We study a pair of anharmonic microwave cavities that is connected by an optical fiber. The photonic spectral density characterizes the evolution of the coupled cavities after the system has been prepared in a Fock or N00N state. We evaluate the photonic spectral density within the recursive projection method and find that the anharmonicity has no substantial effect on the spectral properties. In all cases the photonic spectral density has a Gaussian envelope for large photon numbers.