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...

We study theoretically and experimentally the emergence of supersolid properties in a dipolar Bose-Einstein condensate. The theory reveals a ground state phase diagram with three distinct regimes -a regular Bose-Einstein condensate, incoherent and coherent arrays of quantum droplets. The coherent droplets are connected by a background condensate, which leads -in addition to the periodic density modulation -to a robust phase coherence throughout the whole system. We further theoretically demonstrate that we are able to dynamically approach the ground state in our experiment and that its lifetime is only limited by three-body losses. Experimentally we probe and confirm the signatures of the phase diagram by observing the in-situ density modulation as well as the phase coherence using matter wave interference. arXiv:1901.07982v2 [cond-mat.quant-gas]

We show that for ultracold magnetic lanthanide atoms chaotic scattering emerges due to a combination of anisotropic interaction potentials and Zeeman coupling under an external magnetic field. This scattering is studied in a collaborative experimental and theoretical effort for both dysprosium and erbium. We present extensive atom-loss measurements of their dense magnetic Feshbach-resonance spectra, analyze their statistical properties, and compare to predictions from a random-matrix-theory-inspired model. Furthermore, theoretical coupled-channels simulations of the anisotropic molecular Hamiltonian at zero magnetic field show that weakly bound, near threshold diatomic levels form overlapping, uncoupled chaotic series that when combined are randomly distributed. The Zeeman interaction shifts and couples these levels, leading to a Feshbach spectrum of zero-energy bound states with nearest-neighbor spacings that changes from randomly to chaotically distributed for increasing magnetic field. Finally, we show that the extreme temperature sensitivity of a small, but sizable fraction of the resonances in the Dy and Er atom-loss spectra is due to resonant nonzero partial-wave collisions. Our threshold analysis for these resonances indicates a large collision-energy dependence of the three-body recombination rate.

A supersolid is a counter-intuitive state of matter that combines the frictionless flow of a superfluid with the crystal-like periodic density modulation of a solid 1, 2 . Since the first prediction in the 1950s 3 , experimental efforts to realize this state have focussed mainly on Helium, where supersolidity remains elusive 4 . Recently, supersolidity has also been studied intensively in ultracold quantum gases, and some of its defining properties have been induced in spin-orbit coupled Bose-Einstein condensates (BECs) 5 and BECs coupled to two crossed optical cavities 6, 7 . However, the periodicity of the crystals in both systems is fixed to the wavelength of the applied periodic optical potentials. Recently, hallmark properties of a supersolid -the periodic density modulation and simultaneous global phase coherence -have been observed in arrays of dipolar quantum droplets 8-10 , where the crystallization happens in a self-organized manner due to intrinsic interactions. In this letter, we prove the genuine supersolid nature of these droplet arrays by directly observing the low-energy Goldstone mode. The dynamics of this mode is reminiscent of the effect of second sound in other superfluid systems 11,12 and features an out-ofphase oscillation of the crystal array and the superfluid density. This mode exists only due to the phase rigidity of the experimentally realized state, and therefore confirms the genuine superfluidity of the supersolid. arXiv:1906.04633v1 [cond-mat.quant-gas]

We study theoretically and experimentally the behaviour of a strongly confined dipolar Bose-Einstein condensate, in the regime of quantum-mechanical stabilization by beyond-mean-field effects. Theoretically, we demonstrate that self-organized striped ground states are predicted in the framework of the extended Gross-Pitaevskii theory. Experimentally, by tilting the magnetic dipoles we show that self-organized striped states can be generated, likely in their metastable state. Matter-wave interference experiments with multiple stripes show that there is no long-range off-diagonal order (global phase coherence). We outline a parameter range where global phase coherence could be established, thus paving the way towards the observation of supersolid states in this system.Comment: 9 pages, 7 figure

Dipolar quantum droplets are exotic quantum objects that are self-bound due to the subtle balance of attraction, repulsion and quantum correlations. Here we present a systematic study of the critical atom number of these self-bound droplets, comparing the experimental results with extended mean-field Gross-Pitaevskii equation (eGPE) and quantum Monte-Carlo simulations of the dilute system. The respective theoretical predictions differ, questioning the validity of the current theoretical state-of-the-art description of quantum droplets within the eGPE framework and indicating that correlations in the system are significant. Furthermore, we show that our system can serve as a sensitive testing ground for many-body theories in the near future.