Time of arrival through a quantum barrier

arXiv:quant-ph/9903060v117Mar1999IMAFF-PCC/99-17Timeofarrivalthroughaquantumbarrier1J.Le´on,J.Julve,P.Pitanga2andF.J.deUrr´ıes3InstitutodeMatem´aticasyF´ısicaFundamental,CSICSerrano113-bis,28006MADRID,SpainAbstractWeintroduceaformalismforthecalculationofthetimeofarrivaltatadetectorofparticlestravelingthroughinteractingenvironments.Wedevelopageneralformula-tionthatemploysquantumcanonicaltransformationsfromthefreetotheinteractingcasestocomputet.WeinterpretourresultsforthetimeofarrivaloperatorintermsofaPositiveOperatorValuedMeasure.Wethencomputetheprobabilitydistribu-tioninthetimesofarrivalatadetectorforthoseparticlesthat-aftertheirinitialpreparation-haveundergonequantumtunnelingorreflectionduetothepresenceofpotentialbarriers.Wemakeaextensiveanalysisofseveralcases,themainresultsbeingthepresenceoftheexpectedretardationoradvancementfortransmission,andofnon-foreseentwobumpstructuresforsomecasesofreflection.IMAFF-PCC/99-17March19991WorksupportedinpartbyDGESICPB97-1256.2InstitutodeF´ısica,UFRJ.21945-970RiodeJaneiro-Brasil.3DepartamentodeF´ısica,UniversidaddeAlcal´a,Alcal´adeHenares,Spain.e-mailaddresses:[e-mail],leon@iff.csic.es,[e-mail],[e-mail]1IntroductionInthispaperweworkoutatheoreticalframeworktocomputethetimeinwhichaparticlethatmovesinaninteractingenvironmentarrivesatagivenpoint.Weapplyourresultstoanswerthelongpendingquestion:Howlongdoesittakeforaparticletotunnelacross(orthrough)aquantumbarrier?Intheconstructionofthisframeworkwewillhavetodealwithproblemsofverydifferentkindthatweintroducenow:First,thereisthenatureoftimeinquantummechanics.ItappearsastheexternalevolutionparameterintheSchr¨odingerandHeisenbergequations,commontoboth,sys-temsandobserversalike.However,timearisesinmanyinstances(transitions,decays,arrivals,etc.)asapropertyofthephysicalsystems.TheattemptstopromotetimetothecategoryofobservablerunearlyintotheobstructiondetectedbyPauli[1]:Aselfadjointtimeoperatorimpliesanunboundedenergyspectrum.Thiswassoonrelatedtotheuncer-taintyrelationfortimeandenergy,whosestatusandphysicalmeaninghasbeensubjectofcontroversies[2,3,4,5],andisstillsubjectofelucidationtoday(seeforinstance[6]and[7]).Thequestionremainsunsettledforclosedquantumsystems,speciallyinthecaseofquantumgravity,whoseformulationispervadedbythesocalledproblemoftime[8].Second,thedefinitionofthetime-of-arrival(toa),whichisprobablythesimplestcan-didatetimetobecomeapropertyofthe(arriving)physicalsystem,ratherthanamereexternalparameter.Duetoitsconceptualsimplicity,ithasbeenusedinmanycasestoillustratedifferentproblemsrelatedtotheroleoftimeinquantumtheory.Allcockana-lyzed[9]extensivelythedifficultiesmetbythetoaconcludingtheywereinsurmountable.Thepresentsituationisambiguous.Ontheoneside,therearesoundtheoreticalanaly-sis[10,11]ofthetoashowingthatitcannotbepreciselydefinedandmeasuredinquantummechanics.Thiscontradictsthepossibilityofdevisinghighefficiencyabsorbers[12],thatcouldbeusedasalmostidealdetectorsfortoa[13].Ontheotherhand,thereareex-plicitconstructionsofaself-adjointtoaoperatorforthenon-relativisticfreeparticleinonespacedimension[14],therelativisticfreeparticlein3-D[15],bothavoidingthePauliproblem.Thereisalsoanalternativeformulation[16]asaPositiveOperatorValuedMea-sure(POV).Finally,thetoahasbeenmeasuredinhighprecisionexperiments[17,18]onthearrivaloftwoentangledphotonsproducedbyparametricdown-conversion,oneofwhichhasundergonetunnelingthroughaphotonicbandgap(PBG).Theexperimentalresultsthatshowsuperluminaltunneling,neatlyidentifytheHartmaneffect[19]andtheWignertimedelay[20](orphasetime)asthephysicallyrelevantmechanismsforthetun-nelingtimeandtoarespectively.WhethertheseresultsapplyonlytophotonsandareduetothespecificpropertiesofthePBGused,orcanbeextendedtootherparticlesandbarriers,cannotbedecidedinthelackofasatisfactorytheoryofthetoaataspacepointthroughinteractingenvironments.Thethirdquestionisthusthetunnelingtime,forwhichtherearethreemainproposals.Wignerintroducedthephasetimeinhisanalysis[20]oftherelationshipbetweenretar-dation,interactionrange,andscatteringphaseshifts.ButtikerandLandauerintroducedthetraversaltime[21]intheirstudyoftunnelingthroughatime-dependentbarrier.Soonafter,ButtikerusedtheLarmorprecessionasaclock[22],identifyingthedwell[23],traver-sal,andreflectiontimesasthreecharacteristictimesdescribingtheinteractionofparticleswithabarrier.Recentreviewsthatincludetheseandotherapproaches,discussingtoaand1tunnelingtimesfromamodern,unifiedperspective,canbefoundin[24]and[25].Thelightshedonthesequestionsbythetwophotonexperimentsisrevisedin[26]and[27].Theplanoftheworkisasfollows:InSection2weconstructthegenerictoaformalism,generallyvalid,butstillawaitingthenecessaryconnectionstoeachphysicalsituationofinterest.Thestartingpointisthecaseofthefreeparticle.Asuitablecanonicaltransformation,similartothatusedinscatteringtheory,givesthetoaininteractingenvironments,(alsoatpointswhereV(x)6=0).InSection3wespecializetothecaseofpotentialbarriers,showingtheexistenceoftwocharacteristictimesofarrivalateachpoint,oneforincomingandotherforoutgoingparticles,