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April18,200816:58WSPC-ProceedingsTrimSize:9inx6inqmath10RepeatedInteractionQuantumSystems:DeterministicandRandomAlainJoyeInstitutFourierUniversite´deGrenoble47PB38402SaintMartind’He`res,FranceThispapergivesanoverviewofrecentresultsconcerningthelongtimedy-namicsofrepeatedinteractionquantumsystemsinadeterministicandran-domframework.Wedescribethenonequilibriumsteadystates(NESS)suchsystemsdisplayandwepresent,asamacroscopicconsequence,asecondlawofthermodynamicstheseNESSgiveriseto.Wealsoexplaininsomedetailstheanalysisofproductsofcertainrandommatricesunderlyingthisdynamicalproblem.Keywords:Nonequilibriumquantumstatisticalmechanics,Repeatedinterac-tionquantumsystems,Productsofrandommatrices11.IntroductionandModelArepeatedinteractionquantumsystemconsistsofareferencequantumsubsystemSwhichinteractssuccessivelywiththeelementsEmofachainC=E1+E2+∙∙∙ofindependentquantumsystems.Ateachmomentintime,SinteractspreciselywithoneEm(mincreasesastimedoes),whiletheotherelementsinthechainevolvefreelyaccordingtotheirintrinsicdynamics.ThecompleteevolutionisdescribedbytheintrinsicdynamicsofSandofalltheEm,plusaninteractionbetweenSandEm,foreachm.Thelatterischaracterizedbyaninteractiontimeτm>0,andaninteractionoperatorVm(actingonSandEm);duringthetimeinterval[τ1+∙∙∙+τm11+∙∙∙+τm),SiscoupledtoEmonlyviaVm.Systemswiththisstructureareimportantfromaphysicalpointofview,sincetheyarisenaturallyasmodelsforfunda-mentalexperimentsontheinteractionofmatterwithquantizedradiation.Asanexample,the“Oneatommaser”providesanexperimentalsetupinwhichthesystemSrepresentsamodeoftheelectromagneticfield,whereastheelementsEkdescribeatomsinjectedinthecavity,onebyone,which
April18,200816:58WSPC-ProceedingsTrimSize:9inx6inqmath102interactwiththefieldduringtheirflightinthecavity.Aftertheyleavethecavity,theatomsencodesomepropertiesofthefieldwhichcanbemea-suredontheseatoms14,16Forrepeatedinteractionsystemsconsideredasideal,i.e.suchthatallatomsareidenticalwithidenticalinteractionsandtimesofflightthroughthecavity,correspondingmathematicalanalysesareprovidedin17,7Totakeintoaccounttheunavoidablefluctuationsintheexperimentsetupusedtostudytheserepeatedinteractionsystems,mod-elizationsincorporatingrandomnesshavebeenproposedandstudiedin8and.9Withadifferentperspective,repeatedquantuminteractionmodelsalsoappearnaturallyinthemathematicalstudyofmodelizationofopenquantumsystemsbymeansofquantumnoises,see4andreferencestherein.Any(continuous)masterequationgoverningthedynamicsofstatesonasystemScanbeviewedastheprojectionofaunitaryevolutiondrivingthesystemSandafieldofquantumnoisesininteraction.Itisshownin4howtorecoversuchcontinuousmodelsassomedelicatelimitofadiscretizationgivenbyarepeatedquantuminteractionmodel.Letusfinallymention15forresultsofasimilarflavourinasomewhatdifferentframework.Ourgoalistopresenttheresultsofthepapers7,8and9on(random)repeatedinteractionquantumsystems,whichfocusonthelongtimebe-haviourofthesesystems.Letusdescribethemathematicalframeworkusedtodescribethesequantumdynamicalsystems.Accordingtothefundamentalprinciplesofquantummechanics,statesofthesystemsSandEmaregivenbynormal-izedvectors(ordensitymatrices)onHilbertspacesHSandHEm,respec-tively,3,6a.WeassumethatdimHS<,whiledimHEmmaybeinfinite.ObservablesASandAEmofthesystemsSandEmareboundedopera-torsformingvonNeumannalgebrasMS⊂B(HS)andMEm⊂B(HEm).TheyevolveaccordingtotheHeisenbergdynamicsR3t7→αtS(AS)andtttR3t7→αEm(AEm),whereαSandαEmare-automorphismgroupsofMSandMEm,respectively,seee.g.6Wenowintroducedistinguishedreferencestates,givenbyvectorsψS∈HSandψEm∈HEm.TypicalchoicesforψS,ψEmareequilibrium(KMS)statesforthedynamicsαtS,αtEm,atinversetemperaturesβS,βEm.TheHilbertspaceofstatesofthetotalsystemisthetensorproductH=HS⊗HC,aAnormalizedvectorψdefinesa“pure”stateA7→hψ,Aψi=TrP(%ψA),where%ψ=|ψihψ|.Ageneral“mixed”stateisgivenbyadensitymatrix%=n1pn%ψn,wheretheprobabilitiespn0sumuptoone,andwheretheψnarenormalizedvectors.
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