Three-dimensional vortex and multipole quantum droplets in a toroidal potential

We investigate the existence, stability, and evolution dynamics of three-dimensional quantum droplets (QDs) in binary Bose–Einstein condensates trapped in a toroidal potential. The interplay of competing mean-field (MF) and beyond-MF nonlinear terms (attractive and repulsive ones) with the potential enables the formation of a variety of stable QDs with complex structures, from higher-order vortices and multipoles to necklace complexes. There are two branches of the vortex and multipole QDs with opposite slopes of the norm-vs.-chemical-potential curves. The upper-branch vortex QDs are completely stable, regardless of the value of their topological charge (winding number), m. On the other hand, multipole QDs are stable only near the turning point of the curves. Although the stability region shrinks with the growth of the number of poles, necklace-shaped QDs remain stable for the number up to 16. Applying a torque perpendicular to the original QD's angular momentum, we observe stable precession of vortex QDs. Periodic rotation is also observed when multipole QDs are set in motion by a planar phase torque.