Splitting between quadrupole modes of dilute quantum gas in a two-dimensional anisotropic trap

We consider quadrupole excitations of quasi-two-dimensional interacting quantum gas in an anisotropic harmonic oscillator potential at zero temperature. Using the time-dependent variational approach, we calculate a few low-lying collective excitation frequencies of a two-dimensional anisotropic Bose gas. Within the energy weighted sum-rule approach, we derive a general dispersion relation of two quadrupole excitations of a two-dimensional deformed trapped quantum gas. This dispersion relation is valid for both statistics. We show that the quadrupole excitation frequencies obtained from both methods are exactly the same. Using this general dispersion relation, we also calculate the quadrupole frequencies of a two-dimensional unpolarized Fermi gas in an anisotropic trap. For both cases, we obtain analytic expressions for the quadrupole frequencies and the splitting between them for arbitrary value of trap deformation. This splitting decreases with increasing interaction strength for both statistics. For a two-dimensional anisotropic Fermi gas, the two quadrupole frequencies and the splitting between them become independent of the particle number within the Thomas-Fermi approach. Received 21 September 2001 and Received in final form 9 December 2001