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Applied Mathematics Research eXpress

Annular Bose-Einstein Condensates in the Lowest Landau Level

Nicolas Rougerie

Published Dec 15, 2010• 6 citations
Publisher

Abstract

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.

DOI
10.1093/amrx/abq020

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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.
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.
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.
When Bose-Einstein condensates are rotated sufficiently fast, a giant vortex phase appears, that is the condensate becomes annular with no vortices in the bulk but a macroscopic phase circulation around the central hole. In a former paper [M. Correggi, N. Rougerie, J. Yngvason, Com-munications in Mathematical Physics 303, 451-308 (2011)] we have studied this phenomenon by minimizing the two dimensional Gross-Pitaevskii energy on the unit disc. In particular we computed an upper bound to the critical speed for the transition to the giant vortex phase. In this paper we confirm that this upper bound is optimal by proving that if the rotation speed is taken slightly below the threshold there are vortices in the condensate. We prove that they gather along a particular circle on which they are uniformly distributed. This is done by providing new upper and lower bounds to the GP energy.
The state of a rotating Bose-Einstein condensate in a harmonic trap is modeled by a wave function that minimizes the Gross-Pitaevskii functional. The resulting minimization problem has two new features compared to other similar functionals arising in condensed matter physics, such as the Ginzburg-Landau functional. Namely, the wave function is defined in all the plane and is normalized relative to the L2L^2L2-norm. This paper deals with the situation when the coupling constant tends to 000 (Thomas-Fermi regime) and the rotation speed is large compared with the first critical speed. It is given the leading order estimate of the ground state energy together with the location of the vortices of the minimizing wave function in the bulk of the condensate. When the rotation speed is inversely proportional to the coupling constant, the condensate is confined in an elliptical region whose conjugate diameter shrinks and whose transverse diameter expands as the rotation speed increases.
Rapidly rotating atomic gases provide a platform for studying phenomena akin to type-II superconductors and quantum Hall systems. Recently, these systems have attracted renewed interest due to technological advances in the trap anisotropy control, in-situ observation capabilities, and cooling and rotating complex atomic species such as dipolar gases. Understanding the vortex lattice formation and quantum melting is crucial for exploring quantum Hall physics in these systems. In this paper, we theoretically investigate the vortex lattices in anisotropic quantum gases. We formulate the rotating gas Hamiltonian in the Landau gauge, and consider the effects of additional perturbations such as the trap potential in the lowest Landau level (LLL). Focusing on the gases with short-range interactions, we obtain the many-body Hamiltonian projected to lowest Landau level. We consider the limit of full Bose-Einstein condensation and obtain the governing Gross-Pitaevskii equation to identify the possible vortex phases. We numerically solve the Gross-Pitaevskii equation using the imaginary time evolution, and demonstrate the possible vortex lattices as a function of anisotropy, rotation speed and interaction strength. We show that the number of states with a support in the LLL, which determines the number of vortices, follows a Thomas-Fermi type scaling albeit with slightly different coefficients from the usual condensates.