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Journal of Physics Conference Series

Damping of collective modes in ultracold trapped Fermi gases with in-medium cross section

Mi­chael Urban, Silvia Chiacchiera, D. Davesne, Thomas Lepers

Published Sep 16, 2011• 0 citations
Publisher

Abstract

We study the damping of the quadrupole and scissors modes of ultracold trapped Fermi gases with large negative scattering length at temperatures above the superfluid critical temperature T c within the framework of the Boltzmann equation. The in-medium scattering cross-section is strongly enhanced at temperatures slightly above T c . This has strong effects on the collisional damping of the collective modes and the transition from hydrodynamic to collisionless behaviour as a function of temperature. Within the method of phase-space moments of second order, it was found that the resulting damping rates are in strong disagreement with experimental data from the Innsbruck group, which seemed to favour the free cross section. However, the comparison with numerical solutions of the Boltzmann equation revealed that the method of second-order phase-space moments is insufficient. Here we show that the agreement between the calculation using the in-medium cross-section and the data can be clearly improved by including also fourth-order moments.

DOI
10.1088/1742-6596/321/1/012026

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We study the damping of the quadrupole and scissors modes of ultracold trapped Fermi gases with large negative scattering length at temperatures above the superfluid critical temperature T c within the framework of the Boltzmann equation. The in-medium scattering cross-section is strongly enhanced at temperatures slightly above T c . This has strong effects on the collisional damping of the collective modes and the transition from hydrodynamic to collisionless behaviour as a function of temperature. Within the method of phase-space moments of second order, it was found that the resulting damping rates are in strong disagreement with experimental data from the Innsbruck group, which seemed to favour the free cross section. However, the comparison with numerical solutions of the Boltzmann equation revealed that the method of second-order phase-space moments is insufficient. Here we show that the agreement between the calculation using the in-medium cross-section and the data can be clearly improved by including also fourth-order moments.
We study the transition from hydrodynamic to collisionless behavior in collective modes of ultracold trapped Fermi gases. To that end, we solve the Boltzmann equation for the trapped Fermi gas via the moments method. We showed previously that it is necessary to go beyond second-order moments if one wants to reproduce the results of a numerical solution of the Boltzmann equation. Here, we will give the detailed description of the method including fourth-order moments. We apply this method to the case of realistic parameters, and compare the results for the radial quadrupole and scissors modes at unitarity to experimental data obtained by the Innsbruck group. It turns out that the inclusion of fourth-order moments clearly improves the agreement with the experimental data. In particular, the fourth-order moments reduce the effect of collisions and therefore partially compensate the effect of the enhanced in-medium cross section at low temperatures.
Due to Pauli blocking of intermediate states, the scattering matrix (or TTT matrix) of two fermionic atoms in a Fermi gas becomes different from that of two atoms in free space. This effect becomes particularly important near a Feshbach resonance, where the interaction in free space is very strong but becomes effectively suppressed in the medium. We calculate the in-medium TTT matrix in ladder approximation and study its effects on the properties of collective modes of a trapped gas in the normal-fluid phase. We introduce the in-medium interaction on both sides of the Boltzmann equation, namely, in the calculation of the mean field and in the calculation of the collision rate. This allows us to explain the observed upward shift of the frequency of the quadrupole mode in the collisionless regime. By including the mean field, we also improve considerably the agreement with the measured temperature dependence of frequency and damping rate of the scissors mode, whereas the use of the in-medium cross section deteriorates the description, in agreement with previous work.
We investigate the transition and damping of low-energy collective modes in a trapped unitary Fermi gas by solving the Boltzmann-Vlasov kinetic equation in a scaled form, which is combined with both the T-matrix fluctuation theory in normal phase and the mean-field theory in order phase. In order to connect the microscopic and kinetic descriptions of many-body Feshbach scattering, we adopt a phenomenological two-fluid physical approach, and derive the coupling constants in the order phase. By solving the Boltzmann-Vlasov steady-state equation in a variational form, we calculate two viscous relaxation rates with the collision probabilities of fermion's scattering including fermions in the normal fluid and fermion pairs in the superfluid. Additionally, by considering the pairing and depairing of fermions, we get results of the frequency and damping of collective modes versus temperature and s-wave scattering length. Our theoretical results are in a remarkable agreement with the experimental data, particularly for the sharp transition between collisionless and hydrodynamic behaviour and strong damping between BCS and unitary limits near the phase transition. The sharp transition originates from the maximum of viscous relaxation rate caused by fermion-fermion pair collision at the phase transition point when the fermion depair, while the strong damping due to the fast varying of the frequency of collective modes from BCS limit to unitary limit.
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.
The excitation spectra of the radial quadrupole and scissors modes of ultracold Fermi gases in elongated traps are studied across the BCS superfluid-normal phase transition in the framework of a transport theory for quasiparticles. In the limit of zero temperature, this theory reproduces the results of superfluid hydrodynamics, while in the opposite limit, above the critical temperature, it reduces to the collisionless Vlasov equation. In the intermediate temperature range, the excitation spectra have two or three broad peaks, respectively, which are roughly situated at hydrodynamic and collisionless frequencies, and whose strength is shifted from the hydrodynamic to the collisionless modes with increasing temperature. By fitting the time-dependent quadrupole deformation with a damped oscillation of a single frequency, we can understand the ``jump'' of the frequency of the radial quadrupole mode as a function of interaction strength which has recently been reported by the Innsbruck group.
We study the transition from the hydrodynamic to the collisionless regime in collective modes of three- and two-dimensional Fermi gases by using the semiclassical Boltzmann equation. We use direct numerical simulations as well as the method of phase-space moments to solve the Boltzmann equation and show that the restriction to second-order moments is not accurate enough. By including higher-order moments, we can successfully describe the hydrodynamic to collisionless transition observed in the quadrupole mode in three-dimensional Fermi gases and the frequency shift and damping of the sloshing mode due to the anharmonic shape of the experimental trap potential. In the case of two-dimensional Fermi gases, however, the strong damping of the quadrupole mode observed in a recent experiment remains unexplained.