Numerical investigation of the radial quadrupole and scissors modes in trapped 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 suciently accurate as an average relaxation time is used and the high-order moments are ignored. Here, we propose to 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 nd that the analytical expressions underestimate the damping in the transition regime. In addition, we demonstrate that the numerical simulations are able to provide reasonable predictions for the collective oscillations in the Gaussian potentials.