Phonon spectrum and damping of a dilute quantum degenerate Bose–Fermi mixture at finite temperature

Physics Letters A 340 (2005) 161–169 www.elsevier.com/locate/pla

Phonon spectrum and damping of a dilute quantum degenerate Bose–Fermi mixture at finite temperature M.A. Shahzamanian, H. Yavary ∗ Department of Physics, University of Isfahan, Isfahan 81744, Iran Received 24 August 2004; received in revised form 14 March 2005; accepted 26 March 2005 Available online 8 April 2005 Communicated by A.R. Bishop

Abstract In this Letter we investigated Bose–Fermi gas mixture at finite temperature by using Green’s function method. First for spinpolarized system the temperature effects on the phonon spectrum have been taken into account. Second we consider a dilute mixture of a Bose gas and a two components Fermi gas when both bosons and fermions have undergone superfluid transitions. In this case the interaction between two hyperfine Fermi components is important. In addition we calculate the dispersion of phonon excitations, including correction to the sound velocity and Landau damping. The damping rate varies as T 2 in low temperatures limit. There is also damping at absolute zero temperature due to the interaction between bosons and fermions.  2005 Elsevier B.V. All rights reserved. PACS: 03.75.Fi; 05.30.Fk; 67.55.Jd Keywords: Bose; Fermi; Mixture; Damping; Spectrum

1. Introduction Recent experimental progress in atomically trapped gases has lead to a resurgence of interest in quantum fields. A particular notable feature is the number of systems available ranging from single component Bose gas in the original experiments, where Bose–Einstein condensation was first achieved [1] to a binary Bose mixture [2], a spinor condensate in optical traps [3], and a degenerate Fermi gas [4,5]. Other systems also received much recent attention, in particular, a Bose–Fermi mixture. This last, mentioned system occurs naturally if “sympathetic cooling” is employed to reduce the kinetic energy of the fermions [6]. Direct evaporative cooling is not applicable in a gas of spin-polarized fermions since s-wave interactions are suppressed by the Pauli principle. So the Bose gas, which * Corresponding author.

E-mail address: [email protected] (H. Yavary). 0375-9601/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.03.056

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can be cooled evaporatively, is used as a coolant. The fermionic cloud remains in thermal equilibrium with the cold Bose gas through boson–fermion interactions in the region where both clouds overlap. In this way the fermionic species is cooled sympathetically. The two-component BEC mixture of dilute gases is interesting from both theoretical and experimental viewpoints. Experiments have been conducted to study the creation of topological excitations in two-component BEC gas [7], quantum tunneling effects [8], metastable effects [9], Rabi oscillations [2], the dynamics of component separations [11], and relative phase coherence [12] in a binary mixture of Bose gases. Theoretical work on trapped two-component BEC gases has included the static and stability properties [13,14], the dynamics of the relative phase [15,16], the collective modes [17], and the phase diagrams and collective modes for spinor BEC [18]. As to the Bose–Fermi mixture, the density profiles of the mixture trapped in a harmonic potential at nonzero temperature under the Thomas–Fermi approximation [19] have been investigated, and the effects of phonon exchange on the fermion–fermion interactions strength [20] are discussed at temperatures below the BEC transition. The effect of boson–fermion interactions on the phonon excitation spectrum has been investigated in Refs. [10,21]. In most of the studies on the stability of the Bose–Fermi mixture, the effects of temperature have not been discussed. In the present Letter we consider a homogeneous dilute mixture of a spin polarized Fermi gas and a Bose– Einstein condensed gas at nonzero temperature. We calculate the phonon spectrum, correction of the sound velocity and stability of the system to second order in the Bose–Fermi coupling constant by using the Green’s function method. We also consider a dilute mixture of a Bose gas and a two components Fermi gas when both bosons and fermions have undergone superfluid transitions.

2. Formulation of the problem 2.1. Spin polarized system Our starting point is the second-quantized grand canonical Hamiltonian of interacting Bose and Fermi gases, H = HB + HF + HBF , where

(1)

  1 (0) d 3 r ψˆ B† HB − µB + gB ψˆ B† ψˆ B ψˆ B , 2
  (0) HF = d 3 r ψˆ F† HF − µF ψˆ F ,
HBF = gBF d 3 r ψˆ B† ψˆ B ψˆ F† ψˆ F ,


HB =

(2)

(0) here ψˆ B and ψˆ F denotes boson and fermion field operators with masses mB and mF , respectively. Hα = Tα + Vα (α = B, F), where Tα , Vα and µα are the associated kinetic energy, trapping potential and chemical potential (the subscripts B and F represent boson and fermion, respectively). For weakly interacting dilute gases, the interactions between boson–boson and between boson–fermion atoms are modeled by delta function potential and the interactions among the fermions atoms may be neglected, since the interactions between atoms at very low temperature is suppressed for polarized systems. gB and gBF stand for boson–boson and boson–fermion coupling constant, respectively,

gB =

4πaB h¯ 2 , mB

gBF =

2πaBF h¯ 2 , mBF

aB (aBF ) are the s-wave scattering length between boson and boson (boson and fermion), and mBF = reduced mass of the boson and the fermion.

(3) mB mF mB +mF

is a

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To determine the excitation spectrum of the Bose condensate, we use the following linearization scheme as done in the standard Bogoliubov treatment of dilute and weakly interacting Bose gas to decompose the field operator as ψˆ B (r , t) = φB (r ) + ξˆB (r , t),

(4)

where φB (r ) = ψˆ B (r , t) is the condensate wave function. The bosonic quasiparticle Green’s function at finite temperature (Matsubara Greens function) is given by [22]    D(r τ, r τ  ) = − Tτ ψˆ B (r , τ )ψˆ B† (r  , τ  ) , (5) where Tτ is the time ordered product and τ is the imaginary time. By using Eq. (4) into Eq. (5) we may write D(r τ, r τ  ) = −φB (r )φB∗ (r  ) + D  (r τ, r τ  ),

(6)

where    D  (r τ, r τ  ) = − Tτ ξB (r , τ )ξB† (r  , τ  ) .

(7)

We introduce Heisenberg operators τ

τ

ξB (r , t) = eHeff h¯ ξB (r )e−Heff h¯ ,

τ

τ

ξB† (r , t) = eHeff h¯ ξB† (r )e−Heff h¯ ,

(8)

where Heff = HB + HBF , and ξB (r , t) satisfy field equation as i h¯

  (0) ∂ ξB (r , t) = HB − µB + 2gB nB + gBF nF ξB (r , t) ∂t      + gB φB2 (r )ξB† + gB φB∗ (r ) ξB ξB − ξB ξB  + 2gB φB ξB† ξB − ξB† ξB       + gB ξB† ξB ξB − ξB† ξB ξB + gBF φB ψF† ψF − ψF† ψF + gBF ξB ψF† ψF ,

(9)

here ψF† ψF  = nF is the fermion density, and ψF† ψF = nF + δρF , where δρF is the correction to the fermionic density in the presence of the Bose condensate. Similar expression can be written for ξB† . Eq. (9) is an exact field equation for ξB , but we restrict ourselves in the present Letter to approximation that neglecting terms involving two or more fluctuation operator for ξB and ψF (it is under our consideration and the results will be published elsewhere). In the interaction representation the bosonic Green’s function is defined as [22] D  (r τ, r τ  ) = − where β =

1 kB T

Tτ ξˆB (r , τ )ξˆB† (r  , τ  )S(β) , S(β)

(10)

and the time evolution operator S(β) is defined through the series expansion [22]


β  S(β) = Tτ exp − dτ HBF (τ ) .

(11)

0

In the usual case of a time independent Hamiltonian and homogeneous mixture, we have Fourier representation, 1 −iωn (τ −τ  )  D  (r τ, r τ  ) = e D (r − r , ωn ), β h¯ n
1    ωn ), D  (r − r , ωn ) = (12) d 3 k ei k.(r −r ) D  (k, (2π)3 where ωn =

2πn β h¯

is the even Matsubara frequency for bosons.

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The spectrum of collective excitation is given by the poles of the exact Green’s function  ωn ) = D0 (k,  ωn ) + D0 (k,  ωn )Σ(k,  ωn )D  (k,  ωn ) D  (k,

(13)

 ωn ) is the boson self energy. To second order of HBF the Dyson equation can be written as where Σ(k, 2  ωn ) = D0 (k,  ωn ) + gBF nB D  (k,

where Ek =

εkB   ωn )2 Π0F (k,  ωn ), D (k, Ek 0

(14)

(εkB )2 + 2gB nB εkB , (nB is the condensate density for pure bosonic system) is the usual Bogoliubov  ωn ) the unperturbed quasiparticle retarded boson Green’s function is spectrum for pure boson system, and D0 (k, given by  ω) = D0 (k,

¯

(hω)2

2Ek , − Ek2 + iδ

(15)

 ωn ) is the density–density response function of an ideal Fermi gas defined as and Π0F (k,    ω) = 1 nF (q) 1 − nF (q + k) Π0F (k, v q   1 1 − , × (16) h¯ ω + εF (q) − εF (k + q) + iδ h¯ ω + εF (q + k) − εF (q) − iδ where nF (q) is the Fermi distribution function and δ is a positive infinitesimal.  ω)−1 one gets the following result for the excitation energy to second order of By solving Eq. (14) for D  (k, gBF as  2 2  hω h¯ 2 ω2 = εkB + 2gB nB εkB + 2gBF (17) nB εkB Π0F (k, ¯ = Ek ), where the density response function in the long wavelength limit (k → 0) may be written as [22]          h¯ ω = Ek ) = − mF kF 1 − 1 x ln x + 1  + i π xθ 1 − |x| , Π0F (k, x − 1 2 2 π 2 h¯ 2

(18)

F Ek where x = mh¯ kk and θ is the step function. F This equation shows the effects of fermions on the excitations of the condensate. The real and imaginary parts of the fermionic response function Π0F give rise respectively to a frequency shift and a damping of the quasiparticles. We are interested in the collective modes of the boson system which correspond to low momenta Ek  gB nB and k  kF . The results are qualitatively different depending on whether c0 > vF or c0 < vF , where 1/2 is the sound velocity for pure bosonic system at absolute zero temperature, and v is the c0 = (nB (0)gB m−1 F B ) Fermi velocity. In the regime c0 < vF or kF √8πa1 n 1 Eq. (18) can be approximated by (mB = mF ) B B √   π mF 4πaB nB mF kF F  1+i . Π0 (k, h¯ ω = Ek ) = − (19) 2 mB kF 2π 2 h¯ 2

By using Eq. (17), the phonon excitation energy can be written as   2 √ aBF g2 1/2 nB . h¯ ω = h¯ kc0 1 − N (0) BF − i π 1/2 2gB aB

(20)

In the above equation the real part in the square bracket gives the correction to the Bogoliubov sound velocity. This correction due to the presence of the Fermi gas is small. For example, in the mixture of 6 Li–87 Rb, where the

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interaction scattering length for boson–boson (87 Rb–87 Rb) is taken to be 109a0 (a0 being the Bohr radius), and the boson–fermion (87 Rb–6 Li) scattering length is about 100a0 this correction is nearly 0.01. The imaginary part gives the Landau damping of the phonon excitation due to the collisions with the fermions. The stability condig2 tion requiring that the real part of the excitation energy is positive, corresponding to the condition N (0) gBF <1 B 1/3 2 mF kF 2 h¯ 2 2 2/3 and gB > 0. Using the expression of N (0) = 2 2 this condition gives nF gBF < 3 2mF (6π ) gB , where derived 2π h¯ earlier in Refs. [20,23] using slightly different considerations. If gB < 0, the original bosonic system is unstable and the Bogoliubov mode has an imaginary frequency for long wavelength. This is in fact the case for the 6 Li–7 Li mixture investigated in Ref. [6], where the 7 Li bosons have a negative scattering length aB 1.5 nm. In the opposite regime, c0 > vF or kF √8πa1 n  1, the imaginary part of Π0F vanishes at low momenta, and in B B the low temperature limit when the distribution function becomes a step function Eq. (16) gives mF kF . 2π 2 h¯ 2 By using Eq. (21) into Eq. (17) we have  Ek ) = Π0F (k,

(21)

 2 mF kF 2 h¯ 2 ω2 = εkB + 2gB nB εkB + gBF nB εkB . 2π 2 h¯ 2

(22)

In this regime a propagating bosonic mode exists and the original bosonic mode at ω = (gB nB /mB )1/2 k is pushed upward by the particle–hole modes lying below. Eq. (22) is the same as Eq. (14) in Ref. [21] and Eq. (7) in Ref. [10], but here nB (T ) is a function of temperature and boson–fermion interaction. It is interesting to study the effects of Bose–Fermi interaction and temperature on nB (T ). At finite temperature we may write nB (r ) = nB (r ) + nB (r ),

(23)

where nB (r ) = |φB (r )|2 and   nB (r ) = −D  rτ, rτ + ,

(24)

where τ + denotes the limiting value τ + η as η approaches zero from positive value. A combination of the above equations gives 1 iωn η  e D (r , r , ωn ). nB (r ) = (25) β h¯ n By using Eqs. (14) and (15) into Eq. (25) we obtain
  εkB 1 1 d 3 k εkB + gB nB (0) 2 − g nB (T ) − nB (0) = − n (0) nF ( q ) 1 − nF (k + q) B BF βE 3 k Ek e −1 Ek V q (2π)   εkB + gB nB (0) 1 gB nB (0) . × − Ek2 (Ek + εF (k + q) − εF ( Ek (Ek + εF (k + q) − εF ( q ))2 q ))

(26)

After integrating the first term on the right-hand side of Eq. (26), since the total destiny is independent of temperature, in the limit of low temperatures (T → 0) the condensate must be depleted according to the relation nB (T ) − nB (0) = −

mB

2 (kB T )2 − gBF nB (0)

  εkB 1 nF ( q ) 1 − nF (k + q) Ek V q

12h¯ 3 c0   εkB + gB nB (0) 1 gB nB (0) . × − Ek2 (Ek + εF (k + q) − εF ( Ek (Ek + εF (k + q) − εF ( q ))2 q ))

(27)

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By using Eq. (22) the long wavelength behavior of phonon spectrum is given by h¯ ω ≈ h¯ ck,

(28)

where the sound velocity in the limit of T → 0 and in the presence of Bose–Fermi interaction to second order of aBF is  2  kF aBF mB 2 . (kB T ) + c = c0 1 − (29) 2πaB 24h¯ 3 c0 nB (0) It is noted that, at T = 0 this equation is the same as Eq. (14) of Pu et al. [21]. In their paper, they claimed that there is a dynamical instability of the system for certain parameter regimes. They based their conclusion on an analysis of the dispersion relation of the phonons (Eq. (14)), and conclude that an instability arises if process shown in their Fig. 1 can occur. Their claim is incorrect. The process in their Fig. 1 represents in fact Landau damping of the collective mode but not instability of the system. The process depicted over there absorbs energy from the sound wave [25]. 2.2. Unpolarized spin system Until now we consider a mixture of weakly interacting Bose and Fermi gases at low temperature. Both gases are assumed to be spin polarized, such as would usually be the case in magnetic traps. For a dilute mixture, interaction among the bosons themselves and between the bosons and fermions can be characterized by the scattering lengths aB and aBF in the s-wave channels. However, the fermions do not interact among themselves, since they are spin polarized. Now we assume that the system is not spin polarized, i.e., the interaction between two hyperfine Fermi component (↑, ↓) is important. In this case s-wave collisions between these two different components are allowed and elastic collisions subsequently restore thermal equilibrium of the two components gas at a lower temperature, and both boson and fermion have undergone superfluid transition. The HF and HBF parts of the Hamiltonian may be written as
 † 2  †  (0)  , Hf − µFα ψˆ Fα + gF |ψˆ Fα |2 ψˆ Fα d 3 r ψˆ Fα HF = α

HBF =




† ˆ d 3 r gBFα ψˆ B† ψˆ B ψˆ Fα ψFα ,

(30)

α 2

2

2π h¯ aBFα ¯ Fh here α = (↑, ↓), gF = 4πa mF and gBFα = mBFα , where aB and aBFα are the scattering length of fermion–fermion and boson–fermion interaction, respectively. Not that only interaction between two different hyperfine states is mFα is considered, since Pauli principle prohibits s-wave scattering between two identical fermions. mBFα = mmBB+m Fα the reduced mass and µFα is the chemical potential of the corresponding component. Bogoliubov approximation is then applied to the Bose field and the pairing field is so also separated into its average and fluctuation parts, i.e.,

∆(r , τ ) = ∆0 + δ(r , τ ).

(31)

Now we introduced Heisenberg operators for fermion field τ

τ

ψˆ F↑ (r τ ) = eH h¯ ψˆ F↑ (r )e−H h¯ ,

τ

τ

† † ψˆ F↓ (r τ ) = eH h¯ ψˆ F↓ (r )e−H h¯ .

The single particle Green’s functions are defined as [22]       † (r  τ  ) , F (r τ, r τ  ) = Tτ ψˆ F↓ (r τ )ψˆ F↑ (r  τ  ) , g(r τ, r τ  ) = − Tτ ψˆ F↑ (r τ )ψˆ F↑   †  † F † (r τ, r τ  ) = Tτ ψˆ F↑ (r τ )ψˆ F↓ (r  τ  ) .

(32)

(33)

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In the usual case of a time-independent Hamiltonian the functions g, F and F † depend only on the difference τ − τ  , and it is convenient to introduce the abbreviation ∆(r ) = gF F (r τ, rτ ),

(34)

which defines the gap function ∆(r ). Introduce a two component field operator

 ψˆ F↑ (r τ ) ΨF (r τ ) = ψˆ † (r τ )

(35)

F↓

and a 2 × 2 matrix Green’s function   F (r τ, r τ  ) g(r τ, r τ  )   . g( ˜ r τ, r τ ) = F † (r τ, r τ  ) −g(r τ, r τ  )

(36)

The corresponding equation of motion becomes 

2 2 ∂ − h¯2m∇F − µ↓ + g↓ |φb |2 ∆ h¯ ∂τ   g( ˜ r τ, r τ  ) = hIδ( ¯ r − r )δ(τ − τ ), 2 2 ∂ ∆∗ + h¯2m∇F + µ↑ − g↑ |φB |2 h¯ ∂τ

(37)

where I is the unite matrix. The spectrum of collective excitation is given by the poles of the exact Green’s function  ωn ) + g˜ 0 (k,  ωn )Σ(k,  ωn )g(  ωn ),  ωn ) = g˜ 0 (k, ˜ k, g( ˜ k,

(38)

 ω) is the Fourier transform of g(  ω) denotes the matrix of the self-energy parts. It follows where g( ˜ k, ˜ r , τ ) and Σ(k, that the spectrum is determined by     ωn ) = det g˜ 0 (k,  ωn ) −1 − Σ(k,  ωn ) = 0, det g˜ −1 (k, (39) where the unperturbed Green’s function reads  ωn ) = g˜ 0 (k,

h¯  hω  − |∆0 |2 (i hω ¯ n − ε↓ (k))(i ¯ n + ε↑ (k))



 i hω ¯ n + ε↑ (k) ∆0

 ∆0  , i hω ¯ n − ε↓ (k)

 = h¯ k − µα + gBFα nB and ωn = 2nπ is the even Matsubara frequency. with εα (k) 2mF β h¯ The self energy of the Bose gas at the RPA bubble, is given by  2  2  iωn ) + gBF↓  iωn ) nB ,  iωn ) = gB nB + gBF↑ Π↑ (k, Π↓ (k, Σ(k,

(40)

2 2

(41)

or equivalently the density–density response function of the superfluid Fermi gas
 − nF (Eα (p))  d 3 p nF (Eα (p + k))  iωn ) = − , ΠαF (k, (42) 3 (2π) i hω  + Eα (p)  ¯ n − Eα (k + p)  1 and Eα = εα2 + ∆2 are, respectively, the Fermi distribution function, and energy in where nF (E) = exp(βE)+1  iωn ) and Π(k,  iωn ) are invariant under transformation (k,  iωn ) to (−k,  −iωn ), superfluid state. Note that Σ(k, due to the time reversal symmetry of the problem. By using Eqs. (40)–(42) into Eq. (39) after some algebra the phonon spectrum in the limit of long wavelength h¯ ω  1) can be derived analytically ( kh¯BωT  1 and ∆ 0   h¯ 2 k 2 h¯ 2 k 2 h¯ 2 vF2 k 2 2 2 2 2 , h¯ ω = (43) + 2gB nB − N (0)gBFA nB − gBFC nB N (0) 2 2mB 2mB 6∆0 3

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+g

g

−g

where gBFA = BF↑ 2 BF↓ is the average of boson–fermion interaction and gBFC = BF↑ 2 BF↓ is the deviation of interaction strength from average. g +g g −g It is noted that if the system is spin polarized, i.e., gBFA = gBF↑ = BF↑ 2 BF↑ and gBFC = BF↑ 2 BF↑ = 0 Eq. (43) is the same as Eq. (22). Including the imaginary parts of bubbles, damping of the modes due to boson–fermion coupling can be obtained by finding the poles of Eq. (39). In the long wavelength limit the dispersion relation is found to be in the form ω = (c − iγ )k, where γ denoting the damping rate. By using Eq. (39) we obtain   h¯ 2 k 2 h¯ 2 k 2 h¯ 2 vF2 k 2 2 2 2 2 2 2 + igBFA nB Z1 + igBFC nB Z2 , + 2gB nB − N (0)gBFA nB − gBFC nB N (0) 2 h¯ ω ≈ 2mB 2mB 6∆0 3 (44) where Z1 and Z2 are, respectively, −N (0)πω Z1 = kB T v F k




∞ 2

dE sech

 E , 2kB T

−N (0)πω Z2 = 2kB T vF k

0


∞ 0

  E 2E 2 2 . sech 2kB T E 2 − ∆20

In the limit of low temperatures Eq. (44) can be written as   N (0)πω h¯ 2 k 2 h¯ 2 k 2 h¯ 2 vF2 k 2 2 2 2 2 2 h¯ ω ≈ + 2gB nB − N (0)gBFA nB − gBFC nB N (0) 2 − 4igBFA nB 2mB 2mB vF k 6∆0 3

(45)

(46)

and in the long wavelength limit the above equation can be written in the form ω = (c − iγ )k, where c and γ are respectively given by   2 mB N (0) gBFA , c = c0 1 − − 4 gB2 24h¯ 3 c0 nB (0) 1 2 N (0)π πN(0) 2 γ≈ gBFA nB (0) − gBFA (kB T )2 . mB vF 12h¯ 3 c0 vF

(47)

(48) (49)

At finite temperature, the damping of the excitation is due to the possibility of their colliding with one another. Note that there is also damping at absolute zero (the first term in Eq. (49)).

3. Discussion and conclusion remarks We have calculated the dispersion relation and damping of a dilute mixture of boson and fermion in two cases. First we consider spin polarized Fermi gas. The addition of fermion atoms to superfluid boson modifies the sound velocity and by providing additional degrees of freedom for energy absorption increases the attenuation of sound velocity over its value in pure boson system. The results are qualitatively different depending on whether c0 > vF or c0 < vF . In the regime c0 < vF phonon spectrum includes both real and imaginary parts (Eq. (20)). The real part gives the correction to the Bogoliubov sound velocity, and the imaginary part gives the Landau damping of the phonon excitation due to the collisions with the fermions. In the opposite regime, c0 > vF , the imaginary part of the boson’s dispersion relation vanishes at low momenta, and a propagating bosonic mode exists and the original B nB 1/2 ) k is pushed upward by the particle–hole modes lying below. bosonic mode at ω = ( gm B Our results for phonon spectrum at absolute zero temperature are the same as Pu et al. results [21]. It is noted that Pu et al. [21] claimed that there is a dynamical instability of the dilute Bose–Fermi mixture for certain parameter regimes. They found the dispersion relation for boson (Eq. (14)) and conclude that instability arises if

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process shown in their Fig. 1 can occur. Their claim is incorrect. The process in Fig. 1 represents in fact Landau damping of the collective mode [24]. The process depicted over there absorbs energy from the sound wave. This damping is due to the negative imaginary part, which exists, in long wavelength of dispersion relation, but the mode become unstable when the long wavelength dispersion relation has the positive imaginary part. This gives rise to exponentially growing fluctuations of the system and the ground state become unstable. Second we consider a dilute mixture of boson and two components fermion, i.e., the unpolarized spin system. In this case s-wave collisions between these two different components are allowed and elastic collisions subsequently restore thermal equilibrium of the two components gas at a lower temperature, and both boson and fermion have undergone superfluid transition. The phonon spectrum has additional term due to interaction of boson atoms with the components of fermion atoms (Eq. (43)). The damping rate varies as T 2 in low temperatures limit. There is also damping at absolute zero temperature due to the interaction between bosons and fermions. Damping is found to be small in low temperature region.

References [1] M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, E.A. Cornell, Science 269 (1995) 198. [2] M.R. Matthews, B.P. Anderson, P.C. Haljan, D.S. Hall, M.J. Holland, J.E. Williams, C.E. Wieman, E.A. Cornell, Phys. Rev. Lett. 83 (1999) 3358. [3] J. Stenger, S. Inoye, D.M. Stamper-Kurn, H.J. Miesner, A.P. Chikatur, W. Ketterle, Nature (London) 369 (1998) 345. [4] B. Demarko, D.S. Jin, Science 285 (1999) 1703. [5] B. Demarko, J.L. Bohn, J.P. Burk, M. Holland, D.S. Jin, Phys. Rev. Lett. 82 (1999) 4208. [6] M.O. Mewes, G. Ferrari, F. Schreck, A. Sinatra, C. Salomon, Phys. Rev. A 61 (2000) 011403. [7] M.R. Matthews, et al., Phys. Rev. Lett. 83 (1999) 2498. [8] D.M. Stamper-Kurn, H.J. Miesner, A.P. Chikkatur, S. Inouye, J. Stenger, W. Ketterle, Phys. Rev. Lett. 83 (1999) 661. [9] H.J. Miesner, D.M. Stamper-Kurn, J. Stenger, S. Inouye, A.P. Chikkatur, W. Ketterl, Phys. Rev. Lett. 82 (1999) 2228. [10] S.K. Yip, Phys. Rev. A 64 (2001) 023609. [11] D.S. Hall, M.R. Matthews, C.E. Wieman, E.A. Cornell, Phys. Rev. Lett. 81 (1998) 1543. [12] D.S. Hall, M.R. Matthews, J.R. Ensher, C.E. Wieman, E.A. Cornel, Phys. Rev. Lett. 81 (1998) 1539. [13] T.L. Ho, V.B. Shenoy, Phys. Rev. Lett. 77 (1996) 3276. [14] P. Ohberg, S. Stenholm, Phys. Rev. A 57 (1998) 1272. [15] A. Sinatra, Y. Castin, Eur. Phys. J. D 8 (2000) 319. [16] X.X. Yi, C.P. Sun, Commun. Theor. Phys. 32 (1999) 521. [17] Th. Busch, J.I. Cirac, V.M. Perez-Carcia, P. Zoller, Phys. Rev. A 56 (1997) 2978. [18] T.L. Ho, Phys. Rev. Lett. 81 (1998) 742. [19] K. Molmer, Phys. Rev. Lett. 80 (1998) 1804. [20] M.J. Bijlsma, B.A. Heringa, H.T.C. Stoof, Phys. Rev. A 61 (2000) 053601. [21] H. Pu, W. Zhang, M. Wilkens, P. Meyster, Phys. Rev. Lett. 88 (2002) 070408. [22] A.L. Fetter, J.D. Walecka, Quantum Theory of Many Particle Systems, McGraw-Hill, New York, 1971. [23] L. Viverit, C.J. Pethick, H. Smith, Phys. Rev. A 61 (2000) 053605. [24] E.M. Lifshitz, L.P. Pitaevskii, Statistical Physics, Part II, Pergamon, Oxford, 1980, see, in particular, Chapter 1, Section 4. [25] E.M. Lifshitz, L.P. Pitaevskii, Physical Kinetics, Pergamon, Oxford, 1981, see, in particular, Chapter 3, Sections 30, 35.