Trends in Resonance Energy Shifts and Decay Rates for Bose Condensates in a Harmonic Trap

This is a study of quasi-discrete Bogoliubov quasi-particles in a spherically symmetric harmonic trap. We first evaluate analytically the aymptotic energy shifts of the high energy modes and find them to have \(1/sqrt(n)\) dependence on the number of radial nodes, \(n\), consistent with earlier semiclassical discussions. To address the question of the widths or decay rates, we attempt to clarify previous discussions by deriving an implicit equation for the widths from an assumption of exponential decay. Numerically, we study the trends in the behavior of the widths as a function of temperature, energy, particle number and scattering lengths. In particular, we find that the width due to Landau decay rises rapidly at low \(n\) and then declines, while the Beliaev decay rate rises slowly with \(n\). As temperature goes to zero, Beliaev decay reaches a constant (>0 for n>0), while the Landau decay rate goes to zero. The decay rate is approximately linear in the s-wave scattering length.