and random matrix theory

Wavepacketdynamics,quantumreversibility,andrandommatrixtheoryMoritzHillera,*,DoronCohenb,TheoGeisela,TsampikosKottosa,caMaxPlanckInstituteforDynamicsandSelf-OrganizationandDepartmentofPhysics,UniversityofGo¨ttingen,Bunsenstraße10,D-37073Go¨ttingen,GermanybDepartmentofPhysics,Ben-GurionUniversity,Beer-Sheva84105,IsraelcDepartmentofPhysics,WesleyanUniversity,Middletown,CT06459,USAReceived14June2005;accepted7December2005Availableonline9February2006AbstractWeintroduceandanalyzethephysicsof‘‘drivingreversal’’experiments.Theseareprototypewavepacketdynamicsscenariosprobingquantumirreversibility.Unlikethemostlyhypothetical‘‘timereversal’’concept,a‘‘drivingreversal’’scenariocanberealizedinalaboratoryexperiment,andisrelevanttothetheoryofquantumdissipation.Westudyboththeenergyspreadingandthesurvivalprobabilityinsuchexperiments.Wealsointroduceandstudythe‘‘compensationtime’’(timeofmaximumreturn)insuchascenario.Extensiveeffortisdevotedtofiguringoutthecapabilityofeitherlinearresponsetheoryorrandommatrixtheory(RMT)todescribespecificfeaturesofthetimeevolution.WeexplainthatRMTmodelingleadstoastrongnon-perturbativeresponseeffectthatdiffersfromthesemiclassicalbehavior.2005ElsevierInc.Allrightsreserved.PACS:03.65.w;03.65.Sq;05.45.Mt;73.23.bKeywords:Quantumdissipation;Quantumchaos;Randommatrixtheory0003-4916/$-seefrontmatter2005ElsevierInc.Allrightsreserved.doi:10.1016/j.aop.2005.12.003*Correspondingauthor.E-mailaddress:[e-mail](M.Hiller).AnnalsofPhysics321(2006)1025–1062[1–11].DrivensystemsaredescribedbyaHamiltonianHðQ;P;xðtÞÞ,wherex(t)isatime-dependentparameterand(Q,P)aresomegeneralizedactions.Duetothetimedependenceofx(t),theenergyofthesystemisnotaconstantofmotion.Ratherthesystemmakes‘‘transitions’’betweenenergylevels,andthereforeabsorbsenergy.Thisirreversiblelossofenergyisknownasdissipation.Tohaveaclearunderstandingofquantumdissipationweneedatheoryforthetimeevolutionoftheener-gydistribution.Unfortunately,ourunderstandingonquantumdynamicsofchaoticsystemsisstillquitelimited.Themajorityoftheexistingquantumchaosliteratureconcentratesonunderstand-ingthepropertiesofeigenfunctionsandeigenvalues.Oneofthemainoutcomesofthesestudiesistheconjecturethatrandommatrixtheory(RMT)modeling,initiatedhalfacen-turyagobyWigner[12,13],cancapturetheuniversalaspectsofquantumchaoticsystems[14,15].DuetoitslargesuccessRMThasbecomeamajortheoreticaltoolinquantumcha-osstudies[14,15],andithasfoundapplicationsinbothnuclearandmesoscopicphysics(forarecentreviewsee[16]).However,itsapplicabilitytoquantumdynamicswasleftunexplored[17,18].Thispaperextendsourpreviousreports[10,17,18]onquantumdynamics,bothindetailanddepth.Specifically,weanalyzetwodynamicalschemes:thefirstistheso-calledwave-packetdynamicsassociatedwitharectangularpulseofstrength+whichisturnedonforaspecifiedduration.Thesecondinvolvesanadditionalpulsefollowedbythefirstonewhichhasastrengthandisofequalduration.Wedefinethislatterschemeasdrivingreversalscenario.Weilluminatethedirectrelevanceofourstudywiththestudiesofquan-tumirreversibilityofenergyspreading[10]andconsequentlywithquantumdissipation.Weinvestigatetheconditionsunderwhichmaximumcompensationissucceededanddefinethenotionofcompensation(echo)time.Tothisendwerelybothonnumericalcal-culationsperformedforachaoticsystemandonanalyticalconsiderationsbasedonlinearresponsetheory(LRT).ThelatterconstitutestheleadingtheoreticalframeworkfortheanalysisofdrivensystemsandourstudyaimstoclarifythelimitationsofLRTduetocha-os.OurresultsarealwayscomparedwiththeoutcomesofRMTmodeling.WefindthattheRMTapproachfailsingeneral,togivethecorrectpictureofwave-evolution.RMTcanbetrustedonlytotheextendthatitgivestrivialresultsthatareimpliedbyperturba-tiontheory.Non-perturbativeeffectsaresensitivetotheunderlyingclassicaldynamics,andthereforethehfi0behaviorforeffectiveRMTmodelsisstrikinglydifferentfromthecorrectsemiclassicallimit.Thestructureofthispaperisasfollows:inthenextsection,wediscussthenotionofirreversibilitywhichisrelatedtodrivingreversalschemesanddistinguishitfrommicro-re-versibilitywhichisassociatedwithtimereversalexperiments.InSection3wediscussthedrivingschemesthatweareusingandweintroducethevariousobservablesthatwewillstudyintherestofthepaper.InSection4,themodelsystemsareintroducedandananal-ysisofthestatisticalpropertiesoftheeigenvaluesandtheHamiltonianmatrixispresent-ed.TherandommatrixtheorymodelingispresentedinSubsection4.4.InSection5weintroducetheconceptofparametricregimesandexhibititsapplicabilityintheanalysisofparametricevolutionofeigenstates[19].Section6extendsthenotionofregimesindynamicsandpresentstheresultsoflinearresponsetheoryforthevarianceandthe1026M.Hilleretal./AnnalsofPhysics321(2006)1025–1062survivalprobability.Thelinearresponsetheory(LRT)forthevarianceisanalyzedindetailsinthefollowingSubsection6.1.Inthissubsection,wealsointroducethenotionofrestrictedquantum-classicalcorrespondence(QCC)andshowthat,asfarasthesecondmome