Vortex core structure and global properties of rapidly rotating Bose-Einstein condensates

We develop an approach for calculating stationary states of rotating Bose-Einstein condensates in harmonic traps which is applicable for arbitrary ratios of the rotation frequency to the transverse frequency of the trap ${\omega }_{\perp }$. Assuming the number of vortices to be large, we write the condensate wave function as the product of a function that describes the structure of individual vortices times an envelope function varying slowly on the scale of the vortex spacing. By minimizing the energy, we derive Gross-Pitaevskii equations that determine the properties of individual vortices and the global structure of the cloud. For low rotation rates, the structure of a vortex is that of an isolated vortex in a uniform medium, while for rotation rates approaching the frequency of the trap (the mean-field lowest-Landau-level regime), the structure is that of the lowest $p$-wave state of a particle in a harmonic trap with frequency ${\omega }_{\perp }$. The global structure of the cloud is determined by minimizing the energy with respect to variations of the envelope function; for conditions appropriate to most experimental investigations to date, we predict that the transverse density profile of the cloud will be of the Thomas-Fermi form, rather than the Gaussian structure predicted on the assumption that the wave function consists only of components in the lowest Landau level for a regular array of vortices.

Figure 1

Density within a vortex core in units of the average density in the cell, as a function of the transverse radius in units of the core radius $\ell$: (a) the single vortex form (21); (b) the linear core approximation (22); (c) the lowest-Landau-level structure (30); and (d) the free particle Bessel function $J_1$ (dashed). Curves $a$ and $b$ are calculated for ${\xi }_0=0.1\ell$.

Figure 2

Variation of the core size with rotational velocity, in the linear approximation to the core structure.

Figure 3

Mean-square radius of the core in units of ${\ell }^2$, as a function of rotational rapidity $y$ computed for the linear core approximation. The solid line shows the exact limit for condensation in the lowest Landau level.