Atom-light interactions in optical cavities provide a platform for investigating many-body quantum physics in controlled environments. In particular, they have been proposed for the realization of collective quantum spin models with tunable long-range interactions. Besides the investigation of the rich steady-state phases that can arise due to the interplay between atom-light interactions and dissipation from the cavity, one opportunity offered by these systems is the study of out-of-equilibrium dynamical phases of matter  precluded from existence at equilibrium . Moreover, these phases can display intriguing universal behaviors akin to standard equilibrium phase transitions. Here, we report the observation of distinct dynamical phases of matter in a nearly unitary implementation of the collective XY spin model with transverse and longitudinal fields simulated via an ensemble of ∼10^6 88Sr atoms. The unique properties of our cavity-QED platform allow us to probe thedependence of the associated dynamical phase transitions on parameter space, system size and initial state. In the spirit of quantum simulation our observations can be linked to similar dynamical phases featured in a range of related scystems, including the Josephson effect in superfluid helium , coupled atomic and solid-state polariton condensates, with complementary types of control including the magnitude and sign of Hamiltonian parameters. Moreover, our system offers potential for the generation of metrologically useful entangled states in optical transitions, which can enable real metrological gains via quantum enhancement in state-of-the-art atomic clocks.

All experimental sets include error bars on y axis (1s.d.). Some thoery sets include minum and maximum value of y axis according to shot-to-shot parameter fluctuations. Units specified in header if necessary.

Fig 2a i - One experimental set, one simulation set

Fig 2a ii - One experimental set, one simulation set

Fig 2a iii - One experimental set, one simulation set

Fig 2a iv - One experimental set, one simulation set

Fig 2b - three N-scaled data sets and simulation

Fig 2b - Inset, three experimental sets

Fig 3a - two experimental sets, one simulation set (symmetric) and non-interacting case

Fig 3b - two experimental sets, one simulation set (symmetric) and non-interacting case

Fig 4b i - Two theory curves 

Fig 4b ii - one experimental data, one simulation

Fig 4b iii - one experimental data, one simulation

Fig 4b iv - one experimental data, one simulation

Fig ED2a - one experimental set, one simulation set

Fig ED2b - two experimental sets, one numerical (symmetric), one analytical (symmetric)