy
I he cir< umference which requires Lv > L. This effect is present already in the weak
localization correction to the classical conductivity which as we have seen — aris<”-
from interference between the two opposite directions in which a closed electron path
may be traced out. Now any closed path on the surface of the cylinder may be classilh
according to its winding number n = 0,±l, ±2,... which is the number of times th<-
trajectory winds around the г-axis. This winding number is a topological invarimii
and arbitrary paths with the same value of n can bo continuously deformed into e:i< b
38 Theory of Coherent Transport
other, while paths with different n’s cannot. Since a closed path with winding number
n encloses flux пф, the interference factor between such a path and its time reversed
partner (which has winding number —n) is obviously
ei2?rn0/0o (e-i2?rn<£/0o V _ gi4?rn0/0o
(1.129)
(note that e/h = 2tt/0q). In order to calculate the weak localization correction to
the sheet conductance ct2 = aLx usc the result (1.104) with P(t) = P^O/Lx since
the film thickness Lx is much smaller than Lv. However one has to take into account
that for a nonsimply connected sample like a cylinder, the two-dimensional probability
density at the. origin P2(^) has to be decomposed into contributions over different
winding numbers with relative phases as given in (1.129). As a result this probability
is now
,	1	“	( (nL)2 Длф
(1.130)
Including a factor e due to dephasing, the time integral in (1.104) can be performed
explicitly giving
2e2 °°
- <?2(ф = °) = - ^ 52к0
nL\ / 4тг0\
— cos n--------
J \ Фо J
(1.131)
with 7<0(z) a modified Bessel function. Here we have used that the lower cutoff r
can be taken equal to zero in all terms with n 0 since the singularity at t = 0
is now integrable. The result (1.131) shows that the sheet conductance contains a
flux dependent contribution which is periodic in ф —>• ф + фо/2 in perfect agreement
with the experimental result, [47]. Since the phase coherence length is typically
of the same order as the circumference L, the contribution n = 1 is the dominant
one. Higher harmonics are damped out exponentially like exp(—nL/L^), which is the
probability of retaining coherence over a distance nL. The occurrence of a periodicity
in normal metals which is identical with that appearing in superconductivity where
one has electron pairs with charge —2e seems surprising at first sight. It is simply
related, however, to the peculiar interference structure in weak localization, where
both interfering trajectories enclose the full flux ф (see (1.129) above). Indeed, using
quite general arguments based on gauge invariance, it may be shown that in any ring
geometry with a finite magnetic flux ф all observable quantities д{ф) must be of the
form [10, 49]
= go + 52 fh cos
n=l
(1.132)
i.e. they are periodic under ф ф + irrespective of whether the system is normal
or superconducting. It is important to note that this general result includes both the
case of no flux dependence at all (<;„ = 0 for all n = 1,2,...) or the weak localization
result (1.131) where gn = 0 for all odd n, similar to the situation in superconductivity.
1.3 Weak Localization 39
Now in normal systems one expects that the lowest harmonic n = 1 should be the
dominant one in general. It arises from the interference of two partial waves each of
which anly encloses half of the flux. Indeed this contribution was observed in small
metallic rings with height Lz <C Lv [50]. The long cylinder with Lz » Lv used in the
Sharvin experiment, however, may be viewed as a collection of Lz/Lv > 1 independent
rings. Since the phase 7n of the n-th harmonic in (1.132) is random for each ring, the
flux sensitive terms are usually averaged out in such a geometry. This argument does
not aprly, however, to the weak localization contribution to fji, g.i,. . In fact, since
weak localization always decreases the conductivity (assuming no spin orbit effects),
the coi responding contribution in each of the effective rings has the same sign and
thus survives the averaging. Finally it is instructive to give a derivation of the result
(1.131) by using the expression (1.109) for the weak localization correction in terms of
the eigenvalues An of the diffusion equation for the Cooperon. To do this we note that
the vector potential associated with an infinitely thin flux line along the z-direction f
given tv A — (0/2тгг)е^ with the unit vector tangent to the cylinder r = const.
Since 2rrr —> L and ev ey in our geometry, the eigenvalue equation (1.123) reads
n [Л 4тг0
2
The fact that the film forms a cylinder imposes the boundary condition P(y + L, z', —
P(y, z) which leads to eigenvalues
т'2тг\2 /	ф \ 2	.)
— D ( — j I m + 2— j + DQ,	(1.1'<11
\	фо J
with ni = 0, ±1,... The presence of a finite flux ф thus simply shifts the disci :e
momen ,a Qv according to QyL = 2тг(7п + 2ф/фо). Using (1.109) and performing tw
Qj-integration from (1/Тг)£у. —> ./<1Qz/2tt, the weak localization correction to th
sheet conductance <r2 i-s
e2 /. /	6\2V'12
Acr2 =------- V // + I m + 2^-	(1.135)
\	<?()) J
with b = Т/2тгТ¥>. The summation over in = 0, ±1................ which corresponds to the
discrete angular momentum around the z-axis. can now be transformed to the dual
winding number sum over n = 0, ±1 ... via the Poisson summation formula
<l.r/(.r)e‘tal'r.	(1.13-
II)	11	l>'
In this manner one exactly recovers the result (1.131) of the above- more intuitive
derivati)n. (The contribution n = 0 with zero winding number is formally diverg- : t.
however since the diffusion approximation is only valid on scales larger than the nw w
free path I, the maximum value of Qu is of order 1//. This restricts the m-sum ’’>
m < L/l and thus implies a cutoff |.t| < L/l in the integral over ./ which then precis-
yields tlie result (1.114) for the weak localization correction of a planar film. No su- b
cutoff i< necessary for tin- n 0 contributions, similar to the calculation above).
40 Theory of Coherent Transport
1.3.3 Dephasing
A crucial parameter for the calculation of the weak localization correction to the con-
ductivity is the dephasing time or the associated phase coherence length Lv, which
is equal to (Dr^)1/2 for diffusive motion. In the preceding sections this parameter was
introduced in a phenomenological manner, argueing that an electron looses its ability
for interference exponentially with time on a typical scale In the present section
we want to see how this assumption can be justified by a more microscopic approach
and which are the dominant processes contributing to dephasing.
Following Chakravarty and Schmid [33], this problem is most conveniently treated
by considering the Feynman path representation (1.54) of the semiclassical propaga-
tor К at fixed energy eF. As was explained in section 1.3.1, the weak localization
contribution to the conductivity arises from those terms in A')2 where the electron
and “hole” paths о and [3 are time reversed partners, i.e. ,3 — a. Provided that the
Hamiltonian does not break time reversal invariance (T-invariance). the corresponding
classical actions S obey Sa = Sa. In this case, which is realized for an arbitrary static
scattering potential U(x), there is no dephasing and thus e.g. the weak localization
correction in two dimensions diverges in an infinite sample. In practice, the electron is
always coupled to other degrees of freedom like phonons which destroy T-invariance.
Here it is crucial to understand that although the total electron-phonon Hamiltonian is
T-invariant, this invariance is broken if only the electron’s trajectory is reversed. Thus
phonons lead to dephasing because one is observing the interference1 of the electrons
alone, not that of the whole system. Formally the classical action S[a:(/)] of the total
system for a given electron path x(t) is different from S[x(-1)], where1 the electron
takes the time reversed path. The phase difference
hip = S[x(t)] - S[x{-t)]	(1.137)
between x(t) and x{—t) which enters into the interference contribution just like in
(1.129) is thus both a functional of the electron trajectory and the eliminated degrees
of freedom, e.g. phonons. Since the latter remain unobserved we introduce an average
() which should also include an average over the relevant electronic motion (see below).
The weak localization contribution to (| A'|2) is then formally given by
(|AT)wl = (E « EI--U vn.	(1.138)
a	a
Here, in the second approximate relation, we have assumed that all perturbations which
destroy T-invariance are weak enough that they are negligible for the electron’s dy-
namics at the classical level, which is determined by the incoherent sum of probabilities
|.4„|2. From (1.138) it is now very plausible that «quite generally the quasiprobabil-
ity /’^'(t) introduced in (1.104) is related to the standard classical probability density
F(t) for an electron at the initial point at time t by
P^\t) = F(t)(e^).
(1.139)
1.3 Weak Localization 41
The problem of dephasing is thus formally reduced to identifying processes which lead
to a nonzero value of <p as defined in (1.137) and to calculate the corresponding average
of e"A In particular, it has to be shown explicitely whether for large enough times the
factor (e'*’) can indeed be written in the simple exponential form exp(—t/rv) as was
assumed in (1.106).
A problem closely related to dephasing arises in the theory of pair breaking in con-
ventional s-wave superconductivity, where non T-invariant perturbations like magnetic
impurities lead to a lowering of the critical temperature [48]. In this context, the de-
phasing factor (e’^jjs usually written as the correlation function (/<(t)/<t) of the time
reversal operator К [48,51]. As a simple specific example we consider the case of ;
two-din ensional electron gas in a fixed perpendicular magnetic field В = Be,. Since
the field remains unchanged under a reversal of the electron trajectories, this is indeed
a non d-invariant perturbation although no inelastic collisions or any eliminated <’
grees of freedom are involved here. To calculate the associated phase we note that, a
magnetic field В = V x A gives a contribution — e/J dtv  A to the classical action of’
an elect-on. Thus, since v -> -v under T-reversal, the phase difference between '
oppositely transversed trajectories is simply
2e ft	-	,
= dtv • A =	(1.1 ' '
where A[a;(t)] is the enclosed area of the closed trajectory x(t) and /B = {h/2e.B)1'2
is the mignet.ic length for charge 2c. Provided that, the magnetic field is weak enough
to negle’t the bending of the trajectories on the scale of the mean free path /, the
classical probability P(t) and the related conductivity rr0 are unaffected by the field.
To calct late (e1^) we thus have to average over the random diffusive motion of the
electron in the absence of the magnetic field. Now the probability density p(.4) for the
enclosed area A in a two-dimensional diffusion process in time f is known exactly [52]
and
is just tie associated characteristic function. Evidently this factor decays with a typ-
ical time scale = l2l}/D, although the behaviour is not strictly exponential. It is
straightforward to check that the weak localization correction (1.104) which follow ,
with the expression (1.141), gives the identical result for Ao2(B) as has been deii cd
in the pr 'vious section by different means. In particular the present, formulation sb-
I hat. the1 dephasing due to inelastic processes like phonons does not differ in prii;
from the effect of a fixed magnetic, field. Indeed, as (1.141) shows, the factor (e,lp) may
decay in time even without any inelastic processes just, as a consequence of the diff”
sive random nature of the electronic motion. Inelastic scattering is therefore neith. .
necessarx nor is it, sufficient for dephasing [53].
As a second example for processes which lead to dephasing we consider the influ-
ence of electron-electron interactions. This calculation is bast'd on a remarkable idea
by Altshuler. Aronov and Khinehiitskii [54] that the ability for interference between
42 Theory of Coherent Transport
two partial waves of a single electron is reduced by the interaction with all other elec-
trons, which essentially provide a fluctuating electromagnetic field for the motion of
the electron considered. This fluctuating field may be described by a Gaussian random
vector potential A(a:, t) with zero average and variance (qa = qa/q, for convenience we
choose кв = 1 in the following)
2T
(.4,,.4.3)^^. =	- qaqp.	(1.142)
emF
This result follows from the standard connection between the imaginary part of the
inverse dielectric function	and the fluctuations of the electromagnetic field in
equilibrium [35]. In conductors at low momenta and frequencies, Im£'-1((y,cu) behaves
like w/4tt<7 with a the actual three-dimensional conductivity. The result (1.142) —
which evidently contains only longitudinal fluctuations — is then applicable in the
classical regime [53, 54]
Dq2, — <w< — < — <<?	(1.143)
п T
where, in particular, the energy transfer hu> is smaller than the thermal energy T.
Moreover the dephasing rate I/t^, must itself be smaller than the thermal rate и>т = T/h
which is about HP's at T = IK. (Note that the three-dimensional conductivity a
has dimensions of a frequency and that от « (е2//г.с>р) (A;i-'Z)2 is large compared to one
as long as kpl W 1). Since in contrast to (1.140) the vector potential is now time-
dependent and this dependence is not reversed under x(t) —> x( — t) because it arises
from the background electrons, we have
<p[x] = -<-[ dti u(Zi) [А(ж(|Ь)Л) + А(ж(-МЛ)] •	(1.144)
/г ./о
In order to calculate the dephasing factor (ei¥>) we use the fact that (e1*’) =exp( —(<p2)/2)
for Gaussian random variables with zero mean. The resulting average; (ip2) contains
four terms. The two mixed ones do not give a contribution which goes to infinity as
t no. The two other terms are equal and thus expressing A(x,t) in terms of its
Fourier transform, the result (1.142) for the variance implies that
1	,	e'2 f* . f1 ,	1 v-' fu'r 2T i ,ii i i
= 'M	7------
2	h Jo /о I	2k crur
X E(^.f„(t1)7,eM(G)ei^(tl)-"(,2)l) .	(1.145)
Writing Ea (7«.i'„(f Je'4^*'1' = (it/)’1 deiqx(tl'/dti and similarly for x(t2). partial inte-
grations of the two time integrals give a factor u/2. Using
( exp(iq [a?(G ) - ar(62)])) = exp( - Dq2\t, - t-,\)	(1.146)
1.3 Weak Localization 43
for diff isive motion one obtains
=	de. 1	d	д
ah J-uT 27Г V у q2 Jo Jo	v
Since J‘ dti dt2 /(ti - t2) grows like t duf(u) for large times, we have indeed
the expected behaviour (e'*’) —> exp(-t/r¥,) with a dephasing rate
1 _ ILEye2	(L 1 „	1
tv uh J-шг 2тг V	w2 + (Z)<?2)2'
(1.,
In a three-dimensional system the ^-integration gives a contribution ~ [w]-1/'2 and thus
1
5м
V
З/б?'2 3/2
(Ы)2 Шг
(1.145)
is proportional to T3?2 at low temperatures. This rate is closely related to the lifetime of
an electron (or better quasiparticle) at an energy of about T above the Fermi surface
[54]. Evidently, as T —> 0, the uncertainty in energy is still much smaller than the
energy itself and thus the quasiparticle concept is meaningful. Due to the presence
of disorder which is necessarily connected with the finite conductivity u, however the
standard small lifetime uncertainty proportional to T2 in an ideal Fermi liquid [4,3'!
is changed to a much larger value ~ T3!2. In two dimensions we replace
ly 1 [ d2?
Lj (2tt)2
and thus with g = cr2(/i/e2) the dimensionless sheet conductance, one obtains
1
ijj.r г^т de.
g Jo ш
(1.151
Here ev dently our bare expression for ту1 is logarithmically divergent and the integral
has to l e cutoff at a minimum frequency of order l/ту, itself, since fluctuations' of the
electromagnetic field on time scales slower than are effectively static and thus do not
contribi te to dephasing (note that this would not be true for a static magnetic field
for instance as was shown above, however here we have only longitudinal fluctuations
correspt nding to an effectively static electric field which does not dephase, see also
[54]). For good conductors with g w- 1, the resulting selfconsistent equation for т,г, can
easily Is’ solved by
1
In g
—-u>T
<)
(1.152)
which is linear in T and small compared to hjt as required. In many cases the result
(1.152) >ives the dominant contribution to dephasing in two-dimensional films at low
44 Theory of Coherent Transport
temperatures. In order to determine the relevant film thickness Lz, below which two-
dimensional behaviour is realized, we recall that the replacement (1.150) requires that
all modes with a nontrivial г-dependence are negligible. This is obeyed if Dq2 A>
Wmax = wr for the discrete values qz = mr/Lz. As a result the relevant condition reads
Lz <
W 1
(1.153)
defining a thermal length Lr for diffusive motion. It is interesting to compare this
with the condition Lz L~ appearing in the weak localization context (see (1.113)).
Evidently for effects related to the Coulomb interaction, it is Lp rather than L.~ which
is the relevant length [5]. Finally let us discuss the quasi one-dimensional case of
wires whose transverse dimensions are much smaller than Lp. In this case a one-
dimensional (/-integration in (1.148) gives a contribution ~ |w|~3/2. Cutting off the
frequency integral at umin = r,”1 and determining from the resulting simple self-
consistent equation gives
1
zyld
(8^D\2/3 ,/л
\V2gw)
(1.154)
where g is again t.he dimensionless sheet conductance; and W the width of the wire.
The dephasing rate via electron-electron interactions is thus proportional to Т2?л at
low temperatures and obeys 1/r^, < wp only at temperatures such that ajp > D/gW2.
Now this result seems to indicate that the quasiparticle picture breaks down at very
low temperatures. However, as has been shown recently [55], at these temperatures the
disordered system crosses over to an effective zero dimensional behaviour of a quantum
dot, where the quasiparticle broadening vanishes like T2 just as in an ideal three-
dimensional Fermi liquid (in the limit of a single transverse channel a one-dimensional
interacting electron system is a so-called Luttinger liquid, which is discussed briefly in
the following chapter). Another rather interesting recent application of the ideas above
is the calculation of the dephasing rate in a random magnetic field, which in d — 2
behaves like T1/3 at low temperatures. Such a behaviour is observed in the normal
state of high temperature superconductors and provides evidence for the existence of
anomalous gauge field fluctuations in these systems [56].
Finally we mention that the dephasing due to phonon scattering at low T usually
turns out to be much smaller than that caused by electron-electron interactions. Indeed
as T —> 0, the corresponding rate r~l behaves like Тл in the range ft/kpl -''7'	0 and
like T'1 if T -C d/kpl with 0 a temperature of the order of the Debye temperature. To
obtain these results, it is important to take into account that long-wavelength acoustic
phonons which are the dominant modes as T —> 0 correspond to vibrations of the
whole system, including the impurities. As a result, the phonon contribution to t~1
is suppressed compared to the clean case, contrary to the case of electron-electron
interactions whose effect is enhanced by disorder [57].
1.4 Universal Conductance Fluctuations -45
1.4 Universal Conductance Fluctuations
This section gives an introduction to the theory of Universal Conductance Fluctuate '
(UCF), which were first observed as apparently random but reproducible and sampi;
specific fluctuations of the conductance of mesoscopic systems as a function of magnei ic
field [5f] or Fermi energy [59]. These fluctuations are caused by quantum interference
between the many electron paths which contribute to the conductance in the diffusive
limit. In contrast to weak localization, these interferences are absent in theories whit'
take an average over disorder and thus they are a genuine mesoscopic effect. After a
brief discussion of the crucial concept of self-averaging, we present the basic ideas and
techniques of both the diagrammatic and the random matrix approach which allow to
derive t re specific and largely universal expressions for the variance of the conductance
in metallic samples.
1.4.1 Self-averaging
Usually in systems with uniform, static disorder, one assumes that physically mean-
ingful observables are independent of the specific realization of the disorder and onh
depend on macrosopic parameters like volume, impurity concentration etc. This as-
sumption is based on the argument that a sufficiently large system can always be
viewed as consisting of statistically independent subvolumes of characteristic size L'.
Macroscopic observables which have a well defined meaning also on scales L*, then
take on independent random values in each of the N = (L/L*)d statistically indepen-
dent pieces. By the law of large numbers this implies that any macroscopic (but in;!
necessarily extensive) observable is equal to its average value over all subvolumes wit li
probability one. Therefore, as L 3> L*, the system effectively performs an average-
over ma jy different realizations of the disorder, and there arc no sample specific fluc-
tuations although each sample contains a different impurity configuration. In section
1.2.1 we have seen that under reasonable assumptions on the impurity potential U(x).
self-aver rging indeed occurs for the density of states and thus the complete thermody-
namic behaviour is non-random. Now obviously, self-averaging is strictly speaking only
valid in die thermodynamic limit L —> oo. The crucial question which determines the
relevant size for which finite systems exhibit sample specific fluctuations is therefore
whether we can calculate the characteristic length L* and the precise values of the
fluctuations for the observables of interest. For the* particular case of the conductance
it turns out that this length is equal to either Lp or L^. whatever is shorter. As 7
goes to >ero, therefore, the characteristic size of the statistically independent subvol-
umes diverges, a situation which is similar to the case of a thermodynamic system at a
phase transition. Therefore, eventually L* becomes always of the order of the sample
size. It s then no longer sufficient to describe the observables by their macroscopic
ai’craije value but instead we need their whole probability distributions for a complete
descripti >n. In practice one is often content, with calculating the variance of the dis-
tribution which is sufficient if it is Gaussian as is the case e.g. for the conductance in
the metallic limit [37]. To make contact with the experimentally observed behaviour in
46 Theory of Coherent Transport
one specific sample, one then uses an ergodic hypothesis [11,12] (which can be proven
in particular models, see [37]). This hypothesis states that the variance of the conduc-
tance fluctuations which are induced by changing the Fermi energy or the magnetic
field in one specific sample, is identical with the purely statistical variance obtained
by performing a disorder average over many microscopically different samples at fixed
er or B. Specifically let g = Gfh/e2} be the dimensionless conductance which is large
compared to one in the metallic limit. Consider then the correlation function
E(AE, AB) = (<59(£f, B) 6g(eF + AE,B+ AB))	(1.155)
of the statistical conductance fluctuations 6g = g — (g) at different energies and mag-
netic fields. Since systems with very different values of ep or В should be statistically
independent, this function will decay to zero at large AB A> E, or AB Bc. The
associated energy or field correlation ranges Ec or Bc thus have the property that av-
eraging g(ep,B) for a specific sample over a range AE A> Ec of £1.-values or a range
AB Br of magnetic fields is equivalent to a full statistical average over all impu-
rity configurations at fixed and B. Thus F(A.E, Л.В) determines both the relevant
correlation ranges Ec and Bc and the variance
Vary- (y2)~(y)2 = E(0,0)	(1.156)
of the conductance which is obtained in a specific sample when or В are changed
by values larger than Ec or Bc. In the following section this correlation function will
be calculated at T = 0 with the surprising result that Vary is a universal constant
of order one, independent of the size of the system, provided we are in the metallic
regime. Since (y) ~ Ld~2 in this case (see (1.38)), this implies that at T = 0
Var у
W
1
£2(^-2) '
(1.157)

For one- or two-dimensional systems this ratio does not go to zero as L —> oc and thus
the conductance is not self-averaging in d < 2. In d = 1 the ratio even increases like L2
as long as L is smaller than the one-dimensional localization length <fi(| which may be
rather large if there are many transverse channels. As a result typical conductances mav
differ grossly from their average value even in a regime where the behaviour is metallic.
This becomes worse in the localized regime L > £, where the relative fluctat.ions are
exponentially large in L and it is In у rather than у which has a proper distribution
[60]. For dimensions 2 < d < 4, the conductance is self-averaging although not in
the manner which is expected for a classical variable. Indeed from the Boltzmann-
Drude result cr = ne2r/hl, the conductivity depends on the impurity configuration
only through the collision term which is inversely proportional to tin' impurity density
n.,. For small fluctuations one thus obtains
Var у
Var
(пг)'2
(Г
I L
(1.158)
1.4 Universal Conductance Fluctuations 47
with L* = I as the characteristic size. This is the standard 1/N = (L* / L)d scaling of
the relative fluctuations of a thermodynamic variable which follows from the central
limit theorem for N independent random variables. Now from the theory of weal;
localizauon we know that in quantum transport elastic scattering does not destroy tiw
phase o' the electron. Domains of size I are therefore not independent in a quantum
mechan cal treatment. It is only the much larger phase coherence length Lv beyond
which the electron has forgotten the specific configuration of impurities at which it h.,:,
been scattered. Indeed from the diagrammatic theory of UCF described belov., •
finds that the result (1.158) is valid with L* replaced by min(Lr, Lv), provided <
With increasing temperature and thus decreasing L* the magnitude of the UCF it
decreasing like the standard thermodynamic fluctuations with L* playing the role of
the correlation length. It is important to note that this decay is only algebraic in L//
instead of exponential like in the AB effects (see (1.131)), because the UCF do ,
require -oherence over the entire sample as might naively have been guessed. As a
consequence UCF have even been observed at room temperature!
1.4.2 Diagrammatic theory
As was i oted above, the physical origin of the UCF is the quantum interference between
the man / paths an electron can take between both ends of a sample. Qualitatively this
may be understood by considering the semiclassical expression (1.89) of the square of
the Feyi man propagator К at, fixed energy, which is basically the transmission prob-
ability t.irough the sample. Since	Ap. there are many classical paths a from
0 to L and the interference terms with n в (or /)) have effectively random phases.
Therefore, in the disorder average (|A'|2) which determines the average conductance
(<]), only the diagonal terms /1 = a survive. Now consider the variance of the transmis-
sion probability. Assuming again that the amplitudes Aa are real, i.e. all phase factors
are included in the action Stl, this is simply given by
Var | A'|2 = <| A'|2)2	(1.159)
because ,he phase factor exp(i(5n, - ft, + Sm —	vanishes both for «i =
<‘2 = ft and for щ = ft. o2 = ill- This argument shows that quantum interference
is necessary to have a nonvanishing variance of the transmission probability, althoug
the resell ing value of Var |A'|2 can be expressed solely in terms of the classical average'
(|A’[2). .'sing the Landauer formula for the conductance and some further assump-
tions, these arguments can be used to explain why Var is of order one [21]. In t !)<
following, however, we will indicate how the UCF can be rp/m/Atativelv derived from
a microscopic theory. This derivation has the advantage of exhibiting in a very clear
manner that UCF are independent of system size and art1 closely connected with t'
diffusive nature of transport. Our presentation essentially follows the comprchensivi
paper by Lee. Stone and Fukuyama [61]. The starting point is the GI? expression
(1.83) fo' the T = 0 longitudinal conductivity. Taking a cubic sample with volume
I' = L,,',,!'. and a current in г-direction, the corresponding dimensionless eonduc
48 Theory of Coherent Transport
tance g — (LxLy/к,]о..к:С can be written as
/ h2 \2
9 = 2\MLj ^kzk'zG^{kk’)G^k’k).	(1.160)
The correlation function (1.155) which determines the fluctuations of the conductance
is then given by (for simplicity we first consider the case AT? = 0)
/ ft2 \4
F(A£) - 4 —-	£ klzk'lzk2zk'2z
\ ^'4 T’r / i i r i j f
'	' k^k^k^k^
x	(1.161)
- (Gr(1)Ga(1)} (Gr(2)Ga(2)^>]
in an obvious notation. In terms of diagrams, this expression is equivalent to a product
of two conductivity “bubbles” GRGA, which arise by joining the ends of the GR- and
G'A-lines via the summation over momenta in (1.160). Since the square of the average
conductance is subtracted, only those diagrams contribute to F in which some impu-
rity lines dressing the bare GF connect the two bubbles. As has been discussed in the
weak localization context, in the limit kvl » 1 of weak disorder the dominant contri-
butions are diffusons or Cooperons. Physically they may be interpreted as trajectories
in which two partial waves (the electron and the hole) are scattered at the same se-
quence of impurities, following each others path directly or in time reversed order (see
section 1.3.1). For the fluctuations of the conductance, the leading contribution turns
out to be a two diffuson- or Cooperon-diagrani. In terms of real space trajectories, this
corresponds to two closed electron paths following the same trajectory but starting at
different points [62] (the contribution with a single diffuson gives a subdominant con-
tribution). Let. us first consider two diffusons, which have precisely opposite momenta
</ = k2 — ki and q = k[ — k2 in momentum space. Referring to Fig. 1.10, we see that
this requires к\ — k'.2 = k\ — k2 and thus restricts the summation in (1.161) to three
independent momenta. Choosing the direction of arrows, which determine whether we
have a retarded or advanced GF, as shown in the figure and using the result (1.96) for
the diffuson, we see that the upper half of the1 diagram contributes a factor
,,2 I
— 7^—Gr(2)Ga(1)Gr(2')Ga(1')
tv Uqz — iu’
while the lower part gives
777Г— Gr(1)Ga(2)Gr(1')Ga(2') .
T\ IJq- - iw
Here the GF a»1 now already the disorder averaged. GF (1.60), which are diagonal in
the momentum indices. This takes into account all diagrams in which impurity lines
1.4 Universal Conductance Fluctuations 49
Fig. 1.10 The basic diagram containing two conductivity bubbles for calculating the variance of the
conductance. The hatched areas denote either diffusons or Cooperons.
are attached to each GF in Fig. 1.10 separately (see (1.92) and below for a similar
argumcn,). Moreover we have defined a frequency w by A£? = Itcj, which formally
enters in the same way as in the diffuson or Cooperon contribution to the frequency
dependent conductance. In the present context, however, co has a rather different
physical neaning since we are considering two dc-conductances at. energies which differ
by hoj. S lice the singular behaviour for q,ui -> 0 is already contained in the* expression
for the diffusons, we can take q = w = 0 in the multiplying GF. which implies that
ki m k2, Ц « k!2 and £F + AE « eF there. The contribution Fal(AE) of the diagram
shown in Fig. 1.10 is then
F„i(AE)
= 4

1
(Dq2 - iwj2
(1.162)
where the disorder averaged diagonal GF G>’A are now all taken at the Fermi energv.
Similar to the calculation of weak localization in 1.3.1, the' fact that G*RGA is strong! .
peaked around the Fermi energy, allows one’ to replace the factor (FJ2 by fc2/3. The re
maining integral has already been evaluated in (1.101). With 7 = 7ru2p(tF) = hv^/6D
(see (1.59), (1.61) and the definition of и'2 = /yup), the contribution of this diagram to
t he correlation function of the conductance fluctuations takes the simple form
Ftt,(AE) =
. AE '
7t'2E(.
(1.163)
Here we 1 ave defined a characteristic energy Ev = T1D/L2 which is called the Thouless
energy fo • a diffusive system of size Lz. Physically the Thouless energy is equal to
the quantum mechanical uncertainty which is connected with the1 typical time L'2/D
necessary to diffuse through a sample of size1 L,, i.e. in the direction of the current
flow. As we will see, this energy is precisely the characteristic correlation range E,
introduced above. Thus changing the Fermi energy by values of order E<, effectively
50 Theory of Coherent Transport
leads to a complete rearrangement of the electron trajectories. In terms of the thermal
length Lt introduced in (1.153), the condition Lz <g; Lr is identical with T Er . In
order to obtain the full contribution to F(AE) — which must be real by definition
— from diagrams of the type shown in Fig. 1.10, we must exchange the retarded and
advanced GF, which leads to —iAE —> iAE. In addition we have to specify the discrete
momenta in our sum over q by applying proper boundary conditions for the diffuson.
They are P = 0 at z = 0, Lz where we have a contact to an ideal metallic lead and
duP = 0 at x — 0, Lx and у = 0, LIy which is the conductor-vacuum boundary (sec
(1.107), (1.108)). The corresponding eigenfunctions are
Р(хуг)
/ 7TZ\	/ 1Гу\	/ 7TZ\
cos mx— cos my— sin —
к XLJ yLy)	V
(1.164)
with mx<y = 0,1,2,... and m, = 1,2,3,... The ratio (</2.-/тг)2 in (1.163) is thus
transformed into a dimensionless “eigenvalue”
(1.165)
Defining Xm = Xm — iAE/?r2Ec the complete diffuson contribution to F(AE) can finally
be written as
/ 4 \ 2
Г„(Д£) = (-) £
4	' m
1 1
+ - Re — .
2	A2
mJ
(1.166)
Here the first term is due to diagrams as shown in Fig. 1.10, while the second contri-
bution arises from diagrams in which the vertices at the two loops are attached in a
different manner (see Fig. 5d and Fig. 5e in Ref. [61], note however, that the three
and four diffuson diagrams F2 and F3 in this reference should be omitted, see e.g. [63]).
Instead of diffusons, we can also insert Cooperons into our diagrams. Now as long as
the magnetic field is zero, i.e. T-invariance is not violated, they give an identical contri-
bution Fc. = Eq to the conductance correlation function. At В = (J therefore the result
(1.166) simply has to be doubled to obtain the full F{^.E). Taking AE = 0. we can
now evaluate the variance of the conductance at zero temperature. From the expression
(1.166) it is evident that this variance is independent of the system size, containing only
a weak dependence on its shape through the ratios Lz/Llyr An analytical result can
be obtained for quasi one-dimensional systems, which requires LT,y L:. In this case,
the sum over m, which is generally of the form	reduces to a single
sum over in,. since only = niy = 0 contribute. Thus, using 111	= vr'^DO,
the zero temperature variance in d = 1 and at В = 0 is
Vam; = -|-.	(1.167)
15
In a quasi two-dimensional situation with LT Ly,L: or for a three-dimensional
sample where Lx, Ly and Lz are of the same order, the summations have to be performed
1.4 Universal Conductance Fluctuations 51
numerically. For a square this gives Var g » 0.74 while Var g «1.18 for a cubic sample.
The standard diagrammatic perturbation theory for weakly disordered systems thus
explains not only the quantum corrections to the classical averaged conductance but
also the universal nature Var g = const, of the sample specific conductance fluctuations.
A very simple but formal argument which shows that the variance of the conductance
is independent of the system size L, can be derived from the fact that the singular
dependence on L is due to the square of the diffusion propagator (Dq'2)~2 which appears
in (1.162). Since the classical self-averaging (1.158) is equivalent to Var 9|cI ~ 1/E4 “
we have
Var9~L-/ Л4
Ji/L ql	K
This shows that the //“^-divergence due to the diffusive nature of the electron motion
exactly cancels the vanishing of Var(<?) with the system size if the fluctuations wm
classical, leading to a variance of order one for all d < 4. Let us next discuss the eneig-.
correlat on range Ec. According to the ergodic hypothesis introduced in 1.4.1, Ev is
simply ihe half width of F(AE). Now obviously from |Am|2 = X2n + (AE/Tr'2Ec)'2, th;.-
width it of the order of the Thouless energy Ec = TiD/L2 as was claimed above, h
should be pointed out. though, that F(AE) ~ (Ec/AE)(,“rf)/2 decays rather slowl>
for ДЕ 2> Ec (this dependence is easily derived by going from the discrete summation
over m ,o an integration f d,lm).
In о ir discussion so far, we have assumed that there is no magnetic field В =
AB = (i. In order to extend our results to finite Helds, it is sufficient to consider a
semiclassical approximation, in which the only influence of the magnetic field is an
additional contribution </? = ~(e/h) jadx  A to the phase of the* electron, with a a
given classical path. As was discussed in section 1.3.2, this approximation is valid
long as the cyclotron radius rc is much larger than the mean free path I. Recalling
our qualitative discussion of the transmission probability in terms of the interference
between classical paths a and ft, we see that the diffuson contribution to Var|/Vi2
corespot ds to the special subset o-j = and a2 = Bi of four trajectories cp, d, m
field В uid a2,A at. field В + AB. With A and A + ДА as the corresponding
vector potentials, the field-dependent contribution to the phase of the diffuson is th- n
(omitting the irrelevant overall minus sign)
- \ [ dx  A - I dx  A + I dx • (A + AA) - / dx • (A + ДА)
L/d	J fi\	J0'2
= ~ [ dx-AA.	(1.169)
ii JHi-m
Here* J| - O) is the closed path from 0 to L. via ift and back to 0 via oj. Introducing
the corresponding enclosed flux А0(/Л — oj in the magnetic field AB, we see that
r'[> = 2~Дф(/11 — Oi)/<>(> is gauge invariant and most importantly is /independent
of the magnetic field B. Thus, as has been noted already in section 1.3.2. the diffu-
son is ш affected by a magnetic field and its contribution to the correlation function
52 Theory of Coherent Transport
Bc(T = 0)«g-.
FCFtB(AE, AB) only depends on the difference field AB. As a result, field-dependent
conductance fluctuations can be observed up to fields where rc « I and indeed, experi-
mentally, they have been seen beyond 10 Tesla. The field correlation range Bc at zero
temperature may be easily found from the condition that <pD is of order 2тг for typical
closed paths (3\ — оц. With L2± as the area of the sample which is transverse to the
field, this leads to the simple estimate
(1.170)
The pattern of the aperiodic conductance fluctuations is therefore altered completely
if the field is changed by a value AB such that an additional flux quantum threads
the sample. Let us now consider the contribution of the Cooperons. In the notation
above, the Cooperon corresponds to a choice of paths such that cq = 32 and a2 =
Evaluating the corresponding gauge invariant phase acquired in magnetic fields В and
В + AB according to (1.169), we obtain
<Pc=l[ da: • (2.4 + ДА).	(1-171)
The Cooperon is thus sensitive to both В and AB, with a characteristic factor of two
in the dependence on B, which has already been discussed in the weak localization
context. As has been noted in section 1.3.2, the magnetic field leads to Landau levels
for the eigenvalues of the Cooperon. In our dimensionless units they are
A(C) = (2n + l) (—-)	(1.172)
for a magnetic field in ж-direction. Thus even the lowest eigenvalue n = 0 is large
compared to one if lB <C Lz or В ф0/L'2, while the lowest diffuson eigenvalues remain
of order one. The contribution of the Cooperons to the conductance fluctuations is
therefore quenched already at the very small fields which also destroy weak localization.
For such fields only the diffusons remain and the above values of Var g are reduced by
a factor two (for a discussion how to detect this reduction by noise measurements see
e.g. Ref. [63]).
Finally let us discuss very briefly the influence of finite temperature and dephasing.
According to the general Kubo formula (1.80), the dc-conductance of a system of
noninteracting electrons at T yt Q is given by
!J(T)= /dsf-^W)	(1.173)
>	\ (JS /
where g(s) is the T = 0 conductance with Fermi energy e. The variance of the con-
ductance at finite temperature can thus be written as
F(T) = Var.9|r = I dAB K(AE)F(AE')	(1.174)
_____________________________1.4 Universal Conductance Fluctuations 51
with F &E) the T = 0 correlation function defined above and
К(Д£) = I ds/'(£)/'(£ - ДЕ).	(1.17- -
The fin.te temperature variance is therefore an average over the T = 0 variant ,,
different energies with a smearing function K(AE) which reduces to <5(Д£) at T =
0. For practical calculations К(ДЕ) may be replaced by a box distribution funrii- e
K(AE) ~ 9(T - \&E\)/2T. A quite different mechanism which destroys UCF i.,
dephasiag. Indeed it is obvious that for small values of Lv the interference contributions
to |7<|2 are suppressed and the conductance approaches its classical and nonfluctuating
average value. Following the discussion of weak localization in section 1.3, dephasing
may be introduced in a phenomenological manner by adding a contribution l/rv =
D fto the Cooperon denominator Dq2 - icv. For our dimensionless eigenvalues Xm
this corresponds to the replacement
л x ( L* V
Am -> Am+ — 	(1.176)
\ 'ft л-'ф /
It is then immediately evident that the UCF are strongly suppressed for Lv < Lz. At
this poirt it is important to realize, that in the context of weak localization dephasing
enters only in the Cooperon but not in the diffuson because particle number conserva-
tion is unaffected by rv (a consequence of this is that interaction effects are insensitive
to Lv, ste [5]). In the case of UCF, however, the two loops in Fig. 1.10 represent dif-
ferent systems which are only correlated by the fact that their impurity configurations
are identical. The diffuson there is thus not connected with the density-density corre-
lation function and — for instance — electron-electron interactions within each loop
lead to a finite dephasing rate 1/t^, as discussed in section 1.3.3. The incorporation
of dephasing via (1.176) therefore applies to both diffusons and Cooperons. Now al-
though thermal averaging and dephasing, which may be characterized by characteristic
lengths Ly = (KD/T)l/2 and Lv are quite different physically, their effect on UCF i-
very similar. In particular for d, = 2 or <7 = 3 we have
Var<;|r = (L*/L)^d Var<;|7=o	(1.17/j
with L* = min(Z/7-, Lv) < L the shortest length at which coherence is effectively
broken. The result (1.177) shows that the conductance fluctuations have their fiT
T = 0 valance only at temperatures such that L* 5> 7, while classical averaging set-
in once L* becomes shorter than the system size L (in quasi one-dimensional systems
the situation is more complicated, see [61]). Since L* decreases with a power law as 7
is increas ;d, the UCF only decay algebraically with temperature as was noted above.
The characteristic length L* also determines the correlation energy and field at finite
T, which can simply be estimated as
£(.(T) «W/(L‘)2	BC(T)«AO/(L-)2.	(1.178)
54 Theory of Coherent Transport
1.4.3 Random matrix theory
A very different approach to UCF was suggested independently by Imry [64] and Alt-
shuler and Shklovskii [30]. It is based on the statistical theory of levels in random
matrix ensembles for the Hamiltonian or the transfer matrix in quasi one-dimensional
systems (for reviews see Stone et al. in Ref. [19] and Bcenakker, Ref. [68]). In its
simplest form, it starts from the Thouless formula
9-^	(1-179)
which expresses the dimensionless conductance g of a finite sample as a simple ratio of
two energies:
(i) the energy Ec which is the typical shift of the discrete levels induced by changing
the boundary conditions say from periodic to antiperiodic and
(ii) the average level spacing Д at the Fermi energy.
To understand the origin of this formula, we first note that the average level spacing
A « (p(e^LdyX	(1.180)
is simply obtained from the estimate р(гр)ТйД » 1, which states that in a range Д
around the Fermi energy we have typically one level for a system of size Ld. On the
other hand, the shift of the individual levels due to a change in the boundary conditions
may be estimated by the quantum mechanical uncertainty associated with the time it
takes to move through the sample. For a diffusive system this time is of order L'2/D
and thus Eq ~ hD/L'2 is just the Thouless energy introduced above. By contrast, if
the states at the Fermi energy were localized, Ec ~ exp(—F/£) vanishes exponentially,
i.e. localized states are basically not sensitive to a change in the boundary conditions
[65]. Introducing a scale dependent diffusion constant D(L) via EC(L) = hD(L)/E2,
the relation (1.179) can be rewritten in terms of a generalized Einstein relation
g(L) ~ hp(EV)D(L)Ld-'2	(1.181)
which is the basis of the one-parameter scaling theory of localization [7]. In the metallic
regime of UCF, D(E) is a scale-independent constant and the Thouless formula is
equivalent to the standard Einstein relation (1.6), for a recent more rigorous derivation
see [66]. According to this rather profound relation, the dimensionless conductance
g is simply equal to the number Ar(Ec) of levels within a band of width Ec around
the Fermi energy (N(EC) 1 in the metallic limit). Expressed in these terms, the
statement of UCF is then equivalent to saying that the variance of this number taken
over microscopically different samples is of order one,
VarAr(Er) и 1	(1.182)
1.4 Universal Conductance Fluctuations 55
rather than of order {N(EC)) itself, as would be expected for uncorrelated levels with a
Poisson, an distribution. As has been noted by Altshuler and Shklovskii [30], the UCF
from this point of view are therefore anomalously small fluctuations, while they appear
anomalously large compared with thermodynamic fluctuations (see (1.157), (1.153))'
The basic origin of the fact that Var N(EC) is much smaller than for uncorrelated levels
is the well known phenomenon of level repulsion. Indeed level repulsion suppresses ’ ’
occurrence of very small level spacings s —> 0 and effectively leads to a spectrum i..
which s remains close to the average spacing Д, thus reducing drastically the fluctv
ations in the number of levels within a given energy range. This phenomenon is a.
known as “spectral rigidity”. For a quantitative description, one uses the Wigner-
Dyson approach which assumes that the energy eigenvalues of complex systems may
be approximated by those of random matrices H. Their distributions are fixed by
the requirement of maximum entropy consistent with a given average spacing and the
overall s/mmetries of the physical system which is modelled. The most simple case is
the so-called Gaussian orthogonal ensemble which is defined by the distribution
f N / ч \
р(Я)~ехр --jtrlH2	(1.183)
\ ло /
for real symmetric N x А-matrices H. representing real, time reversal invariant Hamil-
tonians. The parameter Ao is an energy scale which determines the average level density
as we will see below. It may be shown that the distribution (1.183) is in fact the en-
semble with maximum entropy, subject to the constraint of a fixed expectation value
of tr№ = J); E2. Since trH'2 is invariant under orthogonal transformations, the distri-
bution is independent of the choice of the basis and matrices with identical eigenvalues
Et are equiprobable. The factor N in the exponent is chosen in such a way that proper
limiting listributions are obtained as N —» oo. For pedagogical reasons, let us first
discuss t ie case N — 2 of 2 x 2-matrices
Ли /*12
Л12 h-22
(1.184)
with three independent random variables	and Л22. Its eigenvalues are Е2д =
h ± s/2 with h = (Ли + Л22)/2 and spacing s = E2 — E\ — (u2 + 4Л(2)Е2 > ().
where v = Ли — Л22. They are fixed if h and s are given. In order to obtain the
distribution of the eigenvalues alone, the distribution (1.183) has thus to be integrat' d
over i> with Л12 = ±|(s2 — v2)1/2. Noting that dhndh,22 = d/id?.’ and d/ids = dEjdE2.
the probability density p(Ei,E2) for the eigenvalues is given by
p(Ei, E-2) ~ exp
г (ll.
Ац / J-s
dhi2
d.s
= |(E2-E,)expf-^-^y	(1.185)
The factor E-2 — E\ in front of the Gaussian is a special ease of a general factor П,>2(Е, —
Ej) (see 11.189) below) which is due to the Jacobian of the transformation from the
56 Theory of Coherent Transport
2V(W + l)/2 matrix elements to the N eigenvalues, with the angular parameters of the
orthogonal matrix which diagonalizes H being integrated over. From (1.185) it is now
easy to calculate the probability density p(s) for the level spacing s
-	( s2 \
dhp(Ei, E2) ~ sexp I —(1.186)
-OO	\ Aq j
which is linear in s for small spacings. The expression (1.186) is the famous Wigner
surmise for the spacing distribution. It is obviously exact for N = 2 but turns out
to be very close to the true result even for N —> oo, which is not known analytically.
According to this result, the probability to find two levels at close distance s —> 0
vanishes linearly with s, which is the probabilistic statement of level repulsion. The
universality of this behaviour can indeed be understood very easily by considering
the expression s = (v2 + Д/г^)1/2 for the spacing in a 2 x 2-matrix. Indeed, even if
there are many levels N 3> 1, the problem of level crossing as a function of some
parameter reduces to an effective 2 x 2-problem in the vicinity of the (usually avoided)
degeneracy. Therefore, generically, two independent parameters v and hl2 have to
vanish simultaneously to obtain s = 0. Provided that the probability density for v and
/;42 remains finite at the origin, the level spacing distribution
p(s) = У dv У dhi2p(«, hl2) 5 (s - (v2 + 4/i22)1/2)	(1.187)
then vanishes linearly as s —> 0 (this is easy to see if v/s and hi2/s are introduced
as new integration variables). Similarly, we may consider the case where time reversal
invariance is broken, e.g. by a magnetic field. Then the corresponding 2 x 2-matrix is
hermitean rather than real symmetric with spacing
s = [(/ijj — h'22)2 + 4(Re /И2)2 + 4(Im /г^)2] ' 	(1.188)
In this case three parameters have to vanish simultaneously for an accidental degeneracy
,s = 0. By a trivial generalization of the argument in (1.187) it is then obvious that in
the generic case the spacing distribution p(s —> 0) must vanish quadratically. Indeed
arguments of this kind go back to von Neumann and Wigner in 1929 in their study of
degeneracies in complex molecules. Quite generally, it turns out that p[s —> 0) ~ efi
with 0 = 1. 2 or 4 for the so-called orthogonal (0 — 1), unitary (0 = 2) or symplectic
(0 — 4) case is a universal result for the three standard random matrix ensembles.
Their corresponding eigenvalue distributions are given by
p[E1<E2...<EN)^X[(Et-EJ)f}exp(^^E^	t1-189)
l>3	\ Л0 г /
generalizing the result (1.185) to arbitrary 0 and N.
In order to see whether UCF are indeed just a consequence of level repulsion in the
spectrum of the Hamiltonian, one needs to calculate the variance of the number of levels
N(EC) in a given range Ec. According to the diagrammatic calculations this should be
1.4 Universal Conductance Fluctuations 57
a universal constant of order one, depending only on the symmetry parameter /3 (as we
have seen in section 1.4.2 this constant also depends weakly on dimensionality which
never aopears in random matrix theories for the Hamiltonian. This shows that the
simple arguments based on level repulsion cannot explain UCF in detail, but they still
indicate why the phenomenon is independent of any microscopic models). The crucial
technical step allowing to evaluate Var,V(Ec) in the random matrix ensembles is a
theorem due to Dyson and Mehta. It states that the variance of any random variable
which may be written in the form
A = ^a(Ei)	(1.190)
i
depends only on the function a(E) and the symmetry parameter /3 via
VarA =	[ dt|a(t)|2t.	(1.1! ,
7ГР JO
Here a(i) = J dE a(E) exp(iEt) is the Fourier transform of a(E). Random variai .
of the form (1.190) are called linear statistics since different, eigenvalues are not mixed
although the function a(E) may well be nonlinear. The proof of the Dyson-Mel
theorem is rather straightforward. Define the unaveraged density of states
p(E) = £ 6(E - Et)	(1.192 
г
and the averaged spectral correlation function
K(E, E') = (p(E)p(E')} - (p(£))(p(E')).	(1.193)
Now provided that I\(E,E') = K(E — E1) is translation invariant on the scale of
interest, the convolution and Parseval’s theorem give
Var 4 = [°° dE [°° dE' K(E - E')a(E)a(E') = - Г° dt |u(t)|2I\(t)	(1.1941
where /Ш) is the Fourier transform of K(E) (both functions are real and even by defi-
nition). "" о prove the Dyson-Mehta theorem, one thus has to show that К(t) =
More precisely, the behaviour A'(t) ~ |t| is only valid for the relevant times whiih
contribute in (1.194). Indeed it breaks down for very short times of order r and also
for times larger than l/Д because translation invariance of the eigenvalue dist’
tion is n< t, valid for energies smaller than Д or larger than h/т. Therefore (1.19-rj i.->
applicable if a(E) vanishes at large energies and has no structure on small scales of
order A. A detailed discussion of K(t) which is closely related to the probability
performii g periodic motions with period t, is given in Ref. [62]. For further progress,
it is convenient to express the eigenvalue distribution (1.189) in terms of a Boltzmann
distribution in classical statistical mechanics
p(B! ...Ev) = exp(—/3?f)
(1.195)
58 Theory of Coherent Transport
with an energy function
H = -£ In(Ei- E,) + £	(1.196)
i>j	i
This function describes a one-dimensional “Coulomb” gas of particles interacting via
a logarithmic two-particle potential which favors to spread out the particles as far as
possible. This spreading is counterbalanced by a quadratic single particle potential
V'(E) ~ E2 which tends to confine them close to the origin E = 0. In equilibrium, the
total force on each particle must vanish dH/dEi = 0, which gives
52 ---------=		(1.197)
M Et -	E,	dEt	V ’
г	J	4
Upon averaging, this relation turns into an integral equation
Г dE'= V'(E)	(1.198)
J —ОС -Cz — Ej
which determines the average density of states (p(E)) in terms of the confining potential
V(E) (actually this is a kind of mean field approximation which, however, becomes
exact as N —> oo). For the particular case V(E) = NE2/A2 discussed above, this
equation has the exact solution (subject to the normalization condition /' AEp(E) = N)
9N /	F2 \ 1/Z2
(p(E)) = — I 1 -j	(1.199)
тгЛо \	Ao j
for |E| < Ao and zero otherwise. For obvious reasons, this is called Wigner’s semicircle
law. The associated average level spacing in the center of the distribution is A =
(/>(()))' = ttX0/2N. Thus for fixed Ao, the spectrum becomes continuous as N —> oo,
with a continuous intensive density of states (p(E))/N which is normalized to one. For
the Dyson-Mehta theorem it turns out that the detailed form of V(E) is irrelevant, what
only counts is the symmetry parameter /5 which fixes the strength of the logarithmic
interaction. To see this, we express (p(E)) in a thermodynamic form with probability
distribution (1.195). By functional differentiation it is then straightforward to show
that
|^|^ = -Ж(Е,Е').	(1.200)
This relation is in fact well known in the theory of liquids, expressing the pair corre-
lation function in terms of the functional derivative of the density with respect to an
external potential [67]. The associated integral equation
(p(E)) - -ВdE'A'(E,E')T(E')	(1.201)
is trivially solved by Fourier transformation p(f) = —	provided Л'(Е. E')
depends only on E - E'. Noting that -itF'(t) is the Fourier transform of I '(E) and
1.4 Universal Conductance Fluctuations
i?rsign(t) that of the principal value P/E, the Fourier transform of (1.198) leads to
p(t) = -|t|V(t)/7r, again as an identity between distributions. By comparison we have
therefore /<(t) = |<|/тг/3 as required for the proof of the Dyson-Mehta theorem.
The number of eigenvalues N(EC) in a range Ec around the Fermi energy, which we
take a, E — 0, is obviously a random variable of the form (1.190) with a(E) — 1 for
|E| < Ec/2 and zero otherwise. Evaluating its variance according to (1.191), we find
that Var A(EC) diverges unless the integral is cut off at f « l/Д, which arises from the
finite level spacing (see above). As a result we then find
2
VarA(Ec) = — ln(A(Ec))	(1.202)
with (N(Ec')) = Ес/Д the average number of levels. This result exhibits the uni-
versal dependence on /3 and the drastic suppression of the fluctuations compared to
uncorrdated levels (where Var A(EC) ~ (N(EC))) due to level repulsion. Obviousljc
howeveq in order to explain UCF from this point of view via the Thouless formula
(1.179) the logarithm in (1.202) should be absent. The origin of this discrepancy was
resolved by Altshuler and Shklovskii [30], who pointed out that the expression (1.202)
only applies to a closed system. By contrast, UCF necessarily require an open system
in contact with a reservoir. The discrete eigenvalues are therefore broadened bv
an amount of order Ec itself. To incorporate this broadening, the number of levels
within a range Ec around E = 0 has to be calculated using a smooth edge, e.g. by
choosing a(E) = (1 + (E/Ec)2) \ With this choice VarA(Ec) = is indeed a
universal constant of order one as required to explain UCF. Unfortunately, the precise
factor depends on the choice of the smoothing function and is always independent c!
dimensionality as was noted above. A quantitative comparison with the predictions
of the diagrammatic theory is thus not possible. In fact, a more basic problem wit h
the abo'7e simple explanation of UCF is, that the eigenvalues of the Hamiltonian am
not sufficient to determine a transport property like the conductance. Now at least Li
quasi one-dimcnsional systems there is a generalization of random matrix theory which,
is able to describe the statistical properties of the conductance itself. This is based on
the exact two probe Landauer formula (see section 1.5.2 below)
y = 2tr(S214)	(1.203)
for the dimensionless conductance between two leads 1 and 2 (actually this shows
that the theory of UCF as presented here only applies to two probe measurements
and not in a multiprobe situation, as is also evident from the boundary conditions in
the diagrammatic theory). Here the S21 are N x А-matrices with matrix elements
N'2i.(,n which are the quantum mechanical transmission amplitudes for going from modi
<1 — 1,2, .., N in lead 1 to mode b — 1, 2,.... A in lead 2. Since S^Sji is hermitean
and positive, it has A real and positive eigenvalues T\,. .. ,TN which obey 0 < Ti < 1
due to u litarity. In order to determine the statistical properties of the conductan •
</ = 2^2iTi, the crucial problem is to find the proper distribution of the variables 7,
for quasi one-dimensional systems with length L, which is much larger than the uteac
60 Theory of Coherent Transport
free path I but smaller than the localization length £id = Nl. This problem was finally
solved exactly by Beenakker and Rejaei [68], at least for the case (3 — 2. They showed
that in the limit N —> oo, the variables A; = (1 - Т^/Тг e [0,oo) have a distribution
like that in (1.195). However the associated energy function
H = Еи(А;,А7) + £У(Аг)	(1.204)
i>j	i
contains a two particle interaction u, which is not translation invariant and also more
complicated than a simple logarithm. In particular it behaves in the standard way
u(A) = — In A only in the limit A —> 0 of strongly transmitting channels 7) close
to one, while for the weakly transmitting ones it is reduced by a factor of two [68].
The associated average level density is far from uniform and may be expressed as (for
limitations see [69])
ч	(1.205)
4/Ф + 1)
with (g) = 2NI/L the average conductance. Here we have supressed a ^-function
cutoff at 2Amax = exp(L/Z) which is necessary for normalization /0°° dA (p(A)) = N.
For expectation values of observables like g = 2^;(1 + A;)-1 containing decaying
functions of A this cutoff can be omitted, however. The distribution (1.205) shows
that the most probable channels are those with A —> 0, i.e. the strongly transmitting
ones. In Imry’s terminology [64] these are the active transmission channels which give
the dominant contribution to the conductance. In spite of the complicated form of
the energy function (1.204), the analog of the Dyson-Mehta theorem is very similar to
(1.191). Indeed introducing variables xt via 7] = cosh-2 .r,, the variance of any random
variable of the form A = Е)га(з;г) is given by [68]
v„.4 = _Lrdt |а(У,,,	(1206)
~Ni Jo 1 + coth (tt/c/2)
with a(k) = J' dx a(x) exp(ifcx). For the variance of the conductance g ~ 2)Гг7’, we
have a(x) — 2 cosh-2 j: which leads to Var y = 8/15/3, in exact agreement with the
result (1.167) of the diagrammatic calculation in one dimension.
1.5 The Landauer-Biittiker formalism
This final section discusses the scattering theory approach to conductances which is due
to Landauer and Biittiker. We start with an elementary derivation of the Landauer
formula in simple quantum waveguides and a discussion of the crucial role of reser-
voirs. The full Landauer-Biittiker theory for general multiprobe conductors is derived
in section 1.5.2, starting from the microscopic Kubo formula for the nonlocal conduc-
tivity tensor. Finally we briefly discuss the associated resistances and their symmetry
properties.
1.5 The Landauer-Biittiker formalism
Fig. 1.11: Schematic picture of a conductance measurement in an ideal electron waveguide of width
IV (x).
1.5.1 Landauer formula
In section 1.2.3 we have shown that the nonlocal conductivity tensor <t(x,x'), which
generally relates the local current density j(x) to the local electric field Л(.г') in linear
response, can be expressed in terms of current matrix elements between the. exact
single prrticle eigenstates of the unperturbed system. Now in most cases one is not
interested in local currents and fields but only in the global conductances Gmn, .
relate ths net outgoing currents Im in the various leads m to the applied voltages 1
via the generalized Ohm’s law (1.12). Starting with his work in 1957 [15], Landauer’"
crucial idea was, that in the absence of any inelastic scattering these conductances
must be expressablc in terms of the scattering matrix S of the sample. Evidently such a
relation provides a considerable simplification in the calculation of transport properties.
Indeed it reduces the problem to the evaluation of ordinary transmission and reflection
coefficients as in elementary quantum mechanics, rather than the complicated current
matrix elements. For simplicity let us first consider the — apparently — trivial case
of an ideal electron waveguide between two reservoirs (see Fig. 1.11). Suppose that
the electrochemical potential of the left hand side is increased by eV compared to that
on the right. Then obviously a net current will flow from left to right because there
is a finite region in energy between Ep and £p- + eV where right going states on the
left hand side are occupied but no left going ones from the right can compensate their
current as in equilibrium. Let be the discrete energies of transverse motion in the
waveguide. At given total energy
b2k2
+	(1-207)
Z Ivl
which is larger than ea (“open channels”), the longitudinal velocity va = hka/M > 0
in a given channel a is then equal to
"a(f)= [^(e-s«)]1/2 •	(1.208)
62 Theory of Coherent Transport
Similarly the one-dimensional density of states for particles in channel a for each di-
rection is
Pa(£) =
1 dka
2?r de
1
27r7w0(e)
(1.209)
which has the characteristic square root dependence on energy. Now in an ideal wave-
guide, electrons which are injected from the left reservoir can propagate to the right
without reflection. More generally, only a fraction Ta < 1 of the incoming current is
transmitted, while the reflected part is soaked up by the left reservoir. The net current
Ia in each mode a can now be obtained from an elementary counting argument. Indeed,
at T = 0 there is no current from right to left and thus, integrating the density per
energy times the velocity over the relevant energy range, one obtains
Ia = e depn(e)u0(e)T0(£) = — Ta	(1.210)
J€p	ll
(as always, we take the linear response limit V —> 0 and thus only the transmission
probability Ta at the Fermi energy is relevant). Obviously, in the ideal limit Ta = 1,
the electrical current per energy is equal to epa va = e/h, a fundamental constant for
each channel a. Thus there is an equipartition rule [21], which states that the current
in an ideal waveguide is shared equally among all available open channels because
a small velocity is exactly cancelled by a large density of states. Summing over all
transverse channels a = 1,2,... and taking into account a factor of two for the spin
degeneracy, the total current I — £)a Ia = GV is linear in the applied voltage with a
zero temperature conductance
2e2
h
(1-211)
This relation between conductance and transmission probabilities at the Fermi energy
is the simplest version of the Landauer formula. It applies to a two probe situation with
no mixing between different channels. Experimentally this may be realized in a quan-
tum point contact (QPC) [17,18]. This is a narrow constriction in a two-dimensional
electron gas whose length is much smaller than the mean free path. Thus the motion in
each of the constriction channels is ballistic with Ta — 1. The number N of open chan-
nels which contribute to the conductance is determined by the number of transverse
modes obeying ea < Ep at the narrowest point. Provided that the constriction width
IF(.;;) changes slowly on the scale of a Fermi wavelength, the longitudinal motion is adi-
abatic and thus there is no backscattering or mixing of different channels [70, 71]. More
precisely, the condition of adiabaticity is did'2(.r)/Ax <C Ap. With A'(.r) = А:рИ’(ж)/7г
as the local number of transverse channels, this is equivalent to dll' (.i;)/d.z;	iV-1(.r)
[71]. In the absence of backscattering the conductance Gqpc = 2eiN/Ji of a QPC is
then quantized in units of 2e2/h as has been observed experimentally with a precision
of better than 10~2 [17,18]. In order to describe the transition between different values
N and N + 1 as the width of the waveguide is increased, it is necessary to include the
exponentially small but finite transmission probability of closed channels. Moreover Ta
1.5 The Landauer-Biittiker formalism 63
is smaller than one even for open channels, because of the finite probability of reflection
over t ie barrier. Indeed, in a waveguide of varying width, the longitudinal motion is
subject to an effective potential which is just the local subband energy £a(x) due to
the transverse motion and thus has a maximum at the narrowest point. Including the
contin ious increase of Ta from exponentially small values to one as a particular channel
a turns from closed to open, gives a smooth transition between successive quantized
values of the conductance [70].
The surprising prediction that even a perfect waveguide with no scattering has a
conductance which is not infinite, but is equal to N times the universal conductance
2e2//i, has long been considered suspect, in particular before its experimental verifica-
tion! Now the crucial point involved here is, that the simple formula (1.211) applies
to a conductance in which the potential difference V is measured between points way
inside the reservoirs. The finite resistance is thus a boundary or contact resistance
between the perfect conductor and the reservoirs, as was first pointed out by Irnrv
[10]. The fact that this resistance has a universal value which only depends on th”
number of modes entering the reservoir appears surprising. In fact it is consequence ot
our ideilized model of the reservoirs which, however, is usually well realized in actjci!
experiments. Here an ideal reservoir is defined by the following properties [36]:
(i)	it is in equilibrium with a given electrochemical potential /acl (which is differ ” <
fcr the various reservoirs attached to the conductor)
(ii)	it is large enough such that the currents flowing in or out are negligible deep
within the reservoir (i.e. it stays in equilibrium with unchanged p(.| even in the
piesence of current flow)
(iii)	electrons entering a reservoir are not reflected back into the conductor befo
ecuilibration.
As was explained above, with these assumptions the current per energy interval between
two reservoirs is equal to e/h per channel, leading to a universal value of the boundary
resistance. Actually the concept of a contact resistance had been known for a long
time from Sharvin’s work on classical point contacts [72], whose width is much larger
than Ар. Such a contact may be imagined as a hole between two ideal Fermi gases
with dif erent densities. In the process of equilibration, a net diffusion current I flow
between both sides. From simple effusion kinetics for a ballistic classical gas of particb /
which all have a velocity of magnitude vF, one obtains [21]
(1.2L''
7Г
for a hole of width IV in a two-dimensional gas and 6n the difference in areal de:rm ..
Translating this into an electric conductance via 5n = (<9n/<9/i)elz, we obtain a tv. -
dimensicnal Sharvin conductance (note that dn/dp. = M/nti1)
64 Theory of Coherent Transport
Fig. 1.12: A four-probe conductor with
perfect leads 1,.. .,4. Scattering is con-
fined to the hatched area. In each of the
leads there is a cross section Cn with trans-
verse coordinates у and outgoing direction
x.
which already contains the fundamental unit e2//i and — trivially — is linear in the
width of the opening. Quantum mechanically the transverse motion through the hole is
discrete and kpW/~ is precisely the number of transverse channels TV. In this way the
Sharvin conductance (1.213) of a classical point contact directly leads to the universal
boundary resistance in the quantum case. It is obvious that a steady state current I
will flow only as long as 6n or the associated voltage V is kept constant externally.
Moreover the dissipation appearing in such a purely ballistic case is due to the equi-
libration of particles with higher electrochemical potential in the reservoir to which
the current flows. The dissipation associated with a boundary resistance is therefore
exclusively in the reservoirs. Indeed dissipation requires scattering, which is absent
if the motion is completely adiabatic. However at the boundary to the reservoirs the
local number N(x) of transverse channels becomes very large and thus the adiabaticity
condition dIV(x)/dx -C N~i(x) will eventually always break down. An interesting re-
lated question is that of the local electric field E(x) which was discussed in section 1.2.1
for the case of a single impurity. Now in classical transport we have Eci(x) = j(x)/a
locally. In a perfect waveguide, however, the local field is zero even in the presence of a
finite current since there is no scattering. The QPC though is a perfect waveguide only
for those channels which are open at the narrowest point, where Ec\(x) ~ I/W(x} is
maximal. The finite resistance of a QPC is thus related to the scattering of an incoming
wave at the constriction boundaries, while the current is effectively carried by electrons
which are not scattered at all. For a discussion of the local potential distribution in
such a situation see [73-75]. Fortunately for the calculation of the conductances in
the regime of linear transport the precise local fields and the problem of where the
dissipation occurs is irrelevant. In fact this is well known in the context of the residual
resistivity due to defect scattering, where the magnitude of the resistivity depends only
on the elastic scattering cross section and not on the detailed form of the scattering
state, while the existence of a finite dissipation associated with the resistivity requires
reservoirs to maintain a steady state [76].
1.5.2 Scattering theory of multiprobe conductances
In the present section we will derive the general Landauer-Biittiker theory of conduc-
tances from the Kubo-Greenwood expression (1.77) for the nonlocal conductivity. In
1.5 The Landauer-Biittiker formalism 65
its most general form, this derivation is due to Baranger and Stone [36] who were able
to show, how the multiprobe Landauer formula proposed by Biittiker [16] on the basis
of simple counting arguments, follows from a microscopic theory. Biittiker’s crucial
step was to realize that in phase coherent transport, current and voltage probes had to
be treatel on an equal footing. The resulting conductances are therefore determined
by the entire S-matrix, including scattering into all the probes not just those between
the curre it source and sink. A second important point is that standard conductance
measurements are done by externally imposing the current and determining the volt-
age by ecuilibration with a reservoir. In order to be able to define conductances as
a property of the system itself, independent of the precise form of the reservoirs for
which no detailed models exist, it is therefore necessary to separate the system and
its environment. This is done formally by attaching a number n = 1,..., of pe"
feet infinite leads to the conductor, see Fig. 1.12. These leads are not true reservoirs
even in tl e limit of a very large number N 3> 1 of transverse channels, because there
is no scattering there and thus no equilibration. Nevertheless they allow to define
proper scattering theory for the conductor itself and with the assumptions (i)-(iii) of
ideal reservoirs, the associated S-matrix then determines the conductances in an actual
measurement.
Let us start with a discussion of the symmetry of the nonlocal conductivity under
time reversal В —> —В. Introducing the time reversed exact eigenstates a and /?, the
current matrix elements appearing in (1.77) obey [36]
-B) ® ja0(x',-B) = B) ® B).	(1-214)
Since the summation f da f d/5 over the time reversed states is equivalent to one over the
original ores, time reversal effectively exchanges the indices a and /3. This is restored
to the original order by interchanging the spatial arguments x and x' and taking the
transpose )f the tensor product. As a result the generalized Onsager symmetry for the
nonlocal conductivity tensor reads
a(x,x', В) = aT(x',x,—B).	(1.215)
Moreover ,he fact that the first term in the large brackets in (1.77) is even unde-
a o it w file the second principal value term is odd, immediately shows that '
principal value term vanishes at zero magnetic field. At В = 0 therefore, even the
nonlocal dc-conductivity tensor is determined by states at the Fermi energy only. F<”
simplicity, in the rest of this section, we will only discuss the case В = 0. Including a
factor of tv'o for spin and retaining for the moment a small but finite frequency w, the
nonlocal c< nductivity at zero temperature and vanishing magnetic field is then
Re cr(x,x,cj -> 0) = h У da j <1/3 6(ea - eF)<I(^ - en - M j(Ja(r) ® jа0(х'). (1.216)
At zero frequency the nonlocal conductivity obeys
V -a(x,x') =0	(1.217)
66 Theory of Coherent Transport
because VJ/3q(x) = 0 at eq = ep, see (1.74). This simple relation is not valid at finite В
where also ea ep contributes, nevertheless the induced current j(x) obviously must
still obey the current conservation condition Vj(x) = 0 [36]. In order to calculate the
global conductances, which relate the total outgoing currents to the electrochemical
potential differences of the reservoirs, the local electric field is written as E(x') =
— W(i'). The associated potential V(z') is now required to approach finite constants
Vn in the asymptotic regime of lead n. Similar to the behaviour in a true reservoir, the
electric field therefore vanishes asymptotically. As a result, the Vn may be identified
with the electrochemical potentials in the reservoirs, as is necessary for a comparison
with the standard measurement of a conductance. It is important to emphasize that
these assumptions do not describe the real physical situation. Indeed, as was noted
above, infinite perfect leads are not true reservoirs. Moreover the potential drop is
not confined to the sample itself but in general also extends into the reservoirs (in
Ref. [36] an ideal reservoir is defined by (i)-(iii) mentioned above plus the additional
requirement that there is no boundary resistance between the reservoir and the leads.
Such a requirement is only necessary if we imagine that the hypothetical perfect leads
attached to the sample eventually open up to a real reservoir. In actual experiments
there are no perfect leads but only reservoirs are connected to the sample. Thus the
problem of an additional resistance between the leads and the reservoir — which can
never be zero — does not arise. However, as in the case of a quantum point contact
the sample itself may be a perfect though finite lead). In spite of these idealizations,
our model correctly describes the net currents in a situation where the reservoirs have
electrochemical potential differences determined by the Vn. The formal construction
of perfect leads allows to eliminate the reservoirs as far as possible. In fact the oidy
two properties we need to know about them is their electrochemical potential and
the number N of channels connecting the sample with the reservoir, thus fixing the
maximum conductance into each of them.
Using the relation (1.217) and the fact that the normal components cr(x,x')  x' of
the conductivity tensor must vanish at all insulating boundaries, a partial integration
of the local Ohm’s law (1.11) shows that the current density
j(x) = — /d3x' a(x, x1)  W(.r') = — У~] V„ f dy' <r(x, x1) • x (1.218)
J	П J
can be written in terms of the asymptotic voltages Vn and an integration of the normal
components of the nonlocal conductivity tensor over the cross sections Cn. Here x' is
a unit vector parallel to the lead pointing outward, while y' stands for all coordinates
perpendicular to x' (see Fig. 1.12, where the sample is two dimensional, i.e. there is only
one transverse coordinate y). From (1.218) we see that even the local current density
is fully determined by the asymptotic voltages Vn and the nonlocal conductivity tensor
a(x,x') which is a characteristic property of each sample in the absence of current
flow [77]. As a consequence, arbitrary potentials V(x') having the same asymptotic
values Vn give identical current densities j(x). Without restriction of generality, we
can therefore take V(H) to be locally linear implying a constant electric field across the
sample. This justifies the expression (1.78) for the macroscopic conductivity although
1.5 The Landauer-Biittiker formalism

the actual electric field in the presence of current flow is far from uniform, as we hav.
seen ir section 1.2.1. The result (1.218) is only valid at vanishing magnetic field В = 0.
However if we integrate the local current density over the cross section Cm to obtain
the to al outgoing current
Im= [ dyj(x)-x = ^GmnVn	(1.219)
-/Cm	n
in lead m, the resulting expression
Gmn = -[ Ay! Ay' x • tr(i,x') • x'	(1.220)
(-'111 J Cn
for the conductance coefficients is valid for arbitrary В [36]. Equation (1.220) has the
intuitive interpretation that the conductance Gmn is simply the flux of the nonlocal
conductivity tensor <r(x, x') between the leads n and m. Moreover it is evident that
the G„n are completely determined by the asymptotic behaviour of cr(.r, x'), which i
crucial to allow a representation of the Gmn in terms of the S-matrix of the conductor
The local current conservation V  j(x) — 0 immediately implies the global Kirchhoff
law
£/m = 0	(1.221)
m
which must hold for arbitrary Vn. As a result, the Gmn obey the sum rule
EG™ = 0.	(1.22J;
m
The matrix of the conductance coefficients is therefore determined completely by if;
o/fdi agonal elements, since
G-nn —	] Gmn .	(1.22.>j
rn^n
In the following we will show how to express the conductance coefficients (1.220) ii
terms of the complete S-matrix. As a trivial but instructive example we start with
an ideal single channel wire for which the two outgoing perfect leads 1 and 2 are
just the left and right asymptotes of the sample. From our arguments in section
1.5.1, we expect that the associated two terminal conductance is simply G21 = 2e2fh
To see how this result arises from our general expressions (1.216) and (1.220), we
use the exact eigenstates r/Ajx) = e‘*“'r/\/27r of the ideal wire. They are chosen Ь,
obey thj orthonormality relations (1.67) with a continuous index a which is just the
one-dimensional wavevector, i.e. a = k€ (—00,00). The associated current mat!’?;
element; are
68 Theory of Coherent Transport
Inserting this into (1.216) and taking the limit w -> 0, the two delta functions fix
a — к = ±/cf and (3 = k' = ±(kp + w/vp)- For small frequencies we thus obtain
z , x 2e2 a>(x — x1)	z
Re a(x, x , <jj -> 0) = —— cos---------.	(1.225)
h,	'Up
The nonlocal dc-conductivity is therefore simply a(x, x') = 2e2/h which is independent
of x and x', expressing the translation invariance of the ideal wire. Since there is no
transverse integration in this case and x • x' = —1 for the oppositely pointing unit
vectors in leads 1 and 2, the two terminal conductance from (1.220) is
as expected. This trivial example shows that the nonlocal dc-conductivity a{x, x') is a
perfectly well defined quantity even if there is no scattering at all. Moreover the result
(1.225) at finite frequency immediately leads to the infinite macroscopic conductivity
expected in such a situation. Indeed from (1.78), the macroscopic conductivity of the
infinite translation invariant system is given by
Re cr(u>) = f d(r — x') Re a(x, x', w) = тг—— <5(w),	(1.227)
where we have used that -xn/M = 2vp/h for a one-dimensional Fermi gas. Now this
is precisely the conductivity of a perfect conductor which is characterized by a delta
function in Re <r(w) with weight n. Via the Kramers-Kronig relation this implies that
Im <t(cj) = ne2/Мы, i.e. all electrons are freely accelerable. The expression (1.225)
therefore contains both the finite dc-conductance 2c2///, of an ideal lead between two
reservoirs and the perfect conductivity ~ <5(iu) of an infinite sample.
Let us now consider the general case of an arbitrary conductor with leads n =
1,..., Nl (,Nl = 4 in Fig. 1.12) which are all asymptotically identical for simplicity. For
a given energy e the channels a = 1,. .., N are characterized by a positive longitudinal
wave vector ka > 0 as in (1.207) (note that the number of channels in general depends
on energy and N is its value at £ = fp). The asymptotic states in the perfect leads are
now chosen as
^(х)- -=Д=е±!^Ха(/У).	(1.228)
^2xnva
Here the Xa(y) are the eigenfunctions describing the transverse motion in channel a
with normalization
/ dy Ха(у')Ыу') = dab 	(1.229)
The index (±) describes outgoing or incoming plane waves in the perfect leads, with
the prefactor (2xhua)~chosen in such a way that all of the states carry the same
flux independent of the channel index a. Using the waveguide states (1.228), we can
1.5 The Landauer-Biittiker formalism 69
now define exact scattering states i/>a(x) with a continuous index a = e,n,a by their
asymptotic behaviour in lead m
ФсЛХ	(*^m) T 5 ~	(*Um) 	(1.230)
b
The state V>eno(x) describes an incoming wave in channel a of lead n which has on-
going components in arbitrary channels b and leads m. The associated dimensionless
amplitude for being scattered from an to bm is denoted by With our choice of
normalization, the states ^>o(rr) obey
У	= S(a - /3) = rS(e - e')Jni„2dad (1.231)
as in (1.67) (this is easily checked in the case of an ideal multichannel wire). The
scattering states (1.230) are now inserted into (1.216) as an exact orthonormal system
of eigenstates ipa(x) in the absence of a field. Noting that the two delta functions
reduce
/ d« [ d/3 = [ de 52 / de'
J J	J niaJ П2Ь
to a disc rete sum over the lead and channel indices with e = e' fixed at eF, the
conductance coefficients (1.220) can be written as
C - .J, W	r(’i)	о 9O»\
LTmn	/ J 1 пъЬуТца1 nia,ri2b ’	\L.Lo..)
ЩЛ TL?b
Here
= f dy'j^-x'	(1.23.^
j Cn
is the contribution of the states to the outgoing current per energy in lead n.
Inserting the asymptotic behaviour (1.230) of the scattering states cf”u and using the’
orthonormality of the transverse eigenfunctions y, a straightforward but somewhat
t edious calculation gives
^ntl,n2b ~ ^гцЛ^аЬ — 57 Smli ,ca^nn2,cbj 	(1.234)
In particular, taking = n2 = n and a — b we have
(k235)
with Rna = Er |Snn,CH|2 the reflection probability for an incoming mode a in lead n.
The1 result (1.235) has a very simple physical (explanation: the net outgoing current
per energy in lead n associated with an incoming state na is the sum of the negative
70 Theory of Coherent Transport
incoming current plus the total reflected one for that mode (note that the charge is
-e). By a trivial relabeling of the indices in (1.234) we have
^n2b,nia =	^<5n2m<5n2ni<5|x> ~ У? ^mn2,cb^rrm>,ca^ 	(1.236)
In order to check the unitarity of the scattering matrix as introduced by (1.230), we
define N x N matrices Smn for given leads m and n by their matrix elements
(Smn)^ = Smnj>a	(1.237)
in the channel subspace. The complete S-matrix is then an Nl x Nl matrix
< Sn ... SINl \
S= :	:	(1.238)
\S/vLi ... Snlnl/
of N x N submatrices Smn (note that in the case of an ideal single channel wire we
have
The fact that even with no scattering at all this is different from the unit matrix is
due to our definition of S, whose diagonal elements are the reflection instead of the
transmission amplitudes which is common in quantum mechanics textbooks). The
matrix S defined in (1.238) is unitary, i.e.
SS+ = SfS = 1	(1.239)
is a unit matrix. This property is easily verified via the condition of current conservation
which implies that Gmn = 0. Using (1.232), current conservation requires that
(1.240)
m
Inserting (1.236) this is just the	iia-matrix element of 1 — S*S = 0, thus proving the
unitarity of S. The conductance coefficients Gmn are now easily obtained from (1.232)
by inserting (1.234) and (1.236). For the offdiagonal elements m / n the product of
the six Kronecker delta’s does not contribute. Moreover from the general reciprocity
relation [78]
S(B) = ST(-B) Snm,ca(B) = Smn<ac(-B)	(1.241)
which is a consequence of time reversal S*(B) = S-1(—Bj and unitarity S J = S^, the
two contributions quadratic in S are identical at В = 0. Finally, from the contribution
which is quartic in the S-matrix elements, one may seperate off a factor
E2 ^пП2,с'Ь^тп2,сЬ ~ (SS^)nm,c'c
112b
1.5 The Landauer-Biittiker formalism 7 1
which s zero for m n. Introducing a small trace tr in the channel indices vic
fl* (^mn)	' $тп,аа>
a
the final result for the offdiagonal conductance coefficients can be written in the sim.pl-
form
2e2	2e2
&тп = ~~h l^rnn.col — ~h~ tr у^тп^гтт) '	(1.242)
Although our derivation has used properties which are only valid at В = 0, it is
remarkable that the final expression (1.242) remains valid for arbitrary magnetic fields
В [16, 36,78]. The only change at finite В is that the S-matrix is not symmetric
but obeys the general reciprocity relation (1.241). Since the Smn contain scattering
amplitudes at the Fermi energy, the conductance coefficients are completely determined
by states at the Fermi level even at finite magnetic field B, where cr(x, x1) contains
non Fermi surface contributions. The physical interpretation of G(SmnS^mn) is simply
the total quantum mechanical transmission probability (defined also for m = n)
Tmn = tr (SmnS^n)	(1.243)
from lead n to lead m, giving Gmn = 2(e2//i)Tmn for m n. These transmission
probabilities obey the sum rule
Y.T^ = N	(1.244)
m
which fallows from the fact that S^nSmn = 1 is a IV x /V unit matrix by the
unitarity S^S = 1 of S. Using this, the diagonal conductance coefficients (1.223) are
given by
2e2 (	7	2e2
Grm = —I У?Tmn — Tnn j = —j—(Tnn — N).	(1.24 >i
•I \ m	/
The final expression for the outgoing currents Im in a multiprobe conductor in tc
of the asymptotic voltages is therefore
2e2 /	\
7m = I —	] Imn^n j	(1.Z:‘ 1
which is the form first proposed by Biittiker [16] (note that Biittiker’s currents Im air
•incoming instead of outgoing currents and thus his equations differ from ours by
overall minus sign). The factor 2 is due to the spin degeneracy and has to be omitted
in cases where only a single spin orientation contributes, as e.g. in the quantum Hall
effect. At zero temperature, the Tmn are the complete transmission probabilities at the
Fermi energy. Provided that other sources of scattering like phonons are negligible, the
effect of a finite temperature can be trivially included by the replacement,
ТтпЫ [ <& [~^r HmnW	(1.247)
J \ de J
which takes into account the thermal smearing of the Fermi distribution.
12 Theory of Coherent Transport
1.5.3 Symmetry of electrical conduction
For a discussion of the resistances and their symmetry properties which follow from the
general multiprobe conductance theory, we start with the simplest case of a two probe
conductor. In this case the voltages are measured in the same leads 1 and 2 from or
into which current flows. It is convenient to express the general relation (1.246) in a
matrix notation
2e2
/ = G-V = — (T-IVl)-V	(1.248)
relating the current and voltage vectors via a NL x Nl conductance or transmission
probability matrix G or T. Then in a two probe situation we have
r = [Тц - N Тц \ _ c — Тц Tl2 \	/ —Тц T2i \	.
2e2 к Тц T22-N) l Тц -Tn) к Тц -Th)'
Here the first identity is an immediate consequence of the relation (1.222). The second
one is due to the related condition
£Gmn=0	(1.250)
n
which follows from the fact that all currents must vanish if Vn = V is a constant
independent of n (note that at finite magnetic field G is not symmetric and thus (1.250)
is not a trivial consequence of (1.222)). In a two-probe situation, therefore, only a
single transmission probability T21 = tr(S21S21) determines the complete conductance
matrix. The related longitudinal resistance R = Rn,n = h/2e2T2i (see the notation in
(1.252) below) turns out to be symmetric in В in any case. Indeed using the reciprocity
relation (1.241) we have
Tum(B) = Tmn(-B) —> G„,zl(B) - Gmn(-B)	(1.251)
quite generally. The conductance matrix thus obeys G(B) = G'(-B), which is the
macroscopic analogue of the nonlocal Onsager symmetry (1.215). From Tl2 = Тц in
the two probe case, this implies that Тц(-В) = T12(B) = Тц(В) is a symmetric
function of the magnetic field.
A much richer situation arises in the case of four-probe measurements which are
commonly used in experiments. In this case there are various different resistances,
which are defined by the six possibilities to choose two of the leads as current source and
sink while the other two are used as voltage probes, drawing no current. It is important
to realize that these resistances cannot be obtained from a Nl x resistance matrix
R = G“! by simply inverting I = G  V into V = R • I. In fact, due to the conditions
(1.222) and (1.250), the conductance matrix has vanishing determinant and thus is not
invertible. Physically this is a consequene of the freedom to add an arbitrary constant
V to all the voltages which must leave the currents unchanged. As a result, G has
an eigenvector ~ (1,.. ., 1)T with eigenvalue zero and thus can only be inverted in the-
1.5 The Landauer-Biittiker formalism 73
subspace perpendicular to this eigenvector. To be specific, let us define a generali;'
resistance
V -V
.	(1.2o2)
as the ratio of the voltage difference between leads m and n and the current ei>!
in lead 'c and leaving in lead I. Quite generally such a resistance can be classified as
either longitudinal or Hall like, depending on whether the voltage probes are on
same or on opposite sides of a line drawn from the current source to the current slim.
Referring to Fig. 1.12, this implies for instance that /44,23 is a longitudinal and /?13,42 a
Hall like resistance. In order to derive an explicit expression for the Rki,mn in terms of
the elen ents of our conductance matrix, we use the conditions that Ц = -Ik = I while
/m = /« — 0, since the voltage probes m and n draw no current. Moreover, using the
arbitrariness to add a constant to all voltages, we may set Vn = 0 without restriction
of generrlity. Omitting the equation for In = 0, the general relation I = G • V can
then be written as
/-/\	(Vk\
/	= G(nn) • Vt	(1.253)
\ 0 /	\vm)
where G-1111) is the 3x3 matrix which arises from G by omitting the zt-th row and
column. Now this matrix has a nonvanishing determinant
/2с2 \ 3
detG(nn)=(—j D^O	(1.254)
which is independent of n. In fact, if we expand det G = 0 in terms of its i-th row
0 = det G = £ G0(-l)’+> det, G(‘2>	(1.255)
we realize that this is just the condition (1.250) of zero current when all voltages are
equal. A< a result ( — l)‘+j det G^ is a constant independent of i. and j. This constant
cannot b1 zero, otherwise G would have a further but nontrivial eigenvector with eigen-
value zero, which would imply that no currents flow even though the reservoirs are a'
different electrochemical potentials. It is now easy to solve (1.253) for the unknown
I by using Kramers rule which gives
Vm detG<"")
Gkk
Glk
G mk G-ml
Gu
ft TmiTnk — TmkTnl
2 c2	"
(1.25<Q
1
I
0
D
where we have used that hGkk/2e2 = -Tik-Tmk-Tnk and hGu/2e2 = -Tkl-Tml-T„i.
The associated resistance (1.252) is thus given by [16,78] (note again the opposite sign
< onventi< n by Biittiker)
Rkl,mn —
ft TmlTnk  TmkTnl
2c2 D
(1.257)
74 Theory of Coherent Transport
with the dimensionless determinant D defined in (1.254). In order to determine the
symmetry properties of these resistances, we note that D(B) = D(—B) is even in
В which follows by explicitely writing out the determinant and using the symmetry
(1.251). As a result we find that
Rki,mn(-B) = Rmn,ki(B')	(1.258)
again using the symmetry (1.251) of the transmission probabilities. The resistance
is therefore invariant under a reversal of the magnetic field В provided we exchange
the current and voltage leads. The experimental consequences of this relation have
been beautifully demonstrated by Benoit et al. [79]. We will not discuss the many
simple, yet remarkable and very successful applications of this formalism in particular
to ballistic systems or the quantum Hall effect [80]. As a final point, however, it should
be emphasized that — contrary to naive expectations — the four-probe resistances
obtained from (1.257) are not necessarily positive, in agreement with experimental
results [80]. Nevertheless the total dissipated power
P = [ddx [ ddx’ E(x)  <r(x, x1)  E(x’) = - £ Gmn Vn Vm	(1.259)
J '	n,m
is always positive as it should be. Indeed, using (1.222) and (1.250), we may replace
— VnVm by (Vn — Vm)2/2 in (1.259) which leads to
e2
p= T ^(Tmn + Tnm)(Vn- Vm)2.	(1.260)
The dissipated power is thus always positive and connected with the symmetric part
of the transmission probabilities, which are even under В —> -В.
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78 Theory of Coherent Transport
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Bernhard Kramer
Quantization of Transport
In this chapter, the quantization effects in the electronic transport properties of quan-
tum mechanical systems that were discovered during the past two decades are de-
scribed. The fundamentals of their theoretical understanding will be outlined by start-
ing from very elementary concepts and gradually merging into the present status of the
field. Starting from the quantization aspect, the contents of the present chapter repre-
sents in a sense the “contrapunkt” of the chapter 1, where, by starting from the classical
picture, and introducing quantum corrections, the effects of quantum interference on
the electron transport were described.
First we introduce the elements of the electronic structure of systems with reduced
dimensionality. We describe the quantization effects in the electronic spectrum in-
duced b' a magnetic field and by geometrical confinement. Then, in the second and
third section, we will follow the historical development and discuss in some detail the
quantum Hall effect. The final section of the chapter contains the fundamentals of the
quantization of the conductance of quantum wires and point contacts.
2.1 Noninteracting electrons in reduced dimensions
rhe basis of the quantization of the conductance at low temperature is the quantization
of the electron levels in quantum mechanical systems. This may be due to geometries'
confinement or caused by a magnetic field. We start by discussing the latter.
2.1.1 The Landau model
The knowledge of the quantum mechanics of a free electron moving in a two-dimensiona'
plane, su yject to a magnetic field directed perpendicular to the plane, lies at the heart
of the understanding of the integer quantum Hall effect [1-4]. We briefly summarize the
main properties that can also be found to a certain extent in the standard textbooks
of quanta m mechanics [5,6].
The Hamiltonian of an electron with a charge —e in a vector potential A is given
80 Quantization of Transport
by
Я° = 2^*^P + eA^'	(2Л)
Here, m* is the effective mass of the electron, p — (h/i)V the momentum operator,
and В = V x A the magnetic field.
In the Landau gauge,
A=(0, Bx, 0),	(2.2)
we have \py, HQ] = 0. The stationary Schrodinger equation that corresponds to (2.1)
reduces to that of a one-dimensional harmonic oscillator in the ^-direction,
-Q—X + ^(* - A’)2] Vn(x - X) = EnkVn(x - X),	(2.3)
2m* ax^ 2
which is centered at X = l^k. The wave number к characterizes the plane-wave part
of the energy eigenstates of HQ,
tpnk(x,y) = —7=e'kyipn(x - X).	(2.4)
v-Ly
The quantities Q = JhJeB and wc = eB/m* are the magnetic length and cyclotron
frequency, respectively. By applying periodic boundary conditions in the y-direction,
d’{x, у) = #r, у + Ly),	(2.5)
we obtain the allowed wave numbers к = 2^q/Ly, with q = 0, ±1, ±2, ±3,. . .
The spectrum of the harmonic oscillator consists of equidistant energy levels
Enk = hwc(n. + |)	(2.6)
with the corresponding wave functions
1	(	\	(\/2r\
Л1.М = -^_ехр Я„ Г— , (n =0,1.2.3,...).	(2.7)
Утг'/Яе»!	к 2£с/	\ J
They fulfill the boundary conditions
lim y>„(.r) = 0.	(2.8)
|.r|->oo
Since the у-component of the momentum commutes with the Hamiltonian, the
eigenstates of the latter can be chosen to be simultaneously eigenstates of the y-
component of the momentum operator with tik. as the corresponding eigenvalues, as
indicated in (2.4). The Hamiltonian that describes the motion in the .т-direction de-
pends on k only via the center of motion coordinate X (see (2.3)). Therefore, the
2.1 Noninteracting electrons in reduced dimensions
energies En = En(k) are degenerate with a degree per unit area that is given by t’ 
density of the flux quanta in the system.
. eB 1
= T =	(2£”
It can be determined by calculating the trace of the projection operator onto the
subspace that corresponds to a given Landau level,
E/dxVn(^ - X(k)) = ^2 1 =	(2.10)
for very large system size. The filling factor is defined as the ratio between the density
of the electrons, N, and Nb
It is an integer, v — n+ 1, if the states of the lowest n,+ 1 Landau levels are completely
filled with electrons — and the other levels are empty — or, equivalently, if the system
contains an integer number of electrons per flux quantum.
The physical properties are invariant under gauge transformations of the vector
potentia , A = A' + V/, where f is an arbitrary scalar function. Using f(x,y) =
— Bxy/2 one obtains the vector potential in the symmetric gauge,
A = ( — By/2, Bx/2,0).
(2.121
By transforming to polar coordinates (p, ip) one notices that the resulting Hamiltonian

h2	m*uj'2.p2	hu>c d
2m	8 2i up
(2.13)
with the two-dimensional Laplacian V2 = д'2/др2 + p~{d/dp + p~'2d2 / др2, commutes
with the operator of the angular momentum L = (ti/\f(d/dp). Its eigenstates ca
therefore be chosen to be simultaneously eigenstates of lz.

n!
___J:___, I__________—___________nHPw
^H+i у 27t2I”‘! (\m| + n)1.
4f2J "
(2-14)
with n non-negative integer, m integer, hm the eigenvalues of L. the Laguerre polyno-
mials and the energy eigenvalues
„	.	/	1 I nil — rn
Bnl„ = hiv’e I n + - 4---------------------
(2.15)
The lowest Landau level corresponds to the states with n. = 0 and in > 0. For the
higher levels the energies are given by the condition
82 Quantization of Transport
Each of the levels is associated with states with 0 < — m < n', and n' > n > 0 in
addition to the states with non-negative angular momentum m > 0.
By comparing the (2.7) and (2.14), the transformation between the Landau and the
symmetric gauge corresponds to a unitary transformation between different complete
orthonormal basis sets.
The mean area covered by the states which correspond to the lowest Landau level
is given by
тг(р2) = 2лТ2(т + 1).	(2-17)
By counting the number of possible angular momentum states within a system of the
radius R, one concludes that the degeneracy of the level is the same as given above in
the case of the Landau gauge.
2.1.2 Confinement and magnetic field
We have seen above that a magnetic field leads to the quantization of the energies of
a quantum particle moving in two dimensions. In this section, quantization of energy
levels due to geometrical confinement will be discussed, and the influence of a megnetic
field on geometrically confined states will be described.
Harmonic confinement
A most simple model for an electron which is confined geometrically in. say, the x-
direction around x = 0 is given by the above Schrodinger equation (2.3) for Ar = 0. It
describes a particle in the harmonic potential
* 2
V-(x} =	(2.18)
In order to avoid confusion with the cyclotron frequency in (2.3), л,. is here renamed
into w0, a parameter that represents the strength of the potential.
The eigenvalues of the Schrodinger equation corresponding to this potential are
En(k) = En + —,	(2.19)
with En the harmonic oscillator energy levels, (2.6), and the eigenstates are given by
(2.4) and (2.7) for A' ~ 0. We only have to replace uy by w0 and bv ('0 = (h / m* ,
the characteristic length scale of the confining potential. The larger .c(l the smaller Q,
and the stronger confined is the particle. The better the spatial confinement, the larger
is the energetic separation of the states. The characteristic feature' of the harmonic
confinement potential is that tin; energy eigenvalues are equally spaced. It is worth
noticing that the diameter of the “quantum wire” defined by (2.18) is not a constant
but depends on the energy, i.e. ск nV2.
Using the harmonic, confinement model, it is particularly simple to include a mag-
netic field and to st udy the interplay of geometrical and magnetic confinement. In the
2.1 Noninteracting electrons in reduced dimensions 8
Landau gauge, an additional harmonic confinement potential leaves the form of the
Schrodinger equation (2.3) invariant. It leads to the replacements
t П2 ~~ T w0i
LU2
x^x' = xcfi-	(2J-
The eig( nvalues are no longer degenerate
/	1\	/j2Z.2
Е"‘ч^ = №Г+2) + йв)-	!2J!;
with the renormalized effective mass m(B) = m*£l2/u'q. The eigenstates are now given
by
/ X ,	1	( (x-X')2\	/У2(.'С - -V') \
Vnk(x) = Vn(x - Л ) = 7-=^- exp---------—------- Hn ------------- ,	(2.22)
V7rl'2€n! X ^	/	\	/
with the new characteristic length f = (/i/m*Q)V2 which interpolates between f0 and
for small and large magnetic field, respectively.
If a particle is confined within a finite circular region in the two dimensional x-y
plane by a harmonic potential the polar version of the Hamiltonian of the harmonic
oscillatoi (2.13) can be used to obtain the energy spectrum and the eigenstates.
Thus knowing the properties of the quantum mechanical harmonic oscillator pro-
vides not only the solution of the problem of a charged particle moving in a homoge-
neous magnetic field, but also the complete information about the interplay between
the latte - and a harmonic confinement potential. We add here that in many realistic
experimental situations the approximation of harmonic confinement is rather reason-
able sine.! confining potentials in real systems as quantum wires and quantum dots,
are siikx th functions of the coordinates. Therefore, a Taylor expansion around the
minimum of the potential is in order,
V (t) = Vo + - f-ry'j 1:2 +    	(2.23)
2 Vk2A=o
This is especially for the lowest energy levels a good approximation (Fig. 2.1). By
comparing with the above (2.18), we find d2V/d.T2|	= шц).
.Sl/vm/'e well confinement,
l lie harmonic potential is an example of a particularly “soft confinement". One of n>
peculiarities is that the diameter of the system is not very well defined. It is given m
i he width of the region within which the wave functions are non-zero and depends m:
t hi' energv.
84 Quantization of Transport
Fig. 2.1: Example of a confining potential.
For small energies the potential is almost
parabolic. Here, the eigenenergies and cor-
responding states are well described within
the harmonic approximation. For higher en-
ergies, the potential walls become steeper,
thus resembling a square well potential.
The opposite extreme of a geometrical confinement potential is represented by the
box potential discussed as a standard example in elementary quantum mechanics [6, 7],
in one dimension,
vm = voe - |.r|).
(2.24)
For I о co the (symmetric) solution of the corresponding Schrodinger equation is
particularly simple. The energy eigenvalues are
=	(n/0)	(2.25)
2m* \Lx /
and the corresponding states,
г-^~ f sin(7mx/Lx), n even
vr I	(2-26)
' x [ cos(7rn2.-/Lx), n odd
The generalization to higher dimensions is obvious.
For the square well potential, the interplay between confinement and a magnetic
field is highly non-trivial. It leads to subtle effects in the energy spectrum as can
be seen by considering the two-dimensional system studied above1. (2.1), using again
the Landau gauge, (2.2). We assume a finite extension in the .(.-direction, (2.24).
with In —> oc. The Schrodinger equation to be solved turns out to be exactly (2.3).
Since still	= 0 one can again factorize the eigenstates into a plane wave in
the //-direction and a function of only the .(.’-coordinate. (2.4). However, the boundary
conditions in tin1 z-direction are now
Fn \ 2 ) —	\~2~/ —
(2.27)
instead of (2.8). This is equivalent, to assuming infinitely high potential walls at ±1^/2.
Since1 we still have the periodic boundary conditions in the //-direction, the system
2.1 Noninteracting electrons in reduced dimensions
Fig. 2.2: The geometrical equivalent of a two-dimensional system with a perpendicular magnetic field
with periodic boundary conditions in the y- and hard-wall boundary conditions in the т-direction i.s
a cylinder of finite length in the т-direction. The magnetic flux density В points everywhere in the
direction perpendicular to the surface of the cylinder. In the Landau limit, the latter is infinitely long.
consider :d corresponds geometrically to a cylinder with a very large circumference L;/
and of a finite length E, in the ./.'-direction with the magnetic field perpendicular to
the surface of the cylinder (Fig. 2.2)1.
The solutions of (2.3) can now be discussed in terms of the parabolic cylinder
functions [3,8]. For convenience, let us introduce dimensionless variables according to
y/2	y/2	r~	E
( = *Т.	=	=	(2-28)
(<	E	//wt.
We furtl er substitute
£ =	Unk(.v.) =^(.r + ^-) o =	(2.29)
Then, wt obtain from (2.3) the standard form, which is also called Weber’s differential
equation
U” - |	+ a j [/ = 0,	(2.30)
\ 4 j
with the boundary conditions
1 Noto the peculiar properties to be assumed for the magnetic field here. In order to generate sue!,
.1 system physically, a line of magnetic monopoles has to be assumed along the axis of the cylinder
86 Quantization of Transport
If the width of the system is large compared to the magnetic length, A c 2>1, and
in the regions of |^| < A,.r/2 (Landau limit), |&| w Aj/2, and for -> oo (edge state
limit) approximate analytical expressions for the energy spectrum can be derived [9]
(Fig. 2.3) by expanding the solutions of (2.30) in a power series.
In the Landau limit, we obtain as expected,
1 ЧАГ
а/2тгп! \	2 7
(n = 0,1,2,...).	(2.32)
The second term in (2.32) is only a Gaussian exponentially small correction to the
energies (2.6) obtained for Ax —> oo. The corresponding approximate states are given
by the Hermite polynomials (2.7).
If is close to the boundaries of the system, sa Ax./2, one finds
/	3\ V2 (	1\ 2n +1 / Ал ,	,
= [2п + - ±— Г n.+ ---------------7-	(« = 0,1,2,...).	2.33
\	2) тг \	27 n! \	2 7
The corresponding wave functions have considerable amplitude only near the bound-
aries, near ±Лл/2. They are called therefore edge states.
Finally, in the edge state limit, one finds
f2
enk «	(2.34)
4
Here, the wave functions are strongly localized near ±AT/2. When ^k > 0 the corre-
sponding plane wave in the ^-direction has a positive wavevector, whereas for negative
£k the wavevector points into the opposite direction. The edge states have chiral prop-
erties. The magnetic field leads to a strong spatial separation of the energetically
degenerate states. For large width of the system, the edge states localized at ±i\x/2
practically do not overlap. Backward scattering (k —> —A), induced by impurities, for
instance, which is easily possible for one-dimensional plane waves when no magnetic
held is present, becomes negligible for the edge states induced by a strong magnetic
held.
Classically, the Landau limit corresponds to cyclotron orbits that are closed circles
within the system. In the intermediate case, |^.| x A,/2, the cyclotron orbits start
to “skip” along the boundaries in the opposite direction as in the cyclotron orbits. In
the edge state limit, not much of the orbits is left. The particles move practically only
along the boundaries without entering the interior of the system anymore (Fig. 2.3).
In contrast, for the above discussed harmonic confinement potential, the separation
into "bulk” (Landau) and “edge” states is not really possible. Even for fc x (), the
spectrum is changed due to the presence of the confining potential, the distance between
the energies being Ш +w2)1/2, instead of /lay.. In addition, for fc yt () the states
with positive and negative wave vectors are not automatically spatially separated.
2.1 Noninteracting electrons in reduced dimensions 87
Fig. 2.3: Ihe spectrum of eigenvalues, en(&), of an electron confined within a system of finite v n 1' ’
A.,. Classi ;al orbits which correspond to the quantum mechanical states are shown in the upper p.u:
Near the center of the system, where the energies are independent of the quantum state*-- ar<-
approximately Landau states. For |^| ~ Лж/2 and > Ax/2 the classical orbits “skipping” al**!-"
the boundaries correspond to quantum mechanical “edge states” which are well localized near ±A
88 Quantization of Transport
2.1.3 Influence of impurities
The degeneracy of the Landau levels is immediately lifted by any perturbation. A
particular example, namely the confining potential, was discussed above. Two other
examples of such perturbations are a random potential, introduced by defects and impu-
rities, and the interaction between the electrons. They are of fundamental importance,
since they also change the nature of the quantum states qualitatively. Their most
simple consequence is a broadening of the Landau levels due to the lifting of the degen-
eracy. However, they eventually lead to dramatic and fundamental consequences for
the electronic states that are most clearly experimentally observable in the quantum
Hall regime. They are therefore crucial for the physical understanding of the quantum
Hall effect.
In this and the following section we will discuss the influence of the disorder. First,
we deal with the broadening of the Landau bands, before we investigate in the next
section the fundamental effect the disorder has on the nature of the electronic states,
namely localization.
We consider the Hamiltonian of a single electron moving in a homogeneous magnetic
field and in a random potential landscape (Fig. 2.4),
H = HQ + V.	(2.35)
The kinetic energy Ho is given by (2.1), V is the operator of a random potential which is
assumed for simplicity to be Gaussian white noise with vanishing mean. By abreviating
(.T7, %) —> j we can write
V(l) = V(x, у) = У F(x, y)dxdy = 0,	(2.36)
F(1)V(2) = Ях.т/Мх+Су + г/)
= У V(x, y)V(x + £,y + y)dxdy
= Vo^)5(y).	(2.37)
The configurational average is denoted by It is assumed to be equivalent to a
spatial average, though this is not under all circumstance’s justified (see chpater 1).
Any n-point correlation function of the randomness either decouples at most into
sums of products of two-point correlation functions (2.37), if n is even, or vanishes if
n is odd. For instance,
F(l)V(2)V(3)V(4) = 1W + V’(1)V(2) F(3)V(4)
+ V(1)V(3) V(2)V(4) + V(1)V(4) I'(2)F(3).	(2.38)
As an example, we calculate the broadening Г of the Landau bands in the limit of
‘‘weak” randomness,
Г -C hajc.
(2.39)
2.1 Noninteracting electrons in reduced dimensions 89
V12:)
Fig. 2.4: One-dimensional example
of a “white noise” potential V(x)
with vanishing spatial average. The
magnetic length is assumed to be
much larger than the spatial correla
tion length of the random potential.
Then, the disorder-induced coupling between the Landau bands may be neglt. U-d
As /lwc scales with B, but, as we shall see, the broadening of the Landau levels <>iilv
with у/З, the weak-randomness limit can always be reached for a given disorder by
increasing B. It is generally believed that this high-field limit is the relevant case for
the quantum Hall effect.
We apply the perturbation theory discussed in chapter 1, in order to estimate the
width of the Landau bands. The starting point is the resolvent expansion for the
one-elec ;ron Green function [10],
G(c) = -----—----— = Gfiz) + Go(c) + G0(2)I'Gn(c)I G'o(c) + ....	(2.40)
z - ti0 - v
Here, z - E + it is the complex energy variable, and Gn(o) the Green function of the
impertuibed electron. It is diagonal in the Landau representation (2.4),
{nk\G0(z)\n'k’) = —dUi„, 5kik>.	(2.41)
z — Jz„
The average density of states is given by the configurationally averaged one-electron
resolvent
p(E) = lim ~ tr6(E-H)
= —lim lim trim G(E + ie),	(2.42>
where .4 = LcLy is the area of the system. For the unperturbed system, the densii;.
of states consists of delta functions at the Landau energies, with weights given by the
degeneracy per unit area (2.9). In the presence of disorder, the averaged resolvent inns!
be calculated. This is in general a highly non-trivial problem [11] which may be solved
exactly only in special cases as the so-called Lloyd model, where the random potential
is distribited according to a Lorentzian [12], and the above white-noise potential in
t he limit В —> oo [13,14].
90 Quantization of Transport
The lowest non-trivial approximation can be obtained by writing the resolvent
expansion in the position representation, and configurational avaraging. From the
above assumptions about the random potential it follows that all odd order terms
vanish. From the decompositions of the even-n point potential correlation functions,
(2.38), corresponding decompositions of the even-nth order terms in the series are
generated. The fourth order term, for instance, gives rise to four terms, depending
on whether (1 = 2 = 3 = 4), (1 = 2,3 = 4), (1 = 4, 2 = 3) or (1 — 3,2 = 4) (j =
(xj, yj)). They are representative of the different types of contributions that occur quite
generally also in higher order. Figure 2.5 shows their diagrammatic representations.
In the diagrams, lines connecting dots j and к represent unperturbed Green functions
containing sites r, and r^, dots with indices j scatterings at potentials V(rj'). An
integration is understood for each dot in a diagram. The “external electron lines”,
those starting at a dot j but not ending in a dot represent the free particle Green
function with arguments r and r;.
It is obvious that the number of the diagrams grows very rapidly with increasing
order. The three different types of terms can easily be distinguished by their “topol-
ogy”, in the representation of the free particle, where the unperturbed Green function
is diagonal (Fig. 2.5). Diagram (R) can be “cut” into two independent pieces by remov-
ing the free electron line between the two scattering processes. Diagrams of this type
factorize with respect to the spatial integrations into simple matrix products. They
are called reducible diagrams. For instance,
M(R)Hfc') =	(2.43)
with
C'(M) = v02 /О-Ч2) E	(2 44)
71	П1Л-1	2
Diagrams of the form (I), in which scatterings are multiply connected with free
particle lines do not factorize. They are irreducible, and are the main sources of the
difficulties when calculating configurationally averaged Green functions.
The third kind of diagrams, type (S) in Fig. 2.5, can be cut into independent pieces
by removing two free particle lines that connect two different dots. In these terms, the
particle experiences additional scatterings when being scattered from and returning to
a given site. They can eventually be accounted for by replacing all of the free electron
Green functions in the irreducible diagrams by the averaged Green function.
When investigating the higher order diagrams, one observes t hat, one can construct
a geometrical series for the configurationally averaged Green function in the represen-
tation of the unperturbed particle in terms of the sum of the irreducible diagrams.
By inserting the above Landau states (2.4) one notes that the sum of the irreducible
diagrams is diagonal in k. Let, us denote the sum of the irreducible diagrams by the
matrix Sn,n,(fc, z). Then,
(nfc|G(z)|n’k1} = Gn.n'{k, z) 6/^' = (-77-7777—(2/15)
2.1 Noninteracting electrons in reduced dimensions 91
second order diagram
1=2
1=2=3=4
fourth order diagrams
1=2
(S)
Fig. 2.5: Diagrammatic representation of the nonvanishing second and fourth order terms of the re-
solvent expansion. Lincs represent the Green function of the unperturbed electron, dots the scattering
from the pjtcntiaJ. Integration over the independent sites is understood.
92 Quantization of Transport
and
Sn>n-(A:,z)= E (nA:|V|niA:)Gniin2(A;)z)(n2A:|V|n'A:)
П1 ,П2
+ higher order irreducible terms (type I)
= Vo2 E /
П1,П2 71
+ higher order irreducible terms.	(2-46)
The occurence of Gnitnj(k, z), which contains the full self-energy in the denominator,
on the right-hand-side of this expression is due to the presence of the diagrams of the
type (S) (Fig. 2.5) in the configurationally averaged resolvent expansion. The sum of
the irreducible diagrams is often called self-energy of the particle. It represents the shift
of the particle energy, due to the presence of the perturbing potential. The particle
energies are given by the poles of the averaged Green function
det|z0 - En - £n,n'(k,Zo)\ = 0.	(2.47)
In general, and this is also the case in the present example, the self-energy is complex,
and can be thought of as originating in an effective complex, non-Hermitian potential.
In the high magnetic field limit, (2.39), we may assume that = Sn <5nn- up to
corrections of the order У0/Ншс. This can be seen by inspection of the spatial integrals
in (2.46). Then,
эд « v02 / ыi - ч2)!2E k(i - M)l2—f 1 v t \
Ji	Tf	z-E„-En(z)
where is the effective disorder seen by the electron when moving in the presence of
the random potential and a magnetic field. Since
£k(l-M)|2 = A	(2.49)
we find
I/2 — k)2 _ I/2—	/9 rn\
Kft 2%£2 V° h '	(2'O0)
Equation (2.48) is called the self-consistent Born approximation [15.16]. Its solution
is, with the zero of the energy such that En = 0,
Z	IZ^
^n(~) = 2 ± у K2ff-	(2-51)
2.1 Noninteracting electrons in reduced dimensions 93
PttV^/Nb
Fig. 2.6 : The density of states, p.
in the self-consistent Born approxima-
tion (full line) compared with the ex-
act result (shaded region) [13]. The
broadening of the band increases with
the square root of the strength of the
magnetic field.
1	£/2Ve(f
We note that £n(z) is complex as long as
|£| <2Vetf.
(2.5'.?)
These aie the regions where the averaged Green function has a finite imaginary part,
thus yielding the finite density of states which is normalized to NB (Fig. 2.6),
p(E)
Nb
1 -
(2.53)
In the self-consistent Born approximation, the broadening of the Landau bands is
therefore
Г = 4Veff-
(2.54)
It depends on the magnetic field strengths,
(2.55)
The broadening increases with the square root of the magnetic field. Since the ener-
getic distance between the Landau bands increases linearly with the magnetic field, it
is clear that the limit of weak effective disorder, for which the self-consistent Born ap-
proximat on is tailored, can always be achieved by increasing B, as mentioned already
above.
The d agrams that are omitted in the self-consistent Born approximation are relate
to multip’e scattering processes involving more than one site. They lead eventually to
correctior s that are of particular importance in the energy regions of the tails of tho
bands,
Г
2
1^1
(2.56
94 Quantization of Transport
Here, in the region, where the quantum states are well localised within accidental
minima of the potential energy, the present approximation breaks down and other
approaches, as the path-integral method, for instance, have to be used [17,18]. As
metioned above, the shape of a single Landau band in the presence of white-noise
disorder, can in fact be calculated exactly analytically [13,14]. The result is
p(E) =	(2-57)
TrVeff v?r 1 +	(г)
with e = E/2Vefi and the error function
9 rE 0
ф(е) = —= / ef dfi.	(2.58)
у тг Jo
The energy is measured again from the center of the band. The above lowest order
theory yields indeed, as we can see in Fig. 2.6, a remarkably good overall description
of the density of states for energies |E'| < 2 Veff, given the rather crude approximations
involved.
The somewhat astonishing and interesting point to mention here is that it is seem-
ingly not possible to explain the available experimental data on the density of the states
in the quantum Hall regime [19] by using impurity induced broadening of the bands
alone, as indicated by recent theoretical work [20].
2.1.4 Percolation limit and localization
The origin of the discussion of the localization properties of the states in a slowly
varying random potential in the presence of a strong magnetic field traces back to
Tsukada [21] who pioneered what is nowadays called the high-field limit. The key
point is to consider a potential which is spatially correlated with a finite correlation
length,
y(x,y)V(r + e,y+7?) = V0W, tj).	(2.59)
Here, K(fi, rf) = K(r) is the correlation function of the potential. It is assumed to
decay at least exponentially when r = y/^2 + rl2 ~► 00 and to have a range a
(Fig. 2.7). Furthermore, we assume again that
V(x,y) = 0,	(2.60)
and that the distribution function P(V(x,y)) = P(V) is independent of (x,y). contin-
uous and normalized to unity.
Such a spatially correlated potential may be obtained by superposition of impurity
potentials of finite range a. It varies only slowly in space. In the high-held limit, the
motion of the electron can be described by using a quasi-classical approximation. It
must be emphasized that such a potential is much more realistic than the white noise
which was used in the preceding paragraph for electrons in semiconductor inversion
layers because the impurities in these systems are usually not located directly within
2.1 Noninteracting electrons in reduced dimensions 95
Fig. 2.7 : The correlation function
K(r) of a slowly varying random po-
tential. The correlation length a is as-
sumed to be much larger than Ic.
the plane of the inversion layer but at some distance. The electrons feel only the rati-'-r
smooth Coulomb tails of the impurity potentials rather than the rapidly varying inn, r
parts.
When the magnetic field is so large that £c a it is of advantage to introduce the
so callee center coordinates [22]
X = k£2
r = ~^dT’	(2-61)
which obey the commutation relation
[X,Y]~i£2.	(2.62)
They will replace the ordinary spatial coordinates x, у when В —> oo.
In order to make this explicit, and to discuss the consequences, we write solutions ol
the Schrodinger equation corresponding to (2.35) in the representation of the Land, н
states т/у k(x, y) = (xy\nk), (2.4),
Ф(т,у) = ^(х?/|п/г)(п/г|Ф) = E Сп(Х(к))-фпк{х, у).	( 1
пЛ	n,k
1’he Schrodinger equation for the coefficients Cn(A') becomes
E (En5n.n^x,x> + (nA|V|n'X')) Cn,(X') = ECn(X),	(2.6u;
n' X'
with the Landau energies En and the potential matrix elements
(nA|V]n'X') -	[ [ y>n(x - X)V(x,y)ipn'(x - A') exp	• (2.65)
Ljy Jx Jy	у	/
When В —> oo, the potential varies only slowly as a function of the coordinates
as compared with the Landau wave functions. This implies that in the integral in the
96 Quantization of Transport
Schrodinger equation (2.64) the potential V(x, y) may be expanded into a power series.
One can easily verify that
[ dyV(x,y)exp	= V^i^cTV?^ 2тг^<5хл<,	(2.66)
where 6x,xl is the Kronecker delta. Finally, the integration with respect to x may be
approximately evaluated
[ dxV (x,	- X)<pn,(x - X) и V fx, i^-рЙ <5n,n',	(2-67)
J \ GA j	\ GA j
such that we obtain
^enCn(X).	(2.68)
X' = X
This differential equation can be formally solved with the ansatz
C(X) <x exp (-i	dX'k(X')^ .	(2.69)
The function k(X) has to fulfill the condiction
en = V(X,e2ck{X)).	(2.70)
v(x,i£2c-^-\cn(X')
\ GA j
This result tells that only those Landau states contribute to the eigenstates of the
Hamiltonian with disorder which correspond to the equipotential lines defined by (2.70)
(Fig. 2.8). The average density of states in this quasi-classical limit is given by the
probability distribution of the potential,
p(E) = P(E).	(2.71)
The Schrodinger problem in the limit В —> ос has been replaced by the problem of
finding the equipotential lines, which is nothing but a classical percolation problem.
Qualitatively, one can consider a smoothly (on the scale of the magnetic length)
varying landscape of mountains and valleys as it is increasingly filled with water. The
landscape corresponds to our smoothly varying random potential, the water level to
the energy in (2.70). When the water level is low, only the deepest valleys are filled.
The coast lines of these lakes correspond to the closed equipotential lines for E < 0.
When the landscape is almost completely filled, only the highest mountains are above
water level. Again the closed coast lines of these mountains represent equipotential
lines, but for E > 0. In each of these cases it is not possible to cross the entire system
by walking along any of the coast lines. The quantum states which correspond to these
equipotential lines are necessarily “localized” superpositions of Landau states centered
at the locations defined by (2.70). Only at a certain intermediate filling, there is a coast
line which connects the edges of the landscape. In this case, the equipotential line will
2.1 Noninteracting electrons in reduced dimensions
Fig. 2.8: Contour plot of a smoothly varying r?” '
potential landscape [25]. “Valleys”, V(x,y) < ,
indicated by dark, “mountains”, V(x,y) > 0, by liciit
areas. The correlation length of the potential, a. is
here 2/y.
correspond to a quantum state which is a superposition of Landau states centered  ।
positions throughout the whole system and will therefore be “extended".
This argument shows that for sufficiently high magnetic field, there is with vesy
high probability a state in each Landau band which is extended. When the ditribution
of the random potential is symmetric, P(V) = P(—V), the energy of this state imisi
be near the center of the band.
The connection with classical percolation theory may be exploited quantity
In classical site percolation, a lattice site j is occupied with a certain probability p, bT ;
one is interested in the properties of connected clusters of occupied sites as a function
of p. In the present problem the discrete site variables j are replaced by the continue'”
variable r. The probability p is defined by the energy E via
p(E) = fE P{V)dV.	(2.72)
J -oo
A n imber of results may be directly copied from classical percolation theory [23,
24]. First, of all, states which correspond to closed equipotential lines are necessarily
spatially localized (Fig. 2.9). For E 0 all contour lines are closed when the system
is infinitely large. Only at E = 0, which corresponds to the percolation threshold, an
infinite connected cluster, and correspondingly an infinite equipotential line exists. It
follows that all eigenstates are localized, except at E = 0.
The electronic wave functions will typically have a spatial extent - the localization
distance — which is of the order of the mean diameter £(E) of the connected clusters.
From percolation theory it is known that in two dimensions
^(E)«|E|-“	(2.73)
with the universal exponent v = 4/3 [26].
The percolation limit shows very transparently that all electron states are localized
with a tpatial extent which diverges in the centres of the disorder-broadened Landai!
98 Quantization of Transport
£
Fig. 2.9: Grey scale and three dimensional plots of some of the eigenstates of the smoothly varying
random potential shown in Fig. 2.8, as determined by numerical diagonalization of a system of finite
size [25]. They are superpositions of the Landau states that essentially correspond to the equipotential
lines as may be seen by comparing with the potential shown in Fig. 2.8. Light and dark areas denote
large and vanishingly small amplitudes, respectively. The two upper states correspond to energies in
the tails of the band, whereas the two lower are at energies near the band center.
2.1 Noninteracting electrons in reduced dimensions
O'!
bands. This holds true as long as the correlation length a of the potential energy
is finite. No matter how small it is, by increasing В one can then always reach the
percolation limit. As the localization-delocalization transition is a universal critic.J
phenomenon [27] we can suspect that this result will remain valid even when the
condition a 3> lc is not fulfilled.
We cannot, however, assume that the critical exponent v from classical percolation
theory is the correct exponent for the localization length. Important quantum effects,
like tunneling and interference are not included in this approach. It has indeed taken
considerable numerical effort to precisely evaluate the exponent, and to demonstrate
that its numerical value, v = 2.35 ±0.04 is independent of the type of the randomness,
and of the Landau band index [28-31].
In doing so, the localization length was defined as the exponential decay length of
the modulus of the one-electron Green function [27],
1	lG(E;r,r')l
77— hm	—j---------— .	(2.7 I'
£(E)	|г-г'|-+оо |r — r'|	v
The Geen function G(E\r — r') is the position representation of the one-electror.
resolvent
G(E-r, r') =	•	(2.73
Here, 7/ is an infinitesimally small parameter which is added to the energy in or<±.
avoid the singularities at the real energy axis due to the eigenvalues of the. Hamiltonian,
and which has to be taken to zero at the end of the calculation.
This definition makes the close connection between the localization of the ец
states and the transport properties immediately transparent. The modulus of tin-
Green function is the probability that a quantum particle with the energy E travels
within an infinite time interval from the site r to the site r . Only if this probability
remains finite when the distance between the two sites becomes infinite, the states at
energy E are able to contribute to the transport at the absolute zero of the temper-
ature. When the localization length is finite, the probability tends exponentially to
zero. No transport can take place at energies where this happens. Thus, we see that
disordei is able to introduce a qualitatively new feature to the quantum states: they
are elec .rically conducting or insulating depending on whether the localization length
is infini ,e or finite, respectively. The explanation of the quantum Hall effect requires
as we shall shortly see — the existence of both localized, and a certain amount
of exterded, or, equivalently, “conducting” states. Therefore, disorder is one of the
nece.ssaiy ingredients for an understanding of the effect.
Above, in the model of free electrons in a homogeneous magnetic field, we have
considered the influence of the boundary conditions of the energy spectrum and the
eigenstates. In order to discuss the quantum Hall effect we will also have to know
the influence of the disorder on the edge states. We only mention here that there is
sufficien; numerical evidence to assume that the edge states remain extended in the
presence of disorder, and do not mix with the exponentially localized states in tin
100 Quantization of Transport
Fig. 2.10: The influence of the disorder on the energy spectrum ev of a two-dimensional system in
a strong perpendicular magnetic field with a finite extension in the ж-direction (Lx = 26fc). The left
hand side of the figure shows the spectrum with a = 2tc, while the spectrum for a = lc is shown on
the right hand side. For larger correlation, only the bulk states |X„| < Lx/2 are markedly influenced
(after [33]). The coordinates Xv denote the locations of the states.
bulk of the system. Only near the centers of the Landau bands, where the localization
length of the bulk states diverges, edge and bulk states do mix [32,33] (Fig. 2.10).
Qualitatively, this can be easily understood. In order to localize the quasi-one
dimensional edge states, backward scattering is a necessary ingredient [34]. However,
in the energy regions of the edge states, states with fc-vectors in opposite directions
are spatially separated at opposite edges of the system. Thus, impurity scattering
can only cause forward scattering of the electrons. Backward scattering is suppressed
approximately by a factor exp [—(Lx/Q)2], since the range of the edge states in the
^.’-direction is at best of the order of the magnetic length. Therefore, the edge states
are essentially not influenced by impurity scattering, except in regions where backward
scattering is enhanced by tunneling between the edges via localized bulk states near
the centers of the Landau bands.
2.2 The integer quantum Hall effect
2.2.1 Experiment
The quantum Hall effect was discovered by Klaus von Klitzing in 1980, when work-
ing as a guest researcher at the High Magnetic Field Laboratory of the Max-Planck
Gesellschaft in Grenoble [35]. He investigated the electrical transport properties of a
silicon MOSFET subject to a magnetic field of about 18T at a temperature of about
1.5K. He found that the Hall resistance Rh, the ratio between the Hall voltage and the
source-drain current, measured at two contacts perpendicular to the latter, exhibits
2.2 The integer quantum Hall effect 101
Fig. 2.11: The integer quantum Hall effect as discovered in 1980 by K. von Klitzing using a silicon
MOSFET [35]. The Hall voltage Uh and the voltage U at two contacts parallel to the source-drain
current (f = lpA), U, are shown as functions of the voltage Us at the gate of the MOSFET at a
temperatire of T =1.5K and a magnetic flux of В =18T. The insert shows a schematic picture of the
MOSFET with the two dimensional electron gas (circles) underneath the gate electrode (dashed lines)
between the source and drain contacts (shaded areas), the Hall voltage probes and the contacts for
the measurement of the voltage drop parallel to the current. The device was 400pm long and 50pm
wide. The distance between the probes for measuring U was approximately 130pm.
plateaus that were given by integer fractions of h/e2 when the voltage at the gate of
the transistor is varied (Fig. 2.11).
/?н = -4	(.7 = 1,2,3....).	(2.76)
J ez
At the same gate voltages he observed that the magnetoresistance — the ratio between
the voltage measured at two contacts parallel to the direction of the current, and the
current — was extremely small, in some cases even below the limits of detection. Since
the Hall conductance, the inverse of the Hall resistance, was quantized in integer uniis
of c2//i, this was called later on the integer quantum Hall effect.
The integer quantum Hall effect is a genuine quantum transport feature of the tw -
dimensic nal electron gas at the insulator-semiconductor interface in the MOSFET. 1!
occurs whenever the gate voltage is such that the Fermi level is located in a region ci
102 Quantization of Transport
“localized quantum states”, above an integer number of Landau bands, at integer fill-
ing factors. The latter states are, for instance, induced by impurities or other disorder
effects. Surprisingly, in spite of this connection with disorder, the integer quantum Hall
effect provides an extremely precise method to measure the Sommerfeld fine structure
constant, a = (/zoc/2)(e2/7z), in a many-particle system, independently of atomic spec-
troscopy. In addition, it can be used as a standard of the electrical resistance that is
only based on fundamental constants. This was immediately recognized by von Klitz-
ing, and eventually led to the redefinition of the “Ohm” in terms of the von Klitzing
constant Rr = 25812.8085Г2 « e2/h in 1990 [36].
2.2.2 Electrical conductivity
Before we discuss the integer quantum Hall effect in terms of localized and extended
states, we will briefly introduce into the transport theory in high magnetic fields.
In the presence of a magnetic field, and an infinitesimally small electric probe field
E(r) the electrical current density can be written in linear approximation (see also
chapter 1) as
J(r) = У dr'<r(r,r')  E(r').	(2.77)
Equivalently, we can define the electric field as the response to a current density
using the resistivity,
.E(r) = У dr'p(r,r')	(2.78)
The resistivity tensor p is connected with the conductivity tensor via the relation
У dr"cr(r, r")  p(r", r') = <5(r — r').	(2.79)
Quite generally, the conductivity tensor a is anisotropic when a magnetic field is
present, even if the original system without any external fields applied was isotropic.
It is also in general non-local and non-homogeneous. It depends on r and r', in the
quantum limit, when T = 0. Non-locality is a genuine property of quantum transport,
due to the “rigidity” of the quantum states. Non-homogeneity is characteristic for
the transport through confined systems. Equation (2.77) is a non-local relationship
between probe field and current density.
A two-dimensional disordered electron system, in the absence of any special symme-
try, will be homogeneous and isotropic on the average. The configurationally averaged
conductivity depends only on the difference between the positions, r — r'.
With a homogeneous magnetic field perpendicular to the (x, ?/)-plane, the average
conductivity tensor has the form
__ (	XrfiR) \
~ Vyx(B) rnJ/fi )
(2.80)
2.2 The integer quantum Hall effect 10-
The relations
&XX — &yy
Pxx
PxX T Pxy
(2.81)
and
&yX — &xy
Pxy
pL + p'iy
(2.82)
imply a somewhat counter-intuitive result, namely that if crxx —> 0 then also pxx —> 0.
and vi :e versa, a consequence of the fact that j and E are in general not parallel in a
magnetic field.
Using the classical (Drude-Zener) friction model the components of the conductivity
are easily obtained,
_ (To
(Txx i . /	’ (TyX — OJCT(JXX,
1 + (wcr)2
with the Drude conductivity
e2Nr
= ----—
m*
(2.83;
(2.8 P
where г is the mean free time, N the electron density and m* the effective mass.
We nose that for strong magnetic field axx ос t~1 —> 0 when В —» oo. Scattering is
needed in order to obtain transport, in contrast to the limit of a small magnetic liel-1,
Furthe -more, in the friction model,
1
<7ц = —, pxy = — .	(2.S ;
it	<7J{
The Hall resistivity is independent of the scattering time r. Using the filling facts'’
v introduced above, (2.11), we obtain the frequently celebrated result for the sc
classical Hall conductivity
e2
(T\\ = v~^,	(2.8b;
which gives integer multiples of e2/h for integer fillings.
This is, however, not the quantum Hall effect, since that would imply a non-linea-r
dependence of the Hall conductivity on the electron density N or the magnetic field.
and sin .ultaneously the vanishing of the magnetoconductivity.
In crder to achieve a qualitative understanding of the quantum Hall effect, linear
response theory can be used. The Kubo formula for the static conductivity temo
obtained in chapter 1 (cf. (1-76) ff.) can be rewritten by spatially averaging the currem
density and partial integration with respect to the energy in the following form [37, 381.
If
cm = «—lin,‘, У111 т~г / dE-f^
” ’ f ,iG'" , o+ r /'	,|G'
"X U | ~ЛЁЛ>и Im G + P'1 Im G Pl' ~AE
Itl4 Quantization of Transport
Here, f(E) is the Fermi distribution at temperature T, pu the momentum operator in
the direction of v (p = x, y), and the resolvent operators G± — (E ± ir/ — H)~l.
The order of the various limits to be taken are very important. First, in any case,
the thermodynamic limit, LxLy -> oo has to be performed. Only then the imaginary
part of the energy, 77, is allowed to approach zero.
If one considers the quantum Hall effect in a finite system connected with contacts
via external leads to reservoirs, a philosophy applied by Landauer to quantum trans-
port, and later by Biittiker to the Hall effect [39,40], one writes the current Im flowing
through a contact m as linear response to the potentials Vn at the contacts n. As
shown in chapter 1, one can obtain by manipulating the Kubo formula
Im = £ GmnVn ,	(2.88)
n
where the conductance coefficients Gmn are given as integrals over the nonlocal con-
ductivity tensor. The quantum conductance coefficients can here eventually be related
to the quantum transmission probabilities.
In this approach, the thermodynamic limit has been effectively built in from the
very beginning, via ideal leads connected to the system and reservoirs beyond the leads.
Macroscopic, global currents are related to global voltages via global conductances, which
must, nevertheless, be calculated from a microscopic theory. This is in contrast to the
philosophy behind (2.87), which starts from the microscopic theory for the local current
densities and local electric fields. .
The quantum Hall effect shows a remarkable universality and extraordinary pre-
cision. It is observed in large and also very small samples, in a variety of different
materials, at various temperatures and magnetic field strength. We feel that it should
be sensible to consider the slightly more general bulk phenomenon approach. However,
once we consider the thermodynamic limit of a finite system with or without being
connected to the external world via contacts, for instance modelled by boundary con-
ditions, the results should become in any case independent of what happens at the
boundary. A discussion of the theory of the transport in the quantum Hall regime can
be found in [3].
From (2.87) one can obtain another useful expression which is valid for the ther-
modynamic limit of a truncated cylinder geometry at zero temperature [38]. This is a
system with periodic boundary conditions in one, say the у-direction, and with a finite
length in the perpendicular direction. Essentially, the momentum operators in (2.87)
are replaced by the position operators via the commutator relations
= ^-[(G*)"1,^], (v = x,y)	(2.89)
inm
and carrying out the trace in the position representation. After lengthy but straight-
forward algebra one arrives at
" T №01 /im 7^7-/dr- <)(re-<)|G(£’+;r.r')|2.	(2.90)
______________________________2.2 The integer quantum Hall effect 1 05
In the presence of disorder, this expression has to be configurationally averaged.
The components of the conductivity have an intuitive physical meaning. The di-
agonal part, divided by 2т?2 is the mean square distance, an electron with energy E
can diffuse within infinite time, since |G+(E; r, r')|2 is the probability that an electron
travels from r' to r. The off-diagonal part is correspondingly the mean area covered
by the states at energy E. If there are only localized states at E, both, the diffusion
distance and the mean area of the states are finite and both, axx and ayx vanish, due
to the limit у —> 0.
Equation (2.90) is very useful for numerical purposes [41]. It allows also to discuss
some ol the essential features of the quantum Hall effect, as we will see below.
2.2.3 Understanding the integer quantum Hall effect
It is easy to show the connection between the integer multiples of the Hall conductivity
of e2//i and integer filling factors as long as disorder and interactions are completely
absent.
The result obtained above for an infinite system by using a “volume argument”,
(2.86), can also be reproduced for a system of finite extension by using the “edge
state model” [42]. The current is I = eNv. The drift velocity in the eigenstate Vgk
is Vj — h~ldEj(k)/dk. Assume a small Hall voltage Ua to exist accross the system,
produced by a difference in the chemical potentials between the left and the right
edges o:' the system, eUu = p.| — fir. The density of states in this energy interval is
N — [2:гLx|dEj(к)/Лк|eU^, such that the current becomes I = vetU^/h, which
implies exact quantization, as in (2.86).
In this section, we want to generalize this argument in order to provide an under-
standing of the quantization of the Hall conductivity at high magnetic fields in the
presence of disorder. We do not claim that the argument is precise on the experimen-
tally achieved level of l() n. But it shows that quantization of the Hall conductivity in
units of c2//i is possible in spite of the disorder, and that edge states might play an im-
portant role in the regions of the Hall plateaus. In addition, we will discuss some of i he
recently observed scaling properties of the transport in the integer quantum Hall effe< i
with temperature, system size and frequency in the regions between the Hall plateaus.
They stiongly suggest that disorder is necessary for the complete understanding of t>-
effect, a id cannot be neglected.
Quantization of the Hall conductivity
We consider energies well between two successive Landau levels, and evaluate (2.90)
approximately by making use of the properties of the edge states that we derived in the
section at the beginning of this chapter for a system with periodic boundary conditions
in the y-, and of finite length in the .r-direction. In addition, we assume disorder to be
present. However, we assume that it localizes only the states for \k(./\ < Lx/2 (bulk
states), and leaves the edge states, |А.-/'2|	Lx/2, essentially unchanged, as discussed in
the section on the percolation model. Near |M2| « Lx/2 the bulk states become quasi-
extendec since back-scattering is practically absent for the edge states. We consid- !
106 Quantization of Transport
Fig. 2.12: A system that
is spatially confined in the
т-direction with localized
bulk states that are shown
schematically as equipoten-
tial lines of the random
potential, and edge states
extending along the edges.
Energy is such that two
(bulk) Landau bands are oc-
cupied. In this example, two
edge channels contribute to
the transport.
energies well between the centers of the Landau bands.
We decompose the spectral representation of the Green function into contributions
from the bulk and the edge states, \i/) and \nk), respectively,
ф±) = У	+ V------ИИ-------.	(2.91)
1 1 tT E ± - E^ V E ± it/ - Ev	k
The edge states and the corresponding energies are assumed to be not influenced
by the disorder. They are essentially plane waves with wave number к extending
along the edge of the system in the «/-direction. In the .r-direction they art1 very well
localized close to the edges of the system at ±Lx/2. The bulk states, on the other hand,
are assumed to be well localized to finite regions, much smaller than the diameter of
the system (Fig. 2.12). These assumptions are justified for energies well between the
(bulk) Landau bands, the effective disorder sufficiently small and/or the magnetic field
sufficiently high.
In what follows, we will need the following properties of the ^-dependent part of
the edge states and the corresponding energies
En,k = En_k.	(2.92)
dgn* = УЕпек.	(2.93)
dk dk	v ’
У dr.-^/^).;:)^,,/,*,)./;) =	(2.94)
I d:r2#„,±fc(x)|2 « ±y,	Л'(А:)йу.	(2.95)
The last property can be justified for the solutions of Weber’s equation (2.30) when 13
is large and for energies between the Landau bands.
2.2 The integer quantum Hall effect
New let us consider the components of the conductivity, (2.90). They can b<
decomposed into
+ <7^ + <7^ ,	(2.9G
where ab, ae denote the bulk and edge contributions, respectively, and abe contain;
both, bulk and edge states.
Fo • the diagonal part of the conductivity, axx, the integrals on the right hand side
(2.90) converge since the bulk part corresponds to localized states and the edge stT'^
are localized in the ^-direction. Thus, axx vanishes in the limit /) —> 0.
Foe the Hall conductivity, avx, the bulk and the mixed contributions vanish me
portio.ial to /?. The remaining contribution of the edge states can be discussed
rewrit ng (2.90) into
which can be accomplished by using the resolvent identity
G+ - G~ = -2o/|G±|2,
(2^<
and performing the trace in position representation, as before.
Th? second term on the right hand side of this equation gives zero because of (2.95)
and since for vanishing r; the imaginary part G+ — G~ equals —27rici(E — Enk), such
that only wave numbers ±k(E) fulfilling E = Enk contribute.
Foi the trace in the first term one obtains
y, 2i d
Lv d(k — k')
nkk' У v '
sin j(/c - k')Ly 1
(к - к1)
f VnkWVnk'tE) f dx :ry,•*.(.«>„*' (ар	(9g[))
(E — Enk + iq)(E - Enk, - iz/)
For Ly —> oo only the terms к w k' contribute. Thus the first and the second integrals
in the nominator give 1 and sgn(A:)LT/2. respectively. Writing k — k' = q and expanding
the denominator for |</| •?:' A: one obtains
E Enk qank и] E Enk it] у E Euk i//J
where ank is the derivative of Enk with respect to A:. The A’-sunis can be evaluated 5-.
conver ing them to integrals.	~> (^у/^тг) J <ik. One obtains for L,, oo
и,,. _01го4Лспс-} = -(г! /4*
LxLy I	>	2 „ IT J (E - Enk)2 + Г)2
d)<ink
E - Erik - I?) '
108 Quantization of Transport
71+1/2
Fig. 2.13: Qualitative behav-
ior of the magnetoconductivity
axx and the Hall conductivity
axy as a function of the energy
e = Е/Ты,: for various temper-
atures T. When T increases,
T < T' < T", the width of
the peaks in axx increases. The
steepness of the corresponding
steps in axy decreases.
For r) —> 0 we can write further
lim lim —tr {~4y2yG+xG~~] = У | sgn(fc(£))
140 1,400^ i	2 |anA:(E)|
= У (occupied bands)	(2.102)
Tl
since, because of (2.93), the derivative of Enk is positive (negative) for k(E) positive
(negative), and the fc-sum reduces to only the two terms which correspond to the
positive (negative) roots k(E) of Еп^е) = E in the region of the edge states.
Thus, for Fermi energies well between the Landau bands, in the region of well
localized states, the Hall conductivity is quantized in integer multiples of e2//z.
<^=j- (/ = 0,1,2,3,...).	(2.103)
Scaling in the quantum Hall regime
This argument breaks down in the vicinity of the centres of the Landau bands, because
here localized bulk states cannot be separated spatially from the edge states. Here, in
the regions which correspond to well pronounced peaks of the magnetoconductivity axx,
and the Hall conductivity axy changes from one plateau value to another (Fig. 2.13).
Striking scaling behaviors of the transport coefficients with temperature, system size,
and frequency of an applied electromagnetic field are experimentally observed in these
regions [43-45]. In order to understand these effects, the localization model is a useful
starting point, though up to now no quantitative calculation has been successfully
performed that is comparable in accuracy with the experiment.
In the localization model, the result described above for the localization length
in an infinite two-dimensional disordered system subject to a strong perpendicular
2,2 The integer quantum Hall effect
Fig. 2.14: The density of states p(E),
and the localization length £(E) of
the disordered Landau model. One-
electron states are extended in inter-
vals E'c - Ec = ДЕ where £(E) is
larger than the phase-breaking length
magnet c field is very important. The localization length is finite everywhere but with
singulaiities close to the centres of the Landau bands, (2.73), described by a critical
exponent v. This holds at zero temperature. One would therefore expect the peaks in
crxx to be of zero width close to the absolute zero of the temperature, and the steps
in crxy to be infinitely steep, for an infinite system (Fig. 2.13). How can we proceed
to obtain results that are valid for finite temperatures, finite systems sizes, and finite
frequencies, by using this picture?
Stril.tly speaking, the localization model breaks down at finite temperature. The
Hamiltonian should, in addition to the disorder, contain terms which describe the in-
teractions of the electron with each other, and other degrees of freedom as, for instam e,
vibrations of the atoms. These are not accounted for in the non interacting electron
approximation used in the localization theory. One should note that also at T = 0 the
localization model is only valid for non-interacting electrons. No accepted localization
theory exists for interacting electrons in the presence of disorder. Interactions lead to a
destruction of quantum mechanical phase coherence in the one-electron Hilbert space
in which localization of the electron states is defined.
If we assume that the corresponding phase-breaking scattering processes are sufl
ciently rare we can nevertheless use the localization picture as a starting point, and
introduce the effect of the interactions via a phenomenological parameter, a phase-
breaking scattering length L^, which should depend on temperature, frequency and
other parameters, and would roughly be given by the mean distance an electron can
diffuse i i a given quantum state before being scattered inelastically to another one. An
electron state would then appear localized only when its localization length is smaller
than LffT, ui,...) (Fig. 2.14). When H,(E) was larger than Lv the electron would be
scattered by a phase-breaking process before “being able to experience that the stat'-
which it occupies is localized”.
Within each of the Landau bands, at finite temperature, and also at a finite fre-
quency if the applied electromagnetic field, intervals of energies, AE(T, oi...), inn t
exist for which the corresponding one-electron states are effectively delocalized, e-c .
their localization lengths are larger than the phase-breaking length. The states v он c
these in ervals can contribute to transport. Correspondingly, the widths of the - 
110 Quantization of Transport
in oxx are tx &E(T,w...), and the steepness of the steps in the Hall conductivity
a E'c — Ec = AE(T,cii.. .)-1. The corresponding temperature and frequency depen-
dences should be closely related to the dependences of ДЕ on these parameters. In
particular, when the phase-breaking length exceeds the system size, the experimentally
observed widths and steepnesses should saturate, an effect which was indeed observed
in the temperature-dependent scaling of the quantum Hall effect [44], and also when
applying a microwave field [46].
The temperature dependence of ДЕ can easily be calculated when the temperature
dependence of L. is known. Assume oc T~pl'2, where p is of the order 1 and
depends on the nature of the phase-breaking processes. By letting £(E) = LV(T) we
obtain
ДЕ(Т) = const  Tp/2” = const  TK.	(2.104)
When the phase incoherence was solely introduced by an electromagnetic field, the
time scale that was set by the field was ci№. The distance, which the electron could
diffuse within this time interval was ДДа») ex thus p = 1, and к — l/2m The
two phase-breaking lengths have to be combined via the analogon of Matthiesen’s law
r г = r + г / v	(2.105)
since the scattering processes which are involved are incoherent. Presently available
experiments are fully consistent with this approximate picture (Figs. 2.15) [43-45].
2.3 Coulomb interaction and magnetic field
2.3.1 The fractional quantum Hall effect
The discovery of the integer quantum Hall effect stimulated intensive experimental
and theoretical research in practically all of the international research laboratories and
metrological institutes. A particularly important discovery was made only two years
after the first report by the group of D. C. Tsui at Bell Laboratories [47]. Using a
GaAs/AlGaAs heterostructure containing only little disorder, with an extremely high
electron mobility in the electron gas at, the AlGaAs/GaAs-interface, they detected
quantization of the Hall conductivity at magnetic field strengths corresponding to fill-
ing factors v = 1/3, 2/3 when the temperature was much lower than IK. A more
recent experimental result which exhibits structures at several other '‘fractional fill-
ings” is shown in (Fig. 2.16). It turned out that the explanation of these additional
features cannot be understood by using the non-interacting electron model. Coulomb
interaction of the electrons is a necessary prerequisite.
Meanwhile, the fractional quantum Hall effect belongs experimentally and theo-
retically to the most active areas of research, since it provides unique possibilities of
studying features induced by the electron-electron interaction.
2.3 Coulomb interaction and magnetic field
Fig. 2.15: The temperature dependence of the maxima of Apxy/AB (upper three sets of data) and
the widths AB of the peaks in pxx (lower two sets of data). The slopes of the curves are consistei
with к = 0.42 ± 0.04. The data are taken in the regions of the lowest Landau band (spin down, op<
and clostd circles) and the second lowest Landau bands (spin up, open and closed squares and sihi
down, open and closed triangles). Redrawn after [43].
2.3.2 Few electrons with Coulomb interaction
The quantization of the Hall conductivity of the weakly disordered two-dimensional
electron gas in a strong perpendicular magnetic field at filling factors. (2.11), that a'<‘
rational numbers, made it necessary to study interaction of electrons in the presence
a quant, zing magnetic field. Early attempts to explain Hall plateaus at fractional mid
tiples of e2/h within the non-interacting electron model by introducing an additional
periodk potential failed. Only integer Hall plateaus can occur in a non-interacting
quantui i Hall system [49].
At metallic densities, when the mean electron distance rs is smaller than the effective
Bohr rtdius, aB = 4irEE0h2/m*e2 (e relative dielectric constant), screening effects,
collects e excitations, plasmons, and charge density waves have to be considered in the
first ins ,ance, according to the conventional wisdom of textbooks. When the electron
density is low, such that rs > aB, however, the electrons tend to crystallize, due L
the fact that Coulomb repulsion energy, which is proportional to rf1, dominates lie
total energy Eo. In first approximation, it consists of the Coulomb repulsion term, in
addition to the terms that correspond to the zero point motion, and the kinetic energy,
112 Quantization of Transport
Fig. 2.16: The fractional quan-
tum Hall effect. The Hall resisti-
vity (upper curve) of the in-
version layer in a high mobili-
ty AlGaAs/GaAs heterostructure
shows plateaus at magnetic induc-
tions В that correspond to the in-
dicated filling factors u. At the
same filling factors, the magne-
toresistivity (lower curve) shows
minima (redrawn after [48]).
<x rs1,5, and ex rs 2, respectively, [50,51]
. А В c
^o(rs) =-------h -j-g +	. 
rs rsls r/
(2.106)
It is believed that in a magnetic field the tendency to the formation of a Wigner crystal,
or a Wigner glass in the disordered case, is enhanced [52].
The electron density in a typical inversion layer in the region of the quantum Hall
effect is a few 1011 electrons per cm2. This corresponds to a mean electron distance
of rs « lOnm. The effective Bohr radius (m* « O.O7mo, e » 10), is o,b « 7nm.
Thus, we see that the electronic system is in a density region where Coulomb repulsion
cannot be neglected. Early attempts to confirm the existence of a Wigner lattice in
the quantum Hall regime failed [53]. A fresh approach towards the problem of the
interacting electron gas in the presence of a strong magnetic field was necessary. It
turned out, that several very useful statements could be made on the basis of exact
diagonalizations of a few interacting particles. They eventually led to the discovery of
a new kind of state of the electrons in the presence of interactions and a quantizing
magnetic field, the incompressible electron fluid, which is believed to transform into a
Wigner crystal at sufficiently small densities.
Two electrons
It is instructive to study two electrons with Coulomb interaction. The magnetic field is
assumed so strong that the Landau bands do not overlap. This requires » fi2/fc,
2.3 Coulomb interaction and magnetic field 113
the Coulomb repulsion between two electrons in a distance of a magnetic length £c.
The expectation value of the Coulomb energy in the eigenstates which correspond to
the lowest Landau band, (2.14), can easily be calculated (n = 0, symmetric gauge),
Seoul — \ 'Фпт
e2
Фпт
е2Г(т+ 1/2)
fc m!
(2.107)
This is oc m О2 when m is large.
Two electrons can be described by the Hamiltonian [54,55]
Я = 2m7	+	+ ^p2 + еЛ(г2))2] + Hint.	(2.108)
The interaction term can be chosen, for instance, as
Hint = aJ(ri - r2),
for the contact interaction,
н - e'2	1
'nt 4тее0 |fi - т’21 ’
for the Coulomb interaction and
,2	I
H- , = —__________-____p-aln-7-21
111 nt .	I	I t	i
4тгее0 In - r2|
(2.109'
(2.110)
(2.1111
for a scieened Coulomb interaction. We will assume in the following that the interaction
energy depends only on the relative coordinate ri — r2. In most of the interesting cases,
this assumption is fulfilled.
By transforming to relative and center-of-mass coordinates, r and R. respectivelv.
	R=^(.ri+r2), Г = -^=(rj - r2), Г1 = 7тл + Г)’ r'2 =	~ ’
0 _ OR ~	1 / 0	0 \	d _ 1 (_0	 У2	+ dr 2 / ’ Or >/2 \c)r1 Or 2 /
0 drt	1 (JL	э _ 1 f_2_ 2Л y/2 \ OR dr ) Or 2 \/2 \ OR Or J
t he Hamiltonian for the two interacting electrons in the x-y plane, and in the symr
gauge, becomes
H = H0(H) + W0(r) + Hi„t(v/2r).
(2-1 io!
114 Quantization of Transport
where Hq is the unperturbed Hamiltonian (2.13). The first term represents the en-
ergy of the center-of-mass motion in a magnetic field. The second and third terms
correspond to the energy of the relative motion in a magnetic field and a repulsive
potential. Center-of-mass and relative motions are completely decoupled. The total
wave function of the two electrons can be written as a product of the wave functions
of the center-of-mass and the relative motion, ф and ф, respectively,
Ф(Л, r) = ф(Я)ф(г).	(2.114)
Whereas the problem of the center-of-mass motion has been solved above — it is
nothing but the problem of a free single particle in a strong magnetic field — what
remains to be done is to treat the problem of the relative motion. The Schrodinger
equation of two interacting particles has been reduced to a one-particle problem.
Before proceeding further it is useful to consider a few symmetry properties which
follow from the fact that the electrons are Fermions. Upon interchanging the coordi-
nates the sign of the wave function must change,
Ф(Л,г) = -Ф(Л,-г).
(2.115)
Therefore, the wave function that describes the relative motion must be antisymmetric,
U(r) = -ф(-г).
(2.116)
Furthermore, since the interaction potential is radially symmetric, ф can be chosen to
be an eigenfunction of the angular momentum m, and angular and radial parts of the
wave function factorize,
U(r) = 77(r)eim*’,
(2.117)
with m odd. The radial part 77(r) satisfies the Schrodinger equation
1 d277
2 dr2
1 d77	1 /m2	r2	\/2e*2
—i----1" 3 ~2~ + m + ---------------
r dr	2 \ r2	4 r
77 = £77,
(2.118)
with r and E measured in units of lc and respectively, and the effective charge
с* = е/УЗтпЕёо. It can be solved perturbatively for |B| —> oo, when

« 1.
(2.119)
Since according to (2.107) (£illt) ос фВ/т when m is large, and on the other hand
tiujc ос В, this condition can always be fulfilled for sufficiently large B. In this limit,
we can apply first-order perturbation theory. The states are the eigenstates of the
Hamiltonian without interaction, and the corresponding first-order eigenenergies are
r (	1 M ~	/	। f'*2 ! , \
^nm ~ I n £ 2 "I 2	)	’ /9 lUnm) 
(2.120)
2.3 Coulomb interaction and magnetic field 115
For the lowest Landau band, we obtain
1„ е*2Г(т + 1/2)
= ~h^ + —.	(2.121)
2	ml	v '
In orde: to obtain the total energy of the two electrons the energy corresponding to
the cen ,er-of-mass motion, E^, = fiwc/2, has to be added.
The degeneracy of the Landau levels is removed by the interaction. Vanishing
angular momentum corresponds to the state at energy Ыс/2 + с*2^/тг/2fc- When tb-
angular momentum increases, the energy levels converge as m'1|/2 towards ftwc/2. Tie.,
is due tj the fact that the mean distance between the particles increases with y/m. .7
large distances the particles do not feel the interaction anymore. The electrons behav<-
as if they had acquired a “negative binding energy” Eb = —e*2v/7r/2C in the hig.
state where they repel each other most.
N electrons
The transformation to center-of-mass and relative coordinates may also be performed
for Hamiltonians containing N interacting electrons
	N J=l	W +1E v(rJ	“ n).	(2.122;
The ort ic	gonal matrix, OO1 = 1, /	_i_	_1_ Vn	-Tn ~N+l	1	1 y/N 1		7 1	
	y/N(N-l) ()	-N + 2	yv(v-i) 1	V<v(.v--1) 1	(2.123!
О -	x/(/V-l)(/V-2)	y(W-l)(V-2)	v/'(V-l)(lV-2)	
	0	0 т!	1	I x/ti	x/ii	
	0	0	73	73	7	
t ransforms the set of coordinates (/>,.. .. п) into (pw,. . .. pt) according to
(2.124)
By using
(2.125)
116 Quantization of Transport
ЕР? = М,	(2.126)
;=1 j=l
N	N
У^.(РхзУ,1 — Pyjxj) =	“ n>u£j),	(2.127)
j=l	, = i
I E V= ^int(Pi, • • , Pn-i) ,	(2.128)
the Hamiltonian is tranformed to
Я = H0(pN) + 52 Ho(Pj) + #int(P1, • • •,Pn-i) 	(2.129)
j=i
The first term describes again the free center-of-mass motion in a magnetic field,
whereas the second and third terms are due to the relative motions. The Я-particle
problem has been reduced to an (Я — l)-particle problem by this transformation.
We consider three electrons as an example. The Hamiltonian is
2
H = Я0(Рз) + 52 НЛ>,) +	p2),
t=i
(2.130)
with
Рз = “7т(п+г2 + г3),
V ’J
1 ,
Pi = ~ЛГ1 +Г-2- 2r3) ,
VO
(2.131)
and
Pl = 7i(ri ~r'2'1 ’
e*2 Ле*2 Ле*2
+ |Pi + У3р2| + IЛ>р2 ~ Pi |
(2.132)
Again the wave functions of the three particles factorize,
Ф(РЗ=Р2>Р1) = <КРзМР2> Pi) 
(2.133)
Since they must be antisymmetric with respect to permutations of the three particles,
2.3 Coulomb interaction and magnetic field 11 7
ip must fulfill symmetry relations,				
^(P2,P1)	= -^(Р2,-Р1)	(123 -	-> 213)	
0;P2,Pi)	=	~P2)> |(Pi + \/Зр2))	(123 -	-> 132)	
^>2,Pi)	= -^(~5(\/Зр1 + р2),|(р1->/Зр2))	(123 -	-> 321)	(2.134)
<P2, Pi)	= +ip (-|(УЗР1 + p2), -j(pj - \/зр2))	(123 -	231)	
0'P2,P1)	= Ftp (|( A>t — p2), — |(pi + \/Зр2))	(123 —;	► 312) .	
These relations are equivalent to the statement that the probability density is invariant
under rotations about multiples of тг/З.
Also, the Hamiltonian commutes with the total angular momentum,
[H,L]=Q,	< . .
since only differences of coordinates occur in Hj,lt. The Hamiltonian and the to’ !
angular momentum can therefore be diagonalized simultaneously. The eigenvalue,
of L may be used to label the wave functions,
Ф = Ф(М).	(2.136)
The symmetry properties of the states imply that the following functions form a
complete set which may be used to diagonalize the three-particle system [55],
h,m') = | [(z2 +H1)3"1 - (z2 - щ)3"1)] {4 + z^',	(2.137)
where for convenience	+ it/,, F is a normalization factor. The coordinates
are measured in units of the magnetic length fc- These states are eigenstates of the
total angular momentum with eigenvalues M = 3m + 2m'. The diagonalization in the
subspaces that correspond to a given total angular momentum may then be straight-
forwardly performed by computing the roots of
det	- (mml\H-mt\?n"m'”)\ = 0,	(2.138)
since in the one band limit the band center can be assumed to be the zero of energy.
It turns out that in fact the above basis set is already a very good approximation for
the eigeistat.es. The off-diagonal matrix elements of the interaction Hamiltonian an'
tvpicallj ten times smaller than the diagonal elements.
The minimum angular momentum for which a (nondegenerate) solution exists
M = 3 (m. = l,rn.' = 0). Its mean area is about three times smaller than the arc:
of the states that correspond to M = 9. The latter are constructed from the ba1"
states with (m, m') ~ (3,0) and (1,3), and are the energetically lowest states wle !•
118 Quantization of Transport
Fig. 2.17: The energy spectrum of 6 interacting electrons with Coulomb interaction in the lowest
Landau band on a disk with a positive background, v = 1/3. Energy levels are classified according to
their total angular momentum quantum number M [56].
are degenerate when Hint = 0. The average “charge density” in these states is only 1/3
of the density in the state that corresponds to M = 3. Since the angular momentum
is a conserved quantity, and M corresponds to the “area” of the eigenstates, the three
electrons are “incompressible”. This property of the interacting electron system in a
strong magnetic field is now believed to be very important for the understanding of
the quantum Hall effect.
Quite generally, the Hamiltonian may be diagonalized numerically for N particles,
for instance in the properly antisymmetrized basis of the Landau states. Since the total
angular momentum commutes with H, the eigenstates may be chosen as eigenfunctions
of hL. Energy spectra for up to about N = 10 electrons in the lowest Landau band
were obtained by this method. An example of a spectrum is shown in Fig. 2.17 [56].
2.3.3 Fractional quantum Hall states
It is practically impossible to diagonalize the Hamiltonian of many electrons with
interactions. Therefore, approximative methods have been developed, in order to de-
scribe the properties of such a system. The similarity with the effect at integer filling
factors suggests that one of the necessary ingredients for the quantum Hall effect at
fractional filling factors is the presence of a gap in the excitation spectrum of the many-
electron system. Early numerical results indicated non-analytic behavior of the energy
of the ground state and were inconsistent with the results of Hartree-Fock calculations
[53]. Th e search for new types of manv-electron state began. In the following, we will
briefly outline the main ideas without attempting to be complete, as this field is the
subject of current research.
2.3 Coulomb interaction and magnetic field
The Lavghlin state
We use the above results for the quantum states of a small number of electrons, N, in
order to obtain a guess for an approximate ground state of the system described by the
Hamiltorian (2.122) in the thermodynamic limit. We use again the symmetric gauge.
We assu ne also, as before, that the magnetic field is sufficiently large, such that the
independent Landau band approximation can be used. We restrict the discussion to
the lowest one. Conveniently, the one-particle states are written as
{r\0m) = ~ у-- 1  =гтое~^г/4,	(2.139)
'	V2m+lTrm\	V }
where ths coordinates are compiled into a complex variable z = x + ту, and are again
measurer in units of the magnetic length.
A complete set of N-electron states with total angular momentum M =
are the Slater determinants
Ф(1...ЛГ)= E (-l)^n	? ... z>pLlf>A (2
Р(и...^)	^=1 v'2m«+17rm/i!	\ 4M=| J
The above three-particle states are of this form.
Since the ground state in the independent-band approximation is a linear combi-
nation of these Slater determinants, it must be of the form
Ф(1... N) = £[ f(zj - zk) exp E Ы2) 	(2.i41)
The function /(z) must be a polynomial in z. Since Ф should be a Fermion state,
/(z) = -/(-z).	(2.142)
Furthermore, Ф can be chosen to be an eigenstate of the total angular momentum.
Therefore, /(z) has to be homogeneous. The choice
/(z)=z”‘ (nt odd)	(2.143)
is the simplest form which fulfills these requirements. Then, the approximate N-
electron wave function becomes
ФП1(1...Л0= П^-^ГсхрМеЫ2") .	(2.144)
j<fc	\ 4 e=l /
This “Laughlin state” [57] describes a liquid-like system. The two-particle correla-
tion function
(;<т)(г1. z2) = I... I П dr„^m(l . .. N))2	(2.145’1
120 Quantization of Transport
does not show any pronounced structure, except that it vanishes for small distances as
ff2m)(2i,z2) a |zi - z2|2m.	(2.146)
This feature reflects essentially the Pauli principle for the electrons. As the smallest
possible non-trivial value for m is m = 3, g'2 ;(z) ex z6 represents a rather strong
suppression of the probability that “two particles meet” at small distances.
The total angular momentum of the Laughlin state is easily determined, M = Nm .
The average area covered by the electrons is correspondingly given by A = Лг2тгт£2 .
The average electron density is therefore 1/2тгт£2, and the filling factor
A2tt£2 _ 1
A m
(2.147)
In order to make the system electrically stable, one has to introduce a positive back-
ground of exactly the same charge.
It is instructive to consider m — 1 which should correspond to filling factor v = 1,
the completely filled lowest Landau band,
Ф1(1...А) = П Oj-zt)exp	j •	(2.148)
j<k	\ i /
One can prove by complete induction that this state is the Slater determinant of the
filled Landau band.
Considerable numerical work has been performed, in order to confirm that the
Laughlin state is indeed very close to the true ground state, Фш, of the correlated
electrons in the fractional quantum Hall effect [58]. The numerical comparison is done
by considering the overlap integral
(2-149)
It yields very good agreement between the states for N < 8 and rti < 5, within a few
tenths of a percent, typically Cm w 0.999.
At large distances between the particles, the Laughlin state decays as a Gaussian,
due to the strong magnetic field. It shows trivially the same behavior as the true many
electron state very probably does.
The Laughlin state does not correspond to a Wigner crystal or a charge' density
wave. From the Coulomb energy
C’l =	j#? [g!T4r) - 1] ,	(2.150)
one can calculate the energy of the ground state for the Laughlin state, since the kinetic
energy is zero in the one-band approximation [59]. The results clearly demonstrate that
the incompressible electron fluid is energetically more stable than the charge density
wave at the electron densities present in the experiments. This is due to the fact
2.3 Coulomb interaction and magnetic field
that in the incompressible state the electrons repell each other “optimally”, dm
the smallness of g2(r) for small distances. This reduces the Coulomb repulsion energy
below that of a charge density wave state. In addition, it can safely be assumed that
the trm ground-state energy is even lower than that of the Laughlin wave function
which can also be considered as an optimized variational state for the many electron
problem.
A fu/ther argument for the Laughlin state being a good approximation for the true
ground state of the correlated electron system in a strong magnetic field is that for a
contact nteraction,
Я1т <x 6(r),	(2.151)
Фт becomes the exact interacting ground state of the system when zz = 1/m [60].
When one considers only the lowest Landau level, there is an electron hole sym-
metry. Its consequence is that from the knowledge of the state which corresponds to
filling factor v one can in principle find the state that corresponds to the filling factoi
(1 — p) 161,62]. While the former is equivalent to the system with v electrons per
flux quantum, the latter should corresponds to vN holes in a completely filled Lanz'
band, i.e i/ holes per flux quantum.
Elementc ry excitations
The knowledge of the ground state of a many particle system is very important be
also only very restricted information. In order to describe low temperature transport
and dynamical properties one needs knowledge about the elementary excitations, at
the least .hose of lowest energy. Changing the energy of our correlated electron system
can be achieved by changing the mean distance between the electrons, i.e. the mean
area per electron, or, equivalently, the total angular momentum of the system, while
keeping t ie neutralizing background charge distribution fixed. Roughly, expanding a
little fron. filling factor v is equivalent to introducing quasi-holes into the state Ф^. On
the other hand, quasi-electrons are introduced by compression. While the generation of
quasi-holcs requires to introduce additional flux quanta, the creation of quasi-electrons
is achieved by introducing additional electrons.
Mathematically, a quasi-hole can be represented by
Ф+ = A+(zoK(m...~M.	(2.152)
with
Л+Ы = П(~7--о).	(2.153)
j=i
This state will correspond to an effective charge +ne. The state
Ф’Г - [,4+(г0)]тф„	(2.154)
122 Quantization of Transport
is equivalent to an (N + l)-electron state but with one-electron removed from Zq. This
will be equivalent to a quasi-hole with the charge +e. The operator A+(z0) is said “to
create a fractionally charged quasi-hole at z0”-
The ^/-electron wave function with one-electron removed from Zq can be obtained
by operating on Ф^ with the adjoint of A+(zq)
Ф“ = А*(£о)Ф„(£1 . ..zN).	(2.155)
Here, the operator
A“(2o) = П	(2-156)
is the partial derivative operating only on the polynomial part of the wave function,
leaving the Gaussian part as a constant.
The charge of the quasi-particles can be obtained by the following argument. The
average area which is covered by each of the N particles in the state with filling factor
i/ = 1/m is (N — 1 )m2irlf. It is given by the maximum exponent of a given particle
coordinate, say zlt in the Laughlin state, (N - l)m. The corresponding electron charge
density is
-Ne	—e
(N — 1)т2тг^ т2тг£^ ’
since N 1. The average charge density corresponds to that of one-electron within an
area of m flux quanta, 2лТ27п. It has to be compensated by a neutralizing background
charge density of exact!}' the same magnitude. In the above state Ф J the maximum
angular momentum, the average area covered, is increased by 2Tr(f per particle. This
corresponds to a change in the charge density of Др = e/(N — 1);;/‘г2тг/’2 which is
equivalent to a (positive) change in the total charge of
Aq = Др(Д - 1)т2лТ2 =+^.	(2.158)
The m quasi-holes described by (2.154) carry therefore the total charge +c.
By a similar argument one obtains for the average charge of a quasi-electron exci-
tation (2.155) the effective total charge
Ar/ = --.	(2.159)
m
The energies of these quasi-particle states can be calculated by
AE± = (А±(г0)Фг?|Я[ А±(з0)Ф,/) - ('MW.) 	(2-160)
Several quantitative estimates agree at least in the order of magnitude with each other,
AE± « 0.05е7с£(Л [62-66]. With e = 1.6 • IO'19 As, c0 = 9  10~12As/Vm. t 12,
A = 20nm, typical values for an inversion layer in GaAs/AlGaAs in a magnetic field of
2.3 Coulomb interaction and magnetic field 123
a few Tesla, one obtains AE » 5meV, somewhat to large to account for the rather low
critical Temperature (« IK) below which the fractional quantum Hall effect is observed.
One is therefore forced to the conclusion that there must be other, energetically lower
excitations which are relevant for the physics of the fractional quantum Hall system.
Various were proposed in the past, like bound quasi-electron quasi-hole pairs, or “quasi-
excitons’’ [67]. In these pair excitations, the quasi-particles with opposite charges will
be separated by a distance, say r. Their excitation energy will therefore be reduced 1
the “binding energy” of approximately e2/т2еейг as compared to the unbound pair. Il'
more than two quasi-particles participate in a collective state, a single-mode approach
similar in spirit to the one applied successfully to obtain the phonon-roton state in lio
Helium can be attempted [61,64]. If applied directly to the two-dimensional electron
gas in a .strong magnetic field this approach gives the so-called magneto-plasmon mode
at an energy of hwc. Within the Hilbert space of the lowest Landau band, it provid
some idea of the low-lying collective excitations of the system in the region of the
fraction;: 1 quantum Hall effect.
The search for the excitations in the highly correlated fractional Hall system is
presently a very active subject of the research in condensed matter physics. The
picture of the composite Fermions has been developed [68]. Here, the uniform “liquid”
of the electrons in a magnetic field of a strength such that there is an average flux
of ah/e per electron is replaced by a uniform “liquid” of “composite Fermions”. The
latter an essentially electrons each carrying the flux ah/e. In this system the uniform
electron density is required to produce a uniform flux density. A theory of the fractional
quantum Hall states starting from this idea was developed by J. K. Jain [69]. Another
facette which demonstrates the surprisingly large variety of possible collective behaviors
are certa n, most recently proposed topological spin excitations, “Skyrmions” [70].
These recent developments can be pictorially summarized as follows. The filling
factor is rhe ratio between the number of electrons and the number of the flux quanta
in the system. At the filling factors that correspond to the plateaus of the quantum
Hall effec: this ratio is a rational number, one-electron is attached to three flux qunuta
at v — 1 /3, for instance. By increasing the number of the flux quanta slightly, th-
number cf the electrons does not suffice to match the number of flux quanta - a
quasi-hole state is generated. If the number of the electrons is slightly increased, th-'re
are to ma ly electrons to match the number of the flux quanta — this is a quasi-electrmi
state. Th.s idea of characterizing the states by the ratio between the particle number
and the number of the flux quanta can be formalized. This picture forms eventually the
background of the above composite Fermion model [70] which is also useful to discuss
the anomalies appearing at filling factors 1/2, 3/2,...
States With filling factors q/p: the. hierachic.al method
How can we exploit the above ideas in order to obtain states that correspond to filling
factors q/’>, (q, p integer)? The basic idea is to apply the procedure described above
for the electrons to the quasi-particle states that correspond to a certain filling factor
n = 1/p.
124 Quantization of Transport
Let us consider, for instance, quasi-hole states as a starting point. First, we write
the state of Ne interacting electrons in the presence of A'h holes at the positions Q ... Qvh
as
N, Ni.
$₽,{<} (a.. ,zNe) =	...zN') И П e-P
; = 1 fc=l
- E ы'7<2
k=i
(2.161)
where C is a normalization factor, and £* is an as yet undefined parameter, a “magnetic
length” for the quasi-holes. The state Ф does not change sign when the positions of
the holes are interchanged. They are treated here as Bosons. Later, we will generalize
to more general statistical behaviors.
Each electron in this state is attached to p flux quanta, each quasi-hole carries one
flux quantum. The total density of the flux quanta must then be given by
PPe + aph = 5—7 •	(2.162)
2тгс^
The parameter a is here introduced in order to account for the possiblity of treating
quasi-electrons by an analogous ansatz, a = ±1 (+ quasi-holes, — quasi-electrons).
Quasi-holes and electrons are distributed within the same area A,
Ne	7Vh	,
Pe =-7-, Ph = ~Y-	(2.163)
/1	/1
The area of a quasi-hole state (angular momentum Nh) is, on the other hand,
A = Ne2-u^	(2.164)
which implies
1
2?r£*2
(2.165)
This defines the “effective magnetic length”, and, consequently, an “effective magnetic
field”, for the quasi-holes. The distribution of the interacting holes is, in analogy to
the Laughlin state for the electrons, assumed to be given by
Фр,(С1  •  GvJ = П (0 <7P1 exP
1712
4£(‘2
(2.166)
The exponent pi is here even, since we assume the quasi-holes to obey Boson statistics.
The area occupied by this state is
which implies
A = rVhpt27rf’2
27i7‘2ph — — .
Pi
(2.167)
(2.168)
2.3 Coulomb interaction and magnetic field 1~
Inserting (2.165) yields
Pe _
Ph Pi’
and with (2.162) we obtain the filling factor
v = ре2тгЙ =------— .
c p + ~
r pi
The total state that correponds to this filling factor is then given by
pi fe ... z/ve; G •   CnJ = ФР1(С1 • • • Gvh)$p,{<}fe • • • ^ve).
As an example, consider p = 3, pi = 2, a = 1. This yields v = 2/7. When p — 1
pi = 2, / = 2/3.
Analogously, one can arrive at many particle states by starting from states for ih“
quasi-hcles with Fermion symmetry,
(2.169)
(2.170)
(2.17!)
N„ Nh	Г ,V1;
ФР,К}(21..ZN') = С'(С}Фр(х1 ...zNe) life—G) П life—G) exp IGI2/4C2 
J<k	j = l k=l	k~l
(2.172)
The Boson ansatz leads to the states proposed by Halperin [71], whereas the Fermion
ansatz reproduces the states given by Laughlin [67]. The above considerations may also
be generalized to include ansatz states that obey “fractional statistics”,
life ~G)eA
(2.173)
with © тс [68, 72].
Furthermore, these considerations may by continued to produce a hierarchy of
many-pa:ticle states. Quasi-particles on the sth level correspond to particles in the
(s + l)tl level. Let the density of particles in level s be n, and that of the quasi-
particles ns+i. Correspondingly,
and
1
2 tiT's ’
(2.174)
1
2tt£2+1
_	Q.5+1
— H-----------ns+i
7Г
(2.175)
with
a = ±1
(2.17(0
for “holes” and “particles”, respectively. Therefore,
Р'П, T Г*а+1^5+1 ^s — l
(2.177)
126 Quantization of Transport
or
ns	1
(2.178)
For s = 0 we start with electrons, i.e. ©0 = тг, and £0 = 4- Then, as before,
у =--------------щ	(2-179)
Po + 1 + Qi —
no
(2.180)
1
P+-------n?
Pi + a2~
П1
Experimentally, not all of the predicted values of the filling factors are observed as
plateaus in the fractional quantum Hall effect. The field is open for new developments.
The above mentioned “composite Fermion” model is certainly only one, though very
appealing possibility.
The above discussion can certainly not be considered as an “explanation” of the
fractional quantum Hall effect. It provides at best evidence for the rich and unusual
but not forseen properties of the two-dimensional electron gas and in a strong magnetic
field and with interactions. Especially, more work has to be done in order to exploit
the available ideas and to construct a theory of the transport properties.
2.4 Conductance quantization
In this section we want to introduce into the fundamentals of the very basic quantization
phenomena observed in the transport properties of quantum wires without magnetic
field. Again, this has been and still is a broadly investigated subject. More than
about two thousand scientific papers on the subject appeared since the first reports in
1988 [73,74]. Thus, completeness cannot be achieved. Recent proceedings of summer
schools and conferences may serve as guides to the literature [75] in addition to provide
introductions to the various subtopics.
2.4.1 Experimental results
The quantum Hall effect is only the extreme case of the quantization of the transport
properties at low temperatures in quantum systems. That the quantum conductance
can be quantized even without a magnetic field was demonstrated in two independent
experiments in 1988 done at the Technical University of Delft [73] and at the Cavendish
Laboratory in Cambridge [74].
Basically, in the experiments, the inversion layer in a GaAs/AlGaAs heterostructure
was confined via the voltage applied on a split gate into a region of a finite width
2.4 Conductance quantization 127
Fig. 2.18: The quantization of the conductance of a quantum point contact in a two-dimensional
inversion layer in a GaAs/AlGaAs-heterostructure at a temperature of 0.6K. The geometry of ihe
split gate used to model the point contact via an applied voltage is shown in the insert. The me
electron tensity was n = 3.56 x ICL'cm"2. The mobility of the electrons was estimated from ti.-
average resistance per square as 850 000 cm2/Vs at the measuring temperature. The corresponding
mean free path of the electrons is approximately 8.5/zm. The conductance increases as a function о
the split gate voltage in steps that correspond to a conductance of 2e2/h due to the spin degeneracy
(redrawn after [73]).
connecting two relatively large areas of the two-dimensional electron gas. The opening
of the split gate was about 250nm and its length l/rm (Fig. 2.18). When the voltage af
the split gate was varied, the conductance Г through the opening was found to change
according to
2e2
Г = — j, (j integer).	(2.18 H
The phenomenon was immediately related to the quantization of the electron ener-
gies in the point contact due to the lateral confinement. The factor of two is due s
spin deg meracy of the confinement energy. The latter was demonstrated by appl'. ing
a magnetic field, thus lifting the degeneracy via Zeeman splitting.
In the following sections we want to explain how this conductance quantization,
and its 1 miting factors, can be understood.
2.4.2 Conductance quantization in ideal quantum wires
Let us fi st discuss the conductance of an “ideal quantum wire” at zero temperature.
In reality, there are no such things like “ideal” quantum wires, but they can provide a
first idea what the basic mechanism behind the conductance quantization is.
As ai ideal quantum wire, we consider non-interacting electrons that are confined
within a region of a finite width M and of a length L that can be assumed as infinite.
Periodic boundary conditions are assumed in the direction of L, say the ./-direction.
128 Quantization of Transport
The corresponding electron energy eigenvalues and eigenstates are then given, for in-
stance, by (2.25) and (2.26), respectively. The latter have to be multiplied by a plane
wave in, say the г-direction, in order to account for the free motion of the electron.
What is the conductance of such a “wire”?
In order to obtain a first estimate we use a most simple model that starts from a
similar idea as used by Drude when calculating the conductivity of a classical metal,
e2N(Ep)r
T) — ----;---
m*
(2.182)
Here, N(Ep) is the density of the electrons at the Fermi level, m‘ the effective mass,
and т the mean free time between scattering processes. It is assumed that the electrons
move freely between two scattering events under the influence of the electric probe field
E. The current linear in the electric field is
I = eN(Ep)v(E) = a0E .	(2.183)
The stationary drift velocity is obtained from the classical equation of motion of a
particle being freely accelerated by the field within the time interval t.
In order to apply this idea to our ideal quantum wire, we assume that the system
is connected on both sides to “reservoirs” (Fig. 2.19). The latter are assumed to
provide the scattering processes that are necessary to make the current stationary.
No scatterings occur within the wire of length L. The electric field is assumed as
constant within L. Thus, the electrons are ballistically accelerated within L. Since
scattering occurs only in the reservoir regions, L plays the role of the “mean free
path” of the electrons in this particular case. Then, the “mean free time” is given by
т — L/vp = L(m*/2Ep)1^2. Now the electron density is given by N(E) =	n(£,')dE,/
with the electronic density of states n(E') = (т*/2Е,)1^2/тгЛ. This gives for the product
of the density and the “mean free time” Nt = т*Ь/-гЛ, and for the conductivity
сто = 2e2L/h. With the one-dimensional conductance Г = <т0/L we obtain eventually
that Г = 2e2/h. This holds for one subband. If i/ subbands are occupied, the result is
2e2
Г =	(2.184)
Here, the Fermi energies in the different subbands are measured relative to the corre-
ponding band edges. We note that the reason for the conductance quantization is the
exact compensation of the material dependent quantities effective mass, density and
“mean free time” via the energy dependences of the density of states and the Fermi
velocity.
The same result may be obtained by applying linear response theory for the con-
ductance [76]. The quantization is a consequence of a compensation between the one-
dimensional density of states of the subbands and the Fermi velocity.
2.4.3 Adiabatic constrictions
The above qualitative explanation contains a number of assumptions that are certainly
not fulfilled for the constrictions used in the experiment. If the above “ideal quantum
2.4 Conductance quantization 1
Fig. 2.J 9: A transport gedanken experiment done with an ideal quantum wire. The wire (length I.)
is connected on both sides with reservoirs that represent the connections to the electrical circuit. N,;
scatterirgs occur in the wire. The reservoirs contain all of the scattering processes. The electric f;‘ tl
is assumed as constant within L such that the voltage U(x') drops only across the length of t!i< лис.
wire” was a good approximation for the experimental situation, the L should be "meh
larger than the Fermi wavelength Ap. Even then the question arises if the quantum
scattering processes related to the widenings at the ends of the “wire” would contribute
to the resistance. This can be studied quantitatively by considering the model of an
“adiabatic point contact” [78].
We ran expect that a sufficiently smooth variation of the width of the constriction
M(?/) should eventually lead to a reflectionless matching of the electron states with the
(practically infinitely) wide contact regions (Fig. 2.20). We assume that M(0) is the
minimum width.
In о 'der to see this more quantitatively, we consider the Schrodinger equation
— AV>(x, y) = Etp(x, y)
2,m*
with the boundary conditions
,/ M(y) x n
= ±—,У) = o.
Let us write the solution in the form ipn(x, у) = <рПу(х)£(у) with
/ x I 2	.	2;r + M(y)
=	йы~. '
where n — ±1, ±2, ±3,.... This gives the Schrodinger equation
ti2 d2ip(x, y) , . ,,	,	,
—----——A + е„(у)^(.т,у) = М(зМ/) -
(2.185)
(2.186)
(2.187)
(2.1881
130 Quantization of Transport
Fig. 2.20	: The shape of an adiabatic point
contact with a у-dependent width M(y). If
the radius of curvature R is large compared
with the Fermi wavelength Ap transport can
be expected to be quantized.
Fig. 2.21	: The shape of the effective poten-
tial in an adiabatic constriction for transver-
sal quantum numbers n < nmax (dashed
line), n = nmax (full line) and n > nniax
(dotted line). The potential is highest where
the constriction is narrowest. The maximum
value nmax for which the constriction is clas-
sically transparent is given by the condition
that Ep = en(0).
with an effective potential energy (Fig. 2.21)
4тг2Й27г2
2т*М2(у)
(2.189)
The effective potential energy (2.189) represents a scattering barrier of the height
en(0) = 47rn2?i2/2m*№(0). When the spatial variation of M(y) is smooth on the
scale of the Fermi wavelength AF = тг/кр the potential can be considered as semi-
classical. If the Fermi energy Ep < tn(0) the barrier is classically forbidden. Quantum
mechanically, the current is due to tunneling, and exponentially small in this case. Only
for Ep > en(0) quantum reflections can be neglected in first approximation. The critical
value of n, nmax, is obtained by setting approximately Ep « rfkp/T.m' = fn,„ax(0).
^max (fcFM(0)) =
fcFM(0)
2?r
(2.190)
States with n < 7imax are scattering states. They contribute to the current. Semi-
2.4 Conductance quantization 11
Fig. 2.22	: The shape of the step in the con-
ductance Г(Ер) of an adiabatic constriction
(full line). The increase of the conductanc
for E-r < sn as compared with the ideal
conductance step is due to quantum tunnel-
ing through the effective potential barriei,
the decrease above = en corresponds to
quantum reflection above the potential rr
imum.
classically,
-1(S) “p G
as one can verify by inserting into the above (2.188), and neglecting the terms that
contain derivatives of the effective potential.
When the passage through the constriction is semi-classical — this will be the case
for R M(0), where R is the curvature of the constriction at у = 0 — there will
be no mixing between the different “bands”. In this limit, the current as a function
of a vokage drop U across the region of the constriction may bo calculated at у —>
±oo, where the probe voltage is constant (±et7/2), and also the effective potential
energy "anishes. Here, the above considerations for the ideal quantum wire apply.
The conductance is then simply the sum of the conductances of the classically open
“channe s”,
2c2
Г = — nmax(fcFM(0)).	(2.192)
with kp = (2m*Ep/h)1/2. The step-like behavior of the conductance at the energies
which correspond to kpM(0) = тгп appear here as a consequence of the semi-classical
approximation (Fig. 2.22).
By incorporating tunneling and above barrier reflection near the maximum of the ef-
fective barrier one obtains corrections to the discontinuous behavior. The conductance
steps are smoothed. The shape of the step can be calculated [77,78],
2e2	1
ДГ (z) = -------------:----7==—7 ’	(2.193)
1 4- exp [-г^Я/ЛДО)]
with z = kpM(O)/tt — n, and d2M(y)/dy2 = 2/R the curvature of the constricts !
The com ition that the steps are sharp is therefore
тг2У27?/Л/(()) > 1.	(2.1!) •)
132 Quantization of Transport
Note that even when R ~ M(0) the correction is rather small, and the conductance
steps appear practically discontinuous.
Similar results can be obtained by using the exactly solvable model of a hyperbolic
constriction function [79].
We have seen above, how a sufficiently smooth constriction does not affect the
quantization of its conductance. This is due to the absence of mode mixing. For this,
“adiabaticity” of the constricting function is only necessary in the region of the con-
striction. We have also seen in the preceding section that a high magnetic field can
induce quantization of the transport related to e2//i even in the presence of disorder.
The latter certainly does not represent an “adiabatically” varying potential. The mag-
netic field must therefore in a certain sense induce “adiabaticity” across larger scales.
This is indeed the case, as can be discussed in detail for adiabatic point contacts [80].
Experimentally, the magnetic field induced adiabaticity can be studied systematically
on two point contacts in series. When the contacts are such that the edge modes can-
not propagate anymore, due to the occurance of backscattering in the regions of the
constrictions, edge modes that circulate within the island are formed. Then, Aharonov-
Bohm like oscillations with a well defined periodicity may occur, if “global adiabaticity”
is valid.
2.4.4 Disorder
The above discussion of the effect of a smoothly varying potential energy on the quan-
tization of the conductance of a quantum wire shows very clearly that quantization
is preserved as long as no mixing between the different subbands is induced. The
question then remains how the conductance is changed when the potential energy is
such that the inter-band mixing cannot be neglected. This case is of particular impor-
tance for experiments done on very long quantum wires, say of the length of 10/zm or
above [81]. Here, it is practically impossible not to encounter the potentials of at least
a few impurities, which are completely negligible for point contacts in high-mobility
AlGaAs/GaAs samples.
A Hamiltonian in which the mixing between subbands is taken into account is
H =^en(k)\nk)(nk\ + 52 {nk\v\n',k')\nk)(n',k'\.	(2.195)
nyk	n,k,n',k'
Here, en(kn) = en + h2k2/‘2m* and |nA;n) are the energies and states of the subbands
numbered by integers n, respectively. We can here assume the perturbing potential V
to be small, but of a general (random) form, allowing for inter-band mixing as well as
scattering between the states within a given band.
The simplest example one can consider is
{nk\V\n', k') = V?l.n,(fc, fc') = Ц(1 - 5njl,±1),	(2.196)
a potential which is short range in position space and with vanishing diagonal elements.
If |Ц| <<' £,l+1 — £n we can apply perturbation theory in order to calculate the change
______________________________________2.4 Conductance quantization 1
Fig. 2.23	: Anti-resonance like breakdown of
the conductance Г(£?р) of a quantum wire in
the presence of a perturbing potential (full
line) close to the threshold energy of a sub
band, n, induced by scattering between the
sub bands n - 1 and n.
of tiie electronic properties. The lowest-order result is
Sn(E) = |V1|2£
k
1
£) t n+i (^)
It holds for energies near the onset of the subband n + 1, E ~ £„+i, and describes th-,
change in the electronic properties in the subband n due to scattering into and from
the states within the subband n+ 1.
The imaginary part of En(E) is the inverse scattering time r of the scattering
between bands n and n + 1. It is
ImEn(E) = у = m [Vj [2 p,l(1 (Ef).	(2.198)
with p,. + i(E) the density of states of the (n. + l)-st subband. Since that latter diverges
as (E -- £n+i)~1//2 the scattering time becomes very short when the Fermi energy ap-
proaches the threshold of the (n + l)-st subband. This implies that the contribution
of the i-th subband to the conductance is strongly suppressed, as can be seen from
(2.182) In addition to the scattering time, the influence of the perturbation results in
a shift ,o higher energy of the subband (n + 1) relative to the subband n. It is induced
by the : cattering into and from the states within the latter and is given by the real part
of the self-energy En+i(E). This leads to the depletion of the conductance just below
and close to the edge of the subband (n+ 1) which is shown schematically in Fig. 2.23.
This effect is independent of the temperature provided the latter is sufficiently low such
that it :an be observed, since it is caused by a static potential.
Quantitatively, the influence of a random potential on the quantization of the con-
ductance of quantum wires has been the subject of numerous works which reveal clearly
not only the above discussed anti-resonant behavior but provide also insight into th;
transition from quasi-ballistic to weakly localized behavior [76]. While the former re-
gion is characterized by the condition that the mean free path £ is much longer th;
the length of the quantum wire L, within the latter £ < L. At f « £ one observes 
change m the statistical fluctuations of the conductance: while they increase prop. :
tional t i the square of the fluctuations of the potential energy in the quasi-balli
134 Quantization of Transport
region they become approximately constant in amplitude — universal — in the region
of weak localization [82].
2.4.5 Electron-phonon interaction
Not only the scattering at the static potentials of impurities can destroy the exact
quantizationof the conductance of quantum wires, also the scattering due to interac-
tion processes. A characteristic temperature-dependent scattering mechanism is pro-
vided by electron-phonon interaction. The essential feature is here the fact that the
electronic motion is restricted to essentially one dimension, while the phonons are
Z/tree-dimensional. We restrict the following discussion to acoustic phonons which can
be described by a linear dispersion law ay(q) with a sound velocity cs,
w(?) = cs|7|.	(2.199)
We will argue that energy and momentum conservation leads to a characteristic
breakdown mechanism due to the scattering of the one-dimensional electrons with the
three-dimensional acoustic phonons [83]. From the relations
h2k2 h2k'2
+	(2.200)
and
к — к' = |qj cosa,	(2.201)
where a is the angle between the wave vector of the phonons and the :z;-axis, one easily
obtains that phonon scattering becomes particularly effective if
t)F > cs,	(2.202)
with the Fermi velocity of the electron vF = hk^/m*. This is peculiar to the situation
of quantum wires, since here the Fermi velocity can be tuned externally, by adjusting
the voltage at a gate, for instance, or by changing a magnetic field. This enables us
to study the crossover from a region, v-p < cs, where the electron-phonon scattering
rate is exponentially small, proportional Texp (~2m*c2/kBT), to a region where the
scattering rate depends algebraically on the temperature, proportional T3 for instance.
This should lead to a corresponding decrease of the conductance with increasing tem-
perature.
Assuming weak electron-phonon coupling via a deformation potential, the scattering
rate can be calculated explicitly by using second-order perturbation theory. The results
of such a calculation are shown in Fig. 2.24, where tuning a magnetic field was used
to achieve the change of the Fermi velocity [83]. The magnetic field which corresponds
to the crossover velocity cs can clearly be identified. Low magnetic field strength,
В < Вя, corresponds to high Fermi velocity. Here, the inverse of the scattering length,
I = i.>FTph, is large. Such a characteristic energy and temperature dependence has
indeed been observed in an experiment done on long quantum wires [81].
2.4 Conductance quantization 1;’
Fig. 2.24	: The temperature dependence
of the conductance I'Ti-j of a quan*’ ”'
wire induced by the interaction with t
dimensional acoustic phonons with sound ve-
locity cs. With increasing temperature, the
conductance is strongly reduced above a cer-
tain critical Fermi energy which is given by
the condition (2.202). Full line indicates the
conductance at T = 0, T" > T' > 0.
2.4.6 Electron-electron interaction
At. very low temperatures, electron-phonon and electron-electron scattering can be
expected to be the only scattering mechanisms of importance in very clean quantum
wires. Due to the very weak coupling to the phonons in GaAs based systems, one
should expect that eventually the electron-electron Coulomb interaction will dominate
the transport in AlGaAs/GaAs quantum wires. Unfortunately, the presently available
experimental indications are far from convincing [84]. The theoretical treatment of
the inte -action in one dimension is nevertheless extremely instructive. It provides
insight into an interacting system which is considered as a paradigm of non-Fermi
liquid behavior, and which can be treated mathematically exactly. Here1, we considei
t he most elementary spinless case.
The Lutiinger liquid
In two ar d three dimensions, interacting electrons are well described by the Fermi liquid
theory [85]. The electrons are here described as quasi-particles, with the effects of tic
interaction incorporated in their properties as, for instance, an effective mass and
life time. The latter diverges at the Fermi energy. Thus, the concept of quasi-particles
is justified a posteriori. In one dimension, the interaction leads to peculiar properties
that can.tot be described by using the quasi-particle concept. This is related to the
fact, that the Fermi surface, of the free electrons consists only of two points ±fcg. As a
conseque ice, the pair excitation spectrum of the free, non-interacting electrons shows
a gap foi finite wave numbers, in contrast to higher dimensions (Fig. 2.25) [86-88].
The energetically lowest excitations turn out to be collective - so-called charge sound
wave — modes.
The basic assumption of the model is that the spectrum of the non-interacting
electrons may be considered to be a linear function of the wave vector around the
Fermi wavevector ±fcr,
e(k) « hvF(±k - kF).	(2.203)
This may obviously be justified for physical properties for which one has to take into
136 Quantization of Transport
E(q)
E(k)
Fig. 2.25	: The pair excitation spectrum of
non-interacting electrons in one dimension
shows a gap at low energy for finite wave
numbers q (shaded regions), in contrast to
dimensions higher than one.
Fig. 2.26	: Linearization of the energy spec-
trum Elk) of a one-dimensional electron sys-
tem is justified for physical properties which
are determined only by the excitations close
to the Fermi energy Ey.
account only the excitations close to the Fermi energy (Fig. 2.26). The second assump-
tion concerns the scattering processes: in the simplest version of the Luttinger model
only forward scattering is taken into account. Generally, this is not allowed. How-
ever, when being interested only in excitations with a small wave vector q backward
scattering processes that involve wave numbers of the order of 2A:e are not important.
The third assumption, namely extending the dispersion relation to negative energies
and thus creating two branches which correspond to left and right moving particles, is
only a technical trick. It essentially helps to simplify mathematics and again can be
justified for small excitation energies.
The Hamiltonian of the free electrons is
Ho = hvv Y (ak ~ M “ (gUgJo] .	(2.204)
k,a—±
where c^ak, cak create and annihilate, respectively, Fermions with the wave number к
in the left (a = - ) and the right (a = +) branches of the spectrum, {cttk,Yk,} =
 The subtraction of the second term on the right hand side of (2.204) is
2.4 Conductance quantization 137
necessary, in order to compensate for the divergence induced by the inclusion of negative
energies.
The above Hamiltonian can be rewritten in terms of the operators of Bosonic pair
excitations. The starting point is the observation that the Fourier components of the
operators of the densities that correspond to the left and right moving particles
PM = E (с1к+чсак - 54,o{clkcak)o)	(2.205)
nk
obey the exact commutation relations
[/Л1 (?), Pa' (,Q )] — ^a,a'^q,q' „ 	(2.2
Z7T
This can be proven by using the above anti-commutation relations of the Fermion
operate'S and the unboundedness of the spectrum. It is then a straightforward exon
to deme nstrate the validity of the commutation relation
[Ho, pa(q)] = ahvFqpn(q)	(2.207)
by using- (2.206) and the linearity of the spectrum. This shows that the particle-hole
pairs created by pa(q) correspond to eigenstates of the Hamiltonian HQ with energies
avpq. This yields the Bosonic representation
H0 = 5^	£ paWpa(.-q) + £ <
q^0,ct=±	a=±
(2.208)
with Na = pa(q — 0) representing the number of particles added to the ground state
in the branch a. The Hamiltonian of the free electrons can apparently be written
in a Fermionic as well as in a Bosonic form. The two representations are completiiq
equivalent, since they have identical spectra with the same degeneracy of the energy
levels [88,89].
The mteresting point is that the Bosonization technique can be employed in orT i
to diagonalize the Hamiltonian including also the interaction term
= TF E \.Pa(q)pa(-q) + pn(q)p-n{-q)] 	('-
q,a=±
Here, only forward scattering is included. The Fourier transform of the interacti
potentia., V(q), is assumed to be equal for the two possible processes, namely scatters, o
within t re same branch (a —> a) and between different branches (a —> —a) of the
spectrum. Generalizations to different scattering amplitudes can be found in [90-92].
The diagonalization of H = Ho + H-, can be achieved most easily by a standard
Bogolubov transformation,
Pa(q) = Pa(q) cosh (<g(?)) + p--n(q) sinh M<j)).
(2.210)
138 Quantization of Transport
V(<?)
irhvp
K(q) = e2^9' =
and by choosing
-1/2
(2.211)
One obtains
Я = у 52 v(q)Pa(<l)Pa(-Q) + TV [vnN2 + VjJ2] ,	(2.212)
L q^O
the (/-dependent velocity being given by v(q) = hvF/K(q). The operators N = N++N~,
.J — N+ — N_ represent particle and current excitations, respectively, with velocities
t'v = v(0)2/i;p and vj = tip. If the amplitude for the inter-branch scattering was
assumed to be different from the intra-branch scattering, also the current velocity vj
would be renormalized. By introducing normalized Bose operators
b\ =	+ e^P~^	(2-213)
V b\q\
(G(.r) Heaviside function) one obtains the standard form of the Hamiltonian
H = £ h^q)b\bq + ~ \vnN2 + VjJ2]	(2.214)
with the dispersion relation of the pair excitations
W(7)^i;(<?)|<7|.	(2.215)
The latter is shown in Fig. 2.27 for an interaction potential of the form
V(x) - Vo	(2.216)
Here, the inverse of 7 represents the “screening length”, the range of the interaction.
For small wave numbers, the spectrum is renormalized by the interaction, the slope
of the linear increase with the wave number being ?t(0) = vv/K(0). This is the ve-
locity of the charge-sound wave excitations of the model. For large q the slope of the
dispersion shows a crossover u(q) vp since K(q) —> 1. The parameter К = A'(0)
characterizes the interaction. When the interaction is repulsive, V(q = 0) > 0, К < 1.
For attractive interaction, A’ > 1. The non-interacting case corresponds to A' = 1. In
the limit of low excitation energy (q —> 0), A' governs the dynamics. The power-law
decay of most of the correlation functions is determined by K. When A' > 1 precursors
of superconducting behavior are obeserved in some of the correlations.
In principle, the dispersion relation of the excitations can be experimentally de-
termined by using Raman scattering. The characteristic enhancement of the velocity
at small wave numbers predicted by the model with a Coulomb interaction is indeed
consistent with experimental data obtained for a quantum wire with only a single sub-
band occupied [93]. More recent data obtained from quantum wires with more than
2.4 Conductance quantization 139
Fig. 2.27: The dispersion relation u(q') of the pair excitations in the Luttinger model with a repul
interaction potential of a finite range 7 1 for different strengths of the interaction, A'1'.
one subband occupied show also the low-energy charge wave mode which is characteris-
tic for such a system [94]. Although the quantitative interpretation of the experiments
is more complicated in the latter case, due to Raman lines that correspond to inter-
band transitions, these data seem to indicate a rather strong interaction between the
electrons in etched semiconductor quantum wires, in contrast to the transport data.
Conductince of a Luttinger liquid
We are now able to calculate the linear transport properties of the Luttinger system
[95]. As we will see, the de conductance of a quantum wire is completely governed by
the interaction constant. Instead of the quantization of the conductance in units <'
e2/h we will find in the presence of interaction
e2
ДГ = K-.
h
(
z.
First, we identify the operator of the current. From the continuity equation
obtain by integration
/•* , dp(x,t) 1 <9<z:,t)
I dz—----------- = e ' .—----------
-L/2 dt	y/hn dt
(2.218
The current at the site x is given by the change in the total charge between —£/2
(—> — 00 and x within the unit of time. The charge density is here given by the
Fourier transform of (2.205)
b n=± q
(2.219)
140 Quantization of Transport
where a cutoff parameter A has been introduced in order to guarantee convergence.
The time-dependent operators are obtained by using the Heisenberg picture
p(x, t) = e'Ht/hp(x)e
(2.220)
Next, the operator of the external perturbation has to be defined. We will consider
here the case of a quantum wire which is subject to a time- and space-dependent electric
field E’(.r,t) = — dxU(x,t) with the potential energy U(x,t). The coupling term is then
Hexl =e У dxp(x)U(x, t).	(2.221)
Finally, standard linear response theory yields the relation between the average of
the current and the electric field in terms of the complex, nonlocal conductivity a(x, t)
J(x, t) = / da/ / dt'a(x - x’,t — t')E(x', t'),
(2.222)
with
J(x, t) = (j(x, t)} = tr [W(a;, t)j(x, t)],	(2.223)
where W(x,t) is the statistical operator.
The conductivity is given by the “Kubo formula” [96,97] in terms of the equilibrium
retarded current-current correlation function
a(x, w)
i
’ 2
e^ + RM ,
(2.224)
with
i	\ ie2 i •	•	\
R(x, t) = -i©(t)(Ых, t), ДО, 0)]) =	0(f) Ж 0(0, 0)]) .	(2.225)
x	' lift x	'
A rather lengthy but straightforward calculation [95] yields eventually the result
d2
R(x, t) = ~— vF6(x)6(t) -—F(x,t)
П7Г	dt2
(2.226)
with
F(x, t) = -20(t)|y 52 “777 sin
q^O W W
such that from (2.224)
(2.227)
ie2
d((r,cj) = — uF(x, cj).
rift
(2.228)
2.4 Conductance quantization 141
Fig. 2.28: The “intrinsic” ac-conductance,
vF?'(w), of a Luttinger system with an inter-
action of finite range for different strength of
the interaction (as in Fig. 2.27).
The ’requency-dependent conductance is obtained from the absorbed power [98]
pm,
ГМ = Л/2”	<2-22'
where U	dxE(x, co) is the applied voltage with frequency w. The absorber!
power is completely determined by the real part of the conductivity
Re<7(z, w) =	[ dgcos(g.r)[<5(w(g) + w) +	- w)].	(2.230.
2ttzq Jo	'
One finds
ГМ = yUp^Z^M).	(2.231)
h dw
Here we have defined
dxe^E(:r)|2
C/2
a function which depends solely on the shape of the electric field. In a sense, it repre-
sents the “apparatus” of the experiment. For a J-fnnction like electric field we find
(2.232)
t he “intrinsic quantum conductance” of the Luttinger system which is solely determined
by the sf ectrum of the elementary excitations. As seen in Fig. 2.28, the presence of
the crossover frequency in the excitation spectrum leads to a characteristic resona:1'
in the ac-conductance.
In the dc-limit, w —> 0, we find, independent of the shape of the electric field.
the above renormalization of the conductance quantum by the interaction constant.
(2.217).
The above result is interesting from two other points of view [99]. First, for the
ideal Luttinger system which we have discussed so far, the linear response is exact.
142 Quantization of Transport
There are no nonlinear contributions to the current for finite voltages. Second, one can
rigorously prove that the independence of the dc-transport of the shape of the electric
field is also valid in the nonlinear regime in the presence of an additional (spatially
well-localized) potential.
Why is the interaction induced conductance renormalization not observed?
We will see in chapter 3 that in principle the de transport properties of quantum
structures reveal quite strongly the interaction between the electrons. The Coulomb
blockade is the most striking example. We have discussed above that Raman spectra
of quantum wires show clearly that the interaction of the electrons in (long) quantum
wires is quite strong. Nevertheless, the present attempts to observe the interaction-
induced renormalization of the conductance quantum by measuring the dc-conductance
essentially failed to give the predicted result, or at least were not convincing [84]. The
question is why?
There are several possibilities to explain this “almost-null” result. First, due to the
fact that in the dc-transport experiments usually gated structures are used, additional
screening effects may decrease the interaction so strongly, that its effective strength
is practically zero. Second, in the de experiments, relatively short quantum wires are
connected to relatively large contact areas which may also lead to additional screening.
Quantitative estimates of these effects can be done. The results are again not very
convincing in demonstrating the required strong screening.
Another possibility is that the spatial inhomogeneity of the system “quantum wire
and contact areas” could deplete the renormalization of the conductance. One can
assume that the interaction is less important in the regions of the contacts which are
very wide hundreds of micrometers — while in the region of the narrow quantum
wire — diameter only a fraction of a micrometer — the interaction is more important.
If the conductance was dominated by the contact regions, the renormalization of the
conductance would be absent. The issue is presently not settled and a subject of most
careful experimental and theoretical research.
Another remark is in order at this point. The above described integer quantum Hall
effect has been explained in terms of edge states, quasi-one dimensional states induced
by the strong magnetic field, but with chiral properties: the states with positive and
negative wave vectors are spatially separated. Also in the region of the fractional
quantum Hall effect a mapping of the electronic system on a quasi-one dimensional
but interacting system has been achieved — the Luttinger model [70,100]. The role of
the interaction constant К is played by the filling factor v in this case. As quantum
Hall systems are technically very well controlled, this analogy could provide unique
possibilities to detect in a direct experiment some of the predictions of the theory of
the Luttinger model concerning the transport properties in the presence of potential
barriers: at zero temperature even an infinitesimally small barrier would suppress
the linear transport completely [101], due to the pinning of the charge density wave
which forms the ground state of the system. As a consequence, there are peculiar
temperature and frequency dependencies of the conductance of a potential barrier in a
___________________________________________________________2,5 Conclusion 113
one-dimensional interacting electron system which could be experimentally observable
Again, this is not a settled issue and the subject of most exciting and intense prese '
days research [102].
It seems that due to modern device fabrication techniques, we will sooner or lai er
be able to study the effects of interactions in quantum systems in a controlled wav,
although the experimental difficulties are considerable and should not be underesti-
mated.
2.5 Conclusion
The quantum Hall effect was the first of the transport quantization phenomena whiHi
were discovered during the past two decades. Nevertheless, it is undoubtedly Пи’
one which is at best qualitatively understood, although the various theoretical attacks
have eventually led to considerable increase of our understanding of disordered cub
interacting electron systems. It is by now very well established experimentally as -
perfect, very stable and very precise resistance standard. It is this precision which a-
also the challenge for the theory: any explanation must be valid at the level of а rdy r 
uncertainty of 10“9! It is hard to see, how any of the available qualitative pictin- s
described above can fulfill this requirement. Indeed, a great challenge for forthcoiT'
generations of researchers who want to specialize in theoretical (solid state) phys
The situation is considerably better with respect to the transport quantization
induced by geometrical confinement. Here, many features can be treated by elemental"
methoc s — including interaction effects. The field is an ideal playground for those who
intend .o apply quantum mechanics to systems which can be “tailored’’ by experimental
methoc s, and thus constitute the ideal tools for the the application of our fundamental
scientife method: the interplay between experiment and theory that, has been proven
to be ectremely fruitful in the past.
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Gerd Schon
Single-Electron Tunneling
3.1	Introduction
In the It st few years substantial progress has been achieved in micro-fabrication tech-
nology. t has become possible to fabricate in a controlled way metallic tunnel junctions
with capacitances in the range of G = 10~15F. In this case the charging energy asso-
ciated with a single-electron charge, Ec = e*/2С, is of the order of IC 'eV, which
corresponds to a temperature scale Ес/кв « IK. This implies that electron transport
in the sub-Kelvin regime is strongly affected by charging effects. Similar properties
have been observed in semiconductor nanostructures, for instance in quantum dots
in 2-dimmsional electron gases. The Coulomb energy in these systems can be char-
acterized by a capacitance which depends on the size of the dot and also may lie in
the range of 10 l5I? or less. Charging effects play a role in granular materials an ,
ultimately even in molecular systems. Here the capacitance may be as low as 10~18F.
making single-electron tunneling observable even at room temperatures. This opens
spectacular perspectives for future applications.
In this Chapter we will describe single-electron tunneling in the presence of charg-
ing effects. For definiteness we will consider metallic systems with a large density <
quantum states, although the concepts described here are equally important for semi-
conducto - or molecular systems. We will first study in Section 3.2 how the charging
energy depends on the number of electrons and on transport and gate voltages applied
to various parts of the system. The simplest model systems which demonstrate these
features are the so-called ‘‘single-electron box” and the “single-electron transistor”.
We will then derive in Section 3.3 within perturbation theory the single-electron
tunneling rates. In low capacitance systems it is crucial to account for the change in the
charging energy associated with the tunneling process. A master-equation description
accounts for the large-scale features of the current-voltage characteristic of the single-
electron transistor. In the Coulomb-blockade regime, where single-electron tunneling is
suppressed, higher-order processes such as coherent “cotunneling” of electrons through
several junctions become observable.
The mesoscopic junction systems studied here are small such that charging effects
and higher-order quantum processes play a role. On the other hand, they are large
150 Single-Electron Tunneling
enough such that macroscopic current and voltage probes and sources can be coupled
to the system. This makes the mesoscopic system susceptible to the influence of the
electric circuit. It is therefore necessary to include in a complete description the ex-
ternal circuit and to investigate the influence of the electrodynamic environment on
single-electron tunneling (Section 3.4).
We begin by studying tunnel junctions with two normal metal leads (NN). If part of
the system is superconducting further highly interesting effects are found (Section 3.5).
At subgap voltages the single-electron tunneling is suppressed. This makes higher-order
processes such as Andreev reflection in normal-superconductor (NS) junctions the lead-
ing transport process. Furthermore, the energy of excitations created in one tunneling
process can be regained if in a later tunneling process two excitations recombine into
a Cooper pair. This leads to “parity effects”, which distinguish between states with
even or odd electron number in the superconducting electron box. They also lead to
interesting structure in the I-V characteristics of superconducting (NSN or SSS) SET
transistors. The tunneling of Cooper pairs is influenced by charging effects in a similar
way as that of single electrons. However, Cooper pairs can also tunnel coherently.
The charge and the phase difference in a Josephson junction are quantum mechanical
conjugate variables, and the junction is described by a macroscopic Hamiltonian. The
eigenstates in general are superpositions of different charge states. Their properties
can be probed by tunneling of normal electrons. We discuss a model where in an NSS
transistor the Andreev reflection in the NS junction is used as a spectroscopic tool to
detect the coherent Cooper-pair tunneling in the SS junction.
Many of the single-electron effects can be described within simple perturbation
theory. A necessary requirement is that the resistance of the tunnel barriers is high
compared with the quantum resistance /?к = h/e2 = 25.81281... kQ. In order to de-
scribe junctions with lower tunnel resistance a more general formulation is required.
A systematic description of tunneling in systems with strong charging effects is pro-
vided by a path-integral approach. It is a generalization of the formulation developed
for dissipative quantum mechanics and reviewed Chapter 4 of this volume. We first
present (Section 3.6) the imaginary-time path-integral method, which is appropriate
for the description of equilibrium properties, e.g. the Josephson current through SNS
transistors or the influence of charging on the proximity effect. On the other hand, we
will also present and analyze in a real-time approach the time evolution of the density
matrix (Section 3.7). In this approach we can describe systematically higher-order cor-
related tunneling processes, including “inelastic resonant tunneling”. These processes
give rise to a renormalization of system parameters and to life-time broadening effects.
Single-electron effects have been studied now for more than a decade, and a large
number of papers have been devoted to this subject. Where available we ([note review
articles and collections of papers, where further references can be found. The first
article to be mentioned is the one by Averin and Likharev [1], who developed the
perturbation theory of single-electron tunneling and described several applications for
current-biased junctions. Later it became clear that in most experiments voltage-
biased junctions or systems of junctions were used. Initial scepticism against the new
theoretical concepts was quickly overcome when experiments were successful. After an
3.2 Charging energy and single-electron devices 151
(a)
(b)
Fig. 3.1: (a) An overlap junction with an oxide layer, (b) schematic diagram for a tunnel junction,
early experiment by Fulton and Dolan, the important breakthroughs were achieved in
Delft by Mooij and Geerligs and in Saclay by Devoret, Esteve and further members of
these gro ips. Their early work is well summarized in the book Single Charge Tunneling
[2], which contains review articles describing (i) the theory of single-electron tunneling
under the influence of an electrodynamic environment by Ingold and Nazarov, (ii)
some higher-order tunneling processes by Averin and Nazarov, (iii) Coulomb-Blockade
effects in semiconducting nanostructures by van Houten, Beenakker and Staring, and
(iv) properties of junction arrays by Mooij and the author. A collection of research
articles wrs published simultaneously as a devoted issue of Zeitschrift fur Physik [3].
Although much of the theoretical work on single-electron tunneling included also
the effects of superconductivity, it took longer until the experimental situation became
clear. Parity effects related to the presence of a single excitation in a superconducting
island are a particular interesting example. In this context the experimental work C
the groups in Saclay and Harvard should be mentioned; the theory was advanced by
Averin and Nazarov, Glazman and Hekking, and others. The concepts have entered
modern textbooks like that of M. Tinkham [4]. A collection of articles representing
the state of 1994 is contained in the proceedings of the conference Mesoscopic Super-
conductivity [5]. The path-integral formulation of tunneling in systems with strong
charging efects was developed in a collaboration of Ambegaokar, Eckern and the au-
thor [6] and is summarized in the review article with Zaikin [7]. It has been applied
to describe several effects in mesoscopic superconductors in Refs. [8,9]. A systematic
description of tunneling beyond perturbation theory, incl. cotunneling and inelastic
resonant tunneling is presented in Ref. [10].
3.2	Charging energy and single-electron devices
In this Seclion we introduce the concept of a capacitive charging energy and describe
some of the circuits which show single-electron effects. We concentrate on metallic
systems with a large electron density of states. The. number of electrons in an “island”.
i.e. in a pa't of the system which is electrically isolated from the rest of the circuit
is integer, it may change in discrete units by tunneling. On the other hand, we hare
control variables, such as applied gate voltages, which change the polarization cha:.
on the captcitors in a continuous way. Many of the measurable consequences of th<’
single-election effects depend on this interplay of discrete and continuous changes of
the charge. For a further review and extensions of the material covered here the article*
by D. Est.ev? in Ref. [2] is recommended.
152 Single-Electron Tunneling
Fig. 3.2: The single-electron box.
3.2.1 The energy scale
Modern lithographic techniques allow the fabrication of narrow metallic lines with
width down to several lOnm, as well as tunnel junctions in overlap regions of such
lines, as illustrated in Fig. 3.1. Junctions with an area S=(100nm)2 can be produced
reliably. The oxide layer is roughly d = lOA thick, and the dielectric constant of
the oxide is e » 10. Using the classical expression for the capacitance we arrive
at C = eS/(4nrd) « 10-15F. While it was not clear, a priori, whether the classical
expression for the capacitance can be applied at such small length scales, it has been
confirmed by the experimental observation of charging effects. The Coulomb gaps (see
below) in the I-V characteristics are consistent with the simple estimate within a few
percent.
The capacitance introduces an energy scale, the charging energy corresponding to
a single-electron charge -e,
e2
E^2C~	(3.1)
which characterizes all charging effects. It is of the order of £?<; » 10~4cV if the
capacitance is C ~ 10~15F, which corresponds to a temperature Ec/ka « IK. This
implies that in the sub-Kelvin regime the electronic states and transport properties are
significantly affected by charging effects.
3.2.2 Single-electron box
We analyze now the charging energy of simple systems of tunnel junctions. It depends
on the electron number in various parts of the system and the applied voltages. The
first example is the single-electron box, shown schematically in Fig. 3.2. It. consists
of a small metallic island, coupled via a tunnel junction with capacitance C.j to an
electrode and via a capacitor C'c, to a gate voltage source I <;. For F'(; = 0 the lowest
energy state of the system is charge neutral. In this reference state the electrons on
the island compensate the charge of the ions; there are n = 0 excess electrons on the
island. If the gate voltage is turned on the number of excess electrons on the island
can change due to tunneling across the junction in discrete steps to n = ±1. ±2.
While the total number of electrons on an island is integer, the charge is spatially
distributed and in general shifted relative to the positive background. If a voltage is
3.2 Charging energy and single-electron devices 153
Fig. 3.3: The charging energy of a single-electron box as a function of the gate voltage for different
numbers n of electron charges on the island.
applied the surface charges on the capacitor plates, which are of equal magnitude but
opposite sign on the two sides of each junction, are in general non-quantized. They
are determined by the integer n and the non-quantized applied voltage. We obtain
the charging energy from the following elementary arguments: the total excess charge
of the box splits into two parts on the left and right capacitor plate -ne — Qi + Q,.
The corresponding voltage drops add to the applied voltage V(; = Q\/C.i — Qr/C\-,.
and the charging energy is Q'\/2Cj + <2г/2Сц. The relevant free energy is a Legendn-
transform of this energy, which also contains the work done by the voltage source
—I'bQr- Elimination of Q\ and Qr in favor of n and VG yields, up to a contribution
which does not depend on the variable n, the result
^chv^Qc;) — Q/r-j 	(3-2)
Here С == C,] + Cq is the total capacitance of the island. The effect of the voltage
source is contained in the ‘‘gate charge” defined as Qc, = CgIg.
The c rarging energy Ec\, is plotted in Fig. 3.3 as a function of the gate charge for
different, lumbers of excess electrons n. With increasing gate1 voltage, the electron
number o” the lowest energy state increases. It does so in discrete steps from n to n + 1
at the degeneracy points Qa/e = и+ 1/2. Under the same conditions the voltage of the
island Vjsi UK| — dEaJdQc, displays a sawtooth-like dependence on the applied voltage.
At finite temperatures the steps and sawtooth dependence' are washed-out. as fol-
lows from the classical statistical average
ZCh „ = .-00
where Zch is an obvious normalization. The result is displayed in Fig. 3.4 for differed
temperati res. The stepwise increase has been observed experimentally, e.g. by th-.
154 Single-Electron Tunneling
Fig. 3.4 : The average number of electron charges (n) on the island of a single-electron box as a
function of the gate charge (voltage) for different temperatures TjEG = 0 (dashed steps), 0.02, 0.05,
0.1, 0.2, 0.4, and 1 (nearly linear).
Saclay group (see results in Ref. [2]). Their measurement procedure will be discussed
below. The experiments generally agree well with theoretical expectations, if one man-
ages to control heating and the noise from the measurement setup, which usually is at
a temperature higher than that of the cryostat.
3.2.3 Single-electron transistor
Another fundamental example is provided by the single-electron transistor shown in
Fig. 3.5. Hen; an island is coupled via two tunnel junctions to a transport voltage
source V — V'l — Vr such that a current can flow. The island is, furthermore, coupled
capacitively to a gate voltage VG. The charging energy of the system depends again on
the (integer) number of electrons n on the island and the (continuous) voltages. Some
algebra along the lines outlined for the electron box produces again ECh(n,Qc) =
(ne — Qg)'2/(2C). For the transistor C = Cl + Cr + Cg is the total capacitance of the
island, i.e. the sum of the two junction capacitances and the gate capacitance, and all
three voltage sources define the gate charge Qq — c. + ClH. + CrUr.
In a tunneling process, increasing the island charge from to n + 1, the charging
energy changes by
/	1	() , \ l 2
Ecll(?t + 1, Qg) - ECj}(n, Qe,} 1 n + --------J — .	(3.4)
\	2 e / C
These energy differences are equally spaced and can be tuned by the gate voltage.
The situation is illustrated in the energy scheme shown in Fig. 3.6. The differences in
charging energy are plotted in the center. We further display the Fermi levels of the
two leads which are shifted by the applied potentials Vl/r-
For definiteness, we assume that the energy of the electrons in the left, lead is higher
than that in the right lead. Then, at low temperature, tunneling from the left lead to
______________________________3.2 Charging energy and single-electron devices 15.'
Fig. 3.5: The SET transistor.
the isltiid (transition from n to n + 1) is possible if the energy in the left lead is
high erough to compensate for the increase in charging energy of the island
eVn > 7?ch(n + 1,Qg) ~ Е<л\(п, Qq) .	(3-5)
Similarly tunneling from the island (transition from n + 1 to n) to the right lead is
possible at low temperature only if
Ech(n + 1, <?g) Ech{n, Qg) > el-R .	(3.6)
Both conditions have to be satisfied simultaneously in order for a current to flow
throng! the transistor. It is obvious from the figure that at low transport voltages,
depending on the gate voltage VG we may be either in a Coulomb blockade regime or
have a f nite current. By varying the gate voltage we produce the Coulomb oscillations,
i.e. the s-periodic dependence of the conductance on Qq.
Further devices can be constructed (see e.g. Ref. [2]). We mention here the elect: ::
trap, which is similar to the electron box except that it contains at least two junction1-
in series. In contrast to the electron box the trap has metastable charge states. "
traps are combined to build the electron turnstile, which can serve as a current source.
A suitable ac-gate voltage with frequency f allows the controlled transfer of a sin.
electron per cycle. Hence the current is I — ef. Finally we mention single-electr.)'
pumps, where a current is driven by two phase-shifted ac-voltages applied to difb .
ent islands. In this case a current I = ef is transported even at vanishing transper'
voltages. Both devices, in principle, can serve as a current standards, if one manages
to minimize the effect of missed cycles, of thermal fluctuations, and of quantum fluc-
tuations. This requires low frequencies, low temperatures, but also a design (many
junctions) which minimizes higher order quantum tunneling processes.
Man/ properties of the. SET transistor and its extensions can be understood by
consider ng only the energy of the different charge configurations. However, a detailed
understanding of the I-V characteristic requires knowledge of the tunneling rates of
the electrons, which will be the next topic.
156 Single-Electron Tunneling
- E^d.Qo)
Ech(l’QcP ‘ Есь(0>Р(})
E^Qq) - Ech(-1,QG)
EehC-l.QG) -Ech(-2,QG)
eVr
и
Fig. 3.6: The energy differences corresponding to the addition or removal of an electron charge are
shown. They can be shifted by the gate voltage Vq- The Fermi energies of the leads are shifted
relative to each other by the transport voltage V = V"l - I ’n-
3.3 Tunneling rates and I-V characteristics
In this Section, we introduce the Hamiltonian of the SET transistor. Using simple
golden-rule arguments, we derive the rate for the transfer of a single electron charge
across the tunnel barriers. It depends crucially on the change in the charging energy.
The transition rates enter a master equation, from which wc obtain the current-voltage
characteristic. If the tunneling would increase the charging energy it is suppressed at
low temperature, a phenomenon called “Coulomb blockade”. This “orthodox theory”
was developed by Averin and Likharev [1]. In the regime of the Coulomb blockade
higher-order processes gain importance. We describe here “cotunneling” processes (see
e.g. Averin and Nazarov in [2]).
3.3.1 The single-electron tunneling rate
For definiteness, we consider a SET transistor, shown in Fig. 3.5, which consists of
a metallic island coupled via tunneling barriers to two leads and capacitively to an
external gate voltage. Its Hamiltonian is
Н = НЬ + НХ + НК + НЛ + Н,.	(3.7)
Here, Afi = 12/, a fkr'k ack a describes the noninteracting electrons with wave vector к
in the left lead, with similar expressions for the island (with states denoted by (/) and
t.he right lead. We allow that the leads have different electrochemical potentials. The
Coulomb interaction !1,\, is assumed to depend only on the total charge on the island,
as described in the previous Section,
3.3 Tunneling rates and I-V characteristics 1"
The number operator of excess electrons on the island is given by n = c| acq g — Л' F.
where the number of positively charged ions of the island has been subtracted. Ch n! ,-
transfer processes are described by the standard tunneling Hamiltonian, for inst.rnc-<-
tunneling in the left junction between the states к and q by
= 52	+ h.c. .	(3.9)
/с,<7,<7
We determine the transition rates by golden-rule arguments. The rate of tunneling
of an electron (one of many) from one of the states к in the left lead into one of the
availab'e states q in the island, changing the electron number from n to n + 1, is
/-ОО	roc
l\i(«) ='7^— / de*, /	d6,/L(6*;)[l - /i(fQ)]<5(JErh + tq - ck) .	(3.10)
£ -Щ.Ь J-oo	-oo
The crucial point is that the energy, which is conserved as expressed by the J-function,
contains the energies of the electron states ek/q, but also the charging energy. The
latter depends on the electron number and the applied voltages !-(; and V'l/r- In the
process considered it changes by
6Ech = Ech(n+ 1,Qg) - Ech(n,Qc,) - eI'L .	(3.11)
We further introduced the tunnel conductance of the junction
1	4tfc^
	(3.12
rlt.L fi-
ll depends on the tunnel matrix elements Tkq, which here can be considered as con
slants, as well as the densities of states at the Fermi level, /VI/L(0), and the volumes.
of the island and lead. Equivalent expressions apply for the reverse pro-
Fn,(n + 1), changing the island charge from n + 1 to n, and the other tunnel barrier.
In equilibrium the distribution functions are Fermi functions, and the integrals
over the electron states in Eq. (3.10) can be performed explicitly. The resulting “singh -
electron tunneling” (SET) rate is [1]
,	1 Жъ
Гы(п-) =	-------(пГТГт)-------Г '	(3.13)
e27?t,L exp[<5Ech/Z'B7 J - 1
At low temperatures, k[tT < |<5EC11|, if the charging energy wotdd increase in a tunnel-
ing process, the tunneling is suppressed, Г —> 0. This phenomenon is called “Coulomb
blockade1’ of electron tunneling. If the charging energy is decreased the rate is
Гц(«) =	*	for <5Erh <0 ,T -> 0.	(3.14)
e E.t L
At finite temperatures all processes are allowed. The forward and backward rates be-
tween twr states satisfy the detailed balance, condition, Гп|(п)/Г1Е(п+ 1) = e~SEclgkai
158 Single-Electron Tunneling
A familiar limit of what is described above is a single voltage-biased tunnel junction
where 5Ecn is replaced by —eV. In this case (3.13) yields a linear current- voltage
relation, Zt = e[T(V) — Г(-V)] = V/Rt. We can also reverse the argument. The two
requirements - (i) a linear characteristic in the voltage-biased case and (ii) detailed
balance -- uniquely determine the expression for the rate to be of the form (3.13) x.
3.3.2 Master equation for sequential tunneling
Given the electron tunneling rates we can set up a master equation for the probability
p(n,i) to find the island in a state with n electrons. The probability changes by
tunneling in the left and right junctions. Hence
= - [Tli(ti) + ГпДп) + TRI(n) + rIR(n)]p(n,t)
+ [rLi(zJ - 1) + Га1(п - 1)] p(n - 1, t)
+ [rIL(n + 1) + r[R(n+ l)]p(n + 1, t) .	(3.15)
The rates and probabilities also determine the current. In the left junction it is
4(<) = -e £2[Гы(п) - Г1ь(п)]р(п, t) .	(3.16)
n
In most cases we apply dc-voltages and are interested in the dc-current. In this case
we need only the stationary solution of the master equation, and the currents in the
left and right junctions are equal I = ZL = ZR.
As an example we consider a transistor with symmetric bias 14 = ~ hR = V/2.
At low temperatures and low transport voltages (except at symmetry points) only two
different charge states - and those transitions which connect both - have an appreciable
probability. For instance, if ne < QG < (n + l)e we need to consider only p(n) and
p(n + 1) and the two transitions Гы(п) and Гш(п) increasing the island charge from
n to n + 1 electrons, as well as the two reverse transitions ГцДп + 1) and Гп<(п + 1).
The energy changes determining the rates TLi(n) and Fjg(n + 1) are
xzc > L , 1 Qg,c2 eV]	,	.
d£ch = ± (n + ---------->	(3J')
2 e C 2
'Although the I-V characteristic may be linear, the system differs from an Ohmic resistor. For
instance, the noise associated with the stochastic tunneling is shot noise. Assuming a Poissonian
statistics we find for of a voltage-biased junction the power spectrum of current fluctuations
S^= J At ((/(1)Д0))-(/)2)еш, = е2[Г(7) + Г(-1/)] = ^со1ь(™) .
which differs from the Johnson-Nyquist noise of a resistor. Similarly, the current fiuctuations in the
junction of an electron box are [11] Si(w) = (e2/Zch) EjnlWit”) + Г1Ь(п)] ехр[-Е(|,(7|)Д’в'-П, wit5
the rates given by (3.13). In both cases we wrote the classical form, the quantum mechanical form is
obtained by shifting eV or <5Eri, by ±/ia>/2.
3.3 Tunneling rates and I-V characteristics 159
Fig. 3.7: The current of a symmetric transistor is shown as a function of gate and transport volte r."
At low temperatures and low transport voltages VC/e < 1 only two charge states play a role, and the
Coulomb xscillations are clearly demonstrated. At larger transport voltages, more charge states art
involved.
respectively, while for the transitions in the right junction eV is replaced by —eV. To
the 2-state limit the stationary solution of the master equation is
p(n) =	Гпг(п+ i) . p(^n+ i) - j -p(n)	(3.18)
where Г> = Гц(п) + Гш(п) + Г1Ь(п + 1) + Гт(п + 1). The current reduces to
, = __ Гц(п)Г1к(п + 1) - Гм(70Гц,(n + 1)
This exp-fission is readily analyzed by inspection of (3.17). At low temperatures the
tunneling process in the left junction from n to n+ 1, with rate Гц(п), is allowed when
Qg ~ (n + l/2)e > —VC/2. On the other hand, the transition which carries on the
charge tc the right electrode with rate Гц<(п. + 1) is allowed when Qg — (n + l/2)e <
VC/2. Eoth coexist in a window of width CV around Qg = (n + l/2)e. The other
two processes are not allowed simultaneously, and in fact are suppressed in the window
just mentioned. Therefore, at low temperature the current is
4R
4e2
C?V
for
VC Qg	1 VC
--- <------71	< ------ ,
2e “ e-----2 - 2e
(3.20.)
1

while it vanishes outside the window. For simplicity we have assumed in (3.20) that
the two junctions have the same tunneling resistance	— Rtjt-
160 Single-Electron Tunneling
Fig. 3.8: Coulomb staircase: The current of an asymmetric transistor with different tunneling resis-
tances in the two junctions flt.R = 10/4,l is shown as a function of the transport voltage for Qq = 0
(pronounced Coulomb blockade), Qc/e = 0.25 (intermediate), and Qc/e = 0.5 (linear at low voltage).
The current through a symmetric SET transistor is plotted as a function of the
transport and gate voltages in Fig. 3.7. For gate voltages such that Qa/e is close to
an integer, the current vanishes below a threshold bias voltage Ць(<Эс = ne) — e/C.
This is a manifestation of the Coulomb blockade. At non-integcr values of Qa/e the
threshold voltage is lower Vth(Qc) = minn {2|Qg — (''+ l/2)e|/C}. One finds a series
of evenly spaced peaks centered around half-integer values of Qa/e — n + 1/2, each of
parabolic shape as given by Eq. (3.20). These are the so-called “Coulomb oscillations”.
The strong dependence of /(Qg- V") on the gate voltage makes the SET transistor
a highly sensitive “electrometer”. Small changes of polarization charges by fractions
of an electron charge influence a macroscopic measurement current. It has been used,
for instance, to measure the charge in an electron box (ti(Qg)) discussed above.
For larger transport voltages, more charge states play a role even at low tem-
peratures. In order to illustrate this, we consider a junction with symmetric bias
V'L — —VR = V/2 and Qg = 0, where the lowest energy state has и = 0 electrons in the
island. At transport voltages exceeding a threshold Vth 0 = e/C tunneling sets in to a
charge state with n = 1. Above this voltage, the electrochemical potential in the left
lead is sufficient to compensate the increase in charging energy of the island. Since at
the same time this state with n = 1 is unstable against a tunneling process in the right
junction, a current is transported through the system. At the same voltage tunneling
processes involving the state with n = — 1 are possible. At still higher voltages further
charge states |n| > 1 play a role. This leads to a series of voltages Vthn = (2n+ l)e/C,
each marking the threshold above which another pair of charge states becomes popu-
lated and a new channel for the conductance opens. The increase in conductance is
limited, however, due to the normalization condition for the p(n). Still, for suitable
parameters (significantly differing conductances of the two junctions or different ca-
pacitances), the current increases in the shape of a staircase. The phenomenon got
accordingly the name “Coulomb staircase” [12]. The behavior is demonstrated in the
plot of Fig. 3.8.
3.3 Tunneling rates and I-V characteristics 1G1
3.3.3 Cotunneling processes
If sequential single-electron tunneling is suppressed by the Coulomb blockade, high re-
order processes such as coherent “cotunneling” through several junctions become cru-
cial (Averin and Nazarov in Ref. [2]). As a specific example, we consider a SET
transistor, biased such that the current in lowest-order perturbation theory vanishes
(see Fig. 3.7). At low temperatures sequential tunneling is exponentially suppressed in
this regime since the energy of a state with an excess charge on the island lies above
the Ferrni levels of the leads. On the other hand, if a transport voltage is applied, a
higher-order tunneling process transferring an electron charge coherently through the
total system is energetically allowed. In this case the state with an excess electron
charge i:i the island exists only virtually. Standard second-order (or fourth, depending
on the counting) perturbation theory yields the rate
2л-
h
(i|Rt|^)(V’|fl't|/)
Е-ф — E\
2
A(E, - Ef) .
(3.21)
The energy of the intermediate virtual state lies above the initial one, — E\ > 0,
but it enters only into the denominator rather than into the exponent of the sequential
tunneling rate. Hence the higher-order rate is nonzero even at very low temperatures.
Whe i analyzing the process we have to pay attention to the following:
(i) There are actually two channels which add coherently. Either an electron tunnels
first from the left lead onto the island, and then an electron tunnels from the
island to the other lead. In this case the increase in charging energy of the
intermediate state compared with the initial one is 5ЕЪ = Ech(n + 1,QG) -
Eci. (n, Qg) — eVf- Or an electron tunnels first, out of the island to the right lead
and another electron from the left lead replaces the charge. In this case the
increase in energy of the intermediate state is <5Er = Ec|,(n — 1,QG) + cVr -
Eci (n, Qg)- Both amplitudes have to be added before the matrix element is
sqr ared.
(ii) Th з leads contain a macroscopic number of electrons. Therefore, with overwhelm-
ing probability the outgoing electron will come from a different state than tiro
oik which the incoming electron occupies. Hence, after the process an electron-
hols excitation is left in the island, which explains why it is called “inelastic ‘
cot unneling.
Transitions involving different excitations are added incoherently. The resulting rate
for inelastic cotunneling is
n.,t = у 4у~у....... / dQ [ de, f <kql [ с!бИЫ[1 - f(f4)]f(e.q')[l - fQk>)]
aeWtjJiiji Jket Jqei J^'ei Jk'eR
1 1
-----------------1-----------------
(4 + (5El — tk fk' + ЗЕц — e,
(3.22)
162	Single-Electron Tunneling
At T = 0 the integrals can be performed analytically with the result
Tcot
--------------у
27re3/?tiLEtiR
2____SEl6E-r \ / „	/ eV \ \ _
eV6EL + 5ER + eVj П \ SEj )
h / 1
1 \2
— V3 for eV 5El,6Er.
oEr/
(3.23)
At finite temperatures forward and backward processes occur. They obey a detailed
balance relation rcot(—V) = exp(—eV/kBE)rcot(V). The current then is
- isra; U+И)2+v.	(3.24)
In the Coulomb blockade regime of a SET transistor the V3 dependence of the cotun-
neling current has been observed. In systems with N junctions a corresponding 7V-th or-
der process (or 2A-th order, depending on the counting) leads to a current I oc V2W-1.
As an example we consider N = 4 junctions with C = 10~15F and tunneling resistance
Rt. In this case (see D. Esteve in Ref. [2]) Ecot = (2.5 x 10“3/s) (V/pV)7 (kfi/Rt)4.
These cotunneling processes limit the accuracy of the single electron turnstile even
under the most favorable situations, i.e. low T and low frequency, where thermally
activated multi-electron transfer processes and missed cycles play little role.
The expression for the cotunneling rate diverges logarithmically when the interme-
diate and initial or final states are degenerate. This divergence is removed by life-time
broadening effects, which will be derived systematically - together with further effects
- in Section 3.7.
There exists also the process where one electron tunnels through the total system,
leaving no excitations in the island. This process is called “elastic cotunneling”. Its
rate has a small prefactor oc l/fQi-ZVf (0)] (inversely proportional to the number of states
of the island) compared with the inelastic cotunneling rate. On the other hand, it yields
a current which is linear in the applied voltage, which makes it important at very low
voltages and temperatures kRT, eV <C [Ec/fW(0)]^2.
3.3.4 Broadening of the steps
Even at T — 0 where thermal effects are frozen, tunneling of electrons leads to an
uncertainty in their location. This leads to a broadening of the steps in (u(Qg)) in the
electron box. This effect can be estimated in perturbation theory [13]. We start from
the basis states |n;,../°) with total charge n on the island and certain single-electron
states of the lead and the island occupied or empty (indicated by the dots). Due to
tunneling the states are modified. In lowest-order perturbation theory the corrections
= L ~^ + Ech(n + 1, Qg) - Ech(n, Qc)+ 1;
tq + ECh(.n 1, Qg)	Ec^(71,Qg)

(3.25)
3.4 Influence of the electromagnetic environment 163
arise d ie to tunneling from an electron state q of the lead (leaving it empty q) into the
state к of the island, increasing the charge to n + 1, or reversely. The resulting cha-
in the expectation value of the electron number, 6n(QG) = (^4), is
< MQg) = |T|2NlNi f	f de J у------	~.Ж)]------------
J J Ч6*: — £4 + ^ch(n + 1, Qg) — Ech(n, Qg)]2
____________/МДЖ___________________у	..,
[e, - ek + T'chfn - 1,<2g) - Ech(n,Qg)]2 ’
which lor T = 0 reduces to
г (n x _ Дк , Дс1.(» - 1,Qg) ~ ^ch(»,QG)
WG>	Ech(n+l,QG)-EA(n,QG) '	'	’
The result displays several important properties: (i) the expansion parameter is the
dimens onless tunneling conductance R^/R^, where the quantum resistance /?« serves
as refer mce, (ii) tunneling of single electrons leads to logarithmic corrections, (iii) the
perturbation theory fails at the points of degeneracy of the charging energy, QG/e = n+
1 /2. In the last Sections of this Chapter we will present the theoretical framework which
describes tunneling beyond perturbation theory and regularizes these expressions.
3.4 Influence of the electromagnetic environment
So far we have assumed that the electron box or the SET transistor are driven by
ideal voltage sources, and we have considered ideal measurement devices. On the other
hand, in a real experiment the sources are outside the cryostat, some distance away
from th? single-electron device to which they are connected by leads. This introduces
stray capacitances and Ohmic resistors as well as thermal fluctuations. We have to
understand their influence on single-electron tunneling in order to describe a realistic
situation - or to know how to set up an experiment close to ideal. We, therefore, will
consider now a tunnel junction which is connected to an electric circuit described by a
general .mpedance Z(w). A detailed review of this problems has been given by Ingold
and Nazarov in Ref. [2]. It is a specific example of the general problem how to descril»-
dissipat.on in quantum mechanics, which has been addressed for instance by Calden i
and Leggett [14] and which is reviewed in Chapter 4 of this volume. In this appro;: h
the fluctuating linear circuit is modeled by an ensemble of harmonic oscillators.
3.4.1 The model Hamiltonian
The simplest example is a single tunnel junction in series with an impedance Z(u>) and
both driven by a voltage source as shown in Fig. 3.9. The tunnel junction is modeled
in the utual way by a tunneling Hamiltonian. It is coupled to an ensemble of harmonl
oscillators to account for the effect of the impedance. Due to this coupling tunneling
processes in general are accompanied by emission or absorption processes of “photons”.
164 Single-Electron Tunneling
Fig. 3.9: A junction in an external circuit characterized by an impedance Z(w).
We will calculate the tunneling current I(V) as a function of the dc-voltage at the
junction. Because of the voltage drop at the impedance the voltage at the junction
V(t) = V + 6V(t)	(3.28)
is reduced below the applied value, V = Vx — /(V)Z(O). Since this drop depends again
on the current to be determined, we are left - even after we found I(V) - with a self-
consistency problem. Furthermore, the impedance produces current and hence voltage
fluctuations at the junction <5H(t) with (6V(t)} = 0.
Let us recall what is known about the fluctuations of a resistor, or in general of
a linear circuit element with impedance Z(w). For this purpose we ignore tunneling,
which means that the junction is reduced to a capacitor C. Then the balance of currents
in the circuit satisfies (after Fourier transformation)
[zwC + Z^(w)]JV(w) = Jl>) .	(3.29)
The power spectrum of the Gaussian current noise is given by the standard Johnson-
Nyquist relation
[°° 4^-^)е^'>|({<5/(0,й/(/')})
J — oo	2
= RejZ”1)^)} /la’coth (—7—7=) .	(3.30)
Hence the fluctuations of the voltage at the junction are governed by
= Re{Zt(w)} /uucoth (^- \ .	(3.31)
Here Zt(w) is the impedance seen at the site of the junction, i.e. the effect of Z(w) and
the capacitance of the junction shunted in parallel,
Zt(w) = [iwC + Z-»]”1 .	(3.32)
A microscopic description of the system consisting of the junction and the impedance
is provided by the Hamiltonian
H = E (6* + W) 4./1Д + E + E Tk,qCk,aC4-o + h c- +	 (3-33)
fc,(T	k,q,a
3.4 Influence of the electromagnetic environment 165
The first terms describe the right and left electrodes and the tunneling. The last term.
Нъа.л, describes the degrees of freedom responsible for the fluctuations 6V(t). Sir''r'
they result from a linear circuit element they are Gaussian and are in general descrii .
by an ensemble of harmonic oscillators. We set
/	2	\
/ T) 	772	\
#bath = E T2- +	and = 12 CA'W 	(3.3 fl
j \	A	/	J
Here we introduced a phase as the time-integral of the voltage
7i0(t) = У dt'eV(t') = eV t + ft<f</>(t) , Ь6ф(ф) = f dt'e$V(t') ,	(3.35)
which will turn out to be the natural variable. The distribution of the oscillator fre-
quencies Qj and the coefficients have to be chosen appropriately in order to produce
the coriect power spectrum.
Usir g properties of the harmonic oscillators,
(coth coslfif(f - f')] - isin[Q/t - t')] j ,
we find for the Fourier transform of the symmetrized correlation function of 5ф
= J(w)coth (-^E-) •	(3.36)
2	'•Z/vyj '
The coefficient tj and the frequencies of the oscillators enter only in the combination
c^-ti
E		(3.37)
J	* J
We can reproduce the Johnson-Nyquist, correlation functions (3.31) by choosing
A
J(w) = —Re{Zt(w)} .	(3.38)
Tiu)
Technically it is inconvenient to deal with time-dependent energies in the electrodes.
Therefoie, we perform a unitary transformation H = U^H'U — itilPdU/dt, where
(3.39;
In the resulting Hamiltonian H' the electrodes appear in unperturbed form
H' = H(kc\ac.ka + EgTC9^ + E V*’	+ he.) + Hbath ,	(3.40)
/с,<т	Q-.^	k,q.<T
but the tunneling term acquired a time-dependent, phase factor, depending on the
integral of the voltage introduced by (3.35).
,,t
U = exp
166 Single-Electron Tunneling
3.4.2 The single-electron tunneling rate
When evaluating the tunneling rates we have to take into account that a tunneling
process (from state к in one electrode to q in the other) in general is accompanied by
a transition in the bath (X —> X1) as well. Using the golden rule we find the rate for
tunneling in one direction
r+(V) = /de*/dej(e*)[l ~/(e9)]
€ J J
X £ pbath(X)|<X'|e^|X)p(£, + eV + E.v - eq - £X') .	(3.41)
X,X'
Here Pbath(A^) denotes the probability to find the bath in a state X. In thermal equi-
librium and lowest-order in the coupling it is Pbath(A) = (A"| exp[—/3Hbath]|A)/Zbath.
We write 6(ek + eV + Ex - eq - EX’) = J — exp [A(e* + eV + Ex - eq - ^x')*] and
interpret the exponential of the bath energies as the time evolution operators of the
bath. This allows us to express (3.41) as
r+(,/> = Л- /<«•)) /	->"•
X £ PbathWme^d|A')(A'|e^<0’|.Y) .	(3.42)
X,X'
The second line of this expression can be expressed as a bath correlation function
(ei<S,^de-'>50(O)^bath = eWW“<50(O)]50(O))batb =	.	(3.43)
We arrived at the second form using the Bakcr-Hausdorff formula and properties of a
harmonic system. The correlation function K(t), unlike the symmetrized correlation
functions (3.36), depends on the order of the operators. It can be expressed as
K(t) = / d(£Re{Zt(ui)} ( th	[cos(cjf) - 1] - isin(u;t)| .	(3.44)
J—oo CJ Rk [ x2Kq1 '	J
The tunneling rate in forward direction can now be written as
r+(U) = j- Г d£ Г dE'f(E)[l - f(E')]P(E + eV - £') ,	(3.45)
J-co	J—co
where the function P(E) is related to K(t) by
P(E) = -t- [°° dt exp[A'(t) + iEt/ti] .	(3.46)
ztt/z J -co
This completes the derivation. The calculation of the tunneling rate is reduced to
integrations. We will continue with a discussion and derive some limiting results.
3.4 Influence of the electromagnetic environment 167
3.4.3 General properties
The coupling to the environment (bath) is accounted for by the function P(E) in the
integral (3.45). In comparison to the usual expression for the tunneling rate of a voltage
biased junction (see e.g. Eq. (3.10) with <)Ech replaced by eV), P replaces the energy
conser/ing J-function. This can be made apparent also by rewriting the rate (3.45) as
a convolution
r+(V) = I dEr+=0(V - E/e)P(E) .	(3.471
where Г/=()(Е) is given by (3.13). In the absence of the impedance and its fluctuations
i.e. for K(t) = 0, P(E) reduces to a <5-function, and we recover the standard resul*
for the voltage-biased junction. In general, the function P(E) describes the emission
(E > 9) and absorption (E < 0) of energy during a tunneling process due to the
coupling of the electrons to the oscillator bath.
The vanishing of K(t = 0) = 0 implies that the function P(E) is normalized
roc
у ji/;/'(E) . I.	(3. ..
We obtain a second sum rule by taking the derivative of exp[/<(t)], with the resul'
[°° dEEP(E) = ifiE'(O) = h Г	.	(3.49;
J — OQ	J — OO
At 'r = 0 the function P(E) vanishes for negative energies, P(E < 0) = 0, and on!v
the forward tunneling rate is nonzero. From the tunneling rates we obtain the curreni
/(V) = еГ+,
7(V) = 4- Г dE (cV -	•	(3-50)
eRt Jo
which provides a convenient relation
d2/ e
=	О-’"’
At largf voltages, such that eV is larger than the energies where P(E) gives a noticeable
contribution, the limits of integration in (3.50) can be extended to ±oo. In this <-:c-e
the sum rules derived above are sufficient to determine the current-voltage relation
;(V) = iH1'* Й 
The shut of the I-V characteristic is a manifestation of the Coulomb blockade.
At fnite temperatures T 0 the function P(E) obeys a detailed balance rela
P(E)/F(—E) =	. The current then is
1 roc 1 ______
I V)-e(r+(V)-r+(-V)) = — / dE——^EP(eV - E) .	(3.53)
елt, J —oo i — c f
16a Singie-blectron tunneling
Below we will present further analytic and numerical results. For the moment we
only stress that the calculation of I(V) is reduced to integrations. We have to recall,
however, that in Eq. (3.28) we have split the voltage at the junction V(t) = V + 5V(t)
into a de part V and a fluctuating part with vanishing average. There remains the
problem to determine the de part, which differs from the applied voltage Vx due to the
voltage drop at the junction. This in turn is proportional to the current I{V), leading
to the following self-consistency relation
= 0) + V = Vx .
(3.54)
Along the same line we can also describe a current-biased junction with a parallel
Ohmic resistor. Here the imposed current Ix is divided into a current through the
junction I(V) and a current through the resistor, which in turn depends on the voltage
at the junction. Hence
Д V) + = i,
К
(3.55)
In both cases we combine the standard linear circuit description (Kirchhoff’s laws) for
resistors, capacitances, sources, ... with the “black-box” relation /(V) for the junction,
which is assumed to be the only nonlinear element in the circuit. The properties of
the junction depend on the impedance Zt(w) seen at the site of the junction. It is the
same for both examples mentioned above. The current-biased junction with an Ohmic
shunt resistor has been studied by Odintsov [15] and by Panyukov and Zaikin [16], who
arrived at equivalent conclusions as described above.
At this stage we would like to comment on the range of validity of the treatment
presented above. The transition rate was obtained in lowest-order perturbation theory.
This requires that the tunneling conductance l/f?t is low, but the question remains
what is the reference scale. Furthermore, it appears that no assumption was made
about the value of the series impedance Z(u). A systematic analysis of the problem
where the tunneling and the Ohmic resistor are treated on an equal footing (see Section
3.6). yields the requirement Rt Z(0). Obviously, in the extreme limit it is not crucial
to solve the self-consistency relation (3.54). However, in intermediate situations only
the self-consistent calculation produces results with the correct asymptotic behavior.
The analysis presented above can be generalized to more complex systems involv-
ing networks of junctions and general impedances. The basic point is that we treat
the tunneling in lowest-order perturbation theory, i.e. for a tunneling process in one
junction all the other junctions only act as capacitors. This means that the transition
rate in each junction has the form presented above. However, it depends on the spe-
cific impedance Zt(cc) between the two sides of the considered junction, which in turn
depends on the capacitances of all other junctions. The calculation of that impedance
follows the classical electrodynamics rules.
3.4 Influence of the electromagnetic environment 169
3.4.4 The effect of an Ohmic resistor
As an important example we consider a tunnel junction in series with an Ohmic resistor
Z(w) = R, defining the dimensionless conductance
as = R«/R .	(3.56)
The resulting function P(E) is plotted in Fig. 3.10 for different values of as at T -- 0.
and the corresponding current-voltage characteristics in Fig. 3.11. The curves di""'
a pronounced crossover. As R/Rk is increased the peak of P(E) shifts from the c
to Ec, and the I-V characteristics changes from a classical, linear dependence to n
nonlinecr one with a pronounced Coulomb gap.
In the limit of a low impedance environment, R/RK <C 1, the function P(E) redtl
to P(E) -> 5(E). In this case we recover the classical linear I-V characteristic of a
junction driven by a constant voltage source. In the opposite limit of a high impedance
environment, R/Rk 1, the external voltage source and the series resistor act as
a current source, which should lead to Coulomb blockade effects. Indeed at finite
temperatures, kBT » Ti/RC, where it is sufficient to replace Re{Zt(w)} = 7?/(l +
(uiRC)2) -> (-n/C)6(w), we find K(t) = -ir/(CRK) (it + kBTt2/h). Hence, P(E) is
peaked around the Coulomb gap Ec = c2/2C,
R<E) =	’	(3-57)
х/ 4tt EqkqT
and the I-V characteristic shows a Coulomb gap. At very low temperatures, kBT <7
h/RC, t le width of the peak of P(E) is proportional to Ecy/Rs-
We proceed by deriving further asymptotic results for P(E) and the I-V charac-
teristics. At low temperatures K(t) can be expressed by Exponential Integrals
у-А'(т) = — Ге"тЕ1(—-r) -	.	(3.58)
Here we have introduced т ~ t/RC. In the long-time limit r —> oc we have
/<(т) =-— [ln(r)+ 7 + i^ + ...]	(3.59)
L	2 J
where 7 -= 0.5772... is Euler’s constant. From this we obtain P(E) at low energies, up
to a constant which is fixed by the normalization condition. Hence, we have
P(E -4 0+) oc e"2^ P dTT-2/nsciE!i('T/r‘
(,-27/a, ! p E
r(2/os) E L<Xs Ecl
Inserting the expansion into (3.50) we find for T = 0
_	e"2^"’ V Г yr ejV|l2/"b
Г(2 + 2/asjRt
(3.60)
(3.61)
170 Single-Electron Tunneling
Fig. 3.10: The function P(E) at T = 0 for different values of the series resistor. From (a) to (f)
as = Як/Я = 20,3.2,2,1.6,0.4,0.04.
Fig. 3.11: I-V characteristic of a junction in an electric circuit at T = 0 for different values of the
series resistor. From (a) to (g) as = Rr/R = oo, 20,3.2,2,0.4,0.04,0.
3.4 Influence of the electromagnetic environment 171
Fig. 3.12: RLC line
We note that at low temperature, as long as the series resistance does not vanish,
R/ 0, the differential conductance near V = 0 vanishes as a power law. At finite
temperatures, T 0, the conductance is finite. The linear conductance is [16]
d/
dV
1
x ~R
v=o
7Г 2RqT
oig Ec
2/a,
(3..
These examples show that a single tunnel junction only shows Coulomb blockud.
effects i" shunted in series with a resistor exceeding the quantum resistance R^. Th.
difficult to realize in an experiment since a high resistor close to the metal junction cai:
be fabricated only by bringing different materials into good electric contact. Indeed
single-e ectron effects and Coulomb blockade are easier studied in more complex sys-
tems, si ch as SET transistors discussed previously. In this case one junction effectively
acts as i high resistor for the other junctions.
3.4.5 Other environments
Above we considered explicitly the effect of an Ohmic series resistor on the tunnel-
ing. The question arises, how other elements with a different frequency dependence
of the impedance Z(w) influence the tunneling. An example which is important from
a practical point of view is a coaxial line, which can be modeled by an infinite line
of inductances, resistors, and capacitors as shown in Fig. 3.12. When pursuing thi->
question we quickly notice that we have done already most work. Most combinations
of linear elements produce an impedance Z(<u), which is finite at low frequencies. F -r
instance an LC-line has Z(w —> 0) = (L0/C0)1/2, where Lo and Co are the inductan'f
and capacitance per building block. The interesting low-voltage part of the junc';on
I-V chai acteristics depends precisely on this low-frequency impedance. Hence, mos'
the resu ts presented above apply, provided we replace the resistance Я by Z(w
We can expect qualitatively different results only when the impedance does not
approach a constant at low frequencies. Two examples can be mentioned: (i) A single'
LC resonator with resonance frequency Q — LC and Z(w) a <f(cu ± Q). This systi-
is discussed in detail by Ingold and Nazarov in Ref. [2]. (ii) The RC line, i.e. a serie
of resistors and capacitors, shown in Fig. 3.12 with Lq = 0. It has the impedance
•Zrc(w 0) = yjRo/(iwCo).
We now study in more detail the effect of an RC-line. At low frequencies Zt(cu) =
[iuC + Тгдс(ш)]-1 « Zrc(w). Hence, Re{Zt(w)} « уR0/(2wC0), and in the long-time
172 Single-Electron Tunneling
limit
krcW==“2тг\/Ж[1+1 sgnw] • (з-бз)
П у ZC/q JO U '	'	' -ibR V -^0^0
Next we can evaluate
„ ,	1 г00 Г n Ro I 7rt 1	( Et Rq I Trt \
Prc(E) = — dtexp - 2 —-J——- c°s — - 2 -~\^-Er]
'КП, Jo	L /1r V -TWO J	\ n -T1R V -ftoCo /
/ eV0 / eV0\ ,	,, .Ro e2
= Vs^“p(”4e) ”'he" eV°=iRM- (M4)
The RC-line not only introduces a resistance scale but also an energy and voltage scale
Vo. The function Prc(E) has a maximum at E ~ cVq. The resulting I-V characteristic
shows a structure resembling the Coulomb gap discussed above. However, the energy
scale does not depend on the junction but only on properties of the RC-line.
The RC-line sheds light on a fundamental problem. In all real systems the junction
capacitance is shunted by stray capacitors arising from the leads. This raises the
question whether the small junction capacitance C - with large charging energy Eq =
e2/2С and physical consequences on the tunneling  remains observable, or whether
it is masked by the large stray capacitors. It has been conjectured that the tunneling
electron sees only the stray capacitances within a certain ‘horizon’ in space, which
hopefully is small. The question then is, what is the size I of this region, explored by
the tunneling electron. In many cases the Ansatz I = ст where c is the propagation
velocity and т « 7i/max{eV, k^T} appears to work [17]. The model calculation with
spatially distributed capacitances, presented above, yields another limitation. Namely
the effective capacitance is
Ceff = ^-Co .	(3.65)
4rto
Notice that 7?K/R0 is the number of building blocks needed to have a total resistance
of order RK. From (3.65) we see that R^/Ro is also the distance (in units of the
building blocks) up to which the tunneling electron sees the spatially distributed ca-
pacitances. In summary, stray capacitances do influence the tunneling. However, they
are effectively screened by a resistor of the order of the quantum resistance.
3.5 Charging effects and superconductivity
If the electrodes of the junction are superconducting, Cooper pairs can tunnel. At the
same time quasiparticle tunneling is reduced due to the opening of the superconducting
gap. This can make higher-order effects, such as the charge transfer due to Andreev
reflection the dominant process. In low capacitance junctions Cooper-pair and Andreev
tunneling are influenced by charging effects in much the same way as single-electron
3.5 Charging effects and superconductivity 173
tunneling. In systems which contain small superconducting islands “parity effects
may be observable. They arise since single-electron tunneling from the ground staK
where all electrons near the Fermi surface are paired, leads to a state where one er-
electron - the “odd” one - in the island is in an excited state. Its energy lies above
that of the equivalent normal system by the gap Д. Parity effects influence various
physical properties, for instance the state of an electron box or the dissipative and
Josephs m currents through superconducting single-electron transistors. In this Se
tion we discuss examples of these effects on the level of perturbation theory. A more
systematic approach and further results will be presented in the next Section. For an
introduction into superconductivity including some topics of this Section, Tinkham’s
book [4] is recommended. Recent work is presented in the proceedings of the workshop
Mesosccpic Superconductivity [5] and the review article by Bruder [9].
3.5.1 Charging effects on quasiparticle tunneling
If the system, or part of it, is superconducting we have to describe the tunnelme.
of quasiparticles , whose energy depends on the superconducting gap. The rate fci
tunneling is still given by the expression (3.10),
1 roo	COO
Гы(п) =	/ dE dE'
СЧц J—OO	J —oo
xM(EH(E')/l(F)[1 ~ МЕ')]5(8ЕЛ + E'-E),	(3.GG)
with the obvious modification that the energy integrals include the reduced densitc л
of states of lead and island. In ideal systems they take the BCS form
X1/l(E) = e(|E| - A1/L)	|Д|	.	(3.67)
Vs ~ Ai/l
Although the integration can no longer be performed in closed form, the rate can be
expressed in a transparent way
Ги(п) = -Д (1	.	(3.68)
eye/ exp[<5Ech/«BTj - 1
It depends on the change in charging energy given by Eq. (3.11). The function 7t(F)
is the well-known quasiparticle tunneling characteristic (see e.g. Ref. [4] or curve (a) in
Fig. 3.13), which is suppressed at voltages below the superconducting gap(s). Charging
effects reduce the quasiparticle tunneling further. At zero temperature the rate is
nonzero mly if the gain in charging energy compensates the energy needed to create
the excitations, i.e. it sets in with a step at SECh + Aj + Al < 0 if both electrodes an'
superconducting, or proportional to the square root of |<5BCh + Ai/i,|. if the argument is
negative, in NS junctions. The rates approach the normal state result for large energy
differences.
174 Single-Electron Tunneling
2CV/e
Fig. 3.13: I-V characteristic of a superconducting junction in an electric circuit for different val-
ues of the series resistor. From (a) to (k) as = Rk/R = oo, 40,20,8,4, 2, 0.8,0.4, 0.2, 0.04,0. The
superconducting energy gap is Д = 2Eq.
Fluctuations of the electrodynamic environment can be taken into account similar
as in the normal state. The 5-function in Eq. (3.66) has to be replaced by the function
P(E) introduced in the previous Section. The resulting I-V characteristics [18], are
plotted in Fig. 3.13. They show much structure at the sum of gap and charging energy.
3.5.2 Two-electron tunneling, Andreev reflection
In the regime where quasiparticle tunneling is suppressed by the superconducting gap
higher-order processes involving multi-electron tunneling play a role. Cooper-pair tun-
neling is such a process. We will discuss it later. If only one of the electrodes is
superconducting there is still the process of 2-electron tunneling, denoted as Andreev
reflection . In this process an electron approaching from the normal side with energy
below the gap is reflected as a hole, while a Cooper pair propagates into the supercon-
ductor. (Andreev considered a normal metal and a superconductor in good metallic
contact. But his physical picture can be generalized to tunnel junctions.)
For definiteness we consider a SET transistor with a superconducting island and
normal leads (NSN). In order to describe tunneling in this system we have to rewrite the
tunneling Hamiltonian in terms of the Bogoliubov creation and annihilation operators
for the excitations in the superconducting island
#t,L = E TkAUq,</lq,<r + Vq,^-q,-a]ck^ + h.C. .	(3.69)
/с,д,<т
Here, uq,a and vq„ are the standard BCS coherence factors with magnitudes y|(l ± ^-),
3.5 Charging effects and superconductivity 175
and Eq -	+ A2 is the energy of the quasiparticles.
Andreev reflection is a second-order coherent process. In the first part of the tran-
sition one electron is transferred from an initial state, e.g. к f of the normal lead, in
an intermediate excited state q f of the superconducting island. In the second part
of the coherent transition an electron tunnels from k' J. into the partner state —q J,
of the first electron, such that both form a Cooper pair. The final state contains tv.
excitations in the normal lead and an extra Cooper pair in the superconducting island.
The amp itude for this process, to which we add the amplitude of the process in reverse
order, is ;hen given by [19]
= ^2 Tk,qTk' -qUqVq I —	——	— + —-	——	— j .	(3.70)
q	\0r!/ch,l	+	~	Q ЙЬс1, ] + bq — eki)
Here spin	indices have been suppressed	and the	relation vqj = v* has been	used. The
change	in	the charging energy <5Ech,i =	Ech(n +	1, Qc) - Ech(n, Qc) - eV	corresponds
to the virtual intermediate state where one electron has tunneled from the lead (at
voltage V) to the island. Finally, the rate for the Andreev reflection process is
2тг
Гы = ~т- 52 Им'I2	+ 4 + ‘’’•Е'сЬ.г) 	(3-71)
п к,к’
Here, the change in the charging energy <5EC|, 2 = Есъ(п + 2, Qq) — ECh(n, QG) — 2d'
corresponds to the real final state where two electron charges have been added to : hr
superconc noting island.
If we ignore the momentum dependence of the tunneling matrix elements (see bob ’)
the Q-summation in (3.70) can be performed with the result
Ak,k> = тгМ(0)а (	,	(3.72
V-bch,! /
where
a(x) = -	2^. arctan \		(3.73)
7Г V-'r2 - 1 V x + 1
The quasiparticle energy Eq is at least A, and we assumed that the energy of the
intermediate state lies above that of the initial state, A + <5Ech,t >> Q-, tv ~ 0. Andreev
reflection is most important if the gap A is much larger than the relevant energy
differences |6£’ch,i|- In this limit the function (3.73) reduces to a(A/<5Ech,i » 1) « 1.
We, therefore, drop in the following the weak energy dependence contained in the
function c. It has to be taken into account when the energy of the virtual state
coincides with that of the initial state, since a diverges in this case. In the opposite
limit, A + <5ECh,i < 0, single electrons tunnel, and Andreev reflection can be neglected.
If a « I, the integrations in (3.71) can be performed, resulting in
Г’.', /	Eq\\‘2	.	— ...
г n,Qc =-----------------------------?	3-r4
4e2 exp(<5ECh,2/^iU ) - 1
176 Single-Electron Tunneling
Note that this rate coincides in the functional dependence with that for single-electron
tunneling in a normal junction, Eq. (3.13), except that:
(i)	The charge transferred in an Andreev reflection is 2e, and the charging energy
changes accordingly. An important conclusion is that Andreev reflection is also
subject to Coulomb blockade in a way similar to normal-state single-electron
tunneling [20].
(ii)	The effective conductance is of second-order
1 Як
A“4AchJRt2 •
(iii)	We introduced the number of independent parallel channels
1 _ (\(TkiqTkl^)ktk.
^Ch ((\Tk^)k,q)2
(3.75)
(3.76)
which depends on the correlations between the tunnel matrix elements. In the
second-order Andreev process the matrix elements appear in a combination as
shown in the numerator of Eq. (3.76), differing from the square of the expres-
sion determining the normal state conductance 1 / /?t given in the denominator of
Eq. (3.76) 2. For the moment we consider NCh as a fit parameter. Even in small
junctions it turns out (from a comparison of Andreev and normal state conduc-
tance) to be much larger than one. Notice both, the normal state conductance
l/7?t — M:h/-Rt,o and the Andreev conductance Ga <x Мь-^к/^.о are the result
of jVch parallel channels. If we express the second order Andreev conductance by
the normal state conductance l/7?t the factor Ach appears in the denominator.
Since the Andreev reflection rate depends on the charging energy similar as the
normal-state single-electron tunneling rate we expect a similar dependence on gate and
transport voltages as shown in Fig. 3.7, with the obvious rescaling of the conductance
and charge. This is indeed what has been observed in the experiments of the Harvard
group [22].
3.5.3 Parity effects in small superconductors
In a normal-metal electron box, if the applied gate voltage is swept, the electron number
on the island increases in unit steps, and the voltage of the island shows a periodic
saw-tooth behavior. The periodicity in the gate charge is Qq is e. If the island
2The careful reader will notice that the expression (3.76) would be correct if we would not have
performed an integration over in the derivation of the expression for a. Hence the present derivation
is not rigorous. (A separation into magnitude and direction of the momenta, suggested in Ref. [19],
does not account for the relevant difficulty.) A more careful discussion will be presented in Section
3.G, where we will find that the Andreev conductance depends on correlations in space, which extend
over the range of the Cooperon propagator in the normal metal [21].
3.5 Charging effects and superconductivity 177
Fig. 3.14: The lowest energy state of a superconducting single-electron box as a function of the
gate voltage shows a difference between even and odd numbers n of electron charges on the island.
Accordingl/ the island charge is found in a broader range of gate voltages in the even state than in
the odd st; te.
is superconducting, and the gap Д is smaller than the charging energy Ec, then at
low temperatures the charge and the voltage show a characteristic long-short cyclic.
2e-period;c dependence on the induced charge. The effect arises since single-electron
tunneling from the ground state, where all electrons near the Fermi surface, of th;,
superconducting island are paired, leads to a state with one extra electron - the “odd"
one - in in excited state [23]. In a small island, as long as charging effects prevent
further tunneling, the odd electron does not find another excitation for recombination
Hence the energy of this state stays (at least metastable) above that of the equivalent
normal system by the gap energy. Only at larger gate voltages another electron can
enter the island, and the system can relax to the ground state. This behavior repeats
with periodicity 2e in Qc, as displayed in Fig. 3.14.
At low temperatures this even-odd asymmetry has been observed [22,24-26], but
at higher temperatures, above a crossover value Tcr <7 Д, the e-periodic behavior
typical for normal metal electron boxes is recovered. We can explain this crossover by
analyzing the rate of tunneling of electrons between the lead and the island, paying
particular attention to the fate of the “odd” electron [27]. Since at low temperature
single-electron tunneling processes which cost energy are exponentially suppressed the
further fate of the excited “odd” electron gains importance. This single excitation
can tunnel out with a rate 7 which is smaller by a factor 1/Arcff than the rate Г of
the other electrons; in mesoscopic islands Nen is typically of the order of 104 (see
below). On the other hand, in an important range of parameters 7 is not exponentially
suppressed, since the excitation energy of the odd electron is regained if this electron
178 Single-Electron Tunneling
tunnels out. Hence 7 « Ге^каТ/N^f- Parity effects are observable as long as this
single-electron tunneling rate is relevant 7 > Г, from which we obtain the crossover
temperature А:вТсг ss A/ln7Vefj. We will present now the arguments, analyze the rates
in more detail, and use them in the next Subsection to derive the I-V characteristics
of normal-superconducting NSN transistors.
We first consider an electron box with a superconducting island and a normal lead.
If the distribution functions of lead and island are equilibrium Fermi functions, the
rate of tunneling is given by Eq. (3.68). At low temperature the rate Гы is finite only
at voltages where the gain in charging energy (i.e. <5Ech < 0) exceeds the energy of the
excitations (c* > 0,Ep > A) created in the lead and island, i.e. for 8Eci-t 4- A < 0. It is
exponentially suppressed otherwise. The assumption of equilibrium Fermi distributions
is sufficient when we start from the even state. For definiteness let us assume that we
started from n = 0 and that the gate voltage is chosen such that 0 < Qq < e. Hence,
the relevant change in charging energy is <5Ech = Ech(l, Qg) — #ch(0, Qg) and the rate
of tunneling from an even to an odd state is
reo = rL[(n = 0,QG) 	(3.77)
In the odd state the quasiparticle distribution differs from an equilibrium Fermi
function. There is extra charge in the normal component. After thermalization the
distribution of the excitations in the island can be described by a Fermi function,
fSp(e) — [е(Е"'5м^А:в'г+ l]-1, but with a shifted chemical potential [in = Us + 8ц relative
to the condensate 3. The shift in chemical potential is fixed by the constraint to have
one excess electron charge
roc
1 = М(о)17, J °° dE^(E)[fSp(E) - f0(E)] .	(3.78)
This reduces at low temperatures to
8ц = A - k^T\nNeg{T),	(3.79)
where
Weff(T) = M(0)fh\/27rA7B7’	(3.80)
is the number of states in the island available for quasiparticles near the gap [25].
Parity effects are observable as long as the shift of the chemical potential is observable
8ц > k^T. This (again) amounts to the requirement T < Tcr, where the crossover
temperature is
7ltTr - A/lnMff (Tr) 	(3.81)
3A similar phenomenon was described 20 years ago by Tinkharn and denoted as charge (or branch)
imbalance [28]. In those experiments a nonequilibrium state was maintained by a balance of driving
currents and relaxation processes. In the present parity-effect experiments the charge imbalance is
preserved, at least in the sense of a metastable state, by the charging energy which prevents further
electrons from tunneling and the following recombination.
3.5 Charging effects and superconductivity 179
The tunneling rate back from the odd state (here n = 1) to the even state (n = 0)
is given by the expression Гое = Г1Ь,<5м(п = 1,QG) given by (3.66) with the island
distribution function replaced by /<?д(б). For exp(-A/A:BT) « 1 the ratio of the rates
of the two transitions is
poeypeo _ e[Ech(odd)+5p-.E<.h(even)]/kBT — ^F/k^T	•,
i.e. they obey a detailed balance relation, depending on a “free energy” difference
which in addition to the charging energy contains the shift of the chemical potent..u
<5g. This free energy difference coincides with that introduced in Ref. [25].
For t re following discussion it is useful to decompose the rate as
roe = r1L(l,QG) + 7(QG) ,	(3.83j
where Гв, is given by the equilibrium form, analogous to (3.68), and
7(Qg) - ~ Г de, Г ЛЕЩЕ)
J-ac J—oo
- fo(E)] [1 - /о(е*)]<5(бл - E - 6Ech) ,	(3.84)
describes the rate of tunneling of the odd, excited electron only. In the important range
of paramaters A 4- 6ECh > k&T this rate reduces to
7(Cg) = 2е2ад(0)«1 ’	(3'85)
whereas it is exponentially suppressed otherwise. Consistent with the simple picture
outlined before we see that the odd electron tunneling rate 7 contains a small prefactor
1/1Vi(0)Q; as compared with Гт- On the other hand, in an important range of gate
voltages -- since the energy of the excitation in the island is regained in the tunne1'
process - the rate 7 is not exponentially suppressed. Hence it may be larger than Гц,.
Above we described the range 0 < QG < e where tunneling occurs between ..
island states n ~ 0 and n = 1. The range e < QG < 2e can be treated analogously
The tunneling now connects the states n = 1 and n = 2. In this case, except for
the single-electron tunneling processes which create further excitations (described l a
Г), one electron can tunnel into one specific state ( — к, —ст), the partner state of the;
excitatioi (k,cr) which is already present. Both condense immediately; the state with
two excitations only exists virtually. The latter process is described again by 7(QG).
The symmetry implies Feo/o<“(QG) = reo/,oc(2e-QG). Since the properties of the system
arc 2e-pe'iodic in QG, we have provided a complete description for all values of the
gate voltage.
In the following we will consider processes where the sweep rate of the gate voltage
is small compared with the recombination rate of a pair of excitations. Therefore, we
can concentrate at a given gate voltage on the even state (ground state and thermal
distribution of pairs of excitations) and the odd state (one excess charge in an excited
18li	Single-Electron Tunneling
state plus thermal distribution of pairs of excitations). The sequential tunneling of
charges between the island and the lead is described by a master equation for the
occupation probabilities of the even and odd states pe(Qa) and p0(QG)>
= -reo(QG)pe(QG) + HWQg)	(3.86)
with Pe(QG)+Po(QG) = 1- The equilibrium solution ispe(o)(QG) = Гое(ео)(Сс)/Гя((2с),
where rs(QG) = Гое(0с)+Гео(0с)- For Toe Teo we have pe(Qc) ~ 1, i.e. the system
occupies the even state, while for Гео Г°е the island is in the odd state.
The solution of the master equation, combined with symmetry arguments, deter-
mines the crossover value Qcr of the gate charge where the system switches between
the even and the odd state. The condition is pe и po, i.e.
roe(Qcr) « reo(Qcr).	(3.87)
At very low temperatures the switching point is determined by the lowest energy as
shown in Fig. 3.14. At finite, but low temperature we find Qcr(T) = f +	—
А.вГ1п Аея^Т)], where Nen(T) was defined in (3.80). This means the short sections
in Fig. 3.14 get longer until, above Tcr, we have QCT = e/2, and only the e-periodic
behavior known from normal systems is recovered.
3.5.4 I-V characteristics of NSN transistors
The analysis presented above can be extended to describe even-odd effects in SET
transistors with a superconducting island. As a specific example we first consider an
NSN transistor where the energy gap is smaller than the charging energy scale A < Ec-
In this system the important processes are single-electron tunneling processes in the left
and right junction, causing transitions between even and odd states, with rates Г)//06
and Геао/ое which are obvious generalizations of Eq. (3.77) and (3.83). They depend on
the change in charging energies as described in Eq. (3.17), and on the energies of the
excitations created in the island.
These rates enter a master equation. At low T it is sufficient to consider only
one even and one odd state of the island. From the master equation we find again
the crossover gate voltage and temperature, but also the I-V characteristic of the
transistor. In the limit considered (A < Ec) it is
peopoe __ peopoe
I = е(Ое - r°L%) = e -eoJ( Ro roR L 7 	(3.88)
At high temperatures T > Tcr the single-electron tunneling current (3.88) shows
the Coulomb oscillations known from normal systems with parabola-shaped maxima
at the points QG = e/2 + ne with integer n. At low temperature T < Tcr the current
is limited by the odd electron tunneling rate 7 in one of the junctions. In the window
Qcr(T) < Qc < e/2 + AC/e + Qcr/2 < e it is
plateau = t-7 = 2e7?tM(O)Qi	(3'89)
3.5 Charging effects and superconductivity 181
Fig. 3.15: Quasiparticle current in an NSN transistor with Д < Ec as a function of gate and transport
voltage. T1 e parameters are Д = 55geV, Ec = 125geV, and Rt = 25kfl.
and exponentially small outside. A second current plateau exists in the window e <
3e/2 - AC/e - Qcr/2 < QG < 2e - Qcr. Both plateaus create a double structure
which repeats 2e-periodically. For A 4- eV/2 > Ec the two plateaus merge to form a
2e-periodic single plateau structure. An example is shown in Fig. 3.15 with parameters
which are realistic for an experiment on parity effects. In this case the current (3.89)
is of the crder of lOOfA.
In NSN transistors with a larger superconducting gap A > Ec the odd states have
a large energy. Hence a mechanism which transfers two electrons between the normal
metal and the superconductor becomes important. Andreev reflection with rate (3.74)
provides such a mechanism [19]. The master equation description can be generalized to
include also this process. Because of the similarity of the rate for Andreev reflection to
that of single electron tunneling it is clear that the shape of the I-V characteristic due
to Andreev reflection also takes a similar form. At low temperatures a set of parabolic
current peaks is found centered around the degeneracy points QG = ±e, ±3e,... [19]
I^SQg, V) = GA(V- 4^|)0(F - 4^)
\ VC2' v	I' C 2 7
(3Tr
Here 5Qq is <5Qg = QG — e for QG close to e, and similar near the other degeneracy
points.
At larger transport voltages single-electron tunneling sets in, even in the limit
A > Ec, and Andreev reflection becomes blocked; it gets “poisoned” [19]. The reason
is that ab we a threshold voltage the odd state can be reached by a single-electron
182	Single-Electron Tunneling
Fig. 3.16: The current I(Qq,V) through an NSN transistor with A > Eq. The parameters are
chosen to coincide with those of the experiments of Hergenrother et al., Eq = lOOpeV, Д = 245/teV.
tunneling process. This occurs when (e — Qg)2/2С — Q'q/'IC + A < eV/2, which
requires sufficiently large transport voltages, V > Ibison, where
Vpoison = 2 (Ес -	+ A) .	(3.91)
The rate for this transition, from the even to the odd state, is of the order of Гео ~
Gn(V - V)>oison)/e- The state which is reached after such a single-electron tunneling
process is not the ground state. It is energetically favorable that after the first tunneling
process another electron tunnels into the partner state of the excitation which is present
already. The rate for this process is given by 7, which in the considered range of
parameters takes the value given in Eq. (3.85). Typically the rate for the second
transition, from odd to even, is smaller than that of the first processes and, hence,
creates the bottleneck in the sequence of SET processes. The same inequality also
implies that above IZpoison the system is most likely in the odd state, pa/pe ~ Гео/7 2> 1.
Hence the current produced by the cycle is given by Eq. (3.89). (The current due to
Andreev transitions between two odd states is smaller, ^Andreev ~ PeG\V.)
Fig. 3.16 shows the current-voltage characteristic of an NSN transistor with A >
Ec. At small transport voltage the 2e-periodic peaks due to Andreev reflection dom-
inate; they get poisoned above a threshold voltage. The peaks at larger transport
voltages arise from a combination of single-electron tunneling and Andreev reflection
processes. The shape and size of the even-even Andreev peaks and some of the single-
electron tunneling features at higher transport voltages agree remarkably well with
the experiments of Hergenrother et al. [22]. In earlier experiments further odd-odd
Andreev peaks have been observed. They cannot be explained simply by raising the
electron temperature. Their origin, as has been pointed out by Hergenrother et al. [22],
are single-electron transitions induced by the noise of the electromagnetic environment,
which is at a higher temperature than the electron system.
3.5 Charging effects and superconductivity 183
3.5.5 Coherent Cooper-pair tunneling
In “classical” Josephson junctions Cooper pairs can tunnel free of dissipation between
the supei conducting electrodes. The coupling is described by the Josephson energy
—E}Cos</’, which depends on the phase difference across the barrier. The energy
scale Ej -- hlcr/2e is related to the critical current of the junction, which in turn i>i
be expressed by the tunneling resistance of the junction and the energy gap of Сю
superconductor, Icr(T = 0) = тгД/(2е7?е).
Charging introduces quantum effects: The phase difference and the charge on lin:
electrodes, Q, are quantum mechanical conjugate variables. The dynamics of an ideal
Josephson junction is governed by the Hamiltonian
H° =	~ E} cos 95 ’ Q ~ лоГТТТ '	(3-92)
2C	i d(n,ip/2e)
For simplicity we describe here a single junction. The generalization to multi-junction
systems, including gate voltage sources is obvious. An important question is how
dissipatioi due to the flow of normal currents and/or quasiparticle tunneling can be
accounted for, which has been addressed e.g. in Refs. [6,7,14,29]. So-called “macro-
scopic quantum effects” like macroscopic quantum tunneling of the phase, or quantum
coherent oscillations are derived from the Hamiltonian (3.92). Macroscopic quantum
tunneling has been observed in tunnel junctions with small capacitances of the order of
10~12 F. Taese values are orders of magnitude larger than those of the junctions where
single-electron effects play a role.
We now turn to mesoscopic Josephson junctions or junction systems, where the
number of electrons or Cooper pairs in small islands is a relevant degree of freedom.
The charging energy has been discussed in detail above. The Josephson coupling
describes the transfer of Cooper-pair charges in forward or backward direction, and
can be, writ ten in a basis of charge states as
(n|E’.l COS ip|n') = у (<5n',n+2 + <5n',n-2) 	(3.93)
Below vre will first consider situations where Cooper pairs tunnel coherently. This
shows features known from the phenomenon of resonant tunneling. It is nori-dissipative
and hence strongest in situations near degeneracy. We will show how in a supercon-
ducting electron box the steps in the expectation value of the charge on the island arc
broadened by Cooper-pair tunneling. In the next Subsection we will discuss, following
Ref. [30], he w coherent Cooper-pair tunneling can be probed by Andreev reflection and
observed in the dissipative I-V characteristic of an NSS transistor. Further examples of
coherent tunneling of Cooper pairs can be found in the literature. We mention the gate-
voltage dependence of the critical current of SSS or SNS transistors [31-33]. Another
example is ;he combination of coherent Cooper-pair tunneling and dissipative, quasi-
particle tumeling or transitions induced by the environment, which are responsible for
the dissipatve I-V characteristic of SSS transistors [25,34-36].
184	Single-Electron Tunneling
We first consider an electron box with superconducting island and lead, assuming
that the energy gap exceeds the charging energy and that the temperature is low,
A > Ec k&T. In this case, at low voltages quasiparticle tunneling is suppressed, and
the island charge can change only by Cooper-pair tunneling in units of 2e as described
by Eq. (3.93). The tunneling is strong near points of degeneracy. For instance for
Qg « e the states with n = 0 and n = 2 have similar charging energies, and we can
restrict our attention to these two charge states. The coherent tunneling between both
is described by the 2x2 Hamiltonian
( F'ch(O) — Ej/2 \
\ -E}/2 Ел{2) у
(3.94)
This Hamiltonian is easily diagonalized. The eigenstates are
V’o = a|0) + Z?|2> , Vh =/?|0) - a|2)
with coefficients
and energies
^o/i = I [ад) + ад) т y^c2h + ^j]
(3.95)
(3.96)
(3.97)
Here we introduced the difference in charging energy 8Ech = Ech(2) - £ch(0) =
4Ec (Qc/e - !)• The coefficient a is close to unity if the charging energy of the state
|2) lies above that of |0), i.e. for 5Е^ > 0, and vanishes in the opposite limit, while 0
has the complementary behavior.
The expectation value of the charge on the island in the ground state is given by
= 2/32 .
(3.98)
It changes continuously near Qg = e from 0 to 2 in a range of width of order
AQg « Ej/Ec- This has recently been observed experimentally [37]. We note that the
coherent mixing of different charge states due to Cooper-pair tunneling is described by
elementary quantum mechanics (the diagonalization of a 2 x 2 matrix). In contrast, the
perturbative description of single-electron tunneling presented in Section 3.3 diverges
near the degeneracy point and requires a more careful analysis (see Section 3.7).
3.5.6 Andreev spectroscopy of Josephson tunneling
Next we consider an example of coherent Cooper-pair tunneling in an NSS transistor .
Here Cooper pairs can tunnel coherently in the Josephson (SS) junction, which can
be probed by the dissipative current due to Andreev reflection across the NS junction.
Again we restrict ourselves to low temperatures, koT <4C Ej.
In the present example, where we describe coherent Cooper-pair tunneling in the
SS junction in a situation with a nonzero transport voltage we have to account in
________________________________3.5 Charging effects and superconductivity 185
the Hamiltonian for the work done by the voltage sources during the transitions. We,
therefore, keep track also of the number of electrons Nb and NR in the left and right
electrode. A basis set of states is denoted by \NLy n, NR), and the corresponding charg-
ing energy (for symmetric bias W = ~Vr = V/2) is
^h(NL,n,AR) = ^=^-(7VR-O^.	(3.99)
In a situation where only two charge states get appreciably mixed the eigenstates
and energies of the corresponding 2x2 Hamiltonian are
Vo = a|0,0,0) + /3|0,2, -2) ,	= 010,0,0) - a|0,2, -2) ,
Eo/1 = | [ясь(0,0,0) + £ch(0,2, —2) T y<5Ec2h + £2] .	(3.100)
The coefficients coincide with those of the box discussed above, except for the obvious
change of notation, and 5ECh = _ECh(0,2, —2) — Ech(0,0,0).
In the low-bias regime, the dominant mechanism of transport in the NS junction of
the transistor is Andreev reflection. Starting from a state |0, 0, 0) we are led by such ;>
process to the state | — 2,2, 0). The Josephson coupling mixes this state with the state
| — 2, 0, 2). Hence we have to consider a second set of eigenstates
Vo- 2,0,2> 4-/3| — 2,2,0>,	=/?|-2,0,2>-a|-2,2,0) .	(3.101)
The coef.icients a and /? are the same as for the other pair, but the corresponding
energies are shifted E'o^ = £l0/i — 2eV.
Andreev reflection causes transitions between the two sets of eigenstates i/o —> Фо-
The rate for this process can be obtained along the lines described in the previous
Subsection for an NSN transistor. An important modification arises as compared with
Eq. (3.70), since the charge transfer operators pick from the initial state the component
with zero charge on the island, which has amplitude a, and select from the final state
the component with two extra charges, which has amplitude (3. Hence the amplitude for
a Andree', reflection process between the states i/o and Vo with two electrons tunneling
from the states and k', j, of the normal electrode is
Ak„;'^o	52	.-«иЛ ( p ..1 F---H F—W—) 	(3.102)
The energy of the virtual intermediate state | — lfc, 19,0), with one electron added to tlie
island leading a quasiparticle in each electrode, is Ek,Q = Ech( — 1,1,0) -ek + Eq, where ck
and Eq = с2 4- A2]1/2 are the quasiparticle energies in the normal and superconductir
electrode, respectively.
The semination in Eq. (3.102) can be performed, and the rate for the Andreev
reflection process is obtained by the golden rule. After summation over the initial
states к and к' one finds for low temperatures and E’n - E\. = -2eV < 0
=	.	(3.103)
186 Single-Electron Tunneling
The rate is proportional to the product
which displays a typical resonance structure. The Andreev conductance Ga and the
function ад — a (A/[Ech( —1,1,0) — Eo]) have been defined in Eq. (3.75) and (3.73).
Here we assumed that the energy A + £’ch(—1,1,0) of the intermediate state lies above
Eq. If the superconducting gap A is much larger than the charging energies with scale
Ec the function a reduces to а и 1. It diverges if the energy of the virtual state
coincides with that of the initial state. In the other limit, where A 4- Ech(—1,1, 0) lies
below Eq, parity effects play a role (see below).
Andreev reflection processes can also lead to transitions between the other states
introduced above, with rates
TA(V>o -4 Vh) - а4 ao ~ [2eV - (Et - Eo)] ©[2eV - (E, - Eo)] ,
4ez
rA(tin -+ ^) = d4 a? ~ [2eV + (^ - Eo)] ,
4ez
rA (V>! -> ^) = (<W a2^2ev.	(3.105)
The function is defined similar as aQ, but the energy of the initial state Eo is replaced
by Ex.
Below the threshold voltage V < Ць, where
Vth = (E\ - E0)/2e ,	(3.106)
the only transition at low temperatures is the Andreev reflection between the states ipo
and The resulting current, Ires = -2еГА(?/>о	V'o), shows a pronounced resonant
structure due to the overlap of the functions a and /3. The conductance is
r	n2 p2
c"4=c‘tw^ torl'<v;i" |3107)
At higher voltages the Andreev reflection can take the transistor to the excited
state • A master equation yields the probabilities for the ground and excited states
po = Af	7'i > p. = 1 - W ° for v > Hh 	(З.Ю8)
ГА(^о -4- V>i) + rA(V>i -4 <)
The current then is
7 = —2е[Гл(^о ~* V’o) + ГА(^0	V;i)]Po — 2е[ГА(^1 —>	4- Гл(?/’i —> ^o)]Pi 
(3.109)
________________________________3.5 Charging effects and superconductivity 187
Fig. 3.17 I-V characteristic of an NSS transistor. A resonant structure due to Cooper-pair tunneling
is visible i i the dissipative current due to Andreev reflection.
For Д > Ec 4- Ej/2, near the resonance, the difference between an and ai is small. In
this case the current is a sum I — Ires 4- Дь, where Ires follows from (3.107), while
second contribution, which exists only above the threshold V > Vth, is
,	Ej	(2eV)2 - (SEch)2 - E'j
th 16e (SEchy + E2[(6Ech)2 + E2](2eV-6Ech) - E2eV
x© (2eV - y]{&EdlY 4- E2^ .	(3.110)
Also this current contribution has a resonant peak. A plot of the current-voltage-
characterstic as a function of the gate and bias voltage is shown in Fig. 3.17.
When the superconducting gap is smaller than the charging energy A < Eg, parity
effects plsy a role. If the gate voltage is such that the energy of the initial state coincides
with that of the virtual intermediate state the function «о diverges. This happens for
Qg = e ± 8Q*g, where
Clearly, at these points the perturbative treatment of Andreev reflection is no longer
sufficient. However, close to these points we expect a strong increase of the current.
Inside the window e — SQq < Qg < e + 6Qq, the ground state of the transistor is an
odd state with a single quasiparticle present in the island. In this regime the current
is much lower than outside. However, it is difficult to derive a precise value, since a
large number of channels contribute with similar weight.
188 Single-Electron Tunneling
Above a threshold voltage, when (e —Qg)2/2C' —eV/2 + Д < Eg, the odd state can
be reached in a SET process, and the Andreev reflection is again “poisoned” (compare
the NSN transistor). Again the odd electron tunneling creates a bottleneck for the
current, which is small /set = ey = [2e/?tM(0)ni]~1. Once SET is possible the current
related to Andreev reflection processes is negligible.
3.5.7 Incoherent Cooper-pair tunneling
Finally we consider situations where a mechanism is present which destroys the phase
coherence of the quantum mechanical time evolution. In this case Cooper-pair tun-
neling can be treated perturbatively as a stochastic process. A realization of such a
system is a circuit consisting of a Josephson junction in series with a voltage source
and an external impedance Z(u>). It is the same setup as shown in Fig. 3.9 except that
the tunnel junction has superconducting electrodes. The electrodynamic environment
can again be described by a suitable oscillator bath (see Section 3.4). An incoherent
Cooper-pair tunneling process is accompanied by a transition in the bath. In analogy
to Eq. (3.41) the rate for this process can be written as [40]
r+ = E Pbath(X) II2 5(EX - Ex.) .	(3.111)
xx,
Note that the phase in the superconductor, which is related to the voltage by Joseph-
son’s relation Tup ~ 2eV, differs from the phase Кф = eV introduced in Section 3.4 by
a factor 2, which accounts for the difference in charge transferred in the two cases.
The trace over the bath degrees of freedom can be performed, with the result
Г+ = f| [°° dtexp eVxi) (eMt)e“iv,(0)) .	(3.112)
h J—oo \ h /
Proceeding as before we can express the forward tunneling rate for a Cooper pair as
Г+= ^^Р(2еС) ,	(3.113)
where P(2el/) has been defined by
—	1 г 00	Г	P?/i
P(E) = T-/ dt exp 4A"(t) 4-i— .	(3.114)
Z7T/1 </—oo	L	J
This function P(E) differs from P(E) introduced in (3.46) by a factor 22 because of
the difference in the charge transferred. Except for this, much of the discussion given
in Section 3.4 applies here as well. The backward tunneling rate follows simply from
Г~(У) = Г+( — V), and the current is
/ср(V) = 2e[r+(V') - Г-(Т)] = ^i[P(2eV) - P(-2eV)] .	(3.115)
3.6 Effective-action description 189
The result depends strongly on the impedance. At low voltages the expansion
P yielcs /ср ~ y2/''»-'. In a high impedance environment, as = RK/R 1 the
superci rrent has a Gaussian peak at V — e/C.
т	r [ tt2(CV - e)2/2Cl
/СР(E) = /max exp-------------’-L--- .	(3.116)
Ecas J	v '
This feature has been denoted as “Coulomb blockade of Cooper-pair tunneling”. The
predicted voltage dependence has been observed by Kuzmin et al. [41].
3.6 Effective-action description
In the next two sections we will consider several aspects of transport through systems
of junctions in a path-integral formulation. It is a systematic approach to include
dissipation in quantum mechanics (compare Chap. 4 and the work of Caldeira and
Leggett [14,29]). It allows us to go beyond perturbation theory, if
at = RK/(47T2Rt)	(3.117)
is no lor ger small. (For later convenience a factor 1 /4тг2 is included in the definition
of at.) The path-integral formulation displays in a transparent way the interplay
of charging and tunneling phenomena. We will rederive the tunneling rate for sing’
electront, cotunneling and Andreev reflection, describe Cooper-pair tunneling, but also
further effects like the proximity effect and resonant tunneling. We first will review the
imaginary-time approach, starting from the work of Ref. [6,7] and include extensions
discussec in Ref. [8,9]. This approach is appropriate if we are interested in equilibrium
properties such as supercurrents or the proximity effect. But we will also be able to
draw conclusions about the tunneling rates. In Section 3.7 we will then present a reed-
time pat i-integral formulation, which yields directly tunneling rates and currents and
allows us to describe time-dependent and nonequilibrium phenomena.
3.6.1 The effective action in imaginary times
Our aim is to study transport through systems composed of normal metal or super-
conducting tunnel junctions. A typical geometry is the one known from the transistor
consisting of leads and an island, which is coupled capacitively to a gate voltage. The
electrostatic charging energy of the system is given by Eq. (3.2). Tunneling across
the junctions is described by the tunneling Hamiltonian (3.9). We consider “wide”
metallic junctions, which implies that there are many transverse channels. As a result
“inelastic ’ higher-order tunneling processes, involving different electron states for each
step, dorr inate over those higher-order processes which involve the same state repeat-
edly. Acc irdingly, in the effective-action description presented below only simple loop
diagrams have to be retained.
We an: interested in the influence of charging effects on the properties of the junction
system. Since the tunneling of electrons changes the charge on the island, the voltages
190	Single-Electron Tunneling
are fluctuating quantities. They are related by the Josephson relation to the phases of
the superconducting order parameters in the electrodes, <^>j (j = L,R), and that of the
island, p, if they are superconducting. When the electrodes or island are normal we
still define a phase as the integral of the voltage
<^(т) = f (1т'2еУ(т')/Ь ,	(3.118)
Jo
and similar for the electrodes. A voltage drop at a junction interface can be accounted
for by phase factors ехр{±г[9?(т) — 9?j(t)]/2} multiplying the tunneling matrix elements.
For definiteness we study in the following a transistor with normal or superconduct-
ing electrodes and assume that the voltages of the electrodes are fixed. In this case
only fluctuations of the phase of the island need to be considered. After elimination
of the microscopic electronic degrees of freedom the partition function of the junction
system can be expressed as a path integral over this phase [6, 7]
Z — У d<y5(r) exp{-S[9?]//i} .	(3.119)
It is governed by an effective action, which in an expansion in the tunneling matrix
elements can be written as
— Sch + St + Sj 4- ^Andreev + 5sNS •
(3.120)
The first term in the action is the charging energy (3.2), rewritten in terms of the phase
C f h dp\2 h dp
2 \2e dr J G 2e дт
(3.121)
The remaining contributions will be discussed below, term by term.
The question arises, whether the phase is to be viewed as an extended variable
defined in the range —oo < p < oo or whether it is defined on a ring. Both inter-
pretations are possible. The first describes a system where charges can change also in
a continuous fashion, for instance because of the additional flow of Ohmic currents.
The second describes a situation where the charges are quantized, for instance in the
island of a SET transistor [7]. In the latter case the path integral for the partition
function includes a summation over winding numbers p(fifl) = <p(0) -I- 4ттМ, where
M = 0, ±1, ±2,..... (Because of the factor 2 in the definition of p(r) suggested by the
analogy to superconductivity, the ring has circumference 4тг.) The second term in the
charging action (3.121) has a meaning only in the latter case.
3.6.2 Single-particle and Cooper-pair tunneling
The second term in (3.120) describes single-electron tunneling. It is [6,7]
c h [h0 ,	।	l 1 Ч Г^(т) - ^l(t')
= 7«t / dr/ dr Gl(t ~ т )Gi(t - r) cos ---------------------
4 Jo Jo	2
(3.122)
3.6 Effective-action description 191
Fig. 3.18: Different processes contributing to the transport through a SET transistor: a) “bubble"
diagram d< scribing single-electron tunneling, b) Cooper-pair tunneling (if L and I are superconduct-
ing), c) “banana” diagram responsible for Andreev reflection (between a normal I and superconductor
R), d) “sausage” diagram describing the correlated tunneling of a Cooper pair through both junctions
(in an SNS transistor).
which depends on фь = ip - <pL, and similar for the right junction. In a diagrammatic
language, which allows us to keep track of the different contributions to the action, St
corresponds to the “bubble” diagram shown in Fig. 3.18 a. It contains the two diagonal
quasicl ass ical Green functions, Gl/i(t) s i/N(Q) f d3p GL/i(p, t), of the left side and
the island. Their Fourier transforms are given by G(w„) =	+ A2]1/2. In norma,
metals G(r) = ’lifT, while in a superconductor it decays like G(r) к exp(—Д|т|).
At this stage we do not want to present a derivation of the expression for St (which
is given in Refs. [6,7]). Rather we stress the similarity and differences to the term
describing dissipative currents through an Ohmic resistor derived by Caldeira and
Leggett [14] from a harmonic oscillator bath:
(i) If both electrodes are normal the product of Green functions is proportional
l/т2 and coincides with the kernel found for Ohmic dissipation (see Chap. 4).
one or both electrodes are superconducting the kernel depends on the supercon-
ducting gap(s), which accounts for the reduction of the subgap current below a
linear voltage dependence in this case.
(ii) The second difference is the trigonometric dependence on the phase difference in
contrast to a quadratic one of the Caldeira-Leggett action. Also this difference
accounts for different physics. In the present problem the current is due to
discr rte single-electron tunneling rather than a continuous flow of charge through
a res stor.
With suitable analytic continuation we can rederive from St the single-electron or
quasipartide tunneling rates (3.13) and (3.68). Since these results are well-known we
192 Single-Electron Tunneling
do not discuss them here further. Rather we will demonstrate the limiting behavior,
which applies for an ideal SS or SN junctions, where at low voltages, eV Д, and low
temperatures the quasiparticle tunneling is suppressed (vanishingsubgap conductance).
In this case, if the time evolution of the phase is slow on the scale given by the inverse
energy gap ft/Д, the quasiparticle tunneling term 5t can be expanded to quadratic
order in дф/дт. This implies that tunneling effectively renormalizes the charging energy
and hence the capacitance Се« = С + 6C. For an SS junction the result is 6Css =
(3TT2/16)ate2/A [6], for an ideal SN junction the equivalent result is <5Cns — 47rate2/Д.
The next term in the action (3.120) is again of second order in the tunneling,
•5j.l = 7«t / dr/ dr Fl(t - т )Fi(t - t) cos -------------------- .	(3.123)
4 Jo Jo	[2
It involves a product of two off-diagonal quasiclassical Green functions F(r) with
Fourier transform F,(wp) = Д/[и>2 + Д2]1//2. This term is appropriate for an SS in-
terface and describes the Josephson tunneling. Diagrammatically, it corresponds again
to a “bubble” diagram, shown in Fig. 3.18 b, where the two propagators are off-diagonal
Green functions F. If the phases evolve slowly on the scale given by the inverse energy
gap the Josephson coupling can be simplified to
/Л/3
Sj^-Ed drcos[0L(r)] ,	(3.124)
Jo
and in this form is equivalent to the Hamiltonian (3.94).
3.6.3 Higher-order processes
In situations where quasiparticle tunneling is suppressed by the superconducting gap
higher-order processes may be observable. An important example is the correlated
tunneling of two electrons across a junction with a normal and a superconducting
electrode. Such higher-order correlated processes require a careful analysis of the space
correlations of the electron propagators. We, therefore, keep track of the location of
the tunneling process and, accordingly, specify the tunneling Hamiltonian to describe
tunneling at the junction interface (г = 0) only by choosing
Trr = T 6(r - r')5(z) .	(3.125)
Notice that this tunneling matrix element differs from the usual approximation, where
Tk,q is assumed to be independent of к and q and hence Tr r> = T <f(r)<i(r'). Using
(3.125) we can rederive the bubble diagram of Fig. 3.18 a. The comparison of results
in both formulations yields the relation between the old and new matrix elements and
the normal state tunneling conductance
(зл26)
Here A is the junction area and a a numerical coefficient of order one.
3.6 Effective-action description 193
We cm now study higher order terms. In NS junctions with vanishing quasipar-
ticle currant the leading term is the “banana” diagram, shown diagrammatically in
Fig. 3.18 c. Two electron propagators on the normal-metal side are connected to off-
diagonal propagators on the superconductor side (for definiteness we assume that the
lead electrodes are superconducting while the island is a normal metal). This diagram
describes Andreev scattering across the interface. It yields the fourth term in the action
(3.120), which for the right junction becomes [20]
/ h0 rh0 rhff rh(3 f f f	f
= [	dr,'/o
x|f|4FR(p2,Pi;72 -	-T'dF^p^p'^T^ - T2)G^(p'2, р2-,т2 - r2)
z cos	(3127)
Here the modification of the tunneling Hamiltonian is apparent. The propagators
connect the positions p in the junction plane where the tunneling processes occur.
(They are now full Green functions, in contrast to the quasiclassical functions which
appeared in the expressions (3.122) and (3.123).) If we restrict ourselves to low voltage^
eV <S. A the short range of the off-diagonal Green functions F(p, r) in space and time
and their rormalization allow us to write
rh.0	Г f	~
Sa,i= / dr/ dr' / d2p / d2p'|T|4G](p, p';r - t')Gi(p',p;t'- r)
Jo Jo	J A	Ja
xcos[d>R(r) - 0r(t')] •	(3.128)
If the Iliads are made of dirty metals we perform an impurity averaging, introducing
the Cooperon propagator introduced in Chap. 1, K(p,p'\r — t') — (Gi(p,p'-,r —
t')Gi(p' , р.т' — r))imp. It satisfies a diffusion equation and depends on the geometry
of the system. For illustration we consider a very small normal island in the absence of
pair breaking effects such that electrons can propagate phase coherently between each
pair of points. The integral of the Cooperon propagator then reduces to A2/Qi and we
find [20, 21]
rfiP rhi3	|
5a,r = GaFk/ dr/ At'----------------— cos [0r(t) - <^r(t')] ,	(3.129)
Jo Jo (t — г )
where Ga is given by (3.74) with Nch = 1. In general, the Cooperon propaga'
controls the range in space where electron propagation and hence the higher-orc i
tunneling processes - remain correlated. This allows us to evaluate the effective number
of channels introduced by [19]. (The notation is suggested by the following picture:
If g0 denotes the dimensionless conductance per channel, then the total conductance
is g = NChg0 and the effective (dimensionless) Andreev conductance g^A ~ Nchgg —
g2/NCii.) T io problem has been analyzed in Ref. [21] for different junction geometries,
including such geometries where interference effects in a magnetic field play a role.
194	Single-Electron Tunneling
The comparison of (3.128) and the quasiparticle term (3.122) reveals that in the low-
voltage limit Andreev reflection and the associated charge transfer are very similar to
ordinary single-electron tunneling in the normal state. The difference is that instead
of e the charge 2e is transferred as can be seen from the missing factor 1/2 in the
argument of the cosine. This implies that two-electron tunneling is subject to the
Coulomb blockade in much the same way as single-electron tunneling [19, 20].
3.6.4 Josephson current through SNS transistors
The last term in the action (3.120), illustrated by the “sausage” diagram in Fig. 3.18
d, describes the correlated tunneling of two electrons through both junctions. It is
responsible for the Josephson coupling across an SNS structure (with tunnel barriers
between the metals). For vanishing transport voltage = VR = 0 it is
rh.0 chQ f n f	f	f
Ssns = dn dr2 / dr[ / dr2 / d2pt / d2p2 / d2p( / d2p2
Jo Jo Jo Jo	J Aj_,	J Al JAr JAr
x|7j4FL(p2,Pi;72 -	p'l'Ti - т[)Рк(р\,р'2-,т{ - Т^вг(р'2,р2-,Т2 - т2)
(P(ti) - (р(т[) - (р(т2) + (р(т2)
-----------------------------------------<Pl + f/’R
(3.130)
The short range of the off-diagonal Green functions in space and time allows us to set
т2 « n, p2 « px and equivalent relations for the primed coordinates. Hence, (3.130)
can be simplified to
Ssns= dr/ dr'/ d2p d»4^,^ — r')Gi{p', p-,r' -t)
Jo Jo	J Al	J Ar
x cos [<£>(r) - 79(7-') -	+ V’r]  (3.131)
Impurity averaging introduces again the Cooperon propagator through the normal
island.
The term Ssns contributes to the free energy F of the system, from which we obtain
the supercurrent
2e dF
hfidfa ~ Pr)
/ dr dr' d2p
Jo Jo Jal Jar
x(sin[^(7-) - <p(r') -	+ ^R.]>sch 	(3.132)
The supercurrent through a classical SNS structure without charging effects has been
studied before by Aslamazov, Larkin, and Ovchinnikov [42]. Charging effects introduce
the phase correlation function in Eq. (3.132). In an expansion in the tunneling matrix
3.6 Effective-action description 195
Fig. 3.19: Typical path in the charge representation. Single-particle tunneling events occur at times
т and t1 before the system returns to its original state by a 2e-charge transfer at time t0. The charge
paths are periodic with period 0.
elements it is sufficient to evaluate the correlator (expi[ip(r) — <p(r') — ipL +
with the charging energy Sch, given by Eq. (3.121). This is done most easily in the
charge re rresentation. The factor exp{i[<p(r) —	} describes the transfer of a Cooper
pair in ti e left junction, increasing the island charge at time r, while exp{—i[^>(r') —
ipR]} describes the transfer of another Cooper pair in the right junction decreasing the
island charge at time r' (see Fig. 3.19, which visualizes a more complicated process
relevant for the proximity effect). Hence, for vanishing transport voltage, V = 0, we
have
i °°
— VC')] s = 1 У) е-{/3£гЬ(п,<?о) + [£<л(п±2’<?с) + -Е<.|,(п,<?с,)]С--т')}	(3 133)
^Ch n=-oo
where Zrh is an obvious normalization. At low temperatures the sum is dominated by
that 2c chirge transfer process which costs the lowest energy difference Ech (n ± 2, QG)~
Ech (n, Qc ), and we have
(е‘Мт) ^(T,)l)sch ~ max {exp [- (Ech(n ± 2, Qg) - Ech(n, Qg)) (t - r')]j .	(3.134)
If the island is small enough and the external time scales long enough the Cooperon
propagator reduces again to a simple factor A2/fy. In this case we can generalize the
result derived in Ref. [43] to obtain the following expression for the critical current
rcrit
7SNS
2e	A 1 V 1 [ д2
h 16/?t,L/?t,R € 2 °g _тг27’2 + (<JECh(Qc;))2
(3.135)
where <5Er ,(Qq) is the energy difference dominating in Eq. (3.134), and 6e :x 1/QjM(0)
is the level spacing in the island. In the absence of charging effects, <5E)h = 0 we recover
the result of R.ef. [42], which diverges at zero temperature indicating a breakdown of
196 Single-Electron Tunneling
Fig. 3.20: Pair amplitude in a small island induced by the proximity effect. A normal island is
coupled to a bulk superconductor. The gate voltage is chosen such that Qq = 0, and the parameters
are Ес/Д = 0 (upper curve), Ес/Д = 0.23 (middle curve), and Вс/Д — 0.45 (lower curve).
the perturbation theory. Charging effects reduce the supercurrent, and at the same
time regularize the divergence.
To finish our systematic enumeration of fourth-order terms in the tunneling, we
could discuss the inelastic and elastic cotunneling terms [44] using the same technique.
Instead we will first comment on the influence of charging on the proximity effect and
return to cotunneling processes in a real-time formulation of the next Section.
3.6.5 Proximity effect
The essence of the proximity effect is a nonvanishing pair amplitude (V'nV’n) in a normal
metal which is in close neighborhood to a superconductor. If both are separated by
a tunnel junction a nonvanishing pair amplitude arises because of tunneling. Since
tunneling is affected by charging, the proximity effect is as well.
We obtain an expression for pair amplitude in the normal metal by adding a source
term to the Hamiltonian, Яд = fg dr J d3r A(r, т)^м(г, r)^(r, r), and taking the func-
tional derivative т0)^м(г,To)) = — 5ln(Z)/5A(r,tq)|л_0. Proceeding as before
in an expansion in the tunneling we can express this contribution as
Qt 7T^ f d3D	('h0
^(r,T0^(r,T0))^——r ~z dr dr’G(p,T)G(p,-t )F(t - т )
/vN(0) J (2тг)3 Jo Jo
<^(t) + <^(r')
- HTo)
(3.136)
Diagrammatically, (3.136) is shown by the “proximity” diagram, (inset of Fig. 3.20).
We assume that the phase ips of the superconductor remains constant in time. In com-
parison to the classical result for the proximity effect discussed by Azlamazov, Larkin,
and Ovchinnikov [42] there appears a time dependent phase correlation function, which
___________________________________________ 3.6 Effective-action description 197
Qg
Fig. 3.21 Pair amplitude in a small island induced by the proximity effect as a function of the gate
voltage, Q,. The island is superconducting and parity effects reduce the regime of the odd state of
the island. The parameters are Ec/Д = 0.45 (upper curve) and Ec/A = 1.8 (lower curve).
is averaged according to the dynamics of the system. Since we couple to an off-diagonal
Green function the phases add in the exponent.
To lowest order in the tunneling the expectation value in Eq. (3.136) can be eval-
uated wilh the charging energy (3.121), which again is done most conveniently in the
charge reoresentation. The factor ехр[г<р(т)/2] describes the transfer of one charge at
time т and ехр[г<р(т')/2] of another one at time t' before both of them are returned
by exp[— ^(r0)] at time r0, see Fig. 3.19. The result is
(cos --------------------^(t0) )
\	2	/ s,
(3.137)
e-[Bch(n)(/i+r-To) + Bch(n + l)(r'--r) + r;c|l{n+2)(To-T')]
for т < t' < t0. Similar expressions hold for other relations between r, r' and To-
Eq. (3 137) shows that the modification of the pair amplitude depends on the tem-
perature and on the applied gate voltage. Results displaying the influence of charging
effect on the proximity effect are shown in Fig. 3.20 and Fig. 3.21. The gate voltages
can be us'd to modulate the proximity effect and hence the supercurrent in suitable
normal m ?tal-superconductor systems [45].
Summirizing we can say that the effective action displays in a systematic and
transparent way the interplay of charging and transport properties. The latter includes
single-electron and Cooper-pair tunneling, but also several extensions as the proximity
effect, Andreev reflection and the supercurrent through composite structures. We have
reproduce ! several classical results. As long as the tunneling is weak, charging effects
lead to extra phase correlation functions multiplying the classical expressions. Charging
198 Single-Electron Tunneling
suppresses the currents and proximity effects, but at the same time the gate voltage
can be used to modulate these effects.
We considered here “wide” junctions, where a large number Nch 2> 1 of parallel
channels contributes to the transport. As a result, higher order diagrams, such as the
Andreev contribution, carry extra powers of l/jVch as compared with higher powers of
the simple-bubble diagrams St and Sj. From a comparison of the Andreev conductance
and the single-electron current in the experiments of Hergenrother et al. [22], we
conclude that even in these small fabricated junctions Nch is of order 103. Therefore,
wide junctions with conductances as large as 1/7?K are still described by the action
presented above including only simple bubble diagrams. Only where the lowest-order
effects are suppressed by a combination of Coulomb blockade, superconducting gap
or parity effects or where further physical effects such as the Andreev current are of
interest, the appropriate higher-order diagrams need to be considered.
For a renormalization group analysis of the effective-action description of normal
junctions with large conductance we refer to the article [46]. As a result of strong
tunneling various parameters get renormalized, and, e.g., the steps in (n(QG)) in the
electron box are broadened. We will derive several of these results as well as several
extensions in the real-time analysis of the following Section.
3.7 Real-time evolution of the density matrix
From the imaginary-time description presented in the preceding Section we derived
equilibrium properties, such as the supercurrent or the pair amplitude in the proximity
effect. For a systematic analysis of transition rates and I-V characteristics of driven sys-
tems with strong charging effects we present now a real-time path-integral formulation.
We recover the classical (perturbative) description if the resistance Rt of a single barrier
is much higher than the quantum resistance Rk = h/e2, i.e. for at = /?«/(47r2/?t)	1.
In this regime, transport occurs in sequences of uncorrelated tunneling processes with
rates which can be obtained in lowest-order perturbation theory. When the dimen-
sionless conductance at is not small, at very low temperatures or near resonances the
classical description breaks down. Quantum fluctuations and higher-order coherent
tunneling processes become important. We have discussed already cotunneling, where
two electrons tunnel coherently in different junctions, thus avoiding the Coulomb block-
ade. Beyond this resonant tunneling, where electrons tunnel coherently back and forth
between the island and the electrodes, plays a role. This phenomenon is well-known
in situations where the electrons can be treated independent. Here we encounter two
complications. One lies in the fact that the metallic system contains many electrons.
With overwhelming probability different electron states are involved in the different
transitions of the coherent process. The second arises since the Coulomb interaction is
strong and, hence, cannot be accounted for in perturbation theory.
In this Section we study the time evolution of the density matrix and develop a
systematic diagrammatic technique to identify the processes of sequential tunneling,
3.7 Real-time evolution of the density matrix 199
inelastic cotunneling and resonant tunneling. A systematic formulation has been pre-
sented in Ref. [47] where after a separation of charge and fermionic degrees of freedom
a many- oody expansion technique has been used. Here we reformulate it in a real-time
path-integral representation [10]. The latter is familiar from the work of Feynman
and Vernon [48] and Caldeira and Leggett [49] who studied dissipation in quantum
mechanics. Dissipation associated with tunneling of electrons was investigated in Refs.
[6,7]. Aa essential step in the present work is a transformation of the path-integral
description of electron tunneling from a phase to a charge representation [7].
3.7.1 Phase representation
The time evolution of the density matrix involves a forward and a backward propagator,
both of v hich can be expressed as path integrals. After elimination of bath or electronic-
degrees of freedom the two propagators are coupled. This has been described 1
Feynman and others [48,49] for the case where a quantum degree of freedom is coupled
to a harmonic bath. In Refs. [6,7] the equivalent procedure for the case of a tunnel
junction has been described, where the electronic degrees of freedom are eliminated.
For ti ansparency we describe the formalism for a normal SET transistor with ap
plied gats and transport voltages. The time evolution of its reduced density matrix
can be expressed as a path integral over <p(t), which is defined as the integral of the
island voltage in analogy to Eq. (3.118) (in order to avoid confusion we retain the factor
2 in the definition). The phase is the conjugate variable of n, the number of excess
electrons on the island. The phases of the reservoirs r = L, R are fixed by ipr = 2eVrt,
without further fluctuations. Here and in the following we put h = A.-B = 1. The two
time-evol ition operators require that we introduce two variables with a = 1,2 for
the forward or backward branch '. Then we have
p(if! <7if, <F2f) = /'d'Fii d<^2i / WiW /	P^2(t)e1S|*’1'*’2)p(fi; F2i) 	(3.138)
The effect ive action is given by
S'l'Fi, F2] = S<-h['Fi] - SchfFz] + 5t[Fi, F2] •	(3.139;
The first wo terms represent the charging energy
S'chK] = [ dt у	•	(3.14(1)
Jt, [ 2 \ 2e / 2c
In system! with discrete charges, which can be tuned by a gate voltage Q(;, the inte-
grations over the phases Fa include a summation over winding numbers. For instance,
’In orde- to make contact with the classical limit or the Wigner distribution the two variables
and Fr(t) referring to the forward and backward paths, respectively, are frequently replaced by
center of mass and relative coordinates, ф = (fi + </?г)/2 and X ~ Vi ~ V'2- The action in Ref. [5] has
been written explicitly in terms of ф and x- For the present purpose it is more convenient to retain
the original form.
200 aingle-Electron mnneling
in a trace we have	+ 4тгт; m — 0,±l,... [7]. In this case the last term in
SCh does not vanish.
Electron tunneling in junction r, which is assumed to be a ‘wide’ junction with a
large number of parallel channels, is described by [6,7]
St,!#!,!-, <^2,r] = 4?ri	[ dt' ar’a\t - t') cos	,	(3.141)
a,a' = l,2
where фа,т = <pa — is the appropriate phase difference. The tunneling term couples
the forward and backward propagators. This arises in the step where the microscopic
degrees of freedom are traced out. We notice at this stage that the effect of transport
voltages can be absorbed by a shift of the arguments of the tunneling kernels in Fourier
space w —> oj — eV’r. In this case the argument of the cos-function depends only on
^CT(t), and the kernels a(T,cr are given in Fourier space by
=	(3.142)
and
±, 4	,	ai-eVr
a, w = ±at ,-------r------------г---.
’ exp[±/?(w — eVr)] — 1
3.7.2 Charge representation
An important step for a systematic description of tunneling processes is the change
from the phase to a charge representation , accomplished by
p(tf;7Jif,n2f) = p(ii;nii,n2i) / d<pifd^2fd^iid^2i
nii,n2l	J
/W	/V2f	Г	Г
x /	/ Vn^t) / Vn2(t)	(3.144)
x exp
/ i Л	.-or 1	• [ 1. Фa \
z j ( 1) I i q i ^ст,г n 4~ i *^ch |/2<tJ	i /	^<7	1
r__j 2	'	"	“	J	£ /
x exp
-2тг£ £ f dt (lt - t') exp (i ^(4„ . ^'(f)
r <r,<r'=l,2 l' l'	\	2
The forward and backward propagators involve the charging energy exp(±iSCh[n<r]),
where SChizl] = ft' dt(ne — Qg)2/IC.
To proceed we expand the tunneling part of the action exp(iSt) to arbitrary orders
and integrate over tpa. Each of the exponentials exp[±i^tJ(t)/2] causes a change of the
number of electrons on the island by ±1 at time t on the forward or backward branch,
a — 1 or 2, respectively. These changes occur in pairs and are connected by tunneling
lines representing ar’’“ (t — t')- The two correlated transitions can occur on the same
3.7 Real-time evolution of the density matrix 201
or on different branches. The terms of the expansion can be visualized by diagrams;
an example is displayed in Fig. 3.22. It shows the closed time-path corresponding to
the forward propagator from tj to tf (upper line) and the backward propagator from
if to tj (lower line). Vertices, describing tunneling, are connected in pairs by dashed
tunneling lines either within one propagator or between the two propagators.
The Litter are of particular interest. Imagine we started in a diagonal state with
n charges /?(£,) = |n)(n|. Then a transition, described by exp(i[yi,r(t) — <p2,r(t')]/cl
changes the charge on both branches by e due to tunneling in junction r, and the
density matrix acquires a finite value also for states |n + 1). After integrating over the
two timet t and t', limited by tj < t1 < t < tf, we find
(n + 1 |p(tf)|n + 1) = (tf - ti)27rar+(<5E’ch(n)) ,	(3.145)
where 6E,h(n) — Ecb(n + 1) — E'ch(u) (notice that eVr is absorbed in the definition of
ar). Obviously we can interpret the coefficient of the time difference as transition rate,
and indeed we reproduce the well-known single-electron tunneling rate.
3.7.3 Diagrams and rules
The time evolution of the density matrix is visualized in Fig. 3.22. It is expressed by the
sum of all topological distinct diagrams with directed tunneling lines. The diagrams
are evaluated according to the following rules:
1.	Assign charge states n and the corresponding charging energy to each segment of
the propagators. Segments of the forward (backward) propagator between t' and
t > i1 carry factors exp[:FiE’Ch(u)(t — t')].
2.	Each vertex represents an exponential exp[±i^CT(t)/2] of the tunneling contribu-
tion to the action. It changes the charge by An = ±1.
3.	Pairs of vertices are connected by a directed tunneling line a+(t —t') [07 (t —t')] for
the electrodes r = L, R, if the line of is running backward [forward] with respect
to the closed time-path. The charge changes in units A?z = +(—)1 along the
time path by 1 if the tunneling line is directed towards (away from) the vertex.
4.	Each diagram carries an prefactor (—i)M(—l)m, where M is the total number of
vertices and m their number on the backward propagator.
5.	Integ -ate over the internal times and sum over the reservoirs.
In order to calculate stationary transport properties it is convenient to change to
an energy ^presentation. For this purpose we order in each diagram the vertices from
left to right and label them by ij, irrespective on which branch they are. We further
set t, = —co and if =	= 0- We then encounter integrals of the type
Z0	fO	11
dt,... /	dtAfe"(1eiAb’,((2^(1) •е“‘Д£м'м =	------ ------—----------.
-00	+ 7/	—+ 7/
202 Single-Electron Tunneling
Fig. 3.22: Example of a diagram showing various tunneling processes: on the left sequential tunneling
in the left and right junctions, then a term which preserves the norm, next a cotunneling process, and
on the right resonant tunneling processes.
Here Д£) is the difference of the energies of the upper and lower propagator and -
if present - the frequency of the tunneling line within the segment limited by tj and
tj+l. The convergence factor eT,t' (77 —> 0+) describes the adiabatic switching of the
tunneling. The rules in energy representation read:
1.	Draw all topological distinct diagrams. These are the same as in time space. In
addition to the charging energy assigned to the propagators we assign a frequency
w to each tunneling line.
2.	For each segment derived from tj < t < tj+i with j > 1 we assign a resolvent
l/[AEj — izy] where is the difference of the energies of the forward and
backward propagator, plus the sum of the frequencies of the tunneling lines in
the given segment. The latter have to be taken positive for lines from the left to
the right and negative for lines from the right to the left.
3.	The prefactor is given by ( —l)m+;, where m is the total number of vertices on the
backward propagator and I the total number of resolvents.
4.	For each coupling of vertices we write a+(w) [«^(w)], if the tunneling line of
reservoir r is going backward (forward) with respect to the closed time-path.
5.	Integrate over the frequencies of tunneling lines and sum over the reservoirs.
We denote the sum of all diagrams by П"1’",1, where the indices indicate the left
and right charge states on the two branches, <7 = 1,2. They can be expressed as an
iteration in the style of a Dyson equation, illustrated in Figs. 3.23 and 3.24, by the free
propagator H(o)"‘ and an irreducible self-energy part S"'’",1, defined as the sum of all
diagrams where any vertical line cutting through them crosses at least one tunneling
line. Hence
с 5=nw:;<5ni,n/n2,ni + e n-;n4	. (з.ыб)
We start from a density matrix which is diagonal, p(tj, ni, тг2) = p(0)(ni )<5n, iI12. This
means it remains diagonal except during transitions described by S. In this case we can
drop the upper indices and write, for instance, Sn.n, = We identify the solution
3.7 Real-time evolution of the density matrix 203
Ilj	n’| Dj	П] rij	n’’	rij
^2	^2	^2	^2	^2	^2	n2
Fig. 3.2J: The iteration of processes for П, describing the time evolution of the density matrix.
Fig. 3.24: Lowest-order approximation of the self-energy £(l), defined to contain no overlaj .
tunneling hues. Only one representative of each class is shown, the remaining ones are obtained
by changing the direction of the arrows and exchanging the position on the forward and backward
propagator.
of Eq. (3.146) - after multiplication with p<°)(n) from the left - as the stationary
distribution function )>2n//оцл)Пп,п' = pst(n'). Hence
ps4T) =£’(«') + £pst(7t")Sn-',n-n^ .	(3.147)
n"
The last term in this form, ос l/гр, describes a propagation in a diagonal state
with ДЕ == 0. Hence we have £n' pst(n')^n',n = 0, and using symmetry arguments we
can show that £n' ^n,n' = 0- We thus arrive at
o = -Pst(n) £ Sn>n, + £ pst(n')S„n„ ,	(3.148)
n'/n	n'/n
i.e. we reover the structure of a stationary master equation with transition rates
given by In general the irreducible self-energy S yields the rate of all possible
correlated 1 unneling processes. We, furthermore, see that the stationary solution pst(n)
is indepencent of the initial distribution
3.7.4 Simple examples, SET and cotunneling
For illustration, we evaluate all diagrams which contain no overlapping tunneling lines,
as visualized on the left hand side of Fig. 3.22. After each transition the system is in a
diagonal state of the density matrix and a probability can be assigned. These processes
are also described by the master equation. In this simplest case the irreducible self-
energy parts
££ -2ю££(±ад	,	= -2тп££а*(±<5Е±),	(3.149)
r	± r
204 Single-Electron Tunneling
where <5E± = ECh(n ± 1) — Ech(n), coincide with the single-electron tunneling rates
derived within lowest-order perturbation theory above.
In situations where single-electron tunneling is suppressed by Coulomb blockade the
lowest-order contribution to the current arises due to cotunneling. It is described by a
diagram in Fig. 3.22 with tunneling processes in the left and in the right junction, where
the corresponding lines Ol(<l - ^l) and ~ ^k) overlap in time. This means there
is no intermediate state where the density matrix is diagonal, which would describe
sequential tunneling. Performing the integrations we find the cotunneling rate (3.22).
3.7.5 Resonant tunneling
The perturbative approach breaks down at low temperatures or for large values for
the dimensionless conductance at. Specifically we will show that the classical master
equation is valid only for at In (EC/‘2~T)	1, whereas for larger values resonant
tunneling processes become important.
To proceed, we have to find a systematic criterion which diagrams should be retained
and summed. For this, we note that during a tunneling process the reservoirs contain
an electron excitation. Two parallel tunneling lines imply two such excitations. Our
criterion is that we take into account only those matrix elements of the total density
matrix, i.e. reservoirs plus charge states, which differ at most by two excitations in the
leads or (equivalently) in the island. Graphically, this means that any vertical line will
cut at most two tunneling lines.
Furthermore, we will concentrate here on situations where only two charge states
with n = 0,1 need to be considered. This is sufficient when the temperature, the
energy difference of the two charge states Ao = ЕД)!) - ECh(0), and the bias voltage
eV' = eVL — cVr are small compared with Ec, which is the energy associated with
transitions to higher states. The combination of the two restrictions implies that no
diagram contains crossing tunneling lines. In this case we can evaluate the irreducible
self-energy analytically.
In order to sum all diagrams which contain up to two parallel tunneling lines we
introduce a diagram labeled by (г, w) (see Fig. 3.25). It has an open tunneling
line associated with tunneling in the junction r carrying the frequency w. Consequently
it remains in an off-diagonal state at one side. For the two-state problem we need only
<Mr,4 = Co(r,^)	(3.150)
with n = 0,1, for which we can formulate the iteration depicted diagrainmatically in
Fig. 3.25. It yields
d>?i(r,Lj) = тг(се)
a+(w)<5n,o
- ar (w)«5n,i + ar(w) / dw'
ФпФ^Т
uj' — W — U)
(3.151)
Here we encounter the propagator тг(се) = H(1,Jq(u>). It is given by the propagator
П, restricted to first order in the tunneling, starting and ending in an off-diagonal
state. Furthermore, since the parallel tunneling line carries a frequency ш the energies
3.7 Real-time evolution of the density matrix 205
Fig. 3.25: a) Self-consistent equation for ф„(г,ш). A summation over the electrodes r and the
direction cf the tunneling lines is implied, b) Representation of the self-energy l within our
approximation.
of the upper and lower lines are shifted relative to one another. Notice that du
to the restriction to a two-state problem there are no diagrams contained in „
where a tunneling line connects the upper and lower propagator. We can express it
by the first-order self-energy cr(cc) = £(^0’0(cj), which is the off-diagonal version of
the expression known from the first-order discussion, with the added complication
parallel tunneling line with frequency w.
The iireducible self-energy S is obtained from ф(г, w) by connecting the tunneling
line with the appropriate direction to the upper and the lower propagator and adding
both contributions (see Fig. 3.25). We get for n = 0,1
(3.152)
while S,li( follows from the rule Snj0 + Sn,i = 0.
Apply ng our rules for the calculation of the diagrams we find
/ A	1	/ X L ,
г (ал — —--------—------ , <т(ал — - aa> —-----------------------
a; — Aq — cr(w)	J a? — a> — ir/
(3.153)
Here and for the following we introduce the notations а±(а;) —	ar(aj) =
а-+(а;) + <<r(w), a(u>) — а+(а;) + a“(w) = Er<>,5 L and ~	The integral
equation (3.151) is solved by
So,i
= -So,o = 2^i^±
A
Si,о — — Si,i — 2?ri ——	(3.154)
A
with
A± = / do;a±(u')|7r(a>)|2
A = / do;|тг(а?)|2.
(3.155)
Inserting Jiese quantities in the kinetic equation (3.148) we arrive at the stationary
probabilit es Pg1 = A. and Ff1 = A+ . Both are positive and normalized Fost + Pfl ~ 1.
206 Single-Electron Tunneling
3.7.6 The current
The expression for the current at time t in the junction r can be written as
/rst(Z) = 4тг1е [	,	(3.156)
У-oo a
where the expectation value is taken with the reduced density matrix discussed above,
and t = tf. We, therefore, have to evaluate the two real-time correlation functions
describing charge transfer processes at times t and t’
G>(t,t') =-i (e“i*?(t)ei*?(t'))	,	G<(t, t') = i (ei*?(*')e^i*?(t)) .	(3.157)
In the stationary limit the current can be expressed by
Irst = —ie У div |arb(iv)G>(iv) + a7(w)G<(w)|.	(3.158)
We, furthermore, introduce a spectral density for charge excitations
A(w) = -^[G<(m)-G>(iv)].	(3.159)
27Г1
The correlation functions and spectral density can be evaluated in the approximation
which we have used before, with the results
G<>)(w) =(t( 27riJ2ar(iv)/[(t)(iv - eVr)]|тг(ш)|2	(3.160)
Г
and
A(iv) = a(w) |tt(cu)|2 .	(3.161)
The current can then be written as
If - ^4тг2 Idw£	- eVT,) eV,.)] .	(3.162)
These results satisfy the conservation laws and sum rules. The current, is con-
served, J), 7rst — 0, and vanishes in equilibrium when Vr = 0. The spectral density is
normalized f dwA(iv) — 1, and the usual relationships between the correlation func-
tions and the spectral density in equilibrium are reproduced. The classical result can
be recovered if we use the lowest-order approximation in at for the spectral density
.4<°)(ce) = 3(uj — Ao). We conclude with the observation that quantum fluctuations yield
energy renormalization and broadening effects, which manifest themselves in the spec-
tral density via the real and imaginary part of the self-energy o(w) given in Eq. (3.153).
It will be evaluated in the next Section.
3.7 Real-time evolution of the density matrix 207
3.7.7 Charge fluctuations in the single-electron box
In equilibrium when VR = VL, the SET transistor is equivalent to the single-electron
box. The energy difference Д0(<2с) = Ec(l-2CGVG) is tuned by the gate voltage. The
average excess particle number can be obtained from (3.154) and (3.161) (n(<JG)) ---
/dw/(w)A(w).
In the classical limit A(0)(cu) — 6(u - A0(Qg)), and one obtains {nci(QG})
/(Aq(Qg))- It shows a step at Qc = 1/2, which is smeared only by temperature.
At larger values of at or lower temperature we have to include the self-energy cr(w)
(3.153) ir the spectral density (3.161). The two limits, T = 0 and |w| < T, can be
analyzed analytically. In the first case, the spectral density has the form
(3.163)
(3.164)
Ao [u> - Д(ш)]2 + [тгД(ш)й(ш)]2
where
д / ,___ Ao	1 All_____________
Ш 1 + 2at ln(Ec/M) 1 + 7г2й(ш)2 ’ W 1 + 2at ln(Ec/|co|)
The spectral density A(w) has a maximum at the renormalized energy difference A,
which is obtained from the self-consistent solution of
A = Д(Д) , u = a(A).	(3.165)
It further lias a broadening of order тгДа. For 7га С 1 the broadening can be neglected.
In this case our results coincide with the renormalization-group analysis of Refs. [13.
46,50,51].
At finite temperatures |w| < T, we get
4, . Д a w coth (w/2T)	„	_	.
~ [w - Д]2 + [тгасе coth(co/2T)]2 ’ |w| ~ Г (ЗЛ66)
where
Л =	= 1 + 2atln(Fc/27rT) ’ ° =	= Г/2^Ь/Ё//2/Т) '	(3'16^
The broadening is of order 7ratT. It adds in an important way to the thermal smearing
contained in distribution functions if 7rat > 1.
As a consequence of quantum fluctuations the step of the average charge in the
electron box near the degeneracy points is washed out. We neglect broadening effects
(тга << 1) end assume that the energies of the ground state and the first excited state
near the degeneracy point depend symmetrically on the distance. Eq/IT Д/2. In this
case the partition function is Z ~ 2 exp[—EC/4T] cosh(A/27’) and the average excess
charge (n) becomes
<3'168>
208 Single-Electron Tunneling
In contrast to the result of perturbation theory (3.27) we find an anomalous but fi-
nite result. Depending on the temperature we have to choose the appropriate lim-
iting form for Д. At finite temperature the slope at До = 0 is д(п)/дДо|д0=о =
— {4T[l+2at In (Ес/2тгТ)]2}-1. It is reduced compared with the classical result — 1/4T.
This anomalous temperature dependence indicates coherent higher-order tunneling pro-
cesses.
3.7.8 Conductance oscillations in the SET transistor
We will now demonstrate that the linear and nonlinear conductance of a transistor show
pronounced deviations from the classical result, which are observable in an experiment.
The linear conductance G(V —> 0) of a transistor follows from (3.162)
4tt2 r	aR(w)aL(w) , . ,, .
G = - — / dw-------—-----A(w)/ (w)
/tK J	a(w)
(3.169)
Since the derivative of the Fermi function restricts the integration variable to the
regime |w| < T we can use the form (3.166) for the spectral function and perform the
integration [47]. The maximum of the conductance at До = 0 and the width of the
conductance peak are given by
= 2% aR(T)qL(T)
RK 3(T)
7Г
---- arctan
2
/ (ttq(T))2 — 1\
\ 2й(Т)тг J
[<1AOG^O) =
^max
7Г3 T[1 + 2at 1п(Ес/2тгТ)]
7Г — 2 arctan (	~1)
\ 2a(l )n /
(3.170)
Here aL/R(T) is given by (3.167) for the left and right junction and a(T) depends on
the sum of the conductances. We also introduced at — at,R + In the regime
at In (Ec/2ttT) << 1 the height and width of the conductance peak get modified as
2тг2 O-tFi O-t,L	1	7r\[1 , о , EC ]
Gmav ~--------------------------, 'V —J 1 4“	In —“ .	(3.171)
/?к at 1 + 2at In (Ec/2ttT) 2 L	2ttT\
For at —> 0 we recover the classical result for sequential tunneling. The corrections
depend logarithmically on temperature, indicating energy renormalization effects due
to higher-order tunneling processes. For T —> 0 the maximum value as well as the
broadening become Gmsx ~ 1/lnT and 7 ~ JTnT. For at In (Ес/2лТ) ~ 1 finite
life-time effects play a role, which are contained in (3.170).
A pronounced signature of quantum fluctuations is contained in the non-linear dif-
ferential conductance G(V) = dIst(V)/dV. In this case the finite voltage provides an
energy scale eV, and the renormalization and life-time effects are probed over a finite
energy range even at zero temperature. The T — 0 result obtained from Eq. (3.162) is
plotted in Fig. 3.26. For comparison we also show again the result obtained in pertur-
bation theory. In this limit the conductance is nonzero only in the range | До| < e.V/2
3.8 Outlook 20n
Ag _ 2Q<—,/e-1
eV “ eV/Ec
Fig. 3.26: The nonlinear differential conductance at T = 0 as function of the difference in charging
energy between the two lowest branches До = Ec(J ~ 2C'c;Ig), normalized to the transport voltage
V. The parameters are of = eV = 0.05 (f?t,L = f?t,R = Ri), we consider a symmetric bias and chose
(1) eV/Ec = 0.1, (2) eV/Ёс = 0.01, (3) eV/Ec = 0.001. For comparison, (0) shows the result for the
classical c ise obtained from lowest-order perturbation theory.
with vertical steps at the edges. The result of Fig. 3.26 displays clearly the renormalize
tion effects and, moreover, the finite life-time broadening due to the tunneling. Finite
temperatures and Joule heating effects will wash out the effect. However, as long as
the temperature remains lower than eV the quantum effects should be observable.
3.8 Outlook
Here we have described charging effects in normal-metal and superconducting tunnel
junctions and discussed single-electron tunneling. We started from perturbation the-
ory. We included higher-order processes where needed. This includes cotunneling in
the regime of Coulomb blockade and Andreev tunneling in NS junction, where at sub-
gap voltages single-electron tunneling is suppressed. We also described coherent and
incoherent tunneling of Cooper pairs, and we accounted for the effect of the electro-
dynamic environment. In the last two Sections we presented a systematic description
of tunneling in metallic junctions beyond perturbation theory. Using the real-time de-
scription the single-electron and cotunneling rates have been reproduced, but we also
could go beyond and describe for instance inelastic resonant tunneling.
We have restricted ourselves to a discussion of the electron box and single-electron
transistors. More complicated multi-junction systems are interesting too and impoi
taut in various contexts. The electron turnstile which can serve as a current standaoi
210 Single-Electron Tunneling
requires at least 4 junctions. Cotunneling processes which limit the accuracy of this
standard are suppressed if one uses even more junctions. Arrays of junctions show
collective behavior which depends on the competition between charging effects and
tunneling. Another important extension which we had not the space to describe here
are time-dependent perturbations. They produce e.g. side bands in the I-V character-
istics.
Charging effects are also pronounced in the transport through semiconductor nano-
structures, for instance through quantum dots in 2-dimensional electron systems. Here
a new feature enters compared to the metallic case: In a small dot the electron states
are quantized and the energy difference between different levels may become observable.
In fabricated dots the charging energy is still the larger of the energies, but on top of
the Coulomb oscillations further fine-structure related to the discrete energy levels and
the occupation of excited states has been seen.
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Gert-Ludwig Ingold
Dissipative Quantum Systems
4.1	Introduction
Dissipation is a ubiquitous phenomenon in real physical systems. Its nature is made
clear by considering the damped harmonic oscillator, a paradigm for dissipative sys-
tems in the classical as well as the quantum regime. After starting at a nonequilibrium
position, the system will perform damped oscillations and end up in the equilibrium
position. Looking closely, one will notice that even in equilibrium the oscillator co-
ordinate fluctuates. This effect is related to the Brownian motion of a free particle.
Damping and fluctuations are both caused by the coupling of the harmonic oscillator
to other degrees of freedom. A pendulum’s motion is damped because of collisions
with mohcules in the air during the oscillations. For the same reason the pendulum
will fluctuate around its equilibrium position. The identical origin of both effects man-
ifests itself in the fluctuation-dissipation theorem which will be discussed later in this
chapter.
Another system where the coupling to other degrees of freedom plays a prominent
role, is the decay of a metastable state. At not too low temperatures a state in a
potential minimum, which is separated by a barrier from an energetically lower region,
may decay by thermal activation. Here, the other degrees of freedom, which are also
collectively called heat bath or reservoir, provide the system with the necessary energy
to surmount the barrier. At low enough temperatures, the decay of the metastable
state will be dominated by quantum tunneling through the barrier, and one might ask
how the other degrees of freedom influence the tunneling rate.
In this chapter, we will specifically consider the damped harmonic oscillator, the
damped free particle, and the decay of a metastable state to illustrate the techniques
introduced to describe dissipation in quantum mechanics. Of course, there are other
interesting problems like the damped motion in a periodic potential and the dissipativ
two-level system [1]. For a more complete discussion of many aspects of quantum
dissipation, we refer the reader e.g. to the textbook by Weiss [2] which may also serve
as a guide to the literature.
A rather recent area of research activities is concerned with the interplay between
quantum dissipation and chaos which is discussed in Chapter 6. In this context, tech-
niques for the treatment of driven systems discussed in Chapter 5 are required.
214 Dissipative Quantum Systems
4.2	Description of dissipation in quantum mechanics
In classical physics damping may often be described by introducing a velocity propor-
tional term in the equation of motion. For a damped harmonic oscillator with mass M
and frequency wo one has
M (g + 7Q + Wqq) = 0,	(4.1)
where q is the position of the oscillator and 7 is the damping constant. By including a
fluctuating force f (t) on the right-hand-side one arrives at a so-called Langevin equation
which not only describes the damped average motion but also the fluctuations around it.
The velocity proportional damping term is often referred to as Ohmic damping because
an electrical circuit containing a resistance R, an inductance L, and a capacitance C
in series is described by (4.1), where q is the charge on the capacitor, Wq = (LG)-1/2,
and 7 = R/L. Obviously, in a classical description, damping may be introduced in a
rather phenomenological way without necessarily knowing the microscopic details of
the other degrees of freedom and their coupling to the degree of freedom of interest.
In quantum mechanics, the inclusion of dissipation requires more care because
quantum systems are described by a Hamilton operator which in the absence of time-
dependent external forces ensures conservation of energy. Several approaches have
been developed to circumvent this problem [3]. The most successful and rather general
approach, which also catches the physics discussed above, is based on the concept of a
reservoir of other degrees of freedom which will be discussed in the following section.
It will turn out that again knowledge of the microscopic details of the heat bath is
not necessarily needed. Although the system including the reservoir is described by
a time-independent Hamiltonian, the elimination of the bath degrees of freedom gives
rise to dissipation.
4.2.1 Hamiltonian for system and heat bath
We start out with one degree of freedom called system (S) which we view as a particle
of mass M moving in a potential V’(g). The corresponding Hamiltonian reads
n2
Ws = ^ + V(9),	(4.2)
where p is the momentum conjugate to the position q of the particle. The bath (B) is
described by a set of harmonic oscillators
=	+	(4.3)
1=1 X 1	/
System and bath positions are coupled bilinearly according to
N	N 2
=	+	(4.4)
4,2 Description of dissipation in quantum mechanics 215
Here, the last term, which does not depend on the bath coordinates, has to be included
for V(q) .o be the bare potential. This can be seen by considering a free damped
particle, i e. V(q) = 0, for which the Hamiltonian should be translationally invariant.
It is easy to check that this is indeed the case if we set с, = тщш?. This definition of
the coupling constants leaves sufficient free parameters as we will see later. Omission
of the last term in (4.4) would lead to a potential renormalization which would turn
the free particle into a harmonically bound one.
The total Hamiltonian
H = Hs + Яв + #sb	(4.5)
was used I у various authors to study dissipative quantum systems for the special case
of a harmcnic potential У(д) [4]. Probably the first proof that (4.5) in this case leads
to dissipation can be found in the lucid paper by Magalinskii [5]. Later, Zwanzig [6]
within a clissical description generalized the approach to nonlinear potentials. Most of
the early v, ork was aimed at applications in quantum optics and spin relaxation where
the couplir g to the heat bath is usually weak. More recently, Caldeira and Leggett [7]
emphasized that the Hamiltonian (4.5) is also applicable to strongly damped systems
and used it to describe dissipative tunnel systems. In the solid state physics community
it is therefore often referred to as Caldeira-Leggett Hamiltonian.
While the Hamiltonian (4.5) incorporates in some sense the microscopic origin of
dissipation it does not really represent a microscopic model. For example, we could
treat the £C/?-circuit mentioned above quantum mechanically without a microscopic
model of a resistor. Instead, as will become clear from the following discussion, we
may choose the parameters in (4.3) and (4.4) in such a way that we effectively model
the dampiig term in (4.1) by means of the virtual bath oscillators. However, there
are cases where a microscopic derivation is feasible. As an example, we mention the
dissipation caused by quasiparticles in a superconducting tunnel junction [8] which is
discussed ir Chapter 3.
Of course, not every dissipative system may be described in terms of the Hamil-
tonian (4.5'. On the other hand, this model gives a correct description for a wide
variety of qiantum dissipative systems, and maybe equally important, it allows for an
analytical t eatment.
4.2.2 El mination of the heat bath
We now wait to prove that the Hamiltonian (4.5) indeed describes dissipation if we
are interestt d in the system degree of freedom only. To this end, we first write down
the Heisenberg equations of motion which we solve for the reservoir degrees of freedom.
This gives vs an effective equation of motion which may then be compared with the
classical damped equation of motion. Employing the Heisenberg equation of motion
dA
dt

(4.(Ь
216 Dissipative Quantum Systems
we obtain the (operator) equations of motion
1 /dV Л c2 \	1 N
?+ — 5- + (/E—Ч	(4-7)
M \ dq	M	V J
for the system and
Xi+^Xi = — q	(4.8)
rrii
for the bath degrees of freedom. Considering q(t) as given, (4.8) is formally solved by
D' f 0)	C'
z,(t) = a;i(0)cos(wjt) 4- ——-sin(wit)+ / ds—— sin (cdi(t — s)) q(s).	(4.9)
m-itUi	Jo rrii^Ji
We eliminate the bath degrees of freedom by inserting this solution into the inhomo-
geneity of (4.7) and arrive at an effective operator equation for our system. By partial
integration we obtain
rt	dV
Mq + M dsy(t - s)g(s) 4- — = £(t)	(4.Ю)
Jo	dq
where the damping kernel is given by
1 N c2
7(‘) = м£^йс“(“л'	(411)
The inhomogeneity
*i(0)
C* 79(0)^ cos(wjf) 4- sin(wit)
rriiUi )	m,iWi
(4-12)
represents a fluctuating force and depends on the initial conditions of both the system
and the bath. Its equilibrium expectation value (for the definition see (4.38) below)
with respect to the heat bath including the coupling to the system, i.e. Яв 4- /Лв.
vanishes.
4.2.3 Spectral density of bath modes
So far, we have considered a heat bath with a finite number of harmonic oscillators. If
the system is also a harmonic oscillator, we may choose a normal mode representation
and convince ourselves that the whole system will return to its initial state after a
finite time, the Poincare recurrence time [9]. In order to actually describe dissipation,
this time has to be very long which certainly is the case for a continuous distribution
of bath modes. We therefore take the limit TV —> 00 and replace sums by integrals. It
is convenient to introduce the spectral density of bath modes as
N 2
J(w) = 7Т]Г <5(w - Wt).
(4-13)
4.2 Description of dissipation in quantum mechanics 2 > .
This quantity contains information on the frequencies of the modes and their coupling
to the system and is in fact sufficient to characterize the heat bath. By appropriately
choosing the parameters с,,т;, and iu* we may model any spectral density of bath
modes. This is even true if we set сг — тщш? as we did on page 215. In Section 3.4.1 it
is shown how for an electrical circuit the impedance is related to the spectral density
J(w).
With (4.11) and (4.13) we may relate the spectral density of bath modes to the
damping kernel by
Setting 7(w) = Myw, we obtain the Ohmic damping kernel 7(f) = 27<5(f). Since thf
integral in (4.10) ends at s = t, the delta function counts only half, and we thus obtain
the damping term introduced in (4.1). The Ohmic model represents the prototype for
damping and is therefore often used. However, it is not very realistic in its strict form
because the spectral density of bath modes diverges for large frequencies. In practice,
one always has a cutoff which may take different forms. One possibility is the so-called
Drude model where the spectral density
<j2
J(cu) =	(4.15)
behaves ike in the Ohmic case for small frequencies but goes smoothly to zero above
the Drude frequency Together with (4.14), we get for the damping kernel at times
t > 0
7(t) = 7WDe_u,D‘.	(4.16)
The dam oing strength defined as the integral over the damping kernel
roo
7o = / d<7(f)
Jo
(4.17)
yields 70 — 7 like in the Ohmic case. However, the damping kernel (4.16) exhibits
memory c.n the time scale
1
tc -— / dt 17(f)	(4.18)
70 Jo
with tc = in the Drude model. If we are not interested in time scales shorter than
t,:. which usually is the case ifwD represents the largest frequency scale, these memory
effects may often be neglected, and the Ohmic model may be employed instead.
4.2.4 Rubin model
A rather nontrivial damping kernel is obtained by considering a heavy particle (the
system) o' mass M coupled to two semi-infinite chains of harmonic oscillators (the
218 Dissipative Quantum Systems
Fig. 4.1: Mechanical realization of the Rubin model.
heat bath) with masses m and coupling constants K, the so-called Rubin model [10]
shown in Fig. 4.1. The corresponding Hamiltonian is given by
„2	oo 2 Д' oo	Д'
H =	+	+ TT + T	+	“ Г1)2	(4-19)
£1V1	n=l	z n=l	“
where we have taken into consideration only the right part of the oscillator chain. Due
to the symmetry the left chain may later be accounted for in the spectral density J(w)
by an additional factor of two. The Hamiltonian (4.19) couples the bath oscillators
and is therefore not of the form (4.5). However, the reservoir contribution may be
diagonalized by means of the ansatz
xn = \ - [ dksin(kn)x(k).	(4.20)
V 7Г Jo
Together with the relation
+oo	Ч-oo
emk = 2тг 6 (к — 2тгп),	(4-21)
n=-oo	n=-oo
we arrive at the transformed Hamiltonian
н = W + V(9) + i /I dk + 2K Io dk 5(fc)2 sin2(fc/2)
/9”	A
—qd—K / dk x(k) sin(fc) + — q2	(4.22)
V 7Г Jo	2
where p(k) denotes the momentum conjugate to x(k). For the coupling constant cor-
responding to we obtain by comparison with (4.4)
c(k) — К sin(/c).	(4.23)
The frequencies of the bath oscillators form a continuous spectrum with
w(/c) = j, sin(/c/2)
(4.24)
4.3 Density matrices 21'
and a maximum frequency
Inserting these results into (4.13) and accounting also for the left oscillator chain. wc
find for the spectral density of bath oscillators
J(u) -	[ dk SlnA^-^(^ - a;L sin(fc/2)).	(4.2 j
With (4.14), we obtain for the damping kernel
4/C
7(f) = J dfccos2(k/2)cos (a>L£ sin(A:/2)).	(4.27)
The ini egral may be expressed in terms of a Bessel function of the first kind
7(t) = rUL—t-----	(4.28)
where we introduced the mass ratio r = m/M. Making use of the asymptotic expansion
of the Bessel function, we get for the damping kernel at long times
/.x /2wl sin(wL£ ~ tt/4)
tW = r\l —-------------------- for t oo	(4.29)
which i i contrast to the Drude case decays only algebraically. In addition, the damp-
ing kernel is not always positive. Nevertheless, the integrated damping strength, which
equals tcj^, is positive, indicating that the two chains of oscillators indeed cause dis-
sipation. The memory introduced by coupling to the chains is determined by I'-'
characteristic frequency of the chains with rc = wf1 as can be seen by evab -lag
(4.18) for the damping kernel (4.28).
4.3 Density matrices
Before we can continue by considering specific systems, we have to remind ourselves of
some facts from quantum statistical mechanics. In ordinary quantum mechanics a state
is described by a state vector In quantum statistics this is no longer sufficient,
and one has to introduce density matrices of the general form
P =	(4.30)
n,m
where ti e states |n) form a complete set. The density matrix should be normalized
according to
tr(p) =	= 1-	(4.31)
к
220 Dissipative Quantum Systems
Again, the sum runs over a complete set of states. A pure state would be represented
by p = IV’HV’I- However, often only probabilities pn are known with which the states
|n) are found. The lack of phase information then makes it impossible to describe the
state by a state vector. Instead we have to use the density matrix
P = '^,Pn\n){n\.	(4.32)
n
One can show that tr(p2) < 1, where the equality only holds for density matrices
describing pure states.
The most prominent example of a density matrix is the equilibrium density matrix
Pa = y-e’3" =	J2e"^n|n)(n|	(4.33)
where |n) are energy eigenstates and the probabilities pn are given by Boltzmann
weights at inverse temperature /3 = l/kBT. The normalization is provided by the
partition function
Z0 = tr (е"зя) .	(4.34)
The equation of motion for a density matrix is obtained by differentiating (4.30)
with respect to time and employing the time-dependent Schrodinger equation. This
leads us to
p(t) = -i£(t)p(t)	(4.35)
where the Liouville operator
(4.36)
represents a superoperator acting on an operator X. At first sight, (4.35) looks like
a Heisenberg equation of motion. However, there is an extra minus sign and in fact
(4.35) was derived within the Schrodinger picture.
Finally, we may express expectation values of observables through the density ma-
trix as
(A) = tr(M)	(4.37)
where the equilibrium expectation value
И)/з = tr(p^A)	(4.38)
taken with respect to the equilibrium density matrix as indicated by the subscript (3
plays an important role.
4.4 Linear damped systems	22 i
4.4 Linear damped systems
As a f.rst example we discuss one of the simplest dissipative quantum systems, tin
damped harmonic oscillator. The corresponding Hamiltonian is given by (4.2} '
with the potential
,r/ . M .> ,
v(q) =		(4.39)
Because of its linear equation of motion, the problem may be solved exactly not only
in the classical but also the quantum case.
We start out by recapitulating some facts from linear response theory. In particular,
we introduce the response function (Section 4.4.1) and derive the fluctuation-dissipation
theorem (Section 4.4.2). While linear response theory is exact for linear systems like
the damped harmonic oscillator, the concepts presented here may also be useful when
considering nonlinear systems.
The fluctuation-dissipation theorem will allow us to make a connection between
the damped motion as given by the response function and the fluctuations described
by correlation functions. In Section 4.4.3 we discuss in some detail the equilibrium
correlation functions of the damped harmonic oscillator while paying special attention
to the low temperature properties. The free damped particle is obtained as a limiting
case in Section 4.4.4.
4.4.1 Response function
In linear response theory one considers the influence of an external force on the system
to first order in this force. The starting point therefore is the Hamiltonian
H(t) = H - hG(t)G	(4.40)
which in general may depend on time and where the first part is given by (4.5). G
is an operator acting only in the Hilbert space of the system while hG is a c-numbi’r
which s lould vanish for negative times. Such a situation was already discussed in
Section 1.2.3 where the Kubo-Greenwood formula for the conductivity was derived
within 1 near response theory.
Our aim is now to calculate the deviation
A(F(t)) = (F(t)) - (F)^	(4.41)
of the eopectation value of the system operator F in presence of an applied force from
its equil brium value (F)taken for h<; = 0. The time evolution of the expectatio.
value (F(t)) in the Schrodinger picture is governed by the time evolution of the density
matrix according to (4.35). Decomposing the Liouville operator
£ = £0+£i	(4.42)
with
£0X = i[H,X]	(4.43)
222 Dissipative Quantum Systems
and
£1X = -i[G,X]/iG(t),	(4.44)
we may formally solve the equation of motion for the density matrix as
p(t) = e-i£o(p(0) -i ['dsе-^-^Л(s)p(s)	(4.45)
Jo
Assuming that the external force has been absent for negative times, we may replace
the initial density matrix /э(0) by the equilibrium density matrix. Iteration of the
integral equation then yields [11]
p(t)=p/3-i^ dse~'Co^~s'lC1(s)p/j +	(4.46)
Within linear response theory the iteration is carried out only up to first order. As
already mentioned this is exact for linear systems and otherwise represents a good
approximation for sufficiently weak forces, i.e. small ha.
With (4.46) we get for the time dependence of the expectation value of F up to
first order in he
(F(t)) =	/'dstr(Fe-'c^[G,pp])hG(s).	(4.47)
n Jo '	’
By means of the relation
exp(—i£ot)X = exp (——Hot} X exp (—Hot}	(4.48)
\ n / \n /
and the cyclic invariance of the trace, we may switch from the Schrodinger to the
Heisenberg picture by letting the exponential containing the Liouville operator generate
the time dependence of the operator F. This leaves us finally with
A(F(t)) = y‘dstr([F(t- s),G]pp)hG(S).	(4.49)
This result, which establishes a linear relation between the external force ha con-
jugate to the system operator G and the response of the observable F, allows us to
define a response function xfg as
A(F(t)) = [ dsXFG^t - s)hG(s).	(4.50)
Comparison with (4.49) finally yields [11]
XFG(i) = ^([F(t),G]b©(t)	(4.51)
where we introduced a step function 0(t) which makes causality manifest.
4.4 Linear damped systems
As an example, we consider the response function xqq of the damped harmoui
oscillator which may now be obtained by taking the commutator of the equation c
motion (4.10) with g(0). Since the commutator with the noise term (4.12) vanishes
we find that the response function obeys the differential equation
/t
X9?(0 + уо d«7(t - s)xM(s) + u>oX<wW = 0	(4.52
with the initial conditions
Xw(0) = ^<[<?(0), q(0)]>/3 = о	(4.0
and
X<w(0) = Т7г([р(°)л(°)])^ = ^7-	(4Г'Ь
i Vi IL	x Vi
Two comments are now in order. Firstly, (4.52) could have been obtained more di
rectly by making use of (4.50) and (4.10). However, our derivation has lead to rhe
relation (4.51) which will be of importance later on. Secondly, (4.52) is identical to the
corresponding classical equation for the response function. Thus, the classical response
functicn for the damped harmonic oscillator and its quantum version are identical.
This is a direct consequence of the Ehrenfest theorem.
Unfortunately, it is not possible to evaluate the response function for arbitrary
damping kernel in closed form. However, we may determine the corresponding dynamic
susceptibility
Xw(w)=/ dtxM(t)elwt = xw(-iw)	(4.55)
J —oo
where we introduced the Laplace transform
roo
X4q(z)= dte~ztxqq(t)	(4.56)
•/ 0
and made use of causality. From (4.52) together with the initial conditions (4.53) and
(4.54) we find
Х99(т) = -------5---:	-----j-	(4.57)
4	M —w2 — iw7(w) +
A valid damping kernel will lead to poles of (4.57) in the lower half plane to ensure
causality of the response function.
For Ohmic damping with 7(w) = 7 the inverse Fourier transform may be performed,
and one obtains the response function
X„(i) = n~~re~7(/2 sin (tit) 0(t)	(4.50
IVi LU
with a frequency
shifted due to the damping.
224 Dissipative Quantum Systems
4.4.2 Fluctuation-dissipation theorem
We now calculate the power
P =	(4.60)
dissipated due to all external forces hF(t) in order to relate the response function to
dissipation. The change in the expectation value of the observable F is induced by
external forces conjugate to all observables G
hG(t) = hG(a))e~'ut + h*G(Lu)e,ut	(4-61)
which we take to be monochromatic for convenience. From (4.50) we obtain together
with (4.55)
4&(F(f)) = - £) iw [/)G(cu)e^lw(xra(w) - hG(u)e'utxFG(-u)\ 	(4.62)
ar	G
For the dissipated power (4.60) averaged over one period of the driving forces we thus
find
P = 12	(4.63)
F,G
This expression contains the dissipative part of the dynamic susceptibility
Xfg(w) ~ (Xfg(u) ~ Xgf(~^))	(4.64)
which depending on the time reversal symmetry of the observables is given by the
imaginary or real part of the dynamic susceptibility.
Equilibrium fluctuations may be described by equilibrium correlation functions
CFG(t) = (F(t)G(O))^	(4.65)
which are defined according to (4.38). The definition may be used to show stationarity
(F(t)G(O))^ = (F(O)G(-t))^	(4.66)
by exploiting the cyclic invariance of the trace. Using the same technique one can
derive the symmetry
CFG(t) — CGF(—t — ifi/3).	(4-67)
Introducing the spectral function
CFG(u) — [	(4.68)
J — oo
4.4 Linear damped systems 225
this relation turns into
Cfg(w) =	(-w)e/3h“.	(4.69)
It is convenient to introduce the symmetrized and the antisymmetrized equilibrium
correla ,ion functions
SFG(t) = | W)G(O) + G(O)F(t))0	(4.70;
and
AFG(t) = i(F(t)G(0) - G(O)F(t))0,	(4.7,;
respectively. For the Fourier transform of the antisymmetrized equilibrium correlation
function we then find with (4.69)
Afg(u) = —(1 - e~h^')CFG(ai).	(4G2)
Mating use of the result (4.51) of linear response theory, our expression (4.64) for
the dissipative part of the dynamic susceptibility becomes
X^G(iu) = -Afg(u)	(4.73)
or in view of (4.72)
XfgM = “(1 - e-w“)CFG(w).	(4.74)
This relation is called fluctuation-dissipation theorem [12] because it relates the dissipa-
tive part	the dynamic susceptibility to the equilibrium fluctuations described
by its spectral function Gpc(uj).
4.4.3 Correlation functions of the damped harmonic oscillator
The fluctuation-dissipation theorem was derived within linear response theory and is
therefore an exact relation for the damped harmonic oscillator. Following Ref. [13] w<’
may thus use the theorem to determine the equilibrium correlation functions for
system, ft is sufficient to calculate the position autocorrelation function
Gqq(t) = (q(t)q(fiy)0	(4.751
from which we may obtain the other correlation functions
cpq{t) = -C9P(t) = M~Cqq{t)	(4.76)
d2
CPP(t) = -M2—Cqq(t)	(4.77)
226 Dissipative Quantum Systems
by differentiation with respect to time. These expressions may be derived by using
p = Mq and the stationarity (4.66) of equilibrium correlation functions.
The fluctuation-dissipation theorem allows us to express the equilibrium position
autocorrelation function in terms of the imaginary part of the dynamical susceptibility
(4.57). We consider specifically the case of Ohmic damping for which analytical results
may be obtained. After a Fourier transformation the fluctuation-dissipation theorem
reads
_ . . Ti f+ao ,	yu	e~'ut	,
Cqq(t) = ТГ / dw —------7V5------------ал~-	4.78
4 Mtv J-oo (w2 — Wq)2 + 72cj2 1 - e_^n“	7
Making use of the relation
1	11, (h(3u\
1 - e~^ft“ 2	2	\ 2 J	v ’
we decompose the correlation function C4q{fy = Sqq(t) +iAw(t) into its symmetric part
/j r+oo	/|	\
s™(,) = L, d“(^-4)2 + 7v “Л UH '“(“<>	(“ 8»>
and its antisymmetric part
Aw(<) = “~7 /	---775---r-7sin(wt)	(4-81)
w 2тгМ J-oo (w2 - wg)2 + 72w2
which agree with the definitions (4.70) and (4.71), respectively. The antisymmetric
correlation function may be evaluated by complex contour integration yielding
71</Ч(1) = “тггрe“7|i'/2 sin(wl).	(4.82)
£1V1 (jJ
According to (4.71), this correlation function is related to a commutator which should
be zero in the classical limit. This is indeed the case as one can readily see by taking
the limit h —> 0. It should be noted, however, that (4.82) does not contain specific
quantum effects but is directly related to the classical response function (4.58) by
2
W/W ~ ~(4.83)
which is a consequence of (4.51) and (4.71).
For the evaluation of the symmetric correlation function (4.80), we first decompose
the hyperbolic cotangent as
coth f	f 1 + 2 ^2 —A—- ]	(4.84)
\2	)	^p2 + w2/
where we have introduced the so-called Matsubara frequencies
2тг
vn — V7,n-	(4.85)
4,4 Linear damped systems 2'21
It is irteresting at this point to discuss the physical significance of the terms appearing
in (4.1’4). The in the prefactor cancels with the prefactor of the integral (4.80) sc
that tie first term in the brackets of (4.84) is independent of h and proportional t<
the temperature. It therefore represents the classical result. The sum in (4.84), on the
other aand, gives rise to quantum corrections which are present here in contrast to the
antisymmetric correlation function.
The integrand of (4.80) has poles at
w = ±	± i^	(4..
with n defined by (4.59). In view of (4.84), there are additional poles atw = Tir.
Evaluating the integral by complex contour integration one obtains after some stimg’i1
forward calculation for the symmetric correlation function
exp(-2|l|)
> = Ж, »>№)-«(»№) |“"Ь(ВД	+ S™(W2)
O/y OO	,, p-^nHI
" W £ (^ + 4)2-?M '	(4'87)
In the classical limit h —> 0, which here is equivalent to the high temperature limit,
this becomes
= тЛ~2	[cos(wt) + sin(w|t|)] .	(4.88)
Ж P<^q	A L	J
Accord.ng to (4.75) and (4.77), the classical second moments of position and momentum
are the i obtained as
(4-89>
= ЛЛВТ.	(4.90)
Quantum effects come into play if we lower the temperature so that no longer
k^T 3> Iuuq. Since these effects already exist in the undamped case, it, is more inter-
esting to consider another regime which is present for finite damping only. Comparing
the exponential decay of the different terms in (4.87), we notice that the first ter-
determines the long time behavior only if h'y/^k^T < 1, i.e. for sufficiently we,,.,
damping or for sufficiently high temperatures. Otherwise, the exponential decay with
a time (onstant given by the Matsubara frequency V\ will dominate the long time be-
havior. In contrast to the assumption made in standard weak coupling theories, the
decay o' the correlation function is then no longer governed by the damping const;. !
This becomes especially apparent at zero temperature. With decreasing tempera» ee’
228 Dissipative Quantum Systems
the Matsubara frequencies get closer to each other and at zero temperature all of
them contribute. We may then replace the sum in (4.87) by an integral
i- 27 V	= /17 [°° , xe-xt
,3^1. M/3	(p2 + Uq)2 - y2zA ttmJo X (x2 + Wq)2 - y2x2
which may be expressed in terms of exponential integral functions. The leading long
time behavior, however, may easily be obtained by replacing the integrand by its
behavior for small values of the integration variable. One thus finds that all exponential
terms act together to produce an algebraic decay of the correlation function
=	k"4”'	<4'92)
This behavior is not purely academic because it can be found at low but finite temper-
atures at intermediate times before the exponential decay takes over [14].
After the long time behavior we now take a look at the second moments of position
and momentum, i.e. the correlation functions at time zero, at zero temperature. With
(4.87) and (4.91) one finds after evaluation of the integral at t = 0
Zi Г 2	( 'У \ 1
{q2)p = XT7T 1 - -arctan —	(4.93)
2,Мы L 7Г
which for 7 —> 0 reduces to the correct quantum mechanical result for the undamped
oscillator (q2)/s = h/IMujQ. In contrast, the second moment of the momentum diverges
for any finite damping as can be seen from (4.80) by differentiating twice with respect
to time. This ultraviolet divergence is due to the spectral density which for Ohmic
damping increases proportional to frequency. Therefore, one has to introduce a cut-
off for the spectral density like in the Drude model (4.15). The memoryfree Ohmic
damping kernel is then broadened and the divergence removed.
4.4.4 Free damped particle
A free damped particle can be viewed as a particle subject to damping moving in
a harmonic potential with Wq -7 0. In contrast to the damped harmonic oscillator
discussed in the previous section, the particle is no longer bounded and exhibits diffusive
motion.
From (4.80) it is obvious, that the second moment of the position S?Q(0) does not
exist since for the free particle the integrand diverges at small frequencies. This is of
course a consequence of the fact that the particle is not bounded but may be found
everywhere. Therefore, we introduce the mean square displacement
Ф) - ((</(*) -	(4.94)
which is related to the symmetric position autocorrelation function by
s(t) = 2 [(</% - SOT(t)] .	(4.95)
4.5 Short introduction to path integrals 229
For any finite temperature, the long time behavior of s(t) is given by the first term
in (4.87). In the limit u>o —> 0, one finds
s(t) = -;„-t for t —> oo
М/З7
(4.96)
with the leading corrections being time-independent. This represents the classical
diffusive behavior for which a diffusion constant can be defined as
(4
From (/ .96) we thus get the so-called Einstein relation [15]
which in slightly different form appeared already in (1.6). Quantum fluctuations do
not affect this result even for very low temperatures except that the time after which
(4.96) if valid may become quite large. For zero temperature, however, the diffusion
constant vanishes, and thus the mean square displacement no longer increases linearly
for long times. In this case the infinite sum in (4.87) determines the long time behavior.
The latter is obtained from the integral representation (4.91) in the limit w0 —> 0. For
long times the integral is again dominated by small values of the integration variable.
We obtain
(4.99)
In the absence of thermal fluctuations the mean square displacement thus grows only
logarithmically [16].
4.5 Short introduction to path integrals
The path-integral formalism has been quite successful in recent years in the context
of quantum dissipative systems. This formalism was originally invented by Feynman
[17] as an alternative formulation of quantum mechanics which can be shown to bi
equivalen; to the more familiar Schrodinger equation approach. Here, we do not want
to go into the mathematical subtleties of path integrals but give a more practical
introduction as far as it will be needed in the following.
The time evolution of a state |Ф) of an undamped quantum system is governed by
the time-dependent Schrodinger equation
^|Ф) = Н|Ф)
(4.100)
230 Dissipative Quantum Systems
where H is the Hamiltonian describing the system. For a time-independent Hamilto-
nian we may formally integrate (4.100) to obtain the time evolution of the initial state
|Ф(0)) according to
|Ф(£)> = exp	|Ф(0)).	(4.101)
Introducing the propagator in coordinate representation from the initial position q\ to
the final position qt
G(qt, q\, t) = <9f| exp \ \q,),	(4.102)
which was already encountered in the discussion of Green functions in Section 1.2.2,
we may express the time evolution of a state as
Ф(9г, t) = J da G(q{, q., 0Ф(<7Ь 0)	(4.103)
where we have to integrate over all initial positions.
One of the advantages of the path-integral formulation of quantum mechanics is
the absence of operators. Instead, one uses the classical action S[g] which is defined
by the classical Lagrange function L(q, q, t) as
S[g] = [ dsL(q,q,s).	(4.104)
Jo
In contrast to a function like the Lagrange function L which relates a number to a
number, the action is a functional which relates a function to a number. Here, the
function is the path g(s) starting at time s = 0 and ending at s = t. According to
Feynman the propagator (4.102) may now be written as
гчЩ=ч1	/ i \	,
G(<?f,<M) = /	7\ exp(-%]) .	(4.105)
The right-hand-side represents a path integral or functional integral. In analogy to
the difference between a function and a functional, this integral does not run over
an interval as usual, but one has to integrate over all paths satisfying the boundary
conditions g(0) = q\ and q(t) — q.?. To distinguish between integrals and path integrals,
we have replaced the “d” by a “77”. We do not want to prove the validity of (4.105)
here, but refer the reader to the literature [18].
The physical meaning of (4.105) becomes clear by first considering the classical limit
h —> 0. In this case, the integrand will oscillate very rapidly and the contributions of
neighboring paths will cancel. The only exception are extrema of the action, i.e. paths
where its first variation vanishes. This is of course equivalent to Hamilton’s principle
of classical mechanics. Hence, in the classical limit, only the classical paths contribute
to the propagator. In the quantum regime, other paths also contribute and the most
important contributions come from fluctuations around the classical paths. In the
remainder of this section we assume for simplicity that there is just one classical path.
4.5 Short introduction to path integrals
Although we do not have to deal with operators anymore, it can be quite '-md :
evaluate the path integral and to get an exact result for the propagator. Howcv.-:
it is often possible to make a so-called semiclassical approximation where fluctuation
arourd the classical path are treated up to second order. According to our abo\
discussion, the classical path qci gives the dominant contribution to the path inte<” i
It therefore makes sense to expand the paths around the classical path according i.u
q(s) = &i(s)+ £(«)•
(4.106
Since the classical path satisfies the boundary conditions, the fluctuations f(s) have t(
vanish at times s = 0 and t. Accordingly, we expand the action
= Ж1] + [ ds
Jo oq(s
e(s)
7c 1
1 fb f1 62S
+ - ds du .	.	£(s)£(u) + ...
2 Jo Jo 3q(s)6q(u)
(4.107;
where 5 denotes the functional derivative. Since the classical path is a stationary
point of the action, the second term on the right-hand-side of (4.107) vanishes. The
first quantum corrections are thus described by the term which is quadratic in the
fluctuations. For actions which are at most quadratic in the path, e.g. for the fre<
particle and the harmonic oscillator, the expansion will break off after the second
term. Therefore, taking into account only the terms given in (4.107) will then yield
an exact result. In general, however, there will be higher order terms. If h can lu
taken to be small, the exponential in (4.105) represents a Gaussian in the vicinity ol
the classical path. Its width h limits the possible fluctuations to be of order \/Ti er
less. The quadratic term in (4.107) is therefore of order 1 while higher order terms are
smaller by at least a factor \/h. As a consequence, restriction to the quadratic 1<тто
implies a semiclassical approximation.
For damped quantum systems it is not sufficient to consider the propagator becav
we do not deal with pure states. We rather have to treat density matrices, the
important of which is of course the equilibrium density matrix introduced in (4.33). li
we write the equilibrium density matrix in its coordinate representation
PeM) =	(4.10.S)
we notice that this is quite similar to the expression (4.102) for the propagator. A
comparison of the two exponents suggests to interpret the temperature as an imaginary
time t = —ihJ3. By further exploiting this analogy, we may express the equilibrium
density matrix in terms of a path integral. We call this an imaginary-time path integral
in order to distinguish it from the real-time path integral (4.105).
We convert the real-time path integral into an imaginary-time path integral by
considering the action, which for a particle of mass M moving in a potential V(q) i:~.
23 z Dissipative Quantum Systems
given by
Г /	\ 2
S[q,t] = l^dS	-V(q)
Jo 2 yds)
(4.109)
where we specified the final time t as argument of S explicitly for sake of clarity.
Replacing t by — ihfi and substituting s by — it we obtain
S[q, — ift/?] = i / dr
Jo
44-4
(4.110)
It is convenient to rewrite this equation as
S[q, -ihfl] = iSE[g, h(3]
(4.111)
by introducing the so-called Euclidean action
/•Л0
SE[q, h/3] = dr
Jo
44-4
(4.112)
While this looks formally like an action, we note that the Euclidean action describes
the motion of a particle in the inverted potential — V(g). Given the analogy between
the coordinate representations of the propagator (4.102) and the equilibrium density
matrix (4.108), we obtain together with (4.111) the imaginary-time path-integral rep-
resentation of the equilibrium density matrix
1	/1	\
P0(q,q') = T[ P9-exp4-SE[4.
Zfi Q	\ fl /
(4.113)
Here and in the following, we drop again the time argument of the action.
4.6 Dissipation within the path-integral formalism
4.6.1 Influence functional
In Section 4.2.2 we have shown how damping arises from a Hamiltonian description by
coupling to a heat bath. The presence of damping became apparent after eliminating
the bath degrees of freedom and considering the system degree of freedom alone. This
elimination may also be carried out within the path-integral formalism [17].
In the following, we discuss the elimination procedure by considering the equilibrium
density matrix for an Hamiltonian of the form (4.5)
Wp — — exp [—fi (Hs + Hq + Я§в)] •
(4.114)
4.6 Dissipation within the path-integral formalism 233
For N bath oscillators we may express the position representation of the density matrix
in terirs of an (N + l)-fold imaginary-time path integral
W0(q,Xi, q',^ = Y0/ ^ПР^ехР
(4.115)
where the paths run from q = q' and x, = x\ within the imaginary-time interval hfi to
q = q and xt = x,, respectively. In view of (4.2), (4.3), and (4.4) the Euclidean action
may be decomposed according to
Se[<7, X;] = Sse[q] + Sf [xf] + SfB[g, x,]
(4.116)
with
,, г№ [Мл	1
>Ш=/ dr k^ + W) ,
Jo	L 2	J
(4.117)
'o
(4.11K)
and
rag
о
N
dr -д^2ахг + q252
(4.119)

N
N t
We obtain the reduced equilibrium density matrix of the system by tracing out the
bath degrees of freedom
= trB (W0(q, x,,q',x'))
= I T^q J П dx* j П eXP £.])
(4.120)
thereby retaining the correlations between system and bath. The symbol j implies
an integral over paths having the same starting and end point Xj. Together with the
integral on over x, this indeed amounts to taking the trace over the bath variables.
The partition function appearing in (4.120) is given by Z — ZgfZ\X where
= П
1=1
(4.121;
is the pertition function of the uncoupled bath and
2 sinh(/i/3coi/2)
the part tion function of a single bath oscillator.
(4.12
234 Dissipative Quantum Systems
We now have to carry out N path integrals and subsequently N integrations which
can be done since the bath consists of harmonic oscillators coupled linearly to the
system. Note that the following calculation may also be performed if the coupling is
nonlinear in the system coordinate as long as the bath coordinates appear only linearly.
Since the oscillators are not coupled among each other, it is sufficient to carry out the
elimination procedure for one oscillator only and take the product over all oscillators
at the very end. It is thus convenient to rewrite the system density matrix as
PpM) = I Vqexp (-^sW) ^[9]
(4.123)
where we introduced
Л9] = П
(4.124)
and
E;[g] = j dxi Vxtexp	(4.125)
with the action
Jo 2
(4.126)
The functional (4.124) is called influence functional because it contains the complete
information of the influence of the heat bath on the system. In the uncoupled case we
should have F[g] — 1 which is indeed the case because for c, = 0 we have ЕДд] = Z,.
In general, one is not simply interested in the equilibrium density matrix but in the
time evolution of a nonequilibrium density matrix. There the concept of an influence
functional still applies, and it is rather straightforward to appropriately modify the
calculations outlined below. In the simplest case, one neglects the initial correlations
between system and heat bath by assuming that the initial density matrix factorizes
into a nonequilibrium density matrix of the system and the equilibrium density matrix
of the heat bath [19]. More realistic situations including initial correlations may be
treated as well [20]. In any case the calculations will become more tedious compared
to the one which we are going to perform now.
4.6.2 Elimination of the heat bath
We begin the calculation of the influence functional by evaluating the functional integral
in (4.125). One approach to evaluate the trace is to express the most general path in
terms of a Fourier series
+00
*i(O = E	(4.127)
П~~ 00
4.6 Dissipation within the path-integral formalism
The fa:t that this generates also nonperiodic paths which should not be taken int<
accoun; does not represent a problem because a jump in £,(r) will cause an
contribution to the kinetic term in the action. Unwanted paths are therefore automa!
ically s rppressed.
Her;, we take another approach which allows us to demonstrate the general ptoce-
dure of evaluating a path integral within the semiclassical approximation as explained
in Section 4.5. It should be stressed however, that since the action (4.126) is harmonic,
the following calculation is exact.
The first step is to find a solution of the classical equation of motion
^-^xf = -^-q	(4.128)
which is obtained by variation of the action (4.126) with respect to Xj. This is of course
just the imaginary-time version of the equation of motion (4.8). The boundary con-
ditions rj'(O) —	= Xi select the paths needed for tracing out the environment,
leading to the solution of (4.128)
. sinh(wiT) Г Ci гл0 .	,, , .
ж- r = .	хг +------ / dersinh Wj fy? - a ? a
тщ Jr
sinhWfttf — т)1 Г Ci rT ,	. , , x_z J
+—• ’	. k; +------- / dcrsinh^cr)^) .	(4.129)
sintqft/twj L vrti^i Jo	J
Given this solution, we may calculate the classical action. The number of integrals
may be reduced somewhat in the presence of quadratic terms in the potential. We
partially integrate the kinetic energy and make use of the equation of motion (4.128)
to obtain
= у	- ^(0)^(0)]
fhl3 , mt
/ dr V
Jo 2
2
Ci —cl Ci -2
—qxf +
mi	mfuf
(4.l.! .
Inserting (4.129), we get after some algebra
cE,clr- !	COShtfifai) - 1 2
глй sinh(cjir) + sinh[w, (Й/3 — t)]
1 Jo	sinh(fi/3cui)
,2
sinh - t) sinh(wiCT)
,	, <!cr--------- , /j-д- \-------Q{T)q(a)
тцл>г Jo Jo	sinh (при,)
f
2тгцш? Jo
(4.131)
230 Dissipative Quantum Systems
Since according to (4.125) we later on have to integrate over Xi, it is useful to rewrite
(4.131) as
cosh(?l/3wi) — 1
sinh(ft/?cjj)
гЛ/З П
- dr daKi(r - a)q(r)q(u)
Jo Jo
(4.132)
r*2 rtlfj
2т;иг- Jo
where we made frequent use of relations between hyperbolic functions. The quantity
which depends on the path q(r), will drop out in the integration over X{ and is
therefore not needed explicitly. The integral kernel is defined as
h/3
2
„ cosh Ui
Сг________L_
‘imiOJi , , (hBoji
sinh -------
к 2
- =	- r).
(4.133)
The action (4.132) contains local as well as nonlocal terms in the system path q(r).
In order to show that the local terms cancel, it is
(4.133) into a Fourier series
convenient to expand the kernel
cosh cjj
h/3
2
. ,(h/30Ji
sinh -------
к 2
2 +
Wn2
-------e
(4.134)
in the interval 0 < т < h(3 which we are interested in. The local contribution to the
double integral in the action can be identified by using the identity
(4.135)
where the first term on the right hand side represents the nonlocal part while the
second term gives rise to a local part. In the double integral the latter can be written
as
J dr j duKi^r - u) ^q(r)2 + q(ff)2^ = j drq(r)2 J АиКг(а) (4.136)
from which it becomes clear that the strength of the local part is given by the n = 0
Fourier component in (4.134). We therefore decompose the kernel Л'Дт) into a periodic
delta function
9
oo	2
V S(t - пШ =-----
£oo	Wmrf
(4.137)
4.6 Dissipation within the path-integral formalism 237
and a i ew kernel
kt(r) =
c2 +°°	i/2
hfim^2 v2 + cu2
(4.138)
which lesults in a nonlocal contribution to the classical action. The delta function
only gives rise to a local contribution to the second term of (4.132) which could be
interpre ted as potential renormalization but cancels exactly with the third term. This
is in agreement with the reasoning which lead us to include the second term on tlic
right hand side of (4.4).
After this discussion of the classical path, we should consider the fluctuations around
it according to the decomposition (4.106). As mentioned before, the expansion (4.167)
for our problem terminates after the second order and the first order term vanish's
because we expand around the classical path. The second order contribution to the
action is given by
^2)к<]=Г‘1гВ(4.? + ^)	(4.139)
where &(т) represents a fluctuation around the classical path. However, this contri-
bution is independent of and the fluctuations just lead to a numerical factor. This
advantage of expanding around the classical path allows us to proceed by performing
the a^-integration required in (4.125) to obtain
Fi[q]=F^
hit sinh(/i/?w,)
77ljWi(cOSh(fi/?Wi) — 1)
/	} rhg Л0
xexp -— /	<1t / daki(r -
\ in Jo Jo
(4.140)
where is the contribution of the fluctuations which is independent of q and the
coupling constant ct. We may determine this factor without explicitly performing the
path intagral over the fluctuations by considering the uncoupled case where c; = 0 and
therefore /q(r) = 0. According to our discussion on page 234, Fj[rj] should then reduce
to the partition function Zi defined in (4.122). This leads us to
/1 M*3	M/?
Ft[q] = Zi expl-— dr da^r - a)(q(T)q(a)
\ 2/ГЬ Jo Jo
(4.141)
which describes the influence of one bath oscillator on the system.
4.6.3 Evaluation of the fluctuations
While we did not need to evaluate the fluctuation integral to obtain the result (4.14;
it is nevertheless instructive to see how it could be done. We first note that since t'r
238 Dissipative Quantum Systems
fluctuation vanishes at times t = 0 and h{3 the second order contribution (4.139) may
be written as
(4.142)
Jo 2
with the linear operator
d2
5i = -^ + 42.	(4.143)
The normalized eigenfunctions of Si satisfying the boundary conditions are given by
б.пк) =	sin (у r')	(4.144)
у ftp \ 2 J
with the corresponding eigenvalues
i/2
\n = у + w2	(4.145)
where the Matsubara frequencies z/n have been defined in (4.85). Expanding the fluc-
tuation in these eigenfunctions
£i(r) =	(4.146)
n=l
and exploiting their orthogonality, we immediately obtain
=	<4.147)
Z n=l
The functional integral over all fluctuations can now be rewritten in terms of a con-
ventional integral over the expansion coefficients
= J П<1°<ехр -y^A,.no2
j = l	n=l
(4.148)
where J is the Jacobian of the transformation.
While J may be determined [18], it is often useful to take the free particle as a sort
of reference system. The propagator of the free particle
G^t) = (=^) exp (~(q{ - Ч1)Л	(4.149)
\27rnt2 \n 2t	)
may be derived in a number of different ways. Switching to imaginary times, we get
the fluctuation contribution for the free particle
/	\ V2
__________________________4.6 Dissipation within the path-integral formalism 239
On the other hand, we may rewrite (4.148) for the free particle by replacing th
values А,1П by Ao,n = ^2/4. Performing the Gaussian integrals we get
m \ 1/2 00 /An A 1/2
\ tt I ^0,n \
2Trh2/3J
(4.151)
The products of eigenvalues appearing in this result are often referred to in the litera-
ture as determinants of the corresponding operators as a generalization of the case of
finite dimensional matrices. For the interpretation of determinants in the semiclassical
approximation of path integrals it is also instructive to take a look at (6.22).
Th< infinite product in (4.151) may be evaluated with the help of
sinh (ж)
(4.152)
Inserting the result
mui
2тгЙ sinh(/i/3cet)
(4.153)
into (4.140) yields (4.141) as expected.
4.6.4 Effective action
From (4.124) we obtain together with (4.141) the influence functional
F[£] = exp
1 rhg rh0
— dr / dak(r - cr)q(T)q((i)
2n Jo Jo
(4.154)
The nonlocal kernel k(r) is given by the sum over all kernels k^r) defined in (-1.1.”
Introducing the Laplace transform of the damping kernel (4.14)
/•oo	9
7(z) = / dze-zty(t) = — I
Jo	M Jo
dwJ(w) z
7Г Ш Z1 + LJ2
(4.15
and making use of the spectral density of bath oscillators (4.13), we finally obtain fb
kernel
д/ +°°
= z-z 52
TlP nTToo
(4.156)
As the nain result let us note that dissipation may be taken into account within thi
path-integral formalism by adding a nonlocal term to the action of the system under
consideiation. In view of (4.123), we get the imaginary-time effective action for a
damped system
1 th, fl	rKfl
S^[Q] = SE[q] + - dr
2 Jo Jo
dcr^r — о)</('г)д(<т).
(4.157)
24U Dissipative Quantum Systems
We finish this section by discussing the kernel k(r) for a specific damping kernel
7(7-). Since Ohmic damping corresponds to 7(2) = 7, we encounter problems with the
convergence of the sum in (4.156). We therefore consider the damping kernel of the
Drude model which according to (4.16) is described by 7(|z/n|) = 7^d/(^d +l^nl)- Since
the Fourier coefficients of the kernel approach a constant for n —> 00, it is reasonable to
split off a periodic delta function which we defined in (4.137). Expressing the numerator
of 7(|z/n|) as an integral over an exponential function, the summation over n leads to a
geometric sum. This leaves us with
k(r) =	6(r - nh0) -	[ ds——-e^DS	(4.158)
hfi Jo cosh(z/s) - cos(z/r)
where z/ = 2-к/h/3 is the first Matsubara frequency. Since we are only interested in
the case of very large cutoff frequency wD, we expand the integral asymptotically by
partial integration. The kernel then becomes
7Г AV” W	1
k(r) =	52 6(r - nh/3) - ——-----+ OU'r,1 )	(4.159)
n=-oo	\ПР) sin2 ( - I
к 2 J
In the limit of Ohmic damping (wp —> 00), the delta function contribution diverges
in order to ensure nonlocality. This corresponds to the divergence of the potential
renormalization in (4.4). The second term leads to an interaction between paths at
different times. Taking the limit of zero temperature, we find a long range interaction
decaying algebraically like k(r) = —{My/ir)r~2.
4.7 Decay of a metastable state
We now want to make use of what we have discussed in the last two sections by applying
it to the decay of a metastable state in the presence of dissipation. A comprehensive
treatment would be beyond the scope of this chapter. We rather try to convey some
of the key ideas. For details we refer the reader to the literature, e.g. to the review by
Hanggi et al. [21], and mention that other aspects of tunneling are addressed in various
sections of Chapter 3 as well as in Sections 5.6 and 6.3.1.
To be specific, we consider the cubic potential shown in Fig. 4.2 which may be
expressed in the form
(i - -) •
4	\ QoJ
(4.160)
This potential possesses a minimum at q = 0 with a frequency w0 for small oscillations
around this minimum. At gb = 2g0/3 there is a barrier of height VJ, = (2/27)IWiUq(7q.
A harmonic approximation around the barrier yields V(q) = Vb — (Af/2)wb<?2 with a
barrier frequency = w0. Where it is possible we will distinguish between the two
frequencies in order to make the physical origin of the results more transparent.
4.7 Decay of a metastable state 24 I
Fig. 4.2: Cubic potential as defined by (4.160).
We are now interested in the decay of a metastable state initially prepared in t
potential well at q = 0. As we will see below, the decay rate may be calculated fro'” i.v
partition function. Unfortunately, for a cubic potential the partition function ma у Hut
be obtained exactly. We therefore have to resort to the semiclassical approximatio’
In principle, we could proceed as in Section 4.6 and first evaluate the action , ..a
paths starting and ending at a certain point and then integrate over all these points.
Howeve•, for our nonlinear problem this would imply that we include contributions
beyond the semiclassical approximation. Rather it is sufficient to look for extrema
of the action among arbitrary paths and take into account fluctuations around these
extremal paths semiclassically. Then the fluctuations do not have to vanish at the
initial aid final time.
4.7.1 Crossover temperature
The classical equation of motion for a cubic potential in the presence of dissipation is
given by
3	qA
Mqd-Mu%qd +-Mwq—- dcrfc(r - (T)qci(er) = 0	(4.161)
2	q0	Jo
where the dissipative kernel k(r) was defined in (4.156). Since the integral over k{r)
vanishes it is clear that constant solutions at the extrema of the potential, i.e. q — 0
and q — Qb, are solutions of (4.161). For high temperatures (corresponding to short
imaginary times) these two paths are the stationary points of the action. Let us first
consider the path which remains in the potential minimum, i.e. at q — 0. Fluctuations
around this path may be described by a Fourier series
+oo
£(r) = E hT4'"7	(4.162)
7l = -00
242 Dissipative Quantum Systems
where and are complex conjugate in order to ensure a real path. The semiclas-
sical action
SEW = dr
JO
M 2 M 2 2 /	Qcl \	1 fh/3
+ Vwo9d 1--------------+0 / dcrk(r - cr)qcl(r)qcI(cr)
2	2	\	qQ I	L Jo
M	M „ (	Ocl\ о 1
+	+ TW0 1 - 3- К + 9 / dcrk^T ~
2	2 у	Qo J 2 Jo
then becomes
sE =
n=-oo
(4.163)
(4.164)
Obviously, any deviation from the classical path will increase the action and q = 0
represents a minimum of the action.
The same analysis may be performed for q — which yields
Л/й/Ч +°° .
SE = Wb + -r- E + Ы?(Ы) - ^ь) Ы2-	(4-165)
Again, the expression is quadratic in the Fourier components £n but there is one fluctu-
ation mode, Co, which decreases the action. Therefore, q = qt> corresponds to a saddle
point of the action. This should not come as a surprise since this classical path lies at
the top of the barrier. With decreasing temperature one reaches a point where
2/2 + Ы7(Ы) -Wb = 0-	(4.166)
Below this temperature another fluctuation mode becomes unstable indicating the
appearance of an additional solution.
The solution of (4.166) defines a temperature which depends on the damping
strength. For Ohmic damping one obtains
which decreases with increasing damping strength 7. In the case of zero damping this
becomes
To = ^r-	(4-168)
27Г Kq
The latter result can be understood by realizing that a periodic path traversing the
minimum of the inverted potential needs at least the time 2тг/сиь corresponding to one
period in the harmonically approximated potential. Translating this into temperature,
one immediately obtains (4.168). Since the anharmonicity of the potential will increase
the period of finite oscillations, the new solution at temperatures just below To will be
a harmonic oscillation of small amplitude.
___________________________________4.7 Decay of a metastable state 24c
Or the other hand, at zero temperature the periodic path may take infinite time
In this case one can solve (4.161) in the absence of dissipation to obtain
1
9b(t) = Qo—Г27----(4.1C9)
cosh (w0r/2)	4
This z uo-energy path starts at q = 0, traverses the minimum of the inverted potential
to reach q = q0 where it is reflected and retraces its path. Because of the reflection
this solution is often referred to as “bounce”. In the vicinity of q = 0 the motion is
exponentially slow. Therefore, it is not possible to express this solution on the tirn<’
interval from t = 0 to oo. Instead, one uses the time interval starting at t = —
which is obtained by shifting the finite imaginary time interval by — h/3/2 and lettinr
h/3 —> ix).
The temperature TQ leads to a division into two regimes and is therefore often
referred to as crossover temperature. For high temperatures the decay mechanism i;
mainly thermal activation. As temperature is lowered, quantum effects become visible
to a certain extent as we will see below. The appearance of the bounce solution below
the crossover temperature is interpreted in terms of a new decay channel, namely
quantum tunneling which is the dominant effect at very low temperatures.
4.7.2 Imaginary part of the free energy
For temperatures above the crossover temperature we have shown the existence of a
saddle point of the action which is related to the fact that we are treating a poi
containing a barrier. The existence of this saddle point for lower temperatu.
become clear from arguments given in Section 4.7.4. We therefore have to address the
questio i how to properly treat the fluctuations around the saddle point. Fluctua1
modes vith positive eigenvalues can be treated by evaluating Gaussian integrals a.
have seen in Section 4.6.3. This is no longer the case for negative eigenvalues where
the Gaussian integral
(4™)
V 27Г J—<X)	у 2 J
does net exist. On the interval from —oo to zero the problems are due to the semi-
classica' approximation. Taking into account the anharmonicity of the potential would
yield a finite value which to leading order is in fact described by the path q = 0 and
fluctuations around it. The divergence for positive £0 is more serious because the po-
tential on this side is not bounded from below. This difficulty may be removed by an
analytical continuation where one deforms the path of integration in (4.170) to run
into the direction of steepest descent, i.e. it continues from zero along the positive
imaginary axis. By this procedure the integral acquires an imaginary part
f [MJ3	(М0Л 2\\ i
(4.171)
Dissipative quantum oystems
Comparing (4.164) and (4.165), we find that for /3Vb » 1 the imaginary part of the
partition function will be exponentially small compared to the real part. Nevertheless
we have to keep it as the leading imaginary term.
As a consequence of the complex partition function, the free energy
F = -|ln(Z)	(4.172)
also becomes complex. Since at zero temperature the free energy turns into the energy,
this would result in a decay of the probability amplitude of a state proportional to
exp(2Im(F)t/fi). This leads us to conjecture the following expression for the decay
rate [22]
2
Г = —-ImF	(4.173)
n
which expressed in terms of the free energy should also be valid at finite temperatures.
It turns out, however, that above the crossover temperature (4.173) has to be modified
into [23]
2 T
r=---glmF.	(4.174)
n 1
The additional prefactor ensures correct results for very high temperatures, i.e. in
the classical limit. The difference between (4.173) and (4.174) will be motivated in
Section 4.7.4. It should be emphasized that the given arguments do not represent a
proof of the relation between the decay rate and the imaginary part of the free energy
in the dissipative case. However, there exist independent methods employing periodic
orbit [21,24] or real-time path integral techniques [25] which lead to the same results
for the decay rates. For details we refer the reader to the literature.
4.7.3 Above crossover
We now want to calculate the decay rate above the crossover temperature using (4.174).
Within the semiclassical approximation the partition function consists of two contri-
butions
Z = Z0 + Zb	(4.175)
from the potential well and the barrier, respectively. Setting^ = an+ibn with a~n = an
and b_n = — bn to ensure real paths, we obtain by evaluating Gaussian integrals with
the action (4.164)
1 °°	1
-----2	(4-176)
w0 ,7=i I'fi + DDWj + Wo
where N is a normalization factor which will turn out to be irrelevant. With (4.165)
the second contribution to the partition function yields
j oo	p-/?Vb
Zb = .V— П	i	(4-177)
2wb „Vj z/2 + z/„7(z/„) - wg
_________________________________________4.7 Decay of a metastable state 24"
where we made use of (4.171). Using the fact that for fiVb 1 the imaginary part of
the partition function is exponentially small compared to the real part, we obtain fm
the imaginary part of the free energy
* С-Ж TT Уп Vnl(vn) + COp
P 2wb	z/2 + z/n7(z/n) - cog'
(4.178)
This result depends only on the barrier height and the frequencies of small oscillations
around q = 0 and qb but not on the specific form of the potential. This is a consequence
of the fact that the stationary points of the action do not explore the whole potential
and of the semiclassical approximation.
The decay rate according to (4.174) may be expressed as
г = дгс1
where
Гс1 =
h wb
(4.17'T
(4.186)
is the classical rate derived in 1940 by Kramers [26] for not too weak Ohmic da.
so that thermal equilibrium in the well is guaranteed, and
г _ TT I'n + ^nT(^n) +
q „I1! P* +	- Ub
(4.181)
represerts the quantum corrections. The temperature dependence of the exponen-
tial term is the well-known Arrhenius behavior which implies that thermally activated
decay becomes strongly suppressed as temperature is lowered. At sufficiently low tem-
peratures quantum tunneling, which is discussed in the following section, therefore
becomes the dominant process. The factor (4.181) leads to an enhancement of the
thermal decay rate due to quantum corrections to the probability distribution in the
potential well and at the barrier. The expression for /q is diverging as the crossover
temperature is approached from above. Close to the crossover temperature a more
refined treatment taking into account anharmonicities of the potential is required [27].
Below the crossover temperature one has to deal with the bounce solution as we will
discuss in the following section for the example of zero temperature.
4.7.4 Zero temperature
Since we want to elucidate only some keypoints we concentrate on the case, of vanishing
damping. The bounce solution (4.169) represents a stationary solution which in fact
is a saddle point as we will show now. Since the fluctuation contribution to the action
follows from (4.163) as
dry

\ Qo J
(4.182)
246 Dissipative Quantum Systems
we obtain the fluctuation modes as the eigenfunctions of
-ёп+ч2 f i -
\ 9o /
(4.183)
Taking the time derivative of the undamped version of (4.161) we find by comparison
with (4.183) that the time derivative of the bounce
sinh(cuo7"/2)
Qb (t) = -Qo^o—-3------—
cosh (woT/2)
(4.184)
is an eigenfunction with zero eigenvalue. The existence of this zero mode is a conse-
quence of the fact that the position of the bounce in time is arbitrary and that for
small e
Qb (t + e) = qB(r) + eqB(r).
(4.185)
After this discussion, we may write the partition function in the form
Z = Zo (1 + iKh/3e~Sb^	(4.186)
where Zo according to (4.176) is the contribution of the constant path q — 0 and Sb
is the action of the bounce. The constant к is the ratio of the contributions of the
fluctuations around q = дв(т) and q = 0 which we do not want to evaluate explicitly
here. Finally, in writing the factor ft/J we assumed for the moment that the temperature
is small but finite. This factor stems from the integration over the possible positions
of the bounce solution and can be viewed as arising from the zero mode (4.184). This
factor cancels with the temperature factor in the relation between the free energy and
the partition function (4.172). It is actually this integration over the zero mode which
leads to the additional temperature factor in the rate expression (4.174) above the
crossover temperature because in this case a zero mode no longer exists.
From (4.186) one obtains with (4.172) and (4.173) for the decay rate
Г = 2ке“5ь/\	(4.187)
The calculation of the action of the bounce solution is facilitated by noting that this
solution corresponds to zero energy. As a consequence
Sb = drMq2a = 2 I* dqc^2MV(qci),	(4.188)
and we thus recover the exponent for the decay rate known from WKB. For the cubic
potential (4.160) the action takes the value
Sb =	(4.189)
The prefactor к may also be calculated and is found to be in agreement with the
standard WKB result. For further details the reader is referred to the literature [28, 29]
where also results for the weakly damped case can be found [30].
References
Fig. 4.3 : Arrhenius plot for the decay rate i ...
a metastable state. The damping strength fron
the upper to the lower curve takes the valuer
7/2u»q = 0,0.5, and 1. The low temperatun
data were derived from the tables given in [27]
Wc close this chapter by presenting in Fig. 4.3 the decay rate in an Arrhenius
plot for three different values of the damping constant. For high temperatures, i.e.
on tht left side of the plot, the exponential dependence of the rate on the inverse
temperature leads to straight line. Below the crossover temperature the curves flatten
out indicating that the regime of quantum tunneling is reached. The uppermost curve
represents the undamped result while the second and third curve correspond to 7 =
w0 anc; 2w0, respectively. With increasing damping strength the crossover from tie.
classic il to the quantum regime becomes less distinct. More important, the crossover
temperature decreases and the quantum tunneling rate is suppressed. The decay rare
thus approaches the thermal activation rate and we may conclude that dissipation
tends lo make the system more classical.
References
[1]	A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and
W. Zwerger, Rev. Mod. Phys. 59, 1 (1987); 67, 725 (1995).
[2]	U. Weiss, Quantum Dissipative Systems (World Scientific, 1993).
[3]	H. Dekker, Phys. Rep. 80, 1 (1981).
[4]	I. R. Senitzky, Phys. Rev. 119, 670 (1960); 124, 642 (1961);
G. W. Ford, M. Kac, and P. Mazur, J. Math. Phys. 6, 504 (1965);
P. Ullersma, Physica 32, 27, 56, 74, 90 (1966).
[5]	V. B. Magalinskii, Zh. Eksp. Teor. Fiz. 36, 1942 (1959) [Sov. Phys. JETP 9, 1IT1
(1959)].
[6]	R. Zwanzig, J. Stat. Phys. 9 , 215 (1973).
D 10O1 native v^uaiiturn kjj'stems
[7]	A. 0. Caldeira and A. J. Leggett, Phys. Rev. Lett. 46, 211 (1981);
A. 0. Caldeira and A. J. Leggett, Ann. Phys. (N.Y.) 149, 374 (1983), ibid. 153,
445 (1984).
[8]	U. Eckern, G. Schon, and V. Ambegaokar, Phys. Rev. В 30, 6419 (1984).
[9]	P. Mazur and E. Montroll, J. Math. Phys. 1, 70 (1960).
[10]	R. J. Rubin, Phys. Rev. 131, 964 (1963).
[11]	R. Kubo, J. Phys. Soc. Japan 12, 570 (1957);
R. Kubo, Rep. Progr. Phys. 29, 255 (1966).
[12]	H. B. Callen and T. A. Welton, Phys. Rev. 83, 34 (1951).
[13]	H. Grabert, U. Weiss, and P. Talkner, Z. Phys. В 55, 87 (1984).
[14]	R. Jung, G.-L. Ingold, and H. Grabert, Phys. Rev. A 32, 2510 (1985).
[15]	A. Einstein, Ann. Phys. (Leipzig) 17, 549 (1905).
[16]	V. Hakim and V. Ambegaokar, Phys. Rev. A 32, 423 (1985).
[17]	R. P. Feynman, Rev. Mod. Phys. 20, 367 (1948).
[18]	Among the textbooks treating path integrals in detail are e.g.:
R.	P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals
(McGraw-Hill, 1965);
L.	S. Schulman, Techniques and applications of path integration (Wiley, 1981);
H.	Kleinert, Path integrals in quantum mechanics, statistics and polymer physics
(World Scientific, 1995).
[19]	R. P. Feynman and F. L. Vernon, Jr., Ann. Phys. (N.Y.) 24, 118 (1963).
[20]	H. Grabert, P. Schramm, and G.-L. Ingold, Phys. Rep. 168, 115 (1988).
[21]	P. Hanggi, P. Talkner, and M. Borkovec, Rev. Mod. Phys. 62, 251 (1990).
[22]	J. S. Langer, Ann. Phys. (N.Y.) 41, 108 (1967).
[23]	I. Affleck, Phys. Rev. Lett. 46, 388 (1981).
[24]	P. Hanggi and W. Hontscha, Ber. Bunsenges. Phys. Chem. 95, 379 (1991).
[25]	J. Ankerhold, H. Grabert, and G.-L. Ingold, Phys. Rev. E 51, 4267 (1995);
J. Ankerhold and H. Grabert, Phys. Rev. E 52, 4704 (1995).
[26]	H. A. Kramers, Physica 7, 284 (1940).
[27]	H. Grabert, P. Olschowski, and U. Weiss, Phys. Rev. В 36, 1931 (1987).
[28]	I. K. Affleck and F. De Luccia, Phys. Rev. D 20, 3168 (1979).
[29]	S. A. Gurvitz, Phys. Rev. A 38, 1747 (1988).
[30]	E. Freidkin, P. S. Riseborough, and P. Hanggi, J. Phys. C 21, 1543 (1988).
Peter Hanggi
Driven Quantum Systems
5.1	Introduction
During recent years we could bear witness to an immense research activity, both in
experimental and theoretical physics, as well as in chemistry, aimed at understanding
the detailed dynamics of quantum systems that are exposed to strong time-dependent
externa! fields. The quantum mechanics of explicitly time-dependent Hamiltonians
generates a variety of novel phenomena that are not accessible within ordinary station-
ary quantum mechanics. In particular, the development of laser and maser systems
opened .he doorway for creation of novel effects in atoms and molecules, which interaiЛ
with strong electromagnetic fields [1-4]. For example, an atom exposed continuously
to an oscillating field eventually ionizes, whatever the values of the (angular) frequency
u> and tie intensity I of the field is. The rate at which the atom ionizes depends on
both, the driving frequency w and the intensity I. Interestingly enough, in a pioneer-
ing paper by H. R. Reiss in 1970 [5], the seemingly paradoxical result was established
that extremely strong field intensities lead to smaller transition probabilities than m1"
modest ntensities, i.e. one observes a declining yield with increasing intensity,
phenomenon of stabilization that is typical for the above threshold ionization (ATI) is
still actively discussed, both in experimental and theoretical groups [6,7]. Other act!
ities tha, are in the limelight of current topical research relate to the active control of
quantuir processes; e.g. the selective control of reaction yields of products in chemical
reactions by use of a sequence of properly designed coherent light pulses [8].
Our prime concern here will focus on the quantum dynamics of driven bistable sys-
tems. Such systems exhibit an interplay of three characteristic components, (i) non-
linearity, (ii) nonequilibrium behaviour (as a result of the driving), and (iii) quantum
tunneling, with the latter providing a paradigm for quantum coherence phenomena.
We s rail approach this complexity of driven quantum systems in a sequence of
steps. Ir. Sect. 5.2 we introduce archetypal time-dependent interaction schemes such
as the dipole interaction with laser fields or the electron spin resonance system. Sec-
tion 5.3 ntroduces the reader to a variety of tools suitable for tackling the quantum
dynamics of explicitly time-dependent (time-periodic and non-periodic) Schrodinger
equations. Exactly solvable quantum systems with time-dependent potentials are dis-
250 Driven Quantum Systems
cussed in Sect. 5.4. Among these are the quantum mechanics of a two-level system
(TLS) interacting with a circularly polarized laser field. Clearly, the presence of so-
called anti-rotating terms makes most systems inaccessible to analytical closed solu-
tions. Hence, we address with Sect. 5.5 prominent numerical methods for periodically
driven quantum systems. As an application to driven quantum systems, we study in
Sect. 5.6 the phenomenon of coherent tunneling in periodically driven bistable quan-
tum systems. As an intriguing result, we demonstrate therein that an appropriately
designed coherent conyinuous-wave (cw) driving can bring quantum tunneling to an
almost complete standstill (coherent destruction of tunneling [9]). This phenomenon in
turn produces other novel quantum phenomena such as low-frequency radiation and/or
intense, non-perturbative, even-harmonic generation in symmetric systems that possess
an inversion symmetry [10]. The possibility of controlling quantum dynamics by appli-
cation of shape- and phase-designed pulse perturbations is elucidated in Sect. 5.7 with a
time-dependent dipole coupling between two Born-Oppenheirner surfaces. Conclusions
and an outlook are given in the final Sect. 5.8.
5.2	Time-dependent interactions
It is a well-known fact that the time evolution of an isolated quantum system, described
by a Hamiltonian Ho with a discrete spectrum that acts on the space of relevant system
variables x cannot exhibit the type of behaviour usually associated with deterministic
chaos of classical systems. This is so because the time evolution of a quantum state
is almost periodic since it can be expanded in terms of the eigenfunctions i/;n with
eigenvalues En. Only when the spacing between energy levels becomes very small,
the quantum system can imitate various features of the classical behaviour on certain
time scales. It should be noted, however, that even very small quantum systems such
as atoms, quantum dots, molecules, etc., can exhibit a nontrivial behaviour when ex-
posed to intense external fields. Some typical situations are introduced in the following
subsections.
5.2.1 Laser interactions
A vast variety of new nonlinear phenomena such as above-threshold ionization of atoms,
multi-photon dissociation or excitation of atoms or molecules occur in intense laser
fields [1-7]. Usually, the relevant wavelength of the radiation field is far larger than
the size of the quantum system of atomic dimension (long wavelength approximation).
Then, we can invoke in addition the electric-dipole approximation. Given the dipole
moment the interaction energy between the quantum system and the classical
electric field E(t) is given by
V(x, t) — —м(ж) • E(t),
(5.1)
5.2	Time-dependent interactions
which, for a perpetually applied monochromatic field of amplitude Eo and angula
frequency co, reduces to
V(x, t) = -д(ж) • Eosin(wt + ф).	(5.2
In many circumstances only a finite number of quantum levels strongly interaci
under the influence of the time-dependent laser field. This means that a truncation tr
a multi-level quantum system in which only a finite number of quantum states strong!}
interact is adequate. In particular, the truncation to two relevant levels only, i.e., the si
called driven two-level system (TLS), is of enormous practical importance, cf. Sect. 5. i
Setting A = E2 — Ey, this truncation in the energy representation of the ground st '
|1) and excited state |2) is in terms of the Pauli spin matrices az and ax given by
HTLS(t) =	- M-E’o sin(cot + ф)ах,	(Г, V
with = (2|cr|l) being the transition dipole moment. Here we have used a 1
approximation of the field Eo in .redirection. The linearly polarized field in (5
with 2hX = /лЕо, be regarded as a superposition of left and right circularly polarized
radiation, namely setting ф = тг/2 we have
2A cos cot = A exp(—icot) + A exp(icot).	('
Foi the absorption process |l,n) —> |2,n — 1), the term Aexp(—icot) supplies the-
energy ha) to the system. It corresponds to the rotating-wave (RW) term, while the
term A exp(icot) is called the anti-rotating-wave term. This anti-RW term removes the
energy hej from the system, i.e., |1,n) —> |2, n + 1), and is thus energy nonconserving.
Likewise, the process of emission |2,n) —> |l,n + 1) is a RW term, while the second
order process |2,n) —> |l,n — 1) is again an energy nonconserving anti-RW term.
5.2.2 Spin magnetic resonance
In elec.,ron-spin resonance (ESR), nuclear magnetic-spin resonance (NMR) or atomic-
beam spectroscopy, a particle of total angular momentum J = K/2 is placed in both a
static magnetic field Bq in the г-direction, and a time-dependent oscillating magnetic
field 2B\ cos(cot) in ^-direction. The magnetic moment of the particle is p = yj, where
7 is th з gyromagnetic ratio. Therefore the Hamiltonian for the particle in the
time-dependent magnetic field thus reads
#smrW = -p, В = -^hycjzB0 - /iy<rIB1 cos(cot),	(5.51
where । гтх, ay, az) are the Pauli matrices so that the spin is given by her/2. With
A - Й7В0	(5.0;
and
цЕ0 = 2hX = h-yBi,	(5.7:
this Hamiltonian coincides with the laser-driven TLS in (5.3).
2oz Driven Quantum Systems
5.3	Floquet and generalized Floquet theory
5.3.1 Floquet theory
With intense fields interacting with the system, it is well known [11,12] that the semi-
classical theory (treating the field as a classical field) provides results that are equivalent
to those obtained from a fully quantized theory whenever fluctuations in the photon
number (which, for example, are of importance for spontaneous radiation processes)
can safely be neglected. We shall be interested first in the investigation of quantum
systems with their Hamiltonian being a periodic function in time,
H(f) = H(t + T),	(5.8)
where T is the period of the perturbation. The symmetry of the Hamiltonian under
discrete time translations, t —> t + T, enables the use of the Floquet formalism [13].
This formalism is the appropriate vehicle to study strongly driven periodic quantum
systems: Not only does it respect the periodicity of the perturbation at all levels of
approximation, but its use intrinsically avoids also the occurrence of so-called secular
terms, terms that are linear or not periodic in the time variable. The latter character-
istically occur in the application of conventional Rayleigh-Schrodinger time-dependent
perturbation theory. The Schrodinger equation for the quantum system may be written
with the restriction to a one-dimensional system, as
H(x, t) — ift—Ф(а;, i) = 0.	(5-9)
With
H(x,t) = H0(x) + V(x,t), V{x,t) = V(x,t + T),	(5.10)
the unperturbed Hamiltonian HQ(x) is assumed to possess a complete orthonormal
set of eigenfunctions {</?n(z)} with corresponding eigenvalues {f?n}. According to the
Floquet theorem, there exist solutions to (5.9) that have the form (so-called Floquet-
state solution) [13]
tya(x,t) == exp(—ieat/K)$a(x,t),	(5.11)
where Фа(гг, t) is periodic in time, i.e., it is a Floquet mode obeying
<FQ(rr, t) = Фа(а;, t + T).	(5.12)
Here, ea is a real parameter, being unique up to multiples of hw, w = 2тг/Т. It is
termed the Floquet characteristic exponent, or the quasienergy [11,12]. The term
quasienergy reflects the formal analogy with the quasimomentum fc, characterizing the
Bloch eigenstates in a periodic solid. Upon substituting (5.11) into (5.9), one obtains
the eigenvalue equation for the quasienergy ea. With the Hermitian operator
'H(x, t) = H(x, t) —
(5.13)
5.3 Floquet and generalized Floquet theory 251
one finds that
Н(х,1)Фа(х,1) = ea$a(x,t).	(5.14;
We immediately notice that the Floquet modes
Фа'СМ) = Фо(х, t)exp(inwt) = Фап(М)	(5.15)
with r being an integer number n = 0, ±1, ±2,... yields the identical solution to that
in (5.11), but with the shifted quasienergy
fa —> fa' =	+ nhcu = ean.	(5.16)
Hence, the index a corresponds to a whole class of solutions indexed by a' = (a, n),
n — 0, ±1, ±2,.... The eigenvalues {eQ} therefore can be mapped into a first Brillouin
zone, obeying —hw/2 < e < fiw/2. For the Hermitian operator H(x, t) it is convenient
to introduce the composite Hilbert space 71 ® T made up of the Hilbert space 71 of
square integrable functions on configuration space and the space T of functions which
are periodic in t with period T = 2tt/w [14]. For the spatial part the inner product is
definec by
f	&n,m,	(5.1 i )
while the temporal part is spanned by the orthonormal set of Fourier vectors =
exp(inc;t), n = 0, ±1, ±2,..., and the inner product in T reads
1 fT
(m,n) = — J dt exp[\(n - m)ut] = 5щт.	(5.185
Thts, the eigenvectors of H obey the orthonormality condition in the composite
Hilbert space 71 ® T,
| rT fca
<<Фа,(^)|Ф^(<)>> = - / dt ЛхФ*а,(х,1)Фв,(х,/.) = 6a,^ = 5a,e8n^m,	(5.19)
1 JO J—oo
and for n a complete set in 7Z. ® T,
£ £ Ф:п(я, е)Фап(У, t') = S(x - y)5(t - t').	(5.20)
Q П
Note that in (5.20) we must extend the sum over all Brillouin zones, i.e., over all t u
representatives n in a class, cf. (5.16). For fixed equal time t = t', the Floquet mod' s
of the first Brillouin zone TQoC'M) form a complete set in 71,
£ф‘(М)Фа(?М) = 8(x - y).	(’> "
Cl
Clearly, with t' yl t + mT = t (modT), the functions {Ф* (:r, t), Фа(у, t')} do not iC .
an orthonormal set in 71.
254 Driven Quantum Systems
5.3.2 General properties of Floquet theory
With a monochromatic perturbation
V(x,t) = —Sx sin(wt + ф)
(5.22)
the quasienergy ea is a function of the parameters S and ai, but does not depend on the
arbitrary, but fixed phase ф. This is so because a shift of the time origin t0 = 0 —> t0 —
—ф/о) will lift a dependence of ea on ф in the quasienergy eigenvalue equation in (5.14).
In contrast, the time-dependent Floquet function Фа(а;,<) depends, at fixed time, on
the phase. The quasienergy eigenvalue equation in (5.14) has the form of the time-
independent Schrodinger equation in the composite Hilbert space TZ ® T- This feature
reveals the great advantage of the Floquet formalism: It is now straightforward to use
all theorems characteristic for time-independent Schrodinger theory for the periodically
driven quantum dynamics, such as the Rayleigh-Ritz variation principle for stationary
perturbation theory, the von-Neumann-Wigner degeneracy theorem, or the Hellmann-
Feynman theorem, etc.
With H(t) being a time-dependent function, the energy E is no longer conserved.
Instead, let us consider the averaged energy in a Floquet state Фа(з:, t). This quantity
reads
Ha = ^=f dt^Q(;r,t)|#(2:,f)^Q(2:,t))
1 Jo
= еа + ((Фа|1Й^|Фа)).	(5.23)
Invoking a Fourier expansion of the time periodic Floquet function Фа(а;, t) = Y,k ck(x)
exp(—ikait), Y,k f (x)|2 = 1 = Y,k(ck\ck), (5.23) can be recast as a sum over к,
Ha = ea + ^2 Ыш(ск\ск) = J2 (6a + Икш)(ск\ск).	(5.24)
k=~ oo	k=—oo
Hence, Ha can be looked upon as the energy accumulated in each harmonic mode of
^a{x,t) = exp(-ieQt/?i^Q(;z:, t), and averaged with respect to the weight of each of
these harmonics. Moreover, one can apply the Hellmann-Feynman theorem,
= ((ФН|^Н|ФН)).
Setting r — ait and 'Н(х,т) = Н(х,т) — \Ка>д/дт, one finds
(дНУ	d	Id
-к—	=	= -«-к;
\dcd J т	От	a> ot
and consequently obtains [15]
(5.25)
(5.26)
(5.27)
5.3 Floquet and generalized Floquet theory 25
Next we discuss qualitative, general features of quasienergies and Floquet mode
with respect to their frequency and field dependence. As mentioned before, if ea - Ca
possesses the Floquet mode Фа0(^, t), the modes
Ф«0-t ФаА: = Фао(М)ехр(киН), к = 0, ±1, 2,...,	(5.2 '
are also solutions with quasienergies
fak — ea0 + hkcj,	(5.29,
yielding identical physical states,
ФаоСМ) = ехр(-1€аО^/^)Фао(х,О
= ^ak(x,t).	(5.Г"'4
For an interaction S -» 0 that is switched off adiabatically, the Floquet modes and
quasie lergies obey
Фак(я, i) $ak(x>= ФсЛ1) exp(iwfct)	(5.31)
and
fak(S, cj) -5	— Ea + kflLc) ,	(5.3z;
with	denoting the eigenfunctions and eigenvalues of the time-independent
part Hq of the Hamiltonian (5.10). Thus, when S -> 0, the quasienergies depend
linearly on frequency so that at some frequency values different levels intersect.
When 9	0, the interaction operator mixes these levels, depending on the symmetry
properties of the Hamiltonian. Given a symmetry for 7/(a:,Z), the Floquet eigenvalues
fQit can be separated into groups: Levels in each group mix with each other, but do
not interact with levels of other groups. Let us consider levels e°an and e°0k of the same
group at resonances,
Ea + 7ihwres = Ep + /chcures	(5.33)
with <jI3S being the frequency of an (unperturbed) resonance. According to the von-
Neuma in-Wigner theorem [16], these levels of the same group will no longer intersect
for finite S ф 0. In other words, these levels develop into avoided crossings (Fig. 5.1a).
If the levels obeying (5.33) belong to a different group, for example to different gen-
eralized parity states, see below in Sect. 5.5, the quasienergies at finite S 0 exhibit
exact ciossings; cf. Fig. 5.1b.
These considerations, conducted without any approximation, leading to avoided vs.
exact ciossings, determine many interesting and novel features of driven quantum sys-
tems. Ззше interesting consequences follow immediately from the structure in Fig. 5 1
Starting out from a stationary state Ф(гг, t) = ^i(x) exp(—iEit/h) the smooth adia-
batic switch-on of the interaction with w < wres (w > wres) will transfer the system in’
256 Driven Quantum Systems
Fig. 5.1: Quasi-energy dependence on frequency w of a monochromatic electric-dipole perturbation
near the unperturbed resonance wres between two levels. The dashed lines correspond to quasienergies
for S —> 0. In panel (a), we depict an avoided crossing for two levels belonging to the same symmetry
related group number. Note that with finite S the dotted parts belong to the Floquet mode $2m,
while the solid parts belong to the state $in. In panel (b), we depict an exact crossing between two
members of quasienergies belonging to different symmetry-related groups. With S 0, the location
of the resonance generally undergoes a shift <5 = wres(S / 0) - wres(S = 0) (so-called Bloch-Siegert
shift) [17] that depends on the intensity of S. Only for S -+ 0 does the resonance frequency coincide
with the unperturbed resonance wres.
a quasienergy state Ф.о [18]. Upon increasing (decreasing) adiabatically the frequency
to a value w > wres (w < wres) and again smoothly switching off the perturbation, the
system generally jumps to a different state Ф(ж,£) = exp(—i£?2t//i). For exam-
ple, this phenomenon is known in NMR as spin exchange; it relates to a rapid (as
compared to relaxation processes) adiabatic crossing of the resonance. Moreover, as
seen in Fig. 5.1a, the quasienergy б2& and Floquet mode Ф2& as a function of frequency
exhibit jump discontinuities at the frequencies of the unperturbed resonance, i.e., the
change of energy between the two parts of the solid lines (or dashed lines, respectively).
5.3.3 Time-evolution operators for Floquet Hamiltonians
The time propagator A'(t, to), defined by
|Ф(0) = A(t,t0)^(t0)), A(to,to) = l,	(5.34)
assumes special properties when H(t) = H(t + T) is periodic. In particular the prop-
agator over a full period K(T, 0) can be used to construct a discrete quantum map,
propagating an initial state over long multiples of the fundamental period by observing
K(nT,0) = [A(T,0)]n.
(5.35)
5.3 Floquet and generalized Floquet theory 25)
This important relation follows readily from the periodicity of H(t) and its definition
Namely, we find with t0 = 0 (T denotes time-ordering of operators)
K(nT,0) = T exp
n Jo
nT
dt#(f)
— T exp
i " rkT
dtH(t)
which with H(t) = H(t + T) simplifies to
К(nT, 0) = T exp
dtH(t)
n
= T П exp
fc=i
' i
-- dtH(t)
n, Jo
(5.36)
4 [TdtH(t)
It Jo
Because the terms over a full period are equal, they do commute. Hence the time-
ordering operator can be moved in front of a single term, yielding
K(nT,0) = ft T exp
= [K(T, 0)]n.	(5.3-
Likewiie, one can show that with tQ = 0 the following relation holds
K(t + T,T) = K(t,0),	(5.3. ,
which mplies that
K(t + T,0) = K(t,0)K(T,0).	(5.39)
Note that K(t,0) does not commute with K(T, 0), except at times t = nT, so that
(5.39) A'ith the right-hand-side product reversed does not hold. A highly important,
feature of (5.33) — (5.39) is that the knowledge of the propagator over a fundamental
period T = 2tt/cj provides all the information needed to study the long-time dynamics
of periDdically driven quantum systems. That is, upon a diagonalization with an
unitary transformation S
5*А'(7’,0)5 = exp(-iD),	(G.lt’t
with D being a diagonal matrix, composed of the eigenphases {eQT}, one obtains
K(nT,0) = [K(T, 0)]n = Sexp(-inD)Sf.	(t
This re ation can be used to propagate any initial state
|Ф(0)) = £са|Ф„(0)), ca = (ФД0)|Ф(0)>.	(5.42)
258 Driven Quantum Systems
in a stroboscopic manner. Such a procedure generates a discrete quantum map. With
Фа(а:, t — 0) = Фа(а:, t — 0), its time evolution follows from (5.11) as
V(x,t) = 52саехр(-ко^//1)Фа(х, t).	(5.43)
a
With Ф(ж, t) = (x\K(t, 0)|Ф(0)), a spectral representation for the propagator
K(x,t;xo,0) = (x|A"(t, О)|гео),	(5.44)
follows from (5.44) with Ф(х, 0) = 6(x — x0) as
K(x,t;xo,O) = ^2ехр(-и„///1)Фа(ж, г)Ф* (z0,0).	(5.45)
а
This relation is readily generalized to arbitrary propagation times t > s, yielding
K(x,t;y, s) = 52exp(-ieQ(t - з)/П)Фа(х,1)Ф*а(у, s).	(5.46)
a
Equation (5.46) presents an intriguing result, which generalizes the familiar form of
time-independent propagators to time-periodic ones. Note again, however, that the
role of the stationary eigenfunction <pQ(cr) is taken over by the Floquet mode Фа(сг, t),
being orthonormal only at equal times t = s.
5.3.4 Generalized Floquet theory
In the previous subsections we restricted ourselves to pure harmonic interactions. In
many physical applications, e.g. see in [8], however, the time-dependent perturbation
has an arbitrary, for example, pulse-like form that acts over a limited time regime
only. Clearly, in these cases the Floquet theorem cannot readily be applied. This
feature forces one to look for a generalization of the quasienergy concept. Before we
start doing so, we note that the Floquet eigenvalues ean in (5.16) can also be obtained
as the ordinary Schrodinger eigenvalues within a two-dimensional formulation of the
time-periodic Hamiltonian in (5.10). Setting cut = 0, (5.10) is recast as
H(t)= Ho(x,p) + V(x,0(t)).	(5.47)
With 0 = co, one constructs the enlarged Hamiltonian H(x, p; в, pg) = H0(x,p) +
V(x, 0) + cupe, where pg is the canonically conjugate momentum, obeying
(5.48)
The quantum mechanics of H acts on the Hilbert space of square-integrable functions
on the extended space of the ж-variable and the square-integrable periodic functions on
the compact space of the unit circle 0 = 0O +wt (periodic boundary conditions for O').
With V(a;,t) given by (5.22), the Floquet modes Фаь(х,9) and the quasienergies eak
• dH
0 = -— = Ld.
ope
5.3 Floquet and generalized Floquet theory
are tie eigenfunctions and eigenvalues of the two-dimensional stationary Schrod
equat on, i.e., with [0,po\ = if),
’ -ti2 д2	д ]
) Imdx2 +	~ SxSin^ + “ 1Гшдё |	= eak®ak(x’(5-49
This procedure opens a door to treat more general, polychromatic perturbations com
posed of generally incommensurate frequencies. For example, a quasiperiodic pertur-
batior with two incommensurate frequencies oq and w2,
V(x, t) — —xS sin(a>it) — xFsin(w2t),	(5.50,'
can be enlarged into a six-dimensional phase space (m,	; 02,), with =
W]t;02 = w2i} defined on a torus. The quantization of the corresponding momentum
terms yield a stationary Schrodinger equation in the three variables (;r, (d,. 02) with a
corresponding Hamiltonian operator H given by
H - H{x,6i,62)	(5.51)
Oui	UU2
with eigenvalues	and generalized stationary wavefunctions given by the gener-
alized Floquet modes Фа,*,,(ж, #i,$2) = $a,ki,k2 (x; 0| + 2тг; 02 + 2%). We note that witu
quasiperiodic driving the spectrum may become rather complex, consisting generally о I
spectral parts that are pure point, absolutely continuous or even singular continuous.
A general perturbation, such as a time-dependent laser-pulse interaction consists
(via Fourier-integral representation) of an infinite number of frequencies, so that t h<-
above embedding ceases to be of practical use. The general time-dependent Schrodinger
equation
ifl—'tlx, t) = H(x, <)Ф(аг, t),	(Г. .
with ti e initial state given by
Ф(г, to) = Ф0(з|),	(5.5.0
can be solved by numerical means, by a great variety of methods [19-21]. All th
methods must involve efficient numerical algorithms to calculate the time-ordered prop-
agatioi operator K(t, s). Generalizing the idea of Shirley [11] and Sambe [14] foi
time-periodic Hamiltonians, it is possible to introduce a Hilbert space for general time-
dependent Hamiltonians in which the Schrodinger equation becomes time independent.
Following the reasoning by Howland [22], we introduce the reader to the so called (t, t')~
formalism [23].
5.3.5 The (t,
The time-dependent solution
Ф(х,1) = /<(1,<о)Ф(.т,<о)	(5.51)
260 Driven Quantum Systems
for the explicitly time-dependent Schrodinger equation in (5.52) can be obtained as
Ф(М) = ^{x,t',t)\e=t,	(5.55)
where
Ф(x, t', t) = exp	t')(t - io)^ Ф(^, t',to)-	(5.56)
H(x, t') is the generalized Floquet operator
'H(x,t') = H{x,t') -	(5.57)
The time t' acts as a time coordinate in the generalized Hilbert space of square-
integrable functions of x and t', where a box normalization of length T is used for
t' (0 < t' < T). For two functions фа(х, t), фд(х, t) the inner, or scalar product reads
1 rT	roo
{(Фа\Ф(з)) = At' / dx ф'а(х^')ф0(х,?).	(5.58)
1 Jq	J-oo
The proof for (5.55) can readily be given as follows [23]: Note that from (5.56)
1/г^-Ф(а:, t', t) = 'Hix, t') exp[-i71(a;, t')(t — £0)М]Ф(я, t', t0)
= —ih—4>(x, t', t) + H(x, t')9(x, t', t).	(5.59)
Hence,
f д d\
ih I — + — I Ф(ж, t', t) = Hix, t')V(x, t',t).	(5.60)
\dt	dt')
Since we are interested in t' only on the contour t' — t, where dt'/dt = 1, one therefore
finds that
9Ф(а;, t',t)|	d4>(x,t',t)	_ 9Ф(я:,<)	/rm
St' |(,=f + dt t,_t dt
which with (5.60) for t = t' consequently proves the assertion in (5.55).
Note that a long time propagation now requires the use of a large box, i.e. the time
period T must be chosen sufficiently large. If we are not interested in the very-long-time
propagation, the perturbation of finite duration can be embedded into a box of finite
length T, and periodically continued. This so constructed perturbation now implies a
time-periodic Hamiltonian, so that we require time periodic boundary conditions
Ф(а.', t', t) — Ф(ж, t' + T, t),	(5.62)
with 0 < t' < T, and the physical solution is obtained when
t1 = t mod T.
(5.63)
________________________________5.4 Exactly solvable driven quantum systems 26
Stationary solutions of (5.59) reduce to the Floquet states, as found before, namely
Фа(я, t', t) = ехр(-ка^/П)Фо(х, t'),	(5.64,
with Фа(х, t') = Фа(гг, t' + T), and t' = t mod T. We remark that although Ф(з:, t', t)
^a(x, (', t) are periodic in t', the solution Ф(х,<) — Ф(гг, t' = t, t) is generally not time
periodic.
The (f, t')-method hence avoids the need to introduce the generally nasty time-
ordering procedure. Expressed differently, the step-by-step integration that charac-
terizes the time-dependent approaches is not necessary when formulated in the above
generalized Hilbert space where ?{(x,t') effectively becomes time-independent, with t'
acting as coordinate. Formally, the result in (5.59) can be looked upon as quantizing
the new Hamiltonian H, defined by
H(x,p; E,t') = H(x,p, t') - E,	(5.65)
using for the operator E —> E the canonical quantization rule E = ihd/dt; with
[E, t] iti and t</»(t) = This formulation of the time-dependent problem in
(5.52) within the auxiliary t' coordinate is particularly useful for evaluating the state-
to-stat? transition probabilities in pulse-sequence-driven quantum systems [8,23].
5.4 Exactly solvable driven quantum systems
In contrast to time-independent quantum theory, exactly solvable quantum systi ins
with ti ne-dependent potentials are extremely rare. One such class of exactly solv. 1 5
systems are (multidimensional) systems with at most quadratic interactions between
momentum and coordinate operators, e. g. the parametrically driven harminic oscillatm
[24, 25] including generalizations that account for quantum dissipation via bilinear
coupling to a harmonic bath [26], see also chapter 4.
Fur .her, we note that a Hamiltonian part that depends solely on time t can alwa
be absorbed into an overall time-dependent phase of the wavefunction. This is so.
because such an interaction cannot affect the spatial dependence of the wavefunction
5.4.1 Driven quantum oscillators
The Schrodinger equation of a harmonic oscillator with an arbitrary time-dependent
dipole interaction reads
iW(z, t) =
h2 d'2	1	22	r,/ 1 у/ \
—	i - xS(t) 1 Ф
2m ox2 2
Following the reasoning by Husimi [24], this system can be solved explicitly. First "e
introduce the shifted coordinate
x -> у = x - ((/.),
(5.67'
262 Driven Quantum Systems
yielding
ih'Hiy, t) = |i/iC— - —— + -mw02(y + ()2 - (y + <)S(t) j Ф(У, t).	(5.68)
Performing the unitary transformation
Ф(у, t) = ехр{—гт^у/Ь}ф(у, t),	(5.69)
with £(t) obeying the classical equation of motion,
+ tiWqQ = S(t),	(5.70)
the term linear in у vanishes to yield
ih<i>(y,t) =	+ |”H*y2 + L(CC,t)) <№,t).	(5.71)
i £ ill uy	£	I
Here, L(Q, Q, t) is the classical Lagrangian of a driven oscillator,
L =	+ (S(t).	(5.72)
Another unitary transformation
<№, t) = exp |-i dt'L(G £, t') J x(y, t)	(5.73)
reduces the starting equation to the well-known Schrodinger equation of a stationary
harmonic oscillator,
Лу(?М) =	+	(5.74)
I <£iiil (Jу	£	I
In terms of the eigenvalues En = ftw0(n + 1/2), and the well-known harmonic
eigenfunctions <pn, being proportional to the Hermite functions, the solutions of (5.66)
are of the form
Фп(х, t) = <pn (x - <(t))exp Ufm((t)(r~((t))~Ent + [ di' b] ) .	(5.75)
I n L	Jo J J
The set {^„(i-)} forms a complete set in 77; thus any general solution Ф(х, t) can be
expanded in terms of the solutions in (5.75). Next we consider the restriction to a
periodic monochromatic drive
S(t) = Ssin(wi + ф),	(5.76)
A periodic solution фф of (5.70) reads
mQ,(t) = S sin(wt + 0)/(wq — w2).
(5.77)
5.4 Exactly solvable driven quantum systems 26'
The quasienergies {ea} and the Floquet modes $a(x,t) can be deduced from (5.7’
we ad 1 — and subtract — the term that is linearly increasing in time,
t rT 	.S’2
-/	у 2----(5.7«
T Jo	4m(cjg — ш2)
Hence, the quasienergies can readily be read off, to give
S2
€a =	+ 1/2) -   , 2-----rr , a = 0,1, 2,....	(5.?и
- LU2)
with corresponding time-periodic Floquet modes
<ЫМ) = fa - СД*))
x exp
- £0(t)) + ( f dt'L [ dt'L
\Jo 1 Jo
(5.80)
Note that at resonance, w = cjq, the quasienergies in (5.79) are no longer correct. In-
stead, the spectrum assumes an absolutely continuous form [25]. Likewise, the harmoni-
cally driven parabolic barrier (i.e. the inverted harmonic potential UgX2/2 —Wqj:2/2),
can b< treated analogously, with the eigenfunctions tpn becoming parabolic cylinder
functions. The resulting quasienergies are continuous, reading
S2
6q + 4m(aig + w2)
(5.81)
with a € (—oo, oo). Due to the reflection symmetry in (5.66), (5.76), i.e., x -4- —x, t -
t + тг/tj, this continuum {c,,} is doubly degenerate.
5.4.2 Periodically driven two-level systems
The problem of a time-dependently driven two level dynamics is of enormous practi>;.l
importance in nuclear magnetic resonance, quantum optics, or in low tempera, me
glass s/stems, to name only a few. The driven two-level system has a long hisloiv.
and reviews are available [27]. A pioneering piece of work must be attributed !>
Rabi [28] who considered the two-level system in a circularly polarized magnetic fb-’H
— a problem that he could solve exactly, see below. He thereby elucidated how
measuie simultaneously both the sign as well as the magnitude of magnetic moments.
However, as Bloch and Siegert experienced soon after [17], this problem is no longei
exactly solvable in analytical closed form when the field is linearly polarized. ratL r
than circularly. We set for the wavefunction
ф(/) = cj(t)exp(iAt/2/l) f q ) + <"2(t) exp(-iAt/2/i) f ®
(5.82 S
264 Driven Quantum Systems
where |ci(t)|2 + |c2(t)|2 = 1- With 2ЛА = —pE<} and ip = тг/2, yielding a pure cos(wt)
perturbation, the Schrodinger equation has the form
(j / Ci(t) exp(iAt/2/i) \
i/i—
\ c2(t) exp(-iAt/2fi) /
/	—Д/2	— 2/iAcoswt \ / Ci(t) exp(iAt/2/i) \
=	•	(5-83)
\ —2/iAcoscjt	Д/2	/ \ c2(t) exp(—iAt/2ft) J
With hcjo = Д, (5.83) provides two coupled first-order equations for the amplitudes,
— = iA(exp[i(w - w0)t] + exp[-i(w + u0)t] )c2,
= iA(exp[-i(iu — w0)t] + expi(w + w0)t] )ci.	(5.84)
With an additional differentiation with respect to time, and substituting c2 from the
second equation, we readily find that ci(t) obeys a linear second order ordinary dif-
ferential equation with time periodic (T = 2тг/ш) coefficients (Hill equation). Clearly,
such equations are generally not solvable in analytical closed form. Hence, although
the problem is simple, the job of finding an analytical solution presents a hard task! To
make progress, one usually invokes, at this stage, the so-called rotating-wave approxi-
mation (RWA), assuming that и is close to w0 (near resonance), and A not very large.
Then the anti-rotating-wave term exp(i(w + w0)t) is rapidly varying, as compared to
the slowly varying rotating-wave term exp(—i(w — w0)t). Therefore it cannot transfer
much population from state |1) to state |2). Neglecting this anti-rotating contribution,
one has in terms of the detuning parameter 6 = w — wo,
= iA exp(i<5t)c2,	= iA exp( —(5.85)
dt	dt
From (5.83) one finds for cj(t) a linear second-order differential equation with constant
coefficients — which can be solved readily for arbitrary initial conditions. For example,
setting Ci(0) = 1, c2(0) = 0, one obtains
ci (t) = exp(iJt)
c2(t) = exp(-iJt)— sin
(5.86)
where denotes the celebrated Rabi frequency
(<52 + 4A2)1/2.
(5.87)
5.4 Exactly solvable driven quantum systems 2G5
Fig. 5.2 : The population probability of the upper state [c2(t)|2 as a function of time t at resonance
6 = 0 (so id line), versus the non-resonant excitation dynamics (dashed line) at, 6 = 2A yf 0.
The populations as a function of time are then given by
. /м2 MV /2A\2	\
= 5 + 77 cos >
> /м2	/2АА2	, /K\
|c2(*)! = (	} Sin2 .
(5.88)
(5.89)
Note that at short times t, the excitation in the upper state is independent of the de-
tuning, c2(t)|2 —> A2t2 for fit 1. This behavior is in accordance with perturbation
theory, valid at small times. Moreover, the population at resonance w = w0 completely
cycles tl e population between the two states, while with 6 / 0, the lower state is never
completely depopulated, see Fig. 5.2.
Up to now, we have discussed approximate RWA solutions. At this point we remark
that the unitary transformation
Ят = LT'H-r^U, U = exp(iTray/4)
transforms the Hamiltonian in (5.3) into the form
Ну = —-Дсгд. + 2/iA sin(wt + ф)аг.
(5.90,1
(5.91)
This is the appropriate representation for tunneling problems, Hy. Appropriate basis
states aie the “localized” (right/left) wavefunctions |±) = (|1) ±	which are
eigensta .es of az with the eigenvalues ±1. The form given for H-rrs is convenient
266 Driven Quantum Systems
for the description of optical properties such as the dipole moment. We have for the
expectation
mW = trJpTLsa-J = tr{t>T(T2},	(5.92)
where q.„ is the density matrix in the corresponding representation. Note that a static
asymmetry energy can be included if the field assumes a static component, i.e. A sin(wt+
ф) A sin(wt + </>) + Ao.
Explicit results for the time-periodic Schrodinger equation require numerical meth-
ods, cf. Sect. 5.5, one must solve for the quasienergies ean and the Floquet modes
ФО>1(М). Without proof we state here some results that are very important in dis-
cussing driven tunneling in the deep quantum regime. For example, Shirley [11] already
showed that in the high-frequency regime A C rnax[w, (Aw)1/2] the quasienergies obey
the difference
t2,-i ~ fi,i = hwoJo(4A/w),	(5.93)
where Jo denotes the zeroth order Bessel function of the first kind. The sum of the two
quasienergies obeys the rigorous relation [11]
e2n + «и = Ei + T?2 = 0 (mod Aw).	(5.94)
For weak fields, one can evaluate the quasienergies by use of the stationary pertur-
bation theory in the composite Hilbert space H ® T- In this way one finds:
(i)	Exact crossings at the parity forbidden transitions where wH = 2nw, n — 1, 2,...,
so that
e2,i = 0 (mod Aw).	(5.95)
(ii)	At resonance w = wo :
e2,i — ±^w0 ( 1 + —v/l — A2/4wq ] (mod Aw).	(5.96)
(iii)	Near resonance, one finds from the RWA approximation readily the result
£•21 = ±-Aw f 1 н—(mod Aw),	(5.97)
where Q denotes the Rabi frequency in (5.87). Correcting this result for counter-
rotating terms, an improved result, up to order O(A6), reads [27]
1 (
62 i = ±-Aw 1-1—	(mod Aw),	(5.98)
'	\	wj
with the effective Rabi frequency Q
Q2 = <52 +	(5-99)
(w + Wo) (w + Wo)3 * *
5.4 Exactly solvable driven quantum systems
Not:ce that the maximum of the time-averaged transition probability in (5.89)
occurs within RWA precisely at w = w0. This result no longer holds with (5.99)
where the maximum with 9 -> 9 in (5.88) undergoes a shift, termed the Bloch-Siegei t
shift wr,.s / w0 [17,27]. From (392/3w0)A = 0 this shift is evaluated as [17,27].
A2 A4
Wres —	9------1“ —r-
wg 4wJ
(5.100)
This Bloch-Siegert shift presents a characteristic measure of the deviation beyond the
RWA-approximation, as a result of the nonzero anti-rotating terms in (5.84).
Let is next explicitly consider the case pioneered by Rabi [28], a TLS driven in a
spatially homogeneous, circularly polarized external radiation field. This leads to the
Hamiltcnian
H(t) = —	— 26A (o^coswf — OySinwt)
1^ I —wo 4Aexp(iwt)
2 I 4Aexp(—iwt) w0
Absorbiig the phase exp(±iwof) into the time-dependence of the coefficients, i.e., .
ting a12(i) = Ci,2(f) exp(±iwof), we rotate the states around the г-axis by the amount
wt. Witn Sz = huZi/2, one has
6i(f) \	/ i \ / ai(f)
= exp -~Szivt
62(f) /	' h \ «2(f)
ai(t) exp(—iwf/2)
a2(f) exp(+iwf/2)
(5.102)
Upon a substitution of (5.101) and (5.102) into the time-dependent Schrodinger equa-
tion, and collecting all the terms, results in a time-independent Schrodinger equation
for the states (6i(f), 62(f)), which reads
-161 = - j(w - w0)6i + A62,
—lb? ~ Xb[ -f- — (w — ^o)62.	(5.103)
Hence, cne obtains a harmonic oscillator equation for b^t) (and similarly for 62(f)),
/	<52\
б! + I A2 + — I 6j = °.	(5.104)
It describes an oscillation with frequency 1Q = ^/A2 + <52/4, where 9 coincides precisely
with the previously found Rabi frequency in (5.87). With C](0) = аЦО) — 61 (0) —
1, and c2(0) = «2(0) = 62(0) = 0 the populations are given by the corresponding
2t>8 Driven Quantum Systems
relations in (5.88), which in this case are exact. In particular, the transition probability
= |<2|Ф(<)>|2 = |a2(i)|2 =	= |c2(0l2 obeys
W^2(t) = sin2 (^t) .	(5.105)
At resonance, 6 — 0, w = w0, it assumes with fl2 = 4A2 its maximal value. We also
note that the quasienergies are given by the — in this case exact — result in (5.96).
5.4.3 Quantum systems driven by circularly polarized fields
The fact that the time evolution of aTLS in a circularly polarized field can be factorized
in terms of a time-independent Hamiltonian in (5.103) is surprising. We note that this
factorization involves a rotation around the z-axis.
|a(t))	—>•	= exp(-iSzwt//i)|a(t)).	(5.106)
This feature can be generalized to any higher-dimensional system, such as a magnetic
system or a general quantum system that can be brought into the structure which, in
a representation where Jz is diagonal, is of the form
H(t) — Hq(J2) +	— 4A[ Jx cos cot — Jy sin wt].	(5.107)
Here, contains all interactions that are rotationally invariant (Coulomb interac-
tions, spin-spin and spin-orbit interactions). Setting R(t) = exp(—iJzart/Ti) and upon
observing that
R(t) JxR(t)~l = Jx cos cut + Jy sin wt,
RfyJyRtt')"1 = — Jxsinut + Jycoswt,
R(t)JzR{t')~1 = Jz,	(5.108)
one finds upon substituting (5.108) into (5.107)
Я(1) = R(t)H(t)R-\t) =Яо(72)+Я1(Л)-4А.71.	(5.109)
Hence, the transformed Hamiltonian becomes independent of time. With the propa-
gator obeying
= — ^-R~l(t)HR(t)K(t,tQ),
h
we find from
^[R(t)K(t, to)^»] = ~H[R(t)K(t, t0)R-\t0)]	(5.110)
_________________________5.4 Exactly solvable driven quantum systems 26''
where
— Ho +
(5.11
that rhe propagator factorizes into the form
K(t,t0) — exp	exp
(5.112)
Because exp(ij2wt/h) at times t = 2тг/ш equals 1 for integer values of the angular
momentum and —1, for half-integer values, respectively, the propagator in (5.112) can
be recas t into the Floquet form in (5.39),
K(t + nT, 0) = K(t, 0){K(T, 0)]n.
(5.113)
With Jz = ±1, ±2,..., the Floquet form, cf. (5.38), is achieved already with (5.112).
For hal.-integer spin the corresponding Floquet form is obtained by setting for the
propagator
.	\	(i ( т \	( i (л Л \ ,
К(t, t0) = exp - \ JZ + - ujt exp I —- \ H + -w (t - t0)
\ I I \	и I j	\ IL \	£ /
x exp
H = H + w Л
since the first and third contribution are now periodic with period T. Given th"
eigenvalues	of H, the exact quasienergies are given by the relation
= (eQ + hw/2) mod ticj.
(5.115)
The general results derived here carry a great potential for applications involving
time-dependent tunneling of spin in magnetic systems with anisotropy, and strongly
driven molecular and quantum optical systems as well.
In summary, we demonstrated that a periodically driven TLS or a general quan-
tum sys .em of the form in (5.107) — can be solved analytically only when driven by
a circuit rly polarized ac-source. This is the case for the Rabi solution. The situation
changes when we instead consider a infinite number of states or a periodic lattice with
period L, such as a tight-binding Hamiltonian. Then, a linearly polarized dipole inter-
action - [So + S cos(wt)]L £n |n)n(n| yields the exact quasienergy or Floquet states if
S0L = 7ihw, (where n = 0 if So = 0); i.e. if the energy of n photons precisely matches
the energy difference between adjacent rungs of the corresponding Wannier-Stark lad-
der [29, 30]. Also, we mention here that an analytical solution can be constructed when
the above dipole interaction acts in a quantum well that is sandwiched between two
infinitely high walls [31].
270 Driven Quantum Systems
5.5 Numerical approaches to periodically driven
quantum systems
Except for the special cases discussed in Sect. 5.4, exactly solvable quantum systems
with explicitly time-dependent interaction potentials are extremely rare. As demon-
strated with (5.83), this is true already for the periodically driven two-level-system in
a linearly polarized monochromatic field [11] for which no exact closed form solution
can be found. Thus, we generally have to invoke numerical procedures.
5.5.1 Method of Floquet matrix
Since the Hamiltonian H(x, t) and the Floquet modes are time-periodic, we can expand
the Floquet solutions into the Fourier vectors |n), n = 0, ±1,±2,..., such that (t\n) =
exp(inwt),
$a(z,t) = 52 с"(ж) exp(inwt).	(5.116)
n = -00
The functions с"(ж) can be expanded in terms of a complete orthonormal set {</?it(^),
k = 1,..., oo}, yielding in terms of the unperturbed eigenfunctions of Hq(x),
52	(5.117)
k—1 n = -oo
with c2 к = (</?JCa). Hence, in terms of the kets |^*,),	= tpk(x'), the Floquet
equation (5.14) reads
OO OO	OO 00
52 52	exp(inwt) = 52 52 eQc^J^)exp(inwt). (5.118)
A: = ln=-oc	k—1 n=-oo
Setting (ipk|(m| = and multiplying (5.118) with {<Pjrn\ exp(—imut) from the left,
yields after a time-average over one period of driving, the system of equations
OO co
52 52((^HWa^)K,*: = «aCc	(5.119)
n=—oo k— 1
Here we used the scalar-product notation in (5.19). With the definition
Hm~n = -2 [ dtH(t) exp[-i(m - n)wt],	(5.120)
1 Jo
one finds the Floquet-matrix representation for (5.119),
CO oo
52 52 ((Wn|?/F|^n))c^ = e„c- .	(5.121)
k = l П--00
with the Floquet matrix defined by
((^m|?<F|^n)) = {^\Нт~п\<рк) + nhiv6ntm6]tk.	(5.122)
5.5 Numerical approaches to periodically driven quantum systems 27 i
For a sinusoidal perturbation H(t) = Ha - 2hAx sin(u>t + ф), the operator Hm~n takes
on a tri mgular structure
Я =	+ ihAxexp(i^) exp( 1ф))-	(5.123)
Hence, the operator HF has a block-triagonal structure with only the number of angular
frequencies w in the diagonal elements varying from block to block.
The quasienergies {eQ}are now obtained as the eigenvalues of the secular equation
det |77F — el| = 0,	(5.121)
whose block-tri diagonal form provides the quasienergies {ea,n} and eigenvectors '
obeying the periodicity properties
^a,k ta Q T /c/llxJ,	(5.12 • !
{a, n + k\e0tm+k) = {a, п|едга).	(5.12«)
From these solutions, the spectral decomposition in (5.33) and expressions for transition
amplitudes can readily be derived.
Becai se the origin of time can be chosen arbitrarily, the quasienergies do not depend
on the phase ф. In contrast however, the Floquet modes Ф(./', i; ф) depend on the phase.
Keeping r.he time t fixed the variation of ф over the interval of 2тг allows to cover the
time-dependence of the Floquet mode over a whole period T.
5.5.2 Matrix-continued-fraction method
The block-tridiagonal structure of the Floquet matrix can be used to implement an
efficient numerical algorithm, termed matrix continued fraction (MCF) method. Our
starting point is (5.121). Performing the sum over n one finds
OO
(eQ -	- ifiA exp(-i^)c™^ (</фЫ
+i/iA exp(-i</>)c™)?1(</?J|2.-|^.)].	(5.127)
This form can be cast into a tridiagonal recursive relation that reads
G(m, a)c™ + H+c™+1 + Я^с^-* = 0,	(5.128)
where
G(m, a) = Hq- (ca - m/iw)l	(5.129)
and
H± = ехр(МФ)^-
(5.1301
272 Driven Quantum Systems
The recursive matrix equation in (5.128) can be solved by using the ladder operators
C _ jn+1
— l-a i
T_mc~m = c;(m+1),	(5.131)
which are rising (lowering) the index m. The solutions of (5.131) can be given in terms
of a matrix continued fraction, by iterating the recursive solution with m increasing,
Sra-i = -[G(m,a) +Я+Зт]“1Я-
G(m, a) - H+ —-----------------H~
' ’ 1 G(m +	— H+...
T-(m-D = -{С(-т,а)+Н-Т,т]-гН+
=---------------------------j-------------H+.	(5.132)
G(—m, a) - H~ —-------------------H+
v 7	G(-m-l,a)-H- ...
Setting m = 0 yields from (5.128) the linear system of equations
G(0,a)c° +H+Soc°a + H-Toc°a = 0,	(5.133)
composed of both diagonal and — via So, To — also nondiagonal contributions. The
quasienergies follow from the solubility condition,
det[G(0, a) + H+So + H~T0] = 0.	(5.134)
In practice, this system of equations is solved numerically, by evaluating So and To
truncated at some finite value m > 0, i.e. one assumes Sm = 0, T_m = 0 for sufficiently
large rn, such that the result no longer changes significantly with increasing rn. For an
application of this MCF method to the problem of driven tunneling we refer the reader
to the original literature [19].
The above two sections discussed the case of periodic perturbations. A general time-
dependent interaction can be treated similarly — see Sect. 5.3 — by use of the multi-
mode Floquet theory, or the general (t, t')-formalism with the time interval T being
chosen sufficiently large. Time-periodic boundary conditions can usually be assumed
for finite (laser-)pulse interactions also, when the number of oscillations during the pulse
lifetime is large. Alternatively, various direct methods for solving a time-dependent
quantum problem exist. It should be stressed again, that an avoidance of the time-
ordering operator -- via embedding (cf. Sect. 5.3) — results in a great simplification.
Otherwise, the propagator must be split into short segments in which the Hamiltonian
does not change significantly. Some keywords relating to these alternative direct time-
propagation methods are the “split-operator technique” [20], and the “second-order-
difference schemes”. For recent surveys we refer the reader to the reviews in Ref. [21].
5.6 Coherent tunneling in driven bistable systems 273
5.6 Coherent tunneling in driven bistable systems
In this section we address the physics of coherent transport in bistable systems. These
systems are abundant in the chemical and physical sciences. On a quantum mechanical
level of description, bistable, or double-well potentials, are associated with a paradig-
matic coherence effect, namely quantum tunneling. Here we shall investigate the in-
fluence of a spatially homogeneous monochromatic driving on the quantal dynamics in
a symmetric, quartic double well. This archetype system is particularly promising for
studying the interplay between classical nonlinearity — its classical dynamics exhibits
chaotic solutions — and quantum coherence. Its Hamiltonian reads [9,19]
p
H(r,p; t) = --------F Vq(z) + xS sin(cjt + 0),
(5.133)
with the quartic double well potentail
VM -	^02 2 .
W)-	4 x +64£b*
(5.136)
Here m denotes the mass of the particle, cj0 is the classical frequency at the bottom
of each well and Eg the barrier height, and S and u> are the amplitude and angular
frequency of the driving. The number of doublets with energies below the barrier top
is approximately given by D = Eq/TvjJ0. The classical limit hence amounts to D —> с».
For ease of notation, we introduce the dimensionless variables
_ = P
t = CLlOt,
x)
cv = —,
CO0
(5.137)
(5.138)
(5.139)
(5.1401
(5.1 11)
where the overbar is omitted in the following. This is equivalent to setting formally
m = h = jJQ = 1.
As dit cussed in Section 5.3, the symmetry of H(t) reflects a discrete translation
symmetry in multiples of the external driving period T = 2tt/u', i.e., t —> t + nT.
Hence the Floquet operator describes the stroboscopic quantum propagation
K(nT,0) = [K(T,0)]n.
(5.142)
274 Driven Quantum Systems
Besides the invariance under discrete time translations, the periodically driven sym-
metric system exhibits a generalized parity symmetry P,
P : x—x; t^-t + T/2.	(5.143)
This generalized parity can be looked upon as an ordinary parity symmetry in the
composite Hilbert space, 77 ® T- Just as in the unperturbed case with S = 0, this
allows the classification of the corresponding quasienergies ean into an even and an odd
subset. For very weak fields S —> 0, the quasienergies eak follow from (5.32) as
e°ak(S,u)) = Еа + кГш; к = 0, ±1, ±2,...,	(5.144)
with {1FQ} being the unperturbed eigenvalues in the symmetric double well. As pointed
out in (5.16), this infinite multiplicity is a consequence of the fact that there are infinitly
many possibilities to construct equivalent Floquet modes, cf. (5.15): The multiplicity is
lifted if we consider the cyclic quasienergies mod hw. Given a pair of quasienergies ,
fa'.*:', a a', a physical significance can be attributed to the difference ДА: = к' — к.
For example, a crossing eaj- = еа'^+л/с can be interpreted as a (AAi)-photon transition.
With S > 0, the equality in (5.144) no longer provides a satisfactory approximation.
Nevertheless, the driving field is still most strongly felt near the resonances ea<k ~ ea',k'-
The physics of periodically driven tunneling can be qualified by the following two
properties:
(i) First we observe, by an argument going back to von Neumann and Wigner [16],
that two parameters must be varied independently to locate an accidental en-
ergy degeneracy. In other words, exact quasienergy crossings are found at most
at isolated points in the parameter plane (S,w), i.e., the quasienergies exhibit
typically avoided crossings. In presence of the generalized parity symmetry in
(5.143) in the extended space 77 ® T, however, this is true only among states
belonging to the same parity class, or for cases of driven tunneling in presence
of an asymmetry (then (5.143) no longer holds). With the symmetry in (5.143)
present, however, quasienergies associated with eigenstates of opposite parity do
exhibit exact crossings and form a one-dimensional manifold in the (S, cc)-plane,
i.e., {e(S,w)} exhibit an exact crossing along lines. With S —> — S, implying
e(S, w) = t(—S, w), these lines are symmetric around the w-axis.
(ii) Second, the effective coupling due to the finite driving between two unperturbed
levels at the crossing Ea — En, — Akw, as reflected in the degree of splitting of
that crossing at 5	0, rapidly decreases with increasing ДА:, proportional to
the power law SAfc. This suggests the interpretation as a (AA;)-photon transition.
Indeed, this fact can readily be substantiated by applying the usual (Ak)-th order
perturbation theory. As a consequence, for small driving S only transitions with
ДА: a small whole number do exhibit a significant splitting.
5.6 Coherent tunneling in driven bistable systems 2 i J
5.6.1 Limits of slow and fast driving
In the limits of both slow (adiabatic) and fast driving we have a clearcut separation of
time series between the inherent tunneling dynamics and the external periodic driving.
Hence, the two processes effectively uncouple and driven tunneling results in a mere
renormalization of the bare tunnel splitting A. This result can be substantiated by
explicit analytical calculations [19]. Let us briefly address the adiabatic limit, i.e., the
driving frequency cj satisfies ш A. Setting ф = (a>t + ф), the tunneling proceeds in
the adiabatic potential
V(x, ф) = V0(x) + zSsin0.	(5.145)
The use of the quantum adiabatic theorem predicts that Ф(х, t) will cling to the same
instantaneous eigenstates. Thus, the evaluation of the periodic-driving renormalized
tunnel splitting follows the reasoning used for studying the bare tunnel splitting in
presence of an asymmetry ст,
cr = V(r_, Ф) - V(x+, Ф),	(5.146)
with x± denoting the two symmetric unperturbed metastable states. With the instan-
taneous splitting determined by Aff = (A2 + ст2)1/2, the averaging over the phase ф
between [0, 2тг] yields for the renormalized tunnel splitting Aat[(S), the result [19]
Aad(S) = (2A/?r)(l + a)l/2E jy/a/(l + ci)] > A,	(5.147)
with a	and denoting the complete elliptical integral. This shows
that A;ic increases proportional to S2 as a < 1, and is increasing proportional to
S' for n >> 1. Hence, a particle localized in one of the two metastable states will
not stay localized there (this would be the prediction based on the classical adiaba.
theorem) but rather will tunnel forth and back with an increased tunneling frequency
= A?d > A. Obviously, with the slowly changing quantum system passing a ne;
degeneracy (tunnel splitting), the limits h —> 0, w fixed and small (classical adiabatic
theorem) and w —> 0, h fixed (quantum adiabatic theorem) are not equivalent.
The limit of high frequency driving can be treated analytically as well. The unitary
transforn ation
Ф(г, t) — exp ( — i— cos(wt + ф)х ) q(t, t)	(5.148)
\ w	/
describes the quantum dynamics within the familiar momentum coupling in terms of
an electrcmagnetic potential T(t) = -(S/w) cos(wf. + ф), the transformed Hamiltonian
reads
H(x, t) = HQ(x,p) - .4(f)p,	(5.1491
where we have dropped all time-dependent contributions that do not depend on x and
p. Next we remove this T(f)p-term by a Kramers-Henneberger transformation.
g(x, t) = exp f-i [ dt'T(t')p') f(.T,t)	(5.1 Aw
2 /о Driven Quantum Systems
to yield
H(x, t) = -p2 + Vo (x - sin(u>t + ф)} ,	(5.151)
resulting in a removal of the A(t)p-term, and the time-dependence shifted into the
potential V(x, t). After averaging over a cycle of the periodic perturbation we obtain
an effective Hamiltonian
with a frequency-dependent curvature. This large-frequency approximation results in
a high-frequency renormalized tunnel splitting [19],
/ Q / c \ 2\	/ 2 S'2\
Ahf/A = (/ “ ifil? Ы )exp (	- L (5Л53)
Hence, fast driving results in an effective reduction of barrier height, thereby increasing
the net tunneling rate. In conclusion, the regime of adiabatic slow driving and very-
high-frequency driving (away from high-order resonance) can be modeled via a driving-
induced enhancement of the tunnel splitting. A similar shortening of the effective
tunneling duration Ту = тг/Д can be achieved alternatively with an appropriate shaping
of the perturbation amplitude; S —> S(t) = Ssin2(Trt/tp), with tp being the pulse
duration [32].
5.6.2 Driven tunneling near a resonance
Qualitative changes of the tunneling behavior are expected as soon as the driving
frequency becomes comparable to internal resonance frequencies of the unperturbed
double well with energy eigenstates Elt E2,... with corresponding eigenfunctions <pi(.r),
p2(x),... Thus, such resonances occur at ш ~ E3 — E2, E^ — E^, Еъ — E2, ... etc.
A spectral decomposition of the dynamics resolves the temporal complexity which is
related to the landscape of quasienergies planes 6Qj*,(S, w) in parameter space. Most
important are the features near close encounters among the quasienergies. In particular,
two quasienergies can cross one another if they belong to different parity classes, or
otherwise, they form an avoided crossing. The situation for a single-photon transition-
induced tunneling is depicted in Fig. 5.3 at the fundamental resonance w = E>, -E2. For
S > 0, the corresponding quasienergies f2* and £3^-1 form avoided crossings, because
they possess equal parity quantum numbers. Starting from a state localized in the
left well, we depict in Fig. 5.3a the probability to return Рф(/„) = |(Ф(0)|Ф(tn))|2,
= nT. Instead of a monochromatic oscillation, which characterizes the unperturbed
tunneling we observe in the driven case a complex beat pattern. Its Fourier transform
reveals that it is mainly composed of two groups of three frequencies each (Fig. 5.3b).
These beat frequencies can be associated with transitions among Floquet states at the
avoided crossing pertaining to the two lowest doublets. The lower triplet is made up
of the quasienergy differences r.3,1 — «2,0, «2,0 — D,o, e.3-1 - 61,0 ; the higher triplet is
___________________5.6 Coherent tunneling in driven bistable systems 277
n
i
i
i
i
0.02 0.04 0.06 0.08 0.10 0.12
ri
Fig. 5.3: Driven tunneling at the fundamental resonance, = E;, -E2. (a) Time evolution of P*(tn)
over the first 105 time steps; (b) corresponding local spectral two-point correlations P2 M [19]. The
parameter values are D = 2, S = 2 x 10“3, and w = 0.876.
composed of the differences 64,-1 — £3,-1 , £4,-1 — £2,0, £4,-1 — £t,o [19]. An analytical,
weak-field and weak-coupling treatment of a resonantly driven two-doublet system has
been presented with Refs. [33,34].
5.6.3 Coherent destruction of tunneling
A particularly interesting phenomenon occurs if we focus on near-degenerate states that
are tunnel splitted. For example, in the deep quantum regime the two quasienergies
£7 —> e k(S,cv) and E? —> e2i(5,w) the subsets {6i,*,+i(5,02} and {62g-i (5, ur)} belong
to different parity classes so that they can form exact crossings on one-dimensional
manifolds, see below (5.144); put differently, at the crossing the corresponding two-
photon transition that bridges the unperturbed tunnel splitting A is parity forbidden.
To give an impression of driven tunneling in the deep quantal regime, we study how a
state, prepared as a localized state centered in the left well, evolves in time under tin
externa, force. Since this state is approximately given by a superposition of the two
lowest cnperturbed eigenstates, |Ф(0)) ~ (|Ф1) + |Ф2))/\/2, its time evolution is dein-
inated by the Floquet-state doublet originating from |Ф]) and ]Ф2), and the split!in-’
c2 — 6) >f its quasienergies. Then a vanishing of the difference 62,-1 — £1,1 does hav>
an intriguing consequence: For an initial state prepared exactly as a superposition
the corresponding two Floquet states Ф1,](х,<) and Ф2,_i(x, t), ef. (5.11), (5.15), tlw
probabi.ity to return F(tn), probed at multiples of the fundamental driving period
T = 2тг/аг, becomes time independent. This gives us the possibility that tunneling can
be brought to a complete standstill [9,19]. For this to happen, it is necessary that tic
particle does not spread and/or tunnel back and forth during a full cycle of the externa’,
period T after which the two Floquet modes assemble again [35]. Hence, this condition
[9,35], together with the necessary condition of exact crossing between the tunnel-
ing relaied quasienergies 62n,*:-i = 62П-1Л+1, (n: number of tunnel-splitted dublett)
guaranti es that tunneling can be brought to a complete standstill in a dynamically
278 Driven Quantum Systems
Fig. 5.4: Suppression of tunneling at an exact crossing, €2,-i = «1,1  (a) One of the manifolds in the
(S, u)-plane where this crossing occurs (data obtained by diagonalization of the full Floquet operator
for the driven double well are indicated by crosses, the full line has been derived from a two-state
approximation, the arrow indicates the parameter pair for which part (b) of this figure has been
obtained); (b) time evolution of F*(tn) over the first 1000 time steps, starting from an initial state
prepared as a coherent state in the left well.
coherent manner. In Fig. 5.4a we depict the corresponding one-dimensional manifold
of the j-th crossing between the quasienergies that relate to the lowest tunnel dublett.
i.e., M^(S, w), which is a closed curve that is reflection symmetric with respect to
the line S = 0, there a localization of the wave function Ф(з:,1) can occur. A typical
time evolution of P(tn) for a point on the linear part of that manifold is depicted in
panel 4b.
Moreover, a time-resolved study over a full cycle (not depicted) does indeed show
that the particle stays localized also at times t tn. Almost complete destruction of
tunneling is found to occur on for A < cj < E3 — E2. For cj —> /Д — E2, the
strong participation of a third quasienergy mixes nonzero frequencies into the time
dependence so that coherent destruction of tunneling at all times ceases to exist. For
small frequencies, Д/2 < w < A, and corresponding small driving strengths S, as
implied by Af|J,c(5, w), the driven quantum mechanics approaches the unperturbed
quantum dynamics. In particular, it follows from (5.31), (5.32) for w —> A/2 and
S -> 0, Ф1д(т,Л) = exp(iwt), Ф2,-1(:с, t) = tp2(x) exp(—iu/t), that
P(t) = |(Ф(0)|Ф(<))|2 = cos2 (At/2),	w=A/2.	(5.151)
For ш rs A, 6j 1 and e2,-i exhibit an exact crossing. With corresponding Floquet modes
determined from perturbation theory as
$1,1 (z, 0
— [<^>i(x) exp(k^) + i<^2(m)],
ф2,-1См)
— [^2(s)exp(-iwi) + i<^(x)L
(5.155)
5.6 Coherent tunneling in driven bistable systems 279
x
Fig. 5.5: The probability |{Ф(t)|z)|2 at t = 4587 (full line) is compared with the initial sta!
i dashed line, the dotted line depicts the unperturbed symmetric bistable potential). The parameters
are D = 2, 5 = 3.171 x 10-3 and ic = 0-01, i.e., w equals 52.77 times the unperturbed tunnel splitting.
t he result for P{t), with Ф(х, 0) = [</q (ar)+</?2(x)]/\/2, localized in the left well, becomes
P(t) ~ j[3 + cos(2At)], w » A.
4
(5.156)
For larger frequencies obeying A < ш — E^, the Floquet modes can be approxi-
mated by [35]
Ф1д(а;,<) ~ ¥’2(^)1 sin(wt)| — i<pi(r) cos(cjt),
Ф2,-1(х, t) ~ sin(wt)| — i</22Gr) cos(wt).	(5.157)
1 his resulss in a complete localization,
P(t) = 1,	A<w<E3-E2.	(5.158)
I hroughout Eqs. (5.154) — (5.158), we set the initial phase in (5.135) equal to zero.
Starting from a coherent state localized in the left well, taken as the ground state
>f the harmonic approximation, we depict in Fig. 5.5 the spatially resolved tunneling
Ivnamics for |Ф(х, t)|2 at time t = 0 and at time t = 458 T for m = 0.01 = 52.77Д,
md S = 3.17 x 10~3, yielding an exact crossing between fu and £2,-1- For this
mine of 77 — 458, the deviation which originates from small admixtures of higher-
\ ing quasienergy states to the initial coherent state, is exceptionally large. For other
inies the ocalization is even better. It is hence truly remarkable that the coherent
lest ruction of tunneling on M^OC(S, w), with A < ш < E2 — Ei, is essentially not
dfected by the intrinsic time dependence of the corresponding Floquet modes, nor by
he presence of other quasienergy states eQik, a = 3, 4,....
280 Driven Quantum Systems
5.6.4 Two-state approximation to driven tunneling
Additional insight into the mechanism of coherent destruction of tunneling can be
obtained if one simplifies the situation by neglecting all of the spatial information
contained in the Floquet modes Фа(г, t) and restricting the influence of all quasienergies
to the lowest doublet only [35-39]. Such a two-state approximation cannot reproduce
those sections of the localization manifolds that are affected by resonances, e.g. the part
in Fig. 5.3a that bends back to S = 0 for w < E3 — E2. Setting for the transition dipole
moment S{<pi= 2A, we find within the localized basis the TLS Hamiltonian in
(5.91). For the state vector in this localized basis |+) and |—), we set
|Ф(<)) = Ci(t)exp[—i(2A/w) sincui]|—)
+c2(t) exp[+i(2A/w) sin wt]|+).	(5.159)
Given the cos(wt) perturbation, ф = тг/2, we consequently obtain from the Schrodinger
equation for the amplitudes {c1>2(£)} the equation
i^ci,2(^) = -^Aexp[±i(4A/w) sin(wt)]c2,i(t).	(5.160)
For large frequencies » A, we average (5.160) over a complete cycle to obtain the
high-frequency approximation
i^cij2(t) = -|AJo(4A/w)c2.i(t),	(5.161)
where Jo(x) = (х/2тг)^г ds exp [iarsin(ws)], is the zeroth-order Bessel function of the
first kind. This yields a static approximation — which is different from the RWA in
(5.85) — with a frequency-renormalized splitting
A -> J0(4A/w)A.	(5.162)
The static TLS is easily solved to give with c^t = 0) = 1 for the return probability
P(t) the approximate result
P(t) = |C1(t)|2 = cos2(j0(4A/w)At/2).	(5.163)
On the localization manifold M^OC(X, w), we find from (5.162) at the first zero of
Jo(xi) — 0, i.e., 4A/a? — 2.40482..., in agreement with the result in (5.93). On
this manifold, F(t) in (5.163) precisely equals unity, i.e., the effective tunnel splitting
vanishes. Thus one finds a complete coherent destruction of tunneling. This high-
frequency TLS approximation, as determined by the first root of Jq(4A/w), is depicted
in Fig. 5.4a by a solid line. Higher roots yield an approximation for M{o<. with j > 1.
Moreover, we note that Jq{x) ~ x~1^2 as x -> oo. This implies, within the TLS-
approximation to driven tunneling, that tunneling is always suppressed for x > Д
with 4A/w S> 1. An improved formula for P(t) in (5.163), that contains also higher
5.7 Laser control of quantum dynamics 281
odd hai monies of the fundamental driving frequency u> has recently been given in [39].
This driven TLS is closely connected with the problem of periodic, nonadiabatic level
crossing. In the diabatic limit 6 = Д2/(Аш) -> 0, corresponding to a large amplitude
driving, the return probability has been evaluated by Kayanuma [38]. In our notation
and with A > max(w, Д) this result reads
pw~c°4(2b)1/2
у \ 2А7Г /
/ 4A	7r\
sin-------h -
\	4 /
At\
T )
(5.164)
With Jo ~ (2/7rrr)1/2sin(2; 4- тг/4), for x » 1, (5.164) reduces for A/u> » 1 tu
(5.163). Here, the phase factor of тг/4 corresponds to the Stokes phase known from
diabatic level crossing [38], and 4A/cj is the phase acquired during a single crossing
of duration T/2. The mechanism of coherent destruction of tunneling in this limit
A > max(w, Д) hence is related to a destructive interference between transition paths
with 4A/cj = птг + Зтг/4. The phenomenon of coherent destruction of tunneling also
persists >f we use a full quantum treatment for the semiclassical description of the field:
For a quantized electromagnetic field S —> (a+ + a), the quantized ve.rsion of (5.3) reads
H = — ^Д<т2 + aja+a - g(a+ + a)trx.	(5.165)
With (n) = (a+a) the coupling constant g is related to the semiclassical field A by
g^/n = A.
(5.166)
The vanishing of the quasienergy difference is then controlled by the roots of the La-
guerre polynomial Ln of the order of the photon number n [40]. With a large photon
number one recovers with Ln ex Jo, as?:» 1, the semiclassical description. Just as
is the case with the semiclassical description, a rotating-wave approximation of th',
quantum TLS in (5.165), giving the celebrated Jaynes-Cummings model [41], is not
able to reproduce the tunneling-suppression phenomenon.
5.7 Laser control of quantum dynamics
The prev.ous phenomenon of coherent destruction of tunneling is an example of a dy-
namical quantum interference effect by which the quantum dynamics can be mani-
pulated ly an observer. More generally, the dependence of quasienergies on field
strength and frequency can be used to control the emission spectrum by either generat-
ing or by selectively eliminating specific spectral lines. For example, the near crossing
of quasiet.ergies in a symmetric double well generates anomalous low-frequency lines
and — at exact crossing — doublets of intense even-harmonic generation (EHG) [10].
This latter phenomenon is intriguing: A symmetric system possesses inversion sym-
metry so that even harmonics are forbidden by selection rules valid to all orders in
perturbat on theory. EHG thus precisely occurs at the exact crossings where tunnel-
ing can be frozen, so that a dynamically induced static dipole moment is generated.
282 Driven Quantum Systems
This control by a periodic continuous-wave driving can be generalized by recourse to
more complex perturbations. The goal by which a pre-assigned task for the output of
a quantum dynamics is imposed from the outside by applying a sequence of properly
designed (in phase and/or shape) pulse perturbations is known as quantum control
[8]. For example, a primary goal in chemical physics is to produce desired product
yields or to manipulate the atomic and molecular properties of matter [8,42,43]. As
an archetype situation, we present the control of the quantum dynamics of two coupled
electronic surfaces
..d
лм
Фе
Hg -p.(R)E(t) 1 / Ф8
-fl(R)E*(t) He Ц Фе
(5.167)

where R denotes the nuclear coordinates and are the Born-Oppenheimer Hamilto-
nians for the ground- (g) and excited- (e) field free surfaces, respectively. The surfaces
are coupled within the dipole approximation by the transition dipole operator fi(R) and
the generally complex-valued radiation field E(t). Notice that the structure in (5.167)
is identical to that obtained in the driven TLS. Following Kosloff, Hammerich, and
Tannor [43], the rate of change to the ground-state population ng(t) = (Ф6(£)|Фв(г))
is readily evaluated to read
^ = 2МФ8|Ф8)
= -- Im ((Ф8(<)МЯ)|Фе(2>^))•	(5-168)
If we set with C(t) a real-valued function
W) -> E0(t) = (Фс(2НЯ)|Ф8(фС(1),	(5.169)
we can freeze the population transfer (null-population transfer), with (5.169),
77=0,	(5.170)
dt
for all times t. In other words, the population in the ground electronic surface, and
necessarily also the population of the excited surface, remains fixed. If we were to
chose E(t) -> \E0(t), it would cause population to be transferred to the upper state,
while E(t) —> — i£o(t) would dump population down to the groundstate. Hence, by
controlling the phase of a laser, we can control the population transfer at will. This
phenomenon applies equally well to driven tunneling in a TLS. What can we achieve
if we manipulate the amplitude C(t)? The change in the energy of the ground state
surface, which can be varied by exciting specific ground-state vibrational modes, is
obtained as
d£g _ d (Ф8(<)|Н8|Ф8(р)
dt dt (Ф8^)|Ф8^))
(5.171)
______________________________________________5.8 Conclusions and outlook 281
Undei the null-population-transfer condition in (5.169), this simplifies to [43]
IT =	(5.17.
It follows that the sign of C(t) can be used to “heat” or “cool” the ground-stah
wavepacket; the magnitude of C(t) in turn controls the rate of heating (or cooling)
With this scheme of phase and amplitude control of a laser pulse it is possible t(
excite vibrationally the lower state surface while minimizing radiation damage Ian
by ionizing or by dissociating the corresponding quantum system.
5.8 Conclusions and outlook
In this chapter we presented a “tour of horizon” of the physics occurring in driven
quantum systems. The use and advantages of the Floquet-theoretical method and its
generalizations have been discussed. In particular, these methods provide a consistcni
physic il picture for intensity-dependent nonlinear quantum phenomena in terms of Flo
quet modes and energy scales, as determined by corresponding quasienergy differences
Not si rprisingly, exactly solvable quantum problems with time-dependent potentials
are quite rare, Sect. 5.4. The Floquet method can be implemented rather effectively
in numerical calculation schemes, cf. Sect. 5.5, and, most importantly, they are non-
perturoative in nature, applicable to arbitrarily strong fields beyond the conventional
rotatir g-wave schemes. Its use in driven quantum systems results in new phenomev
such as frequency-shifts of resonances (Bloch-Siegert shifts), multi-photon transiti
the result of coherent destruction of tunneling [9,19] and related, the generation cf
low-frequency radiation and intense even-harmonic generation. Finally, we discussed
the application of non-periodic, pulse-designed perturbations to control — a prior’
quantum properties such as the population transfer and reaction yields in laser
quantim processes.
Several topics remained untouched. For example, we mainly restricted the dk
cussior to bound quantum states, to problems with a pure point spectrum f-
quasienergies. Interesting problems occur, however, also for driven quantum transpo..
that irvolves scattering states. Such examples are the quenching of transmission m
potent al driven resonant tunneling diodes [44], or the driven quantum transport ...
a pericdic tight binding model [45,46]. In situations where unbound quantum states
determine the physics (ionization, dissociation, decay of resonances, ас-driven tun-
neling decay, etc.), it is necessary to rotate the coordinates of the Hamiltonian into
the complex plane (complex scaling) [47]. This procedure results in complex-valued
quasienergies. For applications we refer the reader to the references given in [47].
Moreover, the problem of the effect of weak or even strong dissipation on the coher-
ent dynamics of driven systems was also not touched upon. The topic of quantum
dissipation, see chapter 4, extended to driven systems, is a nontrivial task. Now the
bath modes couple resonantly to differences of quasienergies rather than to unper-
turbed energy differences with the latter being of relevance when the time-dependent
284 Driven Quantum Systems
interaction is switched off. Consistent quantitative treatments of dissipation for driven
quantum systems are difficult, but represent a challenging area of timely research. First
interesting accomplishments have been put forward recently in Ref. [48]. Strong driv-
ing and moderate-to-strong dissipation are of particular importance for the intriguing
phenomenon of nonlinear Quantum Stochastic Resonance [49]. Also, we have mainly
addressed the driven dynamics in the deep quantum regime. Characteristic for driven
quantum systems is that these exhibit a chaotic dynamics in the classical limit. For the
phenomena occurring near the border line between quantum and classical dynamics,
where a full semiclassical description is appropriate, the reader is refered to chapter 6
on quantum chaos. With driven quantum systems containing a rich repertory for novel
phenomena, and providing us with the tool to control selectively the quantum dynam-
ics, we hope that the readers become invigorated to extend and enrich the physics of
strongly driven quantum systems with own original contributions.
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Thomas Dittrich
Chaos, Coherence, and Dissipation
6.1 Introduction
The concepts of determinism and predictability have concerned physicist for centuries.
Classical Newtonian mechanics appeared so rigidly deterministic to contemporary
that they saw the universe reduced to clockwork—Laplace invented his demon (181T
to bring this view to the point. The first crack in classical determinism was left by
Boltzmann’s attempt to give thermodynamics a foundation in mechanics, in the late
19th century: The number of degrees of freedom in a macroscopic system is just too
huge tc allow for predictions in microscopic detail, so that the evolution of all variables,
except for a few macroscopic ones, has to be treated as random fluctuations, or noise.
This restriction of determinism, however, may still be regarded as being of a mere!v
practical nature,.brought, about only by the physicist?’ insufficient capability to han-:,
large amounts of information.
The most severe challenge to predictability was posed by the advent of quantum
mechanics, around the turn of the century. According to the traditional Copenhagen
interpretation, there exist objective, fundamental limits of determinism in the sniciu-
scopic lealm. The finite ability of a wavefunction to resolve structures in phase sp:a e
is associated with an inherently stochastic nature of microscopic states, which canrmt
be surmounted by improved experimental techniques or a more complete theoretical
descript ion.
The latest blow at predictability originates in the same decades when quam..
physics was born, but has gained thrust only since powerful computers became avail-
able to 'esearchers: deterministic chaos [1—4]. This concept refers to strongly nonlinear
systems whose dynamics act as natural “phase-space microscopes”, in that they enlarge
phase-space structures, in some of the directions of phase space, by a factor growing
exponentially in time. Such systems are not only found in almost all branches of
physics, but appear to be ubiquitous in nature. Their behavior can be modeled by de-
terministic equations of motion with a few degrees of freedom, yet they are practically
unpredi stable because the amount of information, required to specify the initial state,
increases geometrically with the time span to be bridged: Information, or entropy, is
continuously conveyed from microscopic to macroscopic scales [5]. This “practical inde-
288 Chaos, Coherence, and Dissipation
Fig. 6.1: The Sinai billiard.
terminism within fundamental determinism” is closely related to the unpredictability
of many-body systems due to their large number of degrees of freedom. In fact, it was
a two-dimensional system showing deterministic chaos, the Sinai billiard (Fig. 6.1), for
which ergodicity, a crucial aspect of microscopic disorder, could be proven rigorously
for the first time.
For decades, the microscopic description of matter within the framework of quantum
mechanics was predominantly a formal tool, a theoretical construction necessary to de-
scribe the physics of macroscopic systems in a complete way. This situation has changed
dramatically in recent years. Technological progress has provided us with devices that
allow to isolate, to manipulate, and to measure microscopic systems individually—
traps, single-atom masers, electron turnstiles, to name but a few. Besides their direct
impact on experimental techniques and results, these devices have altered our view on
microscopic systems. The quantum-mechanical wavefunction is no longer a quantity of
merely statistical significance. Rather, it becomes associated with individual systems:
this atom, here and now, in the trap.
With this development, a conceptual question arises and becomes urgent: Does
the “practical indeterminism within fundamental determinism” of classical chaos fit
together with the—supposedly—objective indeterminism of quantum mechanics? Are
they related, independent of one another, or even in mutual contradiction? An intense
research activity is currently devoted to this question, under the somewhat misleading
label “quantum chaos”. Indeed, a preliminary answer suggested by the present state of
knowledge would read, surprisingly: Quantum mechanics is deterministic in a stronger
sense than classical mechanics. In closed systems, it does not allow for quantum chaos
to exist at all!
At this point, a frequently cited but faulty argument ought to be rejected that seems
to corroborate the above statement. It rests on the observation that the Schrodinger
equation is linear, while the equations of motion of chaotic classical systems gain their
instable character only by being highly nonlinear. This view, however, results from
a misconception due to the completely different ways the respective dynamics is rep-
resented: Newton’s and Hamilton’s equations are formulated in terms of trajectories,
6.1 Introduction
while Schrodinger’s equation deals with probability amplitudes. Indeed, if classicai
mechanics is cast in the form closest to the quantum-mechanical language, namely the
representation by a flow of probability density in phase space, governed by the Lion-
ville equation, it appears as perfectly linear as quantum mechanics. An exponential
divergence emerges only upon studying flow lines in phase space.
Rather, the reason that quantum mechanics tends to suppress classical chaos lies in
the fundamental limits quantum theory imposes on the fine-grainedness of structures
in phase space—in short, the information density. As a result, for example, an isolated,
bounded quantum system has a discrete energy spectrum. Its time evolution, starting
from any initial state, can be decomposed into a finite number of discrete frequencies:
it is quasiperiodic. A quasiperiodic dynamics is readily predictable and invertible, with
an error increasing only linearly with time. This phenomenon resembles the fact that
a digital computer, possessing only a finite, if large, number of internal states, must
eventually begin to repeat itself, if it is started with some fixed program and then left
alone. Therefore, it cannot simulate chaotic behavior for an unlimited time.
However, what about Bohr’s correspondence principle, stating that in the limit of
large quantum numbers, the time evolution of quantum-mechanical observables shorn
come c.rbitrarily close to the corresponding evolution of their classical counterparts?
After г 11, quantum theory is more exact and complete than classical mechanics. If it
excludes the existence of chaos, how, then, is the alleged abundance of chaotic system0
in the macroscopic world possible? Part of the solution is that it takes some t'
for the quantum-mechanical coherence effects suppressing chaos to become effect! e.
Until tnen, a quantum system that approaches a chaotic system in its classical limit,
does mimic the corresponding classical dynamics: It shows symptoms of chaos. T1
“break time” that has to elapse before coherence becomes dominant, is proportion .1
to the dimension of the Hilbert space spanned by the states effectively involved in t he
dynamics. By increasing this dimension—which amounts to considering large quantic"
numbers—the break time can be made arbitrarily long. In the above analogy to digi... .
computers, the effective dimension corresponds to the number of internal states: Bv
increasing the memory sufficiently, the initial nonrecurrent phase of a program run is
extended without bound.
Scaling up a system in size without adding more freedoms is, however, only one
facet of the classical limit. To consider a microscopic system as if it were isolated is
an approximation, albeit often a good one. There must be some contact with “the
world out there”, or otherwise we never knew about that system. With the ambient
world, the notion of a macroscopically large number of degrees of freedom enters the
stage again. On the classical level, it is responsible for irreversibility and dissipation.
Its role for quantum mechanics is just as fundamental. There, it counteracts coherence
and thereby tends to reduce quanturn-mechanical, coherent superpositions of states to
classica. alternatives. This process is crucial for quantum measurement, but it al? ‘
takes place wherever microscopic systems are coupled to their environment. It a-i ’
another aspect to quantum chaos: The suppression of chaotic dynamics by quant:
coheren :e is not the last act. Coherence, in turn, is destroyed by the ambient degieos
freedom. In this way, a constituent feature of classical chaos is restored, the product a : 
290 Chaos, Coherence, and Dissipation
of entropy. Translated into the above metaphor, the effect of decoherence corresponds
to connecting the computer to some external, “true” random generator, e.g., a signal
from a radioactive sample or a playful programmer, which every now and then knocks
the program out of its deterministic, repetifive course and, starts it anew.
One basic quantum effect remains unaffected by decoherence: uncertainty. This
becomes particularly relevant if one takes dissipation into account, the second prin-
cipal consequence of the coupling to the environment. In classical chaotic systems,
dissipation lets the dynamics settle in a stationary state, where the phase-space flow
has contracted to a submanifold of smaller, typically fractional dimension, a strange
attractor [1, 3-5]. The corresponding dissipative quantum system approaches a steady
state similar to its classical counterpart. Exact agreement, however, is excluded be-
cause quantum uncertainty is at variance with the infinitely nested, fractal geometry
of strange attractors—the finite value of Planck’s constant imposes a lower bound on
self-similarity in phase space.
Quantum chaos combines the effect of the many degrees of freedom of the macro-
scopic world, quantum-mechanical coherence, and classical instability, as essential ele-
ments. They will be discussed, in the following, in the order in which they affect the
typical time evolution of a classically chaotic quantum system.
6.2 Quasiclassical chaos
The present section is devoted to the initial phase of the time evolution of a classically
chaotic quantum system, where one expects only minor deviations of the quantum
from the classical dynamics. Topics to be addressed are phase-space representations
of the quantum state, approximate methods for reproducing the quantum dynamics
using only classical input (semiclassical methods), features of the quantum dynamics
that bear witness of the underlying classical chaos, and fingerprints of classical chaos
in the quantum-mechanical energy spectra.
In recent years, a number of monographs [6-9], anthologies [10,11], and general
reviews [12-14] have become available which cover broad subsets of these issues.
6.2.1 Time-domain aspects
The characteristic property of classical chaotic systems which immediately comes to
mind, the exponential divergence of initially close trajectories, has no direct quantum-
mechanical counterpart because, strictly speaking, there is no such thing as a trajectory
in quantum mechanics. Other means of detecting symptoms of classical chaos in quan-
tum mechanics are required. A more adequate concept is that of minimal invariant
manifolds in the classical phase space. These are the sets of phase-space points which
remain invariant under the classical time evolution—not pointwise, but as sets—and
cannot be decomposed further. In Hamiltonian systems, the most important types of
invariant manifolds are invariant tori and isolated unstable periodic orbits [13,14].
6.2 Quasiclassical chaos 29
Fig. 6 .2: Motion on a two-dimensional torus in the three-dimensional energy shell of a system witi
two dejrees of freedom (a); the two irreducible types of paths (7j and 72) on the two-dimensimm
torus (э). From [4].
Tori are the hallmarks of regular motion. A system is completely integrable if v
has as many, say f, independent constants of motion as it has degrees of freedom. Tin
typical textbook examples of mechanical systems are of this type, e.g., the harmoni'
oscillator, the pendulum, the Kepler problem. The 2/-dimensional phase space i:
then c ompletely foliated (stratified) into /-dimensional sets with the topology of a
torus (Fig. 6.2). This means that a phase-space point is determined by a vector G
consis .ing of / angles, which describe the motion on a given torus, and an /-dimensional
action vector I, which selects this torus from all the others. The actions are given L;
4 := (Utt)"1 f q -p(q), where 7; denotes one of the / irreducible ways to go around
/-torus (Fig. 6.2). They are uniquely related to the / constants of the motion. If the f
frequencies w = 0 of rotation are all in rational relations to each other, the torus i bci!
is calk d rational (otherwise irrational), and the motion on it disintegrates further 1.
a family of periodic orbits, parameterized by / — 1 initial angles.
Fig. 6.3: Local phase-space coor-
dinates in the vicinity of an iso!"
ted periodic orbit. Here, в de'
the time-like coordinate along .
orbit, and the vector r j. comprises
the remaining coordinates and mo-
menta within the energy shell.
In a completely chaotic system, in contrast, energy is the only conserved quantity,
and a Jingle, typical trajectory explores the whole (2/ — l)-dimensional energy shell:
the system is ergodic. However, within the energy shell, there exist invariant one-
dimensional manifolds in the form of unstable orbits that are closed in phase space
(Fig. 6.3). They correspond to periodic motion with a period T, parameterized by a
single angle 6 — 2-rrt/T (Fig. 6.3). Transverse to each periodic orbit, there are (/ — 1)
stable and (/ — 1) unstable directions within the energy shell. Upon completing one
full cy< le around the orbit, this pattern of stable and unstable directions must return
into itself, but this does not exclude a rotation of the pattern by an integer multiple
292 Chaos, Coherence, and Dissipation
Fig. 6.4: Self-similar structure of phase space, as predicted by the KAM theorem for a non-integrable
perturbation of an originally regular system. From [13].
of 7Г. This integer can be considered as a winding number [15,16]. It is usually
referred to as the Maslov index of the periodic orbit. The f rates of contraction along
the stable, and f rates of stretching along the unstable directions are the eigenvalues
of the stability or monodromy matrix M that describes the evolution of a trajectory
over one period, linearized in the vicinity of the periodic orbit. If det(M) > 0, the
periodic orbit is called hyperbolic, otherwise inverse hyperbolic. The unstable periodic
orbits form one-dimensional families parameterized by the energy, i.e., they are isolated
within a given energy shell. It suggests itself, however, that the energy shell as a whole
forms the closure of the set of all unstable periodic orbits with a given energy and
is uniformly covered by them [17]. For area-preserving maps as a special class of
Hamiltonian dynamical systems, an analogous theorem has indeed been proven.
Completely chaotic (hyperbolic) systems, sometimes referred to as “hard chaos”, are
as untypical as completely integrable systems. The standard situation—“soft chaos” —
is that both types of motion are simultaneously present, each one occupying some
positive fraction of phase space. However, the respective phase-space components are
not distributed like chaotic lakes in a regular landscape, or conversely, regular islands
in a chaotic sea—even if the latter alternative is closer to the truth. Rather, regular
and chaotic areas are intertwined in an extremely intricate manner. The celebrated
KAM (Kolmogorov-Arnol’d-Moser) theorem describes how, with increasing nonlinear-
ity of the dynamics, regular motion is gradually transformed into chaotic motion [13].
Specifically, all the rational tori are immediately destroyed by the slightest deviation
from integrability. Together with a neighborhood of finite extension, they are replaced
by a chaotic layer with an alternating sequence of elliptic (stable) and hyperbolic
(unstable) periodic orbits embedded in it. Applying the KAM theorem also to the
regular islands around the stable “daughter” orbits implies that they must themselves
be surrounded by similar chaotic layers containing “granddaughter tori”, and so on ad
infinitum: A self-similar hierarchy emerges, consisting of regular islands surrounded by
chains of smaller regular islands, etc. (Figs. 6.4,6.5).
6.2 Quasiclassical chaos 291
(с) К = 2.5
(d) К = 4.0
Fig. 6.b: Phase space of the standard map, Eq. (6.76) below, for increasing values of the classical
nonlinearity parameter К = кт. From [2].
An irrational torus is destroyed only when the nonlinearity exceeds some finite
threshold value specific for this torus, and gives way to a fractal invariant manifold
immersed in the chaotic sea, called a cantorus because it forms a Cantor set [18]
(Fig. 6 6). Even when the cantorus has disappeared upon further increase of the
nonlinearity, a distinct structure, called a vague torus, remains [19]. It can be associate-1,
to quantities that form exact constants of the motion in a regular region and are st ill
approximately conserved in the adjacent chaotic region, maintaining a rest of order on
the chaatic side of the regular-chaos border.
The intricate machinery of Hamiltonian chaos just sketched is exclusively assem’ •;
from canonically invariant parts in the 2/-dimensional phase space. Quantum mechan-
ics as it is introduced in textbooks, on the other hand, deals with wavefunctions eitln-i
in configuration or in momentum space. In order to identify quantum-mechanical
294 Chaos, Coherence, and Dissipation
Fig. 6.6: Cantorus occurring in the phase
space of the standard map, Eq. (6.76). From
[18].
0.0	0.5	10
X
counterparts of self-similar island chains, cantori, etc., it would be desirable to have a
representation of quantum mechanics which treats spatial and momentum coordinates
as symmetrically as Hamiltonian mechanics does. Such phase-space representations
do indeed exist. Since the full information on a pure quantum state can be encoded
in a complex function of only f variables, the representation of the same information
by a function on a space of twice that dimension must be redundant and cannot be
unique. This freedom, in turn, can be exploited to devise different types of phase-space
representations, suitable for specific purposes and applications [20,21].
There exist phase-space representations based on pure states, and others based
on the density operator. Of course, the density operator provides the more general
formalism (cf. Section 4.3). It can be translated into a phase-space function by the
following prescription: Associate with an operator A, expressible as a function of the
phase-space vector operator r = (p, q), a function
Aw(r) = tr pl(r)<5(r — r)] .	(6.1)
The 2/-dimensional delta function 8(r — r) is to be read as its Fourier decomposition.
Here, however, a rule is required how to order products of noncommuting operators
like p^q". The Weyl correspondence prescribes specifically to choose the ordering in
such a way that
Av(r) = ^7 tr |A(r) J ^p' J_oadf<l' exP (^(P-P) •?' + (?- q) •Pjjj- (6.2)
The Wigner function (or representation) is defined by inserting A = h~fp. Using the
Baker-Hausdorff transformation together with the fact that all the components of p
and q commute with the commutators [p,,^] =	the momentum integrations
in Eq. (6.2) can be performed to obtain
1 r°°
•Pw(p,q) = T7/ dfq'{q-q'/2\p\q + q'/‘2)eip4/h.	(6.3)
ftJ J —oo
6.2 Quasiclassical chaos 29
In pa rticular, for a pure state p — | ip) (ip |, one has
1 Z*00
•Pw(p, q) = Tf V>*(q + q'/2) V>(q - q'/2) eip'4'/h,	(6,-
with ip(q) = (q\tp).
F: om the definition (6.3), there follow a number of characteristic properties of (h
Wigrer function [20-22]:
(i)	It is real valued (this follows from Eq. (6.3) and the Hermitecity of the densit
operator).
(ii)	It is not positive definite.
(iii)	It is normalized,
/oo	roo
dZP / dfqPw(p,q) = 1.
-oo	J —oo
(iv)	It generates the correct marginal distributions,
[ dfpPw(p,q) — (q\p\q),	[ dfqPw(p, q) = (p\p\p).	(6.C
J —oo	J — oo
(v)	Expectation values are evaluated as phase-space averages: If Xw(p, q) is associ
ated to a given operator A by the Weyl correspondence (6.2), then
л	~	/*00	/*00
{A)=U[Ap]= dfp dfqAw(p,q)Pw(p,q).	(6 7
J — cc	J — oo
(vi)	By inverting the relation (6.3), the density operator can be retrieved fr -ii: th.
Wigner function,
(q|p|q')= Г dWp (,'“,)/ftPw (p,	(6-s
J — oo	\	2 /
From (i) and (v), one might expect that the Wigner function could be interpreted a‘
a quantum-mechanical probability distribution in phase space. Property (ii), however
shows that this is not the case. Rather, the Wigner function contains all the phas,
information of the density operator, and therefore it can give rise to constructive о
destrrctive interference (Fig. 6.7). For this reason, it is sometimes referred to as ;
quasi-probability density. For all finite values of fi, the Wigner function is continuous
In particular, it follows from (iv) that a delta function on a phase-space point is not
admissible as a Wigner function: It would violate the uncertainty principle. Only in
296 Chaos, Coherence, and Dissipation
Fig. 6.7: Wigner representation of an initially
Gaussian minimum-uncertainty wave packet,
after one period of the driving force in a para-
metrically driven quartic oscillator. From [23].
the limit h —> 0, the Wigner function can approach a sum over delta functions at the
phase-space positions of point particles.
A phase-space representation that does have the properties of a classical probabil-
ity density can be constructed, starting from the following consideration: The most
detailed observable phase-space information allowed in quantum mechanics is gained
by an idealized joint measurement of position and momentum, with the best resolu-
tion consistent with the uncertainty principle. Formally, such a measurement amounts
to determining the occupation probability of a coherent state centered at the phase
point r = (p, q). Coherent states are minimum-uncertainty wavepackets that move,
but do not change their shape under the time evolution for a given system. For an
/-dimensional harmonic oscillator, they are Gaussians with a combined momentum-
position variance h!, obtained, e.g., by shifting the ground state from the origin to the
point (p, g),
,7) - ПехР	Re 7, p; - ^hmwi Im 7,	| 0),	(6.9)
where | 0) is the ground state of that oscillator with mass m and frequency in
the spatial direction i. The parameter 7, an /-dimensional complex vector, gives
the position of the coherent state: (pi,<?i) = ([2Ат^г]1'/2 Im7i, [2Й/гпа.’,]1'/2 ReyJ, i =
1,...,/. It is easy to check that I7) is in fact an eigenstate of the annihilation
operator b, with elements Ь, = (тшг/2Ь)1'/2(-;г 4-i(2hmwi)-1/,2pi, of the above oscillator,
with eigenvalue 7, i.e., 6,7) — 7I7). Therefore, it can be written as
I 7) = e“l'1'|2/'2 exp(7 • b^)| 0).	(6.10)
6.2 Quasiclassical chaos 29 /
The Husimi distribution is now defined as the expectation value of the projector <
I T>,
Ph(p,q) = tr|j -y) <7 |p] = ^(tIpIt>-	(6.11)
Its basic properties are [20,21]:
(i)	I, is real and positive definite.
(ii)	I, is normalized,
rOO	roo
/ dfp / d/qPH(p,q) = 1.	(6.12)
(iii)	I, is related to the Wigner function by
Ph(m) = Г d2/7'e-2^^l2Fw(p',q').	(6.13)
\7Г/ J-oo
where (p[, </') = (y/2hrrwi Im 7', yj2h/irwi Re7'), i = 1,..., f.
Properties (i) and (ii) ensure that the Husimi distribution is indeed a proper phase-
space probability density. Property (iii) expresses that it is related to the quantum
state b/ a measurement-like operation. The smoothing with a minimum-uncertainty
Fig. 6.8: Husimi distribution corresponds
to the Wigner function shown in Fig. 6.7.
From [23].
298 Chaos, Coherence, and Dissipation
Gaussian, implied by Eq. (6.13), is sufficient to remove the oscillations of the Wigner
function. In this sense, the Husimi distribution is akin to the squared modulus of the
wavefunction: It has lost all coherence (Fig. 6.8). An important feature of the Husimi
distribution is that there exist f free parameters mcoj in the definition (6.11). They
control the distribution of uncertainty among spatial and momentum coordinates, i.e.,
determine whether the contours of the underlying coherent state are more elongated
in the qi or in the рг direction. This arbitrariness represents a drawback, at times, but
can be made good use of, at other times.
As announced above, it is also possible to devise phase-space representations not
based on the density operator, but on the wavefunction itself. One possible definition
endows the square root of the Husimi distribution with the appropriate phase [24],
V’(t) — e'7'2'/2( T I V’),	(6-14)
where Re 7 ~ q, Im 7 ~ p, as above (note the exponential divergence of the prefactor
with |7|2). It is called the Bargmann representation or coherent-state amplitude. The
Bargmann representation is not defined for density operators that do not correspond
to a pure state.
Phase-space representations allow to compare the quantum to the classical dynamics
on an equal footing. One can go an important step further and try to reconstruct the
quantum dynamics using exclusively classical input. This might appear a paradoxical
task because, after all, classical mechanics is an approximation to quantum mechanics
which is valid only asymptotically in certain limits and therefore, in principle, is just
wrong. On the other hand, classical mechanics offers a very economic and intuitive
description: It can provide a “sceleton” of quantum mechanics.
Attempts to employ classical concepts in quantum mechanics, referred to collec-
tively as semiclassical methods, are as old as the quantum theory itself. The Bohr-
Sommerfeld quantization rule, a precursor of the mature quantum theory, already
belongs to this category. In a more modern rendering, it is known as EBK (Einstein-
Brillouin-Keller) or torus quantization. It rests on the concept of tori: A torus is called
quantizing if it fulfills the condition
I = In= (n + fi,	(6.15)
where I is the vector of f characteristic actions of the torus, p contains the corre-
sponding Maslov indices, and n is a vector of f integers, the quantum numbers. With
each quantizing torus, an energy eigenfunction with an eigenvalue En = H(In) is
associated, which is sharply concentrated on this torus.
Einstein was the first to point out that this quantization condition can be applied
only to completely integrable systems whose entire phase space is foliated by tori [25].
The underlying idea, however, that the scaled classical actions 2irli/h correspond to the
phases of wavefunctions, accumulated along classical paths, is more general. If one adds
to it the notion that the squared modulus of the wavefunction represents a probability
density, one arrives at the WKB (Wentzel-Kramers-Brillouin) approximation [26,27]:
6.2 Quasiclassical chaos 299
Fig. 6.9: Turning points in a binding potential.
Region II is accessible to classical motion at energy
E, regions I and III are not.
A wavefunction can always be written as the product of a real amplitude and a phase
factor,
^q,t) = A(q,t)eis^'h.
(6.16)
Inserting it in this form into the Schrodinger equation and neglecting all terms of order
h or hig'ier yields the equation
H(S(q, t), q) + ^S(q, t) = 0
(6.17)
for the phase S(q,t), where H(p,q) is the Hamiltonian. Equation (6.17) is recogni
as the c.assical Hamilton-Jacobi equation, which shows that S(q, t) can be identified
with the classical action, 5(q, t) = fqdq' -p(q'). Similarly, for the squared amplitu...
p(q, t) - (A(q, t))2, one obtains the continuity equation,
—p(g, i) + V, • \p(q, t)v(q, t) j = 0, v(q, t) = Vp77(p, q),
(6.18)
obeyed by the classical configuration-space density p(q, t). It requires that the density
be inversely proportional to the instantaneous momentum, so that Eq. (6.16) reads
explicitly,
V>(q,t) = A{q0,t0)
|P(<7)I V	J
(6.19)
where q{ is some arbitrary initial point. From this form, it is clear that the WKB
approximation must be modified in the vicinity of turning points, that is, points where
V(q) = E and therefore |p(q)| = 0, corresponding to caustics of the classical motion
(Fig. 6.9). This is not surprising since at caustics, the deBroglie wavelength A(q) ~
2rrfi,[2m(ld(q) — E)]-1/2 diverges. By recurring to the exact solution of the Schrddin " 
equation in a neighborhood of the turning point where the potential can be replaces
a linear approximation, or by switching to the momentum representation which does
not show caustics where the coordinate representation does, one obtains connect
formulae bridging the turning point.
300 Chaos, Coherence, and Dissipation
The divergence of the amplitude near caustics is associated with a sign change of
the denominator in the determinantal prefactor in Eq. (6.19), which can be absorbed
in the phase as a shift by ±тг/2. The accumulated phase, due to these shifts, is taken
into account by the Maslov indices which occurred already in Eq. (6.15) and can be
identified with the winding numbers mentioned above [15,16].
The way it is constructed, the WKB approximation is particularly well suited for
following the time evolution of a given wavefunction over a finite time. A time-evolved
wavefunction is related to its original state by the unitary time-evolution operator
f/(t',t) = f exp ( f dt"H(t")) ,	(6.20)
у ft Jt	)
where T effects time ordering. In the spatial representation, if causality is imposed, it
becomes the outgoing (retarded) propagator,
K+(q', f'i = (Q' I U(t', t) | q)Q(t' - t).	(6.21)
Here, 0(t) denotes the Heaviside step function. The semiclassical propagator derived
directly from the WKB approximation, Eq. (6.19), reads
1	/fl X 1/2
K+(q',t';q,t) — -	det I ] x
4’ 7 (27rih)f/2 Y	\9q')
exp(^Rk(rf,t';q,t) -	•	(6.22)
\n	2 /
It is known as the Van-Vleck propagator [27]. Some features of this formula cannot be
anticipated from the above discussion: The summation over the index k accounts for
the fact that, due to the occurrence of caustics of the classical motion, there can be
several solutions pj.(q) of the classical equations of motion (Fig. 6.10), each with its
specific Maslov index p*, and action
Rk(q',t'-,q,t) = £ dq"  pk(q") - £ dt" H(pk(t"), q(t")-1").	(6.23)
The second term on the right-hand side of Eq. (6.23), involving the Hamiltonian,
appears because the Van-Vleck propagator does not refer to a fixed energy but to a
fixed time interval between initial and final conditions.
It is important to realize that in the derivation of the WKB wavefunction and the
Van-Vleck propagator, no assumption has been made as to the type of the classical mo-
tion: They are valid for integrable and chaotic systems alike. Rather, their limitations
arise upon extending the time span over which they are to be applied. A major prob-
lem is the proliferation of caustics with time, which requires to find a rapidly growing
number of possible classical trajectories connecting the initial with the final position
in the time given.
6.2 Quasiclassical chaos 301
2	0	2	4	-2	0	2	4
X	X
Fig. 6.10: Caustics of classi-
cal regular motion on a torus
in a three-dimensional energy
shell, projected onto coordi-
nate space. In this represen-
tation, the function p(q) pos -
sesses 2, 4, or (in (a)) 6 sheets.
The baselines of the caustics
are repeated schematically in
the respective lower panels.
From [28].
Cau sties come about by the projection of a manifold in phase space onto a subspace,
e.g., on.о configuration space (Fig. 6.10). They do not represent canonical invariants
and disappear once the full phase space is considered. Therefore, it is advantageous
to study also the quantum dynamics in a phase-space representation. For example,
an equation of motion for the Wigner function can readily be derived by invoking the
von-Neumann equation,
=	(6.24)
di n,
and forming the time derivative of the defining Eq. (6.3). The result can be written in
the compact form [20,21]
Q
= {^Mp, 9), Av(p, q, i)}Moyal >	(6.25)
where Hw(jp,q) is the Wigner representation of the Hamiltonian, as obtained from
Eq. (6.2) by setting A(r) = H(r). The Moyal bracket reads explicitly
{Hw,Tw}Moyai = ~^sin (ifl{#w, Fw}Poisson)
= r sin T (v₽2 • Vqi - VPl • VqJ Ffw(ri)Pw(r2; t)|	(6.26)
П, \ Z '	/	1г1=Г2— T
302 Chaos, Coherence, and Dissipation
with т = (p, q), etc. Here, the zeroth-order term in h on the right-hand side generates
the classical time evolution, governed by the Liouville equation, while the subsequent
terms in the expansion of the sine represent quantum corrections of increasing order.
The concept of the semiclassical propagator, introduced above, can readily be car-
ried over to phase-space representations like the Wigner function. The Wigner propa-
gator is defined as the Wigner representation of the quantum-mechanical propagator,
#w(p, q; t) = Г dfq’e-'^K (g - t; g + oY (6.27)
By inserting the Van-Vleck formula (6.22) for the propagator in Eq. (6.27), one obtains
a semiclassical expression which is exclusively built upon canonically invariant phase-
space features and related quantities: Caustics cannot occur from the outset. See
Ref. [29] for a detailed derivation and discussion.
In order to work with semiclassical propagation in the context of quantum chaos,
it is important to have a feeling about its quality, i.e., in particular, for how long in
time it provides a good approximation to the true quantum-mechanical time evolution.
The answer depends on whether one requires the correct reproduction of fine details in
phase space, or merely of the time evolution of global dynamical quantities, like auto-
correlation functions. The mechanism which lets the quantum state eventually deviate
from the intricate complexity of the classical phase space of a chaotic system, can be
elucidated on basis of the equation of motion, (6.25) with (6.26), of the Wigner function
[30]: An initial structure in the Wigner function, say in its momentum dependence,
on the scale Др(0) (e.g., the width of a Gaussian) is stretched out in f — 1 directions
transverse to the chaotic classical flow, and contracted in the other f — 1 transverse
directions by equal amounts, given by the local Lyapunov exponents [3,4]. The con-
traction leads to a variation of the Wigner function on the exponentially shrinking scale
Ap(t) = Др(0)е~Л4, where Л is an effective Lyapunov exponent, and thus to an increase
of the higher derivatives in Eq. (6.26), as dnPw/dpn ~ (Ap(t))-nPw ~ PyjenXt. In this
way, the nth quantum correction term acquires a magnitude comparable to the leading,
classical term as soon as the growth of the derivatives of Pw compensates for the small-
ness of the nonlinearity of the potential, expressed, e.g., by the characteristic lengths
an = (1/'/у<п+1))1/\ This happens at a time
tn«|ln(a-2^P(0)\	(6.28)
A у n }
dubbed the log time. Of course, as is manifest in Eq. (6.28), the time by which a
semiclassical description ceases to be correct depends on the level of approximation,
e.g., the order n of the last term after which an expansion is truncated.
The time evolution of a global dynamical quantity does not generally reflect fine
details of the phase-space structure. In this case, the approximate agreement between
quantum and classical or semiclassical dynamics lasts much longer than would be ex-
pected from the log-time limitation [31]. It is a different mechanism which leads to
deviations here, based on spectral rather than on spatial features. A localized initial
6.2 Quasiclassical chaos 303
state in a bound quantum system can be decomposed into an effectively finite number
of eigenstates, and correspondingly selects a finite number of eigenvalues from a discrete
spectrum (the resulting weighted spectrum is called the local spectrum). These states
and frequencies define its “dynamical repertoire”, rendering quasiperiodic the time de-
pendences of all dynamical quantities derived from it: Quasiperiodic time evolutio;.
is not exactly periodic (because the frequencies involved need not be commensurate'!,
but nevertheless appears “boring” to an observer following it for a long time. The
time scale t* of near self-repetition is determined by the smallest scale in the spectrum,
the typical separation of two neighboring levels. These separations, as will be shown
below, depend on Planck’s constant as hf. For a semiclassical description that d1 1 s
not succeed in reproducing the discreteness of the spectrum, the scale t* ~ defim 3
the uli imate limit of fidelity.
6.2.2 Energy-domain aspects
Classical chaotic behavior can be defined unambiguously only by recurring to infmiie
time series. From this fact, it is to be expected that its traces in the quantum dynamics
are also most clearly represented in features related to the longest time scales, that is in
energy eigenstates and eigenvalues. Indeed, the energy eigenstates are the appropriate
place Ю search for traces of characteristic phase-space structures like chaotic regiom
and regular islands, on the coarsest scales, as well as unstable periodic orbits and tori,
cantor and vague tori, on finer scales.
Th? ЕВК quantization rule, Eq. (6.15), already suggests that in regular regions,
quanti m eigenstates reside on classical tori. More detailed insight is provided by using
the WKB approximation (6.19) for the wavefunction, in the case of regular classical
motion, and studying the corresponding Wigner function, by substituting into Eq. (6.4).
For th? Wigner function associated with the eigenstate labeled by n (cf. Eq. (6.15)).
one obtains to zeroth order in ft [14]
Av,n(p,<7) = tA/W,?) -In).	(6.29'
(27Г)'
It is ccncentrated on the torus labeled by the action In = (n + /z/4)ft, and unifol :: ! 
distributed in the canonically conjugate angle variable в. A more refined semiclas.T !
analys s reveals that the /-dimensional delta function in Eq. (6.29) is in fact an :q-
proxin ation to a smooth distribution which decays rapidly on the side from where i
torus appears convex and exhibits oscillations (fringes) on the opposite side, similar !<>
the os< illations visible in Fig. 6.7.
Th s result shows that one can meaningfully talk of “regular quantum eigenstate.: "
and it is tempting to generalize and postulate “chaotic eigenstates”. By analogy, one
expects them to be concentrated on the invariant manifolds characterizing chaotic
motion, and to be distributed on these manifolds according to the classical invariant
measu e (the distribution that remains unchanged under the time evolution). There
are two types of invariant manifolds which both come into question here: the chaotic
region as a whole, covering a positive fraction of the energy shell, and the unstable
304 Chaos, Coherence, and Dissipation
periodic orbits embedded in it. The periodic orbits, however, cover a zero phase-space
volume. A typical initial condition does not lie on a periodic orbit, so they should not
leave strong traces in quantum states. Indeed, it has been proven that in the classical
limit of certain completely chaotic systems, the Wigner function approaches the form
[14], reminiscent of the microcanonical ensemble,
Pw(p, q) = \dQ/dE\~i6(E - H(p, g)),	(6.30)
where dfi/d£ = / d-fp J d-fq5(E — H(p,q)) denotes the energy-shell volume. This
implies that the configuration-space wavefunction consists of a superposition of plane
waves with fixed wavenumber |Л| = |p|/fi = [2m(E — V^q))]1^2/h, determined by the
local kinetic energy, but with random directions, so that up to statistical fluctuations,
it is isotropic.
To be sure, it is not excluded that a structure forming a set of measure zero in
classical phase space, such as an isolated periodic orbit, becomes visible in quantum
eigenstates, provided the corresponding quantum-mechanical feature does not violate
the uncertainty principle. There is ample evidence that this does indeed occur. In
particular, contrary to the prediction (6.30) for the classical limit, eigenstates of chaotic
systems exhibit clear traces of unstable periodic orbits. They appear as “ridges” of
Fig. 6.11: Scarred energy eigenstates of the stadium billiard (a), represented by node and contour lines
in configuration space. In (b), the scarring periodic orbits states are superposed on the corresponding
states. From [32].
6.2 Quasiclassical chaos
Fig. 6.j 2: Influence of cantori in the classical phase space on quantum transport. A wave packet
launcher, within the strip confined by two cantori (cf. Fig. 6.6) in the phase space of the standard map
(a) does in the long-time average, not penetrate these cantori, up to a leakage orders of magnitude
smaller than the occupation probability within the strip ((b) and (c)). From [33].
the probability distribution along the classical orbit, extending over a distance of the
order of one deBroglie wavelength into the surrounding configuration space, and are
referrec to as scars [32] (Fig. 6.11). There is no simple rule as to which state is or is not
scarred, and by which orbit(s): Some chaotic eigenstates are scarred simultaneously
by several orbits. Conversely, scars may become visible only upon superposition of
several eigenstates neighbored in energy. The number of orbits to leave traces ii
chaotic state increases towards the classical limit. This reconciles the occurrence of
scars with the uniform classical limit (6.30): Classical ergodicity requires that the
set of periodic orbits fill a chaotic region with a density consistent with the classier’
invariant measure within this region, e.g., uniformly in the case of completely cha
regions with constant classical Lyapunov exponents.
Uns .able periodic orbits are not the only type of phase-space fine structure which
affect the quantum dynamics. Another important example are cantori. By forminc
leaky barriers within a chaotic region, they significantly restrict the classical phase-
space flow. This effect not only survives in the corresponding quantum dynamics, but
is even enhanced there, because the self-similar geometry of a cantorus is modified due
to quantum uncertainty (Fig. 6.12): A Cantor set has gaps on all scales. Those gaps,
however, that are too narrow to be resolved quantum mechanically, block the quantum
flow as if they were closed. Therefore, cantori are less penetrable quantum mechanically
than clessically [33]. A similar statement applies to vague tori. In a classical system,
vague tori are not easy to detect, because their influence on the chaotic flow is weak.
Nevertheless, cantori and vague tori are sufficient to support eigenstates which still
retain t re properties of regular states, as discussed above, even if they are localized
within c classically chaotic region. In this way, the quantum-mechanical counterpart.'!
of canton and vague tori smear out the classical borderline between regular and chaotic
and shift it towards the chaotic side: Only in the classical limit, the concepts of regular
and chaotic states represent a sharp, unambiguous distinction.
It is nuch more obvious to search for traces of classical chaos in quantum cigenst.it < s
than in the associated eigenenergies, since for the latter, there is no classical concep*
to immediately compare with. Still, complementary to the discussion of time-domain
306 Chaos, Coherence, and Dissipation
properties, one may ask the following two questions:
(i) How can the spectrum of a given system be obtained, using exclusively informa-
tion from the corresponding classical system, in case it is chaotic?
(ii) Is it possible to tell from the spectrum alone whether the underlying quantum
system is chaotic or integrable in its classical limit?
The basic quantity forming the input for all discussions of the spectrum, is the
spectral density, also called density of states (dos)
OO
d(E) = X^E-En),	(6.31)
n=l
where the En are the eigenenergies. It is assumed that the spectrum is discrete through-
out (other types of spectra will be considered below). Of course, it would help in an-
swering the above questions if the spectral density could be related to some quantity in
the time domain, so that whatever is known about the dynamics of the system could
be translated into spectral features. After all, the eigenenergies are eigenvalues of the
Hamiltonian that generates the dynamics. This allows to write the spectral density as
d(E) = tr [<5(H - E)] .	(6.32)
By evaluating the trace in the spatial representation and writing 5(x — ;r0) = — (1/тг)х
lime^0+ Im((z + ic — io)-1), one obtains
d(E) = ~Im [f°° dsq G+ (q, q; E)] .	(6.33)
The outgoing (retarded) Green function G+(q',q; E), defined by
£+(</,q;E) = (q'|G+(E)|q), G+(E) = lim (E + ic - E)’1,	(6.34)
C-+0+
is related to the outgoing propagator by a Fourier transformation,
G+(q',q;E) =[°° dt e'Et>hK+ (q‘, t\q, 0),	(6.35)
h Jo
so that
d(E) = Г dte'Et/ha+(t),	(6.36)
7Г/1 Jo
where
a+(t) = tr [l7(t,O)0(t)] = У dfqK+(q, t;q, 0).	(6.37)
For the following discussion, we define a(t) = tr[G(t, 0)] so that a(-t) = a*(t) provided
[/(—t, 0) = LE(t,O) exists for t > 0. Eq. (6.37) demonstrates that the time-domain
6.2 Quasiclassical chaos 307
Fig. 6. .3 : Section of the spectrum of Hydrogen in a magnetic field, plotted as a function of a scaled
magnetic field strength at constant energy (a), and section of the Fourier transform of this spectrum
(b). The periodic orbits associated to the three marked resonances in (b) are shown as projections
onto a plane parallel to the magnetic field. From [36].
quantity a(t) is closely related to the diagonal of the propagator. More precisely, i.
gives the fraction of the quantum amplitude that returns to the initial position at time
t, summed over the entire Hilbert space. It may therefore be called a return amplitude
A semiclassical expression for the return amplitude is now established, on bas’-
of Eq. '6.37), by substituting the Van-Vleck formula (6.22) for K(q, t’,q, 0) [7,34,1.
On this level of semiclassical approximation, integrations over quantities with a space-
dependent phase factor, such as the propagator in Eq. (6.37), can be performed 1-
the method of stationary phase. This step results in the condition that the contriL...
ing classical trajectories be closed not only with respect to position, q(t) = q(0), as
is already implied by Eq. (6.37), but also with respect to momentum, p(t) = p(i''
Therefcre, contributions to the semiclassical spectral density are associated with реи-
odic orbits of the classical dynamics. For t —> 0, whatever the dynamics be, all points
in phase space contribute to the spectral density: They form the starting and end
points of “paths of zero length”. Accordingly, the return amplitude can be split into
two components,
a(f) = a0(t) + a(t).	(6.38]
The cortribution of the paths of zero length has the form of a delta function in time,
n0(t) = /t1_/5(t)(|dQ/dE|),	(6.39)
where t ie angle brackets indicate an average over the energy range considered: The
magnitrde of the contribution at t = 0 is simply given by the number of Planck ceils
in the energy shell. (Actually, |dfi/d£j depends strongly on energy: This require.-,
a refinement of the present discussion to be outlined below.) For t > 0, the return
amplitu ie in semiclassical approximation, based on the Van-Vleck propagator, reads
a(t) = 271^2 А7ехр (i^ - ip7^ 6(t - TJ,
3lio Chaos, Coherence, ana Dissipation
r(p)
Aj= ET~E.k	IM’	Si= j.^-P^qY	(6.40)
y|det(Mj - l)| h
This relation has a clear physical interpretation: A classical periodic orbit j contributes
a delta-like spike to the return amplitude when time passes through its period Tj
(Fig. 6.13), with an amplitude Aj determined by the degree of instability of the orbit,
and a phase given by the classical action Sj. Since all points on the orbit return to their
starting position simultaneously, the amplitude of their common coherent contribution
increases with the length of the orbit, which in turn is proportional to the primitive
period T-p\ the time required for a single round trip (repetitions of primitive orbits
are counted as distinct orbits). The amplitude depends on the instability of the orbit
through the monodromy matrix = d(p±{Tj), q_i_(Tj))/d(pj_(0), q±(0)) for deviations
from the orbit in the directions (p±, q±) transverse to it.
Of course, the quality of the semiclassical approximation (6.40) for the return am-
plitude cannot be better than that of the Van-Vleck propagator on which it rests.
As has been discussed above, this means that it can be trusted only for times of the
order of h~a, where presumably a < 1. For this reason, the Fourier transformation
(6.36) which leads back from the return amplitude to the spectral density and involves
times t —» oo, is a subtle task. The contribution from ao(t) is still unproblematic. It
translates into the mean spectral density,
(d(E)} = h~f\d^l/dE{.	(6.41)
Equation (6.41) is referred to as the Thomas-Fermi law. The time scale associated
with the mean level spacing, Л.Е = l/(d(E)), by the energy-time uncertainty relation,
= h/AE = /i1--f|dfi/d£'|,	(6.42)
is called the Heisenberg time. In a bound quantum system, it coincides with the break
time by which the spectral discreteness affects the time evolution. It will become clear
in Section 6.3.2, however, that quantum effects occurring in extended systems can
considerably advance the break time against tH as defined in Eq. (6.42).
The remainder
d(E) = d(E) - (d(E))	(6.43)
represents the fluctuating part of the spectral density. Performing the Fourier trans-
formation of Eq. (6.40) formally, one obtains
^) = |£А,(£)ехрА^-^.	(6.44)
> ' '
Equation (6.44), the energy-domain counterpart of Eq. (6.40), is known as the Gutz-
willer trace formula [7,34,35]. In view of the fact that d(E) contributes the spikes
corresponding to the individual levels, i.e., consists of a train of delta functions, it is
surprising that it should be reproduced by Eq. (6.44), which contains only smooth,
6.2 Quasiclassical chaos 309
Fig. 6.14	: Partial sums of the
trace formula for an integrable sys-
tem, the Morse potential V(x,y) =
- 2e~(r~r°)/<i), r2 -
x2+y2. The sums shown (full lines) in-
clude tori with one of their two wind-
ing numbers not exceeding 4 (a). Ju
(b), 30 (c), or 50 (d). In (d), the'ex-
act quantum eigenenergies are marked
by vertical arrows. The dash-dottd
line is the Thomas-Fermi mean sj
tral density, Eq. (6.41). From [38).
oscillatory terms. That this is feasible, in principle, is demonstrated by a transparent,
analogous case, the decomposition of a periodic delta function into an infinite sum
of harmonic oscillations, <5(imod P) = P~l е27Г1(х/р. However, due to Gibbs'
phenomenon, its convergence is not continuous. Likewise, the limited validity in time
of the se niclassical return amplitude (6.40), casts doubt whether the Gutzwiller formula
can renter the spectrum down to individual levels. In order to do so, the energy-timo
uncertainty relation, > h, would require Eq. (6.40) to be correct up to times
of the order fi-1. According to preliminary evidence, this is not the case. Indee-'
has not yet been possible to reproduce the spectrum of a realistic classically cL._..,..
system, by means of the Gutzwiller formula as it stands, to sufficient resolution such
that all .he individual levels appear as distinct peaks at their correct positions. T'
inverse task, identifying the shortest periodic orbits with their periods and stability
from a given spectrum, using Eq. (6.40) directly, is much less problematic [36,37].
It should be kept in mind that, the way Eq. (6.40) has been derived, it applies only
to the isolated, unstable periodic orbits found in chaotic regions. It has been mentioned
already that periodic motion occurs likewise in regular regions, on rational tori. Indeed,
a similar derivation leads to a semiclassical expression, based on rational tori, for the
spectrum of an integrable system. The outcome must coincide with the well-known
EBK formula (6.15): The energies are given by' En =	where H(I) is the
Hamiltonian in action-angle variables, and the In are the actions of the quantizing tori
discussec above. Remarkably, the two equivalent results relate the energies of a specific
subset of all tori (the quantizing ones) in a highly nonlocal way to properties of another,
seemingly unrelated subset (the rational ones). Fig. 6.14 shows the convergence of the
trace formula for integrable systems in case of the circular Morse potential.
Concluding from the preceding paragraphs, it is clear that the contributions of
regular and chaotic regions to the spectrum are of a different nature. According to
310 Chaos, Coherence, and Dissipation
the EBK quantization rules, tori are labeled by an /-dimensional discrete quantum
number n, and so are the corresponding eigenenergies. To each degree of freedom i,
therefore, a “ladder” of eigenvalues of its own pertains, indexed by the component щ
of n. While each of these ladders is regularly spaced—“stiff”—the typical spacings
of different ladders are independent from each other (except for systems with a high
degree of symmetry like the isotropic harmonic oscillator in two or more dimensions).
As a result, regular spectra (the spectra of integrable systems or of the regular parts
of mixed systems) are composed of several mutually uncorrelated spectral ladders [38].
In chaotic systems, the invariant manifolds are labeled by just a single parameter,
the energy itself, so that the spectrum comprises just a single sequence of levels. The
eigenfunctions associated with these energies are all approximately uniformly spread
over successive, (2/ — l)-dimensional energy shells. In order to remain mutually or-
thogonal, they have to differ markedly, e.g., in their node-line pattern in configuration
space, which leads to a corresponding “interaction” of the eigenenergies. Since it is
caused by the requirement that the eigenstates be mutually orthogonal, this interaction
is repulsive: Chaotic levels tend to repel each other.
The repulsion between chaotic levels does not follow a simple law, so that a statis-
tical description is appropriate. This provides a first clue how to approach the second
question quoted above: Information concerning the global character of the correspond-
ing classical motion has to be extracted from collective, statistical properties of the
whole spectrum or large sections of it, for example, from level correlations [8, 39-41].
The simplest statistical quantity characterizing a spectrum is the mean density
(d(E)), Eq. (6.41). It contains information on the overall size of the accessible phase
space, but not on any finer detail of the dynamics. This information can be separated by
applying a transformation to the spectrum, often referred to as unfolding the spectrum,
that enforces a uniform, dimensionless density (d) = 1 (Fig. 6.15). This is achieved
by replacing the energy, as the independent variable, with the mean accumulated level
number [8, 39-41]
(N(E)) = [E dE' {d(E')) =: r,	(6.45)
^min
so that
= (d(E))^ = (d(EY)	1 = I-	(6.46)
More information on the dynamics is provided by the spectral two-point correlation
function. It can be defined as [8,39-41]
Y2(r) = S(r) -	/ [ drod(ro-^-}d(ro + 7-}\ .	(6.47)
L\r \J- oo \ z/ \	z//
Here, Ar is the size of the spectral sample for which the statistics is evaluated, and
the angle brackets indicate an average obtained, e.g., by shifting the spectral window
along the energy axis. With the conventions introduced in Eq. (6.43), the correlation
_____________________________________________6.2 Quasiclassical chaos 311
Fig. 6.15	: Unfolding of a discrete spectrum with
an inhomogeneous mean density. The smooth line
is the mean spectral staircase.
function is called cluster function. In particular, where У2(г) is negative, there are
more level pairs separated by r than in an uncorrelated spectrum, and vice versa.
The spectral form factor is defined as the Fourier transform of the cluster function,
[8,39-41]
- [ dr У2(г) соз(2тггт-),	(6.48)
The dimensionless time т = t/tn appearing in the argument is canonically conjugate
to r. Inserting Eqs. (6.36) and (6.47), the form factor can be written as [42-45]
b^r) = -1 + (Jd(r)|2^/Дг,	(6.49)
where the quantity a(r) is the return amplitude introduced in Eqs. (6.37), (6.38),
transformed to the new time variable t.
The same strategy that has already been applied to obtain a semiclassical expression
for the spectral density, can now be used for an analogous treatment of the form factor:
By substituting the semiclassical approximation (6.40) for the return amplitude a(r),
a relationship to the set of classical periodic orbits is established [42-45]. In order
to obtain the spectral form factor, another crucial operation is to follow: the step
from the return amplitude to its square, the return intensity |d(r)|2. Semiclassically,
it takes the form of a double sum over periodic orbits. This sum can be separated into
a diagonal part, where only the squared amplitudes of the individual periodic-orbit
terms apoear, without phase factors, and an off-diagonal part, containing all the mix<’
elements responsible for quantum interference,
62(r) =: -1 +	- Tj) +
\ 3
ЕАЛ' °xp - 5y) -i(p7 - б(г-	(6.50,
312 Chaos, Coherence, and Dissipation
Amplitudes Aj and actions Sj are as in Eq. (6.40). The times 7j, ту are the periods of
the periodic orbits, in units of Zh- The delta functions in Eq. (6.50) are only asymp-
totically valid, in the limit of a large spectral window Ar 3> 1, otherwise they achieve
a finite width Ar « l/Ar and height Ar.
One can now argue that the actions of the shorter periodic orbits are sufficiently
well separated so that Sj — Sj> 3> h and the phase factors of the off-diagonal terms,
upon averaging, oscillate too fast to allow for a significant contribution. This argument
is only correct if there is no discrete antiunitary symmetry which lets periodic orbits
appear in pairs with identical actions, amplitudes, and periods (the effect of unitary
symmetries will be discussed in Section 6.3). For example, if the system is invariant
under time reversal, each periodic orbit can be followed in two opposite senses, except
for self-retracing orbits which map onto themselves upon reversal of time. While classi-
cally, the contributions of a symmetry-related orbit pair just add, two amplitudes have
to be added and then squared in the quantum-mechanical case, so that the quantum
probability is enhanced by a factor 22/2 = 2 against the classical, an effect known as
weak localization. This can be taken into account by introducing a degeneracy fac-
tor 7 = 2. The factor r]Pi = Tjp'/tH occurring in the amplitudes Aj (cf. Eq. (6.40))
can likewise be regarded as a degeneracy factor since it comes about by constructive
interference of the contributions from the continuum of points along the orbit.
Introducing the diagonal approximation, the form factor can be reduced to its
diagonal part for sufficiently short times. Up to the two degeneracy factors 7 and
TjP\ which are due to quantum coherence, the diagonal sum as a whole gives the total
classical probability to return, integrated over phase space. This results in a simple
semiclassical expression for the form factor [42-45],
Ыт) = -1 + 7'г-РС1(т<н)-	(6.51)
Equation (6.51) is valid only for the contribution of the chaotic regions to the spec-
trum. Similar arguments allow to derive an analogous semiclassical form factor for the
regular part of phase space. The principal difference is that there, all the points on a
rational torus are periodic points with the same actions, amplitudes, and periods. The
corresponding degeneracy factor is time independent, so that
62(r) = -1 + 7Pci(iTh)	(6.52)
for regular motion.
It is instructive to follow the time evolution of the spectral form factor in qualitative
terms. As discussed above for the return amplitude, contributions to 62(7') arise when-
ever wavepackets, launched anywhere in phase space, return to their starting points.
This is the case whenever т « т3. The corresponding peaks have a finite width reflect-
ing size and shape of the spectral window. The frequency of these recurrences increases
with time because in a chaotic system, the number of periodic orbits with their pe-
riod below some threshold time grows exponentially with that time. Simultaneously,
the periodic orbits become more and more unstable, resulting in increasing values of
I det(MJ(E') — l)| (cf. Eq. (6.44)), so that the amplitudes of their contributions decrease
6.2 Quasiclassical chaos 313
accordingly. The exponential proliferation of the individual spikes lets them eventually
coalesce into a smooth function Fci(t) which no longer depends on the specific set of
periodic orbits characterizing the system at hand (Fig. 6.16). The form of this func-
tion ref ects the competition between increasing density and decreasing amplitude of
the ind vidual contributions. Its outcome is determined by a universal feature of the
classicat chaotic flow, ergodicity. It requires that the flow, after sufficient time has
passed, covers the energy shell uniformly, so that the return probability approaches a
constant [17]. With an appropriate normalization, the uniformity principle amounts
to Pcl(t', -> 1. The time ru by which the return probability assumes its universal form
depends both on the typical width of the periodic-orbit peaks, as determined by the
size of the spectral window, and the rate of their exponential proliferation, which in.
turn is i elated to the classical Lyapunov exponents.
The off-diagonal sum in Eq. (6.50) must be responsible for all quantum coherence
effects besides constructive interference due to degeneracies of the periodic orbits, which
has already been taken into account in the diagonal sum. For r > r* = 1/j, it becomes
dominant, so that a reduction to the diagonal sum is no longer justified. In an exaci
quantum-mechanical treatment, it can be shown that the form factor asymptotically
approaches zero, provided the spectrum is discrete and free of degeneracies. Since the
diagonal approximation rests on the assumption that the actions of the periodic orbits
are statistically independent, this feature indicates the presence of correlations among
the actions of the long periodic orbits [46]. Collecting the various time regimes, one
has [42—15]
Ьг(т) = -1 +
0
(system dependent)
yr
1
(6.53)
for chaotic motion and
Г 0
h2(r) = — 1 + < (system dependent)
1
(6.54)
for regular motion. The essential difference between the two expressions is that in
chaotic cise, the form factor takes negative values of the order of 1 for т^т* к 1. a
phenome ion dubbed the correlation hole. It is absent in the regular case.
Upon transforming the form factors (6.53) and (6.54) back to the cluster function,
this diffe-ence has the following consequences: For chaotic systems, the correlation
hole is reflected in a positive peak of lU'j at r — 0. By the sign convention adopti
(see Eq. (6.47)), this peak indicates negative level correlations, a repulsion of levels. In
regular systems, the cluster function does not deviate significantly from zero, indicating
the absence of spectral two-point correlations. In this way, the semiclassical calculation
confirms he results of the qualitative discussion above.
It has already been mentioned that a typical Hamiltonian system is neither purely
chaotic n >r completely regular, but combines components of both sorts in its phase
314 Chaos, Coherence, and Dissipation
Fig. 6.16	: Typical shape and time scales
of the spectral form factor for a classically
chaotic system without time-reversal invari-
ance. The function plotted is К{т) = Ь?(г) +
1. From [44].
space. Equations (6.53) and (6.54) emerge from parallel semiclassical derivations. This
suggests that in mixed systems, the spectral form factor should comprise an uncorre-
lated component with the weight of the regular parts in phase space, and as many
chaotic components as there are separate chaotic regions in phase space, weighted ac-
cording to their respective share in the full phase-space volume, and superposed as
statistically independent contributions [47]. However, this approach does not take into
account that by way of quantum interference, in particular by tunneling, classically
separate parts of phase space can “communicate” with each other, which leaves its
trace in the spectrum (see Section 6.3 below).
The semiclassical approach is not the only way towards a physical understanding of
the spectra of complex systems. Rather, it is a newcomer in a subject which actually
has its origin in nuclear physics. There, spectra of enormous complexity were observed
experimentally, so that there was no hope of ever reconstructing specific Hamiltonians
to understand them in detail. Instead, the strategy underlying statistical mechanics was
adopted: Only those few facts are required from a statistical ensemble that are known
for certain, while ignorance in other respects translates into unbiased probabilities.
In an analogous way, one devises ensembles qf Hamiltonians where oqiy the basic
global symmetry is specified to restrict an otherwise random distribution of matrix
elements. Just three symmetry classes—“Dyson’s threefold way”—are sufficient to
cover all types of Hamiltonians occurring in nuclear physics [48]: invariance of the
ensemble under orthogonal, unitary, or symplectic transformations, for systems with
time-reversal symmetry, systems in a magnetic field, and systems coupled to a spin,
respectively [8,41,49]. Otherwise, matrix elements are assumed to obey a Gaussian
distribution, so that the corresponding three ensembles are called the GOE (Gaussian
Orthogonal Ensemble), GUE, and GSE.
Many years later, it turned out that the spectra of classically chaotic systems ex-
hibit the same statistical features as those of complex nuclei, and can indeed be classi-
fied according to the same three categories just mentioned [8,41,49]. Random-matrix
theory, as this approach is called, is worked out to such depth and detail that vari-
ous statistical features of spectra can be predicted and compared with experimental
and numerical data. The simplest and most broadly applied concept in this field is
the nearest-neighbor level-spacing distribution P(s). It is different from the two-point
spectral correlation (6.47) in that it excludes all level pairs straddling at least one other
6.2 Quasiclassical chaos 315
level. While P(s) is readily evaluated, the exclusion of interspersed levels involves all
higher spectral correlations and cannot be reduced to two-point correlations. On basis
of P(s), it is easy to identify the spectral fingerprints of integrable and chaotic motion,
respect vely, and of the symmetry classes which occurred already in the semiclassical
analysis above.
The level-spacing distribution for uncorrelated spectra is readily derived [8]: The
probability, given a level at the energy e, to find the next one in the interval [e + ». < '
s + ds] (unfolded energy variables in units of the mean spacing are assumed). : bv
definition, P(s)ds. Alternatively, it is given by the unconditional probability to fi- '
level within [e + s, e + s + ds], regardless of the number of levels in [e, e + s]—i.x cue
absence of correlations, this probability is just ds—times the probability that there :
no level in [e, б+s]. This second factor, in turn, is the complement of P(s), accumulated
over the interval [0,s], i.e., 1 — Jq ds'P(s'). From the condition
P(s)ds = ds(l-^’ds'P(s')) ,	(6.55)
the differential equation
-^P(s) = -P(s)	(6.56)
follows immediately. It is solved by the Poissonian distribution
P(s) = e
(6.57)
1.
Fig. 6.17	: Nearest-neighbor level-spacing
distributions for the 2x2 Gaussian ensem-
bles (cf. Eq. 6.58). From [41].
For small s, s < 1, the probability to find three or more levels within an interval
of length s is negligible, so that near the origin, P(s) ss 1 - l^s). Therefore, the level
repulsion emerging qualitatively from the semiclassical discussion of chaotic spectra is
reflected in a minimum of P(s) at. s = 0, in contrast to the Poissonian distribution
which is leaked at the origin. This is what one indeed finds for the three canonical
ensemble: GOE, GUE, and GSE (Fig. G.17). Furthermore, the degree of repulsion
316 Chaos, Coherence, and Dissipation
depends on the symmetry, as is suggested by the dependence of the semiclassical form
factor on the degeneracy factor 7 (see Eq. (6.52)). The asymptotic form of Pts'), for
small s, is
P(s) = s0,
/3 =	2
GOE,
GUE,
GSE,
(6.58)
1
„ 4
while for large s, the distributions approach Gaussians, P(s) e~c^s2, c(J3) > 0,
which ensures normalizability. By comparison with the corresponding values of 7 in
the orthogonal and in the unitary case, one may formally equate /3 = 2/7.
The behavior of P(s) and ^(s) for small s can be understood by a simple con-
sideration of dimensions. Imagine an ensemble of Hamiltonians to be generated by
the continuous change of an «-dimensional parameter A. Small values of s are then
associated with close encounters of energy hyperplanes over the parameter space. If
level correlations are absent, as is the case for integrable systems, the hyperplanes
just cross each other without interaction, forming exact crossings. Level repulsion, in
contrast, corresponds to a tendency of the hyperplanes to avoid crossings (Fig. 6.18).
Near a crossing of two levels, all other levels being far away, the full Hamiltonian may
approximately be replaced by a two-level system,
Diagonalization of H yields its two eigenvalues
p± = 2^u +	± 4^1! ~ ^г)2 + Я12Я21 	(6.60)
If the Hamiltonian is merely required to be Hermitean, i.e., Hi? = H),. there arc three
independent non-negative terms in the discriminant in Eq. (6.60), (Hn — Н2г)2/4,
(Re//12)2, and (1тЯ12)2, which have to vanish simultaneously to achieve an exact
crossing, E+ = E_ = (Hu +7L22)/2. This case applies to the unitary ensemble. In the
orthogonal ensemble, all Hamiltonians are real symmetric, so that Im(7L12) = 0, and
there are only two independent terms left. In the symplectic ensemble, in contrast,
there is an additional spin degree of freedom, and each entry in the Hamiltonian (6.59)
represents in fact itself a 2 x 2 matrix. An analogous argument then reveals that the
discriminant contains five independent parameters. In geometrical terms, these results
indicate that each avoided crossing is a section through a double cone erected over a
(3 + 1 (-dimensional space (Fig. 6.18) [8,14], with /3 as in Eq. (6.58). This means that
n = /3 + 1 independent parameters must be varied to reach the diabolic point where
the two energy hyperplanes actually touch. By evaluating the mutual separation of
the two surfaces in the vicinity of that point, one obtains the linear, quadratic, and
quartic laws of level repulsion, respectively, stated in Eq. (6.58).
There are various other statistical measures besides the level-spacing distribution,
all of which emphasize specific aspects of the spectrum [8,39-41]. For example, P(.s)
6.3 Chaos and quantum coherence 317
Fig. 6.18	: A diabolic point connecting two energy hyper-
planes, and a nearby avoided crossing (bold curves). The
dashed line in the (a, b) parameter plane defines the sec-
tion through the energy hyperplanes in which the avoided
crossing appears.
is particularly sensitive to the short scales in the spectrum. The spectral rigidity
Лз(£), in contrast, measures the deviation of the spectrum from an equidistant ladder,
and serves to study the spectrum on longer scales. Other measures concern “dynamic"
aspects of the spectrum, such as the distributions of slopes and curvatures, respectively,
of the energy hyperplanes.
6.3 Chaos and quantum coherence
The pr< ceding section has shown that a great deal of the dynamics of a classical' ,
chaotic quantum system can be understood using classical concepts alone. But the
longer the time scales considered are, the more profound modifications of this pictue,
have to be made. Genuine quantum effects can lead to a quantum dynamics radically
different from its classical counterpart. This is as true for chaotic systems as for inte-
grable ones, but the specific manifestations of quantum mechanics may still be distinct
in the t .vo cases. The present section is devoted to the interplay of chaos with two
prominent quantum coherence effects: tunneling and localization. They are in many
respects converse to one another. Tunneling comes about by a discrete symmetry of
the phase space and can be interpreted as a constructive interference of wavefunctions.
It opens the way to classically inaccessible parts of phase space. It facilitates transport.
Localization, in contrast, is caused by destructive interference arising if symmetries are
absent. It restricts the spreading of a wavefunction in a stronger way than classically
expected and thus impedes transport.
6.3.1 Chaotic tunneling
The paradigm of coherent tunneling is the quantum dynamics in a symmetric double-
well potential. There, a separatrix divides the classical phase space into two parts. If
318 Chaos, Coherence, and Dissipation
the energy is below the top of the barrier, the motion consists in oscillation in one of the
wells, analogous to vibration of a pendulum. Above the barrier, the motion encircles
both wells, analogous to libration of a pendulum. A popular model for a symmetric
double well is the Hamiltonian (cf. Section 5.6)
л	1	1
=	+md’-	<661)
It is invariant under reflection with respect to q = 0, i.e., under the operation P:
q —q, corresponding to the quantum-mechanical parity operator, with eigenvalues
±1. Therefore, all eigenstates of Hq\v can be classified as even or odd under P.
The quartic double well, Eq. (6.61), is completely integrable, so that the simple
torus quantization mentioned in Section 6.2 can be applied as a crude approximation.
It predicts that each wavefunction with energy E < 0 comes as a pair | V'i ), | V’r), one
concentrated on a quantizing torus in the left well, the other on its symmetry-related
counterpart in the right well. The eigenfunctions of the parity operator are constructed
as superpositions
ItM =	± I1M)-	(6.62)
According to torus quantization, these states are degenerate in energy. Exact quan-
tization, however, predicts a small but finite splitting i\E = E_ — E+ of these doublets
(Fig. 6.19). The semiclassical treatment can in fact be refined so as to include the en-
ergy splitting. Trajectories that pass underneath the barrier are not allowed classically
because there, the momentum p(q) = ±[2m(E — V(q))]1/2 would be imaginary. In
the semiclassical framework, however, an imaginary momentum makes perfect sense:
Inserting it into the basic expression (6.19) for the wave function in WKB approxima-
tion renders the phase factor exp(i dq'p(q')/^) a decaying (or growing) exponential,
which indeed approximately reproduces the evolution of the wave function under the
barrier. In one dimension, the WKB approximation for '!/’+('?) and ’/’-(</) gives the
q
Fig. 6.19	: Tunneling in the quartic double-
well potential, Eq. (6.61) with D = 2. The
groundstate V’i(q) (solid curve) is symmet-
ric, the first excited state ^>2(9) (dashed) an-
tisymmetric.
6.3 Chaos and quantum coherence 31S
-15-10-5	0	5	10	15
x
-15 -10 -5	0	5	10	15
Fig. 6.20	: Stroboscopic phase-space portraits, at wt = 2im, of the harmonically driven quai tic
double veil, Eq. (6.64) with D = 8, for increasing values of the driving amplitude, S = 0.01 (pane'
(a)), S = 0.05 (b), S = 0.2 (c), and S = 0.5 (d). The dashed figure-eight curve in each panel marhi-
the unperturbed separatrix. From [53].
semickssical energy splitting
EE = —<(0)^(0) = ~ exp , St= [ ' dq \j2ni(y(q) - F),	(6.63)
where ET(q) = (q | фт}, ±a(E) are the two inner turning points of the classical motion,
V(—a) — V(a) = E, and T is its period in either well at energy E = (E_ + E+)/2.
In crder to interpolate the WKB wavefunction across the manifolds that form the
analogues of turning points in higher dimensions, i.e., over confining curves or (hy-
per)surfaces, one has to construct trajectories connecting one allowed region via an
excursion into complex phase space to another one. This task is complicated by the
fact ths t now chaos can come into play. It requires to generalize the concept of tunnel-
ing substantially. Chaotic regions can separate regular parts of phase space from еги к
other like a separatrix, without being tied to a potential barrier.
As t. model to study the quantum-mechanical aspects of this phenomenon, take k-e
double «veil, Eq. (6.61), and “rock it” periodically by adding a spatially constant foi- e
that va des harmonically in time (cf. Section 5.6),
H(p, q; i) = Но\'Лр. q) + Sqcos(wt).	(6.61)
The time dependence increases the dimension of phase space to three (one says sloppily,
there ars degrees of freedom). On first sight, the presence of the driving force should
not allow to talk of tunneling any more, because the driving can supply any amount of
energy necessary to surmount the potential barrier without tunneling. Nevertheless, a
320 Chaos, Coherence, and Dissipation
Fig. 6.21: Overlaps with the chaotic layer in the classical phase space (panel (a)) and tunnel splittings
(b), for the lowest eight doublet states of the harmonically driven quartic double well, Eq. (6.64) with
D = 8, as functions of the driving amplitude. From [53].
dynamical barrier now exists in the form of a chaotic layer between the wells, which
replaces the smooth separatrix for any finite value of S (Fig. 6.20). Furthermore, there
is also a symmetry generalizing the parity invariance of the unperturbed double well.
Thanks to the fact that the driving force chosen in Eq. (6.64) is not only periodic,
/(t + 2tt/w) = /(t), but exhibits the additional symmetry f(t + тг/ш) = — f(t), the
full system is symmetric under the operation Рш: q -л —q, t -> t + тг/w. Thus the
two essential ingredients of coherent tunneling are present: A discrete symmetry, and
mutually inaccessible compartments of the classical phase space which are mapped
onto one another by this symmetry. The present case where the “barrier” consists of
a chaotic region is called dynamical tunneling [50-52].
Of course, a driving force can entail dramatic changes of the tunneling process
which have nothing to do with chaos. They are addressed in Section 5.6. Here, we are
dealing with modifications of the familiar picture of tunneling due to the changes the
classical dynamics undergoes in presence of the driving. There are now two completely
different modes of transport between the two symmetry-related parts of phase space: a
classical one, diffusion along the chaotic layer, and a quantum-mechanical one, coherent
tunneling between the wells. It will be elucidated below that chaotic diffusion and
coherent tunneling can strongly affect one another and lead to a novel, “hybrid” type
of transport.
Eigenstates located on tori deeply inside the regular regions pertaining to the wells,
are not appreciably affected by the presence of the chaotic layer, and continue to
exhibit tunneling in the usual manner. For a state residing amidst the chaotic layer,
in contrast, there is still a counterpart with opposite parity, but this partner is spread
over the same chaotic layer so that their eigenenergies are no longer quasidegenerate.
(Incidentally, the absence of quasidegeneracies of chaotic states in the spectrum of
a mixed system with a discrete symmetry, provides a simple means to separate the
regular eigenstates from the chaotic ones without actually looking at their phase-space
structure: Just filter out the quasidegenerate doublets [51]!)
As has been discussed in Section 6.2, there typically exists a boundary zone of highly
complex structure in classical phase space between a chaotic layer and an adjacent
regular region. Quantum mechanically, the corresponding transition from chaotic to
regular states is continuous, i.e., there exist states of an intermediate character. In this
6.3 Chaos and quantum coherence 321
t
Fig. 6.22: Chaos-assisted tunneling in a peri”''5'
driven pendulum. Left: Crossing between a quasi-
energy doublet pertaining to a pair of regular Flo-
quet states and a single quasienergy pertaining to
a chaotic state. The vertical line marks the center
of the crossing. States A and В have equal parity,
state C has opposite parity relative to A and B. Be
low: Husimi distributions of the states A, C, and
B, respectively, superposed on stroboscopic plots of
the corresponding classical phase space, at 7 = 0.88.
From [57].
P
boundary zone, an interplay between chaotic diffusion and coherent tunneling can arise:
Imagine that a parameter controlling the degree of nonlinearity, e.g., the amplitude A
of the driving in Eq. (6.64), is varied such that the chaotic layer spreads on the expense
of the regular region. As a consequence, the classical boundary zone moves towards
the regular region, and the outermost tori which formerly supported regular states are
resolved in the chaotic layer. This means that the small splittings oc exp( — St)/h
the associated states must gradually widen until they reach the order of the mean lev1!
separation of states in the chaotic layer where level repulsion prevails [53] (Fig. 6.21;.
In the course of their decay, irrational tori take intermediate forms, cantori and vagyy
tori. They support states of almost regular type [19] which still form doublets will
quasidegenerate energies. Their splittings depend sensitively on the overlaps of ike
states with the chaotic layer (Fig. 6.21) [53].
A more complicated type of tunneling involves chaotic states as ‘‘step stones” in
passing through the chaotic layer (Fig. 6.22): Upon variation of the nonlinearity param
eter, it nay happen that a quasidegenerate doublet belonging to a pair of symmetry
related states near the boundary zone approaches, on the energy axis, a single, level
associated with a chaotic state of either parity. The chaotic level will repel that partner
in the djublet that belongs to the same parity class, because the corresponding states
reside о 1 neighboring regions of phase space. Thereby, the doublet is widened. As a
result ol this interaction, the chaotic and the regular state of the same parity exchange
their roles if one follows them from one side of the avoided three-level crossing to the
other. Near the point of closest approach of the two interacting levels, all the three
322 Chaos, Coherence, and Dissipation
states are strongly involved in the dynamics, in contrast to the two-level dynamics
characteristic of ordinary tunneling. Probability is no longer transferred directly from
the vicinity of one torus to that of the symmetry-related partner, but appears tem-
porarily in the chaotic layer. The time scales of this “chaotic three-state tunneling”
are determined by the three level separations near the avoided crossing. They are
vastly larger than the unperturbed tunnel splitting, so that this three-state tunneling
is correspondingly faster than two-state tunneling. In this way, the participation of a
chaotic state can significantly accelerate coherent tunneling (chaos-assisted tunneling
[52-58]).
Tunneling also manifests itself in the spectral statistics. In case it is due to an exact
twofold unitary symmetry like parity, the spectrum can be sorted into two subspec-
tra, corresponding to the two symmetry classes, to remove the signature of tunneling
from the spectral statistics. This will be discussed below. If the symmetry is only
slightly broken, tunneling persists, and with it the presence of quasidegenerate dou-
blets. The contribution from the tunnel splittings strongly increases the probability for
small level separations, corresponding to a negative contribution to the cluster func-
tion (Eq. (6.47)) and a positive peak in P(s) near the origin. This translates into an
overshoot of the form factor b2(r) (Eq. (6.48)), roughly by a factor of 2, on a time scale
1 ^т^1/(<1)ДЕ, where is a typical tunnel splitting. It approaches its asymptote
0 from above and only for times т » l/(d)A£l. The peak formed by the overshoot
and the subsequent decay reflects the positive correlations due to the near degeneracy
of the doublets.
The full spectrum can be related to the set of classical periodic orbits, as explained
in Subsection 6.2.2. An expression analogous to the Gutzwiller trace formula, Eq. (6.44)
should exist also for the partial spectra pertaining to states of only a single parity class:
Constructing symmetry-projected spectral densities, Green functions, etc., involves a
bit of group theory. Formally, the two symmetry classes of the double well and simi-
lar systems with parity conservation correspond to irreducible representations [59] of
the group G of transformations that leave the Hamiltonian invariant. In the present
case, this group consists of two elements, the identity I and the parity P: q -л —q (for
simplicity, we restrict the following discussion to the time-independent case and do not
consider the generalized parity Рш introduced above). By virtue of this symmetry, it
is possible to restrict the full space to a subspace that is invariant under G, e.g. here,
the positive q axis with the origin (times the full momentum space). To each symme-
try class, there belongs a quantum-mechanical operator projecting onto this invariant
subspace [60], Pm — (dm/|G|) £9go	where m counts the irreducible repre-
sentations, g denotes an element of G of which there are |G| all in all, dm and Xm{g) are,
respectively, the dimension and the characters of the mth irreducible representation,
and U(g) is the unitary operator that transforms states as prescribed by g.
Projecting the classical paths in the same way into the invariant subspace leads to
a generalization of the concept of periodic orbits: Orbits that appear periodic upon
this projection can arise also if their endpoints in phase space do not literally coincide,
but are merely related by a group element j / I. In the case G = {I, P}, this means
that also trajectories with endpoints (®,Pf) = (—Qi, Pi) have to be included in the
6.3 Chaos and quantum coherence 32-
symmetry-projected periodic-orbit sums.
Specifically, the semiclassical expressions for the spectral densities pertaining to th
even (+) and odd (“) parity classes, analogous to Eq. (6144)-, read [61]
й±(Е) = ^2 X^MAE) exp	\ .	(6.G-5
Z/i j	у fl	z J
Here, gj is the symmetry relating the endpoints of the orbit j: gj = I if the path i
closec without projection—these are the “ordinary” periodic orbits which contribute
also to the full spectral density. If, on the other hand, gj = P, the path connects th-
two subspaces between which tunneling takes place. These generalized periodic orbit
are typically forbidden in the classical description and contain excursions into comp!.
phase space, as discussed above.
The group characters appearing in Eq. (6.65) are y±(l) = 1 and y±(P) = ±1	.
irreducible representations are one-dimensional). In this way it is ensured that ii.
contributions of the generalized (^ = P) periodic orbits cancel in the trace formula f<>
the full spectral density, d(E) = d+(E) + d~(E). In the same way, the contribution
by no1. truly periodic orbits are removed from the trace formulae for d(E) in all discret-
symmetry groups.
Ccnversely, one may define a tunneling spectral density as </tun(E') = d+(E) — d~(E
[61]. Here, the contributions of the periodic orbits proper cancel. Only those orbit:-
contribute that mediate transitions to the opposite side of the system. The spectra
form factor associated to dtan[E) remains close to —1 in the regime where the usua-
form factor overshoots due to quasidegeneracy, and attains larger values only on tinr.
scales т ^A/{d)AE where tunneling takes place.
6.3.2 Quantum chaos in extended systems
The reflection symmetry discussed in the previous subsection is often present in at<
and small molecules. Extended systems, in addition, typically exhibit discrete trans-
lational symmetries. The attribute “extended” signifies systems composed of a large
numbt r of identical or similar “modules”, “elements”, “subunits”, or “cells”. They ;u i-
realized, e.g., in potentials generated by standing electromagnetic waves, in long <•! b.
molecules like DNA or RNA, and most importantly, in crystalline or near-crystalline
condensed matter.
In the classical dynamics of extended systems, a crucial role is played by a ci- - ,
that is only of minor significance in other systems: diffusion. If the motion within a
single cell includes a chaotic component, and the chaotic regions of all or most of th--
cells are connected to those of the adjacent cells, then the chaotic part of phase space
as a w hole forms a network which allows a chaotic trajectory to link any two cells.
Upon coarsegraining over the spatial and temporal scales characteristic of the motion
within a single cell, the spreading over the chaotic network can be described by the
diffusion law,
(Н) = ж
(6.66)
324 Chaos, Coherence, and Dissipation
Fig. 6.23	: Quasienergy bands pertaining to
chaotic Bloch states of the quantum kicked
rotor, Eq. (6.82), for к = 300 and r = 4тг/75,
corresponding to a periodic potential with a
unit cell that accomodates 75 Bloch states.
where D is the diffusion constant. This diffusion is based on deterministic equations of
motion, as long as the system is Hamiltonian. It is therefore referred to as deterministic,
or chaotic, diffusion.
The quantum dynamics in an extended system depends crucially on the degree of
spatial symmetry. If all cells are exactly identical, that is if the potential is periodic in
space,
V(q + a) = V(q), for all q,	(6.67)
then Bloch’s theorem [62] applies, all eigenstates are extended, and the spectrum con-
sists of continuous bands. If, on the other hand, the extended system is disordered and
translational invariance exists only in an approximate, statistical sense, then Anderson
localization can occur [63]. It is indicated by a point (discrete) spectrum and by eigen-
states decaying exponentially, in the mean, with the distance from a localization center
where the amplitude takes its maximum value. As a result, a wavepacket cannot leave
the neighborhood of the site where it has been prepared. In case there exists a con-
tinuous parameter connecting the periodic to the disordered case, e.g., the strength of
a perturbation introducing the disorder, one typically finds a sharp crossover between
the two regimes, comparable to a phase transition, and at the critical point the states
are neither extended nor localized, but have a fractal structure.
A good deal of the semiclassical concepts developed in Section 6.2 apply, with some
modification, also to quantum chaos in extended systems. For example, in the case of a
periodic potential, it is now the band structure which contains all spectral information.
Energy levels and corresponding eigenstates in a bound system can roughly be classified
as “chaotic” or “regular”. An analogous distinction between chaotic and regular bands
is possible in periodic systems: Bloch states pertaining to chaotic bands support the
/
6.3 Chaos and quantum coherence
quantum-mechanical counterpart of classical deterministic diffusion (Fig. 6.23), state
with .heir energy in a regular band are associated with classical motion that is ei h(
bound to a single cell or traverses the lattice ballistically, similar to the dist:net io
between vibration and libration. In fact, the band structure, with its additional de-
pendence on the Bloch phase, provides much richer information on the correspcw
classical and quantum dynamics than the one-dimensional spectrum of a nonp
system. It will be shown below how this information can be extracted.
At has been discussed in Section 6.2.1, the correspondence principle requires th;',
the qi antum dynamics closely mimic the classical for a finite time span 0 < t id t*, when
t* depends on the degree of disorder and diverges in the classical limit. Accordingly, ii
extended quantum systems in the semiclassical domain, chaotic diffusion occurs durin;,
an initial time regime, irrespective of the large-scale structure of the eigenstates—
extended or localized or fractal. From this initial diffusive phase, a dynamical crossovei
leads into a regime dominated by quantum coherence. There, a wavepacket spread;
ballistically in the periodic case, ceases to spread if localization is effective, or continue;
to dif’use, generally with a different diffusion constant, in the intermediate, critica
case [55,66]. In terms of spectral features, the initial diffusive phase is reflected on
the ccarser energy scales, while the fine spectral details determine the quantum long-
time evolution and long-distance transport. The relationship between dynamical and
spectral properties of extended systems is presently under intense investigation. Onlj
prelininary results can be presented here.
Consider first the simpler case of extended systems with a perfect translationa'
symmetry, as in Eq. (6.67), in one dimension. Assume that the system comprises L
identical unit cells of length a, with periodic boundary conditions at the ends q = (
and q = La: The topology is that of a circle (times additional dimensions). Th
symmetry group then consists of the L translations T/: q —> (q + la) mod La, I = 0
1,..., L — 1. This group has L irreducible representations, corresponding to the Bioci
phase; 0m = 2irm/L, m = 0, 1,..., L-1, of the first Brillouin zone, which parameter
the bands. The corresponding quasimomenta are km = hOm/a.
The construction of symmetry-projected spectral quantities, introduced in the pre
vious subsection for the case of reflection symmetry, is readily extended to an £-f- '' '
discrete translational symmetry. Introducing the group characters ym(/) = exp(i/t„
exp(27 i/m/L), the semiclassical periodic-orbit sum for the fluctuating part of the sp< <•-
tral density pertaining to the Bloch phase 0m reads [60, 67]
= -^52.4j(F)exp	.	(6.6^
Ljll	\ il	Lj	Z /
It comprises orbits that are periodic only modulo the lattice constant a, that is. iIn
endpomt of the orbit j is mapped onto its starting point by a translation Т/ . Tin'
same extra phase 2Trljm/L that enters with the group characters would also arise if ihe
system consisted only of a single cell of length a, again closed to a torus with period!;
boundary conditions, but threaded by a magnetic flux Ф — mhc)Le. The term winding
number for lj thus becomes plausible: It counts the number of times the orbit wind.'
around this flux before closing.
326 Chaos, Coherence, and Dissipation
A symmetry-projected return amplitude, analogous to the one defined in Eq. (6.37),
is obtained by Fourier transforming Eq. (6.68) from the energy to the time domain.
The corresponding periodic-orbit sum is, for m = 0, 1, ..., L — 1,
1	—	/ Q.	/ TD	77 \
•	у П	Lj	Z /
(6.69)
A quantity that allows for a more direct interpretation arises upon Fourier transforming
also with respect to the Bloch phase [68],
1 L~Y
орЛ) = у zL «m(f)e27rl'm/L
L rn=0

'^J2/	-G )	) mod L,
(6.70)
where I — 0, 1, ..., L — 1. The function ai(t) has the meaning of a quantum-mechanical
probability amplitude for being in the initial state, but shifted by I lattice constants, at
time t. As has been discussed in Section 6.2.2, its modulus squared can be associated
to a classical probability (cf. Eq. (6.51)),
= -1 + {|а((т)I2) /Дг « -1 + wPajlrtH),	(6.71)
the prefactor т occurring if the classical dynamics is fully chaotic. The classical return
probability Pci,i(t) is now also generalized to the case that initial and final points are
separated by I unit cells. It will be discussed below.
Since the Bloch phases canonically conjugate to the winding numbers are equivalent
to the presence of a magnetic flux, the degeneracy factor 7; for I / 0 cannot equal 2
throughout the Brillouin zone, even in a system with time-reversal invariance. For
systems with symmetric bands, En(—0m) = Еп{вт} (Fig. 6.23), it reads
(2 Z = 0,
( 1 else.
(6.72)
If L is even, then also 7/72 = 2 (Fig. 6.24), reflecting that orbit pairs with opposite
winding numbers lj = ±L/2 also add coherently to the corresponding amplitudes
“L/zW = a_£/2(t).
By the fact that the motion is globally diffusive, a typical generalized periodic orbit
appearing in Eq. (6.70) must approximately obey the diffusion law, Eq. (6.66). In
order to evaluate Eq. (6.71), we therefore assume that the spreading of the generalized
periodic orbits, as reflected in the time-dependent winding-number distribution Pci,i(t),
follows the same laws as the spreading of a generic, non-periodic trajectory [68]. Ac-
cordingly, we set Pcij (t) = ft^/2	+ la, t; q, 0) « ap(J,a, t; 0, 0), where p(q', t; q, 0) is
the classical propagator or Frobenius-Perron operator from position q to position q' in
6.3 Chaos tod quantum coherence 327
Fig. 6.24	: Winding-number specific spec-
tral form factors for a periodic one-dimen-
sional chain of L = 512 unit cells, in the case
of broken time-reversal invariance, for wind-
ing numbers I = 0,16,32,..., 256. The diffu-
sion constant for the corresponding classical
diffusion is such that To >t'.
time t. An explicit expression for p(q', t; q, 0), in the case of a diffusive dynamics on a
continuous space, is obtained by solving the diffusion equation
d , .	, D d2 , ,
^p{q.t-q^ = -~p{q,t-q,Q}.
For a localized initial state p(q', 0; q, 0) = 6(q' — q), the solution reads
1	/ — (q1 — q)2\
₽(’',;,’0) = 72Ж“рОпГГ
The w inding-number distribution then evolves as (cf. Fig. 6.25 for the case I = 0)
a ( l2a2\
р’м = ^^Ьб1)-	<6-75!
These expressions are valid only if the system is literally of infinite extension. If it
contaii s only a finite, if large, number L of cells, a diffusing trajectory will reach the
boundaries on average at the Thouless time,
to = L2a2/TrD.	(6.iu?
Fig. 6.25	: Time evolution, over the tirsf
103 steps, of the return probability f<"
quantum-mechanical (qm) and clas1-'
versions of the kicked rotor, Eqs. (6. . ....
(6.82), respectively. Parameter values are
К = 20, and т/4тг = 0.05/(>/5 - ]'
the quantum-mechanical case, so that
К= 39.3. The dashed graph repres....
the quantum-mechanical data, reduced by a
factor of 2. The straight line corresponds to
the asymptotic decay ~ tjxfn of the return
probability in the classical case. From [43].
328 Chaos, Coherence, and Dissipation
Fig. 6.26	: Spectral form factor for disor-
dered one-dimensional systems showing clas-
sical diffusion and localized quantum ei-
genstates, in the case of broken time-reversal
invariance. The Thouless time, cf. Eq. (6.78),
increases from то = 0 for the rightmost (full)
curve to a value tq 1 for the leftmost
(dashed) curve.
From to on, diffusion will drive the probability towards a homogeneous distribution
over the system, so that the winding-number distribution approaches a finite constant
rather than decaying to zero. Its short and long-time asymptotes are then given by
7rZ2tD/2L2t)
t to,
t 5> tD.
(6.77)
Inserting Eqs. (6.72) and (6.77) into Eq. (6.71), one finds semiclassical expressions
for the symmetry-projected form factors. The result for I — 0, for example, is (Fig. 6.26)
[43,45,69],
62(7) = 62,o(t) = -1 + j '7)/тто/2
I 7T
T 72td/2,
72td/2	t‘,
(6.78)
where 7 = 70 is the degeneracy factor occurring in Eq. (6.51), and m = to/tib In
contrast to the symmetry-projected form factors with I 0, the case I = 0 applies
also to disordered systems where the concept of finite winding numbers is not defined.
However, in the periodic case (Fig. 6.24), the Heisenberg time used to scale r = t/rii
is defined with respect to the spectral density for a single unit cell, not for the entire
system. The crucial feature in Eq. (6.78) is the initial increase of 62(t) as y/т. It
reflects the classical diffusion and modifies the shape and the size of the correlation
hole exhibited by 62(r) for classically chaotic systems that are not extended.
The diffusive regime where Eq. (6.78) is valid lasts only until either diffusion covers
the entire sample and saturates, which occurs at rD, or quantum coherence becomes
effective at r* and a crossover to localization or ballistic spreading takes place. If the
system is periodic, the break time r* is related by energy-time uncertainty to the mean
level spacing for a unit cell, i.e., the typical band separation. If it is disordered, the
break time is determined by the mean level spacing within a localization neighborhood
(see below). Now r* is independent of r^. The system size and the diffusion constant, or
the typical time a trajectory spends within one unit cell, therefore determine together
whether classical ergodicity or quantum coherence will dominate the long-time limit.
For spatially periodic systems, the case of tunneling in a system with reflection
symmetry gives a hint which behavior to expect for times beyond r*: The gradual
6.3 Chaos and quantum coherence 329
resolution of the fine spectral structure within the bands, eventually revealing tin.
discreteness of the individual levels, will lead to a decay of 62,о(т) by a factor equal
to the quasidegeneracy of the bands, L, on a time scale of the order of Lt*. A full
quantum-mechanical calculation [70] shows that this decay is algebraic, ~ i'r-
The co -responding peak (Fig. 6.24) formed by the initial y/т rise and subsequent <!-- -iv
of the ’orm factor is even more pronounced here than in the spectral correlations C
systems with a twofold unitary symmetry, discussed in the previous subsection. Similar
to the spectral signature of quasidegenerate doublets, this peak reflects the cluster!'
the levels into bands [65,70]. Accordingly, the correlation hole indicating level repulsion
gives way to strong positive correlations, that is, level attraction, on the scale of the
band width.
Consider now a disordered system with localized eigenstates. If the disorder is
sufficiently weak, diffusion can cover the entire system before localization takes over,
that is, tq < t*. The system then behaves more like a bound than like an extended
system, the form factor does not differ appreciably from the one given in Eq. (6.53),
and the spectral properties are essentially those predicted by the canonical random-
matrix ensembles (see Section 6.2): There is linear, quadratic, or quartic level repulsion
according to the antiunitary symmetries present. For strong disorder, in contrast,
г* < ту, so that localization soon inhibits diffusion (Fig. 6.25). Only in this case, the
system can accomodate several localization neighborhoods, the sets of all states whose
overlap with a reference state at the localization center exceeds a certain threshold,
and its extended nature affects the spectrum. As is read off Eq. (6.78), in the limit
t* —> 0 the form factor bcomes (Fig. 6.26) [43,45]
62(7-) =
-1
0
r = 0,
else.
(6.79)
It indicates vanishing two-point correlations, corresponding to a Poissonian spectrum
as it is found for classically integrable systems (see Section 6.2). Indeed, the existence
of many mutually non-overlapping localization neighborhoods has a similar effect on
the spectrum as the separability of the eigenstates into several independent sequence: .
each contributing its own ladder of eigenvalues to the spectrum.
The preceding remarks on the spectral fingerprints of diffusion are valid for one-
dimensional systems only. In higher dimensions, the classical return probability corre-
sponding to free diffusion takes the general form PC| ~ t~^2, if there are f degrees of
freedon . It leads to semiclassical expressions for the short-time evolution of the f
factor [69] which, deviating from Eq. (6.78), show an initial increase as t1-7/2
The dynamical and associated spectral features of extended systems beyond the
crossover to quantum coherent time evolution are not yet understood from a semicl-'
sical point of view. For example, in the spatially periodic case, tunneling from cell 1 ,>
cell is clearly reflected in the band structure [66,67, 70]. Accordingly, orbits connecting
adjacent cells via excursions into complex phase space have to be taken into accoui
in periodic-orbit sums such as Eq. (6.68) .
Even less is known, in terms of periodic orbits, about the interference that leads
to localization in classically chaotic systems. Here, an additional complication comes
330 Chaos, Coherence, and Dissipation
Fig. 6.27: Suppression of chaotic diffusion in the quantum kicked rotor, (a) Typical time evolution
of the mean kinetic energy, over the first 1000 time steps, in the classically chaotic regime, К = 10, for
the quantum-mechanical case, with т/4тг = 0.15/(\/5 — 1) (full line), and the corresponding classical
case (dashed). In (b), the time-reversed quantum and classical maps, respectively, were applied from
n = 501 on.
into play: In three (and more) dimensions, there is a threshold for the disorder in
an extended system below which the eigenstates are extended, despite the fact that
Bloch’s theorem does not apply. This crossover, called the localization-delocalization or
Anderson transition [71], has much in common with a phase transition. It is presently
far from being understood semiclassically.
The last paragraphs of this subsection are devoted to a simple model which never-
theless exhibits many of the aspects, addressed above, of quantum chaos in extended
systems: the kicked rotor. This is a planar rotor driven by sharp pulses of torque
whose strength depends in a nonlinear way on the angular position of the rotor. Its
Hamiltonian reads [2,72,73],
/2 00
= ~ + kcosB У <5(t — nr),	(6.80)
71 —~OO
where 0 is an angle, —тг < 6 < тг, and the canonically conjugate angular momentum
(or action) I, I = 0, ±1, ±2..., is measured in units of h. The classical phase space
therefore has the topology of a cylinder, so that the kicked rotor represents an extended
system with respect to angular momentum, not to its spatial coordinate 0.
If the dynamics is restricted to discrete times tn = пт + e, e -> 0+, it can be
condensed in a map (/n,#n) —> (ln+i, 0n+i),
ln+i = ln + fcsin(0n+1), 0n+1 - (J)n + 7-/n)(mod2tt),	(6.81)
referred to as the standard map. The quantum-mechanical counterpart of this map is
the time-evolution operator propagating the system over a single period of the driving
[72, 73],
U(tn+l,tn) = UKUR, UK = exp(ifccos(9), UK = exp (-irZ2/2).	(6.82)
6.3 Chaos and quantum coherence 33
Fig. 6.28: Typical eigenstates of a mode
closely akin to the quantum kicked rotor
From [75].
Initially, both dynamical systems, Eqs. (6.81) and (6.82), show a diffusive increase of
the argular momentum,
(Gn - lo)2} ~ Dn,	(6.83)
if the effective classical nonlinearity parameter К — кт is large enough so that all ill'
chaotic regions along the cylindrical phase space are connected (Fig. 6.5). This is th;
case for К > Kc « 1. The diffusion constant is then approximately given by [2, 72, 73
Г) k}-/2.	(6.8 Г
While chaotic diffusion in the classical kicked rotor lasts forever, the quantui;
namic; changes its character at a break time n*. For typical parameter values (it will
be specified below what “typical” means here), the diffusion saturates (Figs. 6.25
6.27). Henceforth the wavepacket merely fluctuates around a time-independent nn an
shape that decays exponentially in space from the site Iq where it has been prepared
[73]. r?he reason for this crossover (which was completely unexpected for those who
investigated the quantum kicked rotor for the first time) is that the eigenstates | a) of
the system are themselves exponentially localized [74-76],
where £ denotes the localization length (Fig. 6.28).
Ac ,ually, it is not correct to talk about eigenstates and eigenenergies for a driven
systeir whose energy is not conserved. However, the fact that the driving is peri, i •
in time allows to apply a theorem analogous to Bloch’s: The Floquet theorem imp!i”s
the existence of eigenvalues and eigenstates of the evolution operator (6.82) whic h
preserve many properties of the energy and of its eigenstates. They are referred to as
quasienergies and Floquet states, respectively, see Section 5.3. As a consequence oi
Eq. (6 85), the quasienergies form a pure point spectrum.
Th; disorder responsible for the inapplicability of Bloch’s theorem in ang-.ii.u-
mompntum space, and thus for the absence of transport by tunneling, is not due t1
332 Chaos, Coherence, and Dissipation
the chaotic classical dynamics. It even possesses a spatial symmetry, invariance under
the translation I —> I + Чт^т. This classical “cell size”, however, may or may not
be commensurate with the basic quantized angular-momentum scale Д/ = 1. Only
if 2tt/t is rational, say т/4тг = p/7, p and q integers without common divisor, both
scales are commensurate and the kicked rotor behaves like a system periodic in angular-
momentum space with lattice constant q. In this case, the Floquet states have also
Bloch form with respect to their dependence on I [77]. The general discussion of the
signature of chaotic diffusion in band spectra, cf. Eq. (6.71), applies. Fig. 6.23 shows
an example of a band spectrum for the kicked rotor in the chaotic regime.
The rationals, however, are of measure zero among the real numbers—they are
exceptional. The typical case is that т/4-тг is irrational. The phases due to free rotation
in the factor [7R of the time-evolution operator, Eq. (6.82),
Ф/ = (t/2/2) mod2-7r,	(6.86)
then form an aperiodic sequence which in many respects resembles a sequence of true
random numbers. This quasirandomness is sufficient to suppress tunneling and to
enforce an exponential envelope, Eq. (6.85), of the eigenstates (Fig. 6.28). As this
localization is not a result of static disorder but of a quantum dynamical phase, it is
referred to as dynamical localization [75].
The crossover time n* is related to a characteristic spectral scale by the energy-time
uncertainty relation
n* и 2тг/Де.	(6.87)
Here, the energy scale Де is determined by the mean quasienergy separation for the
selection of Floquet states that contribute to a given wavepacket, i.e., in the local
spectrum at the site where the wavepacket has been prepared. Note that one cannot
sensibly speak of a mean quasienergy spacing without further qualification for a system
with an infinite-dimensional Hilbert space. There, an infinite number of quasienergies,
which are actually phases, is crowded around the unit circle. Since, however, the
eigenstates are exponentially localized with a localization length £, the mean number
of those quasienergy states that participate in an initially localized wavepacket takes a
finite value ~ 2£. This enforces a quasiperiodic time evolution as in a bounded system.
The number of participating states can be considered as the dimension of the restricted
Hilbert space formed by a localization neighborhood. Therefore,
Ac « тг/С, П* « 2£.	(6.88)
One can go even further and relate n' and £ to D: At n*, the diffusive spreading has
reached a scale (cf. Eq. (6.83))
(Gn- -M2)«Dn‘.	(6.89)
Since the wavepacket will not spread appreciably anymore for n > n*, this scale must
coincide with the asymptotic size of the wavepacket,
((/„. - /0)2) « (2f)2,	(6.90)
6.4 Quantum chaos in open systems 333
so thal, with Eqs. (6.88) and (6.89) [74,75],
n* « 2£ » D « fc2/2.	(6.91)
Remarkably, this equation relates the characteristic parameter of a quantum coherence
effect, the localization length, to a purely classical quantity, the diffusion constant.
This is a striking example of how intimately connected the quantum and the classical
dynamics are, even if they appear qualitatively different on long time scales. Indeed,
in the presence of dynamical localization, the classical limit of the quantum dynamics
at a fixed time (a diffusing Gaussian distribution) does not commute with its lone-
time limit for fixed relative h (quasiperiodic fluctuation of an exponentially localized
wavepacket).
6.4 Quantum chaos in open systems
No physical system is strictly closed. Otherwise, we could never learn about its exis-
tence, let alone its internal structure and evolution. The communication of a system
with the ambient world is an essential aspect of its physics, particularly in the micro-
scopic realm.
The following two prototypical physical situations comprise most of the important
cases of interaction with the outside world: In scattering, this interaction is restricted
to a time interval. A probe prepared in a well-defined asymptotic state is sent into
the sys.em, interacts, and emerges in another asymptotic state. The change in state
represents the information gained on the system. It is unitary, if the scattering is
elastic. In the case of a lasting interaction with a large number of “secondary” degrees
of freedom, in contrast, information on the system in focus is conveyed to the secondary
freedon s and encoded in correlations of these freedoms: the paradigm of measurement.
The dynamics of the central system is rendered nonunitary by the presence of the
ambiem freedoms, typically resulting in dissipation.
6.4.1 Chaotic scattering
Chaos is a long-time property. Rigorously speaking, it can be distinguished from a
non-chaotic dynamics only on basis of an infinite time series. How, then, can one talk
of chaos in the context of scattering, where the interaction time is always finite? Tliis
become 5 feasible by considering “true” chaos as a limiting case of chaotic scatter! •
approached for long interaction time. There exist criteria, based on the behavior кл
finite interaction time, when a system can be expected to approach this limit, 'fix'
three following features characteristic of classical chaotic (or irregular) scattering
serve as such criteria [78]:
(i)	The functional dependence of the outgoing on the incoming trajectory (e.g., i’1
two dimensions, the deflection function	where 0.; is the scattering angi"
ai d I\, 9\ are the incoming angular momentum and angle, respectively) varies с,-
al scales of its argument and is approximately self-similar (Fig. 6.29).
334 Chaos, Coherence, and Dissipation
Fig. 6.29: Deflection function for scattering within a waveguide junction with rounded (quarter-circle)
corners. Panels (b) and (c) are successive magnifications of panel (a). From [79].
(ii)	The distribution of the delay time (the time a trajectory is retarded by the
scattering, as compared to free motion with the same initial conditions) follows
an exponential law (Fig. 6.30),
P(<d)
1(,
7
(6.92)
(iii)	There exists a set of unstable periodic orbits trapped within the scatterer. Al-
though not accessible from outside, they form a repellor for the scattering tra-
jectories: The longer a scattering trajectory stays within the scatterer, the closer
it approaches one of these orbits or a succession of several of them. They repre-
sent a self-similar manifold. One speaks of a strange repellor, in analogy to the
strange attractors occurring in dissipative chaotic systems. The decay rate 7 in
Eq. (6.92) is related to the fractal dimension of this repellor.
Fig. 6.30: Delay-time distribution (histo-
gram) for the same scattering system as un-
derlying Fig. 6.29, and exponential law de-
rived from an analytical estimate of the de-
cay rate (straight line). From [79].
The chaotic repellor forms the “backbone”, or, using a more precise, mathematical
term, the closure of the scattering trajectories. It is in this sense that chaos represents
a limiting case of chaotic scattering, and chaotic scattering, in turn, can be considered
as transient chaos.
Both in classical and in quantum scattering, the function which maps the asymp-
totic incoming to the asymptotic outgoing state is of central importance. In quantum
mechanics, this mapping takes the form of a unitary operator,
|Фои:) = 5|Ф1п>	(6.93)
6.4 Quantum chaos in open systems
called the scattering matrix or S matrix. It can be regarded as the limit for tin —> - > .
tout -> oo of the time-evolution operator C4cat(iout, <in), restricted to the Hilbert spwv
spanned by the asymptotically free states. The asymptotic angular momentum is
quantized, /jn(out) = ^in(out)- Scattering channels are labeled by the quantum number
I. Each channel supports two asymptotic states, an incoming and an outgoing one.
The classical limit is approached by going to high channel numbers.
Ako in chaotic scattering, there are two complementary approaches to capture th-
irregular nature of the classical dynamics: the semiclassical description, which allows
to relate the features of individual systems to those of their classical counterparts, and
the random-matrix approach, which emphasizes universal aspects.
Semiclassical scattering is formulated much along the same lines as the semiclas-
sical taeory for bound motion. The starting point is again an approximation to the
propagator, akin to the Van-Vleck propagator (Eq. (6.22)). It allows to express the S
matrix as a sum over classical trajectories [78],
5ц-
= ^2 exP
7T\
(6.9 b
The si in now comprises scattering trajectories which satisfy boundary conditions a-
cording to the channel indices 1,1'. Their incoming angular momentum is I = h’
outgoing value is I' = hl'. The characteristic action in the exponent reads Rj(I, Гj -
- lini|(. >oc dt' [r(t')p(t') +	where r, p, 9, I denote collision coordin •
momentum, angle, and angular momentum, respectively. This action is constc.
so as not to diverge in the asymptotic regime, which is ensured by the vanishing
of p(t) and i(t) for |t| —> oo. The phase д7тг/2 is determined by a Maslov in-
dex. The amplitude is again the square root of a classical transition probability.
A3{I, Г) = (\д2П^д1дГ\/2тгУ/2.
The random-matrix description of chaotic scattering [78, 80] differs from the cor-
respon ling theory for bound systems: We are now dealing with unitary and not with
Hermit ean matrices, characterized by a set of unimodular complex eigenvalues e“*, or
alterne tively by the eigenphases ф, rather than by eigenenergies. Ensembles of ran-
dom u litary matrices analogous to the GO(U,S)E can be constructed, and are called
CO(U,S)E (circular ensembles [8,39-41], because eigenphases ф are defined on the
unit circle). They are statistically invariant under the same transformations as their
Hermitean counterparts and pertain to the same global symmetries (see Section 6.2).
The hypothesis that the S matrices of classically irregular scattering systems can be
modelled by random unitary matrices declares a huge amount of information contained
in the individual matrix elements just noise. This a strong assumption that allows
to predict a number of properties which universally characterize quantum irregular
scattering. An important example is provided by the correlations of the S-matrix
elements as functions of the energy [78,80],
G,r(£) = ^S(>	%
(6.95
336 Chaos, Coherence, and Dissipation
Fig. 6.31: Cross section of a nuclear reaction showing Ericson fluctuations (a), and the corresponding
autocorrelation function (b). The dashed line in (b) is the best fit of a Lorentzian to the data (cf.
Eq. (6.97)). From [81].
By inserting the semiclassical approximation (6.94) and neglecting the off-diagonal
contributions to the double sum over trajectories (see Section 6.2.2), it is possible to
write the correlation function as
roc
С/,г(е) ~ Уо dr{Pl_tII(E, r))B exp(ier/7i),	(6.96)
where the integral goes over the delay time т = BR/дЕ, and Pj^^E, t) denotes the
total classical transition probability from I to I1 with energy E and delay time r.
As pointed out above, chaotic scattering is characterized by a delay-time distribu-
tion ~ е~7(£)т. If the range ДЕ of averaging is chosen smaller than the classical energy
scales over which y(E) varies appreciably, one arrives at (Fig. 6.31) [78,80]
/	. -i
Сц'(б) = С/И0) (l-i-jM .	(6.97)
\	'*7/
This relation is analogous to Eq. (6.49): It connects a purely quantum-mechanical ob-
servable to a global characteristic, 7, of the corresponding classical irregular dynamics.
Here, this is a quantity related to the fractal dimension of a chaotic repellor. Moreover,
7 can also be expressed in terms of characteristic quantities of the periodic orbits that
build up the repellor.
Correlations decaying as a Lorentzian, Eq. (6.97), have been derived for the scat-
tering off atomic nuclei long before the advent of quantum chaos, under the following
assumption: The poles which the S matrix attains if the energy is extended into the
complex plane, have distances Г from the real axis, corresponding to the linewidths
of the associated resonances, that are much larger than their mean spacing AT?, and
are otherwise random. Under this assumption, the contributions of these resonances
overlap strongly on the real axis, and their interference leads to complicated fluctu-
ations (Ericson fluctuations) which can no longer be attributed to individual energy
levels of a similar, but closed system (Fig. 6.31). This is another example for the abil-
ity of chaotic systems to mimic, by their deterministic dynamics, properties which are
otherwise associated with random processes.
Scattering theory represents an extremely general approach, and its applicability
is by no means restricted to “typical” scattering situations, such as the investigation
6.4 Quantum chaos in open systems 337
of nuclei, atoms, molecules, and solid-state samples by means of small projectiles. In
fact, also electronic devices are perfectly apt to being analysed within the framework
of scattering theory. An electronic device has two or more “legs”, the leads by which it
is connected to the rest of a circuit. The nearly free motion of electrons, holes, or other
quasiparticles in these leads represents the asymptotically free states of the scattering
process. The device itself plays the role of the scatterer.
Irregular scattering can be induced in an electronic device, e.g., through the shape
of the potential confining the charge carriers. Electron billiards are two- (or higher-)
dimensional potential boxes (quantum dots) that induce chaotic motion of electrons
by the shape of their boundary [82,83]. They are produced in a semiconductor, either
permanently by micrometer-scale lithography (see insets in Fig. 6.33), or by generating
a correspondingly shaped electrostatic potential of controllable strength. Leads arc
then attached to the billiard in the usual way. Alternatively, one can place obstacles
(antidots) in a broad conducting strip [84]. In a one-dimensional structure, e.g., a
succession of potential barriers, a time-dependent field, such as an ac voltage or a
microwave or laser irradiation, can be sufficient to induce chaotic scattering.
In order that an electronic device fit into the framework of a semiclassical account, w
deterministic quantum scattering, two conditions have to be fulfilled: The wavelength
must be small enough, compared to the relevant spatial scales of the system, so that a
wavepacket can resolve the shape of the confining potential. This condition amounts tn
being close to the classical limit, in the sense outlined in Section 6.2. Another conditi»”1
concerns the dephasing length 1Ф. This is the average path length beyond which pl.. . 
coherence is lost, for example by inelastic scattering events within the device: It nui-»1
be large compared to the spatial scales to enable interference among distinct pa.
Togetl er, these conditions require that
= af « a « 1ф,	(6.9
y/2m*Ep
where Ey, Ap denote Fermi energy and wavelength, respectively, m* is the effective
electron mass, and a is a typical dimension of the billiard or barrier.
The dephasing length should not be confounded with the elastic mean free path
Zei, which refers to elastic scattering off impurities, lattice defects, grain boundaries
roughness of confining surfaces, etc. Imperfections giving rise to elastic scattering :m.
similai, in their effect on spectral and transport properties, to boundary shapes th.;’
induce chaos, but they vary from sample to sample and therefore have to be treated
statist cally. With respect to the density of elastic scatterers, one distinguishes further
betwet n the ballistic regime, where also lei » a, and the diffusive or disordered rc ,
Li < c. In the ballistic regime, the shape of the sample determines the corresp. ...:
classic il dynamics completely, while in the disordered regime, the classical dynamic:-
has a diffusive component (see Subsection 6.3.2).
In in ideal lead of constant width w, the transverse motion is approximately e<"’
alent, to that of a particle in a potential box with steep, infinitely high walls.
eigenfuctions in such a box are iMq±) ~ sin(kj_tqj_), with a momentum quantized m
pn =	= hth/w. For fixed energy—in electronics, the energy of the carriers i-
338 Chaos, Coherence, and Dissipation
of course the Fermi energy EF—quantization of the transverse momentum enforces
a corresponding quantization of the longitudinal momentum, рц/ = (2m E — p2 J1/2 =
Й(|/с|2 — (/тг/ш)2)1/2 (Fig. 6.32). The modes to which the index I refers correspond to the
channels of scattering theory. There is one set of channels for each lead. The number
of channels that can carry a current is restricted by the energy, Zmax = [wV2mE/nh]
([...] denotes the integer part). They are called open channels. The higher-lying
channels have an imaginary longitudinal wavenumber = i((/?r/w)2 — (fcl2)1/2, cor-
responding to waves that decay exponentially (evanescent waves). These channels are
called closed. Transport through a closed channel amounts to tunneling. A complete
quantum-mechanical description must take both channel types into account.
Fig. 6.32: Open and closed channels in a rectangular
waveguide. The quantization of the transverse momen-
tum, fcj_, implies, by energy conservation, a quantization
also of the longitudinal momentum, кц. Open channels
(full horizontal lines) carry running waves, closed chan-
nels (dotted horizontal lines) carry evanescent waves.
Sending individual electrons in and registering the state in which they come out
is not the usual way to operate an electronic device. Rather, one is dealing with
currents entering through one or more of the leads and leaving through others. In this
context, the transmission and reflection amplitudes b,m and rlm, respectively, and the
corresponding probabilities 7},m = |t(jm|2, Ri.m = are the relevant quantities.
Together, the two sets of amplitudes contain the same information as the S matrix.
The relation reads
S-(tr J,'),	(6.99)
where the four blocks are Lx L matrices, if L is the total number of open channels.
A spatial reflection symmetry of the scatterer implies that scattering from the left and
scattering from the right are indistinguishable, so that r = r' and t — t'*. Unitarity of
the S matrix corresponds to probability-flux conservation: All incoming and outgoing
fluxes must sum up to zero.	-
Not even transmission and reflection probabilities are directly measured, but con-
ductances G = (e2where g is dimensionless. Conductance is a concept alien to
everything discussed up to now, because it involves dissipation and therefore is incom-
patible with a unitary dynamics. This conceptual problem can be circumvented by
a trick: The inelastic processes and the electron reservoirs absorbing the dissipated
6.4 Quantum chaos in open systems 339
-0.5	-0.4	-0.3	-0.2	-0.1	0.0	0.1	0.2	0.3	0.4	0.5
BCD
Fig. 6.33: Resistance of ballistic semiconductor quantum dots shaped as (a) a stadium and (b) a
circle, as a function of the perpendicular magnetic field, in the regime of a single open channel. Left
insets: Enlarged sections of the data near zero magnetic field, at 20 mK (full line) and 0.6 К (dashed).
Right insets: electron micrographs of the devices, with 1 pm bar for scale. From [82].
energy ire placed, in a schematic setup, at the far ends of the leads, far away from the
device. In this way, the scattering process proper is not affected by them and can be
treated as if it were unitary. Another difficulty arises by the fact that the simple rela-
tion I -- GV implies a linear response of the current I to the applied voltage V. The
lowest order of a perturbation expansion of the current is indeed linear in the field,
but a perturbative description will eventually fail in a chaotic and therefore highly
nonlineir system.
Keeping these two caveats in mind, one may state the Landauer-Biittiker formula
(cf. See:.ion 1.5) for the conductance from lead о to lead /5 [85,86]
2	(u.illd)
where tie summation goes over all channels I of a and rn. of /3. The factor 2 conus
about b/ spin degeneracy. Eq. (6.100) suggests that chaotic scattering can be reflected
in amperemeter readings (Fig. 6.33)!
Of course, it is an idealization to dislocate all inelastic processes into reservoirs
outside the device. In fact, experiments show that quantum coherence effects, like
weak localization, are actually less pronounced than expected from a theory assuming
340 Chaos, Coherence, and Dissipation
purely elastic scattering. This reduction can be explained by taking residual inelastic
processes within the device into account [87,88]. However, it is possible again to
defer even these internal inelastic processes to a reservoir conceptually separated from
the device [89]: This extra reservoir is coupled directly to the device by a fictitious
“parasitic” ideal lead. The potential of the reservoir is determined by requiring that
no net current flows through the fictitious lead. Its sole effect is to randomize the phase
of all carriers entering and leaving it. It is completely described by the S matrix for
scattering into and out of the fictitious lead. The Landauer-Biittiker formula for the
dimensionless conductance from the real lead a into the real lead (3 is then modified
to [87]
ga,s - 2	"),	(6.Ю1)
\	-* a,0 “I * ф,$ /
where is the total transmission between the two real leads as given by the sum in
Eq. (6.100), and TQ^, Тф<1) are the corresponding transmissions between the fictitious
lead and the real ones. The decay rate due to the phase-breaking processes in the extra
reservoir is [88]
70=T0/tH,	(6.102)
with Тф, the total transmission from the device into the fictitious lead. It is scaled by
tH = 2тг?1/Д, the Heisenberg time within the device with a mean level spacing A. If the
device is a chaotic electron billiard so that the escape times 1/7Q into the real channels
are exponentially distributed (cf. Eq. (6.92)), the escape rates -yQ may be added to уф
to give the total 7cff = 7^> + 7a [88].
By way of Eqs. (6.99) and (6.100), the random-matrix approach to the scattering
matrix can be used to infer the statistics of the conductance and its fluctuations.
Since with present-day technology, electron billiards and similar devices cannot be
manufactured with sufficient accuracy to make a current characteristic reproducible
from one sample to the next, a statistical approach is even unavoidable. For example,
the probability distribution of the dimensionless conductance can be evaluated for the
three canonical circular ensembles [90],
P(g) =	0 < g < 1,	(6.103)
where (3 is as defined in Eq. (6.58). Equation (6.103) is a manifestation of universal
conductance fluctuations (UCF). They have been observed in conductors with dimen-
sions on the scale of microns or below, long before quantum irregular scattering entered
the scene. They appear random, but nevertheless are reproducible for a given sample,
ruling out a thermal origin. In this sense, they are analogous to the Ericson fluctuations
of scattering cross sections observed in nuclear physics, and to the speckle patterns of
laser light reflected from a rough surface. Universal conductance fluctuations can be
ascribed to the elastic scattering off imperfections in the material. It is indeed possi-
ble to reproduce, in a qualitative sense, such fluctuations in ballistic electron billiards
where they reflect the shape rather than the internal disorder.
____________________________________6.4 Quantum chaos in open systems	311
Equation (6.103) implies that a regular dynamics results in stronger fluctuations .4
the conductance than irregular scattering, resembling the case of spectral fluctuation,
(see Section 6.2). Moreover, the presence or absence of time-reversal invariance is agam
reflected in universal exponents of a probability distribution. Incidentally, it is easy t
break time-reversal invariance in electronic devices: In two-dimensional billiards, it is
sufficient to apply a magnetic field perpendicular to the plane of motion. The response
of the current to the field is described by a magnetoresistance. In periodically driven
one-dimensional devices, an asymmetric time dependence of the field has an analogous
effect.
6.4.2 Quantum chaos with dissipation
In the preceding subsection, we have already encountered dissipation as an unavoid-
able aspect of conductance. There, it was still possible to keep it essentially out-
side the consideration. Taking dissipation explicitly into account, however, will up>-s
up new vistas otherwise inaccessible. The discussion in Section 6.3 has shown the;
the strongest, qualitative discrepancies between classical chaos and the correspond! n'
quantum dynamics occur when coherence effects like tunneling and localization d
nate the quantum-mechanical time evolution. In the presence of dissipation, coheivin
effects degrade and give way to an incoherent dynamics closer again to the classical
behavior [30]. Moreover, there are important classical chaotic phenomena involv.
dissipation: strange attractors, fractal basin boundaries, intermittency [3,4], to name
a few. Including dissipation into quantum chaos allows to study all these phenomena
in the mesoscopic realm where quantum coherence is still essential (see, e.g., Ref. [91]
for an eirly review of dissipative quantum chaos in quantum optics).
This requires to switch from the formalism of pure states evolving unitarily, to a
more g( neral representation of the quantum dynamics, based on density operators.
As it is the case with the conservative quantum kinetics and dynamics, establishing
a connection to the classical limit is facilitated by using a phase-space representation.
Not all of them are suitable for this purpose, only those which allow to represent
density operators. This is the case, in particular, for the Wigner function and the
Husimi distribution (see Section 6.2).
The most complete and consistent way to model a dissipative dynamics on the
quantum-mechanical level is to adopt the classical view of a heat bath comprising a
macroscopically large number of degrees of freedom, which acts as a sink for information
and enei gy. Technical problems encountered in the implementation of this concept, and
the tricl s and approximations invented to solve them, are elucidated in Chap. 4 of this
volume. Comprehensive accounts are found in the literature, for example in Ref. [92].
Some aspects are of special relevance in the present context:
A cr rcial step in the formulation of the dissipative quantum dynamics is the elim-
ination of the degrees of freedom of the heat bath, by projection or other techniques.
The reduction to the central system, however, has to be payed by a severe disadvan-
tage: The reservoir remembers for some time the influence it suffered from the centi. :
system. During this time, it can return the received information and in its turn, ex<-
342 Chaos, Coherence, and Dissipation
an influence on the central system which indirectly depends on the state of the central
system in the past. In other words, the reservoir endows the central dynamics with
a memory. As a result, the reduced dynamics cannot be described by a differential
equation which is local with respect to time, i.e., relates the change of state to the
present state alone. Rather, the right-hand side of the equation of motion contains an
integral over time connecting the present to the past.
In many cases, however, it is a good approximation to restrict this memory to a
finite or even infinitesimally short duration. It is referred to as the Markov approxi-
mation [92]. On the classical level, one often describes a chaotic dynamics by a map,
such as Eq. (6.81), connecting two instants separated by a finite time. The quantum-
mechanical analogue of a map, in the conservative case, is the unitary time-evolution
operator, cf. Eq. (6.82). In the case of a dissipative dynamics, a similar device can
only be constructed in conjunction with a Markov approximation. The memory must
not last longer than the time bridged by the quantum map, otherwise the input of the
map, the past, were insufficient to determine its output, the present. A quantum map
for the density operator can then be written in the form [93, 94]
p(t') =G(t',t)p(t),	(6.104)
where G(t', t) is called a superoperator, because it acts on an object which already is an
operator. In order that the basic properties of the density operator be preserved under
the action of G, the superoperator itself has to fulfill a number of formal requirements.
They concern normalization, Hermitecity, and positivity of the density operator, as
well as the preservation of uncertainty [93,94].
The interaction of a system with the ambient degrees of freedom has two principal
consequences: The degradation, and eventual destruction, of phase information in
the central system due to incoherent processes, and the dissipation of energy into the
environment. The destruction of coherence by noise occurs in general on a much shorter
time scale than the global energy loss. In some situations, it is even adequate to forget
about dissipation altogether and to study decoherence alone. This is possible without
going into the cumbersome microscopic modeling of the interaction with a heat bath.
Instead, it suffices to connect the system to a noise source. Formally, this amounts
to the application of a driving force with a random time dependence. In terms of the
Hamiltonian, this takes a form like [95 -98]
Ha(t) = Hdel + ffF/rand(i),	(6.105)
where //det denotes the Hamiltonian for the deterministic dynamics, о controls the noise
strength, and Hra„d is a perturbation with a random (or quasirandom) dependence on
time whose autocorrelation defines the type of noise: white, coloured, etc. [99]. Since
H is now the full Hamiltonian to be considered, without projecting out any degrees of
freedom, the time evolution generated is perfectly unitary. In fact, the same ('fleet can
be achieved by supplying the time-evolution operator directly with a random element,
e.g.. a randomization of the phase1.
= L%ct(Gt)eit0™"‘(,),	(6.106)
6.4 Quantum chaos in open systems
where now e is the noise strength, and </>rand(f) generates a nonglobal random phase. !i.
both formulations, significant physical information can only be extracted by aver;;;'me
over sufficiently many realizations of the noise (whereupon unitarity is lost).
A special source of noise are the round-up errors inherent in all numerical сак ida
tions with fixed precision. Due to the exponential dependence on the initial conditions
of classical chaotic motion, numerical noise can be of some concern in simulations .->*
classical chaos. In comparison, quantum coherence, sensitive as it, is against noi:<'
appears stable since in quantum-mechanical calculations, numerical errors are ampli-
fied only linearly in time. This is demonstrated in Fig. 6.27b, where a noise source
(rounding to 15-digit precision) acting over a certain time span suffices to eradicate the
inversion of the motion in the classical kicked rotor. During the same time, however,
this noise does not produce a visible deviation from the time-reversed evolution in the
corresponding quantum dynamics.
A r oisy driving is not exactly equivalent to the coupling to a heat bath. In fact,
one co ild imagine it to be generated by a single classical degree of freedom, with a
huge irertia so as to prevent any back reaction from the motion of the driven to that of
the driving system, and a sufficiently complex dynamics (actually, a chaotic dynamics
is very well suited for this purpose: see Section 6.5). This “surrogate” can mimic the
effect < ‘f a reservoir on short time scales, but there will be qualitative deviations on
long ti ne scales. Specifically, dissipation cannot occur. To the contrary, a noisy yet
unitarj driving can lead to an unbounded accumulation of energy in the driven system,
in the long-time limit [100].
There exists a third approach to dissipative quantum dynamics, the quantum-stab
diffusion picture, which combines features of the two approaches discussed above [101
10.3]. in its physical content, it is equivalent to a master equation for the density
operator, derived within a Markov approximation. In contradistinction to the mastei
equation, however, it deals with individual pure' quantum states. Of course, a dissipa-
tive dynamics cannot be reduced to the unitary time evolution of pure states. It is her?
replace ! by a stochastic evolution equation (stochastic Schrodinger equation) compil-
ing a d'terministic part that generates the unitary evolution according to the system
Hamiltonian, and a stochastic part that represents the influence of the heat bath, i::
this resoect, the quantum-state diffusion approach is closely related to the* concept of ;
noisy d’iving, Eqs. (6.105) and (6.106). It, is superior to that simple picture, how< :.
in that it does include the effect of energy dissipation as a drift term in the evolute !:
equation. Insofar, it is analogous to the Langevin description of a classical dissipati-.c
dynamics. Accordingly, expectation values must again be calculated by averaging ov< i
a possibly large number of realizations.
In Sections 6.2 and 6.3, it. has already been indicated that some aspects of tl?
crossover from a nearly classical, chaotic time, evolution of a quantum system to a <h
namics dominated by coherence, are generic: They occur under very general conditions
'1'liis applies also to the stages of the dynamics where decoherence and dissipation ex-
ert. their influence. To be. specific, assume a classically chaotic quantum system with
Ohmic dissipation, an energy loss proportional to the speed (of a mass or charge). In
the regime of high heat-bath temperature, the dynamics is described by a transparent
344 Chaos, Coherence, and Dissipation
equation of motion for the Wigner function [30]. It takes the form of a Fokker-Planck
equation [104],
7Г = {H, F\v}Moyal + т'77~ (P-Pw) + —D —Fw-	(6.107)
at	dp	dp dp
The unitary time evolution is generated by the Moyal bracket with the system Hamil-
tonian H(p,q) = p1/2m + Iz(<j) (see Eq. (6.25)). The second and third terms in
Eq. (6.107) correspond to the dissipative reduction of the kinetic energy and to the
incoherent diffusive spreading of the Wigner function, respectively. They are absent in
the conservative case.
The noise that drives the diffusive spreading of the Wigner function stems from
two sources: Thermal noise reflects incoherent transitions induced by the reservoir
due to its thermal excitation. In the case of an Ohmic bath at high temperature, the
diffusion constant for thermal noise is D = 2'утквТ. The proportionality to 7 reflects
the fluctuation-dissipation theorem, see Section 4.4.2. By its temperature dependence,
thermal noise vanishes for T —> 0. But even in the absence of thermal noise, the
coupling to the reservoir will elicit spontaneous transitions which, in the mean, drive
the system towards states of lower energy: Quantum noise persists at T ~ 0, but
disappears in the classical limit. It is a genuine quantum effect. The second term in
Eq. (6.107) generates the drift of the quantum state towards lower energy, a collective
effect of induced as well as spontaneous transitions. The proportionality to momentum
is t he hallmark of Ohmic dissipation; other forms exist as well.
As has been shown in Section 6.2, coherence begins to affect the Wigner function
by the time (cf. Eq. (6.28)) [30]
1 . / O'2n ApciPt (0) \	moi
fcoM « T In -------------- I	(6.108)
A \ Ti )
where an = (V'/V(n+ 0)1/'1 is a length characteristic of the nonlinearity of the potential.
Noise counteracts the development of fine structure in the Wigner function by way of
the diffusion term in Eq. (6.107). It snioothens sharp variations in Fw- This process
is governed by the diffusion equation
(Ap<],n(t))2 = (Ap(0))2 + Eh.	(6.109)
A balance between chaotic contraction and diffusive spreading is reached if
0=4 [(Ap<iet(Q)2 + (Л/Ш(Ш21 = -2A(Ap(U/))2 + F, (6.110)
dr L	J
at a momentum scale [30]
Арт„,~УБ7л.	(6.111)
Smaller scales cannot occur: As soon as the phase-space contraction by the chaotic
flow lias reached the scale Ар||11и. the finest structures in the Wigner function are
6.4 Quantum chaos in open systems 345
smeared out by diffusion. This destruction renders the dynamics irreversible, because
the fine structures are necessary to recover the initial state, upon inverting time so that
contraction and stretching are interchanged.
The time required for the Wigner function to attain structures on the scale Apmin is
obt.aimd by replacing, in Eq. (6.108), the characteristic momentum h/an with Apmin,
1 , / Apdet (0) \
Since the information conveyed to the small scales is irretrievably lost for times t
entropy increases from fdec on.
Quantum coherence effects are suppressed from the beginning by environment-in-
duced decoherence if tdec <7 ZCOh i (< boh -i  •  , cf. Eq. (6.108)). This is the case if
[30]
«(Apmin » Л.	(6.113!
Equatk n (6.113) resembles the Heisenberg uncertainty relation. It suggests to interpn i
the length fz/Apmin as a dephasing length 1Ф (cf. Eq. (6.98)): The system passes from
an approximately reversible classical evolution directly to irreversible behavior if th.-
characteristic scales an of the nonlinearity of the potential exceed 1Ф. In the regime
weak dissipation, f(lec > tCoh,i, there will be an intermediate phase of unitary quant!
median cal evolution with strong coherence effects, as discussed in Section 6.3, following
an initial quasiclassical phase and preceding a slow decay of coherence.
The magnitude of the second term in the evolution equation (6.107), representm
dissipat.on, is independent of that of the diffusion term. For a given diffusion c<
stant D it can take arbitrarily small values, compared to the third term, if mass oi
temperature are sufficiently large. If dissipation is proportional to speed as assumed
in Eq. ((>.107). its influence is also controlled by the momentum scales involved. Sine,
this terr i leads to a contraction of phase space in the p direction, its effect adds to ,
possible contraction by the chaotic flow, hut it is not compensated by an expansion in
other di) ections. The time scale
Q-lax = 1/7	(6.114)
can cxc.e >d all the other time scales discussed above and represents an upper bound for
th<' crosi-over to irreversible evolution. Equation (6.1 11) is a crude simplification since
in general, the relaxation time depends on the initial state’ (see below).
For t > /ГС1М, the system approaches an asymptotic state, which in the absence -.4
an external time dependence is stationary. Since in this state, I here is no net energv
exchangi with the surroundings, it amounts to equilibrium. All spatial and temporal
symmetries of the system are restored in equilibrium. Close to the classical li"
the quantum asymptotic state resembles the asymptotic distribution attained 1л -a
corresponding classical system. It cannot, however, be exactly identical to its dasal
counter]) tri. because both thermal and quantum noise blur the fine structure oi th--
attractor.
346 Chaos, Coherence, and Dissipation
In particular, strange attractors with their fractal geometry are incompatible with
quantum uncertainty. This gives quantum noise a fundamental role in this context:
It compensates for the dissipative contraction of phase space, which otherwise would
reduce the initial volume hf of a quantum phase-space cell to a fraction of that volume
on the time scale 1/7. Thermal noise is of no basic concern here because quantum
uncertainty must, of course, not disappear for T —> 0.
In the long-time limit, noise is the prevailing quantum effect. This suggests a
semiclassical description different from the semiclassical approximations discussed for
the conservative case. In particular, it is not based on the concept of periodic orbits. It
consists in retaining, in Eq. (6.107), only the diffusion term corresponding to quantum
noise, besides the classical Hamiltonian, drift, and thermal-noise terms. It is of first
order in ft, while the discarded quantum corrections introduced with the Moyal bracket,
Eq. (6.26), are at least of the order ft2. The equation of motion for the Wigner function
then reads,
Tjrftvv — {H, Fwjpoisson + 27—pPw + (Dqm +	(6.115)
dt	op dp	op
As Eq. (6.107), it has the form of a Fokker-Planck equation, a classical evolution
equation for a phase-space distribution preserving positivity but, in contrast, to the
Lionville equation, not the phase-space measure. A Wigner function that is initially
positive definite will remain so under the action of Eq. (6.115). This is an artefact of
the approximation. It should not obscure that the Wigner function is not a probability
density and in general, may take negative values. However, its positivity enables to
translate Eq. (6.115) into a semiclassical Langevin equation [99] or, upon integration
over a finite time span, into a stochastic map: a map of the classical coordinates, like
Eq. (6.81), supplied with noise terms of first order in ft [93,94].
Quantum chaos with dissipation can also be considered from a spectral point of
view. The coupling to an environment with a continuous spectrum leads to a finite
linewidth of otherwise discrete levels, inversely proportional to their lifetime. In this
way, those limitations to chaotic behavior are lifted that arise due to the discreteness
of the spectrum of bounded quantum systems: Quasiperiodicity is destroyed. The con-
cept of levels can be preserved in the context of a continuous spectrum by supplying
the energies with an imaginary part. However, it looses its significance if the decay rate
becomes larger than the typical level separation, so that the spectrum is dominated by
overlapping resonances. This is generally the case in chaotic scattering. The character-
istic lifetime, due to dissipation, of an energy eigenstate | о) can roughly be estimat ed
from the drift term in Eq. (6.107) to be t„ ~ l/(7(a| jp| | rr)). The dependence on
the momentum expectation value reflects again the Ohmic type of dissipation assumed
and therefore is not generic.
In the conservative case, random-matrix theory has proven an extraordinarily useful
approach to extract the essentials from the spectra of chaotic quantum systems. It can
be extended to the dissipative case1 by considering the complex energy plain1, instead
of the real energy axis. Ensembles of random non-Hermitean matrices are constructed
by treating the additional free parameters as Gaussian random variables, in tin1 same
6.4 Quantum chaos in open systems
Fig. 6.34: Integrated eigenvalue-spaci
distribution I(S) = f’ ds' P(s') for ge
eral complex random matrices of dimensl
N x N, with N as given in the figure. Ins,
Logarithmic plot of the section marked bj
rectangle in the main figure. The graphs
the inset have approximately slope 4, cor
spending to an increase as s4. From [105
style as for Hermitean matrices. A spectral statistics is then established by using П
Euclidean distance s = ((Re ДЕ)2 + (1тДЕ)2)1/2 as the underlying metric, analogic
to ti e level separation on the real axis.
The canonical ensemble of general complex random matrices is Ginibre’s ensenib!
A central result of the spectral statistics for this ensemble is a cubic distributem
nearest-neighbor level spacings, in the limit of small separation (Fig. 6.34) [105],
F(s)~s3, s«l.	(6’'
(Note that in Fig. 6.34, the integrated spacing distribution I(s) = fg ds' P(s') is pl
whic i according to Eq. (6.116) increases as s4 for s 1.) The characteristic expoic-ii
fl — 3, can again be related to the codimension of exact level crossings in param-1
space (see Section 6.2). Remarkably, this exponent does not depend on the prcsem
or absence of a magnetic field, or coupling to a spin, in the central system, in conti -
to the conservative case [105]
It is instructive to study the succession of distinct phases in the time evolution, <H-
cussed above in a general fashion—reversible near-to-classical behavior, onset of qutu
turn coherence effects, disruption of coherence, irreversible near-to-classical behavio
steady state- -for two representative coherence phenomena, dynamical tunneling an
dynamical localization (see Section 6.3). The models serving here to illustrate then
the harmonically driven double well, Eq. (6.61), and the kicked rotor, Eq. (6.68), r-
spectively, can be coupled to a reservoir to yield a dissipative dynamics, using standai
methods and approximations. One technical aspect of this procedure is worth т<ч
tinning. Both systems are periodically driven and therefore, in the conservative i a
the most adequate framework to describe them is the Floquet formalism (see Sect О
5.3). The coupling to a reservoir does not break the periodicity because it does not it
t,rodice any additional time dependence. However, since the driving may consider: 1 '
alter the. spectrum of the central system, the introduction of the standard Г ‘	,
and rotating-wave) approximations in eliminating the bath degrees of freedom re^’ '
special care (see Section 5.8). Nevertheless, the Floquet formalism can be ext'
the dissipative case. This leads to an enormous gain in simplicity and clarity < Г
resulting equations of motion [97,106,107]. The time evolution starting from an a:-.,
34& OhaQo, coherence, and uissipation
Fig. 6.35: Time evolution of the mean ki-
netic energy, over the first 1000 time steps,
in the classically chaotic regime, К = 10, for
the conservative quantum kicked rotor with
т= 0-15/(\/5 - 1) (graph 1), the cor-
responding dissipative quantum kicked rotor
with damping constants 1 — A - 5 x 10-6
(2), 10-4 (3), 10-3 (4), and the conservative
classical kicked rotor at the same К (graph
5). From [108].
trary initial state is, of course, generally not periodic. Only in the asymptotic state,
the periodicity of the driving is regained.
The decisive property which allows for coherent tunneling to take place in the
presence of a time-dependent driving is the existence of a twofold symmetry of the
driven system, the generalized parity Рш (see Section 6.3). If, as is generally the
case, the coupling to the reservoir does not break this symmetry, tunneling persists
in the presence of dissipation, albeit as a transient phenomenon: It disappears in the
asymptotic state. The time over which tunneling can be observed is determined by the
broadening (or decay rate) ~ l/y of the unperturbed levels. If this linewidth exceeds
the tunnel splitting A, the doublet structure is smeared out, and coherent tunneling
is destroyed from the beginning. Since A is typically extremely small, only very weak
dissipation allows to observe coherent transport.
Transient tunneling does no longer take place between phase-space regions defined
by tori. In the presence of dissipation, their role is taken over by the quantum-
mechanical counterparts of the classical attractors of the contracting phase-space flow.
More precisely, the basins of attraction [3,4] surrounding the classical attractors form
the analogues of the regular regions in conservative systems. Potential or dynamical
barriers are replaced by basin boundaries. Both the attractors and the surrounding
basins share the global symmetries of the flow. In the harmonically driven double well
with dissipation, they either map onto themselves under Рш and then correspond to
libration, or occur in pairs associated to the potential minima: Such pairs of symmetry-
related attractors give rise to transient tunneling across the boundaries between their
basins of attraction. If their geometry, in the classical limit, is fractal, one should
observe tunneling between strange attractors.
For the asymptotic state of the dissipative quantum system, however, coexisting
attractors are no longer distinct, because thermal and quantum noise lead to an equili-
bration of the population concentrated in the various basins of attraction, by inducing
transitions among them. Accordingly, the classical notion of, possibly fractal, basin
6.4 Quantum chaos in open systems 349
Fig. 6.36: Stationary phase-space distribution for the classical dissipative kicked rotor (a) and
quasiprobability distribution for the corresponding dissipative quantum system (b), for the same
nonlinearity К = 5 and damping constant A = 0.3. The classical distribution is plotted as equidistant
sections of the continuous density at constant angular momentum. The quantum-mechanical distri-
bution is the Wigner function, for т/4тг = 0.01, which has support only at discrete lines pf = It From
[108].
boundaries looses its significance in the long-time limit of the quantum dynamics.
While in the case of dynamical tunneling, the coupling to the ambient degrees of
freedom leads to a disruption of coherent transport and thus to a stronger localize.! ii m
on shod time scales, a converse effect of decoherence is expected in systems vd'
dynamical localization: There, the quantum-mechanical suppression of transp<
destruct.ve interference; will be partially or completely lifted by incoherent transition.;
between localized states.
Assuming Ohmic dissipation ~ y|/|, where I denotes angular momentum, the; de ...
rate (linewidth) of a Floquet state | a) in the; kicked rotor is [108]
Г„«7(|/|)„.	(6.117)
Here', у denotes the macroscopic damping constant, as in Eq. (6.107), and (|/|)(l =
|/| |( о 11 )|2 is the expectation of the1 angular-momentum modulus for this state.
For an initial state p(0) = |0) (0| of the dissipative dynamics, corresponding to the
ground-state |0) of the unperturbed rotor, this angular-momentum scale is just the1
localization length £ (see Eq. (6.80)), so that the coherence of this initial state is
destroyed in a characteristic time
«1М	(6.118)
350 Chaos, Coherence, and Dissipation
Only if «dec > n*, the quantum-mechanical crossover time (see Eq. (6.86)), local-
ization will become effective in the first place. In this case, incoherent transitions
take place predominantly into other eigenstates that overlap with the decaying state,
i.e., belong to its localization neighborhood. This will, on average, increase the typ-
ical angular-momentum scale of the decaying states and thus, by Eq. (6.117), of
the decay rate: A self-amplifying cascade of incoherent transitions develops, which
initially leads to a superdiffusive growth of the mean squared angular momentum,
((In — lo)2) ~ 72n2, but saturates in normal diffusion as soon as the angular-momentum
distribution reaches regions of sufficiently high |/| where localization is destroyed from
the beginning (Fig. 6.35) [108]. In the regime of strong dissipation, «dec < !t‘- localiza-
tion does not play any role, and the asymptotic states provide an impressive illustration
of how a “quantum strange attractor” should look like (Fig. 6.36) [108].
6.5 Conclusion
Quantum chaos can be regarded as a laboratory to study the interplay of classical
deterministic chaos, coherent quantum dynamics, and incoherent processes induced
by the ambient degrees of freedom. In the time evolution of a classically chaotic,
dissipative quantum system one expects a succession of time regimes, each of which is
dominated by one of these facets.
In the earliest stage, the quantum-mechanical nature of the system does not yet
become manifest, and the dynamics exhibits the same type and degree of chaos as
corresponds to the limit Й/(characteristic action) —» 0. Quantum coherence modifies
the chaotic classical flow by way of two different mechanisms. As soon as the flow lets
the wavepacket overlap with itself in an increasingly complicated maimer, interference
will distort the finest structures of the classical phase space, on action scab's below h.
This happens on a characteristic time scale ~ log(7i/characteristic action). Simultane-
ously, a discreteness of the quantum-mechanical spectrum prevents a time dependence
of dynamical quantities more complex than quasiperiodic. It is resolved on a time scale
~ h/(mean level separation). Quantum coherence thus imposes fundamental limits on
the complexity both of the phase-space structure and the time evolution. On longer
time scales, it reveals itself in non-classical phenomena such as tunneling and localiza-
tion. Correlations do not decay forever and the dynamics remains reversible -entropy
does not increase. In this sense, conservative quantum systems are less chaotic, more
predictable, than their classical counterparts. The apparent contradiction to the corre-
spondence principle is resolved by recognizing that in classically chaotic Hamiltonian
systems, the long-time limit and the classical limit do not commute.
The interaction with the environment, however weak, again substantially alters
the dynamics. By disrupting the coherence of the unitary time evolution, thermal
and quantum fluctuations drive the dynamics back towards the unpredictability and
irreversibility of the chaotic classical motion. Coherence effects in transport such as
tunneling and localization, in particular, are rendered transients: Besides the limit
6.5 Conclusion
ЗЕ
fi/(c laracteristic action) —> 0, decoherence represents the second crucial ingredient
the quantum-classical correspondence.
In the last stage of the time evolution, the dissipation invariably connected with flu
tuations becomes effective. An approximate, semiclassical description of the quanta
dynamics then no longer requires Hamiltonian, but dissipative, irreversible classic
chaos as its input. On the time scale of classical relaxation, typically much longer tin:
the time needed to destroy quantum coherence, a dissipative system approaches a
asymptotic state where noise is the dominant quantum effect: Dissipation and noi‘
recoi cile the classical with the long-time limit.
Ii each of these dynamical regimes, there exist elements of the classical dymu
ics which recur on the quantum-mechanical level as a sceleton, lending structure ’
the smooth quantum-mechanical functions of phase space and time. During the si u
where decoherence is not yet effective, the invariant manifolds of the classical Haini!»<
nian low leave traces both in quantum-mechanical phase-space structures and specie
The ihort unstable periodic orbits produce scars in the energy eigenfunctions, and
feet the coarse spectral scales. The uniform distribution of the longer periodic <>, .
over ;he energy shell, a consequence of classical ergodicity, is reflected in a repul... .
interaction of the energy levels. The quantum asymptotic state approached in
prese ice of dissipation forms a smooth phase-space distribution concentrated arnrr
the a ,tractor of the corresponding classical dissipative flow, e.g., along the leaves of
strange attractor. Quantum noise smears out the classical fractal geometry on sc:’’’
when self-similarity would become incompatible with the uncertainty principle.
Tlie preceding sections should not create the impression that we have already
comp etc picture of complex dynamics in quantum systems. Open issues reach dov. i
to the very basis of the field. One of the strongest motivations to study quantum cha '
derivi s from the desire to build a sound foundation for quantum statistical mechani':
The famous proof by Sinai that a single circular hard disk in a two-dimensional box i
sufficient to render the motion in the box ergodic, was a breakthrough for the found •
tion of statistical mechanics, on the classical level. The severe restrictions imposed o:
quantum-mechanical chaotic motion by coherence, however, have up to now prevent!'!
a similar proof of ergodicity for a bound quantum system with a few degrees of freedom
A closely related effort is directed at replacing the usual heat reservoirs by '
systems. While in a linear heat bath, a quasicontinuum of harmonic oscillators i
required to achieve an irreversible loss of energy, a few chaotic classical dec,' es '•
freedom can have a similar effect. A corresponding quantum-mechanical chaotic ’
voir works satisfactorily on short time scales. The long-time evolution, however, i:
still p.agued with an incomplete decay of correlations due to the discreteness i.f th?
spectrum [109,110].
Notwithstanding its exotic odour, quantum chaos has important applications. The
is due. in parts, to the ability of chaotic systems to mimic random, disordered phenom-
ena in a completely deterministic and therefore controllable and reproducible manner.
Appliiations require to go from the utterly simplified and abstract models devised tc
study basic questions, to more realistic settings. The rapid technological progress in
the preparation of nanometer-scale structures in semiconductor materials has lent im-
352 Chaos, Coherence, and Dissipation
pact to this trend. The electron billiards mentioned in Section 6.4.1 are a prominent
example. In dealing with microstructured devices, quantum chaos merges with other
issues, such as electron-electron interactions and nonlinear transport in the presence of
strong fields.
Other areas of application are quantum chemistry and solid-state physics. The
extended nature of solid-state systems has profound consequences for dynamical as
well as spectral properties, as has been discussed in Section 6.3. For example, the
deterministic diffusion characterizing the classical chaotic dynamics is clearly reflected
in the band structure. In quantum chemistry, two other aspects of quantum chaos are
in the foreground. In the dynamics of large and complex molecules, the dissipation of
energy from an excited mode into other internal degrees of freedom plays a prominent
role and requires to study dissipative quantum chaos to realistic detail. The means of
choice to feed energy in a controllable way into molecules, is driving by laser pulses.
In this respect, there is a strong overlap with quantum optics. The understanding of
systems which combine a coherent source with a dissipative sink of energy and possess a
chaotic dynamics to “process” the energy flow may eventually allow to control chemical
reactions on the molecular level (cf. Section 5.7).
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Author Index
x.y ndicates that the corresponding name appears in reference у of chapter x as author or edo-n
Abrahams, E. 1.7
Abrimowitz, M. 2.8
Abr kosov, A. A. 1.35, 1.60, 2.34
Acktrhalt, J. R. 6.91
Adachi, S. 6.96
Adhikari, S. K. 1.24
Affleck, I. 4.23
Affleck, I. K. 4.28
Agranovich, V. M. 1.19
Aharonov, Y. 1.53
Aht.ied, H. 1.18, 2.74
Akkermans, E. 1.20, 1.66, 2.92
Altshuler, B. L. 1.5, 1.11, 1.13. 1.19, 1.30,
1.37, 1.42, 1.54, 3.1
Ambegaokar, V. 3.6, 4.8, 4.16
Anastassakis, E. M. 2.46
Amerson, P. W. 1.7, 6.63
Amo, T. 2.15, 2.16
Ankerhold, .1. 4.25
Antonsen Jr., T. M. 6.95
Aoki, H. 2.2
Apcl, W. 2.20, 2.56, 2.66
Arazind, P. K. 5.27
Argunan, N. 1.34, 1.62, 6.46, 6.69
Aroiov, A. G. 1.5, 1.42. 1.47, 1.54, 1.55, 1.56
Aro/as, D. P. 2.68
Ashtroft, N. W. 1.1, 6.62
Ash >ori, R. 2.19
Ask.mazov, L. G. 3.42
Averin, D. V. 3.1, 3.5, 3.23, 3.40, 3.44
Balian, R. 1.2
Ballentine, L. E. 6.54
Bar ingcr, H. U. 1.36, 6.83. 6.87
Bar Irak. A. D. 5.8
Bas in. A. 2.37
Bauernschmit.t, R. 3.43
Bavli, R. 5.10
Beerakker, C. W. J. 1.17, 1.21. 1.68, 2.73, 6.90
Belitz, D. 1.3, 1.43
Ben-Jacob, E. 3.11, 3.12
Benoit, A. D. 1.79
Bensch. F. 5.6
Ber'hmd, P. 2.48
Ber ;maim, G. 1.38
Berrmann. R.. 6.84
Berry, M. V. 6.13, 6.14, 6.22. 6.26, 6 29, 'G
6.42, 6.44, 6.47, 6.110
Betbeder-Matibet, O. 2.37
Biese, G. 2.94
Bishop, D. J. 1.59
Bloch, F. 5.17
Bliimel, R. 6.80, 6.97
Bobbert, P. A. 3.34
Bohigas, O. 6.41, 6.49, 6.51, 6.52
Bonci, L. 6.56
Bonig, L. 1.28
Borkovec, M. 4.21
Bortfeldt, J. 2.36
Brandes, T. 1.51, 2.81, 2.83
Bransden, В. H. 2.7
Brezin, E. 2.14
Brody, T. A. 6.39
Bruder, C. 3.8, 3.9, 3.45
Brumer, P. 5.8
Bubanja, V. 3.18
Buchleitner, A. 6.97
Biittiker, M. 1.16, 1.49, 1.78, 2.40, 6 Gi. 
Bychkov, Yu. A. 2.54
Caldeira, A. O. 3.14, 3.49, 4.7
Calleja, J. S. 2.93
Callen, H. B. 4.12
Casati, G. 5.3, 5.7, 6.11, 6.43, 6.73
Castellani, C. 1.6
Cerdeira, H. 2.75, 3.10, 5.9
Chakraborty, T. 2.59
Chakravarty, S. 1.33, 4.1
Chang, A. M. 2.48
Chang, L. L. 2.19
Chen, C. D. 3.35
Chirikov, В. V. 6.11, 6.43, 6.72, 6.73, 6.74
Chu, S.-I. 5.2, 5.18
Claeson, T. 3.35, 3.41
Clark, T. D. 3.38
Cohen, D. 6.98
Creagh, S. C. 6.15, 6.61
Crommie, M. F. 1.26
Cummings, F. W. 5.41
Dakhnovskii, Yu. 5.10. 5.48
Davis, M. J. 6.50
360 Author Index
de Gennes, P. G. 1.48
Dekker, H. 4.3
Delsing, P. 3.35, 3.41
De Luccia, F. 4.28
den Nijs, M. 2.49
Dennis, B. S. 2.93
Devoret, M. H. 3.2, 3.24, 3.26, 3.32, 3.37
Devreese, J. T. 2.90
Di Castro, C. 1.6
Didsi, L. 6.101
Dietl, T. 2.81
Dion, D. R. 5.27
Dittes, F.-M. 6.46
Dittrich, T. 5.9, 5.19, 5.48, 6.43, 6.45, 6.53.
6.70, 6.94, 6.106, 6.107, 6.108
Dolan, G. J. 1.44
Doniach, S. 1.23
Dorda, G. 2.35
Doron, E. 6.46, 6.79
Dorsey, A. T. 4.1
Drouffe, J. M. 1.41
Dunlap, D. H. 5.45
Duplantier, B. 1.52, 2.26
Dyson, F. J. 6.48
Dzyaloshinski, I. E. 1.35
Eberl, K. 2.94
Eberly, J. H. 5.6
Eckern, U. 3.6, 4.8
Eckhardt, B. 6.12
Econoniou, E. N. 2.10
Efros, A. L. 1.5
Ehrenreich, H. 1.21
Eigler, D. M. 1.26
Eiles, T. M. 3.26
Einstein. A. 4.15, 6.25
Eliashberg, В. M. 2.54
Elliott, R. J. 2.11
Emery, ,1. 2.90
Engel, L. W. 2.45
Ernst, ,J. 6.81
Ertl, T. 6.36
Esteve, D. 3.24, 3.32
Fainshtein, A. G. 5.4, 5.15
Falci. G. 3.18, 3.46
Fazio, R. 3.8, 3.10, 3.45
Feit. M. D. 5.20
Feng, S. 1.14
Feynman. R. P. 3.48. 4.17, 4.18. 4.19
Filipe. A. 3.32
Fisher. D. S. 2.98
Fisher, M. P. A. 2.101. 4.1
Fishman, S. 6.75
Fleck Jr., J. A. 5.20
Floquet, G. 5.13
Flores, J. 6.39
Ford, G. W. 4.4
Ford, J. 6.73
Forster, D. 1.40
Fowler, A. B. 2.16
Foxon, С. T. 2.73
Freidkin, E. 4.30
French, J. B. 6.39
Frenkel, A. 6.79
Friedberg, R. 2.18
Frost, J. E. F. 1.18, 2.74
Fukuyama, H. 1.61, 2.52. 2.82
Gardiner, C. W. 6.99
Garg, A. 4.1
Gavrila, M. 5.6
Geerligs, L. J. 3.34
Geisel, T. 6.33, 6.65
Giannoni, M.-J. 6.10, 6.18, 6.35, 6.41, 6.44,
6.49. 6.78
Girvin, S. 2.4
Girvin, S. M. 2.61, 2.64
Gisin, N. 6.101
Gisselfalt, M. 3.31
Glazman, L. I. 1.70, 2.78, 2.80. 3.13, 3.19,
3.30, 3.31, 3.33
Goda, M. 5.46
Goetsch, P. 5.34
Golubev, D. S. 3.50
Gomez Llorente, .1. M. 5.33. 5.36. 5.40, 6.55
Goni, A. R. 2.93
Gorkov, L. P. 1.8, 1.35
Gornik, E. 2.19
Gossard, A. C. 2.19, 2.47. 6.82, 6.88
Grabert, H. 3.2, 3.51, 4.13, 4.14. 4.20, 1.25,
4.27
Graham, R. 5.34, 6.66, 6.93. 6.94, 6.97. 6.103,
6.107, 6.108
Grambov, P. 2.94
Gredeskul, S. A. 1.29
Grempel. D. R. 6.75
Grifoni, M. 5.48, 5.49
Grigolini, P. 6.56, 6.57
Grinstein, G. 1.10
Grobe, R. 6.105
Gross, D. .1. 2.14
Grossmann, F. 5.9, 5.19, 5.35, 5.47
Guarneri, I. 5.7
Guinea, F. 3.20
Gurvitz, S. A. 4.29
Author Index 36
Gutzwiller, M. C. 6.7, 6.34, 6.35
Haake F. 5.6, 6.8, 6.105
Hajdu J. 2.3
Haken H. 6.8
Hakim, V. 4.16
Haldane, F. D. M. 2.65, 2.88
Halperin, В. I. 2.42, 2.53, 2.59, 2.71
Hamann, D. R. 1.27
Hainmerich, A. D. 5.43
Hangg , P. 4.30
Hangg , P. 4.21, 4.24, 5.9, 5.19, 5.26, 5.35,
5.47, 5.48, 5.49, 6.53, 6.106, 6.107
Hanke W. 1.39
Hannay, .1. H. 6.17
Hansen, J. P. 1.67
Hanson, J. D. 6.95
Harada, Y. 3.35
Hashitsumc, N. 2.22
Hasko. D. G. 1.18, 2.74
Haug, R. J. 2.44
Haussi r, O. 6.81
Haviland, D. B. 3.35, 3.41
Heider rcich, R. 2.89
Heitmann, D. 2.19, 2.94
Hekking, F. W. .1. 3.5, 3.19, 3.21, 3.30
Hellen an, R. H. G. 6.1, 6.14
Heller, E. .1. 6.31, 6.32, 6.50
Herget rot.her, .1. M. 3.22, 3.25
Heuser, M. 2.9
Hibbs, A. R. 4.18
Hillery, M. 6.20
Hillmer, H. 2.81
Hirsch'elder, J. O. 5.27
Ho, T.-S. 5.18
Ho, Y. K. 5.47
Hoile. A. 6.36
Holthrus, M. 5.30, 5.32, 5.46
Honda, T. 2.84
Hone, D. W. 5.30
Hout,si ha. W. 4.24
Hopkins. P. F. 6.82, 6.88
Howland, J. 5.22
Hiibner, R. 6.107
Huckestein, B. 2.28, 2.29, 2.30. 2.31. 2.46
Husimi, K. 5.24
Hwang, .1. С. M. 2.48
lengo, R. 2.72
Ikeda, K. 5.46, 6.96
Imry, 7. 1.10, 1.49. 1.53, 1.55, 1.62. 1.64, 1.71,
6.69
Ince, E. L. 5.13
Ingold, G.-L. 4.14, 4.20, 4.25
looss, G. 6.1, 6.14
lordanskii, S. V. 2.54
Ismail, K. 2.81
Itzykson, C. 1.41, 2.14
Izrailev, F. M. 6.56, 6.64, 6.73, 6.74, 6.77
Jain, .1. K. 2.69
Jaklevic, R. C. 3.12
Jalabert, R. A. 6.83, 6.90
Jauho, A. 5.46
Jaynes, E. T. 5.41
Joachain, C. J. 2.7
Joannopoulos, J. D. 2.46
Jones, G. A. C. 1.18. 2.74
Jonson, M. 2.80, 3.31
Jorna, S. 6.13
Joyez, P. 3.24. 3.32
Jung, P. 5.9, 5.19
Jung, R. 4.14
Кае. M. 4.4
Kaminski, J. Z. 5.0
Kane, C. L. 1.77, 2.101
Kasner, M. 2.56, 2.66
Kastner, M. A. 1.59
Kauppinen, J. P. 3.17
Kawabata, A. 2.79
Kayanuma, Y. 5.38
Keating, J. P. 6.46
Keller, K. 2.94
Kenkre, V. M. 5.45
Kerner, F. H. 5.24
Ketzmerick, R. 6.65
Khrnel’nitskii. D. E. 1.70, 2.78
Khmelnitskii. D. E. 1.8, 1.11, 1.54
Kirkpatrick, T. R. 1.3. 1.43
Kitaev, A. Yu. 6.16
Kittel, C. 1.25
Kleinert. H. 4.18
Koch, S. 2.44
Kohler, S. 6.107
Kohmoto, M. 2.49
Kohn, W. 165
Kolovsky, A. R. 6.66
Konig, J. 3.10
Kopaev, Y. V. 1.39
Korsch, II. J. 5.6, 5.17
Kosloff, D. .1. 5.21
Kosloff. R. 5.21, .5.43
Kotliar, G. 1.6
Kouwenhoven. L. P. 1.17, 2.73
362 Author Index
Kramer, B. 2.20, 2.27, 2.28, 2.32, 2.33, 2.36,
2.38, 2.46, 2.70, 2.75, 2.76, 2.81, 2.83,
2.95, 2.99, 3.10, 5.9
Kramers, H. A. 4.26
Kravtsov, V. E. 1.37
Krummhansl, J. A. 2.11
Kubo, R. 2.22, 2.96, 4.11
Kuchar, F. 2.46, 2.81
Kurdak, C. 2.45
Kuzmin, L. S. 3.41
Lafarge, P. 3.24, 3.32
Laibowitz, R. B. 1.50, 1.58, 1.79
Landau, L. D. 2.5
Landauer, R. 1.9, 1.15, 1.22, 1.49, 1.76, 2.39,
6.85
Landwehr, G. 2.33
Langer, .1. S. 4.22
Larkin. A. I. 1.8. 3.42
Lassnig, R. 2.19
Latka, M. 6.57
Laughlin, R. B. 2.55, 2.57, 2.62, 2.67
Leath, P. L. 2.11
Leboeuf, P. 6.67
Lechner, K. 2.72
Lee, H.-W. 6.21
Lee. K. Y. 2.81
Lee, P. A. 1.3, 1.6, 1.12, 1.14, 1.19, 1.61, 1.77.
2.53, 2.59, 2.82, 2.98, 3.1, 6.71
Leggett, A. J. 3.14, 3.49. 4.1, 4.7
Lerner, I. V. 1.37
Lesovik, G. B. 1.70, 2.78
Levine. R. D. 5.23
Levinson, I. B. 1.73
Lewiner, C. 2.37
Licciardello, D. C. 1.7
Lichtenberg, A. J. 6.2
Licini, J. C. 1.59
Lieb. E. H. 2.87
Lieberman, M. A. 6.2
Lif.shits, I. M. 1.29
Lifshitz, E. M. 2.5
Likharev, К. K. 3.1, 3.39
Lim, K. Y. 2.46
Lin. W. A. 6.54
Littlejohn, R. G. 6.15. 6.27
Lloyd. P. 2.12
Louisell, W. H. 6.92
Lu. J. G. 3.22
Luttinger. .1. M. 2.18, 2.86
Lutz. С. P. 1.26
Maassen v. d. Brink, A. 3.34
MacDonald, A. H. 2.64, 2.70
MacKinnon, A. 2.27, 2.32, 2.38, 2.41
Magalinskii,, V. B. 4.5
Magnus, W. 5.13
Mahan, G. D. 2.97
Main, J. 6.36
Makarov, D. E. 5.40
Manakov, N. L. 5.1, 5.4, 5.15
Maradudin, A. A. 1.19
Marcus, С. M. 6.82, 6.88
Marcus, R. A. 6.28
Martinis, J. M. 3.26
Masek, J. 2.76
Mattis, D. C. 2.87
Matveev, K. A. 3.13, 3.19. 3.31. 3.33
Mayer-Kuckuk, T. 6.81
Mazenko, G. 1.10
Mazur, P. 4.4, 4.9
McDonald, I. R. 1.67
Megaloudis, G. 3.38
Mehlig, B. 6.70
Mehta, M. L. 6.40
Meiscls, R. 2.46
Mello, P. A. 6.39, 6.87
Melngailis. J. 1.59
Menschig, A. 6.84
Mermin, N. D. 1.1, 6.62
Messiah, A. 2.6
Metiu, H. 5.10
Millikan, F. P. 2.102
Milonni, P. W. 6.91
Miyazaki, S. 6.66
Moiseyev, N. 5.6, 5.23, 5.47
Molinari, L. 5-3
Montambaux, G. 1.20, 1.66. 2.92
Montroll, E. 4.9
Mosser, V. 2.19
Mostowski, .1. 5.6
Mottola, E. 3.11
Mouchet, A. 6.67
Mount, К. E. 6.26
Mullen, K. 3.12
Myiake, S. I. 2.22
Nazarov, Y. V. 1.69
Nazarov, Yu. V. 3.21, 3.23. 3.40, .3.41, 3.43.
3.44
Neu, P. 5.40
Newbury, R. 1.18, 2.74
Nightingale, M. P. 2.49
Noid, D. W. 6.28
Nozieres. P. 1.4, 2.37, 2.85
Author Index 36.
O’Coi nell, R. F. 6.20 O’Cornor, P. W. 6.31 Odintsov, A. A. 3.15, 3.34, 3.40, 3.43 OelscHagel, B. 5.48, 6.106 Ohtsuki, T. 2.25, 2.33 Olschowski, P. 4.27 Ono, Y. 2.33 Osheroff, D. D. 1.44 Ott, E. 6.4, 6.95 Ovchinnikov, Yu. N. 3.42 Ovsiannikov, V. D. 5.1, 5.4 Ozorit de Almeida, A. M. 6.6, 6.17	Reichl, L. E. 6.9 Reinhardt, W. P. 6.19 Reiss, H. R. 5.5 Rezayi, E. H. 2.65 Rice, S. A. 5.8 Richter, A. 6.81 Richter, K. 6.84 Rimberg, A. .1. 6.82 Riseborough, P. S. 4.30 R.isken, H. 6.104 Ristow, G. H. 5.30 Ritchie, D. A. 1.18, 2.74 Ritus, V. I. 5.12
Paalai en, M. A. 2.43 Pandev, A. 6.39 Panyu-cov, S. V. 3.16, 3.50 Pastui, L. A. 1.29 Payne M. C. 1.74 Paz, J P. 6.30 Peacot k, D. C. 1.18, 2.74 Pekola, .1. P. 3.17 Poppe ’, M. 1.18, 2.35, 2.74 Percival, I. C. 6.18, 6.101 Perelo nov, A. M. 5.25 Peskin, U. 5.23 Petsch.d, G. 6.65 Pfeifer, P. 5.23 Pfeiffe-, L. N. 2.93 Pichari. J.-L. 1.20, 2.92, 6.90 Pietila.nen, P. 2.59 Pillet, C.-A. 6.100 Pincz.uk, A. 2.93 Pines, D. 1.4 Plata, J. 5.33, 5.36, 5.40. 6.55 Platzn an. P. M. 2.64 Ploog, K. 2.19 Pokrovsky, V. L. 2.60 Pollak M. 1.5 Popov V. S. 5.25 Pranct, H. 3.38 Pranci, R. J. 3.38 Prangt, R. 2.4 Prange, R. E. 6.75 Prokhorov, A. M. 5.7 Pruisk.'ii, A. iM. M. 2.43	Robbins, ,J. M. 6.15, 6.16. 6.60, 6.110 Robnik, M. 6.47 Roncaglia, R. 6.56 Rotvig, .1. 5.46 Rubin, R. .1. 4.10 Rubner, ,1. 6.33 Ruder, H. 6.36 Ryzhkin, I. A. 2.34 Saito, N. 6.23 Saku, T. 2.84 Saleur, H. 2.26 Sarabe, H. 5.14 Sassetti, M. 2.95, 2.99, 5.48 Schanz, H. 6.70 Scharf, R. 6.68 Schlapp, W. 2.46, 2.81 Schlautmann, M. 6.103 Schmid, A. 1.33, 1.57, 3.28 Schmit, C. 6.49 Schoeller, H. 3.10, 3.47 Schon, G. 2.75, 3.5, 3.6, 3.7. 3.8. 3.10, 3. i 3.18. 3.20, 3.27, 3.28. 3.30, 3.34.  Y 3.45, 3.46, 3.17, 4.8, 5.9 Schonhammer, K. 1.28 Schramm, P. 4.20 Schrieffer, J. R. 2.68 Schiiller, C. 2.94 Schulman, L. S. 1.32. 2.17, 1.18 Schulz, H. .1. 2.92 Schuss, Z. 3.12 Schuster, H. G. 6.3 Schweitzer, L. 2.32, 2.38
Rabi, 1. I. 5.28 Radon ;, G. 6.33 Ralph, D. C. 3.22 Ralph, ,1. F. 6.102 Ramakrishnan. T. V. 1.3, 1.7, 6.71 Ramm ;r, J- 157 Rapop jrt. L. P. 5.1. 5.4, 5.15	Schweizer, H. 6.84 Schwinger, .1. 5.28 Scully, M. 6.20 Seiler, R. 2.89 Senitzky, 1. R. 4.4 Scrota, R. A. 1.77 Shahar, D. 2.45
3f Au [nde..
Shakeshaft, R. 5-7
Shao, J. 5.37
Shapiro, M. 5.8
Sharvin, Y. V. 1.47, 1.72
Shaw, R. 6-5
Shekhter, R. I. 1.70, 2.78, 3.19, 3.31, 3.33
Shepelyansky, D. L. 5.7, 6.74, 6.76, 6.77
Shih, M.-L. 6.91
Shirley, J. H. 5.11
Shirts, R. B. 6.19
Shklovskii, В. I. 1.30
Sieber, M. 6.46
Siegert, A. 5.17
Siewert, J. 3.27, 3.36, 3.43
Silbey, R. J. 5.40
Silsbee, R. H. 2.19
Sirko. L. 6.97
Sivan, U. 1.55
Smilansky, U. 1.62, 6.43, 6.46, 6.69, 6.70, 6.78,
6.79, 6.80, 6.97
Smith, H. 5.46
Solyom, J. 2.91
Sommers, H. .1. 6.105
Sondheimer, E. H. 1.23
Spies, L. 2.20
Spiller, T. P. 6.102
Spivak, B. Z. 1.13
Stauffer, D. 2.23
Stegun, I. A. 2.8
Steiger, A. 5.20
Steinebach, C. 2.94
Stern, A. 1.53
Stern, F. 2.16
Stone, A. D. 1.12, 1.14, 1.36, 1.61, 1.63, 2.82,
6.83
Stora, R. 6.1, 6.14
Stormer, H. L. 2.19, 2.47, 2.48
Strasser, G. 2.19
Strinati, G. 1.6
Su. Q. 5.6
Simdaram, 13. 6.68
Tabor, M. 6.38
Takahashi, K. 6.23
Talapov, A. L. 2.60
Talkner, P. 4.13, 4.21
Tannor, D. J. 5.8, 5.42, 5.43
Tarucha, S. 2.84
Tel. T. 6.94
Tersoff. J. 1.27
Thornton. T. J. 1.18, 2.74
Thouless, D. .1. 1.45, 2.49
Tighe, T. S. 3.2.5
Timp, G. 1.80
Tinkham, M. 3.4, 3.22, 3.25, 3.28, 6.59
Toda, M. 6.96
Tomsovic, S. 6.31, 6.51, 6.52, 6.58
Trugman, S. A. 2.24
Tsidil’kovskii, I. 2.51
Tsui, D. C. 2.43, 2.45, 2.47, 2.48
Tsukada, M. 2.21
Tuominen, M. T. 3.22, 3.25
Turnbull, D. 1.21
Uemura, Y. 2.15
Uhlenbrock, A. 2.89
Ullersma, P. 4.4
Ullmo, D. 6.51, 6.52, 6.58
Ulreich, S. 1.75
Umbach, С. P. 1.50, 1.58, 1.79, 2.102
Urbina, C. 3.24
Utermann, R. 6.53
van der Marel, D. 1.17, 2.73
van Houten, H. 1.17, 1.21, 2.73
van Kampen, N. G. 1.31
van Otterlo, A. 3.45
van Wees, B. J. 1.17, 2.73
Vernon Jr., F. L. 3.48, 4.19
Vollhardt, D. 1.39
von Brentano, P. 6.81
von Klitzing, К. 2.1. 2.19, 2.35, 2.44, 6.84
von Neumann, J. 5.16
von Witsch, W. 6.81
Voros, A. 6.10, 6.18, 6.24, 6.35, 6.41, 6.44, 6.78
Wagner, M. 5.31, 5.44
Walther, H. 6.97
Wang, L. 5.37
Wang, X. 3.51
Washburn, S. 1.46. 1.50, 1.58. 1.79
Webb, R. A. 1.19, 1.46, 1.50, 1.58. 1.79, 2.102,
3.1
Wegner, F. 2.13
Wei, H. P. 2.43
Weimaim, G. 2.19, 2.46, 6.84
Weiner, ,1. S. 2.93
Weiss, D. 2.19. 6.84
Weiss, U. 3.29, 4.2, 4.13, 1.27. 5.48
Welge, К. H. 6.36
Welton, T. A. 4.12
Wen, X. G. 2.100
West, B. .1. 6.56, 6.57
West, K. W. 2.93
Wcsterveld, R. M. 6.82. 6.88
Wharain, D. A. 1.18, 2.74
Author Index
Whelan. N. D. 6.61
Widom, A. 3.38
Wiebusch, G. 6.36
Wiedeminn, H. 5.6
Wiegma.m, W. 2.19
Wigner, E. 5.16
Wigner, E. P. 2.50, 6.20
Wilczek, F. 2.68
Wilkinscn, M. 6.109
Williamson, ,J. G. 1.17, 2.73
Winkler, S. 5.13
Wintgen D. 6.37
Wolfle, F. 1.39, 1.56
Wong, S S. M. 6.39
Wrobel, J. 2.81
Wunner, G. 6.36
Yacoby, A. 1.71
Yamada, H. 5.46
Yamada, K. 6.97
Yoshioka D. 2.52, 2.53, 2.58, 2.59, 2.63
Zaikin, A. D. 3.7, 3.16, 3.27, 3.50
Zak, I. 5.29
Zee, A. 2 68
Zeldovitc i, Ya. B. 5.12, 5.18
Zeller, G. 6.36
Zerbe, C. 5.26
Zhao, X. G. 5.39
Ziesemann, M. 2.19
Zimanyi, G. T. 3.46
Zinn-Justin, J. 1.20, 2.92, 6.10, 6.18, 6.35,
6.41, 6.44, 6.78
Zorin, A. B. 3.39
Zurek, W. H. 6.30
Zwanzig, R. 4.6
Zwergcr, W 1.28, 1.51, 1.75, 4.1

Subject Index
above threshold ionization, 250
actior
c assical, 291, 298, 303, 308, 335
elective, 239
Euclidean, 232
action-angle variables, 303, 309, 330
adiabatic theorem, 275
Aharcnov-Bohm effect, 34, 37
ampin ude
n flection, 338
r< turn, 307, 311, 326
tiansmission, 338
Anderson transition, 330
Andreev
rt flection, 174, 193
spectroscopy, 184
antidot, 337
appro: -.imation
diagonal, 312
Markov, 342
semiclassical, 231, 241, 290 319. 322 330,
335, 346
backscattering, 24, 132
band, 324 329, 332
Bargn aim representation, 298
billiard
electron, 336 -341
Sinai, 288
Bloch
pl ase, 325
theorem, 324
Bloch-Siegert shift, 267
Bohr-8 ommerfeld quantization, 298
Boltzmann equation. 1
Boltzmann, Ludwig. 287
Boson. 124, 137
ai satz, 125
statistic, 124
boundary conditions, 80, 84, 85, 104
hard wall, 85
p< riodic, 80, 127, 325
boundary layer, 321
boundary resistance. 63
Brillouin zone. 325
bulk state. 100
ex' ended, 106
Im alized, 100. 106, 108
Caldeira-Leggett Hamiltonian, 21
cantorus, 293, 294, 305. 321
caustic, 299-301
channel, 335, 338
closed, 338
open, 338
chaos
hard, 292
soft, 292
character, 322, 325
charge
fluctuations. 207
representation, 200
charging
effects, 149
energy, 153
chiral states, 86
COE, 335
coherence, 109
coherent
Cooper-pair tunneling, 183
state, 296
amplitude, 298
transport. 273
tunneling, 27.3
complex trajectory. 318. 329
composite Hilbert space, 253, 254
conductance, 79. 131, 338 341
anti-resonance. 133
coefficients. 104
depletion, 133
fluctuation. 1.33
frequency dependent. 140
global, 104
oscillations, 208
quantum, 126. 141
step, 131
confinement, 84
energy. 127
geometrical, 79
harmonic. 82, 83. 86
potential. 84
constriction, 128, 129. 131. 132
Cooper-pair tunneling. 183
Cooperon. 24, 48 50
correlation
function, 224 228
hole, 313, 314, 328
36» Subject Index
cotunneling, 161, 204
Coulomb
blockade, 149, 156
interaction, 135, 139
crossing
avoided, 255 , 276 , 316 , 321
exact, 255, 266, 274, 277, 278, 316
three-level, 321
crossover temperature, 242-247
CSE, 335
CUE, 335
cyclotron
frequency, 80
orbit, 86
damping kernel, 216-219, 239
deBroglie wavelength, 299
decay rate, 244-247, 334, 340, 346, 348, 349
decoherence, 339 -340, 342, 344-345, 347
deflection function, 334
density
mean spectral, 308
of states, 8, 89, 133, 306, 325
fluctuations. 14
local, 10, 332
mean, 308
operator, 294, 341
quasi-probability, 295
spectral, 306, 325
tunneling spectral, 323
density matrix, 198, 219-220
equilibrium, 220, 222, 231-233
dephasing, 40, 41, 339-340, 342, 344-345, 347
length, 337, 345
destructive interference, 281
determinism, 287-290
diagrammatic representation, 90
diffusion, 228-229, 320 , 323, 326 . 331, 344
chaotic, 324
deterministic, 324
diffusion constant, 229, 331, 333
diffuson, 48-50
disorder, 88, 89, 92-94, 96, 97, 99, 102, 105,
108, 109, 132, 324, 327-329, 337
dissipation, 213-247, 283, 289, 338, 341-350
Ohmic, 214, 217, 226, 240, 343
distribution
curvature, 317
delay-time, 334, 336
marginal, 295
nearest-neighbor level-spacing, 314, 315.
347
winding-number, 326
doublet, 318-323, 348
driven
bistable system, 273
parabolic barrier, 263
quantum system
exactly solvable, 261- 269
tunneling, 266, 274, 280, 320
adiabatic limit, 275
two-level system, 251, 263-268
driving
periodic, 319, 331, 337, 341, 347
random, 342
Drude model, 103, 128, 133, 217, 240
dynamic susceptibility, 223-225
Dyson’s threefold way, 314
Dyson-Mehta theorem, 57
EBK quantization, 298. 303. 318
edge state, 86, 99, 106
model, 105
Edwards model, 15
effective action, 239
in imaginary times, 189
in real times, 198
effective-action description of tunneling, 189
Einstein relation, 3, 229
Einstein, Albert, 298
Einstein-Brillouin-Keller quantization, 298,
303, 318
electromagnetic environment, 163
electron turnstile, 155, 288
energy shell, 303, 307
ensemble
circular orthogonal, 335
circular symplectic, 335
circular unitary, 335
Gaussian orthogonal, 55, 314 316
Gaussian symplectic, 314 -316
Gaussian unitary, 314 316
microcanonical, 304
entropy, 345
equation
continuity, 299
master, 158, 343
equipartition rule, 62
ergodicity, 288, 291, 305, 313
Ericson fluctuations, 336
even-odd asymmetry, 177
Fermi
energy, 130, 133, 135, 337, 338
wavelength, 129, 130. 337
wavevector. 135
Subject Index 369
Fermion. 114, 136
ansatz, 125
composite, 123, 126
opeiator, 137
symmetry, 125
Feynmar path, 40
fictitious lead, 339-340
Floquet
exponent, 252, 254, 255, 263, 269, 271
form, 269
mat ix, 270
mode, 252, 253, 255, 258, 263, 278
generalized, 259
state, 252, 254, 331, 332, 349
theorem, 252, 331, 347
theory, 252-261
generalized, 258-259
m ilti-mode, 272
fluctuating force, 216
fluctuation-dissipation theorem, 213, 224-225,
344
flux conservation, 338
Fokker-Planck equation, 344, 346
form factor, spectral, 312
four-prol e measurement, 72
fractal, 290, 292, 293, 324, 333, 334, 341, 346,
348, 349
free dam red particle, 215, 228 229
free energy, imaginary part of, 244
Friedel oscillations, 9, 11, 12
Frobenius-Perron operator, 326
function
cluster, 310, 313, 314, 329
deflection, 333
spec .ral two-point correlation. 310. 313,
311, 329
gate
charge, 153
voltage, 151
Gaussian white noise, 15, 88, 89
Ginibre ensemble, 347
GOE, 311 316
Green function, 15-20, 89, 90. 93, 99. 106, 306
advanced, retarded, 16
group theory, 322, 325
GSE, 314 316
GUE, 314 316
Gutzwilkr trace formula, 308, 322, 325. 326
Hall
conductance, 101
conductivity, 103, 105
plateau, 105
resistance, 101
resistivity, 103
voltage, 105
Hamilton-Jacobi equation, 299
harmonic oscillator
damped, 214, 225-228
driven, 261-263
heat bath, 213, 216, 217, 232, 234, 341
Heisenberg time, 308, 328
Hellmann-Feynman theorem, 254
Hermitecity, 342
higher-order tunneling processes, 189. 192
horizon, 172
Husimi distribution, 297, 341
imaginary part of free energy. 244
impurity, 88, 337
incoherent Cooper-pair tunneling, 188
influence functional. 234. 237, 239
integrable systems. 291, 298, 300, 309, 312,
313, 316
interaction
electron-electron. 3. 41, 110, 135, 352
electron-phonon, 134
invariance
discrete translational. 323 325, 332
time-reversal. 312, 314, 324, 326, 330,
341, 345, 347
invariant
manifold, 290, 303
measure, 303
subspace. 322
irreversibility. 289. 345, 347
Jaynes-Cummings model, 281
Josephson
current, 194
tunneling, 192
kicked rotor
classical, 293. 327, 330 331, 348, 349
quantum-mechanical, 324, 327, 330 333.
348, 349
Kolmogorov-Arnol'd-Moser (KAM) theorem.
292
Kubo-Greenwood formula, 20. 21
Landau
band, 93, 97. 99
energies, 95
energy, 89
gauge1, 80, 84
370 Subject Index
level, 82, 89
model, 80
state, 87, 95, 97, 98
Landauer, 104
dipole, 1113
formula, 7, 62
Landauer-Biittiker
formalism, 60-74
formula, 339-340
Langevin equation, 214, 343, 346
Laplace demon, 287
level
attraction, 322, 329
clustering, 322, 329
repulsion, 55, 310, 313, 321, 329
cubic, 347
linear, 315-316
quadratic, 315-316
quartic, 315-316
limit
classical, 296, 305, 333, 335, 337, 345
high-teinperature, 343
long-time, 328, 333, 343, 345-350
linear response, 20-24, 103, 104, 128, 140,
141, 221- 222, 339
linewidth, 208, 336, 340, 346, 348, 349
Liouville
equation, 289, 302
operator, 220 222
localization, 32, 88, 94, 279
and transport, 99
Anderson, 324, 329
delocalization transition, 99, 330
distance, 97
dynamical, 331 -333, 349
length, 99, 331-333
model, 108, 109
weak. 24 44, 134, 312, 326, 327
long-time dynamics. 257
I.orentzian lineshape, 336
Luttinger liquid, 135
conductance of, 139
elementary excitations, 138, 141
Lyapunov exponent, 302
magnetic
field, 79, 84, 314, 325, 341. 347
flux density, 85
length, 80, 86
magnct.oresist.ance, 36, 339. 341
map, 292, 330, 342
stochastic, 346
Maslov index, 292. 298, 300
matrix
monodromy, 292, 308
scattering, 335, 338
stability, 292, 308
Matsubara frequency, 226-228, 238, 240
mean free path
elastic, 337
inelastic, 337, 345
memory, 342
mesoscopic, 5
metastable state, 213, 240, 241
decay of, 240-247
Moyal bracket, 301, 344, 346
multiprobe
conductor, 64
geometry, 6
noise, 342-345
quantum, 344
thermal, 344
number
level, 310, 311
quantum, 298, 335
winding, 292, 325
odd electron, 177
Onsager symmetry, 1, 65, 72
orbit
hyperbolic, 292
inverse-hyperbolic. 292
periodic, 291, 304, 307, 312, 322
generalized, 322, 325, 326
primitive, 308
self-retracing, 312
parity, 318, 320, 322, 338, 348
parity effects, 176
partition function, 220, 233, 244 246
path integral, 229 234
imaginary-time, 232
real-time, 230
path-integral formulation of tunneling. 189.
240-247
periodic-orbit sum, 308, 322, 325. 326
perturbation
monochromatic, 254
polychromatic, 259
phase-breaking length, 109
phase-coherence length, 31. 10
Poincare recurrence time. 216
point
diabolic. 316, 317
turning, 299, 319
Subject Index
Poissoi
bracket, 301, 346
distribution, 315, 329
positivity, 342
potent al
cubic, 240
dcuble-well, 317-320, 347
M rrse, 309
princip le
co rrespondence, 289, 325
uniformity, 292, 313
probability
reflection, 338
return, 276, 280, 327
transmission, 338-340
propagitor, 16, 230, 256, 268
classical, 326
over full period, 256
quantum-mechanical, 300, 306
spectral representation, 258
stroboscopic, 273
proxim ty effect, 196
quantization
of 'onductance, 126, 127, 132, 134, 139
of energy, 82, 127
of Hall conductivity, 105, 110, 111
of „ransport, 79, 132
toius, 298, 303, 318
quantum
coherence, 273
coi trol, 282
dot, 83, 337, 339
fluctuations, 208
map, 256, 258
measurement, 289, 296, 333
point contact, 7, 62-66, 127
tin noling, 213, 243, 245, 247
wiie, 79, 82, 83. 126, 127, 129, 138
quantum Hall effect. 79, 88, 99
fractional, 110, 118, 123. 142
integer, 100, 101, 105
quasidc >eneracy, 318 -323. 329. 348
quasien irgy, 252, 254, 255, 263, 269. 271. 331
quasimoment.um, 325
quasipe iodicity, 289, 303, 332, 346
quasiraudomness, 332
Rabi frequency, 264. 266, 267
random potential. 8, 14 -15
random matrix theory, 54 60, 314, 329, 335.
340, 346
real-tini ' evolution, 198
regime
ballistic, 337
diffusive, 328. 337
disordered, 337
regular island, 292
representation
irreducible, 322, 325
phase-space, 294
reservoir, see also heat bath, 63, 338, 340
resolvent, 16
response function, 222, 223, 226
rotating-wave, 251
approximation, 264
Rubin model, 218-219
S matrix, 335, 338
scar, 304, 305
scattering, 8-15
chaotic, 333 341
irregular, 333 341
Schrodinger equation, stochastic, 343
self-averaging, 14, 45 -47
self-similarity, 290, 292, 293, 324, 333, 334,
341, 346, 348, 349
separatrix, 317, 320
single-electron
box, 152
pump, 155
transistor, 154, 158
NSN, 180
NSS, 184
SNS, 194
tunneling, 149. 191
tunneling rate, 156
spectral
decomposition, 271
density of bath modes, 216. 217, 219.
form factor. 311, 31 I, 327 329
function, 224. 225
rigidity, 55, 317
spectrum, 136-138
continuous, 346
discrete, 289, 324, 331, 351
energy, 99, 118, 305 317
excitation. 118. 135. 141
local, 303. 332
point, 289, 324. 331, 351
Raman, 142
spin, 314, 339, 347
spontaneous transition. 31 1
stabilization. 219
strange
attractor, 290. 341, 3-16. 349. 350


372 Subject Index
repellor, 334
superoperator, 342
symmetric gauge, 81
symmetry
antiunitary, 312
discrete translational, 323-325, 332
reflection, 318, 320, 322, 338, 348
time-reversal, 312, 314, 324, 326, 330,
341, 345, 347
unitary, 322, 325
thermal
activation, 213, 243, 245
length, 44
Thomas-Fermi law, 308
Thouless
energy, 49
formula, 54
time. 327
time
break, 289, 303, 308, 328, 331, 350
decoherence, 345, 349
delay, 334
log, 302, 344
relaxation, 345
time reversal, 40
time translations, discrete, 252
time-evolution operator, 300, 330, 334, 342
torus, 291, 298, 303, 321, 348
irrational, 291, 293
quantizing, 298, 309
rational, 291, 292, 309, 312
vague, 293, 305, 321
trace formula, 308, 309, 322, 325, 326
transient, 334, 348
(M')-formalism. 259 -261, 272
tunnel splitting, 276, 277, 318 323, 348
renormalized, 275, 276
tunneling, 213, 243, 245, 247. 273, 317-323.
329, 338,348
chaos-assisted, 321, 322
coherent destruction, 250, 277, 280
dynamical, 320
inelastic, 189
inelastic resonant,, 204
quasiparticle, 173
resonant, 204
spectral density. 323
two-level system, 251. 263-268
two-state approximation, 280
UCF. -15-60, 340
uncertainty. 290. 34'2
energy-time, 308, 332
momentum-position, 296, 305, 345, 346
unfolding the spectrum, 310, 311
universal conductance fluctuations, 45-60, 340
Van-Vleck propagator, 300, 302, 307, 335
voltage, alternating-current, 337
von Neumann-Wigner theorem, 254
von-Neumann equation, 301
Ward identity, 23
wave, evanescent, 338
Wentzel-Kramers-Brillouin approximation,
298, 303
Weyl correspondence, 294
Wigner
function, 294, 296, 297, 301, 303. 341,
344, 346, 349
propagator, 302
representation, 294, 296, 297, 301, 303,
341, 344, 346, 349
semicircle law, 58
surmise, 56
WKB approximation, 298, 303