Mean-field regime and Thomas–Fermi approximations of trapped Bose–Einstein condensates with higher-order interactions in one and two dimensions

Abstract

We derive rigorously one- and two-dimensional mean-field equations for cigar- and pancake-shaped Bose–Einstein condensates (BECs) with higher-order interactions (HOIs), which originate from shape-dependent confinement corrections to the effective two-body atomic interaction potential. We show how the higher-order interaction modifies the contact interaction of the strongly confined particles. Surprisingly, we find that the usual Gaussian profile assumption for the strongly confining direction is inappropriate for the cigar-shaped BEC case, and a Thomas–Fermi-type profile should be adopted instead. Based on the derived mean-field equations, the Thomas–Fermi densities are analyzed in the presence of the contact interaction and HOI, and considering the limit of large contact interaction and HOI. For both box and harmonic traps in one, two and three dimensions, we identify the analytical Thomas–Fermi densities, which depend on the competition between the contact interaction and the HOI.