Bose-Fermi mixtures in the molecular limit Andrea Guidini,1 Gianluca Bertaina,2 Elisa Fratini,3 and Pierbiagio Pieri1 1School of Science and Technology, Physics Division, University of Camerino, Via Madonna delle Carceri 9, I-62032 Camerino, Italy 2Dipartimento di Fisica, Universita` degli Studi di Milano, via Celoria 16, I-20133 Milano, Italy 3The Abdus Salam International Centre for Theoretical Physics, 34151 Trieste, Italy (Dated: March 11, 2014) 4 We consider a Bose-Fermi mixture in the molecular limit of the attractive interaction between 1 fermions and bosons. For a boson density smaller or equal to the fermion density, we show ana- 0 lytically how a T-matrix approach for the constituent bosons and fermions recovers the expected 2 physical limit of a Fermi-Fermi mixture of molecules and atoms. In this limit, we derive simple r expressions for theself-energies, the momentum distribution function, and thechemical potentials. a By extending these equations to a trapped system, we determine how to tailor the experimental M parameters of a Bose-Fermi mixture in order to enhance the indirect Pauli exclusion effect on the bosonmomentumdistributionfunction. Forthehomogeneoussystem,wepresentfinallyaDiffusion 8 Monte Carlo simulation which confirms theoccurrence of such a peculiar effect. ] PACSnumbers: 03.75.Ss,03.75.Hh,32.30.Bv,74.20.-z s a g - I. INTRODUCTION bution function. The presence of this region was inter- t n pretedasanindirecteffectonthebosonicdistributionof a the Pauli exclusion principle acting on the unpaired and Bose-Fermimixtureswithatunable boson-fermionat- u compositefermions. Theanalyticexpressionthatwewill q tractionhavebeenobjectofactivetheoretical[1–21]and deriveinthispaperforthemomentumdistributionfunc- . experimental[22–31]investigationoverthelastfewyears. t tion will make suchindirect Pauli exclusion effect on the a Previous theoretical studies of these systems have m bosonic component particularly transparent. shown that for a sufficiently strong attraction between - fermions and bosons, the boson condensation is com- d pletely suppressed in mixtures where the boson density n n is smaller or equal to the fermion density n . This The use of these simple equations will allow us to in- o B F completesuppressionofcondensationoccursevenatzero corporate easily also the effect of an external trapping c [ temperature, and is associated to pairing of bosons with potential. We will calculate then the density profiles fermions into composite fermions. Since the binding oc- andthemomentumdistributionfunctionsforthetrapped 2 curs in a medium, the paired state formed by one bo- system. We will focus in particular in determining the v 6 son and one fermion is influenced by the presence of the idealexperimentalparametersthatmaximizetheindirect 8 remaining particles, and its composite nature can man- Pauli exclusion effect, as to make it possibly observable 6 ifest in appropriate thermodynamic or dynamic quanti- in future experiments with Bose-Fermimixtures. In this 0 ties. Clearly,whentheattractionisincreasedfurther,the respect, we will see that mixtures where the bosons are 1. internaldegreesoffreedomofthecompositefermionsare light compared to the fermions are particularly promis- 0 progressivelyfrozenandthe originalBose-Fermimixture ing. 4 becomes effectively a Fermi-Fermi mixture of molecules 1 and atomic (unpaired) fermions. This kind of evolution : v hasbeenstudiedalreadybyuswithaT-matrixdiagram- The paper is organized as follows. In section II we i matic formalism [12, 16, 20] and with Fixed-node Diffu- derive the asymptotic expressions for the pair propaga- X sion Monte Carlo [19]. tor,self-energies,momentumdistributionsfunctions,and r a Aim of the present paper is to show analytically how chemical potentials, that are obtained in the molecular a T-matrix diagrammatic approach,which is formulated limitoftheBose-FermiattractionbystartingfromtheT- in terms of the constituent bosons and fermions, recon- matrix self-energies. A comparison between the asymp- structs the appropriatedescriptionin terms ofmolecules totic expressionsandthe correspondingT-matrixresults andunpairedfermionswhen the attractionissufficiently is reported in Sec. IIF. In Sec. IIG we present Quan- large. In this limit, we will derive simple expressions tum Monte Carlo estimates for the bosonic momentum for the bosonic and fermionic self-energies, momentum distributionfunction andcomparethem to the T-matrix distribution functions and chemical potentials. Special resultsandasymptoticexpressions. InSec.IIIweinclude attention will be devoted to the momentum distribution the effect of a trapping potential in the asymptotic ex- functions. Indeed,oneveryinterestingfeaturefoundpre- pressions derived in Sec. II, and discuss the visibility of viouslybyusinaBose-Fermimixtureisthepresence,un- theindirectPauliexclusioneffectinBose-Fermimixtures der appropriate conditions, of a region at low momenta ofcurrentexperimentalrelevance. SectionIVpresentsfi- with zero occupancy in the bosonic momentum distri- nally our concluding remarks. 2 II. DERIVATION OF THE ASYMPTOTIC In the above expressions, ω = 2πνT and ω = ν n EQUATIONS IN THE MOLECULAR LIMIT (2n+1)πT,Ω =(2m+1)πT arebosonicandfermionic m Matsubara frequencies, respectively, (ν,n,m being inte- A. Preliminaries ger numbers), while f(x) and b(x) are the Fermi and Bose distribution functions at temperature T [f(x) = (ex/T + 1)−1, b(x) = (ex/T 1)−1]. In Eq. (3) ξs = We consider a mixture of single-component fermions − p andbosons,withtheboson-fermioninteractiondescribed p2/2ms µs is the free dispersion relative to the chemi- − by a contact interaction, as it can be realized with cal potential µs for the species s = B,F, while the bare a (broad) Fano-Feshbach resonance tuning the boson- Green’sfunctionsappearinginEqs.(1)and(2)aregiven fermion scattering length a of an ultracold Bose-Fermi byG0B(k,ων)−1 =iων−ξkB andG0F(k,ωn)−1 =iωn−ξkF. mixture. Wewillbeinterestedinparticularinthemolec- The self-energies (1) and (2) determine the dressed ularlimitofthissystem,namelythelimitwherethebind- Green’s functions G via the Dyson’s equation G−1 = s s ingenergyǫ0 ofthetwo-bodyboson-fermionboundstate Gs0−1−Σs. The dressed Green’s functions Gs, in turn, is the dominant energy scale. For the contact poten- allowoneto calculatethebosonandfermionmomentum tial ǫ0 = 1/(2mra2), where mr = mBmF/(mB+mF) is distribution functions ns(k) through the equations: the reduced mass determined by the boson and fermion masses m and m , and we have set ¯h = 1. The re- pulsive poBtential beFtween bosons, which is necessary for nB(k)= T GB(k,ων)eiων0+ (4) − the stability of the system in the resonance region, can ν X be dropped out from our consideration in the molecular n (k)=T G (k,ω )eiωn0+, (5) F F n limit of interest to the present paper. n X A natural length scale of our system, where fermions are the majority species, is provided by the inverse of from which the boson and fermion number densities are the Fermi wave-vector k (6π2n )1/3 (n being the F F F obtained by integrating over momenta. ≡ fermion number density). One may use then the dimen- sionless coupling parameter g = (kFa)−1 to describe the inTouhrepfurlelvniouumsewriocarklsso[1lu2t,io1n6]o.fHEeqrse.w(1e)−ar(e5)inwtaersetsatcekdleind strength of the interaction. In terms of this parameter, derivinganalyticexpressionsinthemolecularlimitofthe the molecular limit corresponds to the condition g 1, ≫ interaction g. In this limit the binding energy ǫ is the such that the radius of the bound state (which coin- 0 largest energy scale: ǫ T,E , with E k2/(2m ). cideswiththescatteringlengtha,forapositive)ismuch 0 ≫ F F ≡ F F smaller than the average interparticle distance ( k−1). In addition, for the mixtures with nB ≤ nF of interest [Note that in some of our previous works [12, 16∝, 20F] we to the present paper, the bosonic chemical potential µB approaches ǫ in the molecular limit, and is thus large usedadifferentdefinitionofkF (intermsofthetotalden- and negativ−e, 0while the fermion chemical potential re- sity n = n +n , n being the boson number density), F B B mainsofthe orderofthe Fermienergy,andisthen small which coincides with the present one only for n =n .] F B comparedto the binding energy. This hierarchybetween ThethermodynamicandspectralpropertiesofaBose- different energy scales will allow us to derive the asymp- Fermi mixture in the normal phase (i.e. above the con- totic expressions in the molecular limit. densation critical temperature) were studied in our pre- vious works [12, 16, 20] by using a T-matrix approxima- tion for the self-energies. The corresponding equations for the bosonic and fermionic self-energiesΣ and Σ at B F finite temperature read (setting the Boltzmann constant B. The pair propagator k =1): B dP We focus first on the pair propagator Γ(P,Ωm). In Σ (k,ω )= T Γ(P,Ω ) B ν − (2π)3 m order to perform the frequency sum in Eqs. (1) and (2) Z Xm fortheself-energies,weneedtoknowtheanalyticproper- ×G0F(P−k,Ωm−ων) (1) tiesoftheextensionΓ(P,z)ofthepairpropagatortothe dP wholecomplex(frequency)plane. Theanalyticextension Σ (k,ω )= T Γ(P,Ω ) F n (2π)3 m Γ(P,z) is defined by replacing iΩm z on the right- Z Xm hand side of Eq. (3). It is easy to v→erify directly from ×G0B(P−k,Ωm−ωn) (2) Eq.(3)thatΓ(P,z)hasabranch-cutontherealaxisfor Rez 2µ+P2/(2M), where M m +m [32]. In where the pair propagator Γ(P,Ω ) is given by B F m ≥ − ≡ addition,forsufficientlystrongattraction,thepairprop- Γ(P,Ω )= mr + dp agator Γ(P,z) has a pole, which is associated to molec- m − 2πa (2π)3 ular binding. In order to determine this pole, we first (cid:26) Z −1 integrate the terms in Eq. (3) that do not contain the 1−f(ξPF−p)+b(ξpB) 2mr . (3) Fermi or Bose functions [this can be done for Imz = 0 ×" ξPF−p+ξpB−iΩm − p2 #) or for Imz = 0 and Rez ≤ −2µ+P2/(2M)]. The p6 air 3 propagator can be written then The Eq. (13) for the pole determines then the dressed dispersion of the composite-fermion m m3/2 P2 Γ(P,z)= r r 2µ z −(2πa − √2π r2M − − ξ˜CF = P2 µ +Σ , (18) −1 P 2M∗ − CF CF + I (P,z) I (P,z) , (6) B F and the associated composite-fermion Fermi momentum − (cid:27) P , defined by the equation ξ˜CF = 0. Note that the where µ ≡(µB+µF)/2, while IF(P,z) and IB(P,z) are leCaFding order term of the self-enPeCrFgy ΣCF takes into ac- defined by: count the interaction between the molecules and the un- IB(P,z)≡ (2dπp)3 ξF b(+ξpBξ)B z, (7) pBaoirrnedapfeprrmoxioinmsa(twiointhvaalpuperofoxrimthaetemdoelnesciutlye-nfe0µrFm)iwonithsctahte- Z P−p p − tering length aDF: IF(P,z)≡ (2dπp)3 ξF f(ξ+PF−ξBp) z. (8) a = (1+mF/mB)2a. (19) Z P−p p − DF 1+m /2m F B The term I (P,z) is suppressed exponentially by the B Bose function exp( ǫ0/T) and thus does not con- The subleading correction to ΣCF as well as the correc- ∝ − tion (15) to the bare mass of the molecules are instead tributetothepairpropagatorinthemolecularlimit. The term I (P,z) can instead be expanded in powers of ǫ−1, due to the composite nature of the molecules. assumiFng P2/(2M), z ǫ . The leading term is giv0en Finally, the residue w(P) at the pole of Γ is given by 0 | |≪ by: w(P)= lim (z ξ˜CF)Γ(z,P) (20) dp f(ξF) n0 z→ξ˜PCF − P IF0 =Z (2π)3 ǫ0P ≡ ǫµ0F , (9) = 2π 1− 2mπraIF(P,ξ˜PCF) (21) −am2 1 4πaI0 while inclusion of the next-to-leading term yields r − mr F 2π 2πa I0 P2 1+ I0 w , (22) IF(P,z)=IF0−δIF0+ ǫF z+µCF− 2m , (10) ≃−am2r (cid:18) mr F(cid:19)≡− 0 0 (cid:18) B(cid:19) where in the last line we have neglected again terms of where we have introduced the composite-fermion chemi- orderǫ−2. We seein Eq.(22) thatinthe molecularlimit cal potential µ =2µ+ǫ , while 0 CF 0 the dependence on P of the residue is negligible even at 1 dp next-to-leading order. δI0 = f(ξF)p2. (11) F 2m ǫ2 (2π)3 p r 0 Z At T =0, n0 = kµ3F with k =√2m µ , while δI0 = C. Bosonic self-energy and momentum distribution µF 6π2 µF F F F function 3m µ I0/(5m ǫ ). F F F r 0 The pole of Γ is then determined by the equation: ThesumoverthefermionicfrequencyΩ inEq.(1)for m P2 2πa 2 the bosonic self-energy can be performed by transform- z +2µ+ǫ 1 I (P,z) =0, (12) − 2M 0 − m F ing it in a contour integration in the complex z plane, (cid:20) r (cid:21) as usually done when summing over Matsubara frequen- which, by using the expansion(10) and neglecting terms cies (see, e.g., chap. 7 of Ref. [33]). One obtains three of order ǫ−02 at least, yields contributions associated to the different singularities of Γ and G0 in the complex plane: the simple poles of Γ P2 F z = µ +Σ , (13) andG0 andthe integralalongthe branch-cutofΓ(P,z). 2M∗ − CF CF F This integral is, however, suppressed exponentially by ∗ where the molecule effective mass M is given by the Fermi function f(z) which appears when transform- ing the discrete sum in a contour integration. Indeed, 4πam M∗ =M 1+I0 F (14) we have seen above that the branch-cut is present for F m m (cid:18) r B(cid:19) Rez 2µ+P2/(2M). Since 2µ ǫ in the molec- 0 =M 1+ 4 mF(k a)3 (T =0), (15) ular l≥im−it, it follows immediately th≃at−the integral along (cid:20) 3πmB µF (cid:21) the cutis suppressedexponentially at finite temperature (and is vanishing at T =0). while the self-energy correction Σ is CF The contributions from the poles of Γ and G0 yield 4πa F Σ = ǫ (I0 δI0) (16) then CF m 0 F− F r = 4mπran0µF(cid:20)1− 35(kµFa)2(cid:21) (T =0). (17) ΣB(k,ων)=w0Z(2dπP)3 fξ˜PC(ξF˜PC−F)ξ−PF−fk(ξ−PF−iωkν). (23) 4 Note how in the molecular limit the boson self-energy (molecular)limitthiseffectismadeextreme,yieldingfor (23) acquires the form determined by the virtual recom- n < n /2 to a complete suppression of the occupancy B F binationofthe bosonwith afermionto forma molecule, at low momenta. with probability amplitude √w0, followed by the decay of the virtual molecule into its constituent fermions and bosons (with the same probability amplitude). D. Fermionic self-energy and momentum We pass now to the calculation of the boson momen- distribution function tum distribution function, as determined by Eq. (4). We first notice that in the molecular limit we are al- The calculation of the fermionic self-energy from lowed to expand perturbatively the Dyson’s equation Eq. (2) proceeds along the same lines as for the bosonic tGhBer(kef,oωreν)th=er[Gel0Bev(akn,tωrνa)n−g1e−ofΣvBa(luke,sωoνf)]ω−ν1asnindcke2µ/B(2,manBd) siseljfu-esntetrhgey,pwoiltehotfheΓotnolybdeiffceornesnidceertehda,tsiinntcheisthceaspeotlheeroef inside the free boson propagator, are of order ǫ0, while G0 is suppressed exponentially. One obtains then Σ is of order ǫ1/2 (because of the residue w , which is B B 0 0 of order ǫ10/2 ). Σ (k,ω )= w dP f(ξ˜PCF) . (27) The expansion of the Dyson’s equation to first order F n − 0 (2π)3ξ˜CF ξB iω Z P − P−k− n then yields: The presence of µ in the denominator of the expres- B GB(k,ων)≃G0B(k,ων)+G0B(k,ων)2ΣB(k,ων). (24) sion(27)forthe fermionicself-energymakesittobehave in the molecular limit like w ǫ−1 ǫ−1/2. We are al- The first term on the right-hand-side of the above equa- 0 0 ∼ 0 lowed then to expand the Dyson’s equation also for the tion yields again an exponentially small contribution fermionicGreen’sfunction. Beforedoingthis,itisuseful when summed over ω . By inserting the expression (23) ν to introduce a procedure which accelerates the conver- for the self-energy in Eq. (24) and summing over ω one ν gence of the expansion in the fermionic case. Indeed, in gets then: this case, there is a range of k close to the Fermi step, nB(k)=w0 (2dπP)3b(ξ˜PCF−(ξξBPF−+kξ)F[f(ξ˜PCFξ)˜C−Ff)2(ξPF−k)] amnadyobfefcroemqupeanrcaibeslecolorseevetnoszmeraollesructhhatnhaΣtF(Gk0F,(ωkn,)ω,nth)−u1s Z k P−k− P invalidatingtheexpansionoftheDyson’sequationinthis =w dP f(−ξPF−k)f(ξ˜PCF) (25) region. (Inthebosoniccase,forwhichthebosonchemical 0 (2π)3(ξB+ξF ξ˜CF)2 potential is negative and large, the self-energy is instead Z k P−k− P alwaysmuch smallerthan G0 −1.) Beforeexpanding, we =w dP Θ(ξPF−k)Θ(PCF−P) (T =0). thusaddandsubtractintheBdenominatoroftheDyson’s 0Z (2π)3 (ξkB+ξPF−k−ξ˜PCF)2 equation the quantity Σ0F ≡ ReΣRF(kUF,ω = 0), where k corresponds to the position of the Fermi step of G (26) UF F as defined by the equation The expressions (25) and (26) show clearly the effect of the Fermi statistics obeyed by the molecules and un- k2/(2mF)−µF+ReΣRF(k,ω =0)=0, paired fermions on the bosonic momentum distribution function. In particular, at T = 0 the two Θ func- and ΣRF(k,ω) is the analytic continuation of the self- tions in Eq. (26) require simultaneously P < P and energy to the real axis (obtained with the replacement CF fl|oPowr−kbk<o|sok>nµFkcµ−oFnP.cCeAnFst.raaWtrieeosnsue,lets,uthwchehreetnhfoarkteµkFth>at>P, fCoPFr,snu,Bffit(hckie)en=fotlry0- ioloωuwnr.i→nWteeωreh+satv,iek0+UthF)e.n=:In[6πpr2a(nctFic−e,niBn)]t1h/e3,maoslweceuwlairlllismeeitboe-f µF CF mation of the molecules depletes completely the bosonic 1 momentum distribution at low momenta. In particular, GF(k,ωn)= G˜0(k,ω )−1 Σ˜ (k,ω ), (28) by using the asymptotic expressionsfor the chemicalpo- F n − F n tentials derived below, one can see that, to leading or- where G˜0(k,ω )−1 = iω ξ˜F with ξ˜F = k2/(2m ) der in the molecularlimit, kµF correspondsto the radius µ +Σ0F, whilen Σ˜ (k,ω n)−= Σk (k,ω )k Σ0. TheFex−- of the Fermi sphere of the unpaired fermions, with den- F F F n F n − F pansion of the Dyson’s equation (28) improves on that sity n n , while P corresponds to the radius of F B CF − of the original equation. Indeed, in the region where theFermisphereofthecompositefermions,withdensity G˜ (k,ω )−1 is smallor vanishing, Σ˜ (k,ω ) is also van- n . It then follows that the condition n n > n F n F n B F B B − ishing. Inaddition,itiseasytocheckfromEq.(27)that must be fulfilled in order to have k >P , and there- fore the presence of the empty regµioFn at CloFw momenta. Σ˜F(k,ωn) is of order ǫ0−3/2 (while ΣF(k,ωn) is of order Wenotefurtherthatapartialsuppressionofthebosonic ǫ−1/2), thus accelerating the convergence of the expan- 0 momentum distribution at low momenta was found also sion of the Dyson’s equation. We thus have for weakly-interacting Bose-Fermi mixtures in the per- turbativeanalysisofRef.34. Weseethatintheopposite G (k,ω ) G˜0(k,ω )+G˜0(k,ω )2Σ˜ (k,ω ), (29) F n ≃ F n F n F n 5 from which one obtains occurs at a certain critical coupling strength. Whether thistransitioncoincideswiththetransitionfromthecon- n (k)=f(ξ˜F)+T G˜0(k,ω )2Σ˜ (k,ω ) (30) F k F n F n densed phase to the normal one already studied in our Xn previousworks,orinsteadsomewhatanticipatesitwithin =f(ξ˜F)+T G˜0(k,ω )2Σ (k,ω ) ΣFf′(ξ˜F). (31) the condensed phase, is not a priori clear. In order to k F n F n − 0 k n answer this question, one should extend the present dia- X grammaticapproachtothecondensedphase(and/orper- By using Eq. (27), one gets then: form extensive QMC calculations in this phase). Work dP along these lines is in progress. T G˜0(k,ω )2Σ (k,ω )= w f(ξ˜CF) F n F n − 0 (2π)3 P Note also that at large k (i.e. k kF) only the Xn Z fermions belonging to the molecules c≫ontribute to the −f(ξ˜PCF−ξPB−k)+f(ξ˜kF) + f′(ξ˜kF) (32) momentum distribution function. In this case ×" (ξ˜PCF−ξPB−k−ξ˜kF)2 ξ˜PCF−ξPB−k−ξ˜kF# n (k) n φ(k)2 (36) F CF → Note that, neglecting exponentially small terms in the molecularlimit,f(ξ˜CF ξB )=1 f( ξ˜CF+ξB ) 1. where P − P−k − − P P−k ≃ In addition, at T =0, where f′(ξ˜kF)=−δ(ξ˜kF), n dP f(ξ˜CF) (37) CF ≡ (2π)3 P dP f(ξ˜CF)f′(ξ˜F) Z w P k = ΣFδ(ξ˜F), (33) − 0 (2π)3ξ˜CF ξB ξ˜F − 0 k and, neglecting a subleading term in the expression for Z P − P−k− k w , 0 which cancels exactly with the last term on the r.h.s of Eq. (31). At finite T this cancellation holds only ap- 2π 1 3/2 φ(k)= (38) wprhoixcihmiastaenlyy,wthaye dnieffgelirgeinbclee bineitnhgeamtoelremcuolafrolridmeirt.T/ǫ0 , sam2r 2km2r +ǫ0 We thus obtain for the fermionic momentum distribu- istheinternalwavefunctionofthemolecules(asobtained tion function in the molecular limit: fromthesolutionofthetwo-bodyproblem). Notefurther nF(k)=f(ξ˜kF)+f(−ξ˜kF) (2dπP)3(ξ˜CFw0ξfB(ξ˜PCF) ξ˜F)2, cthaantbaetsleaerngeimkmaeldsoianteBly(kf)rocmonEveqr.g(e2s5t)o. nCFφ(k)2, as it Z P − P−k− k (34) E. Chemical potentials which at T =0 becomes: nF(k)=Θ(kUF k) The equations (25) and (34) for the bosonic and − dP w Θ(P P) fermionic momentum distributions (or for their counter- 0 CF +Θ(k k ) − . (35) − UF (2π)3(ξ˜CF ξB ξ˜F)2 parts at zero temperature) can be integrated over k to Z P − P−k− k obtain the boson and fermion density. For given den- One sees clearly from Eq. (34) and (35) that the sities, they can be used then to get the values of the fermionicmomentumdistributionfunctionismadeoftwo chemical potentials. In particular, it can be shown that components: a Fermi distribution function of unpaired the integration over k of Eq. (25) yields fermions and a distribution of fermions which are paired dk withthebosonsinthemolecules. Theoverallmomentum n = n (k) (39) distributionfunctionhasthenastepatamomentumde- B (2π)3 B Z termined by the density of unpaired fermions. This is −3/2 =n +o(ǫ ), (40) CF 0 theexpectedbehaviorinthemolecularlimitoftheBose- Fermi attraction. On the other hand, for a weak Bose- where n depends on the chemical potentials and tem- CF Fermi attraction one expects Fermi liquid theory to be perature through Eq. (37). valid for the Fermi component, predicting a momentum Similarly, the integration over k of Eq. (34) yields distributionfunctionwithastepattheFermimomentum corresponding to the total fermion density. According dk n = n (k) (41) to Luttinger’s theorem, the step remains pinned at the F (2π)3 F Z samemomentumasforthenon-interactingsystem,inde- =n +n +o(ǫ−3/2), (42) pendently of the coupling value. This is precisely what UF CF 0 is found and discussed in Ref. [21]. where Clearly, the only way to allow for such distinct behav- iors for weak and strong attraction is that a quantum- dk n f(ξ˜F). (43) phase transition breaking down the Fermi liquid theory UF ≡ (2π)3 k Z 6 From Eq. (40) one obtains then 0.005 (a) n (k)TMA B µCF =µ0F(T,M∗,nB)+ΣCF, (44) 0.004 nB(nk)1(skt)-TTMMAA F where µ0F(T,M∗,nB) is the chemical potential for a free 0.003 α=0.70 Fermi gas of temperature T, mass M∗, and density nB, n(k) g=2.35 while Eq. (42) yields 0.002 µF =µ0F(T,mF,nF−nB)+Σ0F, (45) 0.001 where we have used Eq. (40) to replace nCF with nB. 0 The equation µ =µ µ ǫ then yields 0 1 2 3 4 5 B CF F 0 − − k µB =µ0F(T,M∗,nB)−µ0F(T,mF,nF−nB) 0.002 (b) n (k)TMA +ΣCF−Σ0F−ǫ0. (46) nB(kB)1st-TMA 0.0015 n (k)TMA A further simplification can be obtained by neglecting F terms of order a2. To this order, one can set M∗ = M, α=0.70 Σ0F = 4mπranB, and ΣCF = 4mπra(nF −nB) in the previous n(k) 0.001 g=3.35 equations for the chemical potentials. At T = 0 one obtains in particular: 0.0005 [6π2(n n )]2/3 4πa F B µ = − + n , (47) F B 0 2m m F r 0 1 2 3 4 5 (6π2n )2/3 4πa k B µ = + (n 2n ) B F B 2M m − r FIG.1. (Coloronline)Bosonicandfermionicmomentumdis- [6π2(n n )]2/3 F− B ǫ , (48) tribution function at T = 0 for a mixture with mB = mF, 0 − 2m − densityimbalanceα=0.70,andcouplingstrengthsg=2.35(a) F (6π2n )2/3 4πa and g=3.35 (b). The numerical results obtained by the T- B µ = + (n n ). (49) matrix self-energy (symbols) are compared with the analytic CF F B 2M m − r expressions in the molecular limit for nB(k) (full curve) and and, at this level of accuracy: nF(k) (dashed curve) derived in the present paper. For the bosonic distributionwe presentalso thenumericalresultsfor the T-matrix approximation expanded to first-order in the k = 2m (µ Σ0) UF F F− F Dyson’s equation (1st-TMA) besides those obtained without =[q6π2(n n )]1/3. (50) expanding it (TMA). The wave-vector k is in units of kF. F− B Notethat for thefermionic momentum distribution boththe The Eqs. (44)-(50) show how the T-matrix self-energy analytic expression and the numerical T-matrix calculation for the constituent bosons and fermions recovers the ex- yield nF(k)=1for k<kUF (out of theverticalrange chosen in the figure). pectedphysicallimitofaFermi-Fermimixtureofdimers (molecules) and unpaired fermions mutually repelling with a scattering length a = γa. The T-matrix ap- DF proximation yields for the proportionality coefficient γ (n n )/(n +n ) = 0.7 for which the indirect Pauli F B F B thevalueγ =(1+mF/mB)2/(1/2+mF/mB),asitcanbe exclu−sion effect can be seen on the bosonic momentum seen by writing the term 4πa/mr as 2πaDF/mDF, where distribution. The two panels correspond to two differ- mDF =MmF/(mF+M) is the reduced mass of a dimer ent coupling values (g = 2.35,3.35). For the bosonic and one fermion. This value for γ is only approximate momentumdistribution,onenoticesthatthe asymptotic and correspondsto a Bornapproximationfor the dimer- expression (26) reproduces well the T-matrix results for fermion scattering. k > 2k as well as the presence of the empty region for F k <∼k P ,butdeviatessomewhatfromtheT-matrix µF− CF resultsforintermediatevaluesofk. Thisdifferenceisdue F. Comparison with the T-matrix results to the fact that the asymptotic expression (26) is ob- tained by expanding the Dyson’s equation to first-order The asymptotic expressions for the momentum dis- (cf. Eq. (24)), an approximation that results to be valid tribution function derived in Secs. IIC and IID can for all k only for rather large values of g. This is con- be compared with the results obtained by the full nu- firmed by the very good agreement with the results ob- merical solutions of the T-matrix set of equations. In tained by expanding the Dyson’s equation and calculat- Fig. 1 we present this comparison at T = 0 for a mix- ingnumericallytheself-energy(circles). Clearly,forboth ture with equal masses and a density imbalance α comparisons the agreement improves when g increases, ≡ 7 plingconsideredinFig.1. Indeed,inderivingEq.(35)we 0.016 nB(nkB)1(skt)-TTMMAA Σh˜avG˜e0exrpatahnedredthtahneΣfulGl 0G,raeetnr’isckfuwnhcticiohn, aGsFnointedterambosvoef, n (k)TMA F F F F F accelerates the convergence of the Dyson’s expansion in 0.012 α=0.25 the fermionic case. n(k) 0.008 g=2.25 As a further check of the asymptotic expressions (24) and(35)wepresentinFig.2thesamecomparisonsasfor Fig. 1, but now for two imbalances α =0.25 and α = 0, 0.004 forwhichtheemptyregionatlowmomentainthebosonic (a) momentum distribution is absent, since P >k . We CF UF 0 0 1 2 3 4 5 notice that even though also for these imbalances the k asymptotic expressionfor the bosonic momentum distri- 0.005 bution deviates more than the fermionic one from the nB(k)TMA T-matrix results, the discrepancy gets smaller when the 0.004 nB(k)1st-TMA density imbalance decreases. This is due to the faster n (k)TMA F convergence of the Dyson’s expansion for the bosonic 0.003 α=0.25 Green’s function at small imbalances. Indeed, one can n(k) g=3.70 see from Eq.(23) for the bosonic self-energy that when 0.002 P and k are comparable, as it happens at small im- CF µF balances, a partial cancellation occurs between the con- 0.001 tributions associated to the two Fermi functions appear- (b) ing in the numerator of Eq. (23), thus making the self- 0 energy small, and the Dyson’s expansionrapidly conver- 0 1 2 3 4 5 k gent. Asamatteroffact,forthedensitybalancedsystem at g =4.0 (panel (c)) one can see that the difference be- 0.007 nB(k)TMA tween the first-order expansion and the full T-matrix is 0.006 nB(k)1st-TMA indeed very small. n (k)TMA 0.005 F Itisinterestingtonotethatinthissymmetriccasewith 0.004 α=0.00 nB = nF (and mB = mF) both the asymptotic expres- n(k) 0.003 g=4.00 sions (24) and (35) and the T-matrix calculations yield slightly different occupation numbers for the boson and 0.002 fermion component. We believe that this is due to the use, in our T-matrix approach, of bare Greens functions 0.001 (c) G0 multiplyingthepairpropagatorintheexpressions(1) 0 and (2) for the boson and fermion self-energies. In par- 0 1 2 3 4 5 k ticular,ifwehadusedadressedfermionGreen’sfunction G inthe placeofG0 inthe expression(1)forthe boson F F FIG.2. (Coloronline)Bosonicandfermionicmomentumdis- self-energy, we would have got a dressed fermion disper- tributionfunctionatT =0foramixturewithmB =mF,den- sion ξ˜PF−k in the place of a bare one in Eq. (24). Since sityimbalanceα=0.25,g=2.25(a)densityimbalanceα=0.25, k = 0 for n = n , one would have that ξ˜F would g=3.70(b),densityimbalanceα=0,g=4.0(c). Thenumer- UF B F P−k be alwayspositive, and the first Θ function appearing in ical results obtained by the T-matrix self-energy (symbols) the numerator of Eq. (24) would be alwaysequal to one, are compared with the analytic expressions in the molecular thus making Eqs. (24) and (35) identical for n = n limit for nB(k)(full curve)and nF(k) (dashedcurve)derived B F and m = m . The use of a bare fermionic Green’s inthepresentpaper. Forthebosonic distributionwepresent B F alsothenumericalresultsfortheT-matrixapproximationex- function in Eq. (24) subtracts instead to the integralde- panded to first-orderin the Dyson’sequation (1st-TMA) be- termining the bosonic distribution function the contri- sidesthoseobtainedwithoutexpandingit(TMA).Thewave- bution of wave-vectors P such that P k < k , with vector k is in units of kF. µF =Σ0F = 4mπranB asitcanbeobtain|ed−from| Eq.µ(F45)for n =n . Thiscontributionvanishesintheextremelimit B F g , but it is still finite at the value of g considered → ∞ albeit rather slowly for the non-expanded T-matrix. inFig.2(c), andaccountsforthe differencesbetweenthe Forthefermionicmomentumdistribution,instead,the fermionicandbosonicdistributions. Theuseofadressed asymptotic equation (35) compares well already with bosonic Greens function in the convolution defining ΣF the non-expanded T-matrix (for this reason we do not wouldproduceinsteadminordifferences,duetothelarge present Dyson-expanded results in this case). One andnegativevalueofµB whichmakesself-energycorrec- sees that the momentum distribution resulting from the tions less important. asymptotic equation (35) reproduces very well the full Note finally that in the previous comparisons we used numericalT-matrixcalculationalreadyatthelowercou- the same chemical potentials µ and µ calculated nu- B F 8 merically within the T-matrix approximation as input function is imaginary-time evolved from Ψ , with the T parameters for the asymptotic equations (26) and (35) constraint on the nodal surface to remain pinned to the ∗ (the remainingparametersM , P andk being fully pointswhereΨ =0,inordertocircumventthefermionic CF UF T determined by the Eqs. (15),(18) and (50)). Alterna- sign problem. We estimate the momentum distribution tively, one could also use the molecular-limitexpressions in VMC with nVMC(k) = Ψ nˆ Ψ / Ψ Ψ , where B h T| k| Ti h T| Ti (47) and (48) for µ and µ as input parameters of the nˆ is the number operator in momentum space aver- B F k analyticcalculations. Thedifferenceinthevaluesissmall agedovermomentumdirection,whileFN-DMC provides and,clearly,progressivelyvanishesasgincreases. Forex- the mixed estimator nDMC(k) = Ψ nˆ Ψ / Ψ Ψ , B h T| k| 0i h T| 0i ample,forg =3.35andα=0.7thediscrepancyamounts where Ψ is the long-(imaginary)-time evolution of Ψ . 0 T to 0.02 %, 0.2 % and 0.8 % for µ , µ and µ , respec- Both the VMC and the DMC estimates are biased by B F CF tively. Ψ ;acommonwayofreducingthe biasistoextrapolate T themviatheformulanEXT =(nDMC)2/nVMC,wherethe B B B dependence on δΨ= Ψ Ψ is second order, provided 0 T − G. Comparison between T-matrix results and δΨ is small. Monte Carlo calculations for the bosonic momentum Following [19], we write the guiding wave function in distribution function the molecular regime as Ψ (R)= Φ (R)Φ (R). Here, T S A Φ is a positive Jastrow function of the particle coordi- S In this section we present a comparison between the nates R = (r1,...,rNF,r1′,...,rNB) and is symmetric T-matrixresultsandVariational(VMC)andFixed-node under exchange of identical particles. We use ΦS(R) = Diffusion (FN-DMC) Monte Carlo simulations obtained ij′fBF(rij′) i′j′fBB(ri′j′) ijfFF(rij),wheretheun- with a novel guiding wave function, which is a suitable primed (primed) coordinates refer to fermions (bosons) Q Q Q symmetrization of the molecular wave function intro- and two-body spherically symmetric correlation func- duced in our previous work [19]. The details of the sim- tions of the interparticle distance are introduced. We ulations arethe same as in[19], exceptfor the trialwave set f = 1, while f is the solution of the two-body BF BB ′ function. Bose-Bose problem with f (L/2)=0; f is described BB FF Addressingthe calculationof the momentum distribu- below. Antisymmetrization in [19] was provided by the tion of the bosons in the molecular regime with Quan- use of a generalized Slater determinant of the following tum Monte Carlo is computationally a very demanding form: problem, due to the need of taking care of the pairing ′ ′ ϕ (1,1) ϕ (N ,1) oofusbloysosynms wmietthrifzeirnmgiownitshinrteospmeoctlectouletsh,ewbhoisleonsiicmucoltoarndei-- (cid:12) K1 ... ·.·.·. K1 ...F (cid:12) (cid:12) (cid:12) natesandantisymmetrizingwithrespecttothefermionic ΦMS(R)=(cid:12)(cid:12)ϕKNM(1,NM) ··· ϕKNM(NF,NM)(cid:12)(cid:12) , (51) coordinates. We thus concentrate on a single choice A (cid:12)(cid:12) ψk1(1) ··· ψk1(NF) (cid:12)(cid:12) of the parameters, namely g = 3, α = 0.7 and equal (cid:12) .. .. .. (cid:12) (cid:12) . . . (cid:12) masses mF =mB =m. We perform our simulation with (cid:12) (cid:12) NF = 40 fermions and NB = 7 bosons. These particle (cid:12)(cid:12) ψkNUF(1) ··· ψkNUF(NF) (cid:12)(cid:12) numbers are chosento reduce partially finite-size effects, where the mo(cid:12)(cid:12)lecular orbitals are defined as ϕ (cid:12)(cid:12)(i,i′) = since the numbers of composite fermions N =N =7 (cid:12) Kα(cid:12) CF B fB(ri ri′ )exp(iKα(ri+ri′)/2), which consist of the and unpaired fermions N = N N = 33 corre- | − | UF F B relative-motionorbitalsf timesthemolecularcenter-of- − B spond to closed shells. Simulations are carried out in a massplanewaveswith K P ,andn =P3 /6π2, cubic box of volume L3 = N /n with periodic bound- | α|≤ CF CF CF F F while for the unpaired fermions k k , with n = α UF UF ary conditions. We model the attractive interaction be- k3 /6π2. The functions f are |cho|se≤n to be the bound tween bosons and fermions with a square-well potential soUlFutions of the two-bodyBBose-Fermi problem up to R¯, wVBi0tFhfirxaeddiubsyRtBhFe sruelcahtiothnaat n=FRRB3BFF(=1−10ta−n7(,κaBnFd)/dκeBpFt)h, em−aβt(cLh−erd))towahefruenRc¯tiaonndalβofarteypvearfiBaat(iro)n=alCpa1r+amCe2t(eer−sβarn+d whereκBF = mVB0FRB2F. Forconsistency,weintroduce fBa′(L/2)=0. the same repulsion between the bosons that we used in ThemolecularorbitalsappearingintheSlaterdetermi- p our previous work [19] (even though here it would not nant (51) are occupied by the bosons in a specific order, be necessary for the stability in the molecular regime). thusthesymmetrizationofthebosoniccoordinatesisnot The repulsion is then modeled by a soft-sphere poten- fulfilled. This isnotaproblemwhencalculatingenergies tial with radius R = 60R and height V0 fixed BB BF BB with Diffusion Monte Carlo, since the DMC pure esti- by the relation a = R (1 tanh(κ )/κ ), where BB BB BB BB mator of the energy does not depend on the trial wave − κBB = mVB0BRB2B; the Bose-Bose scattering length is function (except for the fixed nodal surface), provided set to a =(6π2n )−1/3. there is a finite overlap of the trial wave function with pBB F InbothVMCandFN-DMCthetrialwavefunctionΨ the symmetric ground state. This is the case for a fi- T playsacrucialrole. InVMCthesampledobservablesare nite number of particles using the non-symmetric wave the expectation value of quantum operators in the state function (51). A similar approach has been successfully defined by Ψ . In FN-DMC the amplitude of the wave used in Quantum Monte Carlo studies of the equation T 9 of state of solid 4He [37, 38] with the Nosanow-Jastrow 0.0025 wavefunction [35, 36], where the bosonsare localizedon TMA 1st-TMA specific lattice sites. asymptotic 0.002 Bose symmetry of the trial wave function is, how- VMC DMC ever, crucial when calculating the momentum distribu- tion, which is obtained by a mixed estimator biased by k) 0.0015 α=0.70 the trial wave function. A full symmetrization of the (B n g=3 determinant (51) over all permutations of the bosons is 0.001 not feasible since the number of terms to be summed scales as the factorial of N . For this reason, we resort B 0.0005 to an approximate strong-coupling wave function, where the symmetrization over the bosonic coordinates is per- 0 formedwithinthemolecularorbitalsappearinginasingle 0 0.5 1 1.5 2 2.5 3 3.5 4 determinant: k ϕ (1,R ) ϕ (N ,R ) (cid:12) K1 ... B ·.·.·. K1 ...F B (cid:12) FVIMGC.3a.n(dCoFlNor-DonMliCne)caClcoumlaptaiornissonfobretthweeebnosTo-nmicatmrioxmreensutultms, (cid:12) (cid:12) Φ˜MAS(R)=(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ϕKNψMk1(...1(1,)RB) ··.··.··. ϕKNψMk1(N(...NFF,)RB)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (52) dowriasdvtereri-bveuextcpitooanrndkaetdisgTin=-mu3naitatrsnixdofaαknFd=.t0h.e7.asWymepatlsooticshroewsultthse. fiTrshte- (cid:12) (cid:12) (cid:12)(cid:12) ψkNR(1) ··· ψkNR(NF) (cid:12)(cid:12) (cid:12) (cid:12) where ϕ (i,(cid:12)R )= ϕ (i,i′). (cid:12) any significant change of the kinetic energy. It turns out Kα (cid:12) B i′ Kα (cid:12) that such feeble correlations do not change significantly By expanding the determinant (52) it is easy to show that Φ˜MS(R) Pcan be obtained from the origi- the momentum distribution of the bosons, either. How- A ever, they help in reducing the statistical error of the nal non-symmetric wave function by summing over all simulations;we presenttherefore the results obtainedby possible dispositions with repetition of the bosonic co- using these additional correlations, for which we have ordinates in the non-symmetric wave function (51). smaller error bars. The wave function Φ˜MS(R) contains then all permu- A tations of bosons, as required. It contains however In Fig. 3 we compare the VMC, DMC and T-matrix also additional spurious terms where the same boson resultsfornB(k)atg =3andα=0.7. Evenatthisvalue appears in many different molecular orbitals. For ex- of interaction the VMC estimator gives a finite value of ample, if we had NM = NB = 3 molecular or- n0 = nB(k = 0)/NB ≃ 0.06, while the FN-DMC is able bitals and N = 5 fermions, we would also obtain the to deplete the condensate fraction down to a value com- F ttheremb:osϕoKn11(1′,is1′r)eϕpKe2a(t3e,d1.′)ϕK3(5,1′)ψk1(2)ψk2(4), where pthaatitbolebtwaiitnhinzgeraos(tnraicmtleylyz,enro0c≃on0d.0e0n1s)a.teOwneithcoFuNld-DthMinCk Thesespurioustermstendtoincreasethebosoniccon- and the wave function (52) is in practice impossible, be- densate, because the bosons that are not allotted to the cause of the biased nature of the mixed estimator of the molecularorbitalsareputinaplane-wavestatewithzero momentum distribution. The comparison between the momentum (since their spatial coordinates do not ap- VMC and DMC estimates hints, however,at a complete pear explicitly in these terms). In the above example, depletion of the condensate. This is probably due to the the bosons i′ = 2′,3′ would significantly contribute to ability of the DMC to suppress completely the energeti- the condensatefraction,sincechangingtheircoordinates cally costly spurious terms. would not affect the value of that specific term. These In Fig. 3, we do not report the standard extrapolated termscorrespondalsotothe clusteringofmanyfermions estimator nEXT because the presence of a spurious con- B closetoasinglebosonatadistanceoforderofa ;they densate fraction in the VMC calculation subtracts auto- BF are then significant near resonance, where the molecular matically weight from the rest of the distribution, thus orbitalsareveryloose,whiletheyarestronglysuppressed invalidating the extrapolationprocedure for all values of in the molecular limit due to the Pauli principle, which k (including the values of k where the VMC and FN- forbids the formation of fermion clusters, thereby mit- DMCareclosetoeachother,forwhichtheextrapolation igating the unwanted effect on the bosonic condensate procedurecould appear justified). The DMC calculation fraction. We have tried to suppress further these spu- confirms the suppression of the bosonic momentum dis- rious terms, by introducing a very short-range repulsive tribution at low k, in particular the DMC results ap- Jastrow factor between fermions, with correlations f pear to follow the T-matrix curve from k 1 down to FF ≃ equal to the solution of the two-body problem of a ficti- the value of k where the momentum distribution is pre- tious soft-sphere potential with radius R = R and dicted to vanish according to the T-matrix calculation. FF BF scatteringlengtha =R /10. Theabovevaluesofthe The DMC calculation agrees well with the T-matrix re- FF FF parametersR anda werechosensosmallastoavoid sults also at high momenta (k > 2). Some deviations FF FF ∼ 10 occur in the intermediate region 1 < k < 2, where the 1 DMC seem closer to the first-order expanded T-matrix g=2 curve rather than the full T-matrix curve. We regard 4 0.8 this better agreement with the expanded T-matrix at intermediate k as fortuitous. On the one hand, an ex- 0.6 α=0.70 trapolationoftheVMCandDMCresultswouldincrease the values of the momentum distribution in this region, n(r) making it closer to the T-matrix curve. On the other 0.4 hand, the relative motion molecular orbital f strongly B affects the nodal surface and thus the momentum distri- 0.2 bution. Itcanbearguedthatrefiningitsparametrization nCF(r) nUF(r) would modify the occupation of intermediate momenta. 0 0 0.2 0.4 0.6 0.8 1 1.2 Addressing quantitatively these issues and reducing the errorbars,especiallyfork <k ,wouldrequire,however, r F an extremely large computational effort. FIG.4. (Coloronline) Densityprofilesofcomposite fermions (CF) and unpaired fermions (UF) for a mixture with equal massesandpopulation imbalanceα=0.7forcouplingvalues III. TRAPPED SYSTEM g=2.0,4.0. DensityisinunitsofNFRF3,whiler isinunitsof theFermi radius RF [2EF/(mFω2)]1/2. The equations derived in the previous section for the ≡ momentum distribution functions and for the chemi- cal potentials (and derived quantities, such as P and CF The chemical potentials µ (and thus µ = µ + B,F CF B k ) can be used to describe also a Bose-Fermi mixture UF µ +ǫ ) appearing in Eq.(53) need to be determinedby F 0 trapped in an external potential whenever the particle the number equation, obtained by integrating over r the number is sufficiently large to make a local density ap- corresponding densities n (r). Since in the molecular B,F proximation accurate. For the particle numbers of or- limit all bosons are inside the molecules, it is physically der 105-107 typically used in experiments with ultracold more transparent to work in terms of the molecular and trapped gases this condition is fully satisfied. The effect unpaired fermion densities, n (r)=n (r) and n (r), CF B UF of the trapping potential is then taken into account by respectively. replacing the chemical potentials µ µ V (r) B,F B,F B,F From the Eqs. (47-49) one gets → − wherever they appear in the expressions derived in the previoussectionforhomogeneousgases. Here, VB,F(r)= n (r)= 1 2M[µ V (r) 2πaDFn (r)] 3/2 12ωB,Fr2 istheharmonictrappingpotentialactingonthe CF 6π2{ CF− CF − mDF UF } boson and fermion species, respectively (for definiteness (55) we assume the same trap frequency ω for both species). 1 2πa Thelocalquantitiesderivedinthiswaycanbeintegrated nUF(r)= 6π2{2mF[µF−VF(r)− m DFnCF(r)]}3/2, over r to obtain the corresponding trap-averaged quan- DF (56) tities. We will be interested in particular in the calcula- from which the chemical potentials µ and µ are ob- F CF tion of the trap-averaged momentum-distribution func- tained by fixing the total number of composite fermions tion ntrap(k), with the aim of determining the best con- B NCF =NB and unpaired fermions NUF =NF NB. ditionsfortheobservationofthe“indirectPauliexclusion − Figure 4 reports as an example the density profiles for effect” in trapped gases. The local bosonic momentum a mixture with equal masses and population imbalance distribution function in the molecular limit is then given α (N N )/(N +N ) = 0.7 for two coupling val- F B F B by ≡ − ues g=2.0, 4.0. Here, as for the homogeneous case, we have defined g = (k a)−1 and k = (2m E )1/2, but nB(k,r)=w0Z (2dπP)3[ΘξkB((ξrPF)−+k(ξrPF))−Θk((Pr)C−F(ξr˜PC)F−(rP)])2 (53) wheitrhe wEeFa=re(u6sNinFg)1t/h3Fωe einxatchteretlraatFpiopnedbceatwseFe.enFNaoDteFtahnadt a, as obtained from the solution of the three-body prob- where ξB,F(r) = ξB,F +V (r), while ξ˜CF(r) = P2 lem[39]. Thebehaviorofthedensityprofilesisconsistent k k B,F P 2M − µ + V (r) + Σ (r), with Σ (r) = 2πaDFn (r), withanalogousplotsreportedpreviouslyforFermi-Fermi CF CF CF CF mDF UF mixtures (albeit with equal populations [40, 41]). V (r)=V (r)+V (r) and we have defined the density CF B F Once the chemical potentials are obtained by invert- of unpaired fermions n (r)=n (r) n (r). UF F B − ing the number equations using the above density pro- Thetrap-averagedquantityisthenreadilyobtainedby files, the trap-averagedmomentum distribution function integrating over r: is calculated with Eqs. (53) and (54). Figure 5 reports the trap-averagedbosonic momentum distribution func- ntrap(k)= d3rn (k,r). (54) B B tion n (k) for three different population imbalances at B Z