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Physical Review A

Singularity formation in the Gross-Pitaevskii equation and collapse in Bose-Einstein condensates

Andrei V. Rybin, Ilya P Vadeiko, G. G. Varzugin, JUSSI TIMONEN

Published Feb 12, 2004• 3 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.69.023607

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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.
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.
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 the collapse in a coupled Bose-Einstein condensate of two types of bosons 1 and 2 under the action of a trap using the time-dependent Gross-Pitaevskii equation. The system may undergo collapse when one, two, or three of the scattering lengths aij{a}_{\mathrm{ij}}aij​ for scattering of boson i with j, i,j=1,2,i,j=1,2,i,j=1,2, are negative representing an attractive interaction. Depending on the parameters of the problem a single or both components of the condensate may experience collapse.