Lowest-Landau-level description of a Bose-Einstein condensate in a rapidly rotating anisotropic trap

Abstract

A rapidly rotating Bose-Einstein condensate in a symmetric two-dimensional trap can be described with the lowest Landau-level set of states. In this case, the condensate wave function $\psi (x,y)$ is a Gaussian function of $r^2=x^2+y^2$, multiplied by an analytic function $P\left(z\right)$ of the single complex variable $z=x+iy$; the zeros of $P\left(z\right)$ denote the positions of the vortices. Here, a similar description is used for a rapidly rotating anisotropic two-dimensional trap with arbitrary anisotropy $({\omega }_x∕{\omega }_y⩽1)$. The corresponding condensate wave function $\psi (x,y)$ has the form of a complex anisotropic Gaussian with a phase proportional to $xy$, multiplied by an analytic function $P\left(\zeta \right)$, where $\zeta \propto x+i{\beta }_{-}y$ and $0⩽{\beta }_{-}⩽1$ is a real parameter that depends on the trap anisotropy and the rotation frequency. The zeros of $P\left(\zeta \right)$ again fix the locations of the vortices. Within the set of lowest Landau-level states at zero temperature, an anisotropic parabolic density profile provides an absolute minimum for the energy, with the vortex density decreasing slowly and anisotropically away from the trap center.