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arXiv:1206.5688 · Physical Review A

Trap anharmonicity and sloshing mode of a Fermi gas

Pierre-Alexandre Pantel, D. Davesne, Silvia Chiacchiera, Mi­chael Urban

Published Aug 29, 2012• 16 citations
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Abstract

For a gas trapped in a harmonic potential, the sloshing (or Kohn) mode is undamped and its frequency coincides with the trap frequency, independently of the statistics, interaction, and temperature of the gas. However, experimental trap potentials have usually Gaussian shape and anharmonicity effects appear as the temperature and, in the case of Fermions, the filling of the trap are increased. We study the sloshing mode of a degenerate Fermi gas in an anharmonic trap within the Boltzmann equation, including in-medium effects in both the transport and collision terms. The calculated frequency shifts and damping rates of the sloshing mode due to the trap anharmonicity are in satisfactory agreement with the available experimental data. We also discuss higher-order dipole, octupole, and bending modes and show that the damping of the sloshing mode is caused by its coupling to these modes.

DOI
10.1103/physreva.86.023635

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For a gas trapped in a harmonic potential, the sloshing (or Kohn) mode is undamped and its frequency coincides with the trap frequency, independently of the statistics, interaction, and temperature of the gas. However, experimental trap potentials have usually Gaussian shape and anharmonicity effects appear as the temperature and, in the case of Fermions, the filling of the trap are increased. We study the sloshing mode of a degenerate Fermi gas in an anharmonic trap within the Boltzmann equation, including in-medium effects in both the transport and collision terms. The calculated frequency shifts and damping rates of the sloshing mode due to the trap anharmonicity are in satisfactory agreement with the available experimental data. We also discuss higher-order dipole, octupole, and bending modes and show that the damping of the sloshing mode is caused by its coupling to these modes.
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 develop a hydrodynamic description of the collective modes of interacting liquids in a quasi-one-dimensional confining potential. By solving Navier-Stokes equation we determine analytically the excitation spectrum of sloshing oscillations. For parabolic confinement, the lowest frequency eigenmode is not renormalized by interactions and is protected from decay by the Kohn theorem, which states that center of mass motion decouples from internal dynamics. We find that the combined effect of potential anharmonicity and interactions results in a frequency shift and final lifetime of the Kohn mode. All other excited modes of sloshing oscillations thermalize with the parametrically faster rate. Our results are significant for the interpretation of recent experiments with trapped Fermi gases that observed a weak violation of the Kohn theorem.
We derive analytical expressions for the frequency and damping of the lowest collective modes of a two-dimensional Fermi gas using kinetic theory. For strong coupling, we furthermore show that pairing correlations overcompensate the effects of Pauli blocking on the collision rate for a large range of temperatures, resulting in a rate which is larger than that of a classical gas. Our results agree well with experimental data, and they recover the observed crossover from collisionless to hydrodynamic behavior with increasing coupling for the quadruple mode. Finally, we show that a trap anisotropy within the experimental bounds results in a damping of the breathing mode which is comparable to what is observed, even for a scale-invariant system.
We investigate anomalous damping of the monopole mode of a non-degenerate 3D Bose gas under isotropic harmonic confinement as recently reported by the JILA TOP trap experiment (Lobser et al in preparation). Given a realistic confining potential, we develop a model for studying collective modes that includes the effects of anharmonic corrections to a harmonic potential. By studying the influence of these trap anharmonicities throughout a range of temperatures and collisional regimes, we find that the damping is caused by the joint mechanisms of dephasing and collisional relaxation. Furthermore, the model is complimented by Monte Carlo simulations which are in fair agreement with data from the JILA experiment.
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 consider the collective mode spectrum of a normal Fermi gas in a spherical harmonic trap. Using a self-consistent random-phase approximation, we systematically examine the effects of the two-body interactions on modes of various symmetries. For weak coupling, where the spectrum is shifted very little away from the ideal gas case, a sum-rule approach is shown to work well. For stronger coupling, the interplay between the single particle and collective excitations causes effects such as mode splitting and Landau damping. A finite low-frequency response present at stronger coupling is predicted for the quadrupole mode. Finally, we briefly discuss the effect of a finite temperature on the spectrum, and the excitation of higher collective modes.