Time-of-arrival probabilities and quantum measurem

arXiv:0706.2492v2[quant-ph]1Feb2008Time-of-arrivalprobabilitiesandquantummeasurements:IIApplicationtotunnelingtimesCharisAnastopoulos∗DepartmentofPhysics,UniversityofPatras,26500Patras,GreeceandNtinaSavvidou†TheoreticalPhysicsGroup,ImperialCollege,SW72BZ,London,UKFebruary4,2008AbstractWeformulatequantumtunnelingasatime-of-arrivalproblem:wede-terminethedetectionprobabilityforparticlespassingthroughabarrieratadetectorlocatedadistanceLfromthetunnelingregion.Forthispur-pose,weuseaPositive-Operator-Valued-Measure(POVM)forthetime-of-arrivaldeterminedin[1].Thisonlydependsontheinitialstate,theHamiltonianandthelocationofthedetector.ThePOVMaboveprovidesawell-definedprobabilitydensityandanunambiguousinterpretationofallquantitiesinvolved.Wedemonstratethatforaclassoflocalizedinitialstates,thedetectionprobabilityallowsforanidentificationoftunnelingtimewiththeclassicphasetime.Wealsoestablishlimitstothedefinabil-ityoftunnelingtime.Wethengeneralizetheseresultstoasequentialmeasurementset-up:thephasespacepropertiesoftheparticlesaredeterminedbyanunsharpsamplingbeforetheirattempttocrossthebarrier.Forsuchmeasurementsthetunnelingtimeisdefinedasagenuineobservable.Thisallowsustoconstructaprobabilitydistributionforitsvaluesthatisdefinableforallinitialstatesandpotentials.Wealsoidentifyaregime,inwhichtheseprobabilitiescorrespondtoatunneling-timeoperator.1IntroductionThispaperisacontinuationofRef([1]),inwhichaprocedurewassketchedfortheconstructionofaPositive-Operator-ValuedMeasure(POVM)forthetime-of-arrivalforaparticledescribedbyaHamiltonianˆH.Here,weextendthisPOVMtocoverthecaseofparticlestunnelingthroughabarrier.Thisprocedure∗[e-mail]†[e-mail]1allowsustoprovideanunambiguousdeterminationforthetunneling-timeasitcanbemeasuredintime-of-arrivaltypeofmeasurements.Quantumtunnelingreferstotheescapeofaparticlefromaregionthroughapotentialbarrier,whosepeakcorrespondstoanenergyhigherthanthatcarriedbythetheparticles.Therearetwoimportantquestions(relevanttoexperi-ments)thatcanbeaskedinthisregard.Thefirstis,howlongdoesittakeaparticletocrossthebarrier(i.e.whatisthetunnelingtime?).Thesecondis,whatisthelawthatdeterminestherateoftheparticle’sescapethroughthebarrier?Inthispaper,wedevelopaformalismthatprovidesananswertothesequestionsandweapplyittothefirstone.Theissueofthedecayprobabilitywillbetakenupin[2].Theissueoftunnelingtimehasreceivedsubstantialattentioninthelitera-ture,especiallyafterthe1980’s–seethereviews[3,4].Thereasonisthatthereisanabundanceofcandidatesandadiversityofviewpointswithnoclearconsen-sus.Thereareroughlythreeclassesofapproaches:(i)Wavepacketmethods:onefollowstheparticle’swavepacketacrossthebarrieranddeterminesthetunnelingtimethrougha”delayinpropagation”[5,6],(ii)onedefinessuitablevariablesfortheparticle’spathsandoneobtainsaprobabilitydistribution(oranaverage)forthetransversaltimecorrespondingtoeachpath.Thesepathscanbeconstructedeitherthroughpath-integralmethods[4,7,8],throughBohmianmechanics[9],orthroughWignerfunctions[10],and(iii)theuseofanobserv-ablefortime:thiscantaketheformofanadditionalvariableplayingtheroleofaclock[11,12],orofaformaltimeoperator[13].Ingeneral,thesemethodsleadtoinequivalentdefinitionsandvaluesforthetunnelingtime.1.1OurapproachThebasicfeatureofourapproachtothisproblemisitsoperationalcharacter.Weidentifythetunnelingtimebyconstructingprobabilitiesfortheoutcomeofspecificmeasurements.Weassumethatthequantumsystemispreparedinaninitialstateψ(0),whichislocalizedinaregionononesideofapotentialbarrierthatextendsinamicroscopicregion.AttheothersideofthebarrierandamacroscopicdistanceLawayfromit1,weplaceaparticledetector,whichrecordsthearrivalofparticles.Usinganexternalclocktokeeptrackofthetimetfortherecorder’sclicks,weconstructaprobabilitydistributionp(t)forthetimeofarrival.Thefactthatthedetectorisaclassicalmacroscopicobjectandthatitliesatamacroscopicdistanceawayfromthebarrierallowsonetostate(usingclassicallanguage)thatthedetectedparticlesmusthavepassedthroughthebarrier(quantumeffectslikeaparticlecrossingthebarrierandthenbacktrackingarenegligible).Hence,attheobservationallevel,theprobabilityp(t)containsallinformationaboutthetemporalbehaviorfortheensembleofparticles:allprobabilisticquantitiesreferringtotunnelingcanbereconstructedfromit.Withtheconsiderationsabove,bothproblemsofdeterminingthetunneling1Weexplaininsection2.3thesenseinwhichweemploytheword”macroscopic”.2timeandtheescapeprobabilityasafunctionoftime(see[2])aremappedtothesingleproblemofdeterminingthetime-of-arrivalatthedetector’slocationforanensembleofparticlesdescribedbythewavefunctionψ0att=0andevolvingunderaHamiltonianwithapotentialterm.Tosolvethisproblem,weelaborateontheresultof[1],namelytheconstructionofaPositiveOperatorValuedMeasure(POVM)forthetime-of-arrivalforparticlesforagenericHamiltonianˆH–see[14]and[15]fordefinition,propertiesandinterpretationofPOVMs.ThisPOVMprovidesauniquedeterminationoftheprobabilitydistributionp(t)forthetime-of-arrival.Itisimportanttoemphasizethatbyconstructionp(t)islinearwithrespecttotheinitialdensitymatrix,positive-definit