Bose-Einstein condensation at finite temperatures: Mean field laws with periodic microstructure

TL;DR

This paper develops mean field laws to describe the coexistence of condensed and normal phases in Bose-Einstein condensates at finite temperatures, incorporating spatially varying interactions and microstructured scattering lengths.

Contribution

It derives coupled nonlinear evolution equations for condensate and thermal states, and introduces a homogenization approach for periodic microstructures in scattering lengths.

Findings

Derived mean field equations for finite-temperature BECs.

Established effective equations via homogenization for microstructured interactions.

Provided a framework for analyzing spatially varying interactions in quantum gases.

Abstract

At finite temperatures below the phase transition point, the Bose-Einstein condensation, the macroscopic occupation of a single quantum state by particles of integer spin, is not complete. In the language of superfluid helium, this means that the superfluid coexists with the normal fluid. Our goal is to describe this coexistence in trapped, dilute atomic gases with repulsive interactions via mean field laws that account for a {\em spatially varying} particle interaction strength. By starting with the $N$-body Hamiltonian, $N\gg 1$, we formally derive a system of coupled, nonlinear evolution equations in $3+1$ dimensions for the following quantities: (i) the wave function of the macroscopically occupied state; and (ii) the single-particle wave functions of thermally excited states. For stationary (bound) states and a scattering length with {\em periodic microstructure} of subscale $\epsilon$, we heuristically extract effective equations of motion via periodic homogenization up to second order in $\epsilon$.