  We study the evolution of a photonic state in a coupled pair of anharmonic
cavities and compare it with the corresponding system of two coupled harmonic
cavities. The photons in the anharmonic cavities interact with a two-level atom
and are described within the Jaynes-Cummings model. Starting from an eigenstate
of the Jaynes-Cummings model with N photons, the evolution of this state in the
presence of photon tunneling between the cavities is studied. We evaluate the
spectral density, the dynamics of the return as well as the transition
amplitudes and the probability for the dynamical creation of a N00N state.
