Rapid rotation of a Bose-Einstein condensate in a harmonic plus quartic trap

A two-dimensional rapidly rotating Bose-Einstein condensate in an anharmonic trap with quadratic and quartic radial confinement is studied analytically with the Thomas-Fermi approximation and numerically with the full time-independent Gross-Pitaevskii equation. The quartic trap potential allows the rotation speed $\Omega$ to exceed the radial harmonic frequency ${\omega }_{\perp }$. In the regime $\Omega \gtrsim {\omega }_{\perp }$, the condensate contains a dense vortex array (approximated as solid-body rotation for the analytical studies). At a critical angular velocity ${\Omega }_h$, a central hole appears in the condensate. Numerical studies confirm the predicted value of ${\Omega }_h$, even for interaction parameters that are not in the Thomas-Fermi limit. The behavior is also investigated at larger angular velocities, where the system is expected to undergo a transition to a giant vortex (with pure irrotational flow).

Figure 1

Density profiles of a rotating condensate at $g=80$ and $\lambda =0.5$, for $\Omega =$ (a) 2.0, (b) 2.1, (c) 2.25, (d) 2.5, (e) 3.0, and (f) 3.5. The scale of each figure is $4\times 4$ in units of $d_{\perp }$.

Figure 2

Density profiles of a rotating condensate at $g=1000$ and $\lambda =0.5$, for $\Omega =$ (a) 2.0, (b) 3.0, (c) 3.5, (d) 4.0, (e) 4.5, and (f) 5.0, showing the stable vortice lattice configurations. The scale of each figure is $6\times 6$ in units of $d_{\perp }$.

Figure 3

Phase diagram of a condensate rotating with angular velocity $\Omega$ versus interaction strength $g$, with $\lambda =0.5$, where the solid line is the critical frequency ${\Omega }_h$ [Eq. (12)] derived from our TF analysis. Going from left to right, the first state (denoted VL) is a vortex lattice without a hole, while above ${\Omega }_h$ (solid line) a hole appears (denoted VLH). Then, for $\Omega >{\Omega }_g$ (dashed line) the TF analysis predicts a giant vortex state (denoted GV). In addition, the dotted line indicates the crossover region where the TF approximation fails [defined here by the somewhat arbitrary criterion $d^2\approx 10{\xi }^2$, where $d$ and $\xi$ are given by Eqs. (16, 22)]. For comparison we also show the results of Kavoulakis and Baym [15], where the filled and open circles represent ${\Omega }_h$ and ${\Omega }_g$, respectively. Finally, our numerical results for these quantities are plotted as filled and open triangles.

Figure 4

Energy per particle vs rotation rate $\Omega$, where our numerical results are presented as open circles (vortex-lattice state) and crosses (giant-vortex state). For comparison, the analytical results are plotted as a solid line (Thomas-Fermi vortex lattice with hole) and a dashed line (giant vortex). The inset shows the energies in a small region of the graph around $\Omega =5$. The parameters used here are $g=1000$ and $\lambda =0.5$.

Figure 5

Density profiles of a rotating condensate at $g=1000$ and $\lambda =0.5$, showing the giant-vortex states corresponding to the angular velocities in Fig. 2: $\Omega =\left(a\right)2.0$, (b) 3.0, (c) 3.5, (d) 4.0, (e) 4.5, and (f) 5.0. The scale of each figure is $6\times 6$ in units of $d_{\perp }$.