Quantum droplets may form out of a gaseous Bose-Einstein condensate, stabilized by quantum fluctuations beyond mean field. We show that multiple singly-quantized vortices may form in these droplets at moderate angular momenta in two dimensions. Droplets carrying these precursors of an Abrikosov lattice remain selfbound for certain timescales after switching off an initial harmonic confinement. Furthermore, we examine how these vortex-carrying droplets can be formed in a more pertubation-resistant setting, by starting from a rotating binary Bose-Einstein condensate and inducing a metastable persistent current via a non-monotonic trapping potential.The formation of self-bound droplets is a well-known macroscopic phenomenon. For an exemplary droplet of water, stability and shape rely on the balance of effective forces between its constituent particles -attractive ones that keep it together, and repulsive ones that prevent it from collapse. Their interplay defines the droplets' surface tension, stabilizing the system in a metastable state. Such droplets do not only occur at a macroscopic level, but are ubiquitous also in the quantum realm, where nuclei [1] and superfluid helium droplets [2][3][4] are prominent examples. While these are rather dense and strongly interacting many-body systems, recent experiments with ultra-cold quantum gases of bosonic atoms uncovered a novel type of quantum liquid: Self-bound droplets may form out of a gaseous Bose-Einstein condensate (BEC) of dysprosium [5][6][7][8][9] or erbium [10], atomic species that are known for their strong dipolar interactions [11][12][13]. Similar droplet states have more recently also been realized with binary Bose gases of potassium in different hyperfine states [14,15], where the inter-and intracomponent interactions are short-ranged. These quantum droplets can be large, containing thousands of atoms. Importantly, they are very dilute -by more than eight orders of magnitude when compared with liquid helium [14]. While the discovery with dysprosium [5-7] at first came as a surprise, the binary self-bound droplet states were theoretically predicted a year before [16] for a scenario similar to the experiments with potassium, and also in lower dimensions [17]. That higher-order corrections beyond mean field may lead to self-bound states was discussed earlier in a different setting in Refs. [18,19]. For the dipolar or binary self-bound bosonic systems of 14,15] the physical mechanism of droplet formation is based on tuning the interactions in gas such that only a weak effective attraction remains. While in pure mean field this would lead to a collapse of the system, weak first-order corrections to the mean field energy, often referred to as the Lee-Huang-Yang (LHY)correction [20], can become comparable in size and may thus stabilize the system.

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We investigate a Bose-Bose mixture across the miscible-immiscible phase transition governed by quantum fluctuations in one dimension. We find the recently predicted so-called 'mixed bubbles' as ground states close to the mean-field miscible-immiscible threshold. These bubbles form a pocket of miscibility, separated by one of the components. The collective excitations reflect the symmetry breaking resulting from the bubble formation. The partial miscibility of the system allows for persistent currents in an annular confinement. Intriguingly, the mixed bubble acts like an intrinsic weak link, connecting the rotational behavior of the mixed bubble state to current efforts in atomtronic applications.