搜索
arXiv:cond-mat/0210339v116Oct2002CriticaltemperatureofBose–EinsteincondensationintrappedatomicBose–FermimixturesA.P.Albus1,S.Giorgini2,F.Illuminati3,andL.Viverit2,41Institutf¨urPhysik,Universit¨atPotsdam,D–14469Potsdam,Germany2DipartimentodiFisica,Universit`adiTrento,andIstitutoNazionaleperlaFisicadellaMateria,I–38050Povo(TN),Italy3DipartimentodiFisica,Universit`adiSalerno,andIstitutoNazionaleperlaFisicadellaMateria,I–84081Baronissi(SA),Italy4DipartimentodiFisica,Universit`adiMilano,viaCeloria16,I–20133Milano,Italy(February1,2008)AbstractWecalculatetheshiftinthecriticaltemperatureofBose-Einsteinconden-sationforadiluteBose-FermimixtureconfinedbyaharmonicpotentialtolowestorderinboththeBose-BoseandBose-Fermicouplingconstants.Therelativeimportanceoftheeffectonthecriticaltemperatureoftheboson-bosonandboson-fermioninteractionsisinvestigatedasafunctionoftheparametersofthemixture.ThepossiblerelevanceoftheshiftofthetransitiontemperatureincurrentexperimentsontrappedBose-Fermimixturesisdiscussed.I.INTRODUCTIONTheachievementofBose-Einsteincondensationinultracold,trappeddilutealkaligases,beyondrealizingastrikingandspectacularexperimentalconfirmationofalong-standing,1fundamentalpredictionofquantumtheory,hasinitiatedandstimulatedaswellawholenewfieldofresearchinthephysicsofquantumgasesinconfinedgeometries[1,2].Nowaday,manydifferentatomicspeciesandisotopescanbesuccesfullycooledandtrappedinthegaseousstate,toinvestigateinexceptionalconditionsofpurityandisolationthepropertiesofinteractingBoseandFermisystems,orBose-Fermimixtures.Inparticular,theexperimentalrealizationsoftrappedgaseousmixturesofbosonsandfermionsarebothaninterestingnewinstanceofaquantummany-bodysystemandaveryusefultooltoreachtheregimeofquantumdegeneracyforaFermigasviasympatheticcoolingofthefermionsbythebosons[3–7].Fromatheoreticalpointofview,diluteBose-Fermimixtureshavebeentheobjectofrecentinvestigationsaddressing,forexample,thedeterminationofthedensityprofilesofthetwocomponentsintrappedsystems[8],theproblemofstabilityandphaseseparation[9],andtheeffectofboson-fermioninteractionsonthedynamics[10]andontheground-stateproperties[11]ofthemixture.Boson-fermioninteractionsinaBose-Fermimixturecaninduceanetattractiveinterac-tionbetweenthefermions,thusintroducingafurthermechanismtowardtheachievementoftheBCStransitionintrappedFermigases[12].Inthispaperweaddressthereverseproblem,i.e.howthetransitiontemperatureofBose-Einsteincondensationisaffectedbythepresenceofthefermionsinatrappedmixture.TheshiftofthetransitiontemperatureTcduetointeractionsinapuretrappedBosesystemhasbeencalculatedwithinthemean-fieldapproximationinRef.[13].InthepresentworkweextendtheperturbativemethodsofRef.[13]toobtaintheshiftofTctolowestorderinboththeBose-BoseandBose-Fermicouplingconstants.TothisordertheeffectonTcofboson-bosonandboson-fermioninteractionsareindependentandaddlinearly.Therelativeimportanceofthetwoeffectsdependsontherelevantparametersofthetrappedmixture:numberofbosonsandfermionsinthetrap,ratioofthemassesandoftheoscillatorfrequenciesforthetwospeciesandtheratiooftheBose-BoseandBose-Fermicouplingconstants.Thecalculationiscarriedoutinlocaldensityapproximationwhichisvalidprovidedthatthenumberofbosonsandfermionsinthetrapislarge.Finitesizeeffectshavenotbeenincludedinthepresenttreatment.2Theplanofthepaperisasfollows:inSectionIIwegeneralizetheschemederivedforpureBosesystemsinRef.[13],toincludetheeffectsofboson-fermioninteractions.InSectionIIIwederiveanalyticalresultsfortheshiftofTcinthelimitsofahighlydegenerate(Thomas-Fermi)andaclassical(Boltzmann)Fermigas,andweprovidethefullnumericalsolutionfortheintermediateregimes.Wefinallycomparethetheoreticalpredictionswiththecurrentexperimentalsituationsandwedrawourconclusions.II.THEORYInanon-interactingBosegasconfinedbytheexternalharmonicpotentialVBext(r)=mB(ω2xx2+ω2yy2+ω2zz2)/2,thecriticaltemperatureforBose-Einsteincondensation(BEC)isgivenbykBT0c=¯hωBNBζ(3)!1/3≃0.94¯hωBN1/3B,(1)whereωB=(ωxωyωz)1/3isthegeometricmeanoftheoscillatorfrequenciesandmB,NBarerespectivelytheparticlemassandthenumberofbosonsinthetrap.Theaboveresultisobtainedusingthelocaldensityapproximation(LDA),wherethetemperatureofthegasisassumedtobemuchlargerthanthespacingbetweensingleparticlelevels:kBT≫¯hωx,y,z.Inthiscasethedensityofthermalatomscanbewrittenasn0B(r)=(λBT)−3g3/2(exp{−[VBext(r)−μB]/kBT}),(2)whereλBT=¯h(2π/mBkBT)1/2isthebosonthermalwavelength,andg3/2(x)=P∞n=1xn/n3/2isthestandardBosefunctionoforder3/2.AtT=T0cthebosonchemicalpotentialtakesthecriticalvalueμB=μ0c=0,correspondingtothebottomoftheexternalpotential,andthedensityn0B(0)inthecenterofthetrapsatisfiesthecriticalconditionn0B(0)(λBT0c)3=ζ(3/2)≃2.61holdingforahomogeneoussystem.Finitesizeeffectsmodifythepredictionofthecriticaltemperature(1)resultinginareductionofT0c.Thefirstcorrectionduetothefinitenumberofatomsinthetrapisgivenby[14]:3δTcT0c!fs=−ζ(2)2ζ(3)2/3¯ωBωBN−1/3B≃−0.73¯ωBωBN−1/3B,(3)where¯ωB=(ωx+ωy+ωz)/3isthearithmeticmeanoftheoscillatorfrequencies.InterparticleinteractionshaveaneffectontheBECtransitiontemperatureaswell.Thepresenceofrepulsiveinteractions