Photon cross-correlations emitted by a Josephson junction in two microwave cavities

We study a voltage biased Josephson junction coupled to two resonators of incommensurate frequencies. Using a density approach to analyze the cavity fields and an input-output description to analyze the emitted photonic fluxes and their correlation functions, we have shown, both for infinite and finite bandwidth detectors, that the emitted radiation is non-classical in the sense that the correlators violates Cauchy-Schwarz inequalities. We have also studied the time dependence of the photonic correlations and showed that their line-width becomes narrower with the increase of the emission rate approaching from below the threshold limit.