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In quantum optics, photonic Schr\"odinger cats are superpositions of twocoherent states with opposite phases and with a significant number of photons.Recently, these states have been observed in the transient dynamics ofdriven-dissipative resonators subject to engineered two-photon processes. Herewe present an exact analytical solution of the steady-state density matrix forthis class of systems, including one-photon losses, which are considereddetrimental for the achievement of cat states. We demonstrate that the uniquesteady state is a statistical mixture of two cat-like states with oppositeparity, in spite of significant one-photon losses. The transient dynamics tothe steady state depends dramatically on the initial state and can pass througha metastable regime lasting orders of magnitudes longer than the photonlifetime. By considering individual quantum trajectories in photon-countingconfiguration, we find that the system intermittently jumps between two cats.Finally, we propose and study a feedback protocol based on this behaviour togenerate a pure cat-like steady state.

Exact results for Schr\"odinger cats in driven-dissipative systems and their feedback control