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arXiv:cond-mat/0404112 · Physical Review A

Vortex lattices in the lowest Landau level for confined Bose-Einstein condensates

Nigel R. Cooper, Stavros Komineas, N. Read

Published Sep 17, 2004• 69 citations
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Abstract

We present the results of numerical calculations of the ground states of weakly interacting Bose-Einstein condensates containing large numbers of vortices. Our calculations show that these ground states appear to be close to uniform triangular vortex lattices. However, slight deviations from a uniform triangular lattice have dramatic consequences on the overall particle distribution. In particular, we demonstrate that the overall particle distribution averaged on a length scale large compared to the vortex lattice constant is well approximated by a Thomas-Fermi profile.

DOI
10.1103/physreva.70.033604

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We present the results of numerical calculations of the ground states of weakly interacting Bose-Einstein condensates containing large numbers of vortices. Our calculations show that these ground states appear to be close to uniform triangular vortex lattices. However, slight deviations from a uniform triangular lattice have dramatic consequences on the overall particle distribution. In particular, we demonstrate that the overall particle distribution averaged on a length scale large compared to the vortex lattice constant is well approximated by a Thomas-Fermi profile.
We present an explicit analytical analysis of the ground state of vortex lattice structures, based on a minimization of the Gross-Pitaevskii energy functional in a fast rotating Bose-Einstein condensate between the Thomas-Fermi and the lowest-Landau-level regimes. By a Bogoliubov-like transformation we find that the coarse-grained average of the atomic density varies as ${[1\ensuremath{-}({x}^{2}+{y}^{2})∕{R}_{\ensuremath{\perp}}^{2}]}^{3∕2}$ in the xyxyxy plane, and that the vortices form an almost regular triangular lattice. Our variational calculation is quantitatively and qualitatively consistent with recent analytical, numerical, and experimental data.
We present an analytical solution for the vortex lattice in a rapidly rotating trapped Bose-Einstein condensate in the lowest Landau level and discuss deviations from the Thomas-Fermi density profile. This solution is exact in the limit of a large number of vortices and is obtained for the cases of circularly symmetric and narrow channel geometries. The latter is realized when the trapping frequencies in the plane perpendicular to the rotation axis are different from each other and the rotation frequency is equal to the smallest of them. This leads to the cancellation of the trapping potential in the direction of the weaker confinement and makes the system infinitely elongated in this direction. For this case we calculate the phase diagram as a function of the interaction strength and rotation frequency and identify the order of quantum phase transitions between the states with a different number of vortex rows.
We consider a periodic vortex lattice in a rotating Bose-Einstein-condensed gas, where the centrifugal potential is exactly compensated by the external harmonic trap. By introducing a gauge transformation which makes the Hamiltonian periodic, we numerically solve the two-dimensional (2D) Gross-Pitaevskii equation finding the exact mean field ground state. In particular, we explore the crossover between the Thomas-Fermi regime, holding for large values of the coupling constant, and the lowest Landau level limit, corresponding to the weakly interacting case. Explicit results are given for the equation of state, the vortex core size, as well as the elastic shear modulus, which is crucial for the calculation of the Tkachenko frequencies.
We show that, within mean-field theory, the density profile of a rapidly rotating harmonically trapped Bose-Einstein condensate is of the Thomas-Fermi form as long as the number of vortices is much larger than unity. Two forms of the condensate wave function are explored: (i) the lowest Landau level (LLL) wave function with a regular lattice of vortices multiplied by a slowly varying envelope function, which gives rise to components in higher Landau levels; (ii) the LLL wave function with a nonuniform vortex lattice. From variational calculations, we find it most favorable energetically to retain the LLL form of the wave function but to allow the vortices to deviate slightly from a regular lattice. The predicted distortions of the lattice are small, but in accord with recent measurements at lower rates of rotation.
We study the vortex lattices in a Bose-Einstein condensate in a rotating anisotropic harmonic trap. We first investigate the single-particle wave functions obtained by the exact solution of the problem and give simple expressions for these wave functions in the small anisotropy limit. Depending on the strength of the interactions, a few or a large number of vortices can be formed. In the limit of many vortices, we calculate the density profile of the cloud and show that the vortex lattice stays triangular. We also find that the vortex lattice planes align themselves with the weak axis of the external potential. For a small number of vortices, we numerically solve the Gross-Pitaevskii equation and find vortex configurations that are very different from the vortex configurations in an axisymmetric rotating trap.
We study the ground states of rotating atomic Bose-Einstein condensates with dipolar interactions. We present the results of numerical studies on a periodic geometry which show vortex lattice ground states of various symmetries: triangular and square vortex lattices, ``stripe crystal,'' and ``bubble crystal.'' We present the phase diagram (for systems with a large number of vortices) as a function of the ratio of dipolar to contact interactions and of the chemical potential. We discuss the experimental requirements for observing transitions between vortex lattice ground states via dipolar interactions. We finally investigate the stability of mean-field supersolid phases of a quasi-two-dimensional nonrotating Bose gas with dipolar interactions.