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

TL;DR

This paper rigorously derives mean-field equations for 1D and 2D Bose-Einstein condensates with higher order interactions, revealing the need for Thomas-Fermi profiles and analyzing density distributions under various trapping potentials.

Contribution

It introduces new mean-field equations incorporating higher order interactions and demonstrates the inadequacy of Gaussian profiles, proposing Thomas-Fermi profiles for strongly confined BECs.

Findings

Higher order interactions modify contact interactions in BECs.

Gaussian profiles are inappropriate for cigar-shaped BECs; Thomas-Fermi profiles are better.

Analytical densities depend on the interplay between contact and higher order interactions.

Abstract

We derive rigorously one- and two-dimensional mean-field equations for cigar- and pancake-shaped Bose-Einstein condensates (BEC) with higher order interactions (HOI). 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 presence of the 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.