  We study microwave radiation emitted by a small voltage-biased Josephson
junction connected to a superconducting transmission line. An input-output
formalism for the radiation field is established, using a perturbation
expansion in the junction's critical current. Using output field operators
solved up to the second order, we estimate the spectral density and the
second-order coherence of the emitted field. For typical transmission line
impedances and at frequencies below the main emission peak at the Josephson
frequency, radiation occurs predominantly due to two-photon emission. This
emission is characterized by a high degree of photon bunching if detected
symmetrically around half of the Josephson frequency. Strong phase fluctuations
in the transmission line make related nonclassical phase-dependent amplitude
correlations short lived, and there is no steady-state two-mode squeezing.
However, the radiation is shown to violate the classical Cauchy-Schwarz
inequality of intensity cross-correlations, demonstrating the nonclassicality
of the photon pair production in this region.
