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arXiv:0812.0239 · Physical Review A

Collective modes of trapped Fermi gases with in-medium interaction

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

Published Mar 12, 2009• 27 citations
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Abstract

Due to Pauli blocking of intermediate states, the scattering matrix (or $T$ 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 $T$ 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.

DOI
10.1103/physreva.79.033613

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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 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.
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.
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.
We calculate the effects of two-body interactions on the low-frequency oscillations of a normal Fermi gas confined in a harmonic trap. The mean-field contribution to the collective frequencies is evaluated in the collisionless regime using a sum-rule approach. We also discuss the transition between the collisionless and hydrodynamic regimes with special emphasis on the spin dipole mode in which two atomic clouds occupying different spin states oscillate in the opposite phase. The spin dipole mode is predicted to be overdamped in the hydrodynamic regime. The relaxation time is calculated as a function of temperature and the effects of Fermi statistics are explicitly pointed out.
The viscous relaxation time of a trapped two-component gas of fermions in its normal phase is calculated as a function of temperature and scattering length, with the collision probability being determined by an energy-dependent sss-wave cross section. The result is used for calculating the temperature dependence of the frequency and damping of collective modes studied in recent experiments, starting from the kinetic equation for the fermion distribution function with mean-field effects included in the streaming terms.