Geometry-driven splitting dynamics of a triply quantized vortex in a ring-shaped condensate

TL;DR

The paper addresses how trap geometry controls the splitting of a multiply quantized vortex in a ring-shaped BEC. It combines full 3D GP simulations in cylindrical coordinates with an energy-based and Bogoliubov-stability analysis, aided by truncated-Wigner fluctuations, to predict splitting direction and timescales. The main findings show that splitting preferentially occurs along the trap's long axis due to lower kinetic-energy cost, with near-isotropic traps admitting competing $l_q=3$ and $l_q=2$ modes that yield a transient triangular pattern before anisotropy drives a linear chain; stronger anisotropy or interactions modify these pathways and speeds. Overall, the work demonstrates geometry as a robust control knob for directing the decay of high-charge vortices, enabling geometry-based vortex engineering in quantum gases.

Abstract

We study the splitting dynamics of a triply quantized vortex (TQV) confined in a ring-shaped Bose-Einstein condensate under a weakly elliptical harmonic trap. Using full 3D simulations in cylindrical coordinates, combined with a semi-analytical energy analysis, we show that the vortex preferentially splits along the long axis of the trap, a direction that minimizes the kinetic-energy cost relative to the initial TQV state. Systematic parameter scans reveal that initial quantum fluctuations increase the splitting time and suppress the transient three-core pattern observed in noise-free simulations, whereas stronger nonlinear interactions accelerate the splitting. When the trap is nearly isotropic, the unstable Bogoliubov modes are dominated by both azimuthal quantum number $l_q=3$ and $l_q=2$; this leads to a dynamical sequence where three daughter vortices first form a triangular arrangement, later evolving into a linear chain. For stronger anisotropy, geometric coupling selectively enhances the $l_q=2$ mode, making it the sole dominant channel and resulting directly in linear vortex alignment -- a clear signature of geometry-induced mode competition explained through combined energy-based and Bogoliubov stability analysis. Our results provide a quantitative picture of how trap geometry can steer the instability pathway, splitting time, and final pattern of a multiply quantized vortex, offering a route toward geometry-controlled vortex engineering.

Figures (6)

• Figure 1: (Color online) Representative 3D splitting and subsequent fusion of an initially imprinted $S=3$ vortex in an elliptic trap with anisotropy $\eta=0.1$. Top row: side-view renderings of the condensate isodensity surface $\rho(\mathbf r)=|\psi(\mathbf r)|^{2}$ at $\rho=0.3\%\,\rho_{\max}$. The same surface is shown with reduced opacity in the outer region (gray), while a cropped region around the vortex core is highlighted (yellow) to emphasize the fragmentation and reconnection of the multiply quantized core. Bottom row: corresponding top-view density snapshots used to classify the splitting stage by the number of resolved vortex holes. Times are (a) $t=0~\mathrm{ms}$, (b) $t=0.875~\mathrm{ms}$, (c) $t=0.981~\mathrm{ms}$, (d) $t=1.512~\mathrm{ms}$, (e) $t=1.592~\mathrm{ms}$, and (f) $t=2.122~\mathrm{ms}$.
• Figure 2: (Color online) Splitting time of the TQV. (a) Splitting time $t_s$ as a function of trap anisotropy $\eta$ for a fixed interaction strength $g_0$. Blue squares: dynamics from the mean-field initial state (unseeded). Red circles: dynamics including initial quantum fluctuations via the truncated Wigner approximation (seeded). The schematics in the inset illustrate the characteristic splitting patterns for the near-isotropic limit ($\eta \to 0$, top left) and for finite anisotropy (bottom right). (b) Splitting time $t_s$ versus $\eta$ in the presence of quantum fluctuations for two interaction strengths, $g_0$ (red circles) and $10g_0$ (blue triangles).
• Figure 3: (Color online) Energetic comparison of candidate post-splitting configurations. Top: representative $xy$-plane density profiles for (left) an intact $S=3$ vortex, (middle) a fully split configuration aligned along the $x$ axis, and (right) a fully split configuration aligned along the $y$ axis. Insets show a magnified view of the vortex cores. Bottom: decomposition of the total energy into potential, interaction, and kinetic contributions for the three configurations (semi-analytical evaluation using the corresponding stationary density profiles). The potential and interaction energies are comparable across configurations, whereas the kinetic energy discriminates the splitting direction, favoring the $x$-axis split state for the parameters shown.
• Figure 4: (Color online) Logarithmic scale of the maximum imaginary part (growth rate $\gamma$) of the unstable Bogoliubov eigenfrequencies as a function of the dimensionless interaction strength $\tilde{g}$ for the axisymmetric ($\eta=0$) ring condensate. The curves correspond to different azimuthal quantum numbers of the perturbation: $l_q=2$ (red dashed line with circles) and $l_q=3$ (blue dashed line).
• Figure 5: (Color online) Temporal density isosurface of a ring-shaped condensate with the circulation $S=3$ in a symmetric trap, i.e., $\eta=0$ for $1\%$ of the maximum density and $g=g_0$. Different panels correspond to different time : (a) $t=0~\mathrm{ms}$, (b) $t=0.663~\mathrm{ms}$, (c) $t=1.325~\mathrm{ms}$, (d) $t=1.59~\mathrm{ms}$, (e) $t=2.65~\mathrm{ms}$, and (f) $t=26.474~\mathrm{ms}$.