Validity of Gross–Pitaevskii solutions of harmonically confined BEC gases in reduced dimensions

Abstract

By exact numerical solutions of the Gross–Pitaevskii (GP) equation in 3D, we assess the validity of 1D and 2D approximations in the study of Bose–Einstein condensates confined in harmonic trap potentials. Typically, these approximations are performed when one or more of the harmonic frequencies are much greater than the remaining ones, using arguments based on the adiabatic evolution of the initial approximated state. Deviations from the 3D solution are evaluated as a function of both the effective interaction strength and the ratio between the trap frequencies that define the reduced dimension where the condensate is confined. The observables analyzed are both of stationary and dynamical character, namely, the chemical potential, the wave function profiles, and the time evolution of the approximated 1D and 2D stationary states, considered as initial states in the 3D GP equation. Our study, besides setting quantitative limits on approximations previously developed, should be useful in actual experimental studies where quasi-1D and quasi-2D conditions are assumed. From a qualitative perspective, 1D and 2D approximations certainly become valid when the anisotropy is large, but in addition the interaction strength needs to be above a certain threshold.