Attal s et al (eds ) open quantum systems III recent developments (LNM 1882 2006)(ISBN 3540309934)(325s)

Lecture Notes in Mathematics Editors: J.-M Morel, Cachan F Takens, Groningen B Teissier, Paris 1882 S Attal · A Joye · C.-A Pillet (Eds.) Open Quantum Systems III Recent Developments ABC Editors Stéphane Attal Institut Camille Jordan Universit é Claude Bernard Lyon 21 av Claude Bernard 69622 Villeurbanne Cedex France e-mail: attal@math.univ-lyon1.fr Alain Joye Institut Fourier Universit é de Grenoble BP 74 38402 Saint-Martin d'Hères Cedex France e-mail: alain.joye@ujf-grenoble.fr Claude-Alain Pillet CPT-CNRS, UMR 6207 Université du Sud Toulon-Var BP 20132 83957 La Garde Cedex France e-mail: pillet@univ-tln.fr Library of Congress Control Number: 2006923432 Mathematics Subject Classification (2000): 37A60, 37A30, 47A05, 47D06, 47L30, 47L90, 60H10, 60J25, 81Q10, 81S25, 82C10, 82C70 ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 ISBN-10 3-540-30993-4 Springer Berlin Heidelberg New York ISBN-13 978-3-540-30993-2 Springer Berlin Heidelberg New York DOI 10.1007/b128453 This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Violations are liable for prosecution under the German Copyright Law Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2006 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use A EX package Typesetting: by the authors and SPI Publisher Services using a Springer LT Cover design: design & production GmbH, Heidelberg Printed on acid-free paper SPIN: 11602668 V A 41/3100/ SPI 543210 Preface This volume is the third and last of a series devoted to the lecture notes of the Grenoble Summer School on “Open Quantum Systems” which took place at the Institut Fourier from June 16th to July 4th 2003 The contributions presented in this volume 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 Open Quantum Systems Whereas the first two volumes were dedicated to a detailed exposition of the mathematical techniques and physical concepts relevant in the study of Open Systems with no a priori pre-requisites, the contributions presented in this volume request from the reader some familiarity with these aspects Indeed, the material presented here aims at leading the reader already acquainted with the basics in quantum statistical mechanics, spectral theory of linear operators, C ∗ -dynamical systems, and quantum stochastic differential equations to the front of the current research done on various aspects of Open Quantum Systems Nevertheless, pedagogical efforts have been made by the various authors of these notes so that this volume should be essentially self-contained for a reader with minimal previous exposure to the themes listed above In any case, the reader in need of complements can always turn to these first two volumes The topics covered in these lectures notes start with an introduction to nonequilibrium quantum statistical mechanics The definitions of the physical concepts as well as the necessary mathematical framework suitable for their description are developed in a general setup A simple non-trivial physically relevant example of independent 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-equilibrium steady states, entropy production and other thermodynamical notions introduced earlier The next contribution is devoted to the many aspects of the Fermi Golden Rule used within the Hamiltonian approach of Open Quantum Systems in order to derive VI Preface a Markovian approximation of the dynamics In particular, the weak coupling or van Hove limit in both a time-dependent and stationary setting are discussed in an abstract framework These results are then applied to the case of small systems interacting with reservoirs, within different algebraic representations of the relevant models The links between the Fermi Golden Rule and the Detailed Balance Condition 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 Mechanics The properties of the large time behavior of the dynamics reduced to a subsystem, which is not Markovian in general, are first reviewed Then, the so-called isometric-sweeping decomposition of a dynamical semigroup is presented in an general setup and its links with decoherence phenomena are exposed Applications to physical models such as spin systems or to the unravelling of the classical dynamics in certain regimes are then provided The properties of dynamical semigroups on CCR algebras are discussed in details in the final section The following contribution is devoted to a systematic study of the long time behavior of quantum dynamical semigroups, as they arise in Markovian approximations More precisely, the key notions for applications of stationary states, convergence towards equilibrium as well as transience and recurrence of such quantum Markov semigroups are developed in an abstract framework In particular, conditions on unbounded operators defined in the sense of forms to generate a bona fide quantum dynamical semigroup are formulated, as well as general criteria insuring the existence of stationary states for a given quantum dynamical semigroup The relations between return to equilibrium for a quantum dynamical semigroup and the properties 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 continual measurements in quantum optics, considered as an application of quantum stochastic calculus The basics of this quantum stochastic calculus and the modelization of system-field interactions constructed on it are first explained Then, indirect and continual measurement processes and the corresponding master equations are introduced and discussed Physical interpretations of computations performed within this quantum stochastic modelization framework are spelled out for various specific processes in quantum optics As revealed by this outline, the treatment of the different physical models proposed in this volume makes use of several tools and approximations discussed from a mathematical point of view, both in the Hamiltonian and Markovian approach At the same time, the different mathematical topics addressed here are illustrated by physically relevant applications in the theory of Open Quantum Systems We believe the contact made between the practicians of the Markovian and Hamiltonian during the School itself and within the contributions of these volumes is useful and will prove to be even more fruitful for the future developments of the field Preface VII Let us close this introduction by pointing out that some recent results in the theory of Open Quantum Systems are not discussed in these notes These include notably the descriptions of return to equilibrium by means of renormalization analysis and scattering techniques These demanding approaches were not addressed in the Grenoble Summer School, because a reasonably complete treatment would simply have required too much time We hope the reader will benefit from the pedagogical efforts provided by all authors of these notes in order to introduce the concepts and problems, as well as recent developments in the theory of Open Quantum Systems Lyon, Grenoble, Toulon, September 2005 St´ephane Attal Alain Joye Claude-Alain Pillet Contents Topics in Non-Equilibrium Quantum Statistical Mechanics Walter Aschbacher, Vojkan Jakˇsi´c, Yan Pautrat, and Claude-Alain Pillet Introduction Conceptual Framework Mathematical Framework 3.1 Basic Concepts 3.2 Non-Equilibrium Steady States (NESS) and Entropy Production 3.3 Structural Properties 3.4 C ∗ -Scattering and NESS Open Quantum Systems 4.1 Definition 4.2 C ∗ -Scattering for Open Quantum Systems 4.3 The First and Second Law of Thermodynamics 4.4 Linear Response Theory 4.5 Fermi Golden Rule (FGR) Thermodynamics Free Fermi Gas Reservoir 5.1 General Description 5.2 Examples The Simple Electronic Black-Box (SEBB) Model 6.1 The Model 6.2 The Fluxes 6.3 The Equivalent Free Fermi Gas 6.4 Assumptions Thermodynamics of the SEBB Model 7.1 Non-Equilibrium Steady States 7.2 The Hilbert-Schmidt Condition 7.3 The Heat and Charge Fluxes 7.4 Entropy Production 7.5 Equilibrium Correlation Functions 7.6 Onsager Relations Kubo Formulas 5 10 11 14 14 15 17 18 22 26 26 30 34 34 36 37 40 43 43 44 45 46 47 49 X Contents FGR Thermodynamics of the SEBB Model 8.1 The Weak Coupling Limit 8.2 Historical Digression—Einstein’s Derivation of the Planck Law 8.3 FGR Fluxes, Entropy Production and Kubo Formulas 8.4 From Microscopic to FGR Thermodynamics Appendix 9.1 Structural Theorems 9.2 The Hilbert-Schmidt Condition References Fermi Golden Rule and Open Quantum Systems Jan Derezinski and Rafa Frăuboes Introduction 1.1 Fermi Golden Rule and Level Shift Operator in an Abstract Setting 1.2 Applications of the Fermi Golden Rule to Open Quantum Systems Fermi Golden Rule in an Abstract Setting 2.1 Notation 2.2 Level Shift Operator 2.3 LSO for C0∗ -Dynamics 2.4 LSO for W ∗ -Dynamics 2.5 LSO in Hilbert Spaces 2.6 The Choice of the Projection P 2.7 Three Kinds of the Fermi Golden Rule Weak Coupling Limit 3.1 Stationary and Time-Dependent Weak Coupling Limit 3.2 Proof of the Stationary Weak Coupling Limit 3.3 Spectral Averaging 3.4 Second Order Asymptotics of Evolution with the First Order Term 3.5 Proof of Time Dependent Weak Coupling Limit 3.6 Proof of the Coincidence of Mst and Mdyn with the LSO Completely Positive Semigroups 4.1 Completely Positive Maps 4.2 Stinespring Representation of a Completely Positive Map 4.3 Completely Positive Semigroups 4.4 Standard Detailed Balance Condition 4.5 Detailed Balance Condition in the Sense of Alicki-FrigerioGorini-Kossakowski-Verri Small Quantum System Interacting with Reservoir 5.1 W ∗ -Algebras 5.2 Algebraic Description 5.3 Semistandard Representation 5.4 Standard Representation 50 50 53 54 56 58 58 60 63 67 68 68 69 71 71 72 73 74 74 75 75 77 77 80 83 85 87 88 88 89 89 90 91 93 93 94 95 95 96 Contents XI 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 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 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 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 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 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 Outlook 157 XII Contents References 158 Notes on the Qualitative Behaviour of Quantum Markov Semigroups Franco Fagnola and Rolando Rebolledo 161 Introduction 162 1.1 Preliminaries 164 Ergodic Theorems 165 The Minimal Quantum Dynamical Semigroup 167 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 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 The Convergence Towards the Equilibrium 189 6.1 Main Results 190 6.2 Examples 192 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 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 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 Continual Measurements 230 3.1 Indirect Measurements on SH 230 3.2 Characteristic Functionals 233 3.3 The Reduced Description 241 300 Contents of Volume I 2.4 The C ∗ -algebras CARF (H), CCRF (H) 194 2.5 Leaving Fock space 197 The CCR and CAR algebras 198 3.1 The algebra CAR(D) 199 3.2 The algebra CCR(D) 200 3.3 Schrăodinger representation and Stone – von Neumann uniqueness theorem 203 3.4 Q–space representation 207 3.5 Equilibrium state and thermodynamic limit 209 Araki-Woods representation of the infinite free Boson gas 213 4.1 Generating functionals 214 4.2 Ground state (condensate) 217 4.3 Excited states 222 4.4 Equilibrium states 224 4.5 Dynamical stability of equilibria 228 References 233 Topics in Spectral Theory Vojkan Jakˇsi´c 235 Introduction 236 Preliminaries: measure theory 238 2.1 Basic notions 238 2.2 Complex measures 238 2.3 Riesz representation theorem 240 2.4 Lebesgue-Radon-Nikodym theorem 240 2.5 Fourier transform of measures 241 2.6 Differentiation of measures 242 2.7 Problems 247 Preliminaries: harmonic analysis 248 3.1 Poisson transforms and Radon-Nikodym derivatives 249 3.2 Local Lp norms, < p < 253 3.3 Weak convergence 253 3.4 Local Lp -norms, p > 254 3.5 Local version of the Wiener theorem 255 3.6 Poisson representation of harmonic functions 256 3.7 The Hardy class H ∞ (C+ ) 258 3.8 The Borel transform of measures 261 3.9 Problems 263 Self-adjoint operators, spectral theory 267 4.1 Basic notions 267 4.2 Digression: The notions of analyticity 269 4.3 Elementary properties of self-adjoint operators 269 4.4 Direct sums and invariant subspaces 272 4.5 Cyclic spaces and the decomposition theorem 273 4.6 The spectral theorem 273 Contents of Volume I 301 4.7 Proof of the spectral theorem—the cyclic case 274 4.8 Proof of the spectral theorem—the general case 277 4.9 Harmonic analysis and spectral theory 279 4.10 Spectral measure for A 280 4.11 The essential support of the ac spectrum 281 4.12 The functional calculus 281 4.13 The Weyl criteria and the RAGE theorem 283 4.14 Stability 285 4.15 Scattering theory and stability of ac spectra 286 4.16 Notions of measurability 287 4.17 Non-relativistic quantum mechanics 290 4.18 Problems 291 Spectral theory of rank one perturbations 295 5.1 Aronszajn-Donoghue theorem 296 5.2 The spectral theorem 298 5.3 Spectral averaging 299 5.4 Simon-Wolff theorems 300 5.5 Some remarks on spectral instability 301 5.6 Boole’s equality 302 5.7 Poltoratskii’s theorem 304 5.8 F & M Riesz theorem 308 5.9 Problems and comments 309 References 311 Index of Volume-I 313 Information about the other two volumes Contents of Volume II 318 Index of Volume II 321 Contents of Volume III 323 Index of Volume III 327 Index of Volume I ∗-algebra, 72 morphism, 77 C ∗ -algebra, 71 morphism, 77 C0 semigroup, 35 W ∗ -algebra, 139 ∗-derivation, 132 µ-Liouvillean, 143 Adjoint, Algebra ∗, 72 C ∗ , 71 Banach, 72 von Neumann, 88 Analytic vector, 32, 136, 202 Approximate identity, 84 Aronszajn-Donoghue theorem, 297 Asymptotic abelianness, 229 Baker-Campbell-Hausdorff formula, 193 Banach algebra, 72 Birkhoff ergodic theorem, 125 Bogoliubov transformation, 200 Boltzmann’s constant, 57 Boole’s equality, 302 Borel transform, 249, 261 Bose gas, 140, 145, 177 Boson, 53, 186 Canonical anti-commutation relations, 190 Canonical commutation relations, 50, 190, 192 Canonical transformation, 43 Cantor set, 310 CAR, CCR algebra CARF (h), 195 CCRF (h), 195 quasi-local, 198 simplicity, 199, 200 uniqueness, 199, 200 CAR-algebra, 134, 172 Cayley transform, CCR-algebra, 140, 145, 177 Center, 118 Central support, 118 Chaos, 186 Character, 83 Chemical potential, 58 Commutant, 89 Condensate, 217 Configuration space, 42 Conjugation, 10 Contraction semigroup, 37 Critical density, 227 Cyclic subspace, 22, 295 vector, 22, 195, 273 Deficiency indices, Density matrix, 55, 114, 290 Dynamical system C ∗ , 132 W ∗ , 139 classical, 124 ergodic, 125, 156 mixing, 127, 156 Index of Volume I quantum, 142 Ensemble canonical, 60 grand canonical, 63 microcanonical, 57 Entropy Boltzmann, 57 Enveloping von Neumann algebra, 119 Essential support, 281 Evolution group, 29 Exponential law, 203 Factor, 118 Faithful representation, 80 Fermi gas, 134, 172 Fermion, 53, 186 Finite particle subspace, 192 Finite quantum system, 133 Fock space, 186 Folium, 119 Free energy, 61 Functional calculus, 16, 25, 281 G.N.S representation, 82 Hahn decomposition theorem, 240 Hamiltonian, 290 Hamiltonian system, 43 Hardy class, 258 Harmonic oscillator, 50, 205 Heisenberg picture, 51 Heisenberg uncertainty principle, 49, 290 Helffer-Sjăostrand formula, 17 Hille-Yosida theorem, 37 Ideal left, 84 right, 84 two-sided, 84 Ideal gas, 185 Indistinguishable, 186 Individual ergodic theorem, 125 Infinitesimal generator, 35 Internal energy, 58 Invariant subspace, 22, 272 Invertible, 73 Isometric element, 75 303 Jensen’s formula, 259 Kaplansky density theorem, 111 Kato-Rellich theorem, 285 Kato-Rosenblum theorem, 287 Koopman ergodicity criterion, 129 Koopman lemma, 128 Koopman mixing criterion, 129 Koopman operator, 128 Lebesgue-Radon-Nikodym theorem, 240 Legendre transform, 62 Liouville equation, 43 Liouville’s theorem, 43 Liouvillean, 128, 143, 150, 161, 168 Lummer Phillips theorem, 38 Mean ergodic theorem, 32, 128 Measure absolutely continuous, 240 complex, 239 regular Borel, 238 signed, 239 space, 238 spectral, 274, 280, 295 support, 238 Measurement, 48 simultaneous, 49 Measures equivalent, 280 mutually singular, 240 Modular conjugation, 96 operator, 96 Morphism ∗-algebra, 77 C ∗ -algebra, 77 Nelson’s analytic vector theorem, 32 Norm resolvent convergence, 27 Normal element, 75 Normal form, 143 Observable, 42, 46, 123, 290 Operator (anti-)symmetrization, 187 closable, 5, 268 closed, 2, 268 core, 31, 268 304 Index of Volume I creation, annihilation, 50, 188 dissipative, 37 domain, essentially self-adjoint, extension, field, 192 graph, 3, 268 linear, multiplication, 14, 273 number, 186 positive, 271 relatively bounded, 12, 285 Schrăodinger, 47 self-adjoint, symmetric, trace class, 286 Weyl, 193 Partition function, 61, 64 Pauli’s principle, 54, 191 Perturbation theory rank one, 295 Phase space, 42 Planck law, 226 Poisson bracket, 44 Poisson representation, 256 Poisson transform, 249 Poltoratskii’s theorem, 262, 304 Positive element, 78 linear form, 80 Predual, 90 Pressure, 58 Quantum dynamical system, 142 Quasi-analytic extension, 16 RAGE theorem, 284, 290 Reduced Liouvillean, 161 Representation, 80 Q-space (CCR), 207 Araki-Woods, 224 faithful, 80 Fock, 203 GNS, 120 GNS (ground state of Bose gas), 221 Quasi-equivalent, 206 regular (of CCR), 201 ă Schrdinger, 204 Resolvent, first identity, 4, 268 norm convergence, 27 set, 3, 268 strong convergence, 194 Resolvent set, 73 Return to equilibrium, 127, 230 Riemann-Lebesgue lemma, 241 Riesz representation theorem, 240 Schrăodinger picture, 51 Sector, 186 Self-adjoint element, 75 Simon-Wolff theorems, 300 Spatial automorphism, 133 Spectral averaging, 299 Spectral radius, 74 Spectral theorem, 23, 274, 298 Spectrum, 3, 73, 83, 268 absolutely continuous, 278 continuous, 278 essential, 284 point, 268 pure point, 278 singular, 278 singular continuous, 278 Spin, 53 Standard form, 148 Standard Liouvillean, 150, 168 Standard unitary, 149 State, 81, 198 absolutely continuous, 155 centrally faithful, 118 coherent, 52 disjoint, 119 equilibrium, 124 extremal, 159 factor, 231 faithful, 110, 117 gauge invariant, 173, 212 generating functional, 214 Gibbs, 210 ground (Bose gas), 220 invariant, 141 KMS, 169, 210 local perturbation, 228 mixed, 54 mixing, 232 normal, 92, 112 Index of Volume I orthogonal, 119 pure, 46, 56 quasi-equivalent, 119 quasi-free, 147, 173, 212 relatively normal, 119, 198 tracial, 96 Stone’s formula, 282 Stone’s theorem, 30 Stone-von Neumann uniqueness theorem, 205 Strong resolvent convergence, 194 Support, 117 Temperature, 58, 61 Thermodynamic first law, 58 limit, 184, 197 second law, 58 Topology σ-strong, 111 σ-weak, 87, 111 strong, 86 uniform, 86 weak, 86 weak- , 139 Trotter product formula, 33 Unit, 72 approximate, 84 Unitary element, 75 Vacuum, 186 Von Neumann density theorem, 111 Von Neumann ergodic theorem, 33, 128 Wave operators, 286 complete, 286 Weyl (CCR) relations, 193 Weyl commutation relations, 140 Weyl quantization, 47 Weyl’s criterion, 283 Weyl’s theorem, 286 Wiener theorem, 241, 255 305 Contents of Volume II Ergodic Properties of Markov Processes Luc Rey-Bellet Introduction Stochastic Processes Markov Processes and Ergodic Theory 3.1 Transition probabilities and generators 3.2 Stationary Markov processes and Ergodic Theory Brownian Motion Stochastic Differential Equations Control Theory and Irreducibility Hypoellipticity and Strong-Feller Property Liapunov Functions and Ergodic Properties References 1 4 12 14 24 26 28 39 Open Classical Systems Luc Rey-Bellet Introduction Derivation of the model 2.1 How to make a heat reservoir 2.2 Markovian Gaussian stochastic processes 2.3 How to make a Markovian reservoir Ergodic properties: the chain 3.1 Irreducibility 3.2 Strong Feller Property 3.3 Liapunov Function Heat Flow and Entropy Production 4.1 Positivity of entropy production 4.2 Fluctuation theorem 4.3 Kubo Formula and Central Limit Theorem References 41 41 44 44 48 50 52 56 57 58 66 69 71 75 77 Contents of Volume II 307 Quantum Noises St´ephane Attal 79 Introduction 80 Discrete time 81 2.1 Repeated quantum interactions 81 2.2 The Toy Fock space 83 2.3 Higher multiplicities 89 Itˆo calculus on Fock space 93 3.1 The continuous version of the spin chain: heuristics 93 3.2 The Guichardet space 94 3.3 Abstract Itˆo calculus on Fock space 97 3.4 Probabilistic interpretations of Fock space 105 Quantum stochastic calculus 110 4.1 An heuristic approach to quantum noise 110 4.2 Quantum stochastic integrals 113 4.3 Back to probabilistic interpretations 122 The algebra of regular quantum semimartingales 123 5.1 Everywhere defined quantum stochastic integrals 124 5.2 The algebra of regular quantum semimartingales 127 Approximation by the toy Fock space 130 6.1 Embedding the toy Fock space into the Fock space 130 6.2 Projections on the toy Fock space 132 6.3 Approximations 136 6.4 Probabilistic interpretations 138 6.5 The Itˆo tables 139 Back to repeated interactions 139 7.1 Unitary dilations of completely positive semigroups 140 7.2 Convergence to Quantum Stochastic Differential Equations 142 Bibliographical comments 145 References 145 Complete Positivity and the Markov structure of Open Quantum Systems Rolando Rebolledo 149 Introduction: a preview of open systems in Classical Mechanics 149 1.1 Introducing probabilities 152 1.2 An algebraic view on Probability 154 Completely positive maps 157 Completely bounded maps 162 Dilations of CP and CB maps 163 Quantum Dynamical Semigroups and Markov Flows 168 Dilations of quantum Markov semigroups 173 6.1 A view on classical dilations of QMS 174 6.2 Towards quantum dilations of QMS 180 References 181 308 Contents of Volume II Quantum Stochastic Differential Equations and Dilation of Completely Positive Semigroups Franco Fagnola 183 Introduction 183 Fock space notation and preliminaries 184 Existence and uniqueness 188 Unitary solutions 191 Emergence of H-P equations in physical applications 193 Cocycle property 196 Regularity 199 The left equation: unbounded Gα β 203 Dilation of quantum Markov semigroups 208 10 The left equation with unbounded Gα β : isometry 213 11 The right equation with unbounded Fβα 216 References 218 Index of Volume-II 221 Information about the other two volumes Contents of Volume I 224 Index of Volume I 228 Contents of Volume III 232 Index of Volume III 236 Index of Volume II Adapted domain, 114 Algebra Banach, 156 von Neumann, 157 Algebraic probability space, 154 Banach algebra, 156 Brownian interpretation, 107 Brownian motion, 13 canonical, 14 Chaotic expansion, 106 representation property, 106 space, 106 Classical probabilistic dilations, 174 Coherent vector, 96 Completely bounded map, 162 Completely positive map, 158 Conditional expectation, 173 Conditionally CP map, 170 Control, 24 Dilation, 208 Dilations of QDS, 173 Dynkin’s formula, 21 Elliptic operator, 27 Ergodic, Fock space toy, 84 multiplicity n, 90 Gaussian process, 13 Generator, Gibbs measure, 45 Hăormander condition, 27 Independent increments, 13 Initial distribution, Integral representation, 85 Itˆo integrable process, 99 integral, 15, 99 process, 16 Lyapunov function, 21 Markov process, Martingale normal, 105 Measure preserving, Mild solution, 217 Mixing, Modification, 13 Normal martingale, 105 Feller semigroup strong, weak, First fundamental formula, 186 Obtuse system, 90 Operator process, 185 310 Index of Volume II Operator system, 156 Regular quantum semimartingales, 128 Poisson interpretation, 107 Predictable representation property, 105 Probabilistic interpretation, 87, 107 p-, 88 Probability space algebraic, 154 Process, distribution, Gaussian, 13 Itˆo integrable, 99 Ito, 16 Markov, strong, 20 modification, 13 operator, 185 adapted, 111 path, stationary, Product p-, 89 Poisson, 108 Wiener, 108 Sesqui-symmetric tensor, 91 Spectral function, 48 State normal, 155 Stationary increments, 13 Stinespring representation, 164 Stochastic integral, 15 quantum, 115 Stochastically integrable, 185 Stopping time, 21 Strong Markov process, 20 Structure equation, 107 Quantum dynamical semigroup, 170, 208 minimal, 210 Quantum Markov semigroup, 170, 208 Quantum noises, 111 Quantum probabilistic dilations, 174, 180 Uniform topology, 156 Uniformly continuous QMS, 170 Tensor sesqui-symmetric, 91 Topology uniform, 156 Total variation norm, 12 Totalizing set, 205 Toy Fock space, 84 multiplicity n, 90 Transition probability, Vacuum, 96 von Neumann algebra, 157 Lecture Notes in Mathematics For information about earlier volumes please contact your 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(A )) − η(τ t (αV+ (A )) ) ≤ τ −t ◦ τVt (A) − αV+ (A) , and Assumption (S) yield the statement Open Quantum Systems 4.1 Definition Open quantum systems are the basic paradigms of non-equilibrium quantum. .. Walter Aschbacher et al Fig Junctions V1 , V2 between the system S and subreservoirs Theorem 4.1 Suppose that Assumption (S) holds (i) If there exists a dense set OR0 ⊂ OR such that for all A ∈... statistical mechanics An open system consists of a “small” system S interacting with a large “environment” or “reservoir” R In these lecture notes the small system will be a ? ?quantum dot”—a quantum