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arXiv:cond-mat/0608216 · Physical Review A

Nonuniform vortex lattices in inhomogeneous rotating Bose-Einstein condensates

Gordon Baym, C. J. Pethick, S. Andrew Gifford, Gentaro Watanabe A

Published Jan 3, 2007• 11 citations
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Abstract

We derive a general framework, in terms of elastic theory, for describing the distortion of the vortex lattice in a rotating Bose-Einstein condensate at arbitrary rotation speed and determining the dependence of the distortion on the density inhomogeneity of the system. In the rapidly rotating limit, we derive the energetics in terms of Landau levels, including excitation to higher levels; the distortion depends on the excitation of higher levels as well as on the density gradient. As we show, the dominant effect of higher Landau levels in a distorted lattice in equilibrium is simply to renormalize the frequency entering the lowest Landau level condensate wave function---from the transverse trap frequency, $\ensuremath{\omega}$, to the rotational frequency, $\ensuremath{\Omega}$, of the system. Finally, we show how the equilibrium lattice distortion emerges from elastohydrodynamic theory for inhomogeneous systems.

DOI
10.1103/physreva.75.013602

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We derive a general framework, in terms of elastic theory, for describing the distortion of the vortex lattice in a rotating Bose-Einstein condensate at arbitrary rotation speed and determining the dependence of the distortion on the density inhomogeneity of the system. In the rapidly rotating limit, we derive the energetics in terms of Landau levels, including excitation to higher levels; the distortion depends on the excitation of higher levels as well as on the density gradient. As we show, the dominant effect of higher Landau levels in a distorted lattice in equilibrium is simply to renormalize the frequency entering the lowest Landau level condensate wave function---from the transverse trap frequency, \ensuremathω\ensuremath{\omega}\ensuremathω, to the rotational frequency, \ensuremathΩ\ensuremath{\Omega}\ensuremathΩ, of the system. Finally, we show how the equilibrium lattice distortion emerges from elastohydrodynamic theory for inhomogeneous systems.
"We approach the static elasticity of a vortex lattice using the elastohydrodyamic approach. We derive the elastic Green's function for the vortex lattice, and connect it to the lattice distortion in an inhomogeneous condensate. We find that the longitudinal elastic response of a vortex lattice is screened by the effect of the ""global"" fluid velocity, but the shear response remains the same as conventional elastic systems. Applying the theory of dislocation-mediated melting to the vortex lattice, we find that the lattice melts at a temperature depending only on the shear modulus. Finally, we apply the theory of melting to trapped rapidly rotating BECs."
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.
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 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 derive, from the Gross-Pitaevskii equation, an exact expression for the velocity of any vortex in a Bose-Einstein condensate, in equilibrium or not, in terms of the condensate wave function at the center of the vortex. In general, the vortex velocity is a sum of the local superfluid velocity, plus a correction related to the density gradient near the vortex. A consequence is that in rapidly rotating, harmonically trapped Bose-Einstein condensates, unlike in the usual situation in slowly rotating condensates and in hydrodynamics, vortices do not move with the local fluid velocity. We indicate how Kelvin's conservation of circulation theorem is compatible with the velocity of the vortex center being different from the local fluid velocity. Finally, we derive an exact wave function for a single vortex near the rotation axis in a weakly interacting system, from which we derive the vortex precession rate.
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.