Critical rotation of an annular superfluid Bose-Einstein condensate

We analyze the excitation spectrum of a superfluid Bose-Einstein condensate rotating in a ring trap. We identify two important branches of the spectrum related to outer and inner edge surface modes that lead to the instability of the superfluid. Depending on the initial circulation of the annular condensate, either the outer or the inner modes become first unstable. This instability is crucially related to the superfluid nature of the rotating gas. In particular, we point out the existence of a maximal circulation above which the superflow decays spontaneously, which cannot be explained by invoking the average speed of sound.

Figure 1

Sketch of the system: A Bose gas flowing with a circulation $\ell$ (dashed line) and a Thomas-Fermi width $2R$ is held in a ring trap of radius $r_0$. A defect rotating counterflow at angular velocity $\Omega$ above the critical velocity will induce dissipation.

Figure 2

Excitation spectrum obtained from Eq. (1) (see text for details), with $\ell =0$, $r_0=12$, and $g=9000$. Only the lowest eleven branches are shown. The solid lines are a guide to the eye to distinguish between the different branches. A symmetric spectrum also exists for negative values of $m$. Solid lines: relation dispersion in the surface mode model for the inner (upper blue line) and outer (lower red line) modes.

Figure 3

Inverse of the critical angular velocity as a function of the ring radius $r_0$, at fixed chemical potential [25], as obtained from the full numerical calculation (dots). Inset: critical angular velocity as a function of $\mu$ for $r_0=40$. In both graphs, the solid line is the model of Eq. (4) with $r_e=r_0+R$.

Figure 4

(a) Lower branch of the excitation spectrum for $m<0$, $r_0=12$, $g=9000$, and initial states with increasing circulation, $\ell =0,6,11,12,13,15$. Inset: full spectrum for $\ell =11$ (black solid line) and surface mode model relation dispersion for outer mode (red dashed line) or inner mode (blue dotted line). (b) Radial density profiles of the condensate (black solid line), soundlike mode ($\ell =0$, $m=-1$, green dash-dotted line, scaled $\times 3$ for clarity), inner edge surface mode ($\ell =15$, $m=-14$, blue dotted line), and outer edge surface mode ($\ell =0$, $m=-34$, red dashed line). (c),(d) Corresponding plot of the atomic density with a population of 10$\%$ in the inner mode (c) and the outer mode (d). The size of both images is $40\times 40$ in units of $a_r$. At the periphery of the gas, 14 vortices appear in (c) and 34 antivortices appear in (d).

Figure 5

Critical angular velocity versus initial state circulation, as obtained from the numerical solution (dots), for $r_0=20$ and $g=15000$. Inset: slope of the critical velocity for inner (blue squares) and outer (red dots) modes versus the ring radius, at fixed chemical potential [25]. For both graphs, the lines are the predictions of Eq. (5), for $r_e=r_0+R$ (solid line) and $r_e=r_0-R$ (dashed line).