Variational methods with coupled Gaussian functions for Bose-Einstein condensates with long-range interactions. II. Applications

TL;DR

This paper demonstrates that coupled Gaussian variational methods effectively analyze Bose-Einstein condensates with long-range interactions, capturing both stable and unstable states and providing insights into their stability and bifurcation behavior.

Contribution

It introduces and validates the coupled Gaussian wave packet approach as a superior alternative to direct numerical solutions for studying long-range interacting condensates.

Findings

Coupled Gaussian method accurately reproduces known solutions.

Identifies bifurcation points leading to condensate collapse.

Provides a stable and unstable state analysis framework.

Abstract

Bose-Einstein condensates with an attractive 1/r interaction and with dipole-dipole interaction are investigated in the framework of the Gaussian variational ansatz introduced by S. Rau, J. Main, and G. Wunner [Phys. Rev. A, submitted]. We demonstrate that the method of coupled Gaussian wave packets is a full-fledged alternative to direct numerical solutions of the Gross-Pitaevskii equation, or even superior in that coupled Gaussians are capable of producing both, stable and unstable states of the Gross-Pitaevskii equation, and thus of giving access to yet unexplored regions of the space of solutions of the Gross-Pitaevskii equation. As an alternative to numerical solutions of the Bogoliubov-de Gennes equations, the stability of the stationary condensate wave functions is investigated by analyzing the stability properties of the dynamical equations of motion for the Gaussian variational parameters in the local vicinity of the stationary fixed points. For blood-cell-shaped dipolar condensates it is shown that on the route to collapse the condensate passes through a pitchfork bifurcation, where the ground state itself turns unstable, before it finally vanishes in a tangent bifurcation.