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We investigate the behavior of two coupled non-linear photonic cavities, inpresence of inhomogeneous coherent driving and local dissipations. By solvingnumerically the quantum master equation, either by diagonalizing theLiouvillian superoperator or by using the approximated truncated Wignerapproach, we extrapolate the properties of the system in a thermodynamic limitof large photon occupation. When the mean field Gross-Pitaevskii equationpredicts a unique parametrically unstable steady-state solution, the openquantum many-body system presents highly non-classical properties and itsdynamics exhibits the long lived Josephson-like oscillations typical ofdissipative time crystals, as indicated by the presence of purely imaginaryeigenvalues in the spectrum of the Liouvillian superoperator in thethermodynamic limit.

A dissipative time crystal in an asymmetric non-linear photonic dimer