Vortex signatures in annular Bose-Einstein condensates

We consider a Bose-Einstein condensate confined in a “Mexican hat” or sombrero potential, with a quartic minus quadratic radial dependence. We find conditions under which the ground state is annular in shape, with a hole in the center of the condensate. Rotation leads to the appearance of stable multiply quantized vortices, giving rise to a superfluid flow around the ring. The collective modes of the system are explored both numerically and analytically using the Gross-Pitaevskii and hydrodynamic equations. Potential experimental schemes to detect vorticity are proposed and evaluated, which include measuring the splitting of collective-mode frequencies, observing expansion following release from the trap, and probing the momentum distribution of the condensate.

Figure 1

Density profiles of a condensate, found by numerical solution of the GP equation (20) with $g=1000$ and $\lambda =0.01$ $(\eta =0.726)$ for (a) $\Omega =0.2$ (in units of ${\omega }_{\perp }$), (b) $\Omega =0.4$, and (c) $\Omega =0.5$; (d)–(f) show the corresponding phase profiles for each $\Omega$. The length scale of each box is $28\times 28$ (in units of $d_{\perp }$).

Figure 2

Energy (in units of $\hslash {\omega }_{\perp }$) of the condensate with a central vortex as a function of vorticity $\nu$ for the same parameters as Fig. 1. The circles plot numerical solutions of the GP equation, while the solid line is the analytical estimate (17). Inset: vorticity of the ground state in a frame rotating with angular velocity $\Omega$ (in units of ${\omega }_{\perp }$). Circles are numerical GP solutions, while the solid line plots the analytical result (19).

Figure 3

(a) Frequency (in units of ${\omega }_{\perp }$) of the high-lying $(j=1)$ $m=0$ (solid line) $m=1$ (dashed) and $m=2$ (dotted) modes as a function of $\eta$ for a nonrotating condensate $(\Omega =0)$, from numerical solution of the hydrodynamic equation (25). Also plotted are numerical solutions of the Gross-Pitaevskii equation (3) at $g=1000$ for $m=0$ (◯), $m=1$ (◻), and $m=2$ (▵). (b) Frequencies of the low-lying $(j=0)$ $m=1$ and $m=2$ modes, where the grey lines show the analytical result (28) for each $m$, while the other lines and points are labeled similarly to (a).

Figure 4

(a) Frequencies in the laboratory frame of the low-lying $m=\pm 2$ modes as a function of $\nu$ and $\Omega$, for $g=680$ and $\lambda =0.00322$. The solid lines correspond to numerical solutions of Eq. (24), while the dashed lines show the analytical expression (30) for both modes. (b) The quadrupole moment $⟨x^2-y^2⟩$ (normalized to the mean-squared radius $⟨x^2+y^2⟩$) as a function of time (in units of ${\omega }_{\perp }^{-1}$) given by a numerical simulation of the GP equation in the laboratory frame. The original condensate contains a single $\nu =1$ vortex, and the applied perturbation mainly excites the low-lying $m=\pm 2$ modes.

Figure 5

Density as a function of radial distance through an expanding condensate $(\lambda =0.00322,\eta =0.3)$ with no vortex (solid line), a $\nu =1$ singly quantized vortex (dashed), and a $\nu =2$ doubly quantized vortex (dotted) at (a) $t=0$, (b) $t=4$, (c) $t=8$, (d) $t=12$. The inset in (b) shows the density profiles near to the center, illustrating the differences between the states in this region.

Figure 6

Density of the expanding condensate $(\lambda =0.00322,\eta =0.3)$ at $t=14$ with (a) no vortex $(\nu =0)$, (b) a singly quantized vortex $(\nu =1)$, and (c) a doubly quantized vortex $(\nu =2)$. The length scale of each box is $50\times 50d_{\perp }$.

Figure 7

Scaled momentum distributions $128\lambda {\mid \Phi \left(k\right)\mid }^2∕\left(3\pi \eta \right)$ as a function of $kR$ (where $R=1∕\sqrt{2\lambda }$) for (a) $\nu =0$, and (b) $\nu =1(\lambda =0.00322,\eta =0.3)$. Solid lines plot the (squared) integral (34) calculated with solutions of the Gross-Pitaevskii equation, while the dashed lines show Eq. (35) for the Thomas- Fermi approximation. For comparison, the squared Bessel functions ${\left[J_{\nu }\left(kR\right)\right]}^2$ are plotted with dotted lines.