 The study examines microwave radiation emitted by a Josephson junction connected to a superconducting transmission line and establishes an input-output formalism for the radiation field using perturbation expansion in the junction's critical current. It estimates the spectral density and second-order coherence of the emitted field, showing that two photon emission is predominant at frequencies below the main emission peak at the Josephson frequency. The study also demonstrates the non-classicality of the photon pair production in this region by violating the classical Cauchy-Schwartz inequality of intensity cross-correlations. This article was authored by J. Leppekangers, D. Johansson, M. Marfela, and others.