Matter-wave vortices in cigar-shaped and toroidal waveguides

TL;DR

This paper develops an effective 1D model for vortical states in Bose-Einstein condensates within cigar-shaped and toroidal traps, validating it against 3D simulations and exploring stability and symmetry-breaking phenomena.

Contribution

It derives and validates an accurate 1D nonpolynomial Schrödinger equation for vortices in BECs with both repulsive and attractive interactions, including analytical solutions and stability analysis.

Findings

NPSE accurately predicts vortex density profiles and stability.

Identifies a threshold for symmetry-breaking in toroidal BECs.

Provides analytical formulas for bright solitons in attractive BECs.

Abstract

We study vortical states in a Bose-Einstein condensate (BEC) filling a cigar-shaped trap. An effective one-dimensional (1D) nonpolynomial Schroedinger equation (NPSE) is derived in this setting, for the models with both repulsive and attractive inter-atomic interactions. Analytical formulas for the density profiles are obtained from the NPSE in the case of self-repulsion within the Thomas-Fermi approximation, and in the case of the self-attraction as exact solutions (bright solitons). A crucially important ingredient of the analysis is the comparison of these predictions with direct numerical solutions for the vortex states in the underlying 3D Gross-Pitaevskii equation (GPE). The comparison demonstrates that the NPSE provides for a very accurate approximation, in all the cases, including the prediction of the stability of the bright solitons and collapse threshold for them. In addition to the straight cigar-shaped trap, we also consider a torus-shaped configuration. In that case, we find a threshold for the transition from the axially uniform state, with the transverse intrinsic vorticity, to a symmetry-breaking pattern, due to the instability in the self-attractive BEC filling the circular trap.