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Atom-light interactions in optical cavities have been proposed to realizecollective quantum spin models with tunable long-range interactions, which canfeature exotic out-of-equilibrium phases of matter precluded from existence atequilibrium as well as intriguing universal behaviors. This approach contrastswith previous investigations that primarily focused on the optical nonlinearitycreated by the presence of an atomic ensemble. We report the observation ofdistinct non-equilibrium phases of matter of a collective XY spin model withtransverse and longitudinal fields simulated via an ensemble of $\sim 10^6$$^{88}$Sr atoms. The collective exchange interactions of the model aregenerated by coupling a far-detuned optical cavity mode to a narrow opticalatomic transition. The unique properties of our cavity-QED platform allow us toprobe the different quantum phases that arise from the competition between thetransverse and longitudinal fields and the infinite-range interactions, as wellas the dependence of the associated non-equilibrium phase transitions on systemsize and initial state preparation. Our observations shed light onnon-equilibrium phases featured in a range of related systems, includingJosephson junctions in superfluid helium, as well as coupled atomic andsolid-state polariton condensates, with complementary types of control.Moreover, our platform introduces opportunities for future explorations ofquantum chaos and scrambling dynamics, while also opening a path for the use ofquantum enhancement to improve the performance of state-of-the-art atomicclocks.

Exploring Non-Equilibrium Phases of Matter with a Million Long-Lived Optical Dipoles in a Cavity