A numerical method for generation of quantum noise

arXiv:cond-mat/0303059v1[cond-mat.stat-mech]4Mar2003Anumericalmethodforgenerationofquantumnoiseandsolutionofgeneralizedc-numberquantumLangevinequationDhrubaBanerjeeIndianAssociationfortheCultivationofScience,Jadavpur,Kolkata700032,IndiaBidhanChandraBagDepartmentofChemistry,VisvaBharati,Shantiniketan731235,IndiaSumanKumarBanikIndianAssociationfortheCultivationofScience,Jadavpur,Kolkata700032,India∗DebShankarRay†IndianAssociationfortheCultivationofScience,Jadavpur,Kolkata700032,India(Dated:February2,2008)Basedonacoherentstaterepresentationofnoiseoperatorandanensembleaveragingprocedurewehaverecentlydeveloped[Phys.Rev.E65,021109(2002);ibid.051106(2002)]aschemeforquantumBrownianmotiontoderivetheequationsfortimeevolutionoftrueprobabilitydistributionfunctionsinc-numberphasespace.Weextendthetreatmenttodevelopanumericalmethodforgenerationofc-numbernoisewitharbitrarycorrelationandstrengthatanytemperature,alongwiththesolutionoftheassociatedgeneralizedquantumLangevinequation.ThemethodisillustratedwiththehelpofacalculationofquantummeanfirstpassagetimeinacubicpotentialtodemonstratequantumKramersturnoverandquantumArrheniusplot.PACSnumbers:05.40-a,82.20.-wI.INTRODUCTIONAsystemcoupledtoitsenvironmentisthestandardparadigmforquantumtheoryofBrownianmotion1,2,3,4,5,6,7,8,9,10,11,12.Itsoverwhelmingsuccessinthetreatmentofvariousphenomenainquantumoptics,transportprocessesinJosephsonjunction,coherenceeffectsandmacroscopicquantumtunnelingincondensedmatterphysics,electrontransferinlargemolecules,thermalactivationprocessesinchemicalreactionsisnowwellknownandformsalargebodyofcurrentliterature.Whiletheearlydevelopmentofquantumopticsinitiatedinthesixtiesandseventieswasbasedondensityoperator,semigroup,noiseoperatorormasterequationmethodsprimarilywithinweak-couplingandMarkovapproximations,pathintegralapproachtoquantumBrownianmotionattractedwideat-tentionintheearlyeighties.Althoughthisdevelopmenthadwidenedthescopeofcondensedmatterandchemicalphysicssignificantly,sofarasthelargecouplingbetweenthesystemandtheheatbathandlargecorrelationtimesofthenoiseprocessesareconcernedseveralproblemsstillneedtobeaddressed.First,asearchforquantumanalogueofKramers’equationforanonlinearsystemwhichdescribesquantumBrownianmotioninphasespacehadremainedelusiveandatbestresultedinequationofmotionwhichcontainshigher(thansecond)derivativesofprobabilitydis-tributionfunctions2,13whosepositivedefinitenessisneverguaranteed.Asaresultthesequasi-probabilitydistributionfunctionsoftenbecomesingularornegativeinthefullquantumdomain.Second,althoughlongcorrelationtimesandlargecouplingconstantsaretreatednon-perturbativelyformallyinanexactmannerbypathintegrals,theiranalyticevaluationisgenerallycompletedwithinsemi-classicalschemes1,4,5,11.Thisresultsinsituationswherethetheoryfailstoretainitsvalidityinthevacuumlimit.Third,althoughthenumericaltechniquesbasedonthepathintegralMonteCarloareverysuccessfulinthetreatmentofequilibriumpropertiesitisoftendifficulttoimplementinadynamicalschemebecauseofthewell-knownsignproblemassociatedwithvaryingphasesinaquantumevolutionoverpathsduetotheoscillatingnatureoftherealtimepropagator6,7,10,12.Fourth,thetreatmentofnon-Markoviandynamicswithinaquantumformulationasretardationeffectinamemoryfunctionalisverydifficult,ifnotimpossible,tohandleevennumerically10,12.KeepinginviewofthesedifficultiesitisworthwhiletoaskhowtoextendclassicaltheoryofBrownianmotiontoquantumdomainforarbitraryfrictionandtemperature.Basedonacoherentstaterepresentationofthenoiseoperatorandanensembleaveragingprocedurewehaverecently14,15,16,17developedaschemeforquantumBrownianmotionintermsofac-numbergeneralizedLangevinequation.Theapproachallowsustouseclassicalmethodsofnon-MarkoviandynamicstoderiveexactgeneralizedquantumKramers’equationandotherrelevantquantumanaloguesofclassicaldiffusion,Fokker-PlanckandSmoluchowskiequations.Theobjectofthepresentanalysisistoextendthetreatment(i)todevelopanumericalmethodforgenerationofquantumnoiseasclassicalc-numbers(ii)andto2solvetheassociatedgeneralizedquantumLangevinequation.Thenumericalmethodvalidforarbitrarytemperature,dampingstrengthandnoisecorrelation,isexemplifiedbyanapplicationtocalculationofquantummeanfirstpassagetimeinacubicpotential.Theapproachisclassical-lookinginform,andisindependentofpathintegralquantumMonteCarlotechniquesandtakescareofquantumeffectorderbyorderwithsimplicityandaccuracytoahighdegree.Theoutlayofthepaperisasfollows:Westartwithabriefreviewofc-numberrepresentationofquantumnoiseandtheassociatedquantumLangevinequation,asformulatedrecentlybyus,inSec.II.ThisisfollowedbyanumericalmethodforgenerationofquantumnoiseandsolutionoftheLangevinequationinSec.III.InSec.IVweillustratetheprocedurebycalculatingquantummeanfirstpassagetimeinacubicpotentialwithanemphasisonturn-overfeaturesandArrheniusplotinthermalactivatedprocessesassistedbytunneling.II.ACOHERENTSTATEREPRESENTATIONOFQUANTUMNOISEANDQUANTUMLANGEVINEQUATIONINC-NUMBERSWeconsideraparticleofunitmasscoupledtoamediumcomprisedofasetofharmonicoscillatorswithfrequencyωi.ThisisdescribedbythefollowingHamiltonian:ˆH=ˆp