Monitoring photon entanglement in coupled cavities

We study the dynamics of $N$ photons in a Fock state, initially located inside one cavity, and coupled by an optical fiber to a second cavity. The entanglement of the photons is monitored by projective measurements, repeated with a fixed time step. This approach is applied to the formation of a photonic N00N state. We calculate the probability of the transition of $N$ photons from the left to the right cavity and the probability of the return of $N$ photons to the left cavity under repeated projective measurements. The entanglement is analyzed for the N00N state by its fidelity and its phase sensitivity, while for the entanglement between the states in the two cavities the entanglement entropy is calculated. In addition, we study the monitored evolution of photons in a single cavity, which are coupled to a single qubit, using the Jaynes-Cummings model. Photon entanglement is analyzed in terms of the entanglement entropy. In all these cases we find that entanglement is sensitive to the details of monitoring protocol, which can be used to control photon entanglement for specific applications.