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]

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]

Quantum fluctuations can stabilize Bose–Einstein condensates (BEC) against the mean-field collapse. Stabilization of the condensate has been observed in quantum degenerate Bose–Bose mixtures and dipolar BECs. The fine-tuning of the interatomic interactions can lead to the emergence of two new states of matter: liquid-like self-bound quantum droplets and supersolid crystals formed from these droplets. We review the properties of these exotic states of matter and summarize the experimental progress made using dipolar quantum gases and Bose–Bose mixtures. We conclude with an outline of important open questions that could be addressed in the future.