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arXiv:cond-mat/0208312 · Physical Review A

Collapse of Bose-Einstein condensates with dipole-dipole interactions

Pavel M. Lushnikov

Published Nov 12, 2002• 79 citations
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Abstract

The dynamics of Bose-Einstein condensates of a gas of bosonic particles with long-range dipole-dipole interactions in a harmonic trap is studied. Sufficient analytical criteria are found both for catastrophic collapse of Bose-Einstein condensates and for long-time condensate existence. Analytical criteria are compared with variational analysis.

DOI
10.1103/physreva.66.051601

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The dynamics of Bose-Einstein condensates of a gas of bosonic particles with long-range dipole-dipole interactions in a harmonic trap is studied. Sufficient analytical criteria are found both for catastrophic collapse of Bose-Einstein condensates and for long-time condensate existence. Analytical criteria are compared with variational analysis.
We consider dynamics of Bose-Einstein condensates with long-range attractive interaction proportional to 1/rb1/{r}^{b}1/rb and arbitrary angular dependence. It is shown exactly that collapse of a Bose-Einstein condensate without contact interactions is possible only for $b\ensuremath{\geqslant}2$. The case b=2b=2b=2 is critical and requires the number of particles to exceed a critical value to allow collapse. The critical collapse in that case is a strong one, trapping into the collapsing region a finite number of particles. The case b>2b>2b>2 is supercritical with an expected weak collapse which traps a rapidly decreasing number of particles during approach to collapse. For b<2b<2b<2 a singularity at r=0r=0r=0 is not strong enough to allow collapse but an attractive 1/rb1/{r}^{b}1/rb interaction admits stable self-trapping (a stable three-dimensional soliton solution) even in the absence of an external trapping potential.
We present an action that can be used to study variationally the collapse of Bose-Einstein condensates. This action is real, even though it includes dissipative terms. It adopts long-range interactions between the atoms, so that there is always a stable minimum of the energy, even if the remaining number of atoms is above the number that in the case of local interactions is the critical one. The proposed action incorporates the time needed for the abrupt and delayed onset of collapse, yielding in fact its dependence on the scattering length. We show that the evolution of the condensate is equivalent to the motion of a particle in an effective potential. The particle begins its motion far from the point of stable equilibrium and it then proceeds to oscillate about that point. We prove that the resulting large oscillations in the shape of the wave function after the collapse have frequencies equal to twice the frequencies of the traps. Our results agree with the experimental observations.
Abstract. The motto of this paper is: Let's face Bose-Einstein condensation through nonUnear dynamics. We do this by choosing variational forms of the condensate wave functions (of given symmetry classes), which convert the Bose-Einstein condensates via the time-dependent Gross-Pitaevskii equation into Hamiltonian systems that can be studied using the methods of nonUnear dynamics. We consider in particular cold quantum gases where long-range interactions between the neutral atoms are present, in addition to the conventional short-range contact interaction, viz. gravity-Uke interactions, and dipole-dipole interactions. The residts obtained serve as a useful giude in the search for nonlinear dynamics effects in numerically exact quantum calcinations for Bose-Einstein condensates. A main residt is the prediction of the existence of stable islands as well as chaotic regions for excited states of dipolar condensates, which coidd be checked experimentally. Keywords: Bose-Einstein condensation, Gross-Pitaevskii equation, nonlinear dynamics effects PACS: 03.75.Kk, 34.20.Cf, 02.30.-f, 47.20.Ky 1.