Analytic vortex dynamics in an annular Bose-Einstein condensate

We consider analytically the dynamics of an arbitrary number and configuration of vortices in an annular Bose-Einstein condensate obtaining expressions for the free energy and vortex precession rates to logarithmic accuracy. We also obtain lower bounds for the lifetime of a single vortex in the annulus. Our results enable a closed-form analytic treatment of vortex-vortex interactions in the annulus that is exact in the incompressible limit. The incompressible hydrodynamics that is developed here paves the way for more general analytical treatments of vortex dynamics in non-simply-connected geometries.

Figure 1

(a) The free energy of a vortex in a uniform annular condensate, Eq. (3), with $\xi /b=0.01$, and $a/b=1/8$ (solid line). Also, the Thomas-Fermi result is shown (dotted line) with $r_{\mathrm{trap}}/b=(a/b+1)/2$. (b) The corresponding precession rates around the annulus, Eqs. (9) and (10), respectively. There is a quasistable radius at which $\omega =0$ corresponding to the stationary point of the energy. The dashed-dotted lines correspond to the limit as $a/b\to 0$ of a uniform condensate.

Figure 2

Trajectories of a single vortex. (a) Free-energy contours of two equal vortices such that one vortex is held fixed at $r/b=0.7$ and $\varphi =2\pi /3$ (*) with $a/b=0.05$ [Eq. (6)]. (b) Same as (a) but with opposite vortices. (c) Twelve alternating vortices with 11 held fixed in a necklace of radius $0.9b$, with $a/b=0.8$. (d) Same as (c) but all the vortices are equal. The red thick curves show the condensate boundary.

Figure 3

(a) Lower bound for the lifetime of a vortex drifting towards the inner edge $r=a$. (b) Same as (a) but for a vortex drifting towards the outer edge $r=b$. In both figures, $\xi /b=0.01$, and the unit of time is ${\tau }_0=r_{\mathrm{TF}}^{2}m{b}^4/\left(a_{\mathrm{osc}}^4\hslash \kappa \right)$.

Figure 4

(a) The stream function $\chi$ [i.e., constant-velocity flow lines, Eq. (A8)] for a single vortex (*) at $\rho /b=0.5$ in an annular BEC with $a/b=0.125$. (b) Two (positive) vortices at $\rho /b=0.5$ separated by an angle of $2\pi /9$. (c) Same as (b) but with two opposite vortices. The red circles show the condensate boundaries.