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Journées Équations aux dérivées partielles

Anisotropic Bose-Einstein condensates and metaplectic transformations

Nicolás Lerner

Published Jul 1, 2010• 0 citations
Publisher

Abstract

This conference is a report on a joint work with A. Aftalion and X. Blanc. A detailed paper is available on HAL: Fast rotating condensates in an asymmetric trap ( http://hal.archives-ouvertes.fr/hal-00342278/fr/ ).

DOI
10.5802/jedp.50

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This conference is a report on a joint work with A. Aftalion and X. Blanc. A detailed paper is available on HAL: Fast rotating condensates in an asymmetric trap ( http://hal.archives-ouvertes.fr/hal-00342278/fr/ ).
The structure of the vortex lattice for a fast rotating condensate in a harmonic trap has been studied experimentally and numerically: it is an almost regular hexagonal lattice, with a distortion on the edges. In this paper, we provide rigorous proofs of results announced in [A. Aftalion, X. Blanc, and J. Dalibard, Phys. Rev. A, 71 (2005), p. 023611]. We analyze the vortex pattern in the framework of the Gross–Pitaevskii energy using wave functions in the lowest Landau level. We compute the energy of a regular triangular lattice and of a class of distorted lattices and find the optimal distortion which provides a decay of the wave function similar to an inverted parabola.
The fast rotation of a Bose Einstein condensate in a harmonic trap leads to a hexagonal lattice of vortices, called the Abrikosov lattice. Vortices are described as the zeros of the condensate wave function, which is a solution to a nonlinear Schrödinger equation type. We study the solutions of this equation on a specific eigenspace of the limiting problem, the Lowest Landau Level. Upper and lower bounds for the energy allow to determine the presence of a vortex lattice or the lack of visible vortices, as a function of the anisotropy and the rotation. The study of the critical regime is open and would need some numerical simulations. This constitutes a project in collaboration with Xavier Blanc and Nicolas Lerner.