Oscillations of a Bose-Einstein Condensate Rotating in a Harmonic Plus Quartic Trap

We study the normal modes of a two-dimensional rotating Bose-Einstein condensate confined in a quadratic plus quartic trap. Hydrodynamic theory and sum rules are used to derive analytical predictions for the collective frequencies in the limit of high angular velocities $\Omega$ where the vortex lattice produced by the rotation exhibits an annular structure. We predict a class of excitations with frequency $\sqrt{6}\Omega$ in the rotating frame, irrespective of the mode multipolarity $m$, as well as a class of low energy modes with frequency proportional to $|m|/\Omega$. The predictions are in good agreement with results of numerical simulations based on the 2D Gross-Pitaevskii equation. The same analysis is also carried out at even higher angular velocities, where the system enters the giant vortex regime.

Figure 1

Frequency of the lowest $m=0$ mode as a function of the angular velocity $\Omega$ (in units of the trap frequency) for $g=1000$. The sum rule estimates (9) for $\lambda =0.5$ and $\lambda ={10}^{-3}$ are plotted as solid black and dotted lines, respectively, while the results of solving the GP equation numerically are plotted as solid and open circles. The gray line is the asymptotic prediction $\omega =\sqrt{6}\Omega$ and the dashed lines represent the critical frequencies for hole formation, ${\Omega }_h$, for both values of $\lambda$.

Figure 2

Numerical results for the high-lying (circles) and low-lying (squares) $|m|=1$ (open) and $|m|=2$ (closed) mode frequencies as a function of the angular velocity $\Omega$ (in units of the trap frequency), for $g=1000$ and $\lambda =0.5$. These are compared to the analytical results, where solid lines show the sum rule estimates for the low-lying modes while the gray line is the asymptotic prediction ${\omega }_{\mathrm{H}}=\sqrt{6}\Omega$ expected to hold at large $\Omega$. The vertical dashed line denotes the value of ${\Omega }_h$.