Dynamics of Bose-Einstein condensates in cigar-shaped traps

TL;DR

This paper derives an effective one-dimensional model for cigar-shaped Bose-Einstein condensates, capturing complex phenomena like solitons and waves, and extends known results to high-density regimes with new predictions.

Contribution

It introduces a novel reduction of the Gross-Pitaevskii equation to an effective system that accounts for high-density effects and predicts new soliton solutions.

Findings

Effective 1D equations describe condensate dynamics along the trap axis.

Reproduction of known low-density results and extension to high-density regimes.

Prediction of bright solitons in dense, repulsive condensates.

Abstract

Gross-Pitaevskii equation for Bose-Einstein condensate confined in elongated cigar-shaped trap is reduced to an effective system of nonlinear equations depending on only one space coordinate along the trap axis. The radial distribution of the condensate density and its radial velocity are approximated by Gaussian functions with real and imaginary exponents, respectively, with parameters depending on the axial coordinate and time. The effective one-dimensional system is applied to description of the ground state of the condensate, to dark and bright solitons, to the sound and radial compression waves propagating in a dense condensate, and to weakly nonlinear waves in repulsive condensate. In the low density limit our results reproduce the known formulae. In the high density case our description of solitons goes beyond the standard approach based on the nonlinear Schr\"odinger equation. The dispersion relations for the sound and radial compression waves are obtained in a wide region of values of the condensate density. Korteweg-de Vries equation for weakly nonlinear waves is derived and existence of bright solitons on a constant background is predicted for the dense enough condensate with repulsive interaction between the atoms.