  We consider the non-equilibrium dynamics of a simple system consisting of
interacting spin-$1/2$ particles subjected to a collective damping. The model
is close to situations that can be engineered in hybrid electro/opto-mechanical
settings. Making use of large-deviation theory, we find a Gallavotti-Cohen
symmetry in the dynamics of the system as well as evidence for the coexistence
of two dynamical phases with different activity levels. We show that additional
damping processes smoothen out this behavior. Our analytical results are backed
up by Monte Carlo simulations that reveal the nature of the trajectories
contributing to the different dynamical phases.
