Supersolidity and tunable symmetries with ultracold atoms in optical cavities

By coupling a Bose-Einstein condensate to two optical cavities we gain experimental access to new phases of matter. For large detuning of the cavities from the atomic resonance, the combination of self-organisation processes with $\mathbb{Z}_2$ symmetry in each cavity gives rise to one enhanced U(1) symmetry that corresponds to translational invariance of the atoms in one direction. We report on the observation of a phase transition to a supersolid state that breaks this continuous symmetry, and show spectroscopic measurements of the Nambu-Goldstone and Higgs modes present in this phase. The light fields leaking from the cavities enable real-time surveillance of the system dynamics. Approaching the atomic resonance, the continuous invariance is lifted as cavity-cavity coupling comes into play.