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arXiv:cond-mat/0506452 · Physical Review A

Nonclassical scissors mode of a vortex lattice in a Bose-Einstein condensate

Carlos Lobo, Yvan Castin

Published Oct 17, 2005• 17 citations
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Abstract

We show that a Bose-Einstein condensate with a vortex lattice in a rotating anisotropic harmonic potential exhibits a very low frequency scissors mode. The appearance of this mode is due to the SO(2) symmetry breaking introduced by the vortex lattice. The corresponding Goldstone mode in an isotropic trap becomes, in an anisotropic trap, a scissors mode, whose frequency is finite and tends to zero with the trap anisotropy $ϵ$, with a generic $\sqrt{ϵ}$ dependence. We present analytical formulas giving the mode frequency in the low $ϵ$ limit and we show that the mode frequency for some class of vortex lattices can tend to zero as $ϵ$ or faster. We demonstrate that the standard classical hydrodynamics approach fails to reproduce this low frequency mode, because it does not contain the discrete structure of the vortex lattice.

DOI
10.1103/physreva.72.043606

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We show that a Bose-Einstein condensate with a vortex lattice in a rotating anisotropic harmonic potential exhibits a very low frequency scissors mode. The appearance of this mode is due to the SO(2) symmetry breaking introduced by the vortex lattice. The corresponding Goldstone mode in an isotropic trap becomes, in an anisotropic trap, a scissors mode, whose frequency is finite and tends to zero with the trap anisotropy ϵϵϵ, with a generic ϵ\sqrt{ϵ}ϵ​ dependence. We present analytical formulas giving the mode frequency in the low ϵϵϵ limit and we show that the mode frequency for some class of vortex lattices can tend to zero as ϵϵϵ or faster. We demonstrate that the standard classical hydrodynamics approach fails to reproduce this low frequency mode, because it does not contain the discrete structure of the vortex lattice.
We found an analytical solution for the vortex structure in a rapidly rotating trapped Bose-Einstein condensate in the lowest Landau level approximation. This solution is exact in the limit of a large number of vortices and is obtained for the case of a condensate in a anisotropic harmonic potential. The solution describes as limiting cases both a triangle vortex lattice in the symmetric potential trap and a quasi-one-dimensional structure of vortex rows in an asymmetric case, when the rotation frequency is very close to the lower trapping potential frequency. The shape of the density profile is found to be close to the Thomas-Fermi inverted paraboloid form, except in the vicinity of edges of a condensate cloud.
We consider a finite-temperature Bose-Einstein condensate in a quasi-two-dimensional trap containing a single precessing vortex. We find that such a configuration arises naturally as an ergodic equilibrium of the projected Gross-Pitaevskii equation, when constrained to a finite conserved angular momentum. In an isotropic trapping potential, the condensation of the classical field into an off-axis vortex state breaks the rotational symmetry of the system. We present a methodology to identify the condensate and the Goldstone mode associated with the broken rotational symmetry in the classical-field model. We also examine the variation in vortex trajectories and thermodynamic parameters of the field as the energy of the microcanonical field simulation is varied.
A scissors mode of a rotating Bose-Einstein condensate is investigated both theoretically and experimentally. The condensate is confined in an axisymmetric harmonic trap, superimposed with a small rotating deformation. For angular velocities larger than ${\ensuremath{\omega}}_{\ensuremath{\perp}}/\sqrt{2},$ where ${\ensuremath{\omega}}_{\ensuremath{\perp}}$ is the radial trap frequency, the frequency of the scissors mode is predicted to vanish like the square root of the deformation, due to the tendency of the system to exhibit spontaneous rotational symmetry breaking. Measurements of the frequency confirm the predictions of theory. Accompanying characteristic oscillations of the internal shape of the condensate are also calculated and observed experimentally.
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 rotational motion of an interacting atomic Bose-Einstein condensate confined in a quadratic-plus-quartic potential. We calculate the lowest energy surface mode frequency and show that a symmetric trapped (harmonic and quartic) Bose-Einstein condensate breaks the rotational symmetry of the Hamiltonian when rotational frequency is greater than one-half of the lowest energy surface mode frequency. We argue that the formation of a vortex is not possible in a noninteracting as well as in an attractive Bose-Einstein condensate confined in a harmonic trap due to the absence of the spontaneous shape deformation, but it can occur which leads to the vortex formation if we add an additional quartic potential. Moreover, the spontaneous shape deformation and consequently the formation of a vortex in an attractive system depends on the strengths of the two-body interaction and the quartic potential.
We study rotational motion of an interacting atomic Bose-Einstein condensate confined in a quadratic-plus-quartic potential. We calculate the lowest energy surface mode frequency and show that a symmetric trapped (harmonic and quartic) Bose-Einstein condensate breaks the rotational symmetry of the Hamiltonian when rotational frequency is greater than one-half of the lowest energy surface mode frequency. We argue that the formation of a vortex is not possible in a non-interacting as well as in an attractive Bose-Einstein condensate confined in a harmonic trap due to the absence of the spontaneous shape deformation, but it can occur which leads to the vortex formation if we add an additional quartic potential. Moreover, the spontaneous shape deformation and consequently the formation of a vortex in an attractive system depends on the strengths of the two-body interaction and the quartic potential.