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

TL;DR

The paper addresses how dipolar interactions and beyond-mean-field quantum fluctuations stabilize self-bound vortex quantum droplets in a dipolar Bose-Einstein condensate within a toroidal trap. It solves the extended Gross-Pitaevskii equation with dipole-dipole interactions and the Lee-Huang-Yang correction, using imaginary-time propagation to obtain stationary ring-shaped and multipole droplets, and real-time dynamics to test stability, finding necklace-like density patterns with approximately $n=2S$ lobes for $S\le 6$. The results show that the LHY term and dipolar anisotropy stabilize these complex structures, while centrifugal effects destabilize higher-$S$ states, leading to fragmentation; Gaussian confinement yields geometry-driven differences in morphology, confirming the importance of trap geometry. The findings advance understanding of structured vortex droplets in dipolar condensates and point to possible experimental realization in ultracold dysprosium with implications for vortex matter, supersolidity, and quantum turbulence.

Abstract

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.

Figures (13)

• Figure 1: Isosurface plots illustrating stable ring-shaped and multipole QD density profiles for different winding numbers $S$, with norm $N=2000$, DDI length $a_{\mathrm{dd}}=130.8a_{0}$, scattering length $a=100a_{0}$, and the coefficient of the LHY term $\gamma _{\mathrm{QF}}=2.697\times 10^{7}a_{0}^{5/2}$, confined in the toroidal potential (\ref{['toroid']}). (a) The axisymmetric mode with $S=0$. (b) A dipole QD configuration with $S=1$. (c) A quadrupole QD with four density lobes for $S=2$. The panels in the bottom row display stable higher-order multipole structures with $8$, $10$, and $12$ density lobes, obtained for $S=4$, $S=5$, and $S=6$, respectively. All the stationary states are obtained by means of the ITP method.
• Figure 2: The magnified view of the density distribution in the stable stationary ring-shaped and multipole QDs in the absence of the LHY correction, $\gamma _{\mathrm{QF}}=0$, for different vorticities. Results are shown for $N=5000$ atoms with DDI length $a_{\mathrm{dd}}=130.8a_{0}$ and scattering length $a=100a_{0}$, confined in the toroidal potential (\ref{['toroid']}). Panels correspond to: (a) $S=0$, (b) $S=1$, (c) $S=2$, (d) $S=4$, (e) $S=5$, and (f) $S=6$. The plots display the spatial localization and symmetry of the stable QDs. The stationary states are obtained by means of the ITP method.
• Figure 3: Total energy $E$ as a function of atom number $N$ for different vorticities $S$ in the absence of the LHY correction ($\gamma_{\mathrm{QF}}=0$), the dipolar and scattering lengths being $a_{\mathrm{dd}} = 130.8\,a_{0}$ and $a = 100\,a_{0}$. The stationary states are produced by means of ITP. With the increase of $N$, the energy increases monotonously for given values of $S$. The $S=0$ configuration yields the lowest energy, corresponding to the ground state, while the branches with $S=1,3,5$ correspond to multipole excited states.
• Figure 4: Contour plots of the 2D density projection in the $(x,y)$ plane, corresponding to the stable stationary isosurface profiles, shown in Fig. \ref{['s_0']}, for $N=2000$, $a_{dd}=130.8a_{0}$, $a=100a_{0}$, $\gamma _{QF}=2.697\times 10^{7}a_{0}^{5/2}$ and for different vorticities $S$: (a) $S=0$, (b) $S=1$, (c) $S=2$, (d) $S=4$, (e) $S=5$, and (f) $S=6$. The inset in each panel shows the corresponding phase profile, which reveals the topological structure and phase singularities of each state. All stationary configurations were produced by means of the ITP method.
• Figure 5: a) The chemical potential $\mu$, (b) the total energy $E$, and (c) amplitude $\psi _{\max }$ vs. the number of atoms $N$ in stationary QD states with DDI length $a_{\mathrm{dd}}=130.8a_{0}$, scattering length $a=100a_{0}$, and LHY coefficient $\gamma _{\mathrm{QF}}=2.697\times 10^{7}a_{0}^{5/2}$, for vorticities $S=0,1,3,5$ under the action of the toroidal confining potential (\ref{['toroid']}). All the configurations presented here are stationary solutions obtained by means of the ITP method. As $N$ increases, both $\mu$ and energy increase monotonously for all multipole states, reflecting the increase of the repulsive-interaction energy. In contrast, the amplitude decreases due to the spreading of the respective wavefunctions. States with higher values of $S$ exhibit larger energy and chemical potential for given $N$, and their amplitudes are consistently higher.