Nonlinear dynamics of Bose-condensed gases by means of a low- to high-density variational approach

TL;DR

This paper introduces a variational method using a q-Gaussian wave function to effectively model the dynamics of Bose-condensed gases across different density regimes, simplifying the complex Gross-Pitaevskii equation.

Contribution

It develops a unified variational approach that interpolates between low- and high-density limits, reducing the GP equation to three manageable equations.

Findings

The method accurately reproduces GP equation results.

It bridges the gap between Gaussian and hydrodynamic regimes.

The approach simplifies complex nonlinear dynamics.

Abstract

We propose a versatile variational method to investigate the spatio-temporal dynamics of one-dimensional magnetically-trapped Bose-condensed gases. To this end we employ a \emph{q}-Gaussian trial wave-function that interpolates between the low- and the high-density limit of the ground state of a Bose-condensed gas. Our main result consists of reducing the Gross-Pitaevskii equation, a nonlinear partial differential equation describing the T=0 dynamics of the condensate, to a set of only three equations: \emph{two coupled nonlinear ordinary differential equations} describing the phase and the curvature of the wave-function and \emph{a separate algebraic equation} yielding the generalized width. Our equations recover those of the usual Gaussian variational approach (in the low-density regime), and the hydrodynamic equations that describe the high-density regime. Finally, we show a detailed comparison between the numerical results of our equations and those of the original Gross-Pitaevskii equation.