Supersolid phases of ultracold neutral atoms with light-mediated interactions

Supersolids are characterised by the coexistence of superfluid and crystalline order. Triggered by the first experiments with Helium, efforts have been made to realise this intriguing phase of matter in simpler and more controllable systems. One promising platform involves ultracold neutral atoms strongly coupled to light fields. The latter mediate interactions of very long range favouring the formation of crystalline order. Very recently, two different experiments have indeed observed phases were a Bose-Einstein condensate spontaneously breaks a continuous translation invariance due to light-mediated interactions [Nature 543,87(2017); arXiv:1709.02028]. In this talk, I will discuss our theoretical understanding of these experiments and illustrate how their different properties arise from different underlying spatial symmetries.