Entanglement-based perturbation theory for highly anisotropic Bose-Einstein condensates

TL;DR

This paper develops an analytical perturbation theory based on entanglement concepts to accurately describe the behavior of highly anisotropic Bose-Einstein condensates, including corrections to reduced-dimension models and entanglement quantification.

Contribution

It introduces a novel entanglement-based perturbation approach for anisotropic BECs, enabling correction calculations and entanglement measurement between spatial directions.

Findings

Analytical model agrees well with numerical solutions.

Entanglement remains small even at higher nonlinearities.

Condensate can be approximated by a product wave function.

Abstract

We investigate the emergence of three-dimensional behavior in a reduced-dimension Bose-Einstein condensate trapped by a highly anisotropic potential. We handle the problem analytically by performing a perturbative Schmidt decomposition of the condensate wave function between the tightly confined (transverse) direction(s) and the loosely confined (longitudinal) direction(s). The perturbation theory is valid when the nonlinear scattering energy is small compared to the transverse energy scales. Our approach provides a straightforward way, first, to derive corrections to the transverse and longitudinal wave functions of the reduced-dimension approximation and, second, to calculate the amount of entanglement that arises between the transverse and longitudinal spatial directions. Numerical integration of the three-dimensional Gross-Pitaevskii equation for different cigar-shaped potentials and experimentally accessible parameters reveals good agreement with our analytical model even for relatively high nonlinearities. In particular, we show that even for such stronger nonlinearities the entanglement remains remarkably small, which allows the condensate to be well described by a product wave function that corresponds to a single Schmidt term.