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

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.

Figure 1

Behavior of relevant dimensionless quantities as a function of dimensionless rotation speed $\Omega ∕{\omega }_x$ for the typical anisotropy ${\omega }_y∕{\omega }_x=1.2$. (a) Dimensionless normal mode frequencies ${\omega }_{+}∕{\omega }_x$ and ${\omega }_{-}∕{\omega }_x$ and dimensionless auxiliary frequency $\gamma ∕{\omega }_x$; (b) dimensionless polarization parameters ${\beta }_{+}$ and ${\beta }_{-}$.