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Journal of Statistical Mechanics Theory and Experiment

Study of the scissors mode for a trapped unitary Fermi gas based on hydrodynamics

Zhenbang Lu, Jisheng Chen, Jiarong Li

Published Jan 25, 2012• 4 citations
Publisher

Abstract

The study of the scissors mode gives important information about trapped gases. Based on viscous hydrodynamics, a set of differential equations which describes the dynamics of the trapped unitary Fermi gas is formulated. With relevant initial conditions, the solution of these equations reproduces the angular oscillation, i.e. the scissors mode. Besides the frequency and damping, detailed information on the time evolution of the density and velocity distribution can be obtained. This approach provides an effective way to study the scissors mode.

DOI
10.1088/1742-5468/2012/01/p01020

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The study of the scissors mode gives important information about trapped gases. Based on viscous hydrodynamics, a set of differential equations which describes the dynamics of the trapped unitary Fermi gas is formulated. With relevant initial conditions, the solution of these equations reproduces the angular oscillation, i.e. the scissors mode. Besides the frequency and damping, detailed information on the time evolution of the density and velocity distribution can be obtained. This approach provides an effective way to study the scissors mode.
The scissors mode plays a crucial role in the study of the unitary Fermi gas. In this paper, we simulate the scissors mode by solving the hydrodynamic equations with appropriate initial conditions, and then extract the flow field of the gas. The flow fields in different regimes are found to be essentially different. The characteristic differences in the flow patterns between the superfluid and the normal viscous fluid are presented. Irrotational flow signals superfluidity, while rotational one indicates normal hydrodynamical behavior. These different characteristics can be visualized by the velocity portraits of the flow, which provide an intuitive way to discriminate the states of the Fermi gas.
By means of the Boltzmann-Vlasov kinetic equation we investigate dynamical properties of a trapped, one-component Fermi gas at zero temperature, featuring the anisotropic and long-range dipole-dipole interaction. To this end, we determine an approximate solution by rescaling both space and momentum variables of the equilibrium distribution, thereby obtaining coupled ordinary differential equations for the corresponding scaling parameters. Based on previous results on how the Fermi sphere is deformed in the hydrodynamic regime of a dipolar Fermi gas, we are able to implement the relaxation-time approximation for the collision integral. Then, we proceed by linearizing the equations of motion around the equilibrium in order to study both the frequencies and the damping of the low-lying excitation modes all the way from the collisionless to the hydrodynamic regime. Our theoretical results are expected to be relevant for understanding current experiments with trapped dipolar Fermi gases.
By means of the Boltzmann-Vlasov kinetic equation we investigate dynamical properties of a trapped one-component Fermi gas at zero temperature, featuring the anisotropic and long-range dipole-dipole interaction. To this end, we determine an approximate solution by rescaling both space and momentum variables of the equilibrium distribution, thereby obtaining coupled ordinary differential equations for the corresponding scaling parameters. Based on previous results on how the Fermi sphere is deformed in the hydrodynamic regime of a dipolar Fermi gas, we are able to implement the relaxation-time approximation for the collision integral. Then, we proceed by linearizing the equations of motion around the equilibrium in order to study both the frequencies and the damping of the low-lying excitation modes all the way from the collisionless to the hydrodynamic regime. Our theoretical results are expected to be relevant for understanding current experiments with trapped dipolar Fermi gases.
The analytical expressions for the frequency and damping of the radial quadrupole and scissors modes, as obtained from the method of moments, are limited to the harmonic potential. In addition, the analytical results may not be sufficiently accurate as an average relaxation time is used and the high-order moments are ignored. Here, we numerically solve the Boltzmann model equation in the hydrodynamic, transition, and collisionless regimes to study mode frequency and damping. When the gas is trapped by the harmonic potential, we find that the analytical expressions underestimate the damping in the transition regime. Furthermore, we demonstrate that the numerical simulations are able to provide reasonable predictions for the collective oscillations in the Gaussian potentials. The present method can also be used to study many other problems, e.g. formation of quantum shockwave, expansion of atom cloud, and effective heat conductivity in very elongated traps.
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.