Stationary vortex clusters in nonrotating Bose-Einstein condensates

We investigate the recently found stationary vortex cluster states in dilute atomic Bose-Einstein condensates confined by a nonrotating trap, and also present a stationary three-vortex cluster. We find the stationary states by minimizing directly an error norm for the stationary Gross-Pitaevskii equation, and study the dynamic and energetic stability of the resulting states by solving the corresponding Bogoliubov equations for the elementary excitations. The results are verified by integrating the time-dependent Gross-Pitaevskii equation. Contrary to previously reported results, the stationary states were observed to be both energetically and dynamically unstable. The dynamical decay rate of the clusters is typically very slow, but it should be experimentally observable. The most promising circumstances to experimentally generate and observe these structures and their dynamics is in weakly dissipative condensate systems, using phase-imprinting techniques.

Figure 1

Density profiles of a stationary vortex dipole (a), tripole (b), and quadrupole (c). The separations of the outermost vortices are marked by $d_2,d_3$, and $d_4$ for the dipole, tripole, and quadrupole, respectively. The plus signs in the vortex core denote vortices and minus signs antivortices. The strength of the interactions $\tilde{g}$ is 170 for the vortex dipole and 160 for the tripole and quadrupole.

Figure 2

Energy of the vortex dipole (dashed line), tripole (solid line), and quadrupole (dash-dotted line) as functions of the distances $d_2,d_3$, and $d_4$, respectively. The strength of the interactions is $\tilde{g}=160$ for all the configurations.