Vortex stability in nearly-two-dimensional Bose-Einstein condensates with attraction

Abstract

We perform accurate investigation of stability of localized vortices in an effectively two-dimensional (``pancake-shaped'') trapped Bose-Einstein condensate with negative scattering length. The analysis combines computation of the stability eigenvalues and direct simulations. The states with vorticity $S=1$ are stable in a third of their existence region, $0<N<(1∕3){N}_{\mathrm{max}}^{(S=1)}$, where $N$ is the number of atoms, and ${N}_{\mathrm{max}}^{(S=1)}$ is the corresponding collapse threshold. Stable vortices easily self-trap from arbitrary initial configurations with embedded vorticity. In an adjacent interval, $(1∕3){N}_{\mathrm{max}}^{(S=1)}<N<0.43{N}_{\mathrm{max}}^{(S=1)}$, the unstable vortex periodically splits in twofragments and recombines. At $N>0.43{N}_{\mathrm{max}}^{(S=1)}$, the fragments do not recombine, as each one collapses by itself. The results are compared with those in the full three-dimensional (3D) Gross-Pitaevskii equation. In a moderately anisotropic 3D configuration, with the aspect ratio $\sqrt{10}$, the stability interval of the $S=1$ vortices occupies $\ensuremath{\approx}40%$ of their existence region, hence the two-dimensional (2D) limit provides for a reasonable approximation in this case. For the isotropic 3D configuration, the stability interval expands to 65% of the existence domain. Overall, the vorticity heightens the actual collapse threshold by a factor of up to $2$. All vortices with $S\ensuremath{\geqslant}2$ are unstable.