Quantum mechanics in general quantum systems (III)

arXiv:quant-ph/0601051v526Nov2006Quantummechanicsingeneralquantumsystems(III):opensystemdynamicsAnMinWang∗QuantumTheoryGroup,DepartmentofModernPhysics,UniversityofScienceandTechnologyofChina,Hefei,230026,P.R.ChinaWeinvestigatetheexactsolution,perturbationtheoryandmasterequationofopensystemdy-namicsbasedonourserialstudiesonquantummechanicsingeneralquantumsystems[AnMinWang,quant-ph/0611216,quant-ph/0611217].Inasystem-environmentseparatedrepresentation,ageneralandexplicitsolutionofopensystemdynamicsisobtained,anditisanexactsolutionsinceitincludesallorderapproximationsofperturbation.Intermsofthecut-offapproximationofperturbationandourimprovedschemeofperturbationtheory,theimprovedformoftheperturbedsolutionofopensystemsabsorbingthepartialcontributionsfromthehighorderevenallorderapproximationsisdeduced.Moreover,onlyunderthefactorizinginitialcondition,theexactmasterequationincludingallorderapproximationsisproposed.Correspondingly,theperturbedmasterequationanditsimprovedformdifferentfromtheexistedmasterequationaregiven.Inspecial,theRedfieldmasterequationisderivedoutwithoutusingBorn-Markovapproximation.ThesolutionofopensystemdynamicsintheMilburnmodelisalsogained.Asexamples,Zurekmodeloftwo-stateopensystemanditsextensionwithtwotransversefieldsarestudied.PACSnumbers:03.65.Yz,03.65.Ca,03.65.-wI.INTRODUCTIONArealisticquantumsystemisneverisolated,butisimmersedinthesurroundingenvironment(aliasbath,reservoir)andinteractscontinuouslywithit.Suchasystemwithoutignorablecouplingtotheenvironmentcanbecalledopenquantumsystem.Generally,theenvironmentconsistsofahugenumberofdegreesoffreedom,itisevenwholeoutsideworld(universe)oftheconcerningopenquantumsystem.Infact,wemightnotknowtheexactstateoftheoutsideworld,havingonlysomestatisticalinformationtodescribeit.However,wearereallyinterestedinareliableandeffectivetheoryofopensystemdynamicsundertheinfluenceofitsenvironment.Thebasicideaofquantumtheoryofopensystemsisthoughtofasaninterestingopensystemanditssurroundingenvironmentformatotalcompositesystem,orvisversa,acompositesystemcanbedecomposedintoaninterestingopensystemandasurroundingenvironment.Thekeymattersofquantumtheoryofopensystemsaretodeterminetheinteractionbetweentheopensystemanditsenvironmentandbuildthephysicalmodelsoftheopensystemanditsenvironment.Opensystemdynamicsisjustalawofthisopenquantumsystemhowtoevolutewithtimeanditssolutionatanygiventime.Openquantumsystemanditsdynamicsareveryimportantformanyinterestingquantumtheorybranchessuchasquantumoptics[1,2],condensedmattertheory,quantuminformationandcomputing[3,4],moreconcretely,quantumdecoherence,quantummeasurement[5,6],quantumdissipation[7,8],quantumtransport[9],quantumchaos[10]et.al.Studyofopensystemdynamicsishelpfulforunderstandingsomeveryessentialproblemsinphysics,forexample,thetransitionfromquantumtoclassicalworld.Avarietyofdifferentformaltechniqueshavebeendevelopedandusedindealingwithopenquantumsystems.Fromtheabovereviewsandbooks,theinterestedreadersmaygetthem.Here,weintendtostartwith“thefirstprinciple”ofquantummechanics,thatis,theSchr¨odingerequationorthevonNeumannequation,andthentrytobuildatheoreticalformulismincludingthegeneralandexplicitformsofmotionequation,dynamicalsolution,andperturbationtheoryofopensystems.Itisclearthatsucha“firstprinciple”schememightbenotsuitabletothecaseswhenonecannotclearlyknowtheenvironmentmodeland/orthesystem-environmentcouplingformsincetheenvironmentistoohugeandtoocomplicated.However,ourconclusionsmightbehelpfulforbuildingthemodelsofsuchsomesystems.Moreover,oneofpossiblewaystoavoidthisdifficultyistousetheMilburndynamics[11].Thatis,theenvironmentisseparatedintonear-andremotetwoparts,theHamiltoniansofthenearenvironment(oftenwithfinitedegreeoffreedoms)andthecouplingtotheinterestingopensystemareassumedtobeclearlyknown,buttheinfluenceoftheremote∗Electronicaddress:[e-mail]2environmentontheinterestingsystemisincarnatedbyanextraterminMilburnmotionequationcomparedwiththevonNeumannmotionequation.Similarly,wesuccessfullyobtainthegeneralandexplicitsolutionofMilburndynamicsoftheinterestingsystemaccordingourscheme.Becauseofthedissipativenatureofopensystems,wemustturntothedensitymatrixforaproperdescriptionwhatevertheinitialstateispureormixed.Actually,weareinterestedinthepropertiesofopensystemsonly,itwillbeappropriatetostudythereduceddensitymatrixevolutionwithtimeormotionequationoritssolution.Here,thereduceddensitymatrixdescribingtheopensystemsisobtainedbytracingoutthedegreesoffreedomoftheenvironmentfromthetotal(systemplusenvironment)densitymatrix.Duetothesystemandenvironmentbeingentangledgenerallyinsystemevolutionwithtime,directlysolvingSchr¨odingerequationorvonNeumannequationofthetotalsystemisaformidabletaskbyusingtheexistedmethods.Traditionally,thisproblemisstudiedbyperturbationtheoryinsystem-environmentcouplingscheme.OnesoftentaketheinteractionbetweenthesystemanditsenvironmentasaperturbedpartandthenusetheinteractionpicturetoderiveoutthemasterequationofopensystemsviasomephysicalapproximationssuchasBorn-Markovonesandtheothers.Ifanopensystemisexactlysolvable,thecoupl