  We consider a coupled atom-photon system described by the Tavis-Cummings
dimer (two coupled cavities) in the presence of photon loss and atomic pumping,
to investigate the quantum signature of dissipative chaos. The appropriate
classical limit of the model allows us to obtain a phase diagram identifying
different dynamical phases, especially the onset of chaos. Both classically and
quantum mechanically, we demonstrate the emergence of a steady state in the
chaotic regime and analyze its properties. The interplay between quantum
fluctuation and chaos leads to enhanced mixing dynamics and dephasing,
resulting in the formation of an incoherent photonic fluid. The steady state
exhibits an intriguing phenomenon of subsystem thermalization even outside the
chaotic regime; however, its effective temperature increases with the degree of
chaos. Moreover, the statistical properties of the steady state show a close
connection with the random matrix theory. Finally, we discuss the experimental
relevance of our findings, which can be tested in cavity and circuit quantum
electrodynamics setups.
