Robust quantum-droplet necklace clusters in three dimensions

TL;DR

The paper addresses the challenge of forming and stabilizing self-bound, ring-shaped 3D quantum-droplet necklaces in a free-space binary Bose-Einstein condensate. It adopts a Lee-Huang-Yang corrected Gross-Pitaevskii framework and an energy-minimization approach to predict robust equilibrium ring radii for necklaces of N droplets with a global vorticity M, including phase-imprinted rotation and the concept of supervortices. Key findings show that around the equilibrium radius the necklace can support quasi-stationary rotation with minimal radial pulsations, and robustness increases with N, while off- equilibrium rings contract, expand, or oscillate; supervortex configurations exhibit long-lived dynamics but can be unstable depending on M. These results provide a path toward experimental realization of complex self-sustained 3D quantum states in free space and suggest extensions to dipolar droplet systems.

Abstract

We report the existence of quasi-stable ring-shaped (necklace-shaped) clusters built, in the free space, of 3D quantum droplets (QDs) in a binary Bose-Einstein condensate, modeled by the Gross-Pitaevskii equations with the Lee-Huang-Yang corrections. The QD clusters exhibit diverse dynamical behaviors, including contraction, oscillations, and expansion, depending on the cluster's initial radius. A phase shift between adjacent QDs imparts net angular momentum to the cluster, inducing its permanent rotation. Through the energy-minimization analysis, we predict equilibrium values of the necklace radius that support persistent rotation with negligible radial pulsations. In this regime, the clusters evolve as robust entities, maintaining the azimuthal symmetry in the course of the evolution, even in the presence of considerable perturbations. Necklace "supervortex" clusters, composed of QDs with inner vorticity 1 and global vorticity M, imprinted onto the cluster, may also persist for a long time. The reported findings may facilitate the experimental realization of complex self-sustained quantum states in the 3D free space.

Figures (5)

• Figure 1: (a) and (b): Isosurface plots of $|\psi |=0.15|\psi|_{\mathrm{max}}$, $0.55|\psi |_{\mathrm{max}}$, and $0.95|\psi |_{\mathrm{max}}$ for the QDs with $m=0$ and $m=1$, respectively, and $\mu =-0.14$, which correspond to the points marked in panels (c) and (d). The viewing angles are $135^{\circ }$ in azimuth and $45^{\circ }$ in elevation. (c) Norm $N$ vs. chemical potential $\mu$ and (d) energy $E$ vs. $N$ for QDs with different vorticities $m$. Solid and dashed curves represent stable and unstable branches, respectively.
• Figure 2: Energy $E$ of the necklace clusters composed of $\mathcal{N}$ fundamental QDs at $\mu =-0.14$ as a function of the cluster radius $R$, calculated as per the input ansatz (\ref{['Eq7']}) for $\mathcal{N}=6$ (a) and $\mathcal{N}=9$ (b), for overall charges $M\leq \mathcal{N}/2$.
• Figure 3: The evolution of clusters with $\mathcal{N}=6$. (a) The gradual fusion into a fundamental state elongated along the $z$-direction at $M=0$, $R=12$. (b) Periodic radial oscillations at $M=1$, $R=12$. The cases displayed in panels (a) and (b) correspond to $R>R_{\text{min}}$ [recall $R_{\text{min}}$ is the cluster's radius which provides the energy minimum in Fig. \ref{['Fig2']}]. (c) The quasi-stationary evolution with the minimal radial variation, provided by setting $R=R_{\text{min}}=10$ at $M=1$. (d) The slow expansion at $M=2$, in the case of $R=10<R_{\text{min}}$. The top row shows the variation of the cluster's radius $W_{\text{eff}}$ [defined as per Eq. (\ref{['W']})] as a function of time $t$. The lower rows display isosurface plots of $|\psi |$ at selected times, corresponding to the dots in the top-row plots. Isosurface levels in the 3D figures correspond to $0.15|\psi |_{\mathrm{max}}$ and $0.65|\psi |_{\mathrm{max}}$.
• Figure 4: (a) The metastable evolution of the cluster with $N=5$, $M=1$, $R=8.5$. (b) Periodic radial oscillations of the cluster with $N=7$, $M=1$, $R=12$. (c,d) The robust evolution of the clusters with $M=2$, and $R=14$, composed of $8$ and $9$ droplets, respectively. The top row shows the time dependence of the cluster's radius $W_{\mathrm{eff}}(t)$, which is defined as per Eq. (\ref{['W']}). The lower panels display isosurface plots of $|\psi |$ at selected times, corresponding to the dots in the top-row plots. Isosurface levels in the 3D renderings correspond to $0.15|\psi |_{\mathrm{max}}$ and $0.65|\psi |_{\mathrm{max}}$.
• Figure 5: The evolution of the “ supervortex" clusters composed of vortex QDs with $m=1$, $\mu =-0.14$, $N=6$, and $R=30$. (a) The time dependence of the cluster's radius $W_{\mathrm{eff}}(t)$. (b) The slow expansion of the “ supervortex" with $M=1$. (c,d) The unstable evolution of a “ supervortex" with $M=2$. The isosurfaces in all 3D figures are plotted at $|\psi |=0.05|\psi |_{\mathrm{max}}$ at selected times corresponding to the dots marked in panel (a).