  In an atom-photon interacting system described by Tavis Cummings Hubbard
(TCH) model, we demonstrate the emergence of a quasi-steady state in a
dissipative environment that exhibits intriguing ergodic behavior. The TCH
model undergoes a dissipative transition from normal to superradiant phase
hosting a gapped Higgs and gapless Goldstone modes. However, in a large region
of the phase diagram, the instability of the Goldstone mode leads to the
disappearance of the stable superradiant phase. In this regime, the
decorrelator dynamics reveals light cone spreading of the perturbations and
positive Lyapunov exponent, indicating enhanced fluctuations. Remarkably, a
quasi-steady state emerges under quench dynamics in this unstable regime; in
this state, a class of collective quantities such as site averaged photon
number and atomic excitations approach a steady value, in spite of large
temporal fluctuations in corresponding microscopic variables. This quasi-steady
state describes an incoherent fluid of photons with significant phase
fluctuation. The phase space dynamics reveals a fascinating ergodic behavior in
presence of dissipation, leading to the characterization of the dynamical
variables into two distinct classes. The first class includes site-averaged
photon numbers and atomic excitations; these exhibit a stationary distribution
regardless of the initial condition indicating ergodic behavior. The second
class of variables, particularly those related to phase in contrast, retain
information about the initial conditions, resulting in a violation of
ergodicity for finite size system. Additionally, the dynamical variables of the
ergodic class exhibit fascinating collective scarring phenomenon as the peak of
their distribution is attracted towards the unstable steady state, analogous to
the single particle quantum scar. We discuss the relevance of our findings in
the current experiments.
