Table Of ContentLecture Notes in Mathematics 1882
Editors:
J.-M.Morel,Cachan
F.Takens,Groningen
B.Teissier,Paris
· ·
S. Attal A. Joye C.-A. Pillet (Eds.)
Open Quantum
Systems III
Recent Developments
ABC
Editors
Stéphane Attal Alain Joye
InstitutCamille Jordan Institut Fourier
Université Claude Bernard Lyon 1 Universitéde Grenoble 1
21 av. Claude Bernard BP 74
69622 Villeurbanne Cedex 38402 Saint-Martin d'Hères Cedex
France France
e-mail:[email protected] e-mail:[email protected]
Claude-Alain Pillet
CPT-CNRS, UMR 6207
Université du Sud Toulon-Var
BP 20132
83957 La Garde Cedex
France
e-mail:[email protected]
LibraryofCongressControlNumber:2006923432
MathematicsSubjectClassification(2000):37A60,37A30,47A05,47D06, 47L30,47L90,
60H10,60J25,81Q10, 81S25,82C10, 82C70
ISSNprintedition:0075-8434
ISSNelectronicedition:1617-9692
ISBN-10 3-540-30993-4 SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-30993-2 SpringerBerlinHeidelbergNewYork
DOI10.1007/b128453
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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
InstitutFourierfromJune16thtoJuly4th2003.Thecontributionspresentedinthis
volumecorrespondtoexpandedversionsofthelecturenotesprovidedbytheauthors
tothestudentsoftheSummerSchool.Thecorrespondinglectureswerescheduled
inthelastpartoftheSchool devoted torecentdevelopments inthestudyofOpen
QuantumSystems.
Whereas the first two volumes were dedicated to a detailed exposition of the
mathematicaltechniquesandphysicalconceptsrelevantinthestudyofOpenSys-
tems 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, peda-
gogical efforts have been made by the various authors of these notes so that this
volumeshouldbeessentiallyself-containedforareaderwithminimalpreviousex-
posure to the themes listed above. In any case, the reader in need of complements
canalwaysturntothesefirsttwovolumes.
The topics covered in these lectures notes start with an introduction to non-
equilibriumquantumstatisticalmechanics.Thedefinitionsofthephysicalconcepts
aswellasthenecessarymathematicalframeworksuitablefortheirdescriptionare
developed in a general setup. A simple non-trivial physically relevant example of
independentelectronsinadeviceconnectedtoseveralreservoirsistreatedindetails
inthesecondpartofthesenotesinordertoillustratethenotionsofnon-equilibrium
steady states, entropy production and other thermodynamical notions introduced
earlier.
ThenextcontributionisdevotedtothemanyaspectsoftheFermiGoldenRule
usedwithintheHamiltonianapproachofOpenQuantumSystemsinordertoderive
VI Preface
a Markovian approximation of the dynamics. In particular, the weak coupling or
vanHovelimitinbothatime-dependentandstationarysettingarediscussedinan
abstractframework.Theseresultsarethenappliedtothecaseofsmallsystemsin-
teracting with reservoirs, within different algebraic representations of the relevant
models.ThelinksbetweentheFermiGoldenRuleandtheDetailedBalanceCondi-
tionaswellasexplicitformulasarealsodiscussedindifferentphysicalsituations.
The third text of this volume is concerned with the notion of decoherence,
relevant,inparticular,foradiscussionofthemeasurementtheoryinQuantumMe-
chanics.Thepropertiesofthelargetimebehaviorofthedynamicsreducedtoasub-
system, which is not Markovian in general, are first reviewed. Then, the so-called
isometric-sweepingdecompositionofadynamicalsemigroupispresentedinangen-
eral setup and its links with decoherence phenomena are exposed. Applications to
physicalmodelssuchasspinsystemsortotheunravellingoftheclassicaldynamics
in certain regimes are then provided. The properties of dynamical semigroups on
CCRalgebrasarediscussedindetailsinthefinalsection.
The following contribution is devoted to a systematic study of the long time
behavior of quantum dynamical semigroups, as they arise in Markovian approxi-
mations.Moreprecisely,thekey notions forapplications of stationarystates,con-
vergencetowardsequilibriumaswellastransienceandrecurrenceofsuchquantum
Markov semigroups are developed in an abstract framework. In particular, condi-
tionsonunboundedoperatorsdefinedinthesenseofformstogenerateabonafide
quantum dynamical semigroup are formulated, as well as general criteria insuring
the existence of stationary states for a given quantum dynamical semigroup. The
relationsbetweenreturntoequilibriumforaquantumdynamicalsemigroupandthe
propertiesofitsgeneratorarealsodiscussed.Alltheseconceptsarethenillustrated
byapplicationstoconcretephysicalmodelsusedinquantumoptics.
Thelastnotesofthisvolumeprovideadetailedaccountoftheprocessofcontin-
ualmeasurementsinquantumoptics,consideredasanapplicationofquantumsto-
chasticcalculus.Thebasicsofthisquantumstochasticcalculusandthemodelization
ofsystem-fieldinteractionsconstructedonitarefirstexplained.Then,indirectand
continual measurement processes and the corresponding master equations are in-
troducedanddiscussed.Physicalinterpretationsofcomputationsperformedwithin
thisquantumstochasticmodelizationframeworkarespelledoutforvariousspecific
processesinquantumoptics.
Asrevealedbythisoutline,thetreatmentofthedifferentphysicalmodelspro-
posedinthisvolumemakesuseofseveraltoolsandapproximationsdiscussedfrom
amathematicalpointofview,bothintheHamiltonianandMarkovianapproach.At
the same time, the different mathematical topics addressed here are illustrated by
physically relevant applications in the theory of Open Quantum Systems. We be-
lieve the contact made between the practicians of the Markovian and Hamiltonian
duringtheSchoolitselfandwithinthecontributionsofthesevolumesisusefuland
willprovetobeevenmorefruitfulforthefuturedevelopmentsofthefield.
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
notablythedescriptionsofreturntoequilibriumbymeansofrenormalizationanaly-
sis and scattering techniques. These demanding approaches were not addressed
in the Grenoble Summer School, because a reasonably complete treatment would
simplyhaverequiredtoomuchtime.
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
recentdevelopmentsinthetheoryofOpenQuantumSystems.
Lyon,Grenoble,Toulon, Ste´phaneAttal
September2005 AlainJoye
Claude-AlainPillet
Contents
TopicsinNon-EquilibriumQuantumStatisticalMechanics
WalterAschbacher,VojkanJaksˇic´,YanPautrat,andClaude-AlainPillet ..... 1
1 Introduction .................................................... 2
2 ConceptualFramework........................................... 3
3 MathematicalFramework......................................... 5
3.1 BasicConcepts............................................. 5
3.2 Non-EquilibriumSteadyStates(NESS)andEntropyProduction ... 8
3.3 StructuralProperties ........................................ 10
3.4 C∗-ScatteringandNESS..................................... 11
4 OpenQuantumSystems .......................................... 14
4.1 Definition ................................................. 14
4.2 C∗-ScatteringforOpenQuantumSystems...................... 15
4.3 TheFirstandSecondLawofThermodynamics.................. 17
4.4 LinearResponseTheory ..................................... 18
4.5 FermiGoldenRule(FGR)Thermodynamics .................... 22
5 FreeFermiGasReservoir......................................... 26
5.1 GeneralDescription......................................... 26
5.2 Examples ................................................. 30
6 TheSimpleElectronicBlack-Box(SEBB)Model..................... 34
6.1 TheModel ................................................ 34
6.2 TheFluxes ................................................ 36
6.3 TheEquivalentFreeFermiGas ............................... 37
6.4 Assumptions............................................... 40
7 ThermodynamicsoftheSEBBModel............................... 43
7.1 Non-EquilibriumSteadyStates ............................... 43
7.2 TheHilbert-SchmidtCondition ............................... 44
7.3 TheHeatandChargeFluxes.................................. 45
7.4 EntropyProduction ......................................... 46
7.5 EquilibriumCorrelationFunctions............................. 47
7.6 OnsagerRelations.KuboFormulas. ........................... 49
X Contents
8 FGRThermodynamicsoftheSEBBModel .......................... 50
8.1 TheWeakCouplingLimit.................................... 50
8.2 HistoricalDigression—Einstein’sDerivationofthePlanckLaw.... 53
8.3 FGRFluxes,EntropyProductionandKuboFormulas............. 54
8.4 FromMicroscopictoFGRThermodynamics .................... 56
9 Appendix ...................................................... 58
9.1 StructuralTheorems ........................................ 58
9.2 TheHilbert-SchmidtCondition ............................... 60
References ......................................................... 63
FermiGoldenRuleandOpenQuantumSystems
JanDerezin´skiandRafałFru¨boes ................................... 67
1 Introduction .................................................... 68
1.1 Fermi Golden Rule and Level Shift Operator in an
AbstractSetting ............................................ 68
1.2 ApplicationsoftheFermiGoldenRuletoOpenQuantumSystems . 69
2 FermiGoldenRuleinanAbstractSetting ........................... 71
2.1 Notation .................................................. 71
2.2 LevelShiftOperator ........................................ 72
2.3 LSOforC∗-Dynamics ...................................... 73
0
2.4 LSOforW∗-Dynamics...................................... 74
2.5 LSOinHilbertSpaces....................................... 74
2.6 TheChoiceoftheProjectionP................................ 75
2.7 ThreeKindsoftheFermiGoldenRule ......................... 75
3 WeakCouplingLimit ............................................ 77
3.1 StationaryandTime-DependentWeakCouplingLimit............ 77
3.2 ProofoftheStationaryWeakCouplingLimit.................... 80
3.3 SpectralAveraging.......................................... 83
3.4 Second Order Asymptotics of Evolution
withtheFirstOrderTerm .................................... 85
3.5 ProofofTimeDependentWeakCouplingLimit ................. 87
3.6 ProofoftheCoincidenceofM andM withtheLSO ......... 88
st dyn
4 CompletelyPositiveSemigroups................................... 88
4.1 CompletelyPositiveMaps ................................... 89
4.2 StinespringRepresentationofaCompletelyPositiveMap ......... 89
4.3 CompletelyPositiveSemigroups .............................. 90
4.4 StandardDetailedBalanceCondition .......................... 91
4.5 Detailed Balance Condition in the Sense of Alicki-Frigerio-
Gorini-Kossakowski-Verri ................................... 93
5 SmallQuantumSystemInteractingwithReservoir.................... 93
5.1 W∗-Algebras .............................................. 94
5.2 AlgebraicDescription ....................................... 95
5.3 SemistandardRepresentation ................................. 95
5.4 StandardRepresentation ..................................... 96
Contents XI
6 Two Applications of the Fermi Golden Rule
toOpenQuantumSystems ........................................ 97
6.1 LSOfortheReducedDynamics............................... 97
6.2 LSOfortheLiouvillean ..................................... 99
6.3 Relationship Between the Davies Generator and the LSO
fortheLiouvilleaninThermalCase............................100
6.4 ExplicitFormulafortheDaviesGenerator ......................103
6.5 ExplicitFormulasforLSOfortheLiouvillean...................104
6.6 IdentitiesUsingtheFiberedRepresentation .....................106
7 FermiGoldenRuleforaCompositeReservoir .......................108
7.1 LSOforaSumofPerturbations...............................108
7.2 MultipleReservoirs .........................................109
7.3 LSO for the Reduced Dynamics in the Case of a Composite
Reservoir..................................................110
7.4 LSOfortheLiovilleanintheCaseofaCompositeReservoir ......111
A Appendix–One-ParameterSemigroups.............................112
References .........................................................115
Decoherence as Irreversible Dynamical Process in Open Quantum
Systems
PhilippeBlanchard,RobertOlkiewicz ................................ 117
1 PhysicalandMathematicalPrologue................................118
1.1 PhysicalBackground........................................118
1.2 EnvironmentalDecoherence..................................119
1.3 AlgebraicFramework .......................................120
1.4 QuantumDynamicalSemigroups .............................121
1.5 AModelofaDiscretePointerBasis ...........................123
2 TheAsymptoticDecompositionofT ...............................126
2.1 Notation ..................................................126
2.2 DynamicsintheMarkovianRegime ...........................127
2.3 TheUnitaryDecompositionofT .............................130
2
2.4 TheIsometric-SweepingDecomposition .......................133
2.5 Remarks ..................................................135
3 ReviewofDecoherenceEffectsinInfiniteSpinSystems ...............138
3.1 InfiniteSpinSystems........................................138
3.2 ContinuousPointerStates[10]................................139
3.3 Decoherence-InducedSpinAlgebra[6] ........................143
3.4 FromQuantumtoClassicalDynamicalSystems[38] .............146
4 DynamicalSemigroupsonCCRAlgebras ...........................148
4.1 AlgebrasofCanonicalCommutationRelations(CCR)............148
4.2 PromeasuresonLocallyConvexTopologicalVectorSpaces .......149
4.3 PerturbedConvolutionSemigroupsofPromeasures ..............151
4.4 QuantumDynamicalSemigroupsonCCRAlgebras..............153
4.5 Example:QuantumBrownianMotion..........................155
5 Outlook........................................................157
XII Contents
References .........................................................158
NotesontheQualitativeBehaviourofQuantumMarkovSemigroups
FrancoFagnolaandRolandoRebolledo .............................. 161
1 Introduction ....................................................162
1.1 Preliminaries...............................................164
2 ErgodicTheorems ...............................................165
3 TheMinimalQuantumDynamicalSemigroup........................167
4 TheExistenceofStationaryStates..................................172
4.1 AGeneralResult ...........................................172
4.2 ConditionsontheGenerator..................................174
4.3 Examples .................................................178
4.4 AMultimodeDickeLaserModel .............................178
4.5 AQuantumModelofAbsorptionandStimulatedEmission........182
4.6 TheJaynes-CummingsModel ................................183
5 FaithfulStationaryStatesandIrreducibility..........................184
5.1 TheSupportofanInvariantState..............................184
5.2 SubharmonicProjections.TheCaseM=L(h)..................186
5.3 Examples .................................................188
6 TheConvergenceTowardstheEquilibrium ..........................189
6.1 MainResults...............................................190
6.2 Examples .................................................192
7 RecurrenceandTransienceofQuantumMarkovSemigroups ...........194
7.1 Potential ..................................................194
7.2 DefiningRecurrenceandTransience ...........................198
7.3 TheBehaviorofad-HarmonicOscillator.......................201
References .........................................................203
Continual Measurements in Quantum Mechanics and Quantum
StochasticCalculus
AlbertoBarchielli ................................................ 207
1 Introduction ....................................................208
1.1 ThreeApproachestoContinualMeasurements ..................208
1.2 QuantumStochasticCalculusandQuantumOptics...............208
1.3 SomeNotations:OperatorSpaces .............................209
2 UnitaryEvolutionandStates ......................................210
2.1 QuantumStochasticCalculus.................................210
2.2 TheUnitarySystem–FieldEvolution...........................217
2.3 TheSystem–FieldState......................................223
2.4 TheReducedDynamics .....................................225
2.5 PhysicalBasisoftheUseofQSC .............................228
3 ContinualMeasurements .........................................230
3.1 IndirectMeasurementsonSH ................................230
3.2 CharacteristicFunctionals....................................233
3.3 TheReducedDescription ....................................241
