Ferrofluids exhibit unusual hydrodynamic effects owing to the magnetic nature of their constituents. As magnetization increases, a classical ferrofluid undergoes a Rosensweig instability and creates self-organized, ordered surface structures or droplet crystals. Quantum ferrofluids such as Bose-Einstein condensates with strong dipolar interactions also display superfluidity. The field of dipolar quantum gases is motivated by the search for new phases of matter that break continuous symmetries. The simultaneous breaking of continuous symmetries such as the phase invariance in a superfluid state and the translational symmetry in a crystal provides the basis for these new states of matter. However, interaction-induced crystallization in a superfluid has not yet been observed. Here we use in situ imaging to directly observe the spontaneous transition from an unstructured superfluid to an ordered arrangement of droplets in an atomic dysprosium Bose-Einstein condensate. By using a Feshbach resonance to control the interparticle interactions, we induce a finite-wavelength instability and observe discrete droplets in a triangular structure, the number of which grows as the number of atoms increases. We find that these structured states are surprisingly long-lived and observe hysteretic behaviour, which is typical for a crystallization process and in close analogy to the Rosensweig instability. Our system exhibits both superfluidity and, as we show here, spontaneous translational symmetry breaking. Although our observations do not probe superfluidity in the structured states, if the droplets establish a common phase via weak links, then our system is a very good candidate for a supersolid ground state.

Quantum fluctuations are the origin of genuine quantum many-body effects, and can be neglected in classical mean-field phenomena. Here, we report on the observation of stable quantum droplets containing ∼800 atoms that are expected to collapse at the mean-field level due to the essentially attractive interaction. By systematic measurements on individual droplets we demonstrate quantitatively that quantum fluctuations mechanically stabilize them against the mean-field collapse. We observe in addition the interference of several droplets indicating that this stable many-body state is phase coherent.

Self-bound many-body systems are formed through a balance of attractive and repulsive forces and occur in many physical scenarios. Liquid droplets are an example of a self-bound system, formed by a balance of the mutual attractive and repulsive forces that derive from different components of the inter-particle potential. It has been suggested 1, 2 that self-bound ensembles of ultracold atoms should exist for atom number densities that are 10 8 times lower than in a helium droplet, which is formed from a dense quantum liquid. However, such ensembles have been elusive up to now because they require forces other than the usual zero-range contact interaction, which is either attractive or repulsive but never both. On the basis of the recent finding that an unstable bosonic dipolar gas can be stabilized by a repulsive many-body term 3 , it was predicted that three-dimensional self-bound quantum droplets of magnetic atoms should exist 4, 5 . Here we report the observation of such droplets in a trap-free levitation field. We find that this dilute magnetic quantum liquid requires a minimum, critical number of atoms, below which the liquid evaporates into an expanding gas as a result of the quantum pressure of the individual constituents. Consequently, around this critical atom number we observe an interaction-driven phase transition between a gas and a self-bound liquid in the quantum degenerate regime with ultracold atoms. These droplets are the dilute counterpart of strongly correlated self-bound systems such as atomic nuclei 6 and helium droplets 7 .Liquid droplets of water or helium are formed by the mutual attractive and repulsive forces that are created by the different parts of the inter-particle potential (and are due to covalent or van der Waals attraction and to the electronic Pauli exclusion principle, respectively). Helium droplets in particular have been a focus of research, owing to their interesting quantum nature 8,9 . Droplets can serve as closed, isolated quantum systems with which to probe, for example, superfluidity of mesoscopic ensembles 10 . In the context of ultracold atoms, the observation of an ensemble of stable droplets 11 in a dilute magnetic quantum gas opened up the possibility of a three-dimensional self-bound state 4, 5 . A trapped quantum droplet of magnetic atoms has recently also been observed using erbium atoms 12 . Here we demonstrate the observation of dilute, self-bound liquid droplets in a sample of ultracold bosonic dysprosium atoms, which have a strong longrange magnetic dipolar interaction and a tunable repulsive short-range contact interaction. The interplay between these two interactions can be tuned such that the overall mean field is weakly attractive, but so that the interactions also create quantum depletion and a corresponding many-body repulsion. This repulsion exactly counteracts the attraction when the density of the droplet reaches the stabilization density. We use the word 'liquid' here to describe a state of matter that is defined by the presence of self-bound dro...