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Degenerate atom-molecule mixture in a cold Fermi gas S.J.J.M.F. Kokkelmans1, G.V. Shlyapnikov1,2,3, and C. Salomon1 1Laboratoire Kastler Brossel, Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris 05, France 2FOM Institute AMOLF, Kruislaan 407, 1098 SJ Amsterdam, The Netherlands 3Russian Research Center Kurchatov Institute, Kurchatov Square, 123182 Moscow, Russia 4 (Dated: February 6, 2008) 0 0 We show that the atom-molecule mixture formed in a degenerate atomic Fermi gas with in- 2 terspecies repulsion near a Feshbach resonance, constitutes a peculiar system where the atomic n component is almost non-degenerate but quantum degeneracy of molecules is important. We de- a velop a thermodynamicapproach for studyingthis mixture,explain experimental observations and J predict optimal conditions for achieving molecular BEC. 5 1 PACSnumbers: 05.30.Jp,03.75.-b,05.20.Dd,32.80.Pj,05.45.Yv ] h Interactionsbetweenparticlesplayacrucialroleinthe recombination is efficient [9] and the molecules remain c behaviorofdegenerate quantumgases. Forinstance, the trapped as their binding energy ǫ is smaller than the e B signoftheeffectivemeanfieldinteractiondeterminesthe trap depth [6, 9]. They come to equilibrium with the m stability ofa largeBose-Einsteincondensate(BEC),and atoms, reducing the pressure in the system. - t the shape of such a condensate in a trap can be signifi- Awayfromresonances,the interactionstrengthispro- a cantly altered from its ideal gas form [1]. In degenerate portionaltoa,andisgivenbyg =4π¯h2a/M,withM the t s Fermigasestheeffectsofmeanfieldinteractionsareusu- atom mass. Close to resonance this relation is not valid, . t allylesspronouncedinthe sizeandshapeofthetrapped as the value of a diverges to infinity and the scatter- a | | cloud, and these quantities are mostly determined by ing processstronglydepends onthe collisionenergy. For m Fermi statistics. The strength of the interactions, how- Boltzmann gases, already in the 1930’s, Beth and Uh- d- ever, can be strongly increased by making use of a Fesh- lenbeck[10]calculatedthesecondvirialcoefficientbyin- n bach resonance [2, 3], and then the situation changes. cluding both the scatteringand bound states for the rel- o Recentexperimentspresenttwotypes ofmeasurement ative motion of pairs of atoms [11]. A small interaction- c of the interactionenergyin a degenerate two-component induced change of the pressure in this approach is nega- [ Fermi gas near a Feshbach resonance [4, 5, 6, 7]. At tive on both sides of the resonance [12, 13]. 3 JILA[5]andatMIT [7]the meanfieldenergywasfound However, current experiments are not in the Boltz- v from the frequency shift of an RF transition for one of mann regime. In this letter we show that the atom- 4 the atomic states. The results are consistent with the moleculemixtureformedinacoldatomicFermigas,con- 8 magneticfielddependence ofthe scatteringlengtha,the stitutesapeculiarsysteminwhichtheatomiccomponent 3 energy being positive for a > 0 and negative for a < 0. is almost non-degenerate, whereas quantum degeneracy 8 6 0 In the Duke [4] and ENS [6] experiments with Li, the of the molecules can be very important. This behavior 3 results are quite different. The interaction energy was originates from a decrease of the atomic fraction with 0 obtainedfromthemeasurementofthesizeofanexpand- temperature. It is present even if the initial Fermi gas / ing cloudreleasedfromthe trap. A constantratio of the isstronglydegenerateinwhichcasealmostallatomsare t a interaction to Fermi energy, Eint/EF 0.3, was found converted into molecules. We develop a thermodynamic m ≈− around resonance, irrespective of the sign of a [4, 6]. It approachfor studying this mixture, predictoptimalcon- - wasexplainedinRef.[4]byclaimingauniversalbehavior ditions for achieving molecular BEC, and properly de- d in this strongly interacting regime [8]. The ENS studies scribe the interaction effects as observed at ENS [6]. n o in a wide range of magnetic fields [6] found that Eint We assume that fermionic atoms are in equilibrium c changes to a large positive value when a is tuned posi- withweaklybound(bosonic)moleculesformedinthere- : tive, but only at a field strongly shifted from resonance. combinationprocess. The molecules are treatedas point v i IncontrasttotheJILA[5]andMIT[7]studiesprovid- bosons. Atom-molecule and molecule-molecule interac- X ingadirectmeasurementofthemeanfieldinteractionen- tionsareomitted atfirst, andwill be discussedina later r ergy, the Duke [4] and ENS [6] experiments measure the stage. For a large scattering length a > 0, the binding a influence of the interactions on the gas pressure. An in- energyoftheweaklyboundmoleculesisǫB =h¯2/(Ma2), terpretationoftheENSexperimentinvolvesthe creation andtheir sizeis roughlygivenbya/2. Fortreatingthem ofweaklyboundmoleculesviathree-bodyrecombination as point bosons, this size should be smaller than the at a positive a [6]. Far from resonance, the binding en- meaninterparticleseparation. This requiresthe inequal- ergy of the produced molecules and, hence, their kinetic ity n(a/2)3 < 1, which at densities n 1013 cm−3 is ≈ energy are larger than the trap depth and the molecules satisfied for a<18000a0, and excludes a narrowvicinity escape from the trap. The interaction energy is then of the Feshbach resonance. determined by the repulsive interaction between atoms The presence of molecules reduces the number of par- and is positive [6]. Close to resonance, the three-body ticlesintheatomiccomponentandtoanessentialextent 2 1 relativemomentum of a colliding pair,averagedoverthe (a) 0.8 momentum distribution. In this respect, the interaction problem becomes similar to the calculation of the total 0.6 energy of a heavy impurity as caused by its interactions 0.4 with the surrounding electrons in a metal [19]. This ap- 0.2 proach leads to a relation between the collision-induced 00 0.2 0.4 0.6 0.8 1 shift of the energy levels of particles in a large spherical box, and the scattering phase shift. Adding the integra- 100 tionoverthestatesofthecenterofmassmotionforpairs (b) ofatoms,we findthatthe totalenergyofinteratomicin- 10−1 teraction is equal to Pkk′gkk′ν↑(k,µ,T)ν↓(k′,µ,T)/V, whereν andν areoccupationnumbersofsingle-particle ↑ ↓ 10−2 momentum states, and V is the volume (cf. [19]). The momentum-dependent coupling constant is given by 10−3 0 0.1 T0/.ε2 0.3 0.4 4π¯h2δ(k k′ /2) B gkk′ =− M |k −k′|/2 . (2) | − | FIG.1: Fractionofunboundatomsna/n(lowercurves,bold) and fraction of atoms bound into molecules, 2nm/n, (upper The phase shift δ is expressed through the relative mo- cthuervEesN)Svedrastuas[T20/]ǫ;Bb:)an)Λn3TΛ=3T 1=4.28.,5a,nsdqutahreesvearntdicaclirlcinleesssihnodwi- δme=ntumarcqta=nq|ak.−Ink′t|h/e2laimndittohfeqsacatter1in,gEqle.n(2g)thtraanass- − | | ≪ 2 catetheonsetofmolecularBEC.Dashedcurvesareobtained formsintotheordinarycouplingconstantg =4π¯h a/M. including atom-molecule and molecule-molecule interactions. As we have ν (k,µ,T) = ν (k,µ,T) ν , the total ↑ ↓ k ≡ energy of the atomic component and the number of particles in this component can be written in the form liftsitsquantumdegeneracy. Themolecularchemicalpo- ¯h2k2 gkk′ tential is negative in the absence of atom-molecule and Ea =X νk+X νkνk′; Na =2Xνk. (3) M V molecule-molecule mean field, and thermal equilibrium k kk′ k between atoms and molecules requires a negative chemi- calpotentialfortheatoms. Wethusassumeapriorithat In our mean-field approach, the entropy of the atoms is given by the usual combinatorial expression [18]: the occupationnumbers ofthe states ofatoms aresmall. This proves to be the case at any temperature, except for very low T where the atomic fraction is negligible. Sa = 2X[νklnνk+(1 νk)ln(1 νk)]. (4) − − − Under these conditions we omit pairing correlations be- k tween the atoms, which are important for describing a Equations(3) and (4) immediately lead to an expression crossover from the BCS to BEC regime [14, 15, 16, 17] for the atomic grand potential Ω = E TS µN . a a a a and can be expected even in the non-superfluid state. − − Then,usingtherelationN = (∂Ω /∂µ) ,weobtain a a T,V Assuming equal densities of the atomic components, − for the occupation numbers of atoms: labeled as and , their chemical potentials are µ = ↑ ↑ ↓ µ↓ = µ, where µ is the chemical potential of the system νk =[exp (ǫk µ)/T +1]−1, (5) as a whole. The molecular chemical potential is µ = { − } m 2 2 ǫB +µ˜m, with µ˜m 0 being the chemical potential of where ǫk = h¯ k /2M +Uk, and Uk = Pk′gkk′νk′/V is a−n ideal gasof boson≤s with the mass 2M. The condition the mean field acting on the atom with momentum k. of thermal equilibrium, µ +µ =µ , then reads Accordingly, the expression for the grand potential and ↑ ↓ m pressure of the atomic component reads: 2µ= ǫ +µ˜ . (1) B m − From Eq.(1) we will obtain the number of molecules Nm Ωa =−PaV =X[2T ln(1−νk)−Ukνk]. (6) k andthenumberofatomsN forgiventemperatureT and a total number of atomic particles N = Na+2Nm. This This set of equations is completed by the relation be- requires us to obtain the expression for the occupation tweenthedensityofbosonicmoleculesandtheirchemical numbers of the atoms and the dependence of µ on Na. potential. In the absence of molecular BEC we have: The main difficulty with constructing a thermodynamic approach for the degenerate molecule-atom mix- nm =(√2/ΛT)3/2g3/2(exp(µ˜m/T), (7) ture is related to the resonance momentum-dependent characteroftheatom-atominteractions. Thisdifficultyis where g (x) = ∞ xj/jα, and Λ = (2π¯h2/MT)1/2 is α Pj=1 T circumventedforsmalloccupationnumbersoftheatoms. the thermal de Broglie wavelength for the atoms. For 3 Then, even at resonance, the interaction energy is equal n Λ > 7.38 the molecular fraction becomes Bose- m T to the meanvalue ofthe interactionpotentialforagiven condensed,andwehaveµ˜=0andµ= ǫ /2. Similarly, B − 3 7.5 for values of a as low as possible while still staying at 7 the plateau, as at larger a the interaction between the 6.5 moleculescanreducetheBECtransitiontemperature[1]. 3T Λ m 6 WenowanalyzetheinteractioneffectobservedatENS n for trapped clouds in the hydrodynamic regime [6]. The 5.5 experiment was done near the Feshbach resonance lo- 5 4.5 cated at the magnetic field B0 = 810 G, and the data 0 2000 4000 6000 8000 10000 resultsfromtwotypes ofmeasurementsofthesizeofthe a (a ) 0 cloud released from the optical trap. In the first one, the magnetic field and, hence, the scattering length, are FIG. 2: Molecular degeneracy parameter nmΛ3T underadia- kept the same as in the trap. Therefore, the cloud ex- bnaatniccev.aTrihaetidoanshofedacfuorrv6eLiis,oabsstuaminiendginncΛlu3Td=ing15atcolmos-emtoolerceusole- pands with the speed of sound cs = p(∂P/∂ρ)S, where ρ=mnisthemassdensity. Thespeedc and,hence,the and molecule-molecule interactions. The horizontal dashed s sizeoftheexpandingcloudareinfluencedbythepresence line shows thecritical value for molecular BEC. of molecules and by the interparticle interactions. In the secondtype of measurement,the magnetic field is first rapidly ramped down and the scattering length theenergy,entropy,andgrandpotentialofthemolecules becomes almost zero on a time scale t 2µs. This time ∼ are given by usual equations for an ideal Bose gas [11]. scaleisshortcomparedtothecollisionaltime. Therefore, From Eqs. (1)-(7) we obtain the fraction of unbound thespatialdistributionoftheatomsremainsthesameas atoms n /n and the fraction of atoms bound into in the initial cloud, although the mean field is no longer a molecules, 2n /n, as universal functions of two param- present. Atthesametime,arapiddecreaseofaincreases m eatteorms:icTp/aǫrBticalensd. TnΛhe3T,dewpheenrdeennceisofthaetotmotiacladnednmsitoyleoc-f tth<eb¯hi/nǫdBin,gtheenyercgaynofnmotoaledciualbeastǫiBca.llHyofwolelvoewr,taosathdeeteipmeer ular fractions on T/ǫ for two values of nΛ3 is shownin bo∼und state and dissociate into atoms which acquire ki- B T Fig.1. The molecular fraction increases and the atomic netic energy. Thus the system expands symmetrically as fraction decreases with decreasing T/ǫ . Occupation an ideal gas of N atoms, with the initial density pro- B numbers of the atoms are always small, whereas quan- file. The momentum distribution fk will be a sum of tum degeneracy of molecules is important. The dotted the initial atomic momentum distribution and one that line in Fig.1b indicates the onset of molecular BEC. arisesfromthedissociatedmolecules. Thelatterisfound assuming an abrupt change of a and, hence, projecting This mixture was realized in the ENS experiment [6], where the occupation numbers for the molecules were up to 0.3 and the molecular fraction was exceeding the 0.5 atomic one. In the recent studies [20, 21, 22, 23] al- mostallatomswereconvertedintomoleculesbysweeping 0.4 5000 the magnetic field across the resonance,and at ENS [20] 0.3 a)0 0 the temperature was within a factor of 2 from molecular a ( BEC.Remarkably,onecanmodifythemolecularfraction 0.2 anddegeneracyparametern Λ3 byadiabaticallytuning −5000 m T 0.1 0 500 1000 the atom-atomscatteringlength,asshowninFig.2. The B (G) decrease of a increases the binding energy ǫ and the β 0 B molecular fraction, and thus causes heating [20]. Close −0.1 to resonance, n Λ3 remains almost constant and then m T −0.2 decreases due to heating. Theatom-moleculeandmolecule-moleculeinteractions −0.3 are readily included in our approach for a ΛT, where −0.4 ≪ the corresponding coupling constants are g = 0.9g am −0.5 and gm = 0.3g [26]. In this limit the interactions 600 650 700 750 800 850 900 provide an equal shift of the chemical potential and B (G) single-particle energy ǫ . For the atoms this shift is k FIG. 3: Calculated (solid line) and measured [6] (squares n g/2+n g , where the first term is the atom-atom a m am and crosses) ratio β of the interaction to kinetic energy (see contribution U . For the (non-condensed) molecules the k text). The calculated line for B > 790G is for experimental shift is nagam +2nmgm. The entropy of the mixture is conditions T = 0.9EF = 3.4µK and n = 3×1013cm−3. For givenbythesameexpressionsasintheabsenceofthein- B<700GwetaketheaveragedexperimentalconditionsT = teractions. AsseeninFig.1andFig.2,theatom-molecule 1.1EF = 2.4µK and n = 1.3×1013cm−3. For 700 < B < and molecule-molecule interactions do not significantly 800G, we use the local conditions (see [6]). Inset: Scattering modify our results. From Fig.2 one then concludes that length as a function of magnetic field. the conditions for achieving molecular BEC are optimal 4 the molecular wave function on a complete set of plane topositivevaluesleftfromresonance,ascanbeseenfrom waves. This gives rise to a distribution c(q) for the rela- our calculation in Ref. [6]. This demonstrates that the tive momentum q. The single-particlemomentum distri- apparent field shift from resonance, where a sign-change bution for the atoms produced out of molecules results in the interaction energy is observed, is an indirect sig- then from convoluting c(k k′ /2) with the molecular nature of the presence of molecules in the trap. distribution function ν (|k+−k′)|by integrating over k′. m For high temperatures T E and small binding en- F Onecanestablisharelationbetweentheexpansionveloc- ≫ ergy ǫ T, we find that β has a universal behavior B ity v0 of this non-equilibrium system and the expansion andis pr≪oportionalto the secondvirialcoefficient. How- velocityc0 ofanidealequilibriumtwo-componentatomic ever, this only holds at high temperatures (cf. [13]), and Fermi gas which has the same density and temperature: at low T the molecule-molecule interaction can strongly 4π3nR0Mv0/h¯dkk2fk =R0Mc0/h¯dkk2ν˜k,withν˜k beingthe influence the result. For T approaching the temperature ideal-gas momentum distribution. Using the scaling ap- of molecular BEC, which is T ¯h2n2/3/M 0.2E , the c F proach [24, 25], one can find that in the spherical case atomic fraction is already sma≈ll and the so≈und velocity the velocity c0 coincides with the expansion velocity of c is determinedby the molecularcloud. For a Λ we s T thehydrodynamicFermigasintheabsenceofmean-field find c2 = 0.4T /M +ng /2M, where the seco≪nd term interactions and, accordingly, is given by c20 = 5P0/3ρ, is provsided bycthe molecmule-molecule interaction and is where P0 =2E0/3V is the pressure. omitted in the high-T approach. The ratio of this term The relativedifference between the squaredsize ofthe to the first one is 5(na3)1/3. For B = 700 G at den- expandingcloudinthetwodescribedcasescanbetreated sities of Ref. [6], it∼is equal to 1 and is expected to grow astheratiooftheinteractiontokineticenergyandcalled when approaching the resonance. the interaction shift. This interaction shift is then given Thus, except for a narrow region where na3 1, by the relativedifference betweenthe two squaredveloc- | | ≫ ities: β = [c2 v2]/v2. Our results for this quantity are one can not speak of a universal behavior of the shift β s − 0 0 on both sides of the resonance. The situation depends calculatedforexperimentalconditions andarepresented onpossibilities ofcreatinganequilibriumatom-molecule in Fig.3. The sound velocity c was obtained using the s mixture. Moreover,at low temperatures the universality above developed approach including only atom-atom interactions. Thefieldregionwheren(a/2)3 >1,isbeyond can be broken by the molecule-molecule interactions. the validity of this approachand is shownby the dashed We are grateful to T. Bourdel, J. Cubizolles, C. Lobo, curve. InFig.3wealsoshowourpreviousresultsforfields and L. Carr for stimulating discussions. 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