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