Variational simulation of atom interferometry sequences in the presence of anharmonic potentials

We have studied 1D and 3D atom interferometry (AI) sequences carried out with Bose-Einstein condensates (BECs) in the presence of external potentials consisting of harmonic plus anharmonic terms. The process where a BEC is split into multiple clouds and then recombined to form an interference pattern was numerically simulated by solving the Gross-Pitaevskii equation (GPE). Approximate solutions to this equation were generated using the Lagrangian Variational Method (LVM) whose trial wave function consisted of a sum of Gaussian functions, one for each cloud. The variational parameters for each cloud included time-dependent centers and widths and linear and quadratic phase parameters. We will derive the LVM equations of motion as well as a formula for the phase of the trial wave function. This phase enables the LVM to predict the form of the resulting interference pattern. The variational result is compared to the numerical solution of the GPE to assess the ability of the variational approximation to model the effect of both cubic and quartic anharmonic terms in 1D and quartic terms in the 3D. We will also compare these results with those given by time-dependent perturbation theory [1].