An introduction to decoherence through simple examples

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Niveau: Supérieur, Doctorat, Bac+8
An introduction to decoherence through simple examples P. Degiovanni CNRS-Laboratoire de Physique de l'Ecole Normale Superieure de Lyon, 46, Allee d'Italie, 69007 Lyon, France These notes provides a short introduction to decoherence theory through simple examples such as the imperfect photon box and the influence of collisions on the motion of a quantum particle. The quantum Monte-Carlo method is introduced and used to obtain quantitative results for these systems. These notes are intended to be accessible to undergraduate students having a bachelor level background on quantum mechanics. Version 0.1: March 6, 2006. I. INTRODUCTION One of the most striking difference between the classical and quantum descriptions of the world arises from the enormous amount of perfectly legal quantum states in the latter one compared to the first one. As put forward by Schrodinger, quantum mechanics enables a perfectly civilized cat to be suspended in a weird ”life + death” state. Such a quantum monster is never encountered in our everyday life. Understanding such a simple experimental fact is very important for the status of quantum theory. It turns out that these questions have remained under the carpet for a very long time, even if they were present in the minds of the famous fathers of quantum mechanics, Bohr, Heisenberg and Schrodinger to quote the most famous ones. Even Einstein, a strong opponent to quantum theory, rose this question in his letters to Bohr one year before his death.

  • quantum jumps

  • over all

  • no jump

  • atomic microwave cavity

  • state vector

  • single experiment

  • gap between

  • such

  • open quantum systems

  • photon box


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AnintroductiontodecoherencethroughsimpleexamplesP.DegiovanniCNRS-LaboratoiredePhysiquedel’EcoleNormaleSupe´rieuredeLyon,46,Alle´ed’Italie,69007Lyon,FranceThesenotesprovidesashortintroductiontodecoherencetheorythroughsimpleexamplessuchastheimperfectphotonboxandtheinfluenceofcollisionsonthemotionofaquantumparticle.ThequantumMonte-Carlomethodisintroducedandusedtoobtainquantitativeresultsforthesesystems.Thesenotesareintendedtobeaccessibletoundergraduatestudentshavingabachelorlevelbackgroundonquantummechanics.Version0.1:March6,2006.I.INTRODUCTIONOneofthemoststrikingdifferencebetweentheclassicalandquantumdescriptionsoftheworldarisesfromtheenormousamountofperfectlylegalquantumstatesinthelatteronecomparedtothefirstone.AsputforwardbySchro¨dinger,quantummechanicsenablesaperfectlycivilizedcattobesuspendedinaweird”life+death”state.Suchaquantummonsterisneverencounteredinoureverydaylife.Understandingsuchasimpleexperimentalfactisveryimportantforthestatusofquantumtheory.Itturnsoutthatthesequestionshaveremainedunderthecarpetforaverylongtime,eveniftheywerepresentinthemindsofthefamousfathersofquantummechanics,Bohr,HeisenbergandSchro¨dingertoquotethemostfamousones.EvenEinstein,astrongopponenttoquantumtheory,rosethisquestioninhisletterstoBohroneyearbeforehisdeath.Thereerearetworeasonsforthelonglastingsilenceonthisimportantproblem.ThefirstoneisthegreatsuccessofquantummechanicsinitsorthodoxinterpretationdeveloppedinCopenhagen.Inthisframework,quantummechanicalobjectssuchasthestatevectorrelyonclassicalobserversfortheirinterpretation.TheCopenhageninterpretationsuggeststhatthereisaclassical/quantumboundarysomewherebutmakesitverydifficulttolocalize,assumingthatitcanbe.VonNeumannhimselfidentified”counsciousness”astheoperatorofwavepacketreduction,leavingtheprobleminaverydifficultplacetodophysics.Therefore,thesuccessofquantummechanicsinpredictingquantitativeresultsinmanydifferentareasofphysicsdidnotmotivatethecrowdofphysiciststotacklewithdifficultandunclearfoundationalissues.Thesecondreasonismoreoftechnologicalnature.Probingtheentanglementbetweenaquantumsystemandalltheexternaldegreesoffreedomitcaninteractwithisaveryhardtechnologicalchallenge.Controlledmanipulationofquantumdevicesorsinglequantumobjectshaveonlybeenrealizedinthelast20years.Inthefirst50yearsofquantummechanics,thelackofanyconcreteexperimentalsituationtoanalyzemadeanyprogressondecoherence,quantummeasurementorbasictestsonquantummechanicsverydifficult.ThewholestoryofBellinequalitiesisrathersymptomaticoftheseslowprogresses.Theywerediscoveredin1964[1],putinsuitableformforexperimentalistsinthe1969[2]andreliableexperimentshavebeenconductedinthelate70s[3]andearly80s[4].Thelast20yearsofthe20thcenturyhaveseenanimpressiveprogressontheexprimentalsideaswellasthediscoverythatquantumressourcessuchasentanglementorthesuperpositionprinciplescouldbeusedtoprocessinformationinamoreefficientwaythanclassically.Thecombinationofthesetwoeffectsleadtoarenewalofresearchactivitiesonfundamentalquantumissuessuchasdecoherence.Theaimofthesenotesisnottopresnetanexhaustivereviewofdecoherencetheorynotamagistralcourseforhighlyeducatedadvancedphysicsstudents.Theideaistopresentcasesofstudiesofdecoherencethroughtwosimpleexamplesofopenquantumsystems.Thefirstoneistheimperfectphotonbox,athoughtexperimentdeviceinthe1930swhichisnowavailableandstudiedinseveralphysicslaboratories1.Thesecondexampleconcernstheinfluenceofcollisionsonthequantumpositionofaparticle.Consideringthisexampleindetailwillhelpusunderstandwhythespreadingofthewavepacketforthecenterofmassofeverydaylifeobjectsorevenfornanoparticlescannotbeobserved.1Theimperfectphotonboxisindeedthequantumversionofthedampedharmonicoscillator.
II.ENTANGLEMENTANDDECOHERENCE2AssuggestedfromthediscussionofYounginterferenceexperimentinFeynman’slecturesonquantummechanics[?],thequantumcharacterofaphysicalsystemcanbehiddenifinformationaboutthequantumstateofthesystemiscollectedbyanunobservedenvironmentorbyameasurementappartus.Thisphenomenoniscalleddecoherenceandleadstothedisappearanceofquantuminterferenceeffectsforanyobserverwhocannottrackbackalltheinformationcollectedbythesystem’senvironment.ThediscussionofentangledpairsofphotonssharedbyAliceandBobalreadycontainstheessenceofdecoherence.AssumethatAliceandBobaresharinganEPRpairofqubits:|EPRi=1(|00i+|11i).(1)2IfAliceonlyknowsaboutherqubit,resultsofobservationsthatsheperformsonmultiplerealizationofhersideoftheEPRpaircannotbedescribedusingapurestate.Fromherpointofview,everythinghappensasifshecollectsamixtureofapairoftwoorthogonalstatesofherqubitwithrespectiveprobabilities1/2and1/2.Thisrepresentationisnotunique:anypairofothogonalstatewouldexplainAliceexperimentalresults.Auniformdistributiononthesetofpurestates(Blochsphere)wouldalso.Noquantuminterferencecanbeobservedinanyofinterferenceexperimentshewouldperformonherqubit.Thedisappearenceofquantuminterferencesinaproperlymonitoredinterferometeroccursforthesamereason.Iftheexperimentaldeviceissuchthatthedegreesoffreedomassociatedwiththemotionofthetestparticleintheinterferometergetentangledwithsomequantumsystem(measurementappartusoranythingelse),thenquantuminterferencesarelost.Entanglementisthereasonfordecoherence.Thisexamplestressesadeeprelationbetweendecoherenceandthenotionofcomplementarity,introducedbyBohrintheearlydaysofquantummechanics(1928).Heemphasizedthatparticleandwaveconceptswerecomplementaryinthesensethattheycouldnotbeobservedsimultaneously.Moreprecisly,followinganelectron’strajectoryaroundthenucleus,makesyouunabletoobservetheenergylevelsthatarisefromastationarywavecondition.Reciprocally,observingwelldenedenergylevelsmeansthatthenotionoftrajectoryismeanningless.Whatdeterminesthepredominanceofoneaspectovertheotheristhepreciseexperimentalsetup.Ourbriefdiscussionshowsexactlythis:onceweareabletolabelthetwoquantumalternatives(trajectoriesinaYounginterferenceexperiment)usinganyauxiliarysystem,interferenceeffectsarelost.Beyongthesegeneralities,thequestionofdecoherencedynamicsisveryimportant.Ifweprepareaquantumdeviceinaquantumstate,whatisthetimeneededtodestructitsquantumcharacter?Answeringsuchaquestionrequiresanunderstandingofthesystem’sdynamicsthattakesintoaccounttheinteractionwiththeenvironmentaldegreesoffreedom.Theseinteractionswillentanglethesystemanditsenvironment,leadingtothedecoherenceofthesystemunderstudy.Decoherencethusdependsonthesystemanditscouplingtoitsenvironment.Forthisreason,decoherencephenom-enaarenotexpectedtobeuniversal:theyarestronglydependentontheexperimentalsituationthatweconsider,notasurprisesinceBohr.Butthisobservationshouldbetamperedbynoticingthat,inmanysituations,theenvironmentcanbedescribedusingasimpleeffectivemodel.Formanypracticalpurposes,thedescriptionbyabathofharmonicoscillatorsisveryconvenientandefficient.Thesecouldariseasmicroscopicaldegreesoffreedomsuchaselectromag-neticmodesbutitturnsoutthatmanyweaklycorrelatedanharmonicsystems,weaklycoupledtothesystemwearestudyinginfluenceitasabathofoscillators.Becausethereisahugediversityinthechoiceofthesystem(anditscouplingtotheenvironment),weshallfocusonafewspecificexampleswhichillustrateimportanteideasofdecoherencetheory.InsectionIII,weconsideraclassicsindecoherencestudieswhichisamodelforadampedharmonicoscillator.Asamodelforadissipativeelectromagneticcavity,itisdirectlyrelevantforthedecoherenceexperimentsperformedintheHaroche,RamondandBrunegroupinParis[5].ThenextproblemtobeaddressedinthesenotesistheeffectofcollisionsonthepositionofaquantumparticleV.Thisisveryusefulforunderstandingwhyrealworldobjectsarelocalizedataspecificpositionandnotspreadbyquantumeffects.Suggestionsforfurtherreadingaregivenintheconclusion.III.AMODELFORCAVITYRELAXATIONAnidealelectromagneticcavityisaclosedboxwithperfectlyreflectingwalls.Electromagneticwavesinsideanidealcavitylivethereforeverprovidedthaysatisfycertainstationarityconditions.Themodestructureinsuchacavityisdiscrete,consistingofaseriesofstationarywaveseachofthembehavingasanharmonicoscillator.Sucha