  We investigate the behavior of two coupled non-linear photonic cavities, in
presence of inhomogeneous coherent driving and local dissipations. By solving
numerically the quantum master equation, either by diagonalizing the
Liouvillian superoperator or by using the approximated truncated Wigner
approach, we extrapolate the properties of the system in a thermodynamic limit
of large photon occupation. When the mean field Gross-Pitaevskii equation
predicts a unique parametrically unstable steady-state solution, the open
quantum many-body system presents highly non-classical properties and its
dynamics exhibits the long lived Josephson-like oscillations typical of
dissipative time crystals, as indicated by the presence of purely imaginary
eigenvalues in the spectrum of the Liouvillian superoperator in the
thermodynamic limit.
