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MESOSCOPIC PHYSICS OF ELECTRONS AND PHOTONS

Quantum mesoscopic physics covers a whole class of interference effects related to thepropagation of waves in complex and random media. These effects are ubiquitous inphysics, from the behavior of electrons in metals and semiconductors to the propagation

of electromagnetic waves in suspensions such as colloids, and quantum systems like coldatomic gases. This book is a modern account of the problem of coherent wave propagationin random media.

As a solid introduction to quantum mesoscopic physics, this book provides a unifiedoverview of the basic theoretical tools and methods. It highlights the common aspects ofthe various optical and electronic phenomena involved. With over 200 figures, and exercisesthroughout, the book is ideal for graduate students in physics, electrical engineering, optics,acoustics and astrophysics. It presents a large number of experimental results that cover a

wide range of phenomena from semiconductors to optics, acoustics, and atomic physics. Itwill also be an important reference for researchers in this rapidly evolving field.

Eric Akkermans is Professor of Physics in the Department of Physics at the Technion,Israel Instituteof Technology, Israel. GillesMontambaux isDirecteurdeRechercheattheCNRS, Laboratoire de Physique des Solides, Universit Paris-Sud, France. Their researchinterests include the theory of condensed matter physics, mesoscopic quantum physics, andcoherent effects in the propagation of waves in random media.


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Mesoscopic Physics

of Electrons and Photons

Eric Akkermans

Technion, Israel Institute of Technology

Gilles Montambaux

CNRS, Universit Paris-Sud


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CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, So Paulo

Cambridge University PressThe Edinburgh Building, Cambridge CB2 8RU, UK

First published in print format

ISBN-13 978-0-521-85512-9

ISBN-13 978-0-511-29016-9

E. Akkermans and G. Montambaux 2007

2007

Information on this title: www.cambridge.org/9780521855129

This publication is in copyright. Subject to statutory exception and to the provision ofrelevant collective licensing agreements, no reproduction of any part may take placewithout the written permission of Cambridge University Press.

ISBN-10 0-511-29016-0

ISBN-10 0-521-85512-8

Cambridge University Press has no responsibility for the persistence or accuracy of urlsfor external or third-party internet websites referred to in this publication, and does notguarantee that any content on such websites is, or will remain, accurate or appropriate.

Published in the United States of America by Cambridge University Press, New York

www.cambridge.org

hardback

eBook (NetLibrary)

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Contents

Preface page xiii

How to use this book xv

1 Introduction: mesoscopic physics 1

1.1 Interference and disorder 11.2 The AharonovBohm effect 41.3 Phase coherence and the effect of disorder 71.4 Average coherence and multiple scattering 91.5 Phase coherence and self-averaging: universal fluctuations 12

1.6 Spectral correlations 141.7 Classical probability and quantum crossings 151.7.1 Quantum crossings 17

1.8 Objectives 18

2 Wave equations in random media 31

2.1 Wave equations 312.1.1 Electrons in a disordered metal 312.1.2 Electromagnetic wave equation Helmholtz equation 32

2.1.3 Other examples of wave equations 332.2 Models of disorder 36

2.2.1 The Gaussian model 372.2.2 Localized impurities: the Edwards model 392.2.3 The Anderson model 41

Appendix A2.1: Theory of elastic collisions and single scattering 43A2.1.1 Asymptotic form of the solutions 44A2.1.2 Scattering cross section and scattered flux 46

A2.1.3 Optical theorem 47A2.1.4 Born approximation 51

Appendix A2.2: Reciprocity theorem 54Appendix A2.3: Light scattering 56

A2.3.1 Classical Rayleigh scattering 56

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A2.3.2 Mie scattering 60A2.3.3 Atomphoton scattering in the dipole approximation 61

3 Perturbation theory 70

3.1 Greens functions 71

3.1.1 Greens function for the Schrdinger equation 713.1.2 Greens function for the Helmholtz equation 77

3.2 Multiple scattering expansion 793.2.1 Dyson equation 793.2.2 Self-energy 82

3.3 Average Greens function and average density of states 86AppendixA3.1: Short range correlations 88

4 Probability of quantum diffusion 92

4.1 Definition 924.2 Free propagation 954.3 DrudeBoltzmann approximation 964.4 Diffuson or ladder approximation 974.5 The Diffuson at the diffusion approximation 1024.6 Coherent propagation: the Cooperon 1044.7 Radiative transfer 110Appendix A4.1: Diffuson and Cooperon in reciprocal space 113

A4.1.1 Collisionless probability P0(q, ) 114A4.1.2 The Diffuson 115A4.1.3 The Cooperon 117

Appendix A4.2: Hikami boxes and Diffuson crossings 120A4.2.1 Hikami boxes 120A4.2.2 Normalization of the probability and renormalization of the

diffusion coefficient 125A4.2.3 Crossing of two Diffusons 128

Appendix A4.3:Anisotropic collisions and transport mean free path 132Appendix A4.4: Correlation of diagonal Greens functions 138Appendix A4.5: Other correlation functions 142

A4.5.1 Correlations of Greens functions 142A4.5.2 A Ward identity 145A4.5.3 Correlations of wave functions 145

5 Properties of the diffusion equation 148

5.1 Introduction 1485.2 Heat kernel and recurrence time 149

5.2.1 Heat kernel probability of return to the origin 1495.2.2 Recurrence time 151

5.3 Free diffusion 1525.4 Diffusion in a periodic box 155


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5.5 Diffusion in finite systems 1565.5.1 Diffusion time and Thouless energy 1565.5.2 Boundary conditions for the diffusion equation 1565.5.3 Finite volume and zero mode 1575.5.4 Diffusion in an anisotropic domain 158

5.6 One-dimensional diffusion 1595.6.1 The ring: periodic boundary conditions 1605.6.2 Absorbing boundaries: connected wire 1615.6.3 Reflecting boundaries: isolated wire 1625.6.4 Semi-infinite wire 164

5.7 The image method 165Appendix A5.1: Validity of the diffusion approximation in an infinite

medium 166Appendix A5.2: Radiative transfer equation 168

A5.2.1 Total intensity 168A5.2.2 Diffuse intensity 170A5.2.3 Boundary conditions 172A5.2.4 Slab illuminated by an extended source 175A5.2.5 Semi-infinite medium illuminated by a collimated beam 176

Appendix A5.3: Multiple scattering in a finite medium 177A5.3.1 Multiple scattering in a half-space: the Milne problem 177A5.3.2 Diffusion in a finite medium 180

Appendix A5.4: Spectral determinant 182Appendix A5.5: Diffusion in a domain of arbitrary shape Weyl expansion 184Appendix A5.6: Diffusion on graphs 187

A5.6.1 Spectral determinant on a graph 187A5.6.2 Examples 191A5.6.3 Thermodynamics, transport and spectral determinant 193

6 Dephasing 1956.1 Dephasing and multiple scattering 195

6.1.1 Generalities 1956.1.2 Mechanisms for dephasing: introduction 1966.1.3 The Goldstone mode 199

6.2 Magnetic field and the Cooperon 1996.3 Probability of return to the origin in a uniform magnetic field 2036.4 Probability of return to the origin for an AharonovBohm flux 205

6.4.1 The ring 2066.4.2 The cylinder 208

6.5 Spin-orbit coupling and magnetic impurities 2106.5.1 Transition amplitude and effective interaction potential 2106.5.2 Total scattering time 2126.5.3 Structure factor 214


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6.5.4 The Diffuson 2196.5.5 The Cooperon 2216.5.6 The diffusion probability 2236.5.7 The Cooperon Xc 224

6.6 Polarization of electromagnetic waves 2266.6.1 Elastic mean free path 2276.6.2 Structure factor 2286.6.3 Classical intensity 2316.6.4 Coherent backscattering 233

6.7 Dephasing and motion of scatterers 2346.7.1 General expression for the phase shift 2346.7.2 Dephasing associated with the Brownian motion of the

scatterers 2376.8 Dephasing or decoherence? 238Appendix A6.1:AharonovBohm effect in an infinite plane 240Appendix A6.2: Functional representation of the diffusion equation 242

A6.2.1 Functional representation 242A6.2.2 Brownian motion and magnetic field 244

Appendix A6.3: The Cooperon in a time-dependent field 247Appendix A6.4: Spin-orbit coupling and magnetic impurities,

a heuristic point of view 251A6.4.1 Spin-orbit coupling 251A6.4.2 Magnetic impurities 254

Appendix A6.5: Decoherence in multiple scattering of light by cold atoms 256A6.5.1 Scattering amplitude and atomic collision time 256A6.5.2 Elementary atomic vertex 257A6.5.3 Structure factor 262

7 Electronic transport 270

7.1 Introduction 2707.2 Incoherent contribution to conductivity 273

7.2.1 DrudeBoltzmann approximation 2737.2.2 The multiple scattering regime: the Diffuson 2767.2.3 Transport time and vertex renormalization 278

7.3 Cooperon contribution 2797.4 The weak localization regime 281

7.4.1 Dimensionality effect 282

7.4.2 Finite size conductors 2847.4.3 Temperature dependence 285

7.5 Weak localization in a magnetic field 2867.5.1 Negative magnetoresistance 2867.5.2 Spin-orbit coupling and magnetic impurities 290


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7.6 Magnetoresistance and AharonovBohm flux 2927.6.1 Ring 2937.6.2 Long cylinder: the SharvinSharvin effect 2947.6.3 Remark on the Webb and SharvinSharvin experiments: 0

versus 0/2 2957.6.4 The AharonovBohm effect in an infinite plane 296

Appendix A7.1: Kubo formulae 296A7.1.1 Conductivity and dissipation 296A7.1.2 Density-density response function 301

Appendix A7.2: Conductance and transmission 302A7.2.1 Introduction: Landauer formula 302A7.2.2 From Kubo to Landauer 305A7.2.3 Average conductance and transmission 307A7.2.4 Boundary conditions and impedance matching 311A7.2.5 Weak localization correction in the Landauer formalism 313A7.2.6 Landauer formalism for waves 314

Appendix A7.3: Real space description of conductivity 315Appendix A7.4: Weak localization correction and anisotropic collisions 317

8 Coherent backscattering of light 320

8.1 Introduction 320

8.2 The geometry of the albedo 3218.2.1 Definition 3218.2.2 Albedo of a diffusive medium 322

8.3 The average albedo 3248.3.1 Incoherent albedo: contribution of the Diffuson 3248.3.2 The coherent albedo: contribution of the Cooperon 326

8.4 Time dependence of the albedo and study of the triangular cusp 3308.5 Effect of absorption 333

8.6 Coherent albedo and anisotropic collisions 3348.7 The effect of polarization 336

8.7.1 Depolarization coefficients 3368.7.2 Coherent albedo of a polarized wave 337

8.8 Experimental results 3398.8.1 The triangular cusp 3408.8.2 Decrease of the height of the cone 3418.8.3 The role of absorption 343

8.9 Coherent backscattering at large 3478.9.1 Coherent backscattering and the glory effect 3478.9.2 Coherent backscattering and opposition effect in astrophysics 3488.9.3 Coherent backscattering by cold atomic gases 3498.9.4 Coherent backscattering effect in acoustics 352


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9 Diffusing wave spectroscopy 354

9.1 Introduction 3549.2 Dynamic correlations of intensity 3559.3 Single scattering: quasi-elastic light scattering 3579.4 Multiple scattering: diffusing wave spectroscopy 3589.5 Influence of the geometry on the time correlation function 359

9.5.1 Reflection by a semi-infinite medium 3599.5.2 Comparison between Gr1(T) and c( ) 3619.5.3 Reflection from a finite slab 3639.5.4 Transmission 364

Appendix A9.1: Collective motion of scatterers 367

10 Spectral properties of disordered metals 370

10.1 Introduction 37010.1.1 Level repulsion and integrability 37110.1.2 Energy spectrum of a disordered metal 373

10.2 Characteristics of spectral correlations 37410.3 Poisson distribution 37610.4 Universality of spectral correlations: random matrix theory 377

10.4.1 Level repulsion in 2 2 matrices 37710.4.2 Distribution of eigenvalues for N N matrices 38010.4.3 Spectral properties of random matrices 38210.5 Spectral correlations in the diffusive regime 38510.5.1 Two-point correlation function 38610.5.2 The ergodic limit 39010.5.3 Free diffusion limit 391

Appendix A10.1: The GOEGUE transition 394

11 Universal conductance fluctuations 396


11.2 Conductivity fluctuations 39911.2.1 Fluctuations of the density of states 40211.2.2 Fluctuations of the diffusion coefficient 405

11.3 Universal conductance fluctuations 40611.4 Effect of external parameters 409

11.4.1 Energy dependence 40911.4.2 Temperature dependence 40911.4.3 Phase coherence and the mesoscopic regime 411

11.4.4 Magnetic field dependence 41511.4.5 Motion of scatterers 41811.4.6 Spin-orbit coupling and magnetic impurities 419

Appendix A11.1: Universal conductance fluctuations and anisotropiccollisions 422

Appendix A11.2: Conductance fluctuations in the Landauer formalism 424


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12 Correlations of speckle patterns 427

12.1 What is a speckle pattern? 42712.2 How to analyze a speckle pattern 42812.3 Average transmission coefficient 43312.4 Angular correlations of the transmitted light 435

12.4.1 Short range C(1) correlations 43512.4.2 Long range correlations C(2) 43912.4.3 Two-crossing contribution and C(3) correlation 44112.4.4 Relation with universal conductance fluctuations 445

12.5 Speckle correlations in the time domain 44612.5.1 Time dependent correlations C(1)(t) and C(2)(t) 44712.5.2 Time dependent correlation C(3)(t) 450

12.6 Spectral correlations of speckle patterns 45212.7 Distribution function of the transmission coefficients 453

12.7.1 Rayleigh distribution law 45412.7.2 Gaussian distribution of the transmission coefficient Ta 45512.7.3 Gaussian distribution of the electrical conductance 456

Appendix A12.1: Spatial correlations of light intensity 458A12.1.1 Short range correlations 459A12.1.2 Long range correlations 461

13 Interactions and diffusion 46513.1 Introduction 46513.2 Screened Coulomb interaction 46613.3 HartreeFock approximation 46813.4 Density of states anomaly 470

13.4.1 Static interaction 47013.4.2 Tunnel conductance and density of states anomaly 47513.4.3 Dynamically screened interaction 478

13.4.4 Capacitive effects 48113.5 Correction to the conductivity 48313.6 Lifetime of a quasiparticle 487

13.6.1 Introduction: Landau theory and disorder 48713.6.2 Lifetime at zero temperature 48713.6.3 Quasiparticle lifetime at finite temperature 49413.6.4 Quasiparticle lifetime at the Fermi level 495

13.7 Phase coherence 498

13.7.1 Introduction 49813.7.2 Phase coherence in a fluctuating electric field 49913.7.3 Phase coherence time in dimension d= 1 50213.7.4 Phase coherence and quasiparticle relaxation 50613.7.5 Phase coherence time in dimensions d= 2 and d= 3 50913.7.6 Measurements of the phase coherence time ee 510


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Appendix A13.1: Screened Coulomb potential in confined geometry 512Appendix A13.2: Lifetime in the absence of disorder 514

14 Orbital magnetism and persistent currents 516


14.2 Free electron gas in a uniform field 51814.2.1 A reminder: the case of no disorder 51814.2.2 Average magnetization 52114.2.3 Fluctuations of the magnetization 522

14.3 Effect of Coulomb interaction 52414.3.1 HartreeFock approximation 52514.3.2 Cooper renormalization 52614.3.3 Finite temperature 528

14.4 Persistent current in a ring 52814.4.1 Clean one-dimensional ring: periodicity and parity effects 52914.4.2 Average current 534

14.5 Diffusive limit and persistent current 53614.5.1 Typical current of a disordered ring 53714.5.2 Effect of the Coulomb interaction on the average current 53914.5.3 Persistent current and spin-orbit coupling 54214.5.4 A brief overview of experiments 543

Appendix A14.1:Average persistent current in the canonical ensemble 54515 Formulary 547

15.1 Density of states and conductance 54715.2 Fourier transforms: definitions 54815.3 Collisionless probability P0(r,r, t) 54815.4 Probability P(r,r, t) 54815.5 WignerEckart theorem and 3j-symbols 55115.6 Miscellaneous 552

15.7 Poisson formula 55815.8 Temperature dependences 55815.9 Characteristic times introduced in this book 559

References 561Index 582


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Preface

Wave propagation in random media has been the subject of intense activity for more thantwo decades. This is now an important area of research, whose frontiers are still fuzzy,

and which includes a variety of problems such as wave localization (weak and strong),mesoscopic physics, effects of electronelectron interactions in metals, etc. Moreover, sincemany disorder effects are not truly specific to a given kind of wave, various approaches havebeen developed independently in condensed matter physics, in optics, in atomic physicsand in acoustics.

A large number of monographs or review articles already exist in the literature and theycover in detail various aspects of the field. Our aim is rather to present the basic commonfeatures of the effects of disorder on wave propagation and also to provide the non-specialist

reader with the tools necessary to enter and practice this field of research.Our first concern has been to give a description of the basic physical effects using a

single formalism independent of the specific nature of thewaves (electrons, electromagneticwaves,etc.).Tothispurpose,wehavestartedwithadetailedpresentationofsingle-particleaverage quantities suchas the density of states and elastic collision timeusing the frameworkof the so-called Gaussian model for the two most important examples of waves, namelySchrdinger and scalar Helmholtz wave equations. We have tried, as much as possible, tomake precise the very basic notion of multiple scattering by an ensemble of independent

effective scatterers whose scattering cross section may be obtained using standard one-particle scattering theory.

Nevertheless, the quantities of physical interest that are accessible experimentally andused to describe wave propagation in the multiple scattering regime depend essentially onthe probability of quantum diffusion which describes the propagation of a wave packet. Thisprobability thus plays a central role and Chapter 4 is devoted to its detailed study. We thensee emerging notions such as classical (Diffuson) and coherent (Cooperon) contributionsto the probability, which provide basic explanations of the observed physical phenomena

such as weak localization corrections to electronic transport, negative magnetoresistancein a magnetic field, coherent backscattering of light, as well as universal conductancefluctuations, optical speckles and mesoscopic effects in orbital magnetism.

It thus happens that all these effects result from the behavior of a single quantity, namelythe probability of quantum diffusion. However, in spite of the common background shared

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xiv Preface

by optics and electronics of random media, each of these domains has its own specificitywhich allowsus to develop complementary approaches. For instance, thecontinuous changein the relative phases of electronic wave functions that can be achieved using a magneticfield or a vector potential has no obvious equivalent in optics. On the other hand, it ispossible in optics to change directions of incident and outgoing beams and from this angularspectroscopy to trace back correlations between angular channels.

We have made a special effort to try to keep this book accessible to the largest audience,starting at a graduate level in physics with an elementary acquaintance of quantummechanics as a prerequisite. We have also skipped a number of interesting but perhapstoo specialized issues among which are the study of quantum dots, relations betweenspectral and transport quantities, strong localization and the Anderson metalinsulatortransition, electronic ballistic billiards where quantum complexity does not result fromdisorder but instead from the boundary shape, and metalsuperconductor interfaces. All

these aspects reflect the richness of the field of quantum mesoscopic physics to whichthis book constitutes a first introduction.

A pleasant task in finishing the writing of a book is certainly the compilation ofacknowledgments to all those who have helped us at various stages of the elaboration andwriting, either through discussions, criticisms and especially encouragement and support:O. Assaf, H. Bouchiat, B. Huard, J. Cayssol, C. Cohen-Tannoudji, N. Dupuis, D. Estve, A.Georges, S.Guron, M.Kouchnir, R.Maynard,F. Pichon, H.Pothier, B.Reulet, B.Shapiro,

B. van Tiggelen, D. Ullmo, J. Vidal, E. Wolf. We wish to single out the contribution ofC. Texier for his endless comments, suggestions, and corrections which have certainlycontributedtoimprovethequalityofthisbook.DovLevineacceptedtohelpusintranslatingthe book into English. This was a real challenge and we wish to thank him for his patience.We also wish to thank G. Bazalitsky for producing most of the figures with much dedication.

This venture was in many respects a roller-coaster ride and the caring support ofAnne-Marie and Tirza was all the more precious.

Throughout this book, we use the (SI) international unit system, except in Chapter 13.

The Planck constant is generally taken equal to unity, in particular throughout Chapter4. In the chapters where we think it is important to restore it, we have mentioned this atthe beginning of the chapter. In order to simplify the notation, we have sometimes partiallyrestored in a given expression, especially when the correspondence between energy andfrequency is straightforward.

To maintain homogeneous and consistent notation throughout a book which covers fieldsthat are usually studied separately is a kind of challenge that, unfortunately, we have notalways been able to overcome.

We have chosen not to give an exhaustive list of references, but instead to quotepapers either for their obvious pedagogical value or because they discuss a particular pointpresented for instance as an exercise.


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How to use this book

This book is intended to provide self-contained material which will allow the reader toderive the main results. It does not require anything other than an elementary background

in general physics and quantum mechanics.We have chosen to treat in a parallel way similar concepts occurring in the propagation of

electrons and light. The important background concepts are given in Chapter 4, where thenotion ofprobability of quantum diffusion in random media is developed. This is a centralquantity to which all physical quantities described in this book may be related.

This book is not intended to be read linearly. We have structured it into chapters whichare supposed to present the main concepts, and appendices which focus on specific aspectsor details of calculation. This choice may sometimes appear arbitrary. For example, the

Landauer formalism is introduced in an appendix (A7.2), where it is developed for thediffusive regime, which to our knowledge has not been done in the textbook literature.The standard description of weak localization is presented within the Kubo formalismin the core of Chapter 7, while the Landauer picture of weak localization is developed inAppendix A7.2.

We suggest here a guide for lectures. Although we have tried to emphasize analogiesbetween interference effects in the propagation of electrons and light, we propose twooutlines, for two introductory courses respectively on the physics of electrons and the

physics of light. We believe that, during the course of study, the interested reader will benefitfrom the analogies developed between the two fields, for example the relations betweenspeckle fluctuations in optics and universal conductance fluctuations in electronics.

Quantum transport in electronics

Main course

1 Introduction: mesoscopic physics Provides a unified and general description ofinterference and multiple scattering effects in disordered systems. Introduces the physicalproblems and the main quantities of interest, the different length scales such as the meanfree path and the phase coherence length, the notions of multiple scattering and disorderaverage. Relates the physical properties to the probability of returning to the origin in arandom medium. Notion of quantum crossing. Analogies between electronics and optics.

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2 Wave equations and models of disorder Schrdinger equation for electrons in solidsandHelmholtzequation forelectromagneticwaves.Gaussian,Edwards,Andersonmodelsfor disorder.3 Perturbation theory Presents the minimal formalism of Greens functions necessaryfor the notions developed further in the book. Multiple scattering and weak disorderexpansion.4.14.6 Probability of quantum diffusion Definition and description of essentialconcepts and tools used throughout the book. Iterative structure for the quantumprobability, solution of a diffusion equation. Diffuson and Cooperon contributions.Formalism developed in real space. May also be useful to look at the reciprocal spaceformalism developed in Appendix A4.1.6 Dephasing Proposes a general picture for dephasing and describes severalmechanisms due to electron coupling to external parameters or degrees of freedom:

magnetic field, AharonovBohm flux, spin-orbit coupling and magnetic impurities. Maybe skipped at the introductory level, except for magnetic field and AharonovBohmeffects.7 Electronic transport Deals with calculations of the average conductivity and of theweak localization correction. The latter is related to the probability of return to theorigin for a diffusive particle. Applications to various geometries, plane, ring, cylinder,dimensionality effects. Section A7.2 is a comprehensive appendix on the Landauerformalism for diffusive systems.

10 Spectral properties of disordered metals Generalities on random matrix theory.Spectral correlation functions for disordered systems. The last part requires knowledgeof correlation functions calculated in Appendix A4.4.11 Universal conductance fluctuations Detailed calculation of the conductancefluctuations in the Kubo formalism, using the diagrammatics developed in Chapters 4and 7. Many physical discussions on the role of various external parameters.13 Interactions and diffusion Important chapter on the role of electronelectroninteraction and its interplay with disorder. Density of states anomaly, correction to

the conductivity. Important discussions about lifetime of quasiparticles and phasecoherence time.14 Persistent currents Can be considered optional. Thermodynamics and orbitalmagnetism of mesoscopic systems. Problematics of persistent currents, from the verysimple one-dimensional description to the effect of disorder and interaction.

Optional

5 Properties of the diffusion equation Provides a comprehensive and self-contained

account of properties of the diffusion equation. Diffusion in finite systems, boundaryconditions, diffusion on graphs.Miscellaneous Variousappendicesarebeyondanintroductorylevel,orarenotnecessaryin a first course on mesoscopic physics. They either develop technicalities such as Hikamiboxes(A4.2),Cooperoninatimedependentmagneticfield(A6.3),orimportantextensions


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How to use this book xvii

such as anisotropic collisions developed in A4.3 and their effect on weak localization(A7.4) and universal conductance fluctuations (A11.1). The Landauer formalism fordiffusive systems is developed in A7.2 for the average conductance and the weaklocalization correction and in A11.2 for conductance fluctuations.

Propagation of light in random media

Main course

This course provides a comprehensive introduction to the propagation of light in randommedia. It describes coherent effects in multiple scattering: coherent backscattering,diffusing wave spectroscopy and angular and time correlations of speckle patterns.Compared to coherent electronic transport, this course emphasizes notions specificto electromagnetic waves such as angular correlations of transmission (or reflection)

coefficients in open space geometry, correlation between channels in a wave guidegeometry, as well as the effects of the dynamics of scatterers.14 These chapters are common to the two courses. In addition section 4.6 introducesthe important formalism of radiative transfer which is developed in Appendix A5.2.6 Dephasing Generalities on the mechanism of dephasing. Application to thepolarization of electromagnetic waves, dynamics of the scatterers and dephasingassociated with quantum internal degrees of freedom for the case of scattering of photonsby cold atoms (the last topic is treated in Appendix A6.5).

8 Coherent backscattering of light Physics of the albedo, reflection coefficient of adiffusive medium. Coherent contribution (Cooperon) to the albedo, and its angulardependence. Uses the formalism developed in Chapter 4. Polarization and absorptioneffects (see also section 6.6). Extensive discussion of experimental results and coherentbackscattering in various physical contexts. This chapter relies upon the results ofsection 5.6.9 Diffusing wave spectroscopy An experimental technique routinely used to probe thedynamics of scatterers. Calculations result from a simple generalization of the formalism

of Chapter 4. Interesting conceptually since diffusing wave spectroscopy exhibits thesimplest example of decoherence introduced in a controlled way.Also interesting becausethis method probes the distribution of multiple scattering trajectories (reflection versustransmission experiments). Study of sections 5.6 and 5.7 is recommended.12 Correlations of speckle patterns Analysis of a speckle pattern. Angular correlationsof transmission coefficients. Classification and detailed calculation of the successivecontributions C1, C2 and C3. Simple description in terms of quantum crossings. Rayleighlaw.UseoftheLandauerformalismtorelatespecklecorrelationstouniversalconductance

fluctuations.Optional

5 Properties of the diffusion equation Solutions of the diffusion equation inquasi-one-dimensional geometries, useful for calculations developed in Chapters 8, 9and 12. Important appendix A5.2 on radiative transfer.


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Miscellaneous Various appendices are useful reminders for beginners, for exampleA2.1 on scattering theory and A2.3 on light scattering by individual scatterers (Rayleigh,RayleighGans, Mie, resonant). Other appendices go beyond a course at the introductorylevel, either because they develop additional technicalities such as Hikami boxes (A4.2),useful for the reader interested in detailed calculations of Chapter 12, or because theypresent additional aspects of multiple scattering of light by random media such asspatial correlations of light intensity (A12.1) or anisotropic collisions (A4.3) and theirconsequences. The Landauer formalism for diffusive systems is used extensively inChapter 12 on speckle correlations. Appendix A6.5 gives an overview of the technicaltoolsneededtostudythespecificproblemofmultiplescatteringofphotonsbycoldatoms.

Topics developed in this book. Lines represent logical links between chapters.


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1

Introduction: mesoscopic physics

1.1 Interference and disorder

Wave propagation in a random medium is a phenomenon common to many areas of physics.There has been a recent resurgence of interest following the discovery, in both optics andquantum mechanics,of surprisingcoherent effects ina regime in which disorder was thoughtto be sufficiently strong to eliminate a priori all interference effects.

To understand the origin of these coherent effects, it may be useful to recall some generalfacts about interference.Although quite spectacular in quantum mechanics, their descriptionis more intuitive in the context of physical optics. For this reason, we begin with a discussionof interference effects in optics.

Consider a monochromatic wave scattered by an obstacle of some given geometry, e.g.,a circular aperture. Figure 1.1 shows the diffraction pattern on a screen placed infinitely farfrom the obstacle. It exhibits a set of concentric rings, alternately bright and dark, resultingfrom constructive or destructive interference.According to Huygensprinciple, the intensityat a point on the screen may be described by replacing the aperture by an ensemble of virtualcoherent point sources and considering the difference in optical paths associated with thesesources. In this way, it is possible to associate each interference ring with an integer (theequivalent of a quantum number in quantum mechanics).

Let us consider the robustness of this diffraction pattern. If we illuminate the obstacleby an incoherent source for which the length of the emitted wave trains is sufficientlyshort that the different virtual sources are out of phase, then the interference pattern onthe screen will disappear and the screen will be uniformly illuminated. Contrast this withthe following situation: employ a coherent light source and rapidly move the obstacle inits plane in a random fashion. Here too, the interference fringes are replaced by uniformillumination. In this case, it is the persistence of the observers retina that averages overmany different displaced diffraction patterns. This example illustrates two ways in whichthe diffraction pattern can disappear. In the former case, the disappearance is associatedwith a random distribution of the lengths of wave trains emanating from the source,while in the latter case, it is the result of an average over an ensemble of spatiallydistributed virtual sources. This example shows how interference effects may vanish uponaveraging.

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2 Introduction: mesoscopic physics

Figure 1.1 Diffraction pattern at infinity for a circular aperture.

Let us now turn to the diffraction of a coherent source by an obstacle of arbitrary type. Forinstance, suppose that the obstacle is a dielectric material whose refractive index fluctuatesin space on a scale comparable to the wavelength of the light. The resulting scatteringpattern on a screen placed at infinity, consists of a random distribution of bright and darkareas, as seen in Figure 1.2; this is called a speckle pattern.1 Each speckle associated with

the scattering represents a fingerprintof the random obstacle, and is specific to it. However,in contrast to the case of scattering by a sufficiently symmetric obstacle (such as a simplecircular aperture), it is impossible to identify an order in the speckle pattern, and thus wecannot describe it with a deterministic sequence of integer numbers. This impossibility isone of the characteristics of what are termed complex media.

In this last experiment, for a thin enough obstacle, a wave scatters only once in therandom medium before it emerges on its way to the screen at infinity (see Figure 1.3(a)).This regime is called single scattering. Consider now the opposite limit of an optically

thick medium (also called a turbid medium), in which the wave scatters many times beforeleaving (Figure 1.3(b)). We thus speak ofmultiple scattering. The intensity at a point onthe screen is obtained from the sum of the complex amplitudes of the waves arriving atthat point. The phase associated with each amplitude is proportional to the path length ofthe multiply scattered wave divided by its wavelength . The path lengths are randomlydistributed, so one would expect that the associated phases fluctuate and average to zero.Thus, the total intensity would reduce to the sum of the intensities associated with each ofthe paths.

In other words, if we represent this situation as equivalent to a series of thin obstacles,with each element of the series corresponding to a different and independent realizationof the random medium, we might expect that for a sufficiently large number of such thin

1 These speckles resemble those observed with light emitted by a weakly coherent laser, but they are of a different nature. Herethey result from static spatial fluctuations due to the inhomogeneity of the scattering medium.


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1.1 Interference and disorder 3

Figure 1.2 Speckle patterns due to scattering through an inhomogeneous medium. Here the mediumis optically thick, meaning that the incident radiation undergoes many scatterings before leaving thesample. Each image corresponds to a different realization of the random medium [1].

k

(a) (b)

k

k

L

k

L

Figure 1.3 Schematic representations of the regimes of (a) single scattering, and (b) multiplescattering.

obstacles, the resulting intensity at a point on the screen would average over the differentrealizations, causing the speckles to vanish. This point of view corresponds to the classicaldescription, for which the underlying wave nature plays no further role.

Figures 1.2 and 1.4 show that this conclusion is incorrect, and that the speckles survive,even in the regime of multiple scattering. If, on the other hand, we perform an ensemble


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(a) (b) (c)

Figure 1.4 Averaging. The first speckle pattern (a) represents a snapshot of a random mediumcorresponding to a single realization of the disorder. The other two figures (b and c) correspondto an integration over the motion of scatterers, and hence to a self-average. (Figure courtesy of Georg

Maret.)

average, the diffraction pattern disappears. This is the case with turbid media such as theatmosphere or suspensions of scatterers in a liquid (milk, for example), where the motion ofthe scatterers yields an average over different realizations of the random medium, providedwe wait long enough. The classical approach, therefore, correctly describes the averagecharacteristics of a turbid medium, such as the transmission coefficient or the diffusioncoefficient of the average intensity. It has been employed extensively in problems involving

the radiative transfer of waves through the atmosphere or through turbulent media.This description may be adapted as such to the problem of propagation of electrons in a

metal. In this case, the impurities in the metal are analogous to the scatterers in the opticallythick medium, and the quantity analogous to the intensity is the electrical conductivity. Inprinciple, of course, it is necessary to use the machinery of quantum mechanics to calculatethe electrical conductivity. But since the work of Drude at the beginning of the last century, ithas been accepted that transport properties of metals are correctly described by the disorder-averaged conductivity, obtained from a classical description of the degenerate electron gas.

However, for a given sample, i.e., for a specific realization of disorder, we may observeinterference effects, which only disappear upon averaging [2].

The indisputable success of the classical approach led to the belief that coherent effectswould not subsist in a random medium in which a wave undergoes multiple scattering. Inthe 1980s, however, a series of novel experiments unequivocally proved this view to befalse. In order to probe interference effects, we now turn to the AharonovBohm effect,which occurs in the most spectacular of these experiments.

1.2 The AharonovBohm effect

The Young two-slit device surely provides the simplest example of an interference patternin optics; understanding its analog in the case of electrons is necessary for understandingquantum interference effects. In the AharonovBohm geometry, an infinite solenoid is


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1.2 The AharonovBohm effect 5

ea2i2

ea1i1

Figure 1.5 Schematic representation of the AharonovBohm effect. A flux tube of flux is placedbehind the two slits.

placed between the slits, such that the paths of the interfering electrons are exterior to it, asindicated in Figure 1.5. The magnetic field outside the solenoid is zero, so that classicallyit has no effect on the motion of the electrons.

This is not the case in quantum mechanics where, to calculate the intensity, we must sumthe complex amplitudes associated with different trajectories. For the two trajectories ofFigure 1.5, the amplitudes have the form a1,2 = |a1,2|ei1,2 , where the phases 1 and 2 aregiven by (e is the electron charge):

1 = (0)1 e

1

A dl(1.1)

2 = (0)

2 e

2A dl.The integrals are the line integrals of the vector potential A along the two trajectories and

(0)1,2 are the phases in the absence of magnetic flux. In the presence of a magnetic flux

induced by the solenoid, the intensity I() is given by

I() = |a1 + a2|2 = |a1|2 + |a2|2 + 2|a1a2| cos(1 2)= I1 +I2 + 2I1I2 cos(1 2). (1.2)

The phase difference () = 1 2 between the two trajectories is now modulated bythe magnetic flux

() = (0)1 (0)2 +e

A dl = (0) + 2

0, (1.3)


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B

Figure 1.6 Schematic description of the experiment of Webb et al. on the AharonovBohm effect ina metal. In this experiment, the applied magnetic field is uniform. is the flux through the ring.

where 0 = h/e is the quantum of magnetic flux. It is thus possible to vary continuouslythe state of interference at each point on the screen by changing the magnetic flux .This is the AharonovBohm effect [3]. It is a remarkable probe to study phase coherencein electronic systems [4]. This constitutes an advantage for electronic systems over theiroptical counterparts.2

This effect was observed in the following experiment. A coherent stream of electronswas emitted by an electron microscope and split in two before passing through a toroidalmagnet whose magnetic field was confined to the inside of the torus [6]. Thus, the magneticfield was zero along the trajectories of electrons. However, this experiment was performedin vacuum, where the electrons do not undergo any scattering before interfering. In orderto demonstrate possible phase coherence in metals, in which the electrons undergo manycollisions, R. Webb and his collaborators (1983) measured the resistance of a gold ring [7].

In the setup depicted schematically in Figure 1.6, electrons are constrained to pass throughthe two halves of the ring, which are analogous to the two Young slits, before being collectedat the other end.

The analog of the intensity I() is the electrical current, or better yet, the conductanceG() measured for different values of the magnetic flux . The flux is produced by applyinga uniform magnetic field, though this does not strictly correspond to the AharonovBohmexperiment, since the magnetic field is not zero along the trajectories of electrons. However,the applied field is sufficiently weak that firstly, there is no deflection of the trajectories due

totheLorentzforce,andsecondly,thedephasingofcoherenttrajectoriesduetothemagneticfield is negligible in the interior of the ring. Thus, the effect of the magnetic field may beneglected in comparison to that of the flux. Figure 1.7 shows that the magnetoresistance

2 In a rotating frame, there is an analogous effect, called the Sagnac effect [5].


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1.3 Phase coherence and the effect of disorder 7

(a)

(b)

Figure 1.7 (a) Magnetoresistance of a gold ring at low temperature T = 0.01 K, (b) Fourier spectrumof the magnetoresistance. The principal contribution is that of the Fourier component at 0 = h/e [7].

of this ring is, to first approximation, a periodic function of the applied flux whose periodis the flux quantum 0 = h/e. Indeed, since the relative phase of the two trajectories ismodulated by the flux, the total current, and therefore the conductance of the ring, areperiodic functions of the flux:3

G() = G0 + G cos

(0) + 2 0

. (1.4)

This modulation of the conductance as a function of flux results from the existence ofcoherent effects in a medium in which the disorder is strong enough for electrons to bemultiply scattered. Consequently, the naive argument that phase coherence disappears inthis regime is incorrect, and must be reexamined.

1.3 Phase coherence and the effect of disorder

In the aforementioned experiment of Webb et al., the size of the ring was of the order of

a micrometer. Now we know that for a macroscopic system, the modulation as a functionof magnetic flux disappears. Therefore, there exists a characteristic length such that on

3 We see in Figure 1.7 that the modulation is not purely periodic. This is due to the fact that the ring is not one dimensional.Moreover, multiple scattering trajectories within the same branch may also be modulated by the magnetic field which penetratesinto the ring itself. This is the origin of the low-frequency peak in Figure 1.7(b).


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scales greater than this length, there is no longer any phase coherence. This length, calledthe phase coherence length and denoted L , plays an essential role in the description ofcoherent effects in complex systems.

In order to understand better the nature of this length, it is useful to review some notionsrelated to quantum coherence.4 Consider an ensemble of quantum particles contained ina cubic box of side length L in d dimensions. The possible quantum states are coherentsuperpositions of wave-functions such that the quantum state of the system is coherent overthe whole volume Ld. There are many examples in which quantum coherence extendsup to the macroscopic scale: superconductivity, superfluidity, free electron gas at zerotemperature, coherent states of the photon field, etc.

For the electron gas at finite temperature, this coherence disappears at the macroscopicscale. It is therefore possible to treat physical phenomena such as electrical or thermaltransport, employing an essentially classical approach. The suppression of quantum

coherence results from phenomena linked to the existence of incoherent and irreversibleprocesses due to the couplingof electrons to their environment.This environment consists ofdegreesoffreedomwithwhichtheelectronsinteract:thermalexcitationsoftheatomiclattice(phonons), impurities having internal degrees of freedom, interaction with other electrons,etc. This irreversibility is a source of decoherence for the electrons and its description isa difficult problem which we shall consider in Chapters 6 and 13. The phase coherencelength L generically describes the loss of phase coherence due to irreversible processes.In metals, the phase coherence length is a decreasing function of temperature. In practice,L is of the order of a few micrometers for temperatures less than one kelvin.

None of the above considerations are related to the existence ofstatic disorder of thetype discussed in the two previous sections (e.g., static impurities such as vacancies orsubstitutional disorder, or variation of the refractive index in optics). Such disorder doesnot destroy the phase coherence and does not introduce any irreversibility. However, thepossible symmetries of the quantum system disappear in such a way that it is no longerpossible to describe the system with quantum numbers. In consequence, each observableof a random medium depends on the specific distribution of the disordered potential. On

average, it is possible to characterize the disorder by means of a characteristic length: theelastic mean free path le, which represents the average distance travelled by a wave packetbetween two scattering events with no energy change (see Chapters 3 and 4).

We see, therefore, that the phase coherence length L is fundamentally different from theelastic mean free path le. For sufficiently low temperatures, these two lengths may differby several orders of magnitude, so that an electron may propagate in a disordered mediuma distance much larger than le keeping its phase coherence, so long as the length of its tra-jectory does not exceed L . The loss of coherence, therefore, is not related to the existence

of a random potential of any strength, but rather to other types of mechanisms. It may seemsurprising that the distinction between the effect of elastic disorder described by le and

4 Most of the notions discussed here use the language of quantum mechanics; however, they have more or less direct analogs inthe case of electromagnetic wave propagation.


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1.4 Average coherence and multiple scattering 9

that associated with irreversible processes of phase relaxation was first demonstratedin the relatively non-trivial case of transport in a metal where the electrons havecomplex interactions with their environment. However, the same distinction also appliesto electromagnetic wave propagation in turbid media in the regime of coherent multiplescattering.

1.4 Average coherence and multiple scattering

If phase coherence leads to interference effects for a specific realization of disorder, it mightbe thought that these would disappear upon averaging. In the experiment of Webb et al.described in section 1.2, the conductance oscillations of period 0 = h/e correspond to aspecific ring. If we now average over disorder, that is, over (0) in relation (1.4), we expect

the modulation by the magnetic flux to disappear, and with it all trace of coherent effects.The same kind of experiment was performed in 1981 by Sharvin and Sharvin [8] on a longhollow metallic cylinder threaded by an AharonovBohm flux. A cylinder of height greaterthanL can be interpreted as an ensemble of identical, uncorrelated rings of the type used inWebbs experiment. Thus, this experiment yields an ensemble average. Remarkably, theysaw a signal which oscillated with flux but with a periodicity 0/2 instead of0. How canwe understand that coherent effects can subsist on average?

The same type of question may be asked in the context of optics. If we average a speckle

pattern over different realizations of disorder, does any trace of the phase coherence remain?Here too there was an unexpected result: the reflection coefficient of a wave in a turbidmedium (sometimes called its albedo) was found to exhibit an angular dependence thatcould not be explained by the classical transport theory (Figure 1.8). This effect is knownas coherent backscattering, and is a signature of phase coherence.

These results show that even on average, some phase coherence effects remain. In orderto clarify the nature of these effects, let us consider an optically thick random medium. Itcan be modelled by an ensemble of point scatterers at positionsrn distributed randomly. The

validity of this hypothesis for a realistic description of a random medium will be discussedin detail in Chapters 2 and 3. Consider a plane wave emanating from a coherent source(located outside the medium), which propagates in the medium and collides elasticallywith scatterers, and let us calculate the resulting interference pattern. For this, we studythe complex amplitude A(k,k) of the wave reemitted in the direction defined by the wavevector k, corresponding to an incident plane wave with wave vector k. It may be written,without loss of generality, in the form

A(k,k

)= r1,r2f(r1,r2) e

i(k

r1

k

r2)

, (1.5)

where f(r1,r2) is the complex amplitude corresponding to the propagation between twoscattering events located atr1 andr2. This amplitude may be expressed as a sum of the form

j aj =

j |aj|eij , where each path j represents a sequence of scatterings (Figure 1.9)


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Figure 1.8 Speckle pattern obtained by multiple scattering of light by a sample of polystyrene spheres,as a function of observation angle. The curve in the lower figure represents the intensity fluctuationsmeasured along a given angular direction. The upper figure is obtained by averaging over the positionsof the spheres, and the resulting curve gives the angular dependence of the average intensity. (Figurecourtesy of G. Maret.)

k

r1

r2

k

Figure 1.9 Typical trajectories which contribute to the total complex amplitudef(r1,r2) of a multiply

scattered wave.

joining the points r1 and r2. The associated intensity is given by

|A(k,k)|2 =

r1,r2

r3,r4

f(r1,r2)f(r3,r4) ei(kr1k

r2) ei(kr3kr4) (1.6)

with

f(r1,r2)f(r3,r4) =

j j

aj(r1,r2) a

j (r3,r4) =j j

|aj||aj | ei(jj ). (1.7)

In order to calculate its value averaged over the realizations of the random potential, that is,over the positions of scatterers, it is useful to note that most of the terms in relations (1.6)


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1.4 Average coherence and multiple scattering 11

kr1

rb

ry

r2

ra

rz

k

k

k

r1

rb

ry

r2

ra

rz

(a)

(b)

Figure1.10 Schematicrepresentationofthetwotypesofsequencesofmultiplescatteringsthatremainupon averaging. The first (a) corresponds to the classical average intensity. The second (b), for whichthe two sequences of scattering events are traversed in opposite directions, is the source of the coherentbackscattering effect.

and (1.7) average to zero, provided that the phase j j , which measures the differencein the lengths of the trajectories of Figure 1.9, is random.

In consequence, the only terms which contribute to the average of|A(k,k)|2 are those forwhich the phases vanish. This can only occur for pairs ofidentical trajectories, those which

have the same sequence of scattering events, either in the same or in opposite directions.Such trajectories are schematically represented in Figure 1.10, and correspond to the

sequences

r1 ra rb ry rz r2r2 rz ry rb ra r1.

The fact that the trajectories are identical imposes on us, in particular,r1 =r3 andr2 =r4 forthe former process (same direction) andr1 =r4 andr2 =r3 for the latter (opposite direction)in relation (1.6). These two processes contribute identically to the intensity provided that thesystem is invariant under time reversal. Moreover, the second process gives rise, accordingto (1.6), to an additional dephasing such that the only two non-zero contributions which


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remain upon averaging are

|A(k,k)|2 =

r1,r2

|f(r1,r2)|21 + ei(k+k)(r1r2)

, (1.8)

where denotes averaging over the realizations of the random potential.The essence of the present book is a systematic study of the consequences of the existenceofthesetwoprocesses,whichsurviveuponaveraginginthecourseofmultiplescattering.Theformer process is well known. It may be perfectly well understood using a purely classicaltreatment that does not take into account the existence of an underlying wave equation,since the phases exactly cancel out. In the study of electron transport in metals, this classicalanalysisisperformedintheframeworkoftheBoltzmannequation,whileforelectromagneticwavepropagation,theequivalenttheory,called radiative transfer,wasdevelopedbyMieand

Schwartzchild [9]. Both date from the beginning of the twentieth century.The second term in relation (1.8) contains a phase factor. This depends on the points r1and r2, and the sum over these points in the averaging makes this term vanish in general,with two notable exceptions.

k +k 0. In the direction exactly opposite to the direction of incidence, the intensityis twice the classical value. The classical contribution has no angular dependence onaverage, and the second term, which depends on k + k, gives an angular dependenceto the average intensity reflected by the medium which appears as a peak in the albedo.

This phenomenon was observed first in optics and is known as coherent backscattering;its study is the object of Chapter 8.

In the sum (1.8), the terms for which r1 =r2 are special. They correspond to closedmultiple scattering trajectories. Their contributions to the averaged interference termsurvive even when it is impossible to select the directions k and k. This is the case formetals or semiconductors for which the interference term affects the average transportproperties such as the electrical conductivity. This is the origin of the phenomenon ofweak localization.

1.5 Phase coherence and self-averaging: universal fluctuations

The measurable physical quantities of a disordered quantum system depend on the specificrealizationofthedisorder,atleastsolongasthecharacteristiclengthsofthesystemaresmallerthan the phase coherence length L . In the opposite case, that is, for lengths greater than L ,thephasecoherenceislost,andthesystembecomesclassical,i.e.,thephysicalquantitiesareindependent of the specific realization of the disorder. The physics of systems of size lessthan L

, called mesoscopic systems,5 is thus particularly interesting because of coherence

effects [10, 11]. The physics of mesoscopic systems makes precise the distinction betweenthecomplexityduetodisorderdescribedby le and the decoherence, which depends onL:

disorder (le), loss of symmetry and of good quantum numbers (complexity); loss of phase coherence (L).5 The Greek root o means intermediate.


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1.5 Phase coherence and self-averaging: universal fluctuations 13

Let us now attempt to understand why a disordered quantum system larger than Lexhibits self-averaging, i.e., why its measurable physical properties are equal to theirensemble averages. If the characteristic size L of a system is much greater than L , the

system may be decomposed into a collection ofN =L/L

d 1 statistically independentsubsystems, in each of which the quantum coherence is preserved. A physical quantitydefined in each subsystem will then take on N random values. The law of large numbersensures that every macroscopic quantity is equal, with probability one, to its averagevalue. Consequently, every disordered system of size L L is effectively equivalent toan ensemble average. On the other hand, deviations from self-averaging are observed insystems of sizes smaller than L because of the underlying phase coherence. The study ofthese deviations is one of the main goals of mesoscopic physics. Consider the particularlyimportant example of fluctuations in the electrical conductance of a weakly disordered metal(Chapter 11). In the classical self-averaging limit, for a cubic sample of size L, the relativeconductance fluctuations vary as 1/N:

G2

G 1

N

L

L

d/2(1.9)

where G = G G. The average conductance G is the classical conductance Gcl given byOhms law Gcl = Ld2 where is the electrical conductivity.6 From relation (1.9), wededuce that G2 L

d

4

. For d 3, the fluctuations go to zero in the large scale limit, andthe system is said to be self-averaging. In contrast, for L < L , it is found experimentallythat

G2 constant e2

h. (1.10)

CONDUCTANCE

(e2/h) 88.2

88.0

87.8

87.6

18 4

16

14

12

10

2

0 1 2 3 40 1 2 3 40 1 2 3 4

B (T) B (T) B (T)

0

2

(a) (b) (c)

Figure 1.11 Aperiodic variations in the magnetoconductance of three different systems. (a) A goldring of diameter 0.8 mm, (b) a Si-MOSFET sample, and (c) the result of numerical simulations usingthe disordered Anderson model (discussed in Chapter 2). The conductance varies by several ordersof magnitude from one system to another, but the fluctuations remain of order e2/h [12].

6 The expression Gcl = Ld2 is a generalization to d dimensions of the standard expression Gcl = S/L, for a sample of lengthL and cross section S.


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In the mesoscopic regime, the amplitude of conductance fluctuations is independent of thesize L and of the amount of disorder, and one speaks ofuniversal conductance fluctuations.The variance of the conductance is the product of a universal quantity e2/h and a numericalfactor which depends solely on the sample geometry. This implies that in the mesoscopicregime, the electrical conductance is no longer a self-averaging quantity. This universality isshown in Figure 1.11 where each plot corresponds to a very different system. One essentialcharacteristic of mesoscopic fluctuations is their reproducibility. For a given realization ofdisorder, the dependence of the fluctuations as a function of an external parameter suchas Fermi energy or magnetic field is perfectly reproducible. In this sense, the fluctuationsrepresent, just like speckle patterns in optics, a fingerprintof the realization of the disorder,and uniquely characterize it.

1.6 Spectral correlations

Wehavementionedthesignatureofcoherenteffectsontransportpropertiessuchaselectricalconductance or albedo. For an isolated system of finite size, we may wonder about theeffect of disorder on the spatial behavior of wave functions and on the correlation of eigenenergies. For electromagnetic waves, we are interested in the spectrum of eigenfrequencies.If, for instance, the wavefunctions are stronglyaffectedbythe disorderand are exponentiallylocalized, then the corresponding eigenenergies (or frequencies) may be arbitrarily closeto each other since they describe states for which the spatial overlap is exponentially

small. These wave functions are uncorrelated, as are their energy levels. If, on the otherhand, the wave functions are spatially delocalized over the system and do not exhibit anyspatial structure, which corresponds to a regime we may consider as ergodic, then theimportant spatial overlap of the eigenfunctions induces spectral correlations manifestedby a repulsion of the energy levels. These two extreme situations are very general,and are insensitive to the microscopic details characterizing the disorder. It turns out thatthe spectral correlations present universal properties common to very different physicalsystems. Consider, for example, the probability P(s) that two neighboring energy levels

are separated by s. The two preceding situations are described by two robust limitingcases for the function P(s), corresponding, respectively, to a Poisson distribution for theexponentially localized states, and to a WignerDyson distribution for the ergodic case.These two distributions, represented in Figure 10.1, describe a wide range of physicalproblems and divide them, to first approximation, into two classes, corresponding either tointegrable systems (Poisson) or to non-integrable (also called chaotic) systems (WignerDyson). The latter case may be studied using random matrix theory along the general linesdiscussed in Chapter 10.

Of course, a complex medium exhibits such universal behavior only in limiting cases.From the methods developed in this book, we shall see how to recover certain results ofrandom matrix theory, and to identify corrections to the universal regime. These spectralcorrelations are extremely sensitive to the loss of phase coherence. They thus depend onL and are characteristic of the mesoscopic regime. They are evidenced by the behavior of


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1.7 Classical probability and quantum crossings 15

thermodynamic variables such as magnetization or persistent currents which constitute theorbital response of electrons to an applied magnetic field; this is the object of Chapter 14.

1.7 Classical probability and quantum crossings

Most physical quantities studied in this book are expressed as a function of the productof two complex amplitudes, each being the sum of contributions associated with multiplescattering trajectories:

i

ai

j

aj =

i,j

ai aj. (1.11)

This is the case, for example, of light intensity considered in section 1.4. The combination

of amplitudes (1.11) is related to the probability of quantum diffusion, whose role isessential in characterizing the physical properties of disordered media. This probability,which describes the evolution of a wave packet between any two points r and r, is writtenas the product of two complex amplitudes7 known as propagators or Greens functions.Denoting the average probability by P(r,r), we have

P(r,r)

i,j

ai (r,r)aj(r,r). (1.12)

Each amplitude aj(r,r) describes a propagating trajectory j from r to r, and thus P(r,r)appears as the sum of contributions of pairs of trajectories, each characterized by anamplitude and a phase. This sum may be decomposed into two contributions, one for whichthe trajectories i and j are identical, the other for which they are different:

P(r,r)

j

|aj(r,r)|2 +i=j

ai (r,r)aj(r,r). (1.13)

In the former contribution the phases vanish. In the latter, the dephasing of paired trajectoriesis large and random, and consequently their contribution vanishes on average.8 Theprobability is thus given by a sum of intensities and does not contain any interferenceterm (Figure 1.12):

Pcl (r,r)

j

|aj(r,r)|2. (1.14)

We shall call this classical term a Diffuson. In the weak disorder limit, that is, as long as

the wavelength is small compared to the elastic mean free path le, and for length scales

7 In this introduction, we do not seek to establish exact expressions for the various physical quantities, but simply to discusstheir behavior as a function of multiple scattering amplitudes. We therefore omit time or frequency dependence when it is notessential. More precise definitions are left for Chapters 3 and 4.

8 We show in the following section that the interference terms of expression (1.13) do not vanish completely, and are at the originof most of the quantum effects described in this book.


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aj

ai*

r r'

r'r

(a)

(b)aj

aj*

Figure 1.12 By averaging over disorder, the contribution from pairs of different trajectories (a)vanishes, leaving only terms corresponding to identical trajectories to contribute to the averageprobability (b).

larger than le, the Diffuson is well described by the solution of the diffusion equation t

D

Pcl (r,r, t) = (r r)(t), (1.15)

where D = vle/d is the diffusion constant, v the group velocity of the wave packet, and dthe dimension of space.

One quantity of particular importance is the probability of return to the initial pointPcl (r,r, t), as well as its integral over space Z(t). This last quantity is expressed as afunction of the eigenfrequencies denoted En associated with the diffusion equation (1.15)

Z(t) =

drPcl (r,r, t) =

n

eEnt (1.16)

for t > 0. For example, for a system9 of volume , we have

Z(t) = (4Dt)d/2

. (1.17)

The dependence as a function of the space dimensionality d plays an essential role, and

the physical properties are accordingly more sensitive to the effects of multiple scatteringwhen the dimensionality is small, since the return probability increases with decreasing d.For a finite system of volume = Ld, boundary conditions may play an important role,

since they reflect the nature of the coupling to the environment on which Z(t) depends. Thisintroduces a new characteristic time

D = L2/D (1.18)calledthediffusiontimeorThouless time.Itrepresentsthetimetodiffusefromoneboundary

of the sample to the other. Ift D, the effect of boundaries is not felt, the diffusion is freeand expressed by relation (1.17). If, on the other hand, t D, the entire volume is exploredby the random walk, we are in the ergodic regime, and Z(t) 1. A characteristic energy isassociated with D and called the Thouless energy Ec = /D.9 We ignore boundary effects, and thus are dealing with free diffusion.


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1.7 Classical probability and quantum crossings 17

1.7.1 Quantum crossings

Taking the second contribution of(1.13) to be zero amounts to neglecting all interferenceeffects. In fact, even after averaging over disorder, this contribution is not rigorously zero.There still remain terms describing pairs of distinct trajectories, i = j, which are sufficientlyclosethattheirdephasingissmall.Asanexample,considerthecaseofFigure 1.13(a), wherethe two trajectories in a Diffuson follow the same sequence of scatterings but cross, forminga loop with counter propagating trajectories.10 This notion of crossing is essential becauseit is at the origin of coherent effects such as weak localization, long range light intensitycorrelations, or universal conductance fluctuations. As such, it is useful to develop intuitionabout them. Figure 1.13 shows that one such crossing mixes four complex amplitudes andpairs them in different ways. The crossing, also called a Hikami box, is an object whichpermutes amplitudes [13]. For the induced dephasing to be smaller than 2 , the trajectories

must be as close to each other as possible, and the crossing must be localized in space, thatis, on a scale of the order of the elastic mean free path le. We shall see that the volumeassociated with a crossing in d dimensions is of order d1le. This may be interpreted byattributing a length vt to a Diffuson the object built with paired trajectories propagatingduring a time t, where v is the group velocity, and a cross section d1, giving a volumed1 vt.

(a)

(b)

Figure 1.13 (a) The crossing of trajectories contributing to a Diffuson leads to a new pairing ofamplitudes. (b) The pairing of four amplitudes resulting from the crossing of two Diffusons.

10 These quantum crossings, which exchange twoamplitudes, shouldnot be confused with theself-crossings of a classical randomwalk.


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To evaluate the importance of quantum effects, let us estimate the probability thattwo Diffusons will cross, as in Figure 1.13(b). This probability, for a time interval dt,is proportional to the ratio of the Diffuson volume to the system volume = Ld, that is,

dp

(t)=

d1v dt

1

g

dt

D. (1.19)

In this expression, we have explicitly indicated the diffusion time D = L2/D.Wehavealsointroduced a dimensionless number g, proportional to the inverse ratio of the two volumesd1vD/ . We will show that this number is none other than the classical electricalconductance g = Gcl /(e2/h), in units of the quantum of conductance e2/h (relation 7.22).

When the disordered medium is coupled to leads, the diffusing waves escape fromthe system in a time of the order of D which therefore sets the characteristic time for

diffusivetrajectories.Therefore,theprobabilityforacrossingduringthetime D is inverselyproportional to the dimensionless conductance, namely

p(D) =D

0dp(t) 1

g. (1.20)

This parameter allows us to evaluate the importance of quantum corrections to the classicalbehavior. In the limit of weak disorder le, the conductance g is large, so the crossingprobability, and hence the effects of coherence, are small.

The quantum crossings, and the dephasing which they induce, introduce a correction tothe classical probability (1.14). It is the combination of these crossings, the interferencethat they describe, and the spatially long range nature of the Diffuson, which allows thepropagation of coherent effects over the entire system. These are the effects which lie atthe basis of mesoscopic physics. The simple argument developed here straightforwardlyimplies that the quantum corrections to classical electron transport are of the order Gcl1/g,that is to say, e2/h.

In the limit of weak disorder, the crossings are independent of each other. This allows us

to write successive corrections to the classical probability as a function of the number ofcrossings, that is, as a power series in 1/g.

1.8 Objectives

This book deals with coherent multiple scattering of electronic or electromagnetic wavesin disordered media, in the limit where the wavelength11 = 2/k is small compared tothe elastic mean free path le. This is the limit of weak disorder. It is possible to develop

a general framework for the description of a large number of physical phenomena, whichwere effectively predicted, observed, and explained, by employing a small number of rathergeneral ideas. In this section, we briefly outline these ideas, and indicate the relevant chapterin which these phenomena are discussed.

11 For electrons, is the Fermi wavelength.


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1.8 Objectives 19

Weak localization corrections to the conductivity (Chapter 7) and the coherent

backscattering peak (Chapter 8)

One particularly important example where the notion of quantum crossing appears is thatof electron transport in a weakly disordered conductor. Consider, for example, transport

across a sample of size L. The conductance corresponding to the classical probability is theclassical or Drude conductance Gcl. The quantum correction to the probability leads to acorrection to the conductance.

This correction associated with a single crossing is of order 1/g, but it depends as wellon the distribution of loops, that is, on the closed diffusive trajectories (Figure 1.14), whosenumber is given by the spatial integral (1.16), namely by the integrated probability ofreturning to the origin Z(t). The probability po(D) of crossing the sample with a singlequantum crossing (one loop) is of the form

po(D) 1g

D

0Z(t)

dt

D, (1.21)

where D =L2/D. We obtain the relative correction to the average conductance G = G Gcl as,

G

Gcl po(D) (1.22)

The minus sign in this correction indicates that taking a quantum crossing and therefore aclosed loop into account has the effect of reducing the average conductance. This is calledthe weak localization correction.

(a)

(b)

Figure 1.14 The crossing of a Diffuson with itself (b) leads to a quantum correction to the classicalDrude conductivity (a).


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We note that the two multiple scattering trajectories that form a loop evolve in oppositedirections. If the system is time reversal invariant, the two amplitudes associated withthese trajectories j and jT are identical ajT(r,r) = aj(r,r) so that their product is equal tothe product of two amplitudes propagating in the same direction. If there are processeswhich break this invariance, then the weak localization correction vanishes. This pairing oftime-reversed conjugate trajectories is called a Cooperon.

This pairing closely resembles that described in the optical counterpart of section 1.4 andFigure 1.10(b), which corresponds to time-reversed multiple scattering amplitudes aj(r1,r2)and a

jT(r1,r2). For the geometry of a semi-infinite disordered medium, and a plane wave

incidentalongthedirectionk whichemergesalongk,theaveragereflectedintensity I(k,k)(also called the average albedo) depends on the angle between the directionsk andk. From(1.8), we have

I(k,k)

dr dr Pcl (r,r)1 + ei(k+k)(rr)

(1.23)

We identify |f(r,r)|2 with the Diffuson Pcl (r,r) whose endpoints r and r are taken tobe close to the interface between the diffusive medium and the vacuum. The first term inthe brackets is the phase independent classical contribution, while the interference term hasan angular dependence around the backscattering direction k k. The albedo thereforeexhibits a peak in this direction, called the coherent backscattering peak, whose intensity

is twice the classical value.

Correlations of speckle patterns (Chapter 12)

For a given realization of disorder, the intensity distribution of a light wave undergoingmultiple scattering is a random distribution of dark and bright spots (Figure 1.2) calleda speckle pattern. This interference pattern, arising from the superposition of complexamplitudes, constitutes a fingerprint of the specific disorder configuration. In order tocharacterize a speckle pattern, we can measure the angular distribution of the transmitted

(or reflected) intensity using the geometry of a slab of thicknessL.Inthiscase,onemeasuresthe normalized intensity Tab transmitted in the direction sb and corresponding to a waveincident in the direction sa (see Figure 12.2). On average, the transmission coefficient Tabdepends only slightly on the directions sa and sb,andwedenoteit T. The angular correlationof the speckle is defined by

Cabab = TabTab

T2

, (1.24)

where Tab = Tab T. The fluctuations of the speckle, for a given incidence direction sa,are described by the quantity Cabab = 2Tab/T2 which happens, as we shall see, to be equalto 1, yielding

T2ab

= 2 T2. (1.25)


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1.8 Objectives 21

aa

a'a'

bb

b'

(b)

b'

b'

a

a

a'

(c)

a'

b

b

b'

(a)

(d) (e)

aa

a'

a'

b'

b

b'

b

a

a

a'

a'

b

b

b'b'

aa

a'a'

b

b

b'

b'

Figure 1.15 (a) The angular correlation function of a speckle pattern is built from the product offour complex amplitudes corresponding to four plane waves incoming along directions sa and sa andemergingalong sb and sb .Themaincontributionsareobtainedbypairingtheamplitudestwobytwotoform Diffusons. This gives rise to contributions (b) and (c). Contribution (c), which corresponds to the

correlation function C(1)abab decays exponentially in angle. Contribution (d) contains one quantum

crossing, while (e) has two quantum crossings. In this last case, we note that the corresponding

correlation function has no angular dependence.

This result, which constitutes the Rayleigh law, describes the most visible aspect ofa speckle pattern, namely, its granularity, with relative fluctuations of the order ofunity.

In contrast to the probability (average conductance or light intensity), a correlation

function such as (1.24) is the product offour complex amplitudes (Figure 1.15(a)). Whenaveraging over disorder, the only important contributions are obtained by pairing theseamplitudes so as to form Diffusons or Cooperons. Neglecting, in a first step, the possibilityof a quantum crossing of two Diffusons, there are two possibilities, shown in Figures1.15(b, c). The first is the product of two average intensities Tab and Tab . The second gives


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the principal contribution to the correlation function (1.24), denoted C(1)abab . It is non-zero

only ifsa sa = sb sb and it decays exponentially as a function ofk|sa sa |/L, that is,over a very small angular range.

It is also possible to pair the amplitudes by interposing one or more quantum crossings.This results in corrections to the angular correlation function in powers of 1/g. The firstone, denoted C(2)

abab has a single crossing, and is shown in Figure 1.15(d). The presenceof a crossing imposes constraints on the pairing of the amplitudes, and thus gives rise to adifferent angular dependence. We will show in section 12.4.2 that C(2)

abab decays as a powerofk|sa sa |/L, instead of an exponential decay. It has a weight 1/g 1 as compared tothe term with no crossing.

The contribution C(3)abab , with two crossings is sketched in Figure 1.15(e). Because of the

two-crossing structure, this contribution has no angular dependence, i.e., it yields a uniformbackground to the correlation function. This result is characteristic of coherent multiplescattering, that is to say, of the combined effect of quantum crossings and of their long rangepropagation through the Diffusons. Upon averaging the total angular correlation functionover all directions of incident and emergent waves, only this last contribution survives,which constitutes the analog for waves of universal conductance fluctuations [1416].

Universal conductance fluctuations (Chapter 11)

The considerations obtained for electromagnetic waves are easily transposed to the case ofelectrons in a weakly disordered metal, and in this context lead to conductance fluctuations.In the mesoscopic regime, these fluctuations differ considerably from the classical result:they are universal and of order e2/h (see section 1.5). This results from the existence

of quantum crossings. More precisely, calculation of the fluctuations G2 = G2 G2involves the pairing of four complex amplitudes paired into two Diffusons. Moreover,in the framework of the Landauer formalism (Appendix A7.2), the conductance, in units ofe2/h, is related to the transmission coefficient Tab summed over all incident and emergent

directions a and b. Thus, as for the speckle angular correlations, it can be shown that theterm with no quantum crossing corresponds to G2. The contribution from a single crossing

vanishes upon summation over the emergent directions. In contrast, the term with twoquantum crossings has no angular dependence (Figure 1.15(e)) and it gives a universalvariance G2 proportional to G2

cl/g2 = (e2/h)2.

We note that, as for the weak localization correction, the variance G2 depends on thedistribution ofloops. Here the loops result from two crossings (Figure 1.15(e)). For a loopof length vt, the choice of the relative position of the two crossings introduces an additional

factor d1vt/ t/(gD) in the integral (1.21). We thus deduce that

G2

G2cl

1g2

D0

Z(t)t dt

2D

(1.26)


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1.8 Objectives 23

This expression resembles the relative correction of weak localization (1.21, 1.22), butthe additional factor t has important consequences. The dependence Z(t) td/2 ofthe integrated return probability to the origin implies that weak localization correctionis universal for d < 2, while conductance fluctuations are universal for d < 4.

Dephasing (Chapter 6)

The interference effects discussed earlier result from the existence of quantum crossings.They depend on the coherence of the wave-scatterer system, and may be modified in thepresenceofdephasingprocesses.Suchprocessesarerelatedtoadditionaldegreesoffreedomwhich we may divide into three classes, examples of which are as follows.

External field: uniform magnetic field, AharonovBohm flux.

Degrees of freedom of the scattered wave: electron spin and photon polarization.

Degrees of freedom of the scatterers: magnetic impurities, environment induced by otherelectrons, motion of scatterers, internal quantum degrees of freedom (atomic Zeemansublevels).

Let us first consider the case of multiple scattering of electrons, now in the presenceof a magnetic field. Full coherence implies that time-reversed trajectories have the sameamplitude. This is no longer the case in the presence of a magnetic field, which induces adephasing between conjugate trajectories:

ajT(r,r) = aj(r,r) eij (r,r). (1.27)Using(1.13)andthediscussiononpage 20, thecorrectiontothereturnprobabilityassociatedwith the Cooperon, which we denote Pc, is of the form

Pc(r,r)

j

|aj(r, r)|2 eij (r,r), (1.28)

where j(r,r) is the phase difference accumulated along the closed trajectories. The

dephasing due to a magnetic field is

j(r,r) = 2e

rr

A dl, (1.29)

where the factor 2 comes from the fact that both of the paired trajectories accumulate thesame phase, but with opposite signs, so that their difference adds. The coherent contributionto the return probability is thus affected by this phase factor, and the weak localizationcorrection to the electrical conductance takes the form

G

Gcl

dt Z(t)

ei(t)

(1.30)

whereei(t)

is the average phase factor of the ensemble of trajectories of length vt.


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The magnetic field thus appears as a way to probe phase coherence. In particular, theAharonovBohm effect gives rise to the spectacular SharvinSharvin effect, in which theaverage conductance has a contribution which oscillates with period h/2e (section 7.6.2).To evaluate the coherent contribution in the presence of a magnetic field, we must look forsolutions of the covariant diffusion equation which replaces (1.15)

tD

r + i 2e

A(r)

2P(r,r, t) = (r r)(t). (1.31)

The dephasing (1.29) resulting from the application of a magnetic field changes the phaseaccumulated along a multiple scattering trajectory. In contrast, to describe the couplingto other degrees of freedom, we average locally the relative dephasing between the twocomplex amplitudes which interfere. This results from our incomplete knowledge of theinternal quantum state of the scatterers. The average over the scatterersdegrees of freedomgives rise to an irreversible dephasing, which we describe using a finite phase coherencetime . We show in Chapter 6 that the contributions of the Diffuson and the Cooperon aremodified by a phase factor which in general decreases exponentially with time:

ei(t)

et/ . (1.32)

Here

indicates an average over both disorder and these other degrees of freedom. Thedetermination of the phase coherence time requires the evaluation of the average in (1.32).This notion of dephasing extends to any perturbation whose effect is to modify the phaserelation between paired multiple scattering trajectories. We present such an example justbelow.

Dynamics of scatterers diffusing wave spectroscopy (Chapters 6and9)

When properly understood, a source of dephasing is not necessarily a nuisance, but maybe used to study the properties of the diffusive medium. Thus, in the case of scattering ofelectromagnetic waves, it is possible, by measuring the time autocorrelation function ofthe electromagnetic field, to take advantage of the coherent multiple scattering to deduceinformation about the dynamics of the scatterers which is characterized by a time scale b.In fact, since the scatterer velocity is usually much smaller than that of the wave, if we sendlight pulses at different times, 0 and T, we can probe different realizations of the randompotential. The paired trajectories thus explore different configurations separated by a timeT. This results in a dephasing which depends on the motion of the scatterers during the timeinterval T. The time correlation function of the electric field Eat a point r (with a source at

r0) is of the form

E(r, T)E(r, 0)

j

aj(r0,r, T)a

j (r0,r, 0)

,

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