We investigate the effect of the anisotropy of a harmonic trap on the behavior of a fast rotating Bose-Einstein condensate. Fast rotation is reached when the rotational velocity is close to the smallest trapping frequency, thereby deconfining the condensate in the corresponding direction. We characterize a regime of velocity and small anisotropy where the behavior is similar to the isotropic case: a triangular Abrikosov lattice of vortices, with an inverted parabola profile. Nevertheless, at sufficiently large velocity, we find that the ground state does not display vortices in the bulk. We show that the coarse-grained atomic density behaves like an inverted parabola with large radius in the deconfined direction and keeps a fixed profile given by a Gaussian in the other direction. The description is made within the lowest-Landau-level set of states, but using distorted complex coordinates.
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.
The fast rotation of a Bose Einstein condensate in a harmonic trap leads to a hexagonal lattice of vortices, called the Abrikosov lattice. Vortices are described as the zeros of the condensate wave function, which is a solution to a nonlinear Schrödinger equation type. We study the solutions of this equation on a specific eigenspace of the limiting problem, the Lowest Landau Level. Upper and lower bounds for the energy allow to determine the presence of a vortex lattice or the lack of visible vortices, as a function of the anisotropy and the rotation. The study of the critical regime is open and would need some numerical simulations. This constitutes a project in collaboration with Xavier Blanc and Nicolas Lerner.
A rapidly rotating Bose-Einstein condensate in a symmetric two-dimensional harmonic trap can be described with the lowest Landau-level set of single-particle states. The condensate wave function $\ensuremath{\psi}(x,y)$ is a Gaussian $\ensuremath{\propto}\mathrm{exp}(\ensuremath{-}{r}^{2}/2)$, multiplied by an analytic function
f(z) of the complex variable
z=x+iy. The criterion for a quantum phase transition to a non-superfluid correlated many-body state is usually expressed in terms of the ratio of the number of particles to the number of vortices. Here a similar description applies to a rapidly rotating nonsymmetric two-dimensional trap with arbitrary quadratic anisotropy (${\ensuremath{\omega}}_{x}^{2}<{\ensuremath{\omega}}_{y}^{2}$). The corresponding condensate wave function $\ensuremath{\psi}(x,y)$ is a complex anisotropic Gaussian with a phase proportional to
xy, multiplied by an analytic function
f(z), where $z=x+i{\ensuremath{\beta}}_{\ensuremath{-}}y$ is a stretched complex variable and $0\ensuremath{\leqslant}{\ensuremath{\beta}}_{\ensuremath{-}}\ensuremath{\leqslant}1$ is a real parameter that depends on the trap anisotropy and the rotation frequency. Both in the mean-field Thomas-Fermi approximation and in the mean-field lowest Landau level approximation with many visible vortices, an anisotropic parabolic density profile minimizes the energy. An elongated condensate grows along the soft trap direction yet ultimately shrinks along the tight trap direction. The criterion for the quantum phase transition to a correlated state is generalized (
1) in terms of
N/Lz, which suggests that a nonsymmetric trap should make it easier to observe this transition, or (
2) in terms of a ``fragmented'' correlated state, which suggests that a nonsymmetric trap should make it harder to observe this transition. An alternative scenario involves a crossover to a quasi one-dimensional condensate without visible vortices, as suggested by Aftalion et al., Phys. Rev. A 79, 011603(R) (2009).
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 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 a harmonically confined Bose-Einstein condensate under rotation. Vortex lattice configurations are investigated through a variational approach. Vortices with more than a unit of angular momentum are not stable. We explicitly show that the critical rotational frequency is quite small, as was reported in a recent experiment, with an asymmetric harmonic confining potential. The width of the condensate along the z direction (the anisotropy direction) and in the xy plane and the vortex density are determined as functions of the rotational frequency. The aspect ratio becomes small for large rotational frequencies, making the condensate an effectively two-dimensional system.