Measurement of the Entropy and Critical Temperatur

arXiv:cond-mat/0611566v1[cond-mat.other]21Nov2006MeasurementoftheEntropyandCriticalTemperatureofaStronglyInteractingFermiGasL.Luo,B.Clancy,J.Joseph,J.Kinast,andJ.E.Thomas∗1PhysicsDepartment,DukeUniversity,Durham,NorthCarolina27708-0305,USA∗Towhomcorrespondenceshouldbeaddressed;E-mail:[e-mail].(Submitted21November,2006)Wereportamodel-independentmeasurementoftheentropy,energy,andcrit-icaltemperatureofadegenerate,stronglyinteractingFermigasofatoms.Thetotalenergyisdeterminedfromthemeansquarecloudsizeinthestronglyinteractingregime,wherethegasexhibitsuniversalbehavior.Theentropyismeasuredbysweepingabiasmagneticfieldtoadiabaticallytunethegasfromthestronglyinteractingregimetoaweaklyinteractingregime,wheretheentropyisknownfromthecloudsizeafterthesweep.Thedependenceoftheentropyonthetotalenergyquantitativelytestspredictionsofthefinite-temperaturethermodynamics.StronglyinteractingFermigasesareofgreatinterest,astheyexhibituniversalthermody-namicbehavior,wherethepropertiesareindependentofthedetailsofthemicroscopicinterac-tions(1,2,3,4).Thesegasesprovidemodelsfortestingnonperturbativemany-bodytheoriesin1avarietyoffieldsfromneutronstarsandnuclearmatter(5,6,2,7)toquark-gluonplasmas(8)andhightemperaturesuperconductors(9).Hence,thermodynamicexperimentsonstronglyinteractingFermigasesareofgreatimportance.Instudiesofthethermodynamicsofthesesystems,wherethermometryisdifficult(10),entropymeasurementplaysacentralandfundamentalrole.WereportthemeasurementoftheentropySofastronglyinteractingFermigasasafunctionofitstotalenergyE.TheresultsyieldthetemperatureTviatheelementarythermodynamicrelation1/T=∂S/∂E.Ourexperimentsquantitativelytestrecentpredictionsoftheentropybasedonmicroscopicmany-bodytheory,yieldthedependenceoftheenergyontemperature,anddeterminethecriticaltemperatureforthesuperfluidtransitionwithoutinvokinganyspecifictheoreticalmodel.Strongly-attractiveFermigasesexhibitbothfermionicandbosonicfeatures,andhavebeenstudiedintenselyforseveralyearsintheory(11,12,13,9)andexperiment(1,14,15,16,17,18,19).Measurementsoftheheatcapacity(20)andcollectivemodedampingversusenergy(21)revealtransitionsinbehavior,whichhavebeeninterpretedasasuperfluidtransitioninthissystem(20).Recently,theobservationofvortices(22)hasprovidedadefinitiveproofofasuperfluidphase.However,therehavebeennomodel-independentstudiesofthethermodynamicproperties.AstronglyinteractingFermigasispreparedusinga50:50mixtureofthetwolowesthy-perfinestatesof6LiatomsinanultrastableCO2lasertrapwithabiasmagneticfieldof840GjustaboveabroadFeshbachresonanceatB=834G(23).Thegasiscooledtoquantumdegeneracybyloweringthetrapdepthbyafactorof≃1000(1).Followingforcedevaporation,thetrapdepthU0isrecompressedtoU0/kB=10μK,whichislargecomparedtotheenergyperparticleofthegas.HerekBistheBoltzmannconstant.Afterthisprocedure,theinitialenergyisclosetothatofthegroundstate,asdescribedbelow.Atthefinaltrapdepth,themeasuredtraposcillationfrequenciesinthetransversedirectionsareωx=2π×670Hzandωy=2π×760Hz,whiletheaxialfrequencyisωz=2π×302Hz@840Gandωz=2π×[e-mail]differduetothesmallchangeinthetrappingpotentialarisingfromthebiasmagneticfieldcurvature.ThetotalnumberofatomsN≃1.3(0.2)×105isobtainedfromabsorptionimagesofthecloudusingatwo-levelopticaltransitionat840G.ThecorrespondingFermienergyEFandFermitemperatureTFforanideal(noninteracting)harmonicallytrappedgasatthetrapcenterareEF=kBTF≡~¯ω(3N)1/3,where¯ω=(ωxωyωz)1/3.Forourtrapconditions,weobtainTF≃1.0μK.Thetotalenergyperparticle,E,ofthestronglyinteractinggasismeasuredinamodel-independentwayfromthemeansquaresizeintheaxialdirection(4).Inthisstronglyinteract-ingregime,thezeroenergys-wavescatteringlengthaSislargecomparedtotheinterparticlespacing,whichislargecomparedtotherangeofthetwo-bodyinteraction,sothatthegasisuni-versal(2,1,3).Then,thelocalpressureisP=2E/3,whereEisthelocalenergydensity(3,4).UsingforcebalanceforatrappingpotentialU,∇P+n∇U=0,wherenisthelocaldensity,onethenobtainsthetotalenergyperparticleE=3mω2zhz2i840(1−κ)orEEF=hz2i840z2F(1−κ),(1)wherehz2i840isthemeansquareaxialcloudsizemeasuredat840Gandmisthe6Limass.Here,z2Fisdefinedby3mω2zz2F≡EF,andisweaklydependentonthemagneticfieldthroughthetrapfrequencies.Thecorrectionfactor1−κarisesfromanharmonicity(24)intheshallowtrappingpotentialU0≃10EFusedintheexperiments.Wefindthatκvariesfrom3%atourlowestenergiesto13%atthehighest.Theentropyofthestronglyinteractinggasat840Gisdeterminedusinganadiabaticsweepofthemagneticfieldtoarelativelyweakly-interactingregimeat1200G,whereareferenceentropycanbeestimatedfromthemeansquareaxialcloudsizehz2i1200.At1200G,aS=−2900bohr(23),andkFaS=−0.75forourshallowtrap,withkF=p2mkBTF/~2.At3kFa=−0.75,weexpectthatthedependenceoftheentropyonthecloudsizeshouldbeclosetothatofanidealnoninteractingFermigaswithprimarilyasmallmean-fieldreductioninthegroundstatecloudsize.Thisconjectureissupportedbytheobservedballisticexpansionofthecloudat1200G,evenatourlowesttemperatures,whichshowsthatthegasisnearlynormal.Wealsofindthatthecalculatedidealgasentropydiffersfromamany-bodyresultforkFaS=−0.75(25)bylessthan1%ove