A phase-space study of the quantum Loschmidt Echo

arXiv:quant-ph/0510151v213Nov2006Aphase-spacestudyofthequantumLoschmidtEchointhesemiclassicallimitMoniqueCombescureIPNL,BˆatimentPaulDirac4rueEnricoFermi,Universit´eLyon-1F.69622VILLEURBANNECedex,Franceemail:[e-mail]´epartementdeMath´ematiquesLaboratoireJeanLeray,CNRS-UMR6629Universit´edeNantes,2ruedelaHoussini`ere,F-44322NANTESCedex03,Franceemail:[e-mail]chmidtecho(alsocalled“quantumfidelity”)hasbeenintroducedinordertostudythe(in)-stabilityofthequantumdynamicsunderperturbationsoftheHamiltonian.Ithasbeenextensivelystudiedinthepastfewyearsinthephysicsliterature,inconnectionwiththeproblemsof“quantumchaos”,quantumcomputationanddecoherence.Inthispaper,westudythisquantitysemiclassically(as~→0),takingasreferencequantumstatestheusualcoherentstates.Thelatterareknowntobewelladaptedtoasemiclassicalanalysis,inparticularwithrespecttosemiclassicalestimatesoftheirtimeevolution.Fortimesnotlargerthantheso-called“Ehrenfesttime”C|log~|,weareabletoestimatesemiclassicallytheLoschmidtEchoasafunctionoft(time),~(Planckconstant),andδ(thesizeoftheperturbation).Thewaytwoclassicaltrajectoriesmergingfromthesamepointinclassicalphase-space,flyapartorcomeclosetogetheralongtheevolutionsgovernedbytheperturbedandunperturbedHamiltoniansplayamajorroleinthisestimate.Wealsogiveestimatesofthe“returnprobability”(againonreferencestatesbeingthecoherentstates)bythesamemethod,asafunctionoftand~.121IntroductionThesemiclassicaltimebehaviourofquantumwavepacketshasbeenthesubjectofintenseinterestinthelastdecades,inparticularinsituationswherethereissomehyperbolicityinthecorrespondingclassicaldynamics(Lyapunovexponents)[9],[17],[30].Moreovertheresponseofaquantumsystemtoanexternalperturbationwhenthesizeδoftheperturbationincreasescanmanifestintriguingpropertiessuchasrecurrencesordecayintimeoftheso-calledLoschmidtEcho(or“quantumfidelity”)[7],[8].ByLoschmidtEchowemeanthefollowing:startingfromaquantumHamiltonianˆHinL2(Rd),whoseclassicalcounterpartHhasachaoticdynamics,andaddingtoita“perturbation”ˆHδ=ˆH+δˆV,thenwecomparetheevolutionsintimeU(t):=e−itˆH/~,Uδ(t):=e−itˆHδ/~ofinitialquantumwavepacketsϕsufficientlywelllocalizedaroundsomepointzinphase-space;morepreciselytheoverlapbetweenthetwoevolutions,orratheritssquareabsolutevalue,is:F~,δ(t):=|hUδ(t)ϕ,U(t)ϕi|2ForexampleforquantumdynamicsinHilbertspaceH=L2(Rd),dbeingthespacedimension,ϕcanbechosenastheusualcoherentstates,sincetheyarethequantumwavepackets“asmostlocalizedaspossible”inphase-spaceR2d.Sinceforδ=0,weobviouslyhaveF~,0(t)≡1,andforanyδ,F~,δ(0)=1,thetypeofdecayintofF~,δ(t)sotosaymeasuresthe(in)fidelityofthequantumevolutionwithrespecttoaperturbationofsizeδforgenericinitialwavepacketsϕ.ThenotionofLoschmidtEchoseemstohavebeenfirstintroducedbyPeres([24]),inthefollowingspirit:sincethesensitivitytoinitialdatawhichcharacterizesclassicalchaoshasnoquantumcounterpartbecauseofunitarityofthequantumevolution,atleastthe“sensitivitytoperturbations”oftheHamiltoniancouldreplaceitasacharacterizationofchaoticityinthe“quantumworld”.Abigamountofrecentworkappearedonthesubject,studyinginanessen-tiallyheuristicwaythedecayintimeofF~,δ(t)astincreasesfromzerotoinfinity;someofthemalsostudythispointinrelationshipwiththeimportantquestionofdecoherence.(See[1],[5],[12-14],[18],[22-26],[30-32]).3Inthis“jungle”ofsometimescontradictoryresults,itishardtoseethevariousargumentsinvolved,inparticulartheprecisebehaviourofF~,δ(t)asδ(thesizeoftheperturbation),t(thetime),andofcourse~(thePlanckconstant)arevaried,inparticularinwhichsenseandorderthevariouslimtsδ→0,~→0,t→∞aretaken.AlsoanimportantpointtoconsiderishowF~,δ(t)dependsonthelocationofthephase-spacepointzaroundwhichtheinitialwavepacketϕispeaked(sinceclassicalchaoticitydistinguishesvariouszonesinphase-spacewith“moreorlessregularityproperties”).TheaimofthepresentpaperistostartarigorousapproachofthequestionofsemiclassicalestimateofF~,δ(t),intermsofclassicalcharacteristicsofthe(perturbedandunperturbedclassicalflows),forinitialwavepacketsϕ=ϕzbeingthecoherentstateatphase-spacepointz.Theseestimatesarenon-perturbative,andarecarefullycalculatedintermsofparameters(z,δ,t,~).Themaintoolswehaveusedanddevelopedinthisrespectare1)semiclassicalcoherentstatespropagationestimates([9])2)abeautifulformulainspiredbyB.MehligandM.Wilkinson([22])abouttheWeylsymbolofametaplecticoperator,andthusofitsexpectationvalueincoherentstatesasasimpleGaussianphase-spaceintegral(see[10]wherewehavecompletedtheproofofMehlig-Wilkinson,andtreatedinparticularthecasewherethemonodromyoperatorhaseigenvalue1).Notethatveryrecently,J.BolteandT.SchwaiboldhaveindependentlyobtainedasimilarresultaboutsemiclassicalestimatesoftheQuantumFidelity([2]).Theplanofthispaperisasfollows.Insection2wegivesomepreliminariesabouttheEchoforsuitablequantumobservables,andgivethesemiclassicsofit.InSection3,weconsiderthe(integrable)d=1case,andconsiderthe“returnprobability”inthesemiclassicallimit.Wegiveamathematicalrigorouspresentationofbeautifulresultson“quantumrevivals”obtainedbyphysiciststwentyyear