Indeed, the materialpresented here aims at leading the reader already acquainted with the basics in quantum statistical mechanics, spectral theory of linear operators, C ∗-dynamicalsyste
Trang 1Lecture Notes in Mathematics
Trang 3Library of Congress Control Number:
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Universit Claude Bernard Lyon 1
21 av Claude Bernard
France
69622 Villeurbanne Cedex
Alain JoyeInstitut FourierUniversit de Grenoble 1
BP 74France
Trang 4This volume is the third and last of a series devoted to the lecture notes of theGrenoble Summer School on “Open Quantum Systems” which took place at theInstitut Fourier from June 16thto July 4th2003 The contributions presented in thisvolume correspond to expanded versions of the lecture notes provided by the authors
to the students of the Summer School The corresponding lectures were scheduled
in the last part of the School devoted to recent developments in the study of OpenQuantum Systems
Whereas the first two volumes were dedicated to a detailed exposition of themathematical techniques and physical concepts relevant in the study of Open Sys-
tems with no a priori pre-requisites, the contributions presented in this volume
request from the reader some familiarity with these aspects Indeed, the materialpresented here aims at leading the reader already acquainted with the basics in
quantum statistical mechanics, spectral theory of linear operators, C ∗-dynamicalsystems, and quantum stochastic differential equations to the front of the currentresearch done on various aspects of Open Quantum Systems Nevertheless, peda-gogical efforts have been made by the various authors of these notes so that thisvolume should be essentially self-contained for a reader with minimal previous ex-posure to the themes listed above In any case, the reader in need of complementscan always turn to these first two volumes
The topics covered in these lectures notes start with an introduction to equilibrium quantum statistical mechanics The definitions of the physical concepts
non-as well non-as the necessary mathematical framework suitable for their description aredeveloped in a general setup A simple non-trivial physically relevant example ofindependent electrons in a device connected to several reservoirs is treated in details
in the second part of these notes in order to illustrate the notions of non-equilibriumsteady states, entropy production and other thermodynamical notions introducedearlier
The next contribution is devoted to the many aspects of the Fermi Golden Ruleused within the Hamiltonian approach of Open Quantum Systems in order to derive
Trang 5VI Preface
a Markovian approximation of the dynamics In particular, the weak coupling orvan Hove limit in both a time-dependent and stationary setting are discussed in anabstract framework These results are then applied to the case of small systems in-teracting with reservoirs, within different algebraic representations of the relevantmodels The links between the Fermi Golden Rule and the Detailed Balance Condi-tion as well as explicit formulas are also discussed in different physical situations.The third text of this volume is concerned with the notion of decoherence,relevant, in particular, for a discussion of the measurement theory in Quantum Me-chanics The properties of the large time behavior of the dynamics reduced to a sub-system, which is not Markovian in general, are first reviewed Then, the so-calledisometric-sweeping decomposition of a dynamical semigroup is presented in an gen-eral setup and its links with decoherence phenomena are exposed Applications tophysical models such as spin systems or to the unravelling of the classical dynamics
in certain regimes are then provided The properties of dynamical semigroups onCCR algebras are discussed in details in the final section
The following contribution is devoted to a systematic study of the long timebehavior of quantum dynamical semigroups, as they arise in Markovian approxi-mations More precisely, the key notions for applications of stationary states, con-vergence towards equilibrium as well as transience and recurrence of such quantumMarkov semigroups are developed in an abstract framework In particular, condi-
tions on unbounded operators defined in the sense of forms to generate a bona fide
quantum dynamical semigroup are formulated, as well as general criteria insuringthe existence of stationary states for a given quantum dynamical semigroup Therelations between return to equilibrium for a quantum dynamical semigroup and theproperties of its generator are also discussed All these concepts are then illustrated
by applications to concrete physical models used in quantum optics
The last notes of this volume provide a detailed account of the process of ual measurements in quantum optics, considered as an application of quantum sto-chastic calculus The basics of this quantum stochastic calculus and the modelization
contin-of system-field interactions constructed on it are first explained Then, indirect andcontinual measurement processes and the corresponding master equations are in-troduced and discussed Physical interpretations of computations performed withinthis quantum stochastic modelization framework are spelled out for various specificprocesses in quantum optics
As revealed by this outline, the treatment of the different physical models posed in this volume makes use of several tools and approximations discussed from
pro-a mpro-athempro-aticpro-al point of view, both in the Hpro-amiltonipro-an pro-and Mpro-arkovipro-an pro-appropro-ach Atthe same time, the different mathematical topics addressed here are illustrated byphysically relevant applications in the theory of Open Quantum Systems We be-lieve the contact made between the practicians of the Markovian and Hamiltonianduring the School itself and within the contributions of these volumes is useful andwill prove to be even more fruitful for the future developments of the field
Trang 6Let us close this introduction by pointing out that some recent results in thetheory of Open Quantum Systems are not discussed in these notes These includenotably the descriptions of return to equilibrium by means of renormalization analy-sis and scattering techniques These demanding approaches were not addressed
in the Grenoble Summer School, because a reasonably complete treatment wouldsimply have required too much time
We hope the reader will benefit from the pedagogical efforts provided by allauthors of these notes in order to introduce the concepts and problems, as well asrecent developments in the theory of Open Quantum Systems
Claude-Alain Pillet
Trang 7Topics in Non-Equilibrium Quantum Statistical Mechanics
Walter Aschbacher, Vojkan Jakˇsi´c, Yan Pautrat, and Claude-Alain Pillet 1
1 Introduction 2
2 Conceptual Framework 3
3 Mathematical Framework 5
3.1 Basic Concepts 5
3.2 Non-Equilibrium Steady States (NESS) and Entropy Production 8
3.3 Structural Properties 10
3.4 C ∗-Scattering and NESS 11
4 Open Quantum Systems 14
4.1 Definition 14
4.2 C ∗-Scattering for Open Quantum Systems 15
4.3 The First and Second Law of Thermodynamics 17
4.4 Linear Response Theory 18
4.5 Fermi Golden Rule (FGR) Thermodynamics 22
5 Free Fermi Gas Reservoir 26
5.1 General Description 26
5.2 Examples 30
6 The Simple Electronic Black-Box (SEBB) Model 34
6.1 The Model 34
6.2 The Fluxes 36
6.3 The Equivalent Free Fermi Gas 37
6.4 Assumptions 40
7 Thermodynamics of the SEBB Model 43
7.1 Non-Equilibrium Steady States 43
7.2 The Hilbert-Schmidt Condition 44
7.3 The Heat and Charge Fluxes 45
7.4 Entropy Production 46
7.5 Equilibrium Correlation Functions 47
7.6 Onsager Relations Kubo Formulas 49
Trang 88 FGR Thermodynamics of the SEBB Model 50
8.1 The Weak Coupling Limit 50
8.2 Historical Digression—Einstein’s Derivation of the Planck Law 53
8.3 FGR Fluxes, Entropy Production and Kubo Formulas 54
8.4 From Microscopic to FGR Thermodynamics 56
9 Appendix 58
9.1 Structural Theorems 58
9.2 The Hilbert-Schmidt Condition 60
References 63
Fermi Golden Rule and Open Quantum Systems Jan Derezi´nski and Rafał Fr¨uboes 67
1 Introduction 68
1.1 Fermi Golden Rule and Level Shift Operator in an Abstract Setting 68
1.2 Applications of the Fermi Golden Rule to Open Quantum Systems 69 2 Fermi Golden Rule in an Abstract Setting 71
2.1 Notation 71
2.2 Level Shift Operator 72
2.3 LSO for C0∗-Dynamics 73
2.4 LSO for W ∗-Dynamics 74
2.5 LSO in Hilbert Spaces 74
2.6 The Choice of the ProjectionP 75
2.7 Three Kinds of the Fermi Golden Rule 75
3 Weak Coupling Limit 77
3.1 Stationary and Time-Dependent Weak Coupling Limit 77
3.2 Proof of the Stationary Weak Coupling Limit 80
3.3 Spectral Averaging 83
3.4 Second Order Asymptotics of Evolution with the First Order Term 85
3.5 Proof of Time Dependent Weak Coupling Limit 87
3.6 Proof of the Coincidence of Mstand Mdynwith the LSO 88
4 Completely Positive Semigroups 88
4.1 Completely Positive Maps 89
4.2 Stinespring Representation of a Completely Positive Map 89
4.3 Completely Positive Semigroups 90
4.4 Standard Detailed Balance Condition 91
4.5 Detailed Balance Condition in the Sense of Alicki-Frigerio-Gorini-Kossakowski-Verri 93
5 Small Quantum System Interacting with Reservoir 93
5.1 W ∗-Algebras 94
5.2 Algebraic Description 95
5.3 Semistandard Representation 95
5.4 Standard Representation 96
Trang 9Contents XI
6 Two Applications of the Fermi Golden Rule
to Open Quantum Systems 97
6.1 LSO for the Reduced Dynamics 97
6.2 LSO for the Liouvillean 99
6.3 Relationship Between the Davies Generator and the LSO for the Liouvillean in Thermal Case 100
6.4 Explicit Formula for the Davies Generator 103
6.5 Explicit Formulas for LSO for the Liouvillean 104
6.6 Identities Using the Fibered Representation 106
7 Fermi Golden Rule for a Composite Reservoir 108
7.1 LSO for a Sum of Perturbations 108
7.2 Multiple Reservoirs 109
7.3 LSO for the Reduced Dynamics in the Case of a Composite Reservoir 110
7.4 LSO for the Liovillean in the Case of a Composite Reservoir 111
A Appendix – One-Parameter Semigroups 112
References 115
Decoherence as Irreversible Dynamical Process in Open Quantum Systems Philippe Blanchard, Robert Olkiewicz 117
1 Physical and Mathematical Prologue 118
1.1 Physical Background 118
1.2 Environmental Decoherence 119
1.3 Algebraic Framework 120
1.4 Quantum Dynamical Semigroups 121
1.5 A Model of a Discrete Pointer Basis 123
2 The Asymptotic Decomposition of T 126
2.1 Notation 126
2.2 Dynamics in the Markovian Regime 127
2.3 The Unitary Decomposition of T2 130
2.4 The Isometric-Sweeping Decomposition 133
2.5 Remarks 135
3 Review of Decoherence Effects in Infinite Spin Systems 138
3.1 Infinite Spin Systems 138
3.2 Continuous Pointer States [10] 139
3.3 Decoherence-Induced Spin Algebra [6] 143
3.4 From Quantum to Classical Dynamical Systems [38] 146
4 Dynamical Semigroups on CCR Algebras 148
4.1 Algebras of Canonical Commutation Relations (CCR) 148
4.2 Promeasures on Locally Convex Topological Vector Spaces 149
4.3 Perturbed Convolution Semigroups of Promeasures 151
4.4 Quantum Dynamical Semigroups on CCR Algebras 153
4.5 Example: Quantum Brownian Motion 155
5 Outlook 157
Trang 10References 158
Notes on the Qualitative Behaviour of Quantum Markov Semigroups Franco Fagnola and Rolando Rebolledo 161
1 Introduction 162
1.1 Preliminaries 164
2 Ergodic Theorems 165
3 The Minimal Quantum Dynamical Semigroup 167
4 The Existence of Stationary States 172
4.1 A General Result 172
4.2 Conditions on the Generator 174
4.3 Examples 178
4.4 A Multimode Dicke Laser Model 178
4.5 A Quantum Model of Absorption and Stimulated Emission 182
4.6 The Jaynes-Cummings Model 183
5 Faithful Stationary States and Irreducibility 184
5.1 The Support of an Invariant State 184
5.2 Subharmonic Projections The Case M = L(h) 186
5.3 Examples 188
6 The Convergence Towards the Equilibrium 189
6.1 Main Results 190
6.2 Examples 192
7 Recurrence and Transience of Quantum Markov Semigroups 194
7.1 Potential 194
7.2 Defining Recurrence and Transience 198
7.3 The Behavior of a d-Harmonic Oscillator 201
References 203
Continual Measurements in Quantum Mechanics and Quantum Stochastic Calculus Alberto Barchielli 207
1 Introduction 208
1.1 Three Approaches to Continual Measurements 208
1.2 Quantum Stochastic Calculus and Quantum Optics 208
1.3 Some Notations: Operator Spaces 209
2 Unitary Evolution and States 210
2.1 Quantum Stochastic Calculus 210
2.2 The Unitary System–Field Evolution 217
2.3 The System–Field State 223
2.4 The Reduced Dynamics 225
2.5 Physical Basis of the Use of QSC 228
3 Continual Measurements 230
3.1 Indirect Measurements on S H 230
3.2 Characteristic Functionals 233
3.3 The Reduced Description 241
Trang 11Contents XIII
3.4 Direct Detection 247
3.5 Optical Heterodyne Detection 252
3.6 Physical Models 257
4 A Three–Level Atom and the Shelving Effect 258
4.1 The Atom–Field Dynamics 259
4.2 The Detection Process 262
4.3 Bright and Dark Periods: The V-Configuration 264
4.4 Bright and Dark Periods: The Λ-Configuration 267
5 A Two–Level Atom and the Spectrum of the Fluorescence Light 269
5.1 The Dynamical Model 270
5.2 The Master Equation and the Equilibrium State 274
5.3 The Detection Scheme 277
5.4 The Fluorescence Spectrum 282
References 288
Index of Volume III 293
Information about the other two volumes 297
Contents of Volume I 298
Index of Volume I 302
Contents of Volume II 306
Index of Volume II 309
Trang 12“F Brioschi”
Piazza Leonardo da Vinci 32
20133 Milano, Italye-mail: franco.fagnola@polimi.it
Rafał Fr ¨uboes
Department of MathematicalMethods in Physics
Warsaw University, Ho˙za 7400-682, Warsaw, Polande-mail: fruboes@fuw.edu.pl
Robert Olkiewicz
Institute of Theoretical PhysicsUniversity of Wrocław
pl M Borna 950-204 Wrocław, Polande-mail: rolek@ift.uni.wroc.pl
Trang 13XVI List of Contributors
Trang 14Walter Aschbacher1, Vojkan Jakˇsi´c2, Yan Pautrat3, and Claude-Alain Pillet4
1
Zentrum Mathematik M5, Technische Universit¨at M¨unchen,
D-85747 Garching, Germany
e-mail: hbar@ma.tum.de
2
Department of Mathematics and Statistics, McGill University,
805 Sherbrooke Street West, Montreal, QC, H3A 2K6, Canada
e-mail: jaksic@math.mcgill.co
3 Laboratoire de Math´ematiques, Universit´e Paris-Sud,
91405 Orsay cedex, France
e-mail: yan.pautrat@math.u-psud.fr
4
CPT-CNRS, UMR 6207, Universit´e du Sud,
Toulon-Var, B.P 20132, 83957 La Garde Cedex, France
e-mail: pillet@univ-tln.fr
1 Introduction 2
2 Conceptual Framework 3
3 Mathematical Framework 5
3.1 Basic Concepts 5
3.2 Non-Equilibrium Steady States (NESS) and Entropy Production 8
3.3 Structural Properties 10
3.4 C ∗-Scattering and NESS 11
4 Open Quantum Systems 14
4.1 Definition 14
4.2 C ∗-Scattering for Open Quantum Systems 15
4.3 The First and Second Law of Thermodynamics 17
4.4 Linear Response Theory 18
4.5 Fermi Golden Rule (FGR) Thermodynamics 22
5 Free Fermi Gas Reservoir 26
5.1 General Description 26
5.2 Examples 30
6 The Simple Electronic Black-Box (SEBB) Model 34
6.1 The Model 34
6.2 The Fluxes 36
Trang 152 Walter Aschbacher et al.
6.3 The Equivalent Free Fermi Gas 37
6.4 Assumptions 40
7 Thermodynamics of the SEBB Model 43
7.1 Non-Equilibrium Steady States 43
7.2 The Hilbert-Schmidt Condition 44
7.3 The Heat and Charge Fluxes 45
7.4 Entropy Production 46
7.5 Equilibrium Correlation Functions 47
7.6 Onsager Relations Kubo Formulas 49
8 FGR Thermodynamics of the SEBB Model 50
8.1 The Weak Coupling Limit 50
8.2 Historical Digression—Einstein’s Derivation of the Planck Law 53
8.3 FGR Fluxes, Entropy Production and Kubo Formulas 54
8.4 From Microscopic to FGR Thermodynamics 56
9 Appendix 58
9.1 Structural Theorems 58
9.2 The Hilbert-Schmidt Condition 60
References 63
1 Introduction
These lecture notes are an expanded version of the lectures given by the second and the fourth author in the summer school ”Open Quantum Systems” held in Grenoble, June 16–July 4, 2003 We are grateful to St´ephane Attal and Alain Joye for their hospitality and invitation to speak
The lecture notes have their root in the recent review article [JP4] and our goal has been to extend and complement certain topics covered in [JP4] In particular, we will discuss the scattering theory of non-equilibrium steady states (NESS) (this topic has been only quickly reviewed in [JP4]) On the other hand, we will not discuss the spectral theory of NESS which has been covered in detail in [JP4] Although the lecture notes are self-contained, the reader would benefit from reading them in parallel with [JP4]
Concerning preliminaries, we will assume that the reader is familiar with the material covered in the lecture notes [At, Jo, Pi] On occasion, we will mention or use some material covered in the lectures [D1, Ja]
As in [JP4], we will work in the mathematical framework of algebraic quantum statistical mechanics The basic notions of this formalism are reviewed in Section 3
In Section 4 we introduce open quantum systems and describe their basic properties The linear response theory (this topic has not been discussed in [JP4]) is described
Trang 16in Subsection 4.4 The linear response theory of open quantum systems (Kubo mulas, Onsager relations, Central Limit Theorem) has been studied in the recentpapers [FMU, FMSU, AJPP, JPR2].
for-The second part of the lecture notes (Sections 6–8) is devoted to an example.The model we will discuss is the simplest non-trivial example of the Electronic
Black Box Model studied in [AJPP] and we will refer to it as the Simple Electronic Black Box Model (SEBB) The SEBB model is to a large extent exactly solvable—
its NESS and entropy production can be exactly computed and Kubo formulas can
be verified by an explicit computation For reasons of space, however, we will notdiscuss two important topics covered in [AJPP]—the stability theory (which is es-sentially based on [AM, BM]) and the proof of the Central Limit Theorem Theinterested reader may complement Sections 6–8 with the original paper [AJPP] andthe recent lecture notes [JKP]
Section 5, in which we discuss statistical mechanics of a free Fermi gas, is thebridge between the two parts of the lecture notes
Acknowledgment The research of V.J was partly supported by NSERC Part of
this work was done while Y.P was a CRM-ISM postdoc at McGill University andCentre de Recherches Math´ematiques in Montreal
2 Conceptual Framework
The concept of reference state will play an important role in our discussion of equilibrium statistical mechanics To clarify this notion, let us consider first a clas-sical dynamical system with finitely many degrees of freedom and compact phase
non-space X ⊂ R n The normalized Lebesgue measure dx on X provides a physically
natural statistics on the phase space in the sense that initial configurations sampledaccording to it can be considered typical (see [Ru4]) Note that this has nothing to
do with the fact that dx is invariant under the flow of the system—any measure of the form ρ(x)dx with a strictly positive density ρ would serve the same purpose.
The situation is completely different if the system has infinitely many degrees of
freedom In this case, there is no natural replacement for the Lebesgue dx In fact,
a measure on an infinite-dimensional phase space physically describes a dynamic state of the system Suppose for example that the system is Hamiltonian
thermo-and is in thermal equilibrium at inverse temperature β thermo-and chemical potential µ.
The statistics of such a system is described by the Gibbs measure (grand canonicalensemble) Since two Gibbs measures with different values of the intensive ther-
modynamic parameters β, µ are mutually singular, initial points sampled according
to one of them will be atypical relative to the other In conclusion, if a system hasinfinitely many degrees of freedom, we need to specify its initial thermodynamicstate by choosing an appropriate reference measure As in the finite-dimensionalcase, this measure may not be invariant under the flow It also may not be uniquelydetermined by the physical situation we wish to describe
Trang 174 Walter Aschbacher et al.
The situation in quantum mechanics is very similar The Schr¨odinger tation of a system with finitely many degrees of freedom is (essentially) uniquelydetermined and the natural statistics is provided by any strictly positive density ma-trix on the Hilbert space of the system For systems with infinitely many degrees
represen-of freedom there is no such natural choice The consequences represen-of this fact are ever more drastic than in the classical case There is no natural choice of a Hilbertspace in which the system can be represented To induce a representation, we mustspecify the thermodynamic state of the system by choosing an appropriate referencestate The algebraic formulation of quantum statistical mechanics provides a math-ematical framework to study such infinite system in a representation independentway
how-One may object that no real physical system has an infinite number of degrees offreedom and that, therefore, a unique natural reference state always exists There arehowever serious methodological reasons to consider this mathematical idealization.Already in equilibrium statistical mechanics the fundamental phenomena of phasetransition can only be characterized in a mathematically precise way within such anidealization: A quantum system with finitely many degrees of freedom has a uniquethermal equilibrium state Out of equilibrium, relaxation towards a stationary stateand emergence of steady currents can not be expected from the quasi-periodic timeevolution of a finite system
In classical non-equilibrium statistical mechanics there exists an alternativeapproach to this idealization A system forced by a non-Hamiltonian or time-dependent force can be driven towards a non-equilibrium steady state, providedthe energy supplied by the external source is removed by some thermostat This
micro-canonical point of view has a number of advantages over the canonical,
in-finite system idealization A dynamical system with a relatively small number ofdegrees of freedom can easily be explored on a computer (numerical integration, it-eration of Poincar´e sections, ) A large body of “experimental facts” is currentlyavailable from the results of such investigations (see [EM, Do] for an introduction
to the techniques and a lucid exposition of the results) From a more theoreticalperspective, the full machinery of finite-dimensional dynamical system theory be-
comes available in the micro-canonical approach The Chaotic Hypothesis
intro-duced in [CG1, CG2] is an attempt to exploit this fact It justifies phenomenologicalthermodynamics (Onsager relations, linear response theory, fluctuation-dissipationformulas, ) and has lead to more unexpected results like the Gallavotti-Cohen Fluc-tuation Theorem The major drawback of the micro-canonical point of view is thenon-Hamiltonian nature of the dynamics, which makes it inappropriate to quantum-mechanical treatment
The two approaches described above are not completely unrelated For ple, we shall see that the signature of a non-equilibrium steady state in quantummechanics is its singularity with respect to the reference state, a fact which is wellunderstood in the classical, micro-canonical approach (see Chapter 10 of [EM])
exam-More speculatively, one can expect a general equivalence principle for dynamical
(micro-canonical and canonical) ensembles (see [Ru5]) The results in this directionare quite scarce and much work remains to be done
Trang 183 Mathematical Framework
In this section we describe the mathematical formalism of algebraic quantum tistical mechanics Our presentation follows [JP4] and is suited for applications tonon-equilibrium statistical mechanics Most of the material in this section is wellknown and the proofs can be found, for example, in [BR1, BR2, DJP, Ha, OP, Ta].The proofs of the results described in Subsection 3.3 are given in Appendix 9.1
sta-3.1 Basic Concepts
The starting point of our discussion is a pair (O, τ), where O is a C ∗-algebra with
a unit I and τ is a C ∗-dynamics (a strongly continuous group R t → τ t of
∗-automorphisms of O) The elements of O describe physical observables of the
quantum system under consideration and the group τ specifies their time evolution.
The pair (O, τ) is sometimes called a C ∗-dynamical system.
In the sequel, by the strong topology onO we will always mean the usual norm
topology of O as Banach space The C ∗-algebra of all bounded operators on a
Hilbert spaceH is denoted by B(H).
A state ω on the C ∗-algebraO is a normalized (ω(I) = 1), positive (ω(A ∗ A) ≥
0), linear functional onO It specifies a possible physical state of the quantum
me-chanical system If the system is in the state ω at time zero, the quantum meme-chanical expectation value of the observable A at time t is given by ω(τ t (A)) Thus, states
evolve in the Schr¨odinger picture according to ω t = ω ◦ τ t The set E( O) of all
states onO is a convex, weak-∗ compact subset of the Banach space dual O ∗ofO.
A linear functional η ∈ O ∗ is called τ -invariant if η ◦ τ t = η for all t The set of
all τ -invariant states is denoted by E( O, τ) This set is always non-empty A state
ω ∈ E(O, τ) is called ergodic if
Let (H η , π η , Ω η) be the GNS representation associated to a positive linear
func-tional η ∈ O ∗ The enveloping von Neumann algebra of O associated to η is
Mη ≡ π η(O) ⊂ B(H η ) A linear functional µ ∈ O ∗ is normal relative to η or
µ( ·) = Tr(ρ µ π η(·)) Any η-normal linear functional µ has a unique normal
exten-sion to Mη We denote byN η the set of all η-normal states µ µ ⊂ N η
A state ω is ergodic iff, for all µ ∈ N ω and A ∈ O,
Trang 196 Walter Aschbacher et al.
For this reason ergodicity is sometimes called return to equilibrium in mean; see
[Ro1, Ro2] Similarly, ω is mixing (or returns to equilibrium) iff
lim
|t|→∞ µ(τ
t (A)) = ω(A),
for all µ ∈ N ω and A ∈ O.
Let η and µ be two positive linear functionals in O ∗ , and suppose that η ≥ φ ≥ 0
for some µ-normal φ implies φ = 0 We then say that η and µ are mutually singular (or orthogonal), and write η ⊥ µ An equivalent (more symmetric) definition is:
η ⊥ µ iff η ≥ φ ≥ 0 and µ ≥ φ ≥ 0 imply φ = 0.
Two positive linear functionals η and µ in O ∗are called disjoint ifN η ∩N µ =∅.
If η and µ are disjoint, then η ⊥ µ The converse does not hold— it is possible that
η and µ are mutually singular but not disjoint.
To elucidate further these important notions, we recall the following well-knownresults; see Lemmas 4.1.19 and 4.2.8 in [BR1]
Proposition 3.1 Let µ1, µ2∈ O ∗ be two positive linear functionals and µ = µ
(iii) The GNS representation(H µ , π µ , Ω µ ) is a direct sum of the two GNS
repre-sentations(H µ1, π µ1, Ω µ1) and ( H µ2, π µ2, Ω µ2), i.e.,
H µ=H µ1⊕ H µ2, π µ = π µ1⊕ π µ2, Ω µ = Ω µ1⊕ Ω µ2.
Proposition 3.2 Let µ1, µ2∈ O ∗ be two positive linear functionals and µ = µ
1+
µ2 Then the following statements are equivalent:
(i) µ1and µ2are disjoint.
(ii) There exists a projection P in π µ(O) ∩ π µ(O) such that
Let η, µ ∈ O ∗ be two positive linear functionals The functional η has a unique
decomposition η = η n + η s , where η n , η s are positive, η n s ⊥ µ The
uniqueness of the decomposition implies that if η is τ -invariant, then so are η nand
Trang 20(Hun, πun) is a faithful representation It is called the universal representation of
O Mun ⊂ B(Hun) is its universal enveloping von Neumann algebra For any ω ∈ E( O) the map
πun(O) → π ω(O)
πun(A) → π ω (A),
extends to a surjective∗-morphism ˜π ω : Mun → M ω It follows that ω uniquely
extends to a normal state ˜ω( ·) ≡ (Ω ω , ˜ π ω(·)Ω ω) on Mun Moreover, one easilyshows that
Ker ˜π ω={A ∈ Mun| ˜ν(A) = 0 for any ν ∈ N ω }. (1)Since Ker ˜π ω is a σ-weakly closed two sided ideal in Mun, there exists an orthog-
onal projection p ω ∈ Mun∩ M
un such that Ker ˜π ω = p ωMun The orthogonal
projection z ω ≡ I − p ω ∈ Mun∩ M
unis called the support projection of the state
ω The restriction of ˜ π ω to z ωMun is an isomorphism between the von Neumann
algebras z ωMunand Mω We shall denote by φ ωthe inverse isomorphism
Let now η, µ ∈ O ∗be two positive linear functionals By scaling, without loss
of generality we may assume that they are states Since ˜η is a normal state on Mun
it follows that ˜η ◦ φ µ is a normal state on Mµ and hence that η n ≡ ˜η ◦ φ µ ◦ π µ
defines a µ-normal positive linear functional on O Moreover, from the relation
φ µ ◦ π µ (A) = z µ πun(A) it follows that
η n (A) = (Ω η , ˜ π η (z µ )π η (A)Ω η ).
Setting
η s (A) ≡ (Ω η , ˜ π η (p µ )π η (A)Ω η ),
we obtain a decomposition η = η n + η s To show that η s ⊥ µ let ω be a µ-normal
positive linear functional on O such that η s ≥ ω By the unicity of the normal
extension ˜η sone has ˜η s (A) = ˜ η(p µ A) for A ∈ Mun Since πun(O) is σ-strongly
dense in Munit follows from the inequality ˜η s ◦πun ≥ ˜ω◦πunthat ˜η(p µ A) ≥ ˜ω(A)
for any positive A ∈ Mun Since ω is µ-normal, it further follows from Equ (1) that ω(A) = ˜ ω(πun(A)) = ˜ ω(z µ πun(A)) ≤ ˜η(p µ z µ πun(A)) = 0 for any positive
A ∈ O, i.e., ω = 0 Since ˜π η is surjective, one has ˜π η (z µ) ∈ M η ∩ M
η and, by
Proposition 3.2, the functionals η n and η sare disjoint
Two states ω1 and ω2 are called quasi-equivalent if N ω1 = N ω2 They arecalled unitarily equivalent if their GNS representations (H ω j , π ω j , Ω ω j) are unitar-
ily equivalent, namely if there is a unitary U : H ω1 → H ω2such that U Ω ω1= Ω ω2
and U π ω1(·) = π ω2(·)U Clearly, unitarily equivalent states are quasi-equivalent.
If ω is τ -invariant, then there exists a unique self-adjoint operator L on H ωsuchthat
LΩ ω = 0, π ω (τ t (A)) = e itL π ω (A)e −itL
Trang 218 Walter Aschbacher et al.
We will call L the ω-Liouvillean of τ
The state ω is called factor state (or primary state) if its enveloping von Neumann
algebra Mωis a factor, namely if Mω ∩ M
ω=CI By Proposition 3.2 ω is a factor
state iff it cannot be written as a nontrivial convex combination of disjoint states
This implies that if ω is a factor state and µ is a positive linear functional in O ∗,
then either ω
Two factor states ω1 and ω2 are either quasi-equivalent or disjoint They are
quasi-equivalent iff (ω1+ ω2)/2 is also a factor state (this follows from Theorem
4.3.19 in [BR1])
The state ω is called modular if there exists a C ∗ -dynamics σ ωonO such that
ω is a (σ ω , −1)-KMS state If ω is modular, then Ω ωis a separating vector for Mω,
and we denote by ∆ ω , J and P the modular operator, the modular conjugation and
the natural cone associated to Ω ω To any C ∗ -dynamics τ on O one can associate a
unique self-adjoint operator L on H ω such that for all t
π ω (τ t (A)) = e itL π ω (A)e −itL , e−itL P = P.
The operator L is called standard Liouvillean of τ associated to ω If ω is τ -invariant, then LΩ ω = 0, and the standard Liouvillean is equal to the ω-Liouvillean of τ
The importance of the standard Liouvillean L stems from the fact that if a state
η is ω-normal and τ -invariant, then there exists a unique vector Ω η ∈ Ker L ∩ P
such that η( ·) = (Ω η , π ω(·)Ω η) This fact has two important consequences On one
hand, if η is ω-normal and τ -invariant, then some ergodic properties of the quantum
dynamical system (O, τ, η) can be described in terms of the spectral properties of L; see [JP2, Pi] On the other hand, if Ker L = {0}, then the C ∗ -dynamics τ has
no ω-normal invariant states The papers [BFS, DJ2, FM1, FM2, FMS, JP1, JP2, JP3,
Me1, Me2, Og] are centered around this set of ideas
In quantum statistical mechanics one also encounters L p -Liouvilleans, for p ∈
[1, ∞] (the standard Liouvillean is equal to the L2-Liouvillean) The L p-Liouvilleans
are closely related to the Araki-Masuda L p -spaces [ArM] L1and L ∞-Liouvilleanshave played a central role in the spectral theory of NESS developed in [JP5] The use
of other L p-Liouvilleans is more recent (see [JPR2]) and they will not be discussed
in this lecture
3.2 Non-Equilibrium Steady States (NESS) and Entropy Production
The central notions of non-equilibrium statistical mechanics are non-equilibriumsteady states (NESS) and entropy production Our definition of NESS followsclosely the idea of Ruelle that a “natural” steady state should provide the statis-
tics, over large time intervals [0, t], of initial configurations of the system which are
typical with respect to the reference state [Ru3] The definition of entropy tion is more problematic since there is no physically satisfactory definition of theentropy itself out of equilibrium; see [Ga1, Ru2, Ru5, Ru7] for a discussion Ourdefinition of entropy production is motivated by classical dynamics where the rate
produc-of change produc-of thermodynamic (Clausius) entropy can sometimes be related to the
Trang 22phase space contraction rate [Ga2, RC] The latter is related to the Gibbs entropy(as shown for example in [Ru3]) which is nothing else but the relative entropy withrespect to the natural reference state; see [JPR1] for a detailed discussion in a moregeneral context Thus, it seems reasonable to define the entropy production as the
rate of change of the relative entropy with respect to the reference state ω.
Let (O, τ) be a C ∗ -dynamical system and ω a given reference state The NESS
associated to ω and τ are the weak- ∗ limit points of the time averages along the
trajectory ω ◦ τ t In other words, if
ω t ≡ 1t
t
0
ω ◦ τ s ds,
then ω+ is a NESS associated to ω and τ if there exists a net t α → ∞ such that
ω t α (A) → ω+(A) for all A ∈ O We denote by Σ+(ω, τ ) the set of such NESS.
One easily sees that Σ+(ω, τ ) ⊂ E(O, τ) Moreover, since E(O) is weak-∗
com-pact, Σ+(ω, τ ) is non-empty.
As already mentioned, our definition of entropy production is based on the
con-cept of relative entropy The relative entropy of two density matrices ρ and ω is
defined, by analogy with the relative entropy of two measures, by the formula
It is easy to show that Ent(ρ |ω) ≤ 0 Let ϕ i an orthonormal eigenbasis of ρ and
by p i the corresponding eigenvalues Then p i ∈ [0, 1] andi p i = 1 Let q i ≡
(ϕ i , ω ϕ i ) Clearly, q i ∈ [0, 1] andi q i = Tr ω = 1 Applying Jensen’s inequality
Hence Ent(ρ |ω) ≤ 0 It is also not difficult to show that Ent(ρ|ω) = 0 iff ρ = ω;
see [OP] Using the concept of relative modular operators, Araki has extended the
notion of relative entropy to two arbitrary states on a C ∗-algebra [Ar1,Ar2] We referthe reader to [Ar1, Ar2, DJP, OP] for the definition of the Araki relative entropy and
its basic properties Of particular interest to us is that Ent(ρ |ω) ≤ 0 still holds, with
equality if and only if ρ = ω.
In these lecture notes we will define entropy production only in a perturbative
context (for a more general approach see [JPR2]) Denote by δ the generator of the group τ i.e., τ t= etδ , and assume that the reference state ω is invariant under τ For
V = V ∗ ∈ O we set δ V ≡ δ + i[V, ·] and denote by τ t
V ≡ e tδ V the corresponding
perturbed C ∗ -dynamics (such perturbations are often called local, see [Pi]) Starting with a state ρ ∈ N ω , the entropy is pumped out of the system by the perturbation V
at a mean rate
Trang 2310 Walter Aschbacher et al.
−1
t (Ent(ρ ◦ τ t
V |ω) − Ent(ρ|ω)).
Suppose that ω is a modular state for a C ∗ -dynamics σ t ω and denote by δ ω the
generator of σ ω If V ∈ Dom (δ ω), then one can prove the following entropy balance
is the entropy production observable (see [JP6, JP7]) In quantum mechanics σ V
plays the role of the phase space contraction rate of classical dynamical systems(see [JPR1]) We define the entropy production rate of a NESS
thermodynamic fluxes across the system produced by the perturbation V and the
positivity of entropy production is the statement of the second law of namics
thermody-3.3 Structural Properties
In this subsection we shall discuss structural properties of NESS and entropy duction following [JP4] The proofs are given in Appendix 9.1
pro-First, we will discuss the dependence of Σ+(ω, τ V ) on the reference state ω On
physical grounds, one may expect that if ω is sufficiently regular and η is ω-normal, then Σ+(η, τ V ) = Σ+(ω, τ V)
Theorem 3.1 Assume that ω is a factor state on the C ∗ -algebra O and that, for all
Trang 24The second structural property we would like to mention is:
Theorem 3.2 Let η ∈ O ∗ be ω-normal and τ
V -invariant Then η(σ V ) = 0 In
particular, the entropy production of the normal part of any NESS is equal to zero.
If Ent(η |ω) > −∞, then Theorem 3.2 is an immediate consequence of the
entropy balance equation (3) The case Ent(η |ω) = −∞ has been treated in [JP7]
and the proof requires the full machinery of Araki’s perturbation theory We will notreproduce it here
If ω+ is a factor state, then either ω+ + ⊥ ω Hence, Theorem 3.2
yields:
Corollary 3.1 If ω+is a factor state and Ep(ω+) > 0, then ω+⊥ ω If ω is also a
factor state, then ω+and ω are disjoint.
Certain structural properties can be characterized in terms of the standard
Li-ouvillean Let L be the standard Liouvillean associated to τ and L V the standard
Liouvillean associated to τ V By the well-known Araki’s perturbation formula, one
has L V = L + V − JV J (see [DJP, Pi]).
Theorem 3.3 Assume that ω is modular.
(i) Under the assumptions of Theorem 3.1, if Ker L V = {0}, then it is
one-dimensional and there exists a unique normal, τ V -invariant state ω V such that
Σ+(ω, τ V) ={ω V }.
(ii) If Ker L V ={0}, then any NESS in Σ+(ω, τ V ) is purely singular.
(iii) If Ker L V contains a separating vector for M ω , then Σ+(ω, τ V ) contains a
unique state ω+and this state is ω-normal.
3.4 C ∗-Scattering and NESS
Let (O, τ) be a C ∗ -dynamical system and V a local perturbation The abstract C ∗
-scattering approach to the study of NESS is based on the following assumption:
Assumption (S) The strong limit
The map α+V is an isometric∗-endomorphism of O, and is often called Møller
morphism α+V is one-to-one but it is generally not onto, namely
Trang 2512 Walter Aschbacher et al.
If the reference state ω is τ -invariant, then ω+ = ω ◦ α+
V is the unique NESS
associated to ω and τ V and
w∗ − lim
t→∞ ω ◦ τ t
V = ω+.
Note in particular that if ω is a (τ, β)-KMS state, then ω+is a (τ V , β)-KMS state.
The map α+V is the algebraic analog of the wave operator in Hilbert space tering theory A simple and useful result in Hilbert space scattering theory is theCook criterion for the existence of the wave operator Its algebraic analog is:
scat-Proposition 3.3. (i) Assume that there exists a dense subset O0⊂ O such that for
0
[V, τ t
Then Assumption (S) holds.
(ii) Assume that there exists a dense subset O1⊂ O such that for all A ∈ O1,
exists SinceO0is dense and τ −t ◦ τ t
V is isometric, the limit exists for all A ∈ O,
and α+V is a∗-morphism of O To prove Part (ii) note that the second estimate in
(8) and (6) imply that the norm limit
Trang 26Until the end of this subsection we will assume that the Assumption (S) holds
and that ω is τ -invariant.
Let ˜ω ≡ ω O+ and let (H ω˜, π ω˜, Ω ω˜) be the GNS-representation ofO+sociated to ˜ω Obviously, if α+V is an automorphism, then ˜ω = ω We denote by
as-(H ω+, π ω+, Ω ω+) the GNS representation ofO associated to ω+ Let L ω˜ and L ω+
be the standard Liouvilleans associated, respectively, to (O+, τ, ˜ ω) and (O, τ V , ω+)
Recall that L ω˜ is the unique self-adjoint operator onH ω˜such that for A ∈ O+,
L ω˜Ω ω˜ = 0, π˜ω (τ t (A)) = e itL ω˜π ω˜(A)e −itL ω˜,
and similarly for L ω+
Proposition 3.4 The map
U π ω˜(α+V (A))Ω ω˜ = π ω+(A)Ω ω+,
extends to a unitary U : H ω˜ → H ω+which intertwines L ω˜ and L ω+, i.e.,
U L ω˜ = L ω+U.
Proof Set π ω˜(A) ≡ π ω˜(α+V (A)) and note that π ω ˜(O)Ω ω˜ = π ω˜(O+)Ω ω˜, so that
Ω ω˜is cyclic for π ω˜(O) Since
ω+(A) = ω(α V+(A)) = ˜ ω(α+V (A)) = (Ω ω˜, π ω˜(α+V (A))Ω ω˜) = (Ω ω˜, π ω˜(A)Ω ω˜),
(H ω˜, π ω ˜, Ω ω˜) is also a GNS representation ofO associated to ω+ Since GNS resentations associated to the same state are unitarily equivalent, there is a unitary
rep-U : H ω˜ → H ω+such that U Ω ω˜= Ω ω+and
U π ω ˜(A) = π ω+(A)U.
Finally, the identities
U e itL˜ω π ω ˜(A)Ω ω˜ = U π ω˜(τ t (α+V (A)))Ω ω˜ = U π ω˜(α+V (τ V t (A)))Ω ω˜
= π ω+(τ V t (A))Ω ω+= eitL ω+ π ω+(A)Ω ω+
= eitL ω+ U π ω˜(A)Ω ω˜,
yield that U intertwines L ω˜and L ω+
We finish this subsection with:
Trang 2714 Walter Aschbacher et al.
Proposition 3.5. (i) Assume that ω˜ ∈ E(O+, τ ) is τ -ergodic Then
Proof We will prove the Part (i); the proof of the Part (ii) is similar If η ∈ N ω,
then η O+∈ N ω˜, and the ergodicity of ˜ω yields
η(τ t (α+V (A))) dt = ˜ ω(α+V (A)) = ω+(A).
This fact, the estimate
η(τ t
V (A)) − η(τ t (α+V (A))) ≤ τ −t ◦ τ t
V (A) − α+
V (A) ,
and Assumption (S) yield the statement
4 Open Quantum Systems
4.1 Definition
Open quantum systems are the basic paradigms of non-equilibrium quantum tical mechanics An open system consists of a “small” systemS interacting with a
statis-large “environment” or “reservoir”R.
In these lecture notes the small system will be a ”quantum dot”—a quantummechanical system with finitely many energy levels and no internal structure ThesystemS is described by a finite-dimensional Hilbert space H S =CN and a Hamil-
tonian H S Its algebra of observablesO S is the full matrix algebra M N(C) and its
dynamics is given by
τ S t (A) = e itH S Ae −itH S = etδ S (A),
where δ S(·) = i[H S , · ] The states of S are density matrices on H S A convenient
reference state is the tracial state, ω S(·) = Tr(·)/ dim H S In the physics literature
ω S is sometimes called the chaotic state since it is of maximal entropy, giving the
same probability 1/ dim H S to any one-dimensional projection inH S
The reservoir is described by a C ∗-dynamical system (O R , τ R) and a reference
state ω R We denote by δ R the generator of τ R
The algebra of observables of the joint systemS + R is O = O S ⊗ O R and
its reference state is ω ≡ ω S ⊗ ω R Its dynamics, still decoupled, is given by τ t=
τ t
S ⊗ τ t
R Let V = V ∗ ∈ O be a local perturbation which couples S to the reservoir
Trang 28R The ∗-derivation δ V ≡ δ R + δ S + i[V, · ] generates the coupled dynamics τ t
V
onO The coupled joint system S + R is described by the C ∗-dynamical system
(O, τ V ) and the reference state ω Whenever the meaning is clear within the context,
we will identifyO S andO Rwith subalgebras ofO via A ⊗ I O R , I O S ⊗ A With a
slight abuse of notation, in the sequel we denote I O R and I O S by I.
We will suppose that the reservoirR has additional structure, namely that it
con-sists of M parts R1, · · · , R M , which are interpreted as subreservoirs The voirs are assumed to be independent—they interact only through the small systemwhich allows for the flow of energy and matter between various subreservoirs.The subreservoir structure ofR can be chosen in a number of different ways and
subreser-the choice ultimately depends on subreser-the class of examples one wishes to describe One
obvious choice is the following: the j-th reservoir is described by the C ∗-dynamicalsystem (O R j , τ R j ) and the reference state ω R j, andO R = ⊗O R j , τ R =⊗τ R j,
ω = ⊗ω R j [JP4, Ru1] In view of the examples we plan to cover, we will choose amore general subreservoir structure
We will assume that the j-th reservoir is described by a C ∗-subalgebraO R j ⊂
O R which is preserved by τ R We denote the restrictions of τ R and ω RtoO R j by
τ R j and ω R j Different algebrasO R j may not commute However, we will assumethatO R i ∩ O R j =CI for i = j If A k, 1≤ k ≤ N, are subsets of O R, we denote
byA1, · · · , A N the minimal C ∗-subalgebra ofO Rthat contains allA k Withoutloss of generality, we may assume thatO R=O R1, · · · , O R M .
The systemS is coupled to the reservoir R j through a junction described by a self-adjoint perturbation V j ∈ O S ⊗ O R j (see Fig 1) The complete interaction isgiven by
and a state ω on O is time reversal invariant if ω ◦ r(A) = ω(A ∗ ) for all A ∈ O An
open quantum system described by (O, τ V ) and the reference state ω is called time
reversal invariant (TRI) if there exists a time reversal r such that ω is time reversal
invariant
4.2 C ∗-Scattering for Open Quantum Systems
Except for Part (ii) of Proposition 3.3, the scattering approach to the study of NESS,described in Subsection 3.4, is directly applicable to open quantum systems Con-cerning Part (ii) of Proposition 3.3, note that in the case of open quantum systems
the Møller morphism α+V cannot be onto (except in trivial cases) The best one mayhope for is thatO+ = O R , namely that α+V is an isomorphism between the C ∗-dynamical systems (O, τ V) and (O R , τ R) The next theorem was proved in [Ru1].
Trang 2916 Walter Aschbacher et al.
Fig 1 Junctions V1, V2between the systemS and subreservoirs.
Theorem 4.1 Suppose that Assumption (S) holds.
(i) If there exists a dense set O R0 ⊂ O R such that for all A ∈ O R0 ,
-KMS for some inverse temperature β, then ω+is a (τ V , β)-KMS state.
Proof The proof of Part (i) is similar to the proof of the Part (i) of Proposition 3.3.
The assumption (10) ensures that the limits
β+V (A) = lim
t →∞ τ
t
V ◦ τ −t (A),
exist for all A ∈ O R Clearly, α+V ◦ β+
V (A) = A for all A ∈ O Rand soO R ⊂
Ran α+
Trang 30To prove Part (ii) recall thatO S is a N2-dimensional matrix algebra It has abasis{E k | k = 1, · · · , N2} such that τ t (E k) = eitθ k E k for some θ k ∈ R From
Assumption (S) and (11) we can conclude that
belongs to the commutant ofO SinO Since O can be seen as the algebra M N(O R)
of N × N-matrices with entries in O R, one easily checks that this commutant ispreciselyO R
Part (iii) is a direct consequence of the first two parts
4.3 The First and Second Law of Thermodynamics
Let us denote by δ j the generator of the dynamical group τ R j (Recall that thisdynamical group is the restriction of the decoupled dynamics to the subreservoir
R j ) Assume that V j ∈ Dom (δ j ) The generator of τ V is δ V = δ R + i[H S + V, · ]
and it follows from (9) that the total energy flux out of the reservoir is given by
Besides heat fluxes, there might be other fluxes across the systemS + R (for
example, matter and charge currents) We will not discuss here the general theory ofsuch fluxes (the related information can be found in [FMU, FMSU, TM]) In the rest
of this section we will focus on the thermodynamics of heat fluxes Charge currentswill be discussed in the context of a concrete model in the second part of this lecture
We now turn to the entropy production Assume that there exists a C ∗-dynamics
σ t
RonO R such that ω R is (σ R , −1)-KMS state and such that σ R preserves each
subalgebraO R Let ˜δ j be the generator of the restriction of σ RtoO R and assume
Trang 3118 Walter Aschbacher et al.
that V j ∈ Dom (˜δ j) The entropy production observable associated to the
pertur-bation V and the reference state ω = ω S ⊗ ω R , where ω S(·) = Tr(·)/ dim H S,
In fact, it is not difficult to show that Ep(ω+) is independent of the choice of the
reference state of the small system as long as ω S > 0; see Proposition 5.3 in [JP4].
In the case of two reservoirs, the relation
(β1− β2)ω+(Φ1) = β1ω+(Φ1) + β2ω+(Φ2)≤ 0,
yields that the heat flows from the hot to the cold reservoir
4.4 Linear Response Theory
Linear response theory describes thermodynamics in the regime where the “forces”driving the system out of equilibrium are weak In such a regime, to a very goodapproximation, the non-equilibrium currents depend linearly on the forces The ul-timate purpose of linear response theory is to justify well known phenomenologicallaws like Ohm’s law for charge currents or Fick’s law for heat currents We are stillfar from a satisfactory derivation of these laws, even in the framework of classicalmechanics; see [BLR] for a recent review on this matter We also refer to [GVV6]for a rigorous discussion of linear response theory at the macroscopic level
A less ambitious application of linear response theory concerns transport ties of microscopic and mesoscopic quantum devices (the advances in nanotechnolo-gies during the last decade have triggered a strong interest in the transport properties
proper-of such devices) Linear response theory proper-of such systems is much better understood,
as we shall try to illustrate
In our current setting, the forces that drive the systemS + R out of equilibrium
are the different inverse temperatures β1, · · · , β Mof the reservoirs attached toS If
all inverse temperatures β j are sufficiently close to some value βeq, we expect linear
Trang 32response theory to give a good account of the thermodynamics of the system near
thermal equilibrium at inverse temperature βeq
To emphasize the fact that the reference state ω = ω S ⊗ ω R depends on the β j
we set X = (X1, · · · , X M ) with X j ≡ βeq− β j and denote by ω Xthis reference
state We assume that for some > 0 and all |X| < there exists a unique NESS
ω X+ ∈ Σ+(ω X , τ V ) and that the functions X → ω X+ (Φ j ) are C2 Note that ω0+
is the (unique) (τ V , βeq)-KMS state onO We will denote it simply by ω βeq
In phenomenological non-equilibrium thermodynamics, the duality between the
driving forces F α , also called affinities, and the steady currents φ α they induce isexpressed by the entropy production formula
α
F α φ α ,
(see [DGM]) The steady currents are themselves functions of the affinities φ α =
φ α (F1, · · · ) In the linear response regime, these functions are given by the relations
γ
L αγ F γ ,
which define the kinetic coefficients L αγ
Comparing with Equ (13) and using energy conservation (12) we obtain in ourcase
Thus X j is the affinity conjugated to the steady heat flux φ j (X) = ω X+ (Φ j) out
ofR j We note in particular that the equilibrium entropy production vanishes The
kinetic coefficients L jiare given by
Trang 3320 Walter Aschbacher et al.
Similarly, (15) and the second law (13) imply that the quadratic form
Linear response theory goes far beyond the above elementary relations Its true
cornerstones are the Onsager reciprocity relations (ORR), the Kubo dissipation formula (KF) and the Central Limit Theorem (CLT) All three of them
fluctuation-deal with the kinetic coefficients The Onsager reciprocity relations assert that the
matrix L jiof a time reversal invariant (TRI) system is symmetric,
The Kubo fluctuation-dissipation formula expresses the transport coefficients of
a TRI system in terms of the equilibrium current-current correlation function
C ji (t) ≡ 1
2ω βeq(τ V t (Φ j )Φ i + Φ i τ V t (Φ j )), (18)namely
L ji= 12
The Central Limit Theorem further relates L jito the statistics of the current ations in equilibrium In term of characteristic function, the CLT for open quantumsystems in thermal equilibrium asserts that
where the covariance matrix D jiis given by
D ji = 2 L ji
If, for a self-adjoint A ∈ O, we denote by 1 [a,b] (A) the spectral projection on the
interval [a, b] of π ω βeq (A), the probability of measuring a value of A in [a, b] when
the system is in the state ω βeqis given by
Probω {A ∈ [a, b]} = (Ω ω , 1 [a,b] (A) Ω ω ).
Trang 34It then follows from (20) that
e−x2/2L2jj dx.
(21)This is a direct translation to quantum mechanics of the classical central limit the-
orem Because fluxes do not commute, [Φ j , Φ i] = 0 for j = i, they can not be
measured simultaneously and a simple classical probabilistic interpretation of (20)
for the vector variable Φ = (Φ1, · · · , Φ M) is not possible Instead, the quantum
fluctuations of the vector variable Φ are described by the so-called fluctuation gebra [GVV1, GVV2, GVV3, GVV4, GVV5, Ma] The description and study of the
al-fluctuation algebra involve somewhat advanced technical tools and for this reason
we will not discuss the quantum CLT theorem in this lecture
The mathematical theory of ORR, KF, and CLT is reasonably well understood inclassical statistical mechanics (see the lecture [Re]) In the context of open quantumsystems these important notions are still not completely understood (see however[AJPP, JPR2] for some recent results)
We close this subsection with some general comments about ORR and KF.The definition (18) of the current-current correlation function involves a sym-
metrized product in order to ensure that the function C ji (t) is real-valued The
cor-responding imaginary part, given by
Trang 3522 Walter Aschbacher et al.
and ORR (17) follows from KF (19)
In the second part of the lecture we will show that the Onsager relations and theKubo formula hold for the SEBB model The proof of the Central Limit Theoremfor this model is somewhat technically involved and can be found in [AJPP]
4.5 Fermi Golden Rule (FGR) Thermodynamics
Let λ ∈ R be a control parameter We consider an open quantum system with
cou-pling λV and write τ λ for τ λV , ω λ+ for ω+, etc
The NESS and thermodynamics of the system can be described, to second
or-der of perturbation theory in λ, using the weak coupling (or van Hove) limit This
approach is much older than the ”microscopic” Hamiltonian approach discussed sofar, and has played an important role in the development of the subject The classi-cal references are [Da1, Da2, Haa, VH1, VH2, VH3] The weak coupling limit is alsodiscussed in the lecture notes [D1]
In the weak coupling limit one “integrates” the degrees of freedom of the voirs and follows the reduced dynamics ofS on a large time scale t/λ2 In the limit
reser-λ → 0 the dynamics of S becomes irreversible and is described by a semigroup,
often called the quantum Markovian semigroup (QMS) The generator of this QMS
describes the thermodynamics of the open quantum system to second order of turbation theory
per-The “integration” of the reservoir variables is performed as follows As usual,
we use the injection A → A⊗I to identify O Swith a subalgebra ofO For A ∈ O S
Obviously, T λ t is neither a group nor a semigroup Let ω S be an arbitrary reference
state (density matrix) of the small system and ω = ω S ⊗ ω R Then for any A ∈ O S,
ω(τ0−t ◦ τ t
λ (A ⊗ I)) = Tr H S (ω S T λ t (A)).
In [Da1, Da2] Davies proved that under very general conditions there exists a linear
map KH:O S → O S such that
lim
λ →0 T
t/λ2
λ (A) = e tKH(A).
The operator KH is the QMS generator (sometimes called the Davies generator)
in the Heisenberg picture A substantial body of literature has been devoted to the study of the operator KH (see the lecture notes [D1]) Here we recall only a fewbasic results concerning thermodynamics in the weak coupling limit (for additional
Trang 36information see [LeSp]) We will assume that the general conditions described inthe lecture notes [D1] are satisfied.
The operator KH generates a positivity preserving contraction semigroup on
O S Obviously, KH(I) = 0 We will assume that zero is the only purely imaginary
eigenvalue of KHand that Ker KH = CI This non-degeneracy condition can be
naturally characterized in algebraic terms, see [D1,Sp] It implies that the eigenvalue
0 of KHis semi-simple, that the corresponding eigenprojection has the form A →
Tr(ω S + A)I, where ω S +is a density matrix, and that for any initial density matrix
ω S,
lim
t→∞ Tr(ω S
tKH(A)) = Tr(ω S + A) ≡ ω S + (A).
The density matrix ω S + describes the NESS of the open quantum system in the
weak coupling limit One further shows that the operator KHhas the form
where K H,j is the QMS generator obtained by considering the weak coupling limit
of the coupled systemS + R j , i.e.,
where τ λ,j is generated by δ j + i[H S + λV j , · ].
One often considers the QMS generator in the Schr¨odinger picture, denoted KS
The operator KSis the adjoint of KH with respect to the inner product (X, Y ) =
Tr(X ∗ Y ) The semigroup e tKS is positivity and trace preserving One similarly
Recall our standing assumption that the reservoirsO R j are in thermal equilibrium
at inverse temperature β j We denote by
ω β= e−βH S /Tr(e −βH S ),
the canonical density matrix ofS at inverse temperature β (the unique (τ S , β)-KMS
state onO S) Araki’s perturbation theory of KMS-states (see [DJP,BR2]) yields that
for A ∈ O S,
ω β j ⊗ ω R j (τ0−t ◦ τ t
λ,j (A ⊗ I)) = ω β j (A) + O(λ),
uniformly in t Hence, for all t ≥ 0,
ω (etK H,j (A)) = ω (A),
Trang 3724 Walter Aschbacher et al.
and so K S,j (ω β j ) = 0 In particular, if all β j ’s are the same and equal to β, then
ω S+ = ω β
LetOd ⊂ O S be the∗-algebra spanned by the eigenprojections of H S.Od
is commutative and preserved by KH, K H,j , KSand K S,j [D1] The NESS ω S+ commutes with H S If the eigenvalues of H S are simple, then the restriction KH
Odis a generator of a Markov process whose state space is the spectrum of H S.This process has played an important role in the early development of quantum fieldtheory (more on this in Subsection 8.2)
We now turn to the thermodynamics in the weak coupling limit, which we will
call Fermi Golden Rule (FGR) thermodynamics The observable describing the heat flux out of the j-th reservoir is
which is the first law of FGR thermodynamics
The entropy production observable is
Trang 38non-Let us briefly discuss linear response theory in FGR thermodynamics using thesame notational conventions as in Subsection 4.4 The kinetic coefficients are givenby
are proven in [LeSp]
Finally, we wish to comment on the relation between microscopic and FGRthermodynamics One naturally expects FGR thermodynamics to produce the first
non-trivial contribution (in λ) to the microscopic thermodynamics For example, the following relations are expected to hold for small λ:
so far carried out only for a few models FGR thermodynamics is very robust andthe weak coupling limit is an effective tool in the study of the models whose micro-scopic thermodynamics appears beyond reach of the existing techniques
We will return to this topic in Section 8 where we will discuss the FGR dynamics of the SEBB model
Trang 39thermo-26 Walter Aschbacher et al.
5 Free Fermi Gas Reservoir
In the SEBB model, which we shall study in the second part of this lecture, thereservoir will be described by an infinitely extended free Fermi gas Our description
of the free Fermi gas in this section is suited to this application
The basic properties of the free Fermi gas are discussed in the lecture [Me3] and
in Examples 4.6 and 5.6 of the lecture [Pi] and we will assume that the reader is miliar with the terminology and results described there A more detailed expositioncan be found in [BR2] and in the recent lecture notes [D2]
fa-The free Fermi gas is described by the so called CAR (canonical tion relations) algebra The mathematical structure of this algebra is well understood(see [D2] for example) In Subsection 5.1 we will review the results we need Sub-section 5.2 contains a few useful examples
anticommuta-5.1 General Description
Let h and h be the Hilbert space and the Hamiltonian of a single Fermion We will always assume that h is bounded below Let Γ −(h) be the anti-symmetric Fock
space over h and denote by a ∗ (f ), a(f ) the creation and annihilation operators for
a single Fermion in the state f ∈ h The corresponding self-adjoint field operator
τ t (A) ≡ e itdΓ (h) Ae −itdΓ (h)
The pair (CAR(h), τ ) is a C ∗-dynamical system It preserves the Fermion number
in the sense that τ tcommutes with the gauge group
ϑ t (A) ≡ e itdΓ (I) Ae −itdΓ (I)
Recall that N ≡ dΓ (I) is the Fermion number operator on Γ − (h) and that τ and ϑ
are the groups of Bogoliubov automorphisms
τ t (a#(f )) = a#(eith f ), ϑ t (a#(f )) = a#(eit f ).
To every self-adjoint operator T on h such that 0 ≤ T ≤ I one can associate a
state ω T on CAR(h) satisfying
ω (a ∗ (f )· · · a ∗ (f )a(g )· · · a(g )) = δ det{(g , T f )}. (29)
Trang 40This ϑ-invariant state is usually called the quasi-free gauge-invariant state erated by T It is completely determined by its two point function
ω T is a factor state It is modular iff Ker T = Ker (I − T ) = {0} Two states
ω T1and ω T2are quasi-equivalent iff the operators
T11/2 − T 1/2
are Hilbert-Schmidt; see [De, PoSt, Ri] Assume that Ker T i = Ker (I − T i) ={0}.
Then the states ω T1and ω T2are unitarily equivalent iff (31) holds
If T = F (h) for some function F : σ(h) → [0, 1], then ω T describes a free
Fermi gas with energy density per unit volume F (ε).
The state ω T is τ -invariant iff T commutes with e ith for all t If the spectrum of
h is simple this means that T = F (h) for some function F : σ(h) → [0, 1].
For any β, µ ∈ R, the Fermi-Dirac distribution ρ βµ (ε) ≡ (1 + e β(ε−µ))−1
induces the unique β-KMS state on CAR(h) for the dynamics τ t ◦ ϑ −µt This state,
which we denote by ω βµ, describes the free Fermi gas at thermal equilibrium in the
grand canonical ensemble with inverse temperature β and chemical potential µ The GNS representation of CAR(h) associated to ω T can be explicitly com-
puted as follows Fix a complex conjugation f → ¯ f on h and extend it to Γ −(h)
Denote by Ω the vacuum vector and N the number operator in Γ −(h) Set
H ω T = Γ −(h)⊗ Γ − (h),
Ω ω T = Ω ⊗ Ω,
π (a(f )) = a((I − T ) 1/2 f ) ⊗ I + (−I) N ⊗ a ∗( ¯T 1/2 f ).¯