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arXiv:cond-mat/0507714 · Physical Review A

Temperature dependence and finite-size effects in collective modes of superfluid-trapped Fermi gases

M. Grasso, Elias Khan, Mi­chael Urban

Published Oct 26, 2005• 19 citations
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Abstract

We discuss collective monopole and quadrupole excitations of a collisionless gas of trapped Fermionic atoms in the superfluid BCS phase, comparing the fully microscopic Bogoliubov--de Gennes and quasiparticle random-phase approximation method with widely used semiclassical methods. In particular, the microscopic treatment allows us to address the questions of temperature dependence and nontrivial dependence on the trap parameters, which cannot be answered within the semiclassical approach. The main result concerning the temperature dependence is a strong Landau damping at intermediate temperature, which disappears in the limits of zero and critical temperature. However, even at zero temperature, considerable deviations from superfluid hydrodynamics are found if the trap frequency is of the same order of magnitude as the pairing gap.

DOI
10.1103/physreva.72.043617

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We discuss collective monopole and quadrupole excitations of a collisionless gas of trapped Fermionic atoms in the superfluid BCS phase, comparing the fully microscopic Bogoliubov--de Gennes and quasiparticle random-phase approximation method with widely used semiclassical methods. In particular, the microscopic treatment allows us to address the questions of temperature dependence and nontrivial dependence on the trap parameters, which cannot be answered within the semiclassical approach. The main result concerning the temperature dependence is a strong Landau damping at intermediate temperature, which disappears in the limits of zero and critical temperature. However, even at zero temperature, considerable deviations from superfluid hydrodynamics are found if the trap frequency is of the same order of magnitude as the pairing gap.
At finite temperature, the hydrodynamic collective modes of superfluid trapped Fermi gases are coupled to the motion of the normal component, which in the BCS limit behaves like a collisionless normal Fermi gas. The coupling between the superfluid and the normal components is treated in the framework of a semiclassical transport theory for the quasiparticle-distribution function, combined with a hydrodynamic equation for the collective motion of the superfluid component. We develop a numerical test-particle method for solving these equations in the linear-response regime. As a first application we study the temperature dependence of the collective quadrupole mode of a Fermi gas in a spherical trap. The coupling between the superfluid collective motion and the quasiparticles leads to a rather strong damping of the hydrodynamic mode already at very low temperatures. At higher temperatures the spectrum has a two-peak structure, the second peak corresponding to the quadrupole mode in the normal phase.
We derive semiclassical transport equations for a trapped atomic Fermi gas in the BCS phase at temperatures between zero and the superfluid transition temperature. These equations interpolate between the two well-known limiting cases of superfluid hydrodynamics at zero temperature and the Vlasov equation at the critical one. The linearized version of these equations, valid for small deviations from equilibrium, is worked out and applied to two simple examples where analytical solutions can be found: a sound wave in a uniform medium and the quadrupole excitation in a spherical harmonic trap. In spite of some simplifying approximations, the main qualitative results of quantum mechanical calculations are reproduced, which are the different frequencies of the quadrupole mode at zero and the critical temperature and strong Landau damping at intermediate temperatures. In addition we suggest a numerical method for solving the semiclassical equations without further approximations.
There are several situations in which the collective motion of a superfluid Fermi gas is coupled to quasiparticle excitations. Quasiparticle excitations are, for example, present at finite temperature, but also at zero temperature in the case of a rotating trap. In this paper, I present a semiclassical description of such systems, where the superfluid component is described by hydrodynamics while the normal component, formed by the quasiparticles, is described by a transport equation similar to the Vlasov equation. The coupling between the two components generates a strong damping of the collective modes. As applications of the semiclassical theory, the case of a rotating trap and several collective modes (quadrupole and scissors modes) of the trapped gas are discussed.
We study collective excitations of a trapped Fermi gas at finite temperature using the Bogoliubov--de Gennes mean-field theory in conjunction with the generalized random-phase approximation. The collective excitations are analyzed through the density response of a monopole excitation. We show the appearance of several modes at the normal-fluid--superfluid phase-transition temperature: Higgs mode--like pairing amplitude modulations, analogues of the second sound in the trapped gas, and an edge mode that is also the strongest mode in the response.
Recently it was shown that the Landau-Khalatnikov two-fluid hydrodynamics describes the collision-dominated region of a trapped Bose condensate interacting with a thermal cloud. We use these equations to discuss the low frequency hydrodynamic collective modes in a trapped Bose gas at finite temperatures. We derive variational expressions based on these equations for both the frequency and damping of collective modes. A new feature is our use of frequency-dependent transport coefficients, which produce a natural cutoff by eliminating the collisionless low-density tail of the thermal cloud. Above the superfluid transition, our expression for the damping in trapped inhomogeneous gases is analogous to the result first obtained by Landau and Lifshitz for uniform classical fluids. We also use the moment method to discuss the crossover from the collisionless to the hydrodynamic region. Recent data for the monopole-quadrupole mode in the hydrodynamic region of a trapped gas of metastable 4He{}^{4}\mathrm{He}4He is discussed. We also present calculations for the damping of the analogous m=0m=0m=0 monopole-quadrupole condensate mode in the superfluid phase.
We study the phononic collective modes of the pairing field and their corresponding signature in both the order-parameter and density response functions for a superfluid Fermi gas at all temperatures below Tc{T}_{c}Tc​ in the collisionless regime. The spectra of collective modes are calculated within the Gaussian pair fluctuation approximation. We deal with the coupling of these modes to the fermionic continuum of quasiparticle-quasihole excitations by performing a nonperturbative analytic continuation of the pairing field propagator. At low temperature, we recover the known exponential temperature dependence of the damping rate and velocity shift of the Anderson-Bogoliubov branch. In the vicinity of Tc{T}_{c}Tc​, we find analytically a weakly damped collective mode whose velocity vanishes with a critical exponent of 1/21/21/2, and whose quality factor diverges logarithmically with ${T}_{c}\ensuremath{-}T$, thereby clarifying an existing debate in the literature [Andrianov and Popov, Theor. Math. Phys. 28, 829 (1976); Ohashi and Takada, J. Phys. Soc. Jpn. 66, 2437 (1997)]. A transition between these two phononic branches is visible at intermediary temperatures, particularly in the BCS limit where the phase-phase response function displays two maxima.