Small numbers of vortices in anisotropic traps

We investigate the appearance of vortices and vortex lattices in two-dimensional, anisotropic, and rotating Bose-Einstein condensates. Once the anisotropy reaches a critical value, the positions of the vortex cores in the ground state are no longer given by an Abrikosov lattice geometry but by a linear arrangement. Using a variational approach, we determine the critical stirring frequency for a single vortex as well as the equilibrium positions of a small number of vortices.

Figure 1

(Color online) Numerical simulations for a ${}^{87}\text{R}\text{b}$ gas of $N={10}^7$ atoms with $a_{sc}=4.67\times {10}^{-9}\text{m}$ and trapping frequency $\omega =20\pi {\text{s}}^{-1}$. (a) Standard deviation ${\sigma }_y$ of the vortices from the $x$ axis for $1\le \lambda \le 2$. For $\lambda \ge 1.9$, the vortices form a linear crystal and ${\sigma }_y=0$. [(b)–(d)] Ground states for a condensate with parameters as above in a trap with aspect ratio of $\lambda =2.5$. Each plot spans a range of $-36\mu \text{m}<x<36\mu \text{m}$ and $-11\mu \text{m}<y<11\mu \text{m}$. (b) Ground-state density for condensate rotating at $\Omega =0.7\omega$ showing six vortices. (c) Ground-state density for condensate rotating at $\Omega =0.65\omega$ showing five vortices. (d) Phase of the cloud shown in (c). The correspondence between the phase singularities and the density dips is indicated.

Figure 2

Critical frequency for a single vortex as a function of aspect ratio $\lambda$. Stirring frequencies are defined in terms of the trapping frequency in the soft direction $\omega$. For stirring frequencies greater than $\omega$, the cloud becomes unstable. $N=1\times {10}^5$, $\omega =20\pi {\text{s}}^{-1}$, and $a_{sc}=4.67\times {10}^{-9}\text{m}$ is the $s$-wave scattering length for ${}^{87}\text{R}\text{b}$.

Figure 3

Distance of each vortex in a vortex pair from the center of the trap as a function of aspect ratio. The results are shown for a gas of $N=1\times {10}^5$ atoms of ${}^{87}\text{R}\text{b}$. The trapping frequency is given by $\omega =20\pi {\text{s}}^{-1}$ and we display three different rotational frequencies $\Omega =0.4\omega$, $0.6\omega$, and $0.8\omega$. The regions in which the two-vortex state is stable is marked by the black part of the graphs. Note that two vortices may be stable for a range of stirring frequencies and aspect ratios which leads to the indicated overlap.

Figure 4

Distance of each vortex in a vortex pair from the center of the trap as a function of the stirring frequency. The system parameters are the same as in Fig. 3 and we display four different aspect ratios. The stable regions are again the parts marked by the black lines.

Figure 5

Distance of outer vortices from the central vortex in the three-vortex case as a function of anisotropy for a range of stirring frequencies. The system parameters are the same as in the two-vortex case.

Figure 6

Distance of outer vortices from the central vortex in the three-vortex case as a function of the stirring frequency for a range of anisotropies. The system parameters are the same as in the two-vortex case.

Figure 7

(Color online) Trajectories of the single vortices over a time interval of $t=3.5\text{s}$. One can see that the outer vortices move around their equilibrium positions (white dots) but do not pass each other out. The central vortex remains stationary throughout. The system parameters are the same as in the figures above.