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Wave-ParticleFluctuations,Coherence,andBose-EinsteinCondensationN.M.ChaseSchoolofArtsandSciencesMassachusettsCollegeofPharmacyandHealthSciencesBoston,MA02115,USAemail:Norma.Chase@mcphs.edu2AbstractByextendingEinstein’sseparationofwaveandparticlepartsofthesecondorderthermalfluctuationtoencompass“generalizedfluctuations”inanyBosefield,P.E.Gordonhasproposedalternativedefinitionsfornthordercoherenceandnthordercoherentstates;astatewhichiscoherenttoordernisonewhichcontainsno“wave”contributionstoitsnumberfluctuations,forall()ΔNmm#n.ThispaperbeginswithaproofoftheequivalenceofGordon’scoherenceconditionstothoseofGlauber;wethenusetheequivalentconditionstoderiveaoneparameterfamilyofstateswhicharecoherenttofiniteorder.Themainpointofthispaperistoexploresomeofthephysicalinsightstobegainedbyextendingdualismtohigherorders.RecentexperimentshaveexaminedaspectsofthecoherenceofBose-Einsteincondensates.Ithasbeenarguedthatthecondensatestateiscoherentto(atleast)secondorthirdorder,butthecoherencepropertiesofBose-Einsteincondensatesremainsomewhatcontroversial.UsingprobabilitydistributionsdevelopedbyM.O.ScullyandV.V.Kocharovskyet.al.,weapplyGordon’sdualisticexpressionofthecoherenceconditionstoinvestigatecoherencepropertiesinBose-Einsteincondensation(BEC).Vianumericalcalculations,wepresentagraphicalsurveyofwave-likeandparticle-likefluctuationsincondensedanduncondensedfractions.Nearthecriticalpoint,wefindaverymarkedpeakintheratioofnthorderwavetonthorderparticlefluctuations.Notsurprisingly,then-pointcorrelationsbetweenthepositionsofcondensateatomsalsopeaknearthecriticaltemperatureandthisapparentlymirrors,toallhigherorders,thewell-knownrelationbetweentheintegralofthe2-pointcorrelationfunctionoveracertainvolumeandthermsfluctuationinthenumberofparticlesinthatvolume.PACS03.65-Quantumtheory,quantummechanics31IntroductionEinstein’sworks[1]onthenatureofthefluctuationsinblack-bodyradiation()ΔNNN22=+(1909)andinthemolecularquantumgas(1924,1925)wereturningpointsintheearlydevelopmentofquantumtheory.Forblack-bodyradiation,heshowedthatwhilethesecondtermwouldbeexpectedforasystemofelectromagneticwaves,thefirsttermwouldarisefromasystemofindependentparticles.Thisseparationprovidedoneofthefirstclearindicationsofthedualnatureoflight.Inlaterworks,Einsteinextrapolatedthissamedualisticinterpretationtothesecondorderfluctuationsinamolecularquantumgas.HenotedthatwhilefirsttermwouldbeexpectedforthefamiliarPoissonianfluctuationof(distinguishable)particles,thepresenceofthesecondtermindicatedthatmattershoulddemonstrate“novel”wave-likeproperties.Inmorerecenttimes,Glauber[2]hasgivenusthedefinitionsofthefullycoherentstateandfiniteordersofcoherence.Afullycoherentstateistheclosestquantumanalogofa“definite”classicalwave;correlationsreducetoPoissonianformtoallorders.Inastatewhichiscoherenttoonlyafiniteorder,correlationsreducetoPoissonianformtoallordersuptoandincludingn,theorderofcoherence.Gordon[3]hasproposedtheextensionofEinstein’sseparationofthesecondorderwaveandparticlefluctuationsinthermalradiationtoencompasshigherorderfluctuations(momentsaboutthemean)inanyBosefield.Intermsofthisdualisticseparation,Gordonproposedanalternativedefinitionoftheconditionsfornthordercoherenceanddemonstrateditsvalidityupto4thorder.WebeginthispaperbyprovingtheequivalenceofGordon’scoherenceconditionstothoseofGlauber.WethenapplyGordon’sdualisticrepresentationofthecoherenceconditionstoderiveaoneparameterfamilyofstateswhicharecoherenttofiniteorder[4,5].Withthisbackgroundinplace,weproceedtothemainpointofthispaper,afirstexaminationofsomeofthephysicalimplicationsofextendingdualismtohigherorders.RecenthistoricexperimentshaveexaminedaspectsofthecoherenceofBose-Einsteincondensates[6].Whileithasbeenarguedthatthestateofthecondensateiscoherentto(atleast)secondorthirdorder,thehigherordercoherencepropertiesofthecondensateremaincontroversial[7-12].M.O.Scully,Vl.V.Kocharovsky,andV.V.Kocharovsky[9-12]havedevelopednon-equilibriumapproachestothedynamicsandstatisticsofanidealN-atomBosegas3coolingviainteractionwithathermalreservoirandhavetherebyobtainedequilibriumdistributionsforthenumberofatomsinthegroundstatesofdifferentmodeltraps.ThroughacomprehensiveexaminationoftheequilibriumfluctuationsandcumulantsoftheBose-Einsteincondensate,[12]showsthatforanidealBosegasinanytrapthedistributionofnoncondensedatomsbecomesPoissonian(coherent)atverylowtemperatureswhileits“mirrorimage”distribution,thatofthecondensate,doesnot.Inthispaperweusetheprobabilitydistributionsdevelopedin[9,10]toexplorethetemperaturedependenceofcoherencepropertiesinBEC.Basedonnumericalcalculationswepresentagraphicalsurveyofwave-particlefluctuationsinbothcondensateandnoncondensate.Discussionisrestrictedtothequasithermalandlowtemperatureapproximationsfortheisotropicharmonictrapandhomogeneousgasinaboxmodelspresentedin[10].Inthequasithermalapproximationof[10],wefindaverymarkedpeakintheratioofnthorderwavetonthorderparticlefluctuationsnearthecriticaltemperature..Notsurprisingly,n-pointcorrelationsbetweenthepositionsofcondensateatomslikewisepeaknearthecriticalpo