Single and multiple vortex rings in three-dimensional Bose-Einstein condensates: Existence, stability, and dynamics

In the present work, we explore the existence, stability, and dynamics of single- and multiple-vortex-ring states that can arise in Bose-Einstein condensates. Earlier works have illustrated the bifurcation of such states in the vicinity of the linear limit for isotropic or anisotropic three-dimensional harmonic traps. Here, we extend these states to the regime of large chemical potentials, the so-called Thomas-Fermi limit, and explore their properties such as equilibrium radii and inter-ring distance for multi-ring states, as well as their vibrational spectra and possible instabilities. In this limit, both the existence and stability characteristics can be partially traced to a particle picture that considers the rings as individual particles oscillating within the trap and interacting pairwise with one another. Finally, we examine some representative instability scenarios of the multi-ring dynamics, including breakup and reconnections, as well as the transient formation of vortex lines.

Figure 1

The top row illustrates stationary single (left) and opposite-charge double (right) VRs for $\mu =12$ in an isotropic trap with ${\omega }_z={\omega }_r=1$. The panels depict isocontour plots for the density (red), and the cores of the VRs are highlighted by green (dark) surfaces corresponding to isocontours of the smoothed norm of the vorticity (curl of the fluid velocity). The bottom row depicts the trajectories for one VR (left) and two oppositely charged VRs (right) in an isotropic trapping with ${\omega }_z={\omega }_r=1$ and $\mu =50$. Using the cylindrical symmetry of the setup, the trajectories are depicted in the $(r,z)$ plane, where $r$ is the radius of the VR and $z$ is its vertical axis coordinate. The thick solid trajectories correspond to the particle picture (PP) prediction of Sec. 2b, while the thin solid trajectories correspond to numerical simulations of the GPE of Eq. (1) integrated in reduced cylindrical coordinates. The outermost semicircle corresponds to the TF radius. For full 3D numerics showing dynamic and symmetry-breaking instabilities, we refer the reader to the results in Figs. 4, 5, 7 and 8.

Figure 2

Equilibrium positions for one and two opposite-charge VR configurations as a function of the chemical potential $\mu$. The top and bottom panels show the equilibrium radius $r_c$ and axial location $z_c$ as a function of the dimensionless (by $\hslash {\omega }_r$) chemical potential $\mu$ for the particle picture (PP; dashed line) and the GPE (solid line) for the single VR (top) and double VR (bottom) for ${\omega }_z={\omega }_r=1$. The top panel includes (thin horizontal line) the asymptotic prediction for $\mu \to \infty$ given in Eq. (11). All quantities in this and subsequent figures are dimensionless; see text.

Figure 3

BdG spectrum for two VRs as a function of the dimensionless chemical potential $\mu$ for ${\omega }_z={\omega }_r=1$. Mode contributions are shown via green (light) points if stable and red (dark) points if unstable. The black circle and triangle represent the frequencies for the normal-mode vibrations around the stationary state depicted, respectively, in Figs. 4 and 5.

Figure 4

The first $n=0$ excitation of the two-VR state. Black [red (gray)] represents the upper (lower) ring. The open circles (and corresponding vertical dashed lines) indicate four time points to demonstrate the motion of the rings via their (a) time-$z$, (b) time-$r$, and (c) ($r,z$) trajectories. The solid curves are fits to the data, while the arrows in (c) indicate time $t_a$ and the subsequent direction of core motion. Time and space are measured in units of $1/{\omega }_r$ and harmonic oscillator length $a_r=\sqrt{\hslash /m{\omega }_r}$, respectively.

Figure 5

The second $n=0$ excitation of the two-vortex-ring state. Curves and labels have the same meaning as in Fig. 4. These rings, as was shown in Fig. 4 for the first excitation, have azimuthal symmetry and cores that remain circular and coaxial with the $z$ axis throughout the motion. Note that in (b) the $r$ motions of the two rings are in phase and the data points overlap. As explained in the caption of Fig. 4, $\mu \sim 16$. Time and space are measured in units of $1/{\omega }_r$ and harmonic oscillator length $a_r=\sqrt{\hslash /m{\omega }_r}$.

Figure 6

The opposite-charge three-VR configuration for $\frac{3}{2}{\omega }_z={\omega }_r=1$. The top four panels show the modulus $\left|\mathrm{\Psi }\right|$ (left) and argument $\theta$ (right) of a triple-vortex-ring state for a dimensionless chemical potential $\mu =12$; the first row illustrates the $z=0$ plane, while the second row illustrates the $x=0$ plane. The middle panel shows an isocontour density plot. The bottom panel depicts the BdG spectrum for the three-VR configuration as a function of the dimensionless chemical potential $\mu$. Notation is the same as in Fig. 3. Notice the higher multiplicity of unstable modes and especially the persistent, albeit weakening, instability due to an eigenmode resulting from the collision of two modes around $\mu =5$.

Figure 7

Evolution of an unstable double-vortex ring for chemical potential $\mu =10$. Six representative snapshots during the time evolution of the state are given at times $t$ = (a) 20, (b) 40, (c) 45.4, (d) 68, (e) 71.2, and (f) 74.8 (units of $1/{\omega }_r$). Notice the intense quadrupolar undulation of the rings leading to their breakup and then reconnection with a perpendicular axis of symmetry, as well as an example of their breakup into vortex lines along the $z$ direction.

Figure 8

As in Fig. 7 but for the unstable evolution of the three-vortex-ring state in the case of $\mu =9$. The six representative snapshots shown during the time evolution correspond in this case to $t$ = (a) 20, (b) 49.4, (c) 52.4, (d) 53.3, (e) 54.9, and (f) 56.0 (units of $1/{\omega }_r$). Notice how the deformation of the rings leads, in this case, to the joining of the two lower ones into a single entity [see (d)] before the subsequent evolution to a vortex tangle.