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arXiv:0801.3571 · Physical Review A

Lowest-Landau-level vortex structure of a Bose-Einstein condensate rotating in a harmonic plus quartic trap

Xavier Blanc, Nicolas Rougerie

Published May 29, 2008• 23 citations
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Abstract

We investigate the vortex patterns appearing in a two-dimensional annular Bose-Einstein condensate rotating in a quadratic plus quartic confining potential. We show that in the limit of small anharmonicity the Gross-Pitaevskii energy can be minimized among the lowest-Landau-level wave functions and use this particular form to obtain theoretical results in the spirit of Aftalion et al. [Phys. Rev. A 73, 011601(R) (2006)]. In particular, we show that the vortex pattern is infinite but not uniform. We also compute numerically the complete vortex structure: It is an Abrikosov lattice strongly distorted near the edges of the condensate with multiply quantized vortices appearing at the center of the trap.

DOI
10.1103/physreva.77.053615

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We investigate the vortex patterns appearing in a two-dimensional annular Bose-Einstein condensate rotating in a quadratic plus quartic confining potential. We show that in the limit of small anharmonicity the Gross-Pitaevskii energy can be minimized among the lowest-Landau-level wave functions and use this particular form to obtain theoretical results in the spirit of Aftalion et al. [Phys. Rev. A 73, 011601(R) (2006)]. In particular, we show that the vortex pattern is infinite but not uniform. We also compute numerically the complete vortex structure: It is an Abrikosov lattice strongly distorted near the edges of the condensate with multiply quantized vortices appearing at the center of the trap.
A rotating superfluid such as a Bose-Einstein condensate is usually described by the Gross-Pitaevskii (GP) model. An important issue is to determine from this model the properties of the quantized vortices that a superfluid nucleates when set into rotation. In this paper we address the minimization of a two dimensional GP energy functional describing a rotating annular Bose-Einstein condensate. In a certain limit it is physically relevant to restrict the minimimization to the Lowest-Landau-Level, that is the first eigenspace of the Ginzburg-Landau operator. Taking the particular structure of this space into account we obtain theoretical results concerning the vortices of the condensate. We also compute the vortices' locations by a numerical minimization procedure. We find that they lie on a distorted lattice and that multiply quantized vortices appear in the central hole of low matter density.
For a fast rotating condensate in a harmonic trap, we investigate the structure of the vortex lattice using wave functions minimizing the Gross-Pitaevskii energy in the lowest Landau level. We find that the minimizer of the energy in the rotating frame has a distorted vortex lattice for which we plot the typical distribution. We compute analytically the energy of an infinite regular lattice and of a class of distorted lattices. We find the optimal distortion and relate it to the decay of the wave function. Finally, we generalize our method to other trapping potentials.
A rotating superfluid such as a Bose–Einstein condensate (BEC) is usually described by the Gross-Pitaevskii (GP) model. An important issue is to determine from this model the properties of the quantized vortices that a superfluid nucleates when set into rotation. In this paper we address the minimization of a two-dimensional GP energy functional describing a rotating annular BEC. In a certain limit, it is physically relevant to restrict the minimization to the lowest Landau level, that is, the first eigenspace of the Ginzburg–Landau operator. Taking the particular structure of this space into account, we obtain theoretical results concerning the vortices of the condensate. We also compute the vortices’ locations by a numerical minimization procedure. We find that they lie on a distorted lattice and that multiple quantized vortices appear in the central hole of low matter density.
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.
We consider a rotating Bose-Einstein condensate confined in combined harmonic and quartic traps, following recent experiments [V. Bretin, S. Stock, Y. Seurin, and J. Dalibard, Phys. Rev. Lett. 92, 050403 (2004)]. We investigate numerically the behavior of the wave function which solves the three-dimensional Gross Pitaevskii equation and analyze in detail the structure of vortices. For a quartic-plus-harmonic potential, as the angular velocity increases, the vortex lattice evolves into a vortex array with hole. The merging of vortices into the hole is highly three dimensional, starting from the top and bottom of the condensate to reach the center. We also investigate the case of a quartic-minus-harmonic potential, not covered by experiments or previous numerical works. For intermediate repulsive potentials, we show that the transition to a vortex array with hole takes place for lower angular velocities, when the lattice is made up of a small number of vortices. For the strong repulsive case, a transition from a giant vortex to a hole with a circle of vortices around is observed.
We study the normal modes of a two-dimensional rotating Bose-Einstein condensate confined in a quadratic plus quartic trap. Hydrodynamic theory and sum rules are used to derive analytical predictions for the collective frequencies in the limit of high angular velocities Omega where the vortex lattice produced by the rotation exhibits an annular structure. We predict a class of excitations with frequency sqrt[6]Omega in the rotating frame, irrespective of the mode multipolarity m, as well as a class of low energy modes with frequency proportional to |m|/Omega. The predictions are in good agreement with results of numerical simulations based on the 2D Gross-Pitaevskii equation. The same analysis is also carried out at even higher angular velocities, where the system enters the giant vortex regime.