Vortices in Attractive Bose-Einstein Condensates in Two Dimensions

The form and stability of quantum vortices in Bose-Einstein condensates with attractive atomic interactions is elucidated. They appear as ring bright solitons, and are a generalization of the Townes soliton to nonzero winding number $m$. An infinite sequence of radially excited stationary states appear for each value of $m$, which are characterized by concentric matter-wave rings separated by nodes, in contrast to repulsive condensates, where no such set of states exists. It is shown that robustly stable as well as unstable regimes may be achieved in confined geometries, thereby suggesting that vortices and their radial excited states can be observed in experiments on attractive condensates in two dimensions.

Figure 1

A quantum vortex of winding number $m=1$ in free space: (a) attractive case; (b) repulsive case. A radially excited state: (c) the first excited state in the attractive case; (d) in the repulsive case, a radially excited state requires an infinite number of nodes and asymptotically resembles the Coulomb function [26]. The radial dependence of the order parameter of an infinitely extended condensate is depicted. Note that all axes are dimensionless.

Figure 2

The development of excited states of a quantum vortex in an attractive BEC. (a) For small $a_0$, no quantum vortex is formed. (b) As $a_0\to a_0^{\mathrm{vortex}}$ from below (solid black curve) or above (dashed red curve), the oscillations are pushed out towards $\chi =\infty$. For $a_0=a_0^{\mathrm{vortex}}$ a node appears at $\chi =\infty$, and a true vortex appears. (c) For still larger $a_0$, the node moves inwards. (d) As $a_0$ approaches a second critical value, the oscillations are again pushed out, until a second node appears at $\chi =\infty$. In this way one obtains the first excited state. An infinite sequence of excited states can be constructed in this way. Note that the case of an infinitely extended condensate of winding number $m=1$ is depicted and all axes are dimensionless.

Figure 3

Stable regimes of quantum vortices in confined attractive BECs. Shown are the chemical potential spectra and linear instability frequencies as a function of the nonlinearity for (a)–(b) the first excited state of the Townes soliton, and (c)–(d) the first excited state of a singly quantized vortex $m=1$, all in a harmonic potential. In (b),(d) the winding number of the instability mode is denoted by $q$ and the instability time given by $2\pi /\mathrm{Im}({\Omega }_q)$. Note that all axes are scaled to harmonic oscillator units.