Quantized optical mode in a phase-switching feedback loop

We investigate the steady state of quantized monochromatic cavity mode excited by coherent pumping and placed in a feedback loop that changes the mode phase by $\pi$ upon each detection of the photon that left the cavity. Such type of feedback is used for fighting decoherence of Schrödinger cat states. Results of numerical calculations of the Glauber $P$-function for different detunings of mode frequency from pumping frequency are presented. Qualitative explanation for certain peculiarities of the behavior of the $P$-function is provided. We also suggest a simplified model that gives accurate description of the behavior of the system in the case where external coherent pumping is far detuned from the cavity eigenfrequency.