Collisionless Modes of a Trapped Bose Gas

Abstract

Since the experimental realization of a Bose condensate in trapped alkali vapors, there has been renewed interest into the subject of degenerate quantum gases. Topics like the equilibrium properties of the condensate, the dynamics of condensate formation, topological defects, and collective excitations have been studied extensively. Of particular interest are the collective excitations in the collisionless regime, because in this regime, where the mean free path of the Bogoliubov quasiparticles is much larger than the wavelength of the collective exitations, there seems to be a discrepancy between the experimental observations and theoretical calculations. At temperatures far below the critical temperature T _c, measurements of the low-lying collective excitations [1,2] are in excellent agreement with theoretical calculations solving the Gross-Pitaevskii equation [3, 4, 5, 6], which describes the condensate dynamics at zero temperature. At higher temperatures there is a considerable noncondensate fraction, and one has to include the mean-field interaction of the thermal cloud into the evolution equation for the condensate wave function. Theoretical calculations solving the resulting nonlinear Schrödinger equation predict almost no temperature dependence of the lowest exitation frequencies [7, 8], whereas experiments clearly show a large temperature dependence [9]. This might partly be explained by including in the effective interaction between two colliding particles the many-body effect of the surrounding gas on the collisions, which causes the effective two-particle interaction to become strongly temperature dependent [10]. The frequencies of the low-lying modes will therefore also depend on temperature [11]. However, in these approaches the nonlinear Schrödinger equation describes the dynamics of the condensate in the presence of a static noncondensed cloud. As a result they violate the Kohn theorem, which states that there should always be three center-of-mass modes with the trapping frequencies. Clearly, this violation is caused by the fact that we also have to describe the time evolution of the thermal cloud. Hence we propose to describe the collective excitations in the collisionless regime by a nonlinear Schrödinger equation for the condensate wave function that is coupled to a collisionless Boltzmann equation describing the dynamics of the noncondensed atoms[12, 13]. This resolves our problem, because the resulting theory can be shown to contain the Kohn modes exactly. A full solution of the collisionless Boltzmann equation for the distribution function of the Bogoliubov quasiparticles is rather complicated. Therefore we apply as a first step in this article the Hartree-Fock approximation for the quasiparticle dispersion. This is appropriate in the most interesting region, near the critical temperature T _C, where the mean-field interaction of the condensate is small compared to the average kinetic energy of the noncondensed cloud. We then determine the eigenfrequencies of the low-lying modes by a variational approach and compare these to the experiments.