Reflection and Transmission in a Neutron-Spin Test

arXiv:quant-ph/9903009v12Mar1999February1,2008BA-TH/316-99WU-HEP-99-1ReflectionandTransmissioninaNeutron-SpinTestoftheQuantumZenoEffectKenMachida,(1)HiromichiNakazato,(1)SaverioPascazio,(2)HelmutRauch(3)andSixiaYu(4)(1)DepartmentofPhysics,WasedaUniversityTokyo169-8555,Japan(2)DipartimentodiFisica,Universit`adiBariandIstitutoNazionalediFisicaNucleare,SezionediBari,I-70126Bari,Italy(3)Atominstitutder¨OsterreichischenUniversit¨atenStadionallee2,A-1020,Wien,Austria(4)InstituteofTheoreticalPhysics,AcademiaSinica,Beijing100080,P.R.ChinaAbstractThedynamicsofaquantumsystemundergoingfrequent“measurements,”leadingtotheso-calledquantumZenoeffect,isexaminedonthebasisofaneutron-spinexperimentrecentlyproposedforitsdemonstration.Whenthespatialdegreesoffreedomareduelytakenintoaccount,neutron-reflectioneffectsbecomeveryimportantandmayleadtoanevolutionwhichistotallydifferentfromtheidealcase.PACS:03.80.+r;03.65.Bz;03.65.Nk;04.20.Cv1IntroductionAquantumsystem,preparedinastatethatdoesnotbelongtoaneigenvalueofthetotalHamiltonian,startstoevolvequadraticallyintime[1,2].Thischaracteristicbehaviorleadstotheso-calledquantumZenophenomenon,namelythepossibilityofslowingdownthetemporalevolution(eventuallyhinderingtransitionstostatesdifferentfromtheinitialone)[3].Theoriginalproposalsthataimedatverifyingthiseffectinvolvedunstablesystemsandwerenotamenabletoexperimentaltest[4].However,theremarkableidea[5]touseatwo-levelsystemmotivatedaninterestingexperimentaltest[6],revitalizingadebateonthephysicalmeaningofthisphenomenon[7,8].Thereseemtobeacertainconsensus,nowadays,thatthequantumZenoeffect(QZE)canbegivenadynamicalexplanation,involvingonlyanexplicitHamiltoniandynamics.Itisworthemphasizingthatthediscussionofthelastfewyearsmostlystemmedfromexperimentalconsiderations,relatedtothepracticalpossibilityofperformingexperimentaltests.Someexamplesaretheinterestingissueof“interaction-free”measurements[9]andtheneutron-spintestsoftheQZE[8,10].Inpracticalcases,onecannotneglectthepresenceoflossesandimperfections,whichobviouslyconspireagainstanalmost-idealexperimentalrealization,moresowhenthetotalnumberof“measurements”increasesabovecertaintheoreticallimits.Theaimofthepresentpaperistoinvestigateaninteresting(andoftenoverlooked)featureofwhatwemightcallthequantumZenodynamics.Weshallseethataseriesof“measurements”(vonNeumann’sprojections[11])doesnotnecessarilyhindertheevolutionofthequantumsystem.Onthecontrary,thesystemcanevolveawayfromitsinitialstate,provideditremainsinthesubspacedefinedbythe“measurement”itself.Thisinterestingfeatureisreadilyunderstandableintermsofrigoroustheorems[2],butitseemstousthatitisworthclarifyingitbyanalyzinginterestingphysicalexamples.Weshallthereforefocusourattentiononanexperimentinvolvingneutronspin[8]andshallseethatinfactthisenablesustokilltwobirdswithonestone:notonlythestateoftheneutronundergoingQZEwillchange,butitwilldosoinawaythatclarifieswhyreflectioneffectsmayplayasubstantialroleintheexperimentanalyzed.Intheneutron-spinexampletobeconsidered,theevolutionofthespinstateishinderedwhenaseriesofspectraldecompositions(inWigner’ssense[12])isperformedonthespinstate.No“observation”ofthespinstates,andthereforenoprojection`alavonNeumannisrequired,asfarasthedifferentbranchwavesofthewavefunctioncannotinterfereafterthespectraldecomposition.Needlesstosay,theanalysisthatfollowscouldbeperformedintermsofaHamiltoniandynamics,withoutmakinguseofprojectionoperators.However,weshalluseinthispaperthevonNeumanntechnique,whichwillbefoundconvenientbecauseitshedslightonsomeremarkableaspectsoftheZenophenomenonandhelpstopindownthephysicalimplicationsofsomemathematicalhypotheseswithrelativelylessefforts.Thepaperisorganizedasfollows.Webrieflyreview,inthenextsection,theseminaltheoremfortheshort-timedynamicsofquantumsystems,provedbyMisra1andSudarshan[2].Itsapplicationtotheneutron-spincaseisdiscussedinSec.3.InSecs.4and5,unlikeinpreviouspapers[8,10],weshallincorporatethespatial(1-dimensional,forsimplicity)degreesoffreedomoftheneutronandrepresentthembyanadditionalquantumnumberthatlabels,roughlyspeaking,thedirectionofmotionofthewavepacket.AmorerealisticanalysisispresentedinSec.6.Finally,Sec.7isdevotedtoadiscussion.SomeadditionalaspectsofouranalysisareclarifiedintheAppendix.2MisraandSudarshan’stheoremConsideraquantumsystemQ,whosestatesbelongtotheHilbertspaceHandwhoseevolutionisdescribedbytheunitaryoperatorU(t)=exp(−iHt),whereHisasemi-boundedHamiltonian.LetEbeaprojectionoperatorandEHE=HEthesubspacespannedbyitseigenstates.Theinitialdensitymatrixρ0ofsystemQistakentobelongtoHE.IfQislettofollowits“undisturbed”evolution,undertheactionoftheHamiltonianH(i.e.,nomeasurementsareperformedinordertogetinformationsaboutitsquantumstate),thefinalstateattimeTreadsρ(T)=U(T)ρ0U†(T)(2.1)andtheprobabilitythatthesystemisstillinHEattimeTisP(T)=TrhU(T)ρ0U†(T)Ei.(2.2)Wecallthisa“survivalprobability:”itisingeneralsmallerthan1,sincetheHamil-tonianHinducestransitionsoutofHE.Weshallsaythatthequantumsystemshas“survived”ifitisfoundtobeinHEbymeansofasuitablemeasurementprocess[13].Assumethatweperformameasurementattimet,i