A three-stage analysis of standing waves in an atomic Bose-Einstein Condensate

Abstract

We present three ways to identify the mode functions and frequencies of standing waves in a Bose-Einstein Condensate (BEC). Firstly, we treat an easier system in prolate spheroidal coordinates, which has already been solved, and we use this knowledge to implement our boundary conditions, which can be done in a rather satisfying way. Secondly, we assume a cylindrical-shaped BEC in order to get exact solutions of a more complicated (but still incomplete) system. This results in the same mode functions as used before in literature as a variational Ansatz, and therefore supports this Ansatz. Finally, we will use perturbation theory on the entire system to determine the first-order corrections on the frequencies.

When we compare our results with simulation data, we find that the frequencies (up to first order) are in the right regime, but in order to improve the accuracy, a different strategy will be needed, since the determination of the first-order mode functions (and hence the higher-order frequencies) seems to require degenerate perturbation theory.