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arXiv:cond-mat/0101160

Singularity Formation and Collapse in the Attractive Gross-Pitaevskii Equation

Andrei V. Rybin, G. G. Varzugin, JUSSI TIMONEN

Published Jan 11, 2001• 1 citation
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Abstract

A generic mechanism of collapse in the Gross-Pitaevskii equation with attractive interparticle interactions is gained by reformulating this equation as Newton's equation of motion for a system of particles with a constraint. 'Quantum pressure' effects give rise to formation of a potential barrier around the emerging singularity, which prevents a fraction of the particles from falling into the singularity. For reasonable initial widths of the condensate, the fraction of collapsing particles, which are thereby removed from the condensate, is found to be a 'universal' number $\simeq 0.7$.

DOI
10.48550/arxiv.cond-mat/0101160

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•Jan 1, 2002•0 citations
A generic mechanism of collapse in the Gross-Pitaevskii equation with attractive interparticle interactions is gained by reformulating this equation as Newton's equation of motion for a system of particles with a constraint. 'Quantum pressure' effects give rise to formation of a potential barrier around the emerging singularity, which prevents a fraction of the particles from falling into the singularity. For reasonable initial widths of the condensate, the fraction of collapsing particles, which are thereby removed from the condensate, is found to be a 'universal' number ≃0.7\simeq 0.7≃0.7.
A generic mechanism of collapse in the Gross–Pitaevskii equation with attractive interparticle interactions is gained by reformulating this equation as Newton's equation of motion for a system of particles with a constraint. 'Quantum pressure' effects give rise to formation of a potential barrier around the emerging singularity, which prevents a fraction of the particles from falling into the singularity. For reasonable initial widths of the condensate, the fraction of collapsing particles for spherically symmetric traps is found to be consistently about 0.7.
We study a mechanism of collapse of the condensate wave function in the Gross-Pitaevskii theory with attractive interparticle interaction. We reformulate the Gross-Pitaevskii equation as Newton's equations for the particle flux and introduce a collapsing fraction of particles. We assume that the collapsing fraction is expelled from the condensate due to dissipation. Using this hypothesis we analyze the dependence of the condensate collapse on the initial conditions. We found that for a properly chosen negative scattering length the remnant fraction becomes larger when the initial aspect ratio is increased.
We study the mechanisms of collapse of the condensate wave function in the Gross-Pitaevskii theory with attractive interparticle interaction. We reformulate the Gross-Pitaevskii equation as Newton's equations for a flux of particles, and introduce the collapsing fraction of particles. We assume that this collapsing fraction is expelled from the condensate due to dissipation. Using this hypothesis we analyze the dependence of the collapse behavior on the initial conditions. We find that, for a properly chosen negative scattering length, the remnant fraction of atoms becomes larger when the initial aspect ratio of the condensate is increased.
This project aims to study the behavior of a cloud of Rubidium-85 atoms under an attractive interaction potential when used to create a Bose-Einstein condensate. Efforts to describe the dynamics of an attractive condensate using only the Gross-Pitaevskii equation have been unable to describe all of the features of such a system, and so the mathematical approach used is a perturbative method of corrections to the Gross-Pitaevskii equation. A system of analytical equations to describe a condensate is derived herein, which is then arranged into a form suitable for numerical solution. The actual calculation is not performed, but details of the computational procedure are provided, as well as an analysis of the difficulties inherent in the calculation.