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By using a mean-field approximation which describes the coupled oscillations of condensate and noncondensate atoms in the collisionless regime, Landau damping in a dilute dipolar Bose–Fermi mixture in the BEC limit where Fermi superfluid is treated as tightly bounded molecules, is investigated. In the case of a uniform quasi-two-dimensional (2D) case, the results for the Landau damping due to the Bose–Fermi interaction are obtained at low and high temperatures. It is shown that at low temperatures, the Landau damping rate is exponentially suppressed. By increasing the strength of dipolar interaction, and the energy of boson quasiparticles, Landau damping is suppressed over a broader temperature range.
In the last few years, ultracold atoms have appeared as an inimitable tool to investigate quantum many-body systems. This new research area, with the unique possibility of tuning the system features, allows researchers to study the new conditions in condensed matter or low temperature physics. There are a large number of parameters such as temperature, number of atoms, trap potentials, interactions between atoms, which can be controlled in an ultracold system.
Significant progress in understanding the phenomena of BEC for bosons and the BCS-BEC crossover for fermions are among the key successes of the field.[1,2] On the other hand, many-body Bose–Fermi mixtures of particles with different quantum statistics, are believed to exhibit very different behavior from pure Bose and Fermi systems. In addition, preparing and studying the mixtures of bosons and fermions experimentally in the quantum degenerate regime, lead to direct experimental tests of theoretical predictions for these mixtures.
Low-lying condensate oscillations of a trapped Bose–Fermi mixture have been studied at finite temperature in the collisionless regime by Liu and Hu.[3] They studied the behaviors of monopole and quadrupole excitations as a function of the coupling of condensate, noncondensate, and degenerate Fermi gas. In a Bose–Fermi superfluid mixture the quasiparticles possess a special property. Recently, a mixture of Bose and Fermi superfluids preparing bosonic 7Li atoms and two spin components of fermionic 6Li atoms, has been realized by ENS group.[4] In the ENS experiment, the Fermi superfluid was controlled from the BCS to the BEC region via Feshbach resonance. In a Bose–Fermi superfluid mixture, there are one gapped fermionic mode of Cooper pair breaking in the Fermi superfluid and two gapless bosonic modes. The two bosonic modes are Goldstone modes of Bose and Fermi superfluids. The ENS experiment leads to many theoretical studies on this new superfluid mixture.[5–11]
At low temperatures and low densities in these systems, where the mean free path of the elementary excitations becomes comparable to the size of the system, collisions play a minor role and the appropriate regime is the collisionless regime. In this regime, the interactions between quasiparticles can be characterized by the familiar Bogoliubov dispersion and determine many features of these systems. Landau–Beliaev damping is one of the widely known effects in these systems. In a damping mechanism, called Beliaev damping, an excitation of the system decays into two states with smaller energies and momenta.[12,13] Beliaev damping can happen even at T = 0. At finite temperatures, a mechanism of damping (known as Landau damping) occurs by the process of a collective mode absorbed by a quasiparticle that is then turned into another quasiparticle. Landau damping becomes the dominant damping mechanism in higher temperatures.[14,15]
The effect of quasiparticle interactions in a Bose–Fermi superfluid mixture has been investigated by Shen and Zheng.[10] They have shown that the Landau damping of the Bose superfluid due to its interaction with the Fermi superfluid is exponentially suppressed and on the BCS and the BEC side of the Fermi superfluid, the damping rate has a distinct threshold manner and momentum dependence. The effect of quasiparticle interaction in a Bose–Fermi superfluid mixture, has been investigated by Zheng and Zhai.[11] They found that at low temperatures, the Landau damping is a constant near the threshold momentum on the BCS side, while it increases rapidly on the BEC side. In fact, the fermion quasiparticles are located near the Fermi surface on the BCS side, while such a restriction does not exist on the BEC side. This restriction causes different damping rate behavior on the BCS and the BEC side of the Fermi superfluid.
The development of experimentally realized dipolar quantum gases such as the heteronuclear 40K87Rb molecules with electric dipoles[16–18] and spinor 87Rb condensates with magnetic dipoles,[19] has aroused the interest in cold-atom mixtures involving dipolar condensates. These experimental studies have now expanded from alkali elements to the transition element, 52Cr,[20] 164Dy,[21] and 168Er[22] with large magnetic dipoles. Mixtures of two different fermionic species further allow the creation of polar molecules, each of which has a long-range dipole–dipole interaction.[17,23] The mixture 6Li−40K is a prime candidate for these studies. 6Li−40K are the only stable fermionic alkali isotopes and thus belong to the experimentally best-mastered class of atoms. Moreover, both species have bosonic isotopes which can also be used to create boson-fermion gases. Furthermore, the difference in the electronic structure between the two species is large, leading to a large electric dipole moment for heteronuclear diatomic 6Li40K molecule.[24] Thus, it is conceivable that the dipolar Bose–Fermi and Fermi–Fermi mixtures can be realized in cold-atom experiments not too far down the road.
Landau damping of excitations in a quasi-2D dipolar Bose gas has been studied in conformity with the time-dependent mean-field formalism by Natu and Wilson.[25] They calculated the Landau damping rates for phonons at low and high temperatures. The effect of long-range 1/r interaction on the Landau damping in a Bose–Fermi superfluid mixture has also been studied.[26]
To the best of our knowledge, the effect of dipole–dipole interaction between particles on the Landau damping of a Bose–Fermi superfluid mixture has not been studied. In this paper, we study the Landau damping in a mixture of dipolar Bose–Fermi superfluid gas. We consider a case in which the Fermi superfluid is in the strongly interacting regime and on the BEC side, whilst the Bose superfluid is in the weakly interacting regime. On the BEC side we can consider the Fermi superfluid as tightly bounded molecules and treat it as a molecular BEC. Using the time-dependent mean-field approach according to the Popov approximation,[27] the damping coefficients are calculated. Finally, for a 2D dipolar Bose–Fermi mixture, the temperature dependence of Landau damping due to boson–fermion interaction is calculated and the results are plotted for different values of dipole–dipole interaction. We think our results are useful for theoretical and experimental research on the effects of dipole–dipole interaction on the Landau damping of a dipolar Bose–Fermi mixture.
The Hamiltonian of a trapped mixture of bosons and molecules in the BEC limit where Fermi superfluid is treated as tightly bounded molecules, reads
![](cpb_25_12_126701/cpb161364ieqn59.gif)
![](cpb_25_12_126701/cpb161364ieqn60.gif)
![](cpb_25_12_126701/cpb161364ieqn61.gif)
![](cpb_25_12_126701/cpb161364ieqn57.gif)
![](cpb_25_12_126701/cpb161364ieqn58.gif)
The Heisenberg equation of motion for the bosonic field operator reads
By considering the Fermi superfluid on the BEC side as molecular condensate, in the usual treatment for Bose systems with broken gauge symmetry, the bosonic (molecular) operators can be written as:
![](cpb_25_12_126701/cpb161364ieqn62.gif)
![](cpb_25_12_126701/cpb161364ieqn63.gif)
In the small amplitude regime and small fluctuations of the condensate, normal and anomalous densities around their equilibrium values can be written respectively as
Substituting Eqs. (
In general, we define ntot b(m) (
The second part of Hamiltonian (
It is convenient to diagonalize the Hamiltonian
![](cpb_25_12_126701/cpb161364ieqn64.gif)
![](cpb_25_12_126701/cpb161364ieqn3.gif)
![](cpb_25_12_126701/cpb161364ieqn4.gif)
![](cpb_25_12_126701/cpb161364ieqn5.gif)
![](cpb_25_12_126701/cpb161364ieqn6.gif)
The resulting Hamiltonians describe a noninteracting gas of Bogoliubov quasiparticles
![](cpb_25_12_126701/cpb161364ieqn7.gif)
In the presence of dipole–dipole interactions, the problem of solving nonlocal Bogoliubov equations becomes computationally intensive.[29–31] The normal and anomalous densities in terms of the Bogoliubov operators ai(t) (di(t)) and
Physically,
In terms of
![](cpb_25_12_126701/cpb161364ieqn19.gif)
![](cpb_25_12_126701/cpb161364ieqn20.gif)
![](cpb_25_12_126701/cpb161364ieqn21.gif)
![](cpb_25_12_126701/cpb161364ieqn22.gif)
![](cpb_25_12_126701/cpb161364ieqn23.gif)
![](cpb_25_12_126701/cpb161364ieqn24.gif)
![](cpb_25_12_126701/cpb161364ieqn25.gif)
![](cpb_25_12_126701/cpb161364ieqn26.gif)
By disregarding the coupling terms between the condensate and noncondensate atoms and molecules (bosons) in Eq. (
![](cpb_25_12_126701/cpb161364ieqn27.gif)
![](cpb_25_12_126701/cpb161364ieqn28.gif)
![](cpb_25_12_126701/cpb161364ieqn29.gif)
![](cpb_25_12_126701/cpb161364ieqn30.gif)
Multiplying Eqs. (
By defining the matrix elements as
Equations (
The damping rate is given by the imaginary part of ω. Landau damping comes from the process that a quasiparticle with the frequency ω0 is absorbed by a thermal excitation Ei and the thermal excitation jumping to the energy Ej = Ei + ω0. This thermal process can be written as
Equations (
In this section we focus on Landau damping due to the boson–molecule interactions in a homogeneous quasi-2D mixture of a dipolar Bose and superfluid Fermi gas. Quasi-2D gas is fabricated experimentally by a tight confinement along the axial (z) direction, the homogeneous quasi-2D gas is created when the trap potential in the x–y plane is eliminated. Because Bose gas is very dilute in the experiment, Landau damping due to the interaction between bosons can be disregarded in this case.
For a homogeneous system the condensate wave function at equilibrium is constant
![](cpb_25_12_126701/cpb161364ieqn36.gif)
![](cpb_25_12_126701/cpb161364ieqn37.gif)
![](cpb_25_12_126701/cpb161364ieqn38.gif)
![](cpb_25_12_126701/cpb161364ieqn39.gif)
![](cpb_25_12_126701/cpb161364ieqn40.gif)
![](cpb_25_12_126701/cpb161364ieqn41.gif)
By inserting Eqs. (
![](cpb_25_12_126701/cpb161364ieqn42.gif)
For a dilute Bose gas, where the healing length of the bosons is much larger than their wavelength, the spectrum of the elementary excitations can be considered as a free-particle energy with quadratic dispersion, i.e.,
![](cpb_25_12_126701/cpb161364ieqn48.gif)
In Fig.
In Fig.
![]() | Fig. 2. Variations of energy with x for different values of ratio ![]() |
The temperature dependences of the I(τ) for different values of y = q/kF are plotted in Fig.
![]() | Fig. 3. Variations of integral I(τ) with τ = kbT/εF for fixed ![]() |
We should confirm that our calculations of temperature dependence in Landau damping are only reliable at low temperatures, because the condensate fraction in the Bose superfluid will decrease with the increase of temperature. For a general analysis of temperature dependence, the changing of the condensate fraction should be included in calculations. Anyway, at low temperatures the condensate fraction is not sensitive to the temperature and our result is reliable in this limit.
In this paper, damping of low-energy excitations in a Bose–Fermi superfluid mixture with dipole–dipole interactions is studied at finite temperature in a collisionless regime. By using the time-dependent mean-field approach based on the Popov approximation, a set of coupled equations for the condensate and noncondensate components of bosons and molecules is obtained. An explicit expression for damping rate is derived, which in the absence of the boson–molecule interaction coincides with the finding of Ref. [25] and in the absence of the dipole–dipole interaction coincides with the finding of Ref. [10]. It is shown that the temperature dependence of Landau damping due to the boson–molecule interaction in a homogeneous 2D dipolar Bose–Fermi superfluid is linear at high temperatures and at low temperatures, exponentially suppressed. The temperature dependences of the Landau damping for different values of q/kF and
In future it will be possible to realize a mixture of Bose and Fermi superfluids with dipolar atoms and our result should be observed in the experiment.
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