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Finite temperature excitations of a trapped Bose-Fermi mixture


Finite temperature excitations of a trapped Bose-Fermi mixture
Xia-Ji Liu1, 2 and Hui Hu3
LENS, Universit` a di Firenze, Via Nello Carrara 1, 50019 Sesto Fiorentino, Italy 2 Institute of Theoretical Physics, Academia Sinica, Beijing 100080, China Abdus Salam International Center for Theoretical Physics, P. O. Box 586, Trieste 34100, Italy (Dated: February 2, 2008)
1

3

arXiv:cond-mat/0307638v1 [cond-mat.soft] 25 Jul 2003

We present a detailed study of the low-lying collective excitations of a spherically trapped BoseFermi mixture at ?nite temperature in the collisionless regime. The excitation frequencies of the condensate are calculated self-consistently using the static Hartree-Fock-Bogoliubov theory within the Popov approximation. The frequency shifts and damping rates due to the coupled dynamics of the condensate, noncondensate, and degenerate Fermi gas are also taken into account by means of the random phase approximation and linear response theory. In our treatment, the dipole excitation remains close to the bare trapping frequency for all temperatures considered, and thus is consistent with the generalized Kohn theorem. We discuss in some detail the behavior of monopole and quadrupole excitations as a function of the Bose-Fermi coupling. At nonzero temperatures we ?nd that, as the mixture moves towards spatial separation with increasing Bose-Fermi coupling, the damping rate of the monopole (quadrupole) excitation increases (decreases). This provides us a useful signature to identify the phase transition of spatial separation.
PACS numbers: PACS numbers:03.75.Kk, 03.75.Ss, 67.60.-g, 67.40.Db

I.

INTRODUCTION

The impressive experimental achievement of BoseEinstein condensation (BEC) in the bosonic systems 87 Rb [1], 23 Na [2], and 7 Li [3] has initiated and stimulated a whole new ?eld of research in the physics of quantum atomic gases. Recently, several groups have extended these experiments to the case of trapped Bose-Fermi mixtures, in order to employ the “sympathetic cooling” to reach the regime of quantum degeneracy for the Fermi gas. As a ?rst step, stable Bose-Einstein condensates immersed in a degenerate Fermi gas have been realized with 7 Li in 6 Li [4], 23 Na in 6 Li [5], and very recently with 87 Rb in 40 K [6]. In the possible next step, investigations of the thermodynamics, collective many-body e?ects and other properties could be available soon in these systems. Especially interesting is the behavior of low-energy collective excitations, since the high accuracy of frequency measurements and the sensitivity of collective phenomena to the interatomic interaction make them good candidates to unravel the dynamical correlation of the many-body system. On the theoretical side, several analyses have been presented for low-lying collective excitations of a trapped Bose-Fermi mixture. Collective modes in the collisionless limit, where the collision rate is small compared with the frequencies of particle motion in traps, have been considered by the sum-rule approach [7], by the scaling theory [8], or in the random-phase approximation [9, 10]. In the collision-dominated regime collective oscillations have been discussed by Tosi et al. in Refs. [11, 12], and by the authors in Ref. [8]. These investigations have mainly concentrated at zero temperature using the standard two-?uid model for the condensate and the degenerate Fermi gas. However, the realistic experiment is most

likely carried out at relatively higher temperatures, where the condensate oscillates in the presence of a considerably large fraction of above-condensate atoms. It thus seems timely to develop an extension of these theories to the ?nite temperature. In the present paper we investigate the low-lying collective excitations of a spherically trapped Bose-Fermi mixture at ?nite temperature in the collisionless regime. We con?ne ourselves to the collective modes of the condensate, i.e., the density oscillations of the condensate. We ?rst calculate the mode frequencies by using the simplest temperature-dependent mean-?eld theory — the Popov version of the Hartree-Fock-Bogoliubov (HFB) theory — that has been generalized by us to a trapped BoseFermi mixture to study its thermodynamics [13]. For a purely Bose gas, it is well known that the HFB-Popov theory includes only the static mean-?eld e?ects of the noncondensate atoms [14] and thus predicts the correct mode frequency only at temperatures T < ? T0 = 0.65Tc [15], where Tc is the critical temperature of BEC. Above T0 the noncondensate component becomes considerably large and its dynamics should be treated on an equal footing with that of the condensate [16, 17, 18, 19, 20]. In our case of Bose-Fermi mixtures, the situation is more crucial. Due to the large number of fermions, the coupled dynamics of the condensate, noncondensate, and degenerate Fermi gas has to be taken into account even at zero temperature. In this paper, we shall treat it perturbatively in the spirit of the random phase approximation (RPA) and linear response theory. We derive the explicit expression for the frequency shift and damping rate arising from the coupled dynamics, which in the absence of the Bose-Fermi interaction coincides with the ?nding of Ref. [16]. Based on this expression and the static HFB-Popov theory, we present a detailed numer-

2 ical study of the monopole and quadrupole condensate oscillations against the Bose-Fermi coupling. The dipole excitation is also studied and found to be consistent with the generalized Kohn theorem. The paper is organized as follows. In the next section we derive the theory used in this paper. In Sec. III we apply this theory to mixtures of 41 K?40 K and 87 Rb?40 K, and calculate the dispersion relation of the monopole and quadrupole excitations as a function of the Bose-Fermi coupling. The behavior of monopole and quadrupole modes against temperature is also discussed in detail. Finally, section IV is devoted to conclusions.
II. FORMULATION

gbb = 4π ? h2 abb /mb and gbf = 2π ? h2 abf /mr , to the lowest order in the s-wave scattering length abb and abf , with mr = mb mf /(mb + mf ) being the reduced mass.
A. time-dependent mean-?eld approximation

In this section we ?rst generalize a time-dependent mean-?eld scheme developed by Giorgini [16] to BoseFermi mixtures, and derive the equation for the smallamplitude oscillations of the condensate. Since the formalism of this time-dependent mean-?eld approximation for an inhomogeneous interacting Bose gas has already been presented in detail in Ref. [16], here we shall merely concentrate on the key points, and indicate the necessary modi?cation in the presence of the fermionic component. By means of the RPA and linear response theory, we further consider the ?uctuations of the noncondensate and of the degenerate Fermi gas induced by the condensate oscillations. The back action of these ?uctuations on the condensate motion is then calculated perturbatively to second order in the interaction coupling constant to obtain the explicit expression for frequency shifts and damping rates. Our starting point is the trapped binary Bose-Fermi mixture that is portrayed as a thermodynamic equilibrium system under the grand canonical ensemble whose thermodynamic variables are Nb and Nf , respectively, the total number of trapped bosonic and fermionic atoms, T , the absolute temperature, and ?b and ?f , the chemical potentials. In terms of the creation and annihilation bosonic (fermionic) ?eld operators ψ + (r, t) and ψ (r, t) (φ+ (r, t) and φ(r, t)), the density Hamiltonian of the system takes the form (in units of h ? = 1, and all ?eld operators depend on r and t) H = Hb + Hf + Hbf , Hb = ψ + ? Hf = φ+ ? ▽ gbb + + b + Vtrap (r) ? ?b ψ + ψ ψ ψψ, 2 mb 2 ▽2 f + Vtrap (r) ? ?f φ, 2 mf (1)
2

According to the usual treatment for Bose system with broken gauge symmetry, we shall apply the decomposi?(r, t), where Φ(r, t) ≡ ψ (r, t) tion: ψ (r, t) = Φ(r, t) + ψ represents a time-dependent condensate wave function and allows us to describe situations in which the system is displaced from equilibrium and the condensate is oscillating in time. With this respect the average ... is intended to be a non-equilibrium average, while time-independent equilibrium averages will be indicated in this paper with ?(r, t) plays the the symbol ... 0 . The ?eld operator ψ role of excitations out of the condensate, and by de?ni?(r, t) = 0. This ansatz is tion satis?es the condition ψ then inserted in the equation of motion for ψ (r, t): i ? ψ = ?t ? ▽2 b + Vtrap (r) ? ?b ψ 2 mb (2)

+gbb ψ + ψψ + gbf ψφ+ φ.

Taking a statistical average over Eq. (2) and setting ?+ (r, t)ψ ?(r, t)ψ ?(r, t) and the triplet average values ψ ?(r, t)φ+ (r, t)φ(r, t) to zero [22] thus leads to the folψ lowing equation of motion for the condensate wave function i ? Φ(r, t) = ?t ? ▽2 b + Vtrap (r) ? ?b Φ(r, t) 2 mb
2

+gbb |Φ(r, t)| Φ(r, t) + 2gbb n ? (r, t)Φ(r, t) +gbb m ? (r, t)Φ? (r, t) + gbf nf (r, t)Φ(r, t)(3) , where the densities are de?ned, respectively, as n ? (r, t) ≡ + ? ? ? ? ψ (r, t)ψ (r, t) , m ? (r, t) ≡ ψ (r, t)ψ (r, t) , and nf (r, t) ≡ φ+ (r, t)φ(r, t) . Under the stationary condition, we replace Φ(r, t), n ? (r, t), m ? (r, t) and nf (r, t) by their equilibrium values Φ0 (r) ≡ ψ (r) 0 , n ? 0 (r) ≡ + 0 ? (r)ψ ?(r) , m ?(r)ψ ?(r) ψ ? (r) ≡ ψ and n0 (r) ≡ φ+ (r)φ(r) 0 , respectively. This yields the generalized time-independent Gross-Pitaevskii (GP) equation for Bose-Fermi mixtures [13] ▽2 b n0 + gbf n0 + Vtrap ? ?b + gbb n0 + 2? f Φ0 = 0 , 2 mb (4) 2 where n0 (r) = |Φ0 (r)| is the condensate density. In the above equation, we already use the Popov prescription: m ? 0 (r) = 0, which amounts to neglect the e?ects arising from the equilibrium anomalous density [25]. We are interested in the small amplitude oscillations of the condensate, which is only slightly displaced from its ?
0 0 f

Hbf = gbf ψ + ψφ+ φ.

Here we consider a spherically symmetric system, with b,f 2 static external potentials Vtrap (r) = mb,f ωb,f r2 /2, where mb,f are the atomic masses, and ωb,f are the trap frequencies. The interaction between bosons and between bosons and fermions are described by the contact potentials and are parameterized by the coupling constants

3 stationary value Φ0 (r): Φ(r, t) = Φ0 (r) + δ Φ(r, t), where δ Φ(r, t) is a small ?uctuation. This small oscillations can consequently induce small ?uctuations of the densities around their equilibrium values: n ? (r, t) = n ? 0 (r)+δ n ? (r, t), m ? (r, t) = δ m ? (r, t), and nf (r, t) = n0 ( r ) + δnf (r, t). The f time-dependent equation for δ Φ(r, t) is then obtained by linearizing the equation of motion (3) i ? δ Φ(r, t) = Lδ Φ(r, t) + gbb n0 (r)δ Φ? (r, t) ?t +2gbb Φ0 (r)δ n ? (r, t) + gbb Φ0 (r)δ m ? (r, t) +gbf Φ0 (r)δnf (r, t), (5) where δω represents the shift in the real part of the frequency and γ is the damping rate. The correction of the wave functions in Eq. (9) is chosen to be orthogonal to the unperturbed Bogoliubov quasiparticle wave functions, dr (u? (r) δ Φ1 (r) ? v ? (r)δ Φ? 1 (r)) = 0. (11)

Inserting this perturbation ansatz into Eq. (7) and its adjoint, we multiply the ?rst equation by u? (r) and the latter by v ? (r), and integrate over space. By using Eq. (11) and the normalization condition dr (u? (r) u(r) ? v ? (r)v (r)) = 1, (12)

where we have introduced the Hermitian operator L ≡? ▽ b 0 + Vtrap (r) ? ?b + 2gbb n0 b (r) + gbf nf (r), (6) 2 mb
2

we get the following relation for the eigenfrequency correction: δω ? iγ = drΦ0 (r) [2gbb (u? (r) + v ? (r)) δ n ? (r) +gbb (u? (r)δ m ? (r) + v ? (r)δ m ? ? (r)) ? ? + gbf (u (r) + v (r)) δnf (r)] . (13)

and n0 ? 0 (r), the total density of bosons. b (r) = n0 (r) + n In Eq. (5), the terms containing δ n ?, δm ? and δnf account for the dynamic coupling between the condensate and the ?uctuations of the noncondensate component and of the degenerate Fermi gas. Assuming that the condensate oscillates with frequency ω : δ Φ(r, t) = δ Φ(r)e?iωt and δ Φ? (r, t) = δ Φ? (r)e?iωt (note that δ Φ(r) and δ Φ? (r) are independent), and consequently δ n ? (r, t) = δ n ? (r)e?iωt , ?iωt δm ? (r, t) = δ m ? (r)e and δnf (r, t) = δnf (r)e?iωt , one ?nds ωδ Φ(r) = Lδ Φ(r) + gbb n0 (r)δ Φ? (r) +2gbb Φ0 (r)δ n ? (r) + gbb Φ0 (r)δ m ? (r) +gbf Φ0 (r)δnf (r).

In the next subsection, based on the RPA and linear response theory, we will derive the explicit expressions for δn ? (r), δ m ? (r), δ m ? ? (r) and δnf (r), which are induced by the condensate oscillations. An alternative way to get these expressions in case of pure Bose gases has been outlined by Giorgini in Ref. [16]. Our derivation presented below is somewhat simpler and more transparent in physics.
B. RPA and linear response theory

(7)

In the absence of coupling terms, Eq. (7) and its adjoint are formally equivalent to the time-independent Bogoliubov-deGennes (BdG) equations [13]
B ωi

ui (r) ?vi (r)

=

L gbb n0 (r) gbb n0 (r) L

ui (r) vi (r)

,

Let us consider the interaction terms in the density Hamiltonian (1) that couples the condensate wave function to the noncondensate component and the degenerate Fermi gas Hint = gbb 2
2 ?+ ?(r, t) dr 4 |Φ(r, t)| ψ (r, t)ψ

(8) which de?ne the Bogoliubov quasiparticle wave functions B ui (r) and vi (r) with excitation energies ωi . This equivalence is not surprising since the Bose broken symmetry leads quite generally to the one-one correspondence between the small oscillations of the condensate and the single-quasiparticle wave functions [26]. For the purpose of solving Eq. (7), to leading order of the two coupling constants [27], we thus can select the Bogoliubov quasiparticle wave functions corresponding to the low-energy collective mode that we are interested, and set accordingly δ Φ0 (r) = u(r), δ Φ? 0 (r) = v (r) and ω = ω0 . The ?rst-order correction due to the ?uctuations δ n ? (r), δm ? (r) (and its complex conjugate), and δnf (r), can be calculated by expanding δ Φ(r) δ Φ? (r) = u(r) v (r) + δ Φ1 (r) δ Φ? 1 (r) , (9) (10)

?(r, t)ψ ?(r, t) + Φ2 (r, t)ψ ?+ (r, t)ψ ?+ (r, t) +Φ?2 (r, t)ψ +gbf dr |Φ(r, t)|2 φ+ (r, t)φ(r, t). (14)

Consistent with setting the triplet averages to zero in derivation of Eq. (3), we have dropped the terms linear in Φ(r, t). In the spirit of RPA, by linearizing the above interaction Hamiltonian, we identify the perturbation induced by small amplitude-oscillations of the condensate (the e?iωt dependence is not shown explicitly):
perb Hint = gbb

?+ (r, t)ψ ?(r, t) drΦ0 (r) 2 (u(r) + v (r)) ψ

?(r, t)ψ ?(r, t) + u(r)ψ ?+ (r, t)ψ ?+ (r, t) +v (r)ψ +gbf drΦ0 (r) (u(r) + v (r)) φ+ (r, t)φ(r, t (15) ),

ω = ω0 + δω ? iγ,

4 where to the leading order we have replaced δ Φ(r, t) one ?nds to second order of gbb and gbf , and δ Φ? (r, t), respectively, by u(r)e?iωt and v (r)e?iωt . |Aij |2 B 2 Within the linear response theory, the ?uctuations are fiB ? fj δω ? iγ = 4gbb B ? ωB ω + + ωi given by j ij ?? ? ? ? ? 2Φ0 (u + v ) χn χn χn δn ? ?n ? ?m ? ?m ?+ |Bij |2 B 2 ?? ? ? δm 1 + fiB + fj +2gbb Φ0 v χm χm ? ? = gbb ? χm ?n ? ?m ? ?m ?+ B + ωB ω + ? ωi j ij Φ0 u χm δm ?? ? +n ? χm ? +m ? χm ? +m ?+ ? 2 (16) ?ij B ? and ? + B + ωB ? ω + ω j i δnf = gbf dr′ χf (r, r′ ; ω ) Φ0 (r′ ) (u(r′ ) + v (r′ )) . (17) Here we de?ne χαβ Φ0 u ≡ dr′ χαβ (r, r′ ; ω ) Φ0 (r′ )u(r′ ) and χαβ Φ0 v ≡ dr′ χαβ (r, r′ ; ω ) Φ0 (r′ )v (r′ ) in Eq. (16), where the indices α, β = n ? , m, ? or m ? + . χαβ and χf are the usual two-particle correlation functions for the Bose and Fermi gas [28]. By using Wick’s theorem, they can be easily expressed in terms of the quasiparticle energies and wave functions. For instance, for χn ?n ? , with the help of the Bogoliubov transformation we can write B B ? iωi t ?(r, t) = in terms of αi e?iωi t + vi (r)? α+ ψ i e i ui (r)? the Bogoliubov quasiparticle operators α ? i and α ?+ i . It is then straightforward to obtain χn ?n ? =
(1) χn ?n ? (1) χn ?n ? 2 +gbf F fiF ? fj ij

|Cij | , F ? ωF + ω + ωi j

2

(21)

?ij , and Cij are, where the matrix elements Aij , Bij , B respectively, given by Aij =
? ? drΦ0 u ui u? j + vi vj + vi uj ? ? +v ui u? j + vi vj + ui vj

,

Bij =

? ? ? ? ? drΦ0 u u? i vj + vi uj + ui uj ? ? ? ? ? +v u? i vj + vi uj + vi vj

(r, r ; ω ) +



(2) χn ?n ?

,

(r, r ; ω ) ,
B fiB ? fj



(18) ,

=
ij

? ? ? (u? i uj + vi vj ) ui uj + vi vj B ? ωB ω + + ωi j

?ij = B

drΦ0 [u (ui vj + vi uj + vi vj ) +v (ui vj + vi uj + ui uj )] ,

(2) χn ?n ?

1 = 2 ?

? ? ? (ui vj + vi uj ) u? i vj + vi uj ij ? u? i vj B + ωB ω + ? ωi j

B 1 + fiB + fj

Cij =

drΦ0 (r) (u (r) + v (r)) ?i (r) ?? j (r) . (22)

+

? ? vi uj

(ui vj + vi uj ) 1 +

fiB

+

B fj

B + ωB ω + + ωi j

,

where ω + = ω + i0+ . We have used the abbrevia? ? ′ ? ′ B tion: u? i uj ui uj = ui (r)uj (r)ui (r )uj (r ), etc., and fi = α ?+ ?i i α
0

= 1/ eβωi ? 1 is the Bose-Einstein distribu(1) (2)

B

tion function with β = 1/kB T . χn ?n ? correspond, ?n ? and χn respectively, to the excitation of single quasiparticles and of pairs of quasiparticles. For χf , we have χf =
ij
F

F fiF ? fj

′ ? ′ ?? i (r)?j (r)?i (r )?j (r ) F ? ωF ω + + ωi j

,

(19)

where fiF = 1/ eβωi + 1 is the Fermi-Dirac distribution, and the single-particle wave function ?i (r) satis?es the stationary Schr¨ odinger equation [13] ? ▽2 f F + Vtrap (r) ? ?f + gbf n0 b (r) ?i = ωi ?i . (20) 2 mf
C. eigenfrequency correction

Eqs. (21) and (22) are the main result of this section. Without the fermionic component, these equations coincides with the ?nding obtained by Giorgini (the Eqs. (39) and (40) in the second paper of Ref. [16] ) as they should be. The last term in the right-hand side of Eq. (21) is novel and arises from the many possibilities of independent particle-hole excitations [29]. This mechanism is known as Landau damping due to the Bose-Fermi coupling. On the other hand, the ?rst and second terms in the right-hand side of Eq. (21) correspond, respectively, to the Landau and Beliaev processes due to the interaction between bosons [16]. One of the advantages of our derivation presented here is that to obtain the eigenfrequency correction we don’t need to impose any constraint used in solving the equilibrium problem, i.e., the Popov prescription m ? 0 (r) = 0. For a pure Bose gas at high temperatures close to Tc , it might be reasonable to use the Hartree-Fock spectrum for χαβ . As a result, only χn ?n ? is nonzero and the eigenfrequency correction reads
2 δω ? iγ = 4gbb

dr

dr′ Φ0 (r) (u? (r) + v ? (r))

′ ′ ′ ′ × χn (23) ?n ? (r, r ; ω ) Φ0 (r ) (u (r ) + v (r )) ,

Substituting the ?uctuations (16) and (17) into Eq. (13), and using the explicit expressions for χαβ and χf ,

which is identical to the ?nding of Reidl et al. obtained by using the dielectric formalism (the Eq. (52) in Ref. [20]), if one notices that δnc (r) = Φ0 (r) (u (r) + v (r)).

5 It should be noted that the second term in the righthand side of Eq. (21), corresponding to the Beliaev process, is ultraviolet divergent. This re?ects the fact that the contact interaction is an e?ective low-energy interaction invalid for high energies. One way to remove this divergence is to express the coupling constant gbb in terms of the two-body scattering matrix obtained from the Lippman-Schwinger equation. This renormalization scheme has been put forward in Ref. [16], however, only valid in the thermodynamic limit, where the ThomasFermi approximation can be implemented [16]. In our derivation, one can explicitly show that such divergence ′ comes from the two correlation functions: χm ?m ? + (r, r ; ω ) ′ and χm ? +m ? (r, r ; ω ). One thus may wish to remove the divergence by regularizing χm ?m ? + and χm ? +m ? in real space in a way similar to that described in Refs. [30, 31]. In this paper, for simplicity we shall neglect the second term in the right-hand side of Eq. (21), since it is always very small compared with other two terms at all temperatures. This treatment is well justi?ed by the excellent agreement between the experimental result [32] and the theoretical prediction by Reidl et al. [20] for a pure Bose gas, where in the theoretical calculations the Beliaev process is completely ignored. There is one last technical issue to resolve: concerning the damping rate, the terms in Eq. (21) involve a sum over many δ functions in energy, which, if interpreted exactly, will tend to be null for discrete quasiparticle states. Here we shall adopt the strategy of Ref. [33] and use an expression with a Lorentz pro?le factor in place of the energy δ function
III. NUMERICAL RESULTS

In this work we analyze the low-lying condensate oscillations of a Bose-Fermi mixture for varying Bose-Fermi coupling constant and temperature in an isotropic harmonic trap, for which the order parameter Φ0 (r), the Bogoliubov quasiparticle amplitudes ui (r) and vi (r), and the orbits ?i (r) can be classi?ed according to the number of nodes in the radial solution n, the orbital angular momentum l, and its projection m. For this sake, we apply the theory developed in the proceeding section to mixtures of 41 K?40 K and 87 Rb?40 K. For the former system, we discuss some generic properties of Bose-Fermi mixtures. In this case the rigid oscillation of the center of mass, or the dipole mode, will also be an eigenstate of the many-body system. As a result, the oscillation frequency will be ?xed at the bare trapping frequency, regardless of any interactions. The ful?llment of this property, which is usually referred to as the generalized Kohn theorem, thus provides us a stringent test on the correctness of our results. The second choice of 87 Rb?40 K mixture corresponds to a speci?c example available in present experiments [6]. In this case we include explicitly the mass di?erence and the di?erent oscillator frequencies of the trapping potentials for the two species, and build the possible experimental relevance of our results.
A.
41

K?40 K

2 δω ? iγ = 4gbb ij 2 +gbf ij

B |Aij | fiB ? fj

2

B ? ω B + iγ ω0 + δω + ωi j F |Cij | fiF ? fj 2

F ? ω F + iγ ω0 + δω + ωi j

,(24)

which can be solved iteratively for δω and γ . This expression can be formally obtained from Eq. (21) by assuming that the perturbed resonance frequency ω is distributed over a range of values characterized by a Lorentz pro?le with a width γ . The structure of our calculation is then as follows: First we solve the unperturbed equilibrium problem for ω0 , u(r) and v (r). This step requires solving a closed set of Eqs. (4), (8), and (20), which we have referred to as the “HFB-Popov” equations for dilute Bose-Fermi mixtures. We already have reported on our self-consistent algorithm in Ref. [13] for this problem. As a result, B , we have all the necessary inputs, namely, Φ0 (r), ωi F ωi , ui (r) and vi (r), and ?i (r) for performing the second step: the use of Eq. (24) in connection with the matrix elements Aij and Cij de?ned in Eq. (22).

We ?rst consider a mixture of 2 × 104 41 K (boson) and 2 × 104 40 K (fermion) atoms with the following set of parameters: mb = mf = 0.649 × 10?25 kg, ωb = ωf = 2π × 100 Hz, abb = 286a0 = 15.13 nm [34], where a0 = 0.529 ? A is the Bohr radius. We also express the lengths and energies in terms of the characteristic oscillator length ab h/mb ωb )1/2 and characteristic trap ho = (? energy h ? ωb , respectively. In Figs. (1a) and (1b), we present, respectively, our results for the monopole (l = 0) and quadrupole (l = 2) condensate oscillations at a very low temperature T = 1/3 0 0 0.01Tc , where Tc = 0.94ωb Nb /kB ≈ 122 nK is the critical temperature for an ideal Bose gas in the thermodynamic limit. The mode frequencies, in units of the bare trapping frequency, are plotted as a function of the BoseFermi coupling constant measured relative to the BoseBose coupling constant, κ = gbf /gbb . The lines with open circles show the unperturbed frequencies obtained from the HFB-Popov equations, ω0 , while the lines with solid circles denote the values after correction, ω = ω0 + δω . For comparison, the predictions of the scaling theory at zero temperature are also plotted by the dashed lines [8]. At this low temperature, the eigenfrequency shift δω arises mainly from the dynamics of the degenerate Fermi gas. For small values of |κ| < ? 1, δω is negligibly small due to square dependence on the Bose-Fermi coupling constant. However, as |κ| increases δω becomes remarkable. In particular, the corrected frequency for the

6
3.0 2.8 2.2
1.4

3.0

2.2

ωD / ωb

δω / ω0

0 .00

1.0

ωM / ωb

ωQ / ωb

ωM / ωb

2.6

1.8

0.8

2.6

ωQ / ωb

-0 .05 -0 .10 -2 0 2 4 6

1.8

-2

0

2.4 2.2 2.0

1.6 1.4 1.2

κ

2

4

δω / ω 0

2.0

1.2

2.8

(a)

0 .05

2.0

(b)

0 .00 -0 .05 -0 .10 -0 .15 -2 0 2 4 6

2.4 2.2 2.0

κ

1.6 1.4 1.2

κ

(a)
-2 0

(b)
-2 0

κ

2

4

6

κ

2

4

6

-2

0

κ

2

4

6

-2

0

κ

2

4

6

FIG. 1: The dispersion relation of the monopole (l = 0) and quadrupole (l = 2) condensate oscillations for a mixture composed of 2 × 104 41 K (boson) and 2 × 104 40 K (fermion) atoms 1/3 0 0 at T = 0.01Tc , where Tc = 0.94ωb Nb /kB ≈ 122 nK is the critical temperature for an ideal Bose gas in the thermodynamic limit. The mode frequencies, in units of the bare trapping frequency, are plotted as a function of the reduced Bose-Fermi coupling, κ = gbf /gbb . The lines with open circles show the unperturbed frequencies calculated by the static HFB-Popov equations ω0 , while the lines with solid circles denote ω = ω0 + δω . For comparison the predictions of the scaling theory in Ref. [8] are also plotted by the dashed lines. The inset in (b) shows the dispersion relation of the dipole excitation. The other parameters used in the numerical calculation are: mb = mf = 0.649 × 10?25 kg, ωb = ωf = 2π × 100 Hz, and abb = 286a0 = 15.13 nm, where a0 = 0.529 ? A is the Bohr radius.

0 FIG. 2: The same as in FIG. 1, but for T = 0.75Tc . In the insets the solid and dashed lines show, respectively, the fractional shift δω/ω0 due to the ?rst and second terms in Eq. (24).

0.3

0.3

(a)
γbb
0.2

(b)
γQ / ωb
γbf γbb
0.2

γM / ωb

γbf

0.1

0.1

0.0

-2

0

κ

2

4

6

0.0

-2

0

κ

2

4

6

quadrupole mode decreases with increasing Bose-Fermi coupling constant up to κ ≈ 3, at which a sharp upturn occurs. This sharp dip is accompanied by a dramatic increase of damping rates (not shown in the ?gure), and can be well interpreted as a signal to approach the spatial separation (demixing) point of the two species. On the contrary, the unperturbed quadrupole frequency ω0 has a qualitatively di?erent behavior against κ: it shows a parabolic dependence with a minimum located around κ = 0.5. In spite of its large value at |κ| > ? 1, δω is still much smaller than unperturbed frequency ω0 for all the calculated points. Therefore, the criterium for the applicability of the perturbation theory in Eqs. (9) and (10) is justi?ed. In the inset of Fig. (1b) we also show the result for the dipole mode (l = 1). As we can see, although ω0 deviates signi?cantly from the bare trapping frequency ωb for relatively small values of κ, ω still persists at ωb < over a wide range of κ (i.e., ?2 < ? κ ? +3). This is consistent with the generalized Kohn theorem. As a result, the correctness of our theory and numerical calculations is partly checked. The eigenfrequency shifts are a?ected by the temperature. In Fig. 2, we report the mode frequencies against 0 κ at a high temperature T = 0.75Tc , where the condensate oscillates in the presence of a large fraction of abovecondensate atoms. Compared with the results for the low temperature, the eigenfrequency shifts are considerably

FIG. 3: The damping rates of the monopole (a) and 0 quadrupole condensate oscillations (b) at T = 0.75Tc . The solid and dashed lines correspond to the contribution from the Landau process due to the Bose-Bose interaction and due to the Bose-Fermi coupling, respectively. The line with solid circles is the total contribution.

reduced. The sharp dip at κ ≈ 3 for the quadrupole mode also becomes much broader. Moreover, in the absence of the Bose-Fermi coupling, the eigenfrequency shift is nonzero for the quadrupole mode. This is caused by the dynamics of the noncondensate component as shown in the inset of Fig. (2b), where the solid and dashed lines depict, respectively, the fractional shift δω/ω0 due to the Bose-Bose interaction and Bose-Fermi coupling (or, in other words, due to the ?rst and second terms in Eq. (24)). The damping rate of condensate oscillations at ?nite temperature deserves its own study. In Figs. (3a) and (3b), we show, respectively, our predictions on the damping rates of monopole and quadrupole oscillations at 0 T = 0.75Tc . The lines with solid circles are the sum over two contributions: one is the Landau damping due to the Bose-Bose interaction (the solid lines), γbb , and the other is the Landau damping due to the Bose-Fermi coupling (the dashed line), γbf . For the monopole mode, the essential feature is the decrease of the damping rate

7
3 2 1 10 γM, ij
3 3

(a)
ωM κ = -2

ωD / ωb

0.3 0.2 0.1 10 γQ, ij 0.3 0.2 0.1 3.0 0.0 1.0 1.5
ωQ ωQ

(b)
κ = -2

2.8

1.8

(a)
2.6

1 .4 1 .2 1 .0 0 .8 0 .0 0 .5 1 .0

(b)
1.6

ωM / ωb

3 2 1 0 1.5 2.0
κ = +6 ωM

T / TC
2.4

0

ωQ / ωb
1.4 1.2 0.0

κ = +6

2.2 0.0

0.5

1.0
0

0.5

1.0
0

ωij

2.5

ωij

2.0

2.5

T / TC

T / TC

γM / ωb

0.1

γQ / ωb

FIG. 4: The damping strength γij de?ned in Eq. (26) as a function of the transition frequencies ωij (in units of ωb ) bb allowed by the selection rules. (a) γij for the monopole exbf citation and (b) γij for the quadrupole excitation. In each subplot, upper and lower panels correspond to the case of κ = ?2 and κ = +6, respectively. In the former case the mixture is in mixed regime, while in the latter the bosonic and fermionic density pro?les separate in space. The arrows point to ω = ω0 + δω .

FIG. 5: The dispersion relation of the monopole (a) and quadrupole excitations (b) against the reduce temperature 0 T /Tc at κ = ?2. The inset in (a) shows the dispersion relation for the dipole mode.
0.2 0.2

(a)
γbb γbf
0.1

(b)
γbb γbf

as κ increases towards the demixing point. This decrease is mainly attributed by γbb , and re?ects the reconstruction of the bosonic monopole-excitation spectrum across the demixing point. To better illustrate this point, we rewrite the expression for γ in the following form γ = γbb + γbf ,
2 γbb = 4gbb ω0 ij 2 γbf = gbf ω0 ij bf γij bb γij

0.0 0.0

0.5

1.0
0

0.0 0.0

0.5

1.0
0

(25) γ/π
B ? ωB ω0 + δω + ωi j 2

T / TC

T / TC

+ γ2 ,

,

FIG. 6: The same as in FIG. 5, but for the damping rates.

γ/π
F ? ωF ω0 + δω + ωi j 2

+ γ2

where the “damping strength”
bb γij = bf γij

π 2 B , |Aij | fiB ? fj ω0 π 2 F = , |Cij | fiF ? fj ω0

(26)

have the dimensions of a frequency. In Fig. (4a), we plot bb B B γij against the transition frequency ωij = ωj ? ωi >0 allowed by the selection rules for κ = ?2 and κ = 6. For the latter value of κ, the overlap between the bosonic and fermionic cloud is very small, and the mixture is deep into the demixing regime. Compared with the mixing case of κ = ?2, the region of transition frequencies at κ = 6 narrows, and its center moves to the low-energy side. Contrarily the calculated ω0 + δω has a blue shift and is completely out of the transition region. As a consequence, the condensate oscillation is not damped by the Landau process due to the Bose-Bose interaction. For the quadrupole mode, instead we observe that the damping rate increases as the mixture moves towards the demixing point with increasing κ. This trend comes from the

increase of γbf , and re?ects, on the other hand, the reconstruction of the fermionic quadrupole-excitation spectrum. As shown in Fig. (4b), with increasing Bose-Fermi bf coupling the damping strength γij becomes larger and denser. Accordingly, the condensate oscillation is heavily damped by generating many particle-hole excitations. The last study in this subsection concerns the temperature dependence of the eigenfrequency shifts and damping rates at a speci?c Bose-Fermi coupling constant. In Figs. (5a) and (5b), we report, respectively, our results for the monopole and quadrupole mode frequencies as 0 a function of the reduced temperature T /Tc at κ = ?2. The corresponding damping rates are shown in Figs. (6a) and (6b). The ful?llment of the generalized Kohn theorem is checked in the inset of Fig. (5a), where the calculated dipole frequency is very close to ωb (or, more precisely, 0.97 ≤ ωD /ωb ≤ 1.0) for all the temperatures considered. At this speci?c value of κ, one can see that both the monopole and quadrupole frequencies have a downshift with increasing temperature, analogous to the results obtained for a pure Bose gas [16, 20]. In addition, the behavior of the damping rates is also qualitatively similar [20, 33, 35].

8
3.0 2.2

δω / ω 0

2.8 2.6

0 .00

2.0 1.8

-0 .05 -0 .10

ωM / ωb

ωQ / ωb

-5

0

2.4 2.2 2.0

κ

5

10

δω / ω0

(a)

0 .05

(b)

0 .0 0 -0 .0 5 -0 .1 0 -0 .1 5 -5 0 5 10

1.6 1.4 1.2

κ

-5

0

κ

5

10

-5

0

κ

5

10

FIG. 7: The dispersion relation of the monopole and quadrupole condensate excitations for a mixture consisting 0 of 2 × 104 87 Rb and 2 × 104 40 K atoms at T = 0.75Tc , where 0 Tc ≈ 112 nK. The other parameters used in the calculation are: mb = 1.45× 10?25 kg, ωb = 2π × 91.7 Hz, mf /mb = 0.463, ωf /ωb = 1.47, and abb = 99a0 = 5.24 nm.
0.15 0.15

(a)
γbb

(b)
γQ / ωb
γbf
0.10

0 where Tc ≈ 112 nK. Both the monopole and quadrupole frequencies decrease slowly with increasing κ up to κ ≈ 5. Above this value the frequencies gradually rise up. In the whole region of κ, the variation of frequencies due to Bose-Fermi coupling is small. However, it is still possible to be detected by the accurate frequency measurement. For instance, at κ = ?6, we ?nd that the values of the relative variation (ωκ=?6 ? ωκ=0 ) /ωκ=0 for the monopole and quadrupole modes are, respectively, 2.8% and 7.6%, well within the experimental resolution. The damping rate of the monopole and quadrupole modes at the same temperature is shown in Figs (8a) and (8b), respectively. The behavior of the damping rate against κ, that is, the decrease (increase) of the monopole (quadrupole) damping rate across the demixing point κ ≈ 5, is very similar to that in Figs. (3a) and (3b), except that the overall magnitude is two times smaller. This behavior together with the slow rise up of the mode frequency around κ ≈ 5 thus may provide us a useful signal to locate the onset of the phase transition of spatial separation.

γbb γbf

γM / ωb

0.10

IV.
0.05

CONCLUDING REMARKS

0.05

0.00

-5

0

κ

5

10

0.00

-5

0

κ

5

10

FIG. 8: The same as in FIG. 7, but for the damping rates. B.
87

Rb?40 K

We now turn to consider a 87 Rb?40 K mixture composed of 2 × 104 bosonic and 2 × 104 fermionic atoms under the conditions appropriate to the LENS experiments [6]. As in experiment, we introduce the quantities α = mf /mb = 0.463 and β = ωf /ωb = 1.47 to parameterize the di?erent mass and di?erent trapping frequency of the two species, which satisfy the constraint αβ 2 = 1 since both bosons and fermions experience the same trapping potential. In addition, we take the s-wave Bose-Bose scattering length abb = 99a0 = 5.24 nm [36], and ?x the trapping frequency ωb = 2π × 91.7 Hz, which is the geometric average of the axial and radial frequencies of Ref. [6]. The s-wave Bose-Fermi scattering length is varying, and in the experiment it can be conveniently tuned by the Feshbach resonance [37]. Notice that the calculations presented here are restricted to the isotropic traps, opposite to the cylindrical symmetric traps used in experiments. As a result, our results are only useful in a qualitatively level. In Fig. 7, we plot the frequencies for the monopole and 0 quadrupole oscillations as a function of κ at T = 0.75Tc ,

In this paper we have developed a theory for studying the low-lying condensate oscillations of a spherically trapped Bose-Fermi mixture at ?nite temperature in the collisionless regime. In this theory, the unperturbed mode frequency is ?rstly calculated within the static Hartree-Fock-Bogoliubov-Popov approximation. The frequency correction, arising from the coupled dynamics of the condensate, noncondensate, and degenerate Fermi gas, is then taken into account perturbatively by means of the random phase approximation. We have applied our theory to the mixtures of 41 K?40 K and 87 Rb?40 K, and have studied the dispersion relation of the monopole and quadrupole condensate excitations as a function of the Bose-Fermi coupling at various temperatures. The correctness of our theory and numerical calculations is partly checked by the ful?llment of the generalized Kohn theorem for the dipole excitation. At a relatively high temperature we ?nd that, as the mixture moves towards demixing point with increasing Bose-Fermi coupling, the damping rate of the monopole (quadrupole) excitation increases (decreases). This behavior provides us a possible signature to identify the phase transition of spatial separation.

Acknowledgments

We are very grateful to Dr. M. Modugno and Dr. G. Modugno for simulating discussions. X.-J.L was supported by the K.C.Wong Education Foundation, the Chinese Research Fund, and the NSF-China under Grant No. 10205022.

9

[1] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science 269, 198 (1995). [2] K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995). [3] C. C. Bradley, C. A. Sackett, J. J. Tollett, R. G. Hulet, Phys. Rev. Lett. 75, 1687 (1995). [4] A. G. Truscott, K. E. Strecker, W. I. McAlexander, G. B. Partridge, and R. G. Hulet, Science 291, 2570 (2001); F. Schreck, L. Khaykovich, K. L. Corwin, G. Ferrari, T. Bourdel, J. Cubizolles, and C. Salomon, Phys. Rev. Lett. 87, 080403 (2001). [5] Z. Hadzibabic, C. A. Stan, K. Dieckmann, S. Gupta, M. W. Zwierlein, A. Gorlitz, and W. Ketterle, Phys. Rev. Lett. 88, 160401 (2002). [6] G. Roati, F. Riboli, G. Modugno, and M. Inguscio, Phys. Rev. Lett. 89, 150403 (2002); G. Modugno, G. Roati, F. Riboli, F. Ferlaino, R. J. Brecha, and M. Inguscio, Science 297, 2240 (2002). [7] T. Miyakawa, T. Suzuki, and H. Yabu, Phys. Rev. A 62, 063613 (2000). [8] X.-J. Liu and H. Hu, Phys. Rev. A 67, 023613 (2003). [9] P. Capuzzi and E. S. Hern? andez, Phys. Rev. A 64, 043607 (2001). [10] T. Sogo, T. Miyakawa, T. Suzuki, and H. Yabu, Phys. Rev. A 66, 013618 (2002). [11] A. Minguzzi and M. P. Tosi, Plys. Lett. A 268, 142 (2000). [12] P. Capuzzi, A. Minguzzi, and M. P. Tosi, LANL Preprint cond-mat/0301522 (2003). [13] H. Hu and X.-J. Liu, LANL Preprint cond-mat/0302570 (2003); to be published in Phys. Rev. A. [14] A. Gri?n, Phys. Rev. B 53, 9341 (1996); D. A. Hutchinson, E. Zaremba and A. Gri?n, Phys. Rev. Lett. 78, 1842 (1997). [15] R. J. Dodd, M. Edwards, C. W. Clark, and K. Burnett, Phys. Rev. A 57, R32 (1998). [16] S. Giorgini, Phys. Rev. A 61, 063615 (2000); ibid. 57, 2949 (1998). [17] M. J. Bijlsma and H. T. C. Stoof, Phys. Rev. A 60, 3973 (1999); U. Al Khawaja and H. T. C. Stoof, ibid. 62, 053602 (2000). [18] E. Zaremba, A. Gri?n, and T. Nikuni, Phys. Rev. A 57, 4695 (1998); E. Zaremba, T. Nikuni, and A. Grif?n, J. Low Temp. Phys. 116, 277 (1999); T. Nikuni, E. Zaremba, and A. Gri?n, Phys. Rev. Lett. 83, 10 (1999). [19] P. O. Fedichev, G. V. Shlyapnikov, and J. T. M. Walraven, Phys. Rev. Lett. 80, 2269 (1998); P. O. Fedichev and G. V. Shlyapnikov, Phys. Rev. A 58, 3146 (1998). [20] J. Reidl, A. Csord? as, R. Graham, and P. Sz? epfalusy, Phys. Rev. A 61, 043606 (2000). [21] J. E. Williams and A. Gri?n, Phys. Rev. A 63, 023612 (2001); ibid. 64, 013606 (2001). [22] The terms cubic in the ?eld operators in the equation of motion (2) take the form: ψ + ψψ = |Φ|2 Φ + ? + Φ2 ψ ?+ + 2Φψ ?+ ψ ? + Φ? ψ ?ψ ?+ ψ ?+ ψ ?ψ ?, and ψφ+ φ = 2 |Φ|2 ψ + + ? Φφ φ + ψφ φ. From the semiclassical point of view, the ?+ ψ ?ψ ? (or ψφ ? + φ), accubic product of the operator, ψ counts for the collisions involving the condensate and noncondensate atoms (or fermions) [18]. In the collisionless regime we may assume that these products have only

[23] [24] [25]

[26]

[27] [28] [29]

[30]

[31]

[32] [33] [34] [35] [36]

a negligible e?ect on the dynamics of the condensate and we may safely set the triplet average value to zero: ?+ (r, t)ψ ?(r, t)ψ ?(r, t) = 0 and ψ ?(r, t)φ+ (r, t)φ(r, t) = ψ 0. On the other hand, for a pure Bose gas the damping due to collisions has been discussed by Williams and Grif?n in Ref. [21]. For typical trap parameters, it is found to be 2 or 3 times smaller than the Landau damping arising from the dynamical mean-?eld e?ect as discussed in the present paper. The collisional frequency shift is also considered in these papers, and found to vanish in the ?rst order perturbation treatment. V. N. Popov, Functional Integrals and Collective Excitations (Cambridge University Press, Cambridge, 1987). S. Giorgini, L. P. Pitaevskii, and S. Stringari, Phys. Rev. A 54, R4633 (1996); Phys. Rev. Lett. 78, 3987 (1997). This approximation was ?rst used by Popov in the study of a homogeneous gas to discuss the ?nite temperature region close to the Bose-Einstein transition [23]. More recently, the Popov approximation has been used extensively in the study of properties of magnetically trapped Bose gases at ?nite temperature [14, 15, 24]. This Popov approximation gives a gapless spectrum of elementary excitations at long wavelengths and formally reduces to the Bogoliubov approximation at zero temperature, where n ? 0 (r) also becomes negligible. While at high temperatures, it approaches the ?nite-temperature Hartree-Fock spectrum. Therefore it is expected to give a reasonable ?rst approximation for the excitation spectrum in Bose gases at all temperatures [14]. A. Gri?n, Excitations in a Bose-Condensed Liquid (Cambridge, New York, 1993), Chap. 5, and references therein. As we shall see, if we take gbb and gbf as small parameters, δn ? (r), δ m ? (r), and δnf (r) are an order smaller than δ Φ(r). A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems (McGraw Hill, New York, 1971). In other words, it arises from the process of one quantum of the condensate oscillation ω being absorbed by a F fermion below the Fermi level with energy ωi , which is turned into a fermion above the Fermi level with energy F F + ω. = ωi ωj G. M. Bruun, Y. Castin, R. Dum, and K. Burnett, Eur. Phys. J. D 9, 433 (1999); G. M. Bruun and B. R. Mottelson, Phys. Rev. Lett. 87, 270403 (2001). That is, the divergence is removed by the substitutions, ′ ′ irr ′ χm ?m ? + (r, r ; ω ) ? δ (r ? r )G0 (r), and ?m ? + (r, r ; ω ) → χm ′ ′ ′ irr χm ? +m ? (r, r ; ω ) → χm ? +m ? (r, r ; ω ) ? δ (r ? r )G0 (r). Here Girr 0 (r) is the singular part of the ideal single-particle Green’s function G0 (r + x/2, r ? x/2; ω = 0) that diverges as 1/x for x → 0 [30]. D. S. Jin, M. R. Matthews, J. R. Ensher, C. E. Wiemann, and E. A. Cornell, Phys. Rev. Lett. 78, 764 (1997). K. Das and T. Bergeman, Phys. Rev. A 64, 013613 (2001). R. C? ot? e, A. Dalgarno, H. Wang, and W. C. Stwalley, Phys. Rev. A 57, R4118 (1998). M. Guilleumas and L. P. Pitaevskii, Phys. Rev. A 61, 013602 (2000). E. G. M. van Kempen, S. J. J. M. F. Kokkelmans, D. J. Heinzen, and B. J. Verhaar, Phys. Rev. Lett. 88, 093201 (2002).

10
[37] A. Simoni, F. Ferlaino, G. Roati, G. Modugno, and M. Inguscio, Phys. Rev. Lett. 90, 163202 (2003).


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