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Lecture Notes in Mathematics 1881 Editors: J.-M.Morel,Cachan F.Takens,Groningen B.Teissier,Paris · · S. Attal A. Joye C.-A. Pillet (Eds.) Open Quantum Systems II The Markovian Approach 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 3-540-30992-6 SpringerBerlinHeidelbergNewYork ISBN-13 978-3-540-30992-5 SpringerBerlinHeidelbergNewYork DOI10.1007/b128451 Thisworkissubjecttocopyright. 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Typesetting:bytheauthorsandSPI Publisher Services usingaSpringerLATEXpackage Coverdesign:design&productionGmbH,Heidelberg Printedonacid-freepaper SPIN:11602620 V A 41/3100/ SPI 543210 Preface Thisvolumeisthesecondinaseriesofthreevolumesdedicatedtothelecturenotes of the summer school “Open Quantum Systems” which took place in the Institut FourierinGrenoble,fromJune16thtoJuly4th2003.Thecontributionspresentedin thesevolumesarerevisedandexpandedversionsofthenotesprovidedtothestudents during the school. After the first volume, developing the Hamiltonian approach of openquantumsystems,thissecondvolumeisdedicatedtotheMarkovianapproach. Thethirdvolumepresentsbothapproaches,butattherecentresearchlevel. Openquantumsystems Aquantum open systemisaquantum systemwhich isinteracting withanother one.Thisisageneraldefinition,butingeneral,itisunderstoodthatoneofthesys- tems is rather “small” or “simple” compared to the other one which is supposed to behuge,tobetheenvironment,agasofparticles,abeamofphotons,aheatbath... Theaimofquantumopensystemtheoryistostudythebehaviourofthiscoupled systemandinparticularthedissipationofthesmallsysteminfavourofthelargeone. Oneexpectsbehavioursofthesmallsystemsuchasconvergencetoanequilibrium state,thermalization...Themainquestionsonetriestoanswerare:Isthereaunique invariant state for the small system (or for the coupled system)? Does one always convergetowardsthisstate(whatevertheinitialstateis)?Whatspeedofconvergence canweexpect?Whatarethephysicalpropertiesofthisequilibriumstate? Onecandistinguishtwoschoolsinthewayofstudyingsuchasituation.Thisis trueinphysicsaswellasinmathematics.Theyrepresentingeneral,differentgroups ofresearcherswith,uptonow,ratherfewcontactsandcollaborations.Wecallthese twoapproachestheHamiltonianapproachandtheMarkovianapproach. In the Hamiltonian approach, one tries to give a full description of the coupled system. That is, both quantum systems are described, with their state spaces, with their own Hamiltonians and their interaction is described through an explicit inter- actionHamiltonian.OnthetensorproductofHilbertspacesweendupwithatotal Hamiltonian, and the goal is then to study the behaviour of the system under this dynamics.ThisapproachispresentedindetailsinthevolumeIofthisseries. VI Preface In the Markovian approach, one gives up trying to describe the large system. The idea is that it may be too complicated, or more realistically we do not know it completely. The study then concentrates on the effective dynamics which is induced on the small system. This dynamics is not a usual reversible Hamiltonian dynamics, but is described by a particular semigroup acting on the states of the smallsystem. BeforeenteringintotheheartoftheMarkovianapproachandallitsdevelopment, in the next courses, let us have here an informal discussion on what this approach exactlyis. TheMarkovianapproach WeconsiderasimplequantumsystemH whichevolvesasifitwereincontact withanexteriorquantumsystem.Wedonottrytodescribethisexteriorsystem.Itis maybetoocomplicated,ormorerealisticallywedonotquiteknowit.Weobserveon theevolutionofthesystemHthatitisevolvinglikebeingincontactwithsomething else,likeanopensystem(byoppositionwiththeusualnotionofclosedHamiltonian systeminquantummechanics).Butwedonotquiteknowwhatiseffectivelyacting onH.WehavetodealwiththeefffectivedynamicswhichisobservedonH. Bysuchadynamics,wemeanthatwelookattheevolutionofthestatesofthe system H. That is, for an initial density matrix ρ at time 0 on H, we consider the 0 stateρ attimetonH.Themainassumptionhereisthatthisevolution t ρ =P (ρ ) t t 0 isgivenbyasemigroup.Thisistosaythatthestateρ attimetdeterminesthefuture t statesρt+h,withoutneedingtoknowthewholepast(ρs)s≤t. EachofthemappingP isageneralstatetransformρ (cid:1)→ρ .Suchamapshould t 0 t be in particular trace-preserving and positivity-preserving. Actually these assump- tionsarenotquiteenoughandthepositivity-preservingpropertyshouldbeslightly extendedtoanaturalnotionofcompletelypositivemap(seeR.Rebolledo’scourse). Weendupwithasemigroup(P ) ofcompletelypositivemaps.Undersomecon- t t≥0 tinuityconditions,thefamousLindbladtheorem(seeR.Rebolledo’scourse),shows thattheinfinitesimalgeneratorofsuchasemigroupisoftheform (cid:2) (cid:3) (cid:1) 1 1 L(ρ)=i[H,ρ]+ L ρL∗− L∗L ρ− ρL∗L i i 2 i i 2 i i i forsomeself-adjointboundedoperatorH onHandsomeboundedoperatorsL on i H.Theevolutionequationforthestatesofthesystemcanbesummarizedinto d ρ =L(ρ ). dt t t This is theso-called quantum master equation inphysics. Itisactually the starting point in many physical articles on open quantum systems: a specific system to be studiedisdescribedbyitsmasterequationwithagivenexplicitLinbladgeneratorL. Preface VII ThespecificformofthegeneratorLhastounderstoodasfollows.Itissimilarto thedecompositionofaFellerprocessgenerator(seeL.Rey-Bellet’sfirstcourse)into afirstorderdifferentialpartplusasecondorderdifferentialpart.Indeed,thefirstterm i[H, ·] istypicalofaderivationonanoperatoralgebra.IfLwerereducedtothattermonly, thenP =etLiseasilyseentoactasfollows: t P (X)=eitHXe−itH. t Thatis,thissemigroupextendsintoagroupofautomorphismsanddescribesausual Hamiltonian evolution. In particular it describes a closed quantum system, there is noexteriorsysteminteractingwithit. Thesecondtypeoftermshavetobeunderstoodasfollows.IfL=L∗then 1 1 LXL∗− L∗LX − XL∗L=[L,[L,X]]. 2 2 It is a double commutator, it is a typical second order differential operator on the operatoralgebra.Itcarriesthediffusivepartofthedissipationofthesmallsystemin favoroftheexterior,likeaLaplacianterminaFellerprocessgenerator. When L does not satisfy L = L∗ we are left with a more complicated term whichismoredifficulttointerpretinclassicalterms.Ithastobecomparedwiththe jumpingmeasureterminageneralFellerprocessgenerator. Now,thatthesemigroupandthegeneratoraregiven,thequantumnoises(seeS. Attal’s course) enter into the game in order to provide a dilation of the semigroup (F.Fagnola’scourse).Thatis,onecanconstructanappropriateHilbertspaceF on which quantum noises dai(t) live, and one can solve a differential equation on the j spaceH⊗F whichisoftheformofaSchro¨dingerequationperturbedbyquantum noisesterms: (cid:1) dU =LU dt+ KiU dai(t). (1) t t j t j i,j This equation is an evolution equation, whose solutions are unitary operators on H⊗F,soitdescribesaclosedsystem(ininteractionpictureactually).Furthermore itdilatesthesemigroup(P ) inthesensethat,thereexistsa(pure)stateΩ onF t t≥0 suchthatifρisanystateonHthen <Ω, U (ρ⊗I)U∗Ω>=P (ρ). t t t This is to say that the effective dynamics (P ) we started with on H, which we t t≥0 didnotknowwhatexactexteriorsystemwasthecauseof,isobtainedasfollows:the smallsystemHisactuallycoupledtoanothersystemF andtheyinteractaccording to the evolution equation (1). That is, F acts like a source of (quantum) noises on H. The effective dynamics on H is then obtained when averaging over the noises throughacertainstateΩ. VIII Preface ThisisexactlythesamesituationastheoneofMarkovprocesseswithrespectto stochasticdifferentialequations(L.Rey-Bellet’sfirstcourse).AMarkovsemigroup isgivenonsomefunctionalgebra.Thisisacompletelydeterministicdynamicswhich describes an irreversible evolution. The typical generator, in the diffusive case say, containstwotypesofterms. Firstorderdifferentialtermswhichcarrytheordinarypartofthedynamics.Ifthe generatorcontainsonlysuchtermsthedynamicsiscarriedbyanordinarydifferential equationandextendstoareversibledynamics. Second order differential operator terms which carry the dissipative part of the dynamics.Thesetermsrepresentthenegativepartofthegenerator,thelossofenergy infavorofsomeexterior. But in such a description of a dissipative system, the environment is not described. The semigroup only focuses on the effective dynamics induced on some systembyanenvironment.Withthehelpofstochasticdifferentialequationsonecan give a model of the action of the environment. It is possible to solve an adequat stochastic differential equation, involving Brownian motions, such that the result- ing stochastic process be a Markov process with same semigroup as the one given atthebegining.Suchaconstructionisnowadays naturalandoneoftenuseitwith- out thinking what this really means. To the state space where the function algebra acts,wehavetoaddaprobabilityspacewhichcarriesthenoises(theBrownianmo- tion).Wehaveenlargedtheinitialspace,thenoisedoesnotcomenaturallywiththe function algebra. The resolution of the stochastic differential equation gives rise to a solution living in this extended space (it is a stochastic process, a function of the Brownian motions). It is only when avering over the noise (taking the expectation) thatonerecoverstheactionofthesemigrouponthefunctionalgebra. Wehavedescribedexactlythesamesituationasforquantumsystems,asabove. Organizationofthevolume Theaimofthisvolumeistopresentthisquantumtheoryindetails,togetherwith itsclassicalcounterpart. ThevolumeactuallystartswithafirstcoursebyL.Rey-Belletwhichpresentsthe classical theory of Markov processes, stochastic differential equations and ergodic theoryofMarkovprocesses. ThesecondcoursebyL.Rey-Belletappliesthesetechniquestoafamilyofclas- sicalopensystems.Theassociatedstochasticdifferentialequationisderivedfroman Hamiltoniandescriptionofthemodel. ThecoursebyS.Attalpresentsanintroductiontothequantumtheoryofnoises andtheirconnectionswithclassicalones.Itconstructsthequantumstochasticinte- grals and proves the quantum Ito formula, which are the cornerstones of quantum Langevinequations. R.Rebolledo’scoursepresentsthetheoryofcompletelypositivemaps,theirrep- resentationtheoremsandthesemigrouptheoryattachedtothem.Thisendsupwith thecelebratedLindblad’stheoremandthenotionofquantummasterequations. Preface IX Finally,F.Fagnola’scoursedevelopsthetheoryofquantumLangevinequations (existence, unitarity) and shows how quantum master equations can be dilated by suchequations. Lyon,Grenoble,Toulon Ste´phaneAttal September2005 AlainJoye Claude-AlainPillet Contents ErgodicPropertiesofMarkovProcesses LucRey-Bellet.................................................... 1 1 Introduction .................................................... 1 2 StochasticProcesses ............................................. 2 3 MarkovProcessesandErgodicTheory.............................. 4 3.1 Transitionprobabilitiesandgenerators ......................... 4 3.2 StationaryMarkovprocessesandErgodicTheory................ 7 4 BrownianMotion................................................ 12 5 StochasticDifferentialEquations................................... 14 6 ControlTheoryandIrreducibility .................................. 24 7 HypoellipticityandStrong-FellerProperty........................... 26 8 LiapunovFunctionsandErgodicProperties.......................... 28 References ......................................................... 39 OpenClassicalSystems LucRey-Bellet.................................................... 41 1 Introduction .................................................... 41 2 Derivationofthemodel........................................... 44 2.1 Howtomakeaheatreservoir ................................. 44 2.2 MarkovianGaussianstochasticprocesses....................... 48 2.3 HowtomakeaMarkovianreservoir ........................... 50 3 Ergodicproperties:thechain ...................................... 52 3.1 Irreducibility............................................... 56 3.2 StrongFellerProperty....................................... 57 3.3 LiapunovFunction ......................................... 58 4 HeatFlowandEntropyProduction ................................. 66 4.1 Positivityofentropyproduction............................... 69 4.2 Fluctuationtheorem......................................... 71 4.3 KuboFormulaandCentralLimitTheorem...................... 75 References ......................................................... 77 XII Contents QuantumNoises Ste´phaneAttal.................................................... 79 1 Introduction .................................................... 80 2 Discretetime ................................................... 81 2.1 Repeatedquantuminteractions................................ 81 2.2 TheToyFockspace......................................... 83 2.3 Highermultiplicities ........................................ 89 3 Itoˆ calculusonFockspace ........................................ 93 3.1 Thecontinuousversionofthespinchain:heuristics .............. 93 3.2 TheGuichardetspace ....................................... 94 3.3 AbstractItoˆ calculusonFockspace............................ 97 3.4 ProbabilisticinterpretationsofFockspace ...................... 105 4 Quantumstochasticcalculus ...................................... 110 4.1 Anheuristicapproachtoquantumnoise ........................ 110 4.2 Quantumstochasticintegrals ................................. 113 4.3 Backtoprobabilisticinterpretations ........................... 122 5 Thealgebraofregularquantumsemimartingales...................... 123 5.1 Everywheredefinedquantumstochasticintegrals ................ 124 5.2 Thealgebraofregularquantumsemimartingales................. 127 6 ApproximationbythetoyFockspace............................... 130 6.1 EmbeddingthetoyFockspaceintotheFockspace............... 130 6.2 ProjectionsonthetoyFockspace ............................. 132 6.3 Approximations ............................................ 136 6.4 Probabilisticinterpretations .................................. 138 6.5 TheItoˆ tables .............................................. 139 7 Backtorepeatedinteractions ...................................... 139 7.1 Unitarydilationsofcompletelypositivesemigroups.............. 140 7.2 ConvergencetoQuantumStochasticDifferentialEquations........ 142 8 Bibliographicalcomments ........................................ 145 References ......................................................... 145 Complete Positivity and the Markov structure of Open Quantum Systems RolandoRebolledo ................................................149 1 Introduction:apreviewofopensystemsinClassicalMechanics......... 149 1.1 Introducingprobabilities..................................... 152 1.2 AnalgebraicviewonProbability.............................. 154 2 Completelypositivemaps......................................... 157 3 Completelyboundedmaps ........................................ 162 4 DilationsofCPandCBmaps...................................... 163 5 QuantumDynamicalSemigroupsandMarkovFlows .................. 168 6 DilationsofquantumMarkovsemigroups ........................... 173 6.1 AviewonclassicaldilationsofQMS .......................... 174 6.2 TowardsquantumdilationsofQMS ........................... 180 References ......................................................... 181