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Fluid Dynamics Research

Gross–Pitaevskii dynamics of Bose–Einstein condensates and superfluid turbulence

Malek Abid, Cristián Huepe, Stéphane Métens, Caroline Nore, Chi-Tuong Pham, Laurette S. Tuckerman, Marc Brächet

Published Oct 15, 2003• 61 citations
Publisher

Abstract

The Gross–Pitaevskii equation, also called the nonlinear Schrödinger equation (NLSE), describes the dynamics of low-temperature superflows and Bose–Einstein Condensates (BEC). We review some of our recent NLSE-based numerical studies of superfluid turbulence and BEC stability. The relations with experiments are discussed.
DOI
10.1016/j.fluiddyn.2003.09.001

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The Gross–Pitaevskii equation (GPE) describes the dynamics of superflows and Bose–Einstein Condensates (BEC) at very low temperature. When a truncation of Fourier modes is performed, the resulting truncated GPE (TGPE) can also describe the correct thermal behavior of a Bose gas, as long as all relevant modes are highly occupied [M.J. Davis, S.A. Morgan, K. Burnett, Simulations of Bose fields at finite temperature, Phys. Rev. Lett. 87 (16) (2001) 160402]. We review some of our groupʼs recent GPE- and TGPE-based numerical studies of superfluid dynamics and BEC stability. The relations with experiments are discussed.
The Gross-Pitaevskii equation, or more generally the nonlinear Schrödinger equation, models the Bose-Einstein condensates in a macroscopic gaseous superfluid wave-matter state in ultra-cold temperature. We provide analytical study of the NLS with L2 initial data in order to understand propagation of the defocusing and focusing waves for the BEC mechanism in the presence of electromagnetic fields. Numerical simulations are performed for the two-dimensional GPE with anisotropic quadratic potentials.
A variational technique is applied to solve the time-dependent nonlinear Schr\"odinger equation (Gross-Pitaevskii equation) with the goal to model the dynamics of dilute ultracold atom clouds in the Bose-Einstein condensed phase. We derive analytical predictions for the collapse, equilibrium widths, and evolution laws of the condensate parameters and find them to be in very good agreement with our numerical simulations of the nonlinear Schr\"odinger equation. It is found that not only the number of particles, but also both the initial width of the condensate and the effect of different perturbations to the condensate may play a crucial role in the collapse dynamics. The results are applicable when the shape of the condensate is sufficiently simple.
Summary form only given, as follows. The physical basis of the theory being used to describe Bose-Einstein condensed (BEC) gases will be described. The simplest and indeed most successful approach uses mean field theories of the phenomena. In the case of the lowest temperatures reached, this theory reduces to the study of the Gross-Pitaevskii nonlinear Schrodinger equation. I will describe the extensions of mean field theory to finite temperatures and show how dynamics and damping can be included. I will address the links to superfluidity and time-evolving order parameters in general.