Three-dimensional parallel vortex rings in Bose-Einstein condensates

We construct three-dimensional structures of topological defects hosted in trapped wave fields, in the form of vortex stars, vortex cages, parallel vortex lines, perpendicular vortex rings, and parallel vortex rings, and we show that the latter exist as robust stationary, collective states of nonrotating Bose-Einstein condensates. We discuss the stability properties of excited states containing several parallel vortex rings hosted by the condensate, including their dynamical and structural stability.

Figure 1

(Color online) Isosurface plots of the vortex “cores” of some examples of three-dimensional topological defects in the noninteracting limit $UN=0$. (a) Vortex star given by Eq. (5), (b) parallel vortex rings [Eq. (6a)], (c) perpendicular vortex rings [Eq. (6b)], (d) two antiparallel vortex lines in an asymmetric trap [Eq. (7)], and (e) a vortex ring in an asymmetric trap [Eq. (8)]. In the latter two cases, we plot also the shape of the atomic cloud to stress the asymmetry [for (d) ${\lambda }_x={\lambda }_z=1$, ${\lambda }_y=2$, and for (e) ${\lambda }_x={\lambda }_y=1$, ${\lambda }_z=2$].

Figure 2

(Color online) Three and four parallel vortex ring states in the noninteracting $UN=0$ [(a),(b)] and the strongly interacting limit $UN\approx \mathrm{10\hspace{0.17em}000}$ [(c),(d)]. Features as in Figs. 1d, 1e.

Figure 3

(Color online) (a),(b) Dynamical stability of the two PVR state in the strongly interacting limit, $UN\approx \mathrm{10\hspace{0.17em}000}$, under a noise level of 20%. (c), (d) Dynamical instability of the PRV in the moderate interacting limit, $UN\approx 1000$, under a noise level of 10%.

Figure 4

(a),(b) Stability of the three PVR state against “needle”-like perturbation along the $z$ axis. In (a) $t=0$ and in (b) $t=60$. (c),(d) Stability against noise of the three PVR state. In (c) $t=0$ and in (d) $t=60$. Here $UN\approx \mathrm{10\hspace{0.17em}000}$.

Figure 5

(Color online) Structural stability of the two PVRs against trap perturbations in the vortex-rings plane as $\Delta {\lambda }_x^2∕{\lambda }_x^2=2\%$ (upper row) and in the plane perpendicular to the vortex-rings plane as $\Delta {\lambda }_z^2∕{\lambda }_z^2=10\%$ (bottom row). Here $UN\approx \mathrm{10\hspace{0.17em}000}$.