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Lecture 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 Thisworkissubjecttocopyright. Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violations areliableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com (cid:1)c Springer-VerlagBerlinHeidelberg2006 PrintedinTheNetherlands Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. Typesetting:bytheauthorsandSPI Publisher Services usingaSpringerLATEXpackage Coverdesign:design&productionGmbH,Heidelberg Printedonacid-freepaper 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 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, pedagogical 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