Toroidal Confinement and Beyond: Vorticity-Defined Morphologies of Dipolar $^{164}$Dy Quantum Droplets

We investigate the formation, stability, and dynamics of 3D ring-shaped and multipole vortical quantum droplets (QDs) in non-rotating dipolar Bose-Einstein condensates held in a toroidal trapping potential. The QD dynamics are investigated in the framework of the extended Gross-Pitaevskii equation, which includes long-range dipole-dipole interactions (DDI) and the beyond-mean-field Lee-Huang-Yang (LHY) term, revealing the emergence of self-bound states. Stable stationary solutions for multipole QDs with different values of the topological charge (vorticity $S$) are shaped as necklace-like modes, with the number of \textquotedblleft beads" (multipole’s order) $n=2S$, up to $S=6$. The stability area of the multipoles shrinks with the increase of $S$. For higher values of $S$ the centrifugal effect associated with the phase winding destabilizes the annular density and drives the formation of fragmented multipole droplet states. The dependence of the chemical potential, total energy and peak density on the norm (number of particles) and $S$ is produced. These findings uncover the stabilizing effect of the LHY correction and DDI anisotropy in maintaining complex QD states in the non-rotating configurations.

💡 Research Summary

This paper investigates the formation, stability, and dynamics of three‑dimensional ring‑shaped and multipole vortex quantum droplets (QDs) in a non‑rotating dipolar Bose‑Einstein condensate of $^{164}$Dy atoms confined by a toroidal trapping potential. The authors employ the extended Gross‑Pitaevskii equation (EGPE), which incorporates the long‑range dipole‑dipole interaction (DDI) and the beyond‑mean‑field Lee‑Huang‑Yang (LHY) quantum‑fluctuation term. The DDI length is set to $a_{dd}=130.8,a_0$, the contact scattering length to $a=100,a_0$, and the LHY coefficient to $\gamma_{\rm QF}=2.697\times10^{7}a^{5/2}$. Lengths are scaled by $l_0=\hbar/(m\omega_0)$ with $\omega_0=2\pi\times61,$Hz, giving $l_0\approx1.01,\mu$m.

The toroidal potential is modeled as
(V(\mathbf r)=-p\exp