  The Dicke model -- a paradigmatic example of superradiance in quantum optics
-- describes an ensemble of atoms which are collectively coupled to a leaky
cavity mode. As a result of the cooperative nature of these interactions, the
system's dynamics are captured by the behavior of a single mean-field,
collective spin. In this mean-field limit, it has recently been shown that the
interplay between photon losses and periodic driving of light-matter coupling
can lead to time-crystalline-like behavior of the collective spin. In this
work, we investigate whether such a Dicke time crystal is stable to
perturbations that explicitly break the mean-field solvability of the
conventional Dicke model. In particular, we consider the addition of
short-range interactions between atoms, which breaks the collective coupling
and leads to complex many-body dynamics. In this context, the interplay between
periodic driving, dissipation and interactions yields a rich set of dynamical
responses including long-lived and metastable Dicke time crystals, where losses
can cool down the many-body heating resulting from the continuous pump of
energy from the periodic drive. Specifically, when the additional short-range
interactions are ferromagnetic, we observe time crystalline behavior at
non-perturbative values of the coupling strength, suggesting the possible
existence of stable dynamical order in a driven-dissipative quantum many-body
system. These findings illustrate the rich nature of novel dynamical responses
with many-body character in quantum optics platforms.
