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Trapped ions are a versatile platform for the investigation of quantummany-body phenomena, in particular for the study of scenarios where long-rangeinteractions are mediated by phonons. Recent experiments have shown that thetrapped ion platform can be augmented by exciting high-lying Rydberg states.This introduces controllable state-dependent interactions that are independentfrom the phonon structure. However, the many-body physics in this newlyaccessible regime is largely unexplored. We show that this system grants accessto generalized Dicke model physics, where dipolar interactions between ions inRydberg states drastically alter the collective non-equilibrium behavior. Weanalyze and classify the emerging dynamical phases and identify a host ofnon-equilibrium signatures such as multi-phase coexistence regions andphonon-lasing regimes. We moreover show how they can be detected andcharacterized through the fluorescence signal of scattered photons. Our studythus highlights new capabilities of trapped Rydberg ion systems for creatingand detecting quantum non-equilibrium phases.

Exploring non-equilibrium phases of the generalized Dicke model with a trapped Rydberg ion quantum simulator