Weakly Interacting Bose Mixtures at Finite Temperature Bert Van Schaeybroeck Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven, Celestijnenlaan 200 D, B-3001 Leuven, Belgium. (Dated: January20,2009) Motivated by the recent experiments on Bose-Einstein mixtures with tunable interactions we study repulsiveweaklyinteractingBosemixturesatfinitetemperature. WeobtainphasediagramsusingHartree- Focktheorywhicharedirectlyapplicabletoexperimentally trappedsystems. Almostallfeatures ofthe 9 diagrams can be characterized using simple physical insights. Our work reveals two surprising effects 0 whicharedissimilartoasystematzerotemperature. Firstofall,nopurephasesexist,thatis,ateachpoint 0 inthetrap,particlesofbothspeciesarealwayspresent.Second,evenforveryweakinterspeciesrepulsion 2 whenfullmixingisexpected,condensateparticlesofbothspeciesmaybepresentinatrapwithoutthem n beingmixed. a J PACSnumbers:03.75.Hh,03.75.Mn,67.60.Bc,67.85.Fg 0 2 Introduction – Since the early realization of mixtures trapped situations, and yet, independent of the specific ] r of Bose-Einstein condensates (BECs) ten years ago [1], trapping parameters. We realize this by drawing the dia- e a host of experiments revealed the rich physics of two- grams as a function of the chemical potentials of the two h component Bose [2, 3] and Fermi [4] systems. Whereas specieswhiletakingthetemperaturetobefixed. Thesedi- t o most experiments use mixtures of hyperfine states of like agrams are apt for gases confined by a smoothly varying . t atomic species, it is also possible for unlike species to be potentialforthenthe useofalocaldensityapproximation a m trappedsimultaneously. Moreoverboth systems allow the is appropriate and an effective chemical potential can be use of Feshbach resonances by means of which the inter- takenaslocallyconstantateachpositioninthetrap. Rather - d atomic interactions can be tuned to arbitrary values and thangivingexactdiagramsusingspecificparameters[17], n which revolutionizedthe physicsof ultracoldgases. Such wefocushereontheirbasicunderstandingbyhighlighting o resonances were observed for binary Bose gases consist- thegenericfeatures;moreelaborateworkwillbepresented c ingof87Rb 41K[5],87Rb-133Cs[6]and85Rb-87Rb[7,8] in a forthcoming paper. We argue that our diagrams can [ mixtures;cl−earphasesegregationwasachievedforthelast be almost fully characterized, based on the knowledge of 1 mixture by changing the intraspecies interactions of the single-species Bose systems at finite temperature and the v 85Rbparticles[8]. BECmixtureatzerotemperature. 8 4 To date theoretical works on BEC mixtures predomi- 0 nantly focussed on static and dynamic properties at zero Equation of state — Consider two Bose species, la- 3 temperature [9]. Interesting questions, however, can be belled 1 and 2, at temperature T and at chemical poten- 1. posed concerning the finite temperature regime: for in- tials µ1 and µ2. The particles of species i and j inter- 0 stance, will, upon increasing the interspecies repulsion, act weakly via s-wave scattering, quantified by a posi- 9 phasesegregationforthermallydepletedparticlessetinat tive scattering length a and a coupling constant G = 0 thesametimeasforcondensateparticles? Doesincreasing 2πℏ2a (m−1 + m−1)ij(henceforth i,j = 1,2). Weijuse : ij i j v the temperature induce an increased tendency to mix the theself-consistentHartree-Fock(HF)equationsofstateas i species? Can pure phases still exist? The issue of phase first introduced by Huang et al. [11]. This very model is X segregationatnonzerotemperaturewasalreadyaddressed knownto givegoodagreementwith experiments[12]and r a in Ref. 10 where the authors calculated the conditions for evenwithquantumMonteCarlosimulations[13]. TheHF aninstabilitytooccurinavolumewithfixedparticlenum- modeltreatsnoncondensedparticlesasfreeparticleswitha bers. AlsothecaseofaBosemixturesinatrapatnonzero meanfieldchemicalpotentialshift,andtakesonlythermal temperaturewasbrieflydiscussed. and no quantum fluctuations into account. The HF grand In this Letter, we aim at clarifying the possible phase potential per unit volume Ω is expressed in terms of the structures in traps of Bose mixtures at finite temperature. condensate densities n and thermally depleted densities ci The presented phase diagrams are directly applicable to n [3,11]: di Ω = g5/2 eβµ0i kBT +µ0n +G n2ci +2n n +n2 µ (n +n ) +G (n +n )(n +n ), i=1,2"− (cid:0) λ3i (cid:1) i di ii(cid:18) 2 ci di di(cid:19)− i ci di # 12 c1 d1 c2 d2 X 2 whereλ = 2πℏ2/(m k T),β = 1/k T andg (x) = i i B B l ∞ s−lxl. Thefirsttwotermsareduetotheentropyand s=1 p the kinetic energy of the depleted particles. The very last Ptermcouplesthetwospeciesbyrepulsiveinteractions. At fixedchemicalpotential,minimizationofΩwithrespectto thedensitiesn andn yieldstheself-consistentHartree- ci di Fockequationsofstate: µ = µ0 +2G (n +n )+G (n +n ), (1a) 1 1 11 c1 d1 12 c2 d2 µ = µ0 +2G (n +n )+G (n +n ), (1b) 2 2 22 c2 d2 12 c1 d1 and, µ0 = G n whenever n = 0. The depleted i − ii ci ci 6 densities on the other hand are governedby minimization withrespecttoµ0: i n = g (eβµ0)/λ3. (2) di 3/2 i i FIG. 1: (Color online) Generic phase diagram for two Bose For most systems of ultracold gases only few length- speciesatfixedtemperatureincaseofstronginterspeciesrepul- sion (∆ < 0) as a function of the chemical potentials of both scales are relevant. These include the particle de Broglie wavelength λ , the interparticle distance n−1/3 and lastly species. TheBEC1(green) phaseconsistsofcondensate parti- i i clesofspecies1andthermallydepletedparticlesofspecies1and the scattering lengths a . The HF theory as introduced ij 2andviceversafortheBEC2(blue)phase.The(grey)phaseNO above is valid in case of weak interactions i.e. for small BECconsistsofdepletedparticlesofbothspecies.Thechemical values of a /λ and when single-particle excitations pre- potentialµC isgivenbyEq.(6)andthelocusofthetriplepoint vail i.e. whijen in a λ2 1 [2]. Also, the theory fails TPisgiveniinEq.(7)whilethefirst-orderphaseboundaries are i ij i ≪ veryclosetothetransitionpointwherefluctuationsbecome describedbyEqs.(8),(9)and(10). dominant. In nowadaysexperiments,thescatteringlength takesvaluesoforderoftennanometersandbelow,whereas Single-species at finite T — Consider now a gas of theinterparticledistancesandtheBrogliewavelengthsare speciesiatfixedtemperature. Onecanthinkofthechem- typicallyoftheorderofmicronsandhigher. Itfollowsthat ical potentialµ as a measurefor the particle density: the the necessaryconditionsfor the HF theory to be valid are i higher µ , the higher the density of species i. For large fulfilled[18]. i and negativevaluesof µ , the system is dilute and the in- We proceed by first discussing BEC mixtures at zero i teraction energy small. On the other hand, above a cer- temperatureandasinglespeciesBosegasatfinitetemper- tain “critical” positive chemical potential a transition to a ature,sincethesesystemsprovideallingredientsnecessary phase with nonzero condensatedensity sets in; by the HF fortheunderstandingoftheresultsobtainedfurther. approach,thiscriticalvalueisattainedwhenµ0 = 0such BEC mixture at T = 0 — At zero temperature, the HF i that[3][19]: theory reduces to Gross-Pitaevskii (GP) theory for a spa- 3 tially homogeneoussystem. Accordingto GP theory, two µC = 2G ζ(3/2)/λ , (6) i ii i regimescanbedistinguished,dependingontheparameter with ζ the Riemann zeta function. Note that µC de- i ∆definedas[9]: pends on temperature via λ and from the scaling µC i i ≃ k T(a /λ )weconcludethatµC k T. ∆ = G G /G2 1. (3) B ii i i ≪ B 11 22 12 − TheintroductionofasecondBosegaswillshiftthecriti- When ∆ < 0, the inter-species repulsion is sufficiently calchemicalpotentialtoahighervaluethanµC;thiscanbe i ascribedtotheinterspeciesrepulsionwhichtendstorarify strong to induces phase segregation and solely BEC 1 or thegasofspeciesi. BEC2 mayexistasgroundstates, thatis, species1and2 PhaseDiagramfor ∆ < 0 andfiniteT — Uponwork- areimmiscible. Onereadilyfindsthefollowingconditions: ing at fixed particle numbers as in Ref. 10, the onset of phasesegregationcanbenaturallyprobedbythepresence BEC1whenµ > µ G /G , (4a) ofadensityinstability. Atfixedchemicalpotentialsonthe 1 2 11 22 other hand, this method is inadequate as an instability in- BEC2whenµ < µ pG /G . (4b) 1 2 11 22 dicates a spinodal (line) rather than the transition itself in Ontheotherhand,when∆ > 0,thpeinterspeciesrepulsion caseitisoffirstorder. Instead,wehavecomparednumer- is weak and a mixed phase of nonzero density of BEC 1 icallythegrandpotentialsfortheeightpossiblethermody- andBEC2,heredenotedasBECMIX,mayappearasthe namicalphasesandtheirassociatedmultiplesolutions. We groundstate: arrive at the generic phase diagrams for Bose mixtures at finitetemperatureinFigs.1and2forthecases∆ < 0and BECMIXwhenG /G < µ /µ < G /G . (5) 12 22 1 2 11 12 ∆ > 0respectively. 3 Identicaltothe behavioratT = 0, nomixingofunlike condensate particles is possible at finite T when ∆ < 0. OnlythreephasesappearinFig.1,noneofwhicharepure; that is, even for extremely strong interspecies repulsion, depletedparticlesofbothspeciesarepresentinallphases. Accordingly, BEC 1 denotes the phase composed of con- densateparticlesofspecies1anddepletedparticlesofboth species,andviceversaforBEC2. TheNOBECphasede- notesthephasewiththermalparticlesofspecies1and2. A triple point TP is present in the phase diagram of Fig. 1. As checked numerically, its locus (µTP,µTP) can 1 2 befoundbysettingµ0 = µ0 = 0inEq.(1): 1 2 µTP = µC +µCG /2G , (7a) 1 1 2 12 22 µTP = µC +µCG /2G . (7b) 2 2 1 12 11 FIG.2: (Coloronline)ThesameappliesasinFig.1exceptthat NotethatinunitsofµC andµC,thepositionofpointTPis weakinterspeciesinteractionsareconsidered(∆>0).The(dark temperatureindepende1nt. 2 grey) BEC MIX phase consists of condensed and depleted par- ticlesofbothspecies. Theredarrowed lineindicatesapossible Consider now the phase boundary between BEC 1 and tracingofthediagramasafunctionofthe(radial)positioninthe 2;forlargeandpositivevaluesofbothchemicalpotentials, trap. Thepoint Bindicatesthechemical potentialsatthecenter the interaction energy per particle will eventuallybecome ofthetrap. muchlargerthanthethermalenergyandsoasymptotically the T = 0 behavior is expected. Indeed, we find that in accordwithEq.(4)theBEC1-2phaseboundarysatisfies: Eq.(5)impliesthattheBECMIXphaseisdelimitedbythe phaseboundaries µ /µ = G /G whenbothµ ,µ . (8) 1 2 11 22 1 2 → ∞ µ /µ = G /G Moving alongpthe NO BEC-BEC 1 boundary away from 1 2 11 12 whenbothµ ,µ . (11) TP, the density of species 2 decreases; in the limit of µ /µ = G /G 1 2 → ∞ (cid:26) 1 2 12 22 µ ,onlyparticlesofspecies1remainsuchthatthe 2 → −∞ ThelocationoftriplepointTP2altersuponvaryingtem- single-componentvalueµ µC isattained. Theassoci- 1 → 1 perature and ∆. For small and positive values of ∆, TP2 atedasymptoticdecayisexponentialastheNOBEC-BEC enters at large and positive values of the chemical po- 1phaseboundaryisdescribedby: tentials while TP2 coalesces with TP1 only in the limit µ = µC +G eµ2/kBT/λ3 whenµ . (9) ∆ . This leads us to a remarkableconclusion: even 1 1 12 2 2 → −∞ → ∞ whenthezerotemperatureconditionforspeciesmixingis BecauseµC k T,itfollowsthattheasymptoticconver- satisfied, at finite temperature, two-phase equilibrium be- 2 ≪ B genceis slow ifµ2 is expressedin units of µC2. The same tweenBEC1and2isstillallowed. Changingthetemper- lineofargumentwithinterchangedindices1and2applies ature also induces a displacement of TP2 with respect to tothedemarcationlinebetweenBEC2andNOBEC: TP1. Intuitively one expects that increasing the tempera- µ = µC +G eµ1/kBT/λ3 whenµ . (10) ture favors mixing. Yet, condensateparticles carry no en- 2 2 12 1 1 → −∞ tropy andwe find rather the opposite: increasingthe tem- Inoutline, wehaveshownthatonecanqualitativelyes- peraturesuppressesmixing. In particular,thedistancebe- tablish the phasediagramforthe case∆ < 0 withoutex- tweenTP1andTP2(inunitsofµC)increasesuponincreas- i plicitly performing any numerics. Indeed, we found that ingthetemperature. theexpressionsforthetriplepoint(Eq.(7)),andthephase In the ideal case of fully suppressed interspecies inter- boundaries(Eqs.(8)-(10))agreewellwiththenumerically actions(∆ )the triple pointsTP1 andTP2 coincide → ∞ exactresults. and, as expected, the phase boundaries are the horizontal Phase Diagram for ∆ > 0 at finite T— At zero tem- andverticallinesµ = µC andµ = µC. 1 1 2 2 perature mixing of unlike condensate particles is possible To summarize, the ∆ > 0 phase diagram is similar to when ∆ is positive. We find the same behavior at finite the∆ < 0diagramsbutismarkedbytheappearanceofan T. This is shown in Fig. 2 where BEC MIX consists of additionalphaseconsistingofmixedBECspecies, andan condensedand depleted particles ofboth species. The bi- additionaltriplepointTP2. Despitetheknownasymptotic furcationoftheBEC1-2linebringsalongtheappearance behavior of the phase boundaries around the BEC MIX ofanewtriplepointTP2(TP1isidenticaltoTPofFig.1). phase (see Eq. (10)), the locus of TP2 depends in a non- For the aforementionedreasons, the T = 0 physics is re- trivial way on temperature and ∆ and must therefore be coveredforlarge and positivechemicalpotentials. Hence determinednumerically. 4 Discussion — In order to extract from Figs. 1 and 2 thespeciessuchthat,spatiallytracingthetrapisequivalent thepossiblephasestructuresappearinginanexperimental to exploring the phase diagram. We highlight the impor- trap, the trapping potentials U are required. In most ex- tanceofthetriplepointsinourdiagramsandfindthat,sur- i perimentsalocaldensityapproximationorThomas-Fermi prisingly,nopurephasesexist,and,increasingthetemper- approximation is justified such that at each position r an aturetendstosuppressthemixingofcondensateparticles. effective chemical potential µ (r) = µ U (r) may be assumedconstant;µ (r)ismaiximalatthie−ceniterandmin- i imal at the edge of a trap. In case both species are har- Acknowledgement— The authorthanksLev Pitaevskii, monically confined or U (r) = m ω2r2/2, the chemical JosephIndekeuandAchilleasLazaridesforusefulsugges- i i i potentials probe our phase diagrams following a straight tions. B.V.S.isaPostdoctoralFellowofFWO. line: m ω2(µ (r) µ ) =m ω2(µ (r) µ ). 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Nevertheless, our theory still applies to the ex- sis,almostallpropertiesofourphasediagramscanbeex- 2 periments ofRef.8sincethelargestvalues ofn a λ are pressedin terms ofknownresults forsingle-speciesgases around0.1. i ij i at finite temperature and for binary mixtures at zero tem- [19] Small corrections apply near the critical points, due to the perature. This means that it is possible to establish the first-ordernatureofthetransition;asmentionedbefore,the phase diagrams (aside from the triple point TP2 in case theoryisnotaccurateveryclosetothetransitionpoint. ∆ > 0)withoutperforminganynumerics. Thephasedia- [20] Thisveryeffectthereforevanishesatzerotemperaturewhen gramsaredrawnasafunctionofthechemicalpotentialsof thedepleteddensitiesarequenched.