Bogoliubov excitation spectrum of an elongated condensate from quasi-one-dimensional to three-dimensional transition

TL;DR

This paper investigates the excitation spectra of elongated Bose gases during the transition from quasi-one-dimensional to three-dimensional regimes, using an efficient numerical method and analyzing its accuracy and limitations.

Contribution

It introduces a computationally efficient approach to study the Bogoliubov spectra across the 1D to 3D transition and assesses the method's validity and the nature of the crossover.

Findings

Effective 1D-GP equation works well for small atom numbers

The transition from 1D to 3D is not smooth in the model

The 1D-GP equation fails for large atom numbers

Abstract

The quasiparticle excitation spectra of a Bose gas trapped in a highly anisotropic trap is studied with respect to varying total number of particles by numerically solving the effective one-dimensional (1D) Gross-Pitaevskii (GP) equation proposed recently by Mateo \textit{et al.}. We obtain the static properties and Bogoliubov spectra of the system in the high energy domain. This method is computationally efficient and highly accurate for a condensate system undergoing a 1D to three-dimensional (3D) cigar-shaped transition, as is shown through a comparison our results with both those calculated by the 3D-GP equation and analytical results obtained in limiting cases. We identify the applicable parameter space for the effective 1D-GP equation and find that this equation fails to describe a system with large number of atoms. We also identify that the description of the transition from 1D Bose-Einstein condensate (BEC) to 3D cigar-shaped BEC using this equation is not smooth, which highlights the fact that for a finite value of $a_\perp/a_s$ the junction between the 1D and 3D crossover is not perfect.