  We study theoretically electromagnetic radiation emitted by inelastic
Cooper-pair tunneling. We consider a dc-voltage-biased superconducting
transmission line terminated by a Josephson junction. We show that the
generated continuous-mode electromagnetic field can be expressed as a function
of the time-dependent current across the Josephson junction. The leading-order
expansion in the tunneling coupling, similar to the "$P(E)$-theory", has
previously been used to investigate the photon emission statistics in the limit
of sequential (independent) Cooper-pair tunneling. By explicitly evaluating the
system characteristics up to the fourth-order in the tunneling coupling, we
account for dynamics between consecutively tunneling Cooper pairs. Within this
approach we investigate how temporal correlations in the charge transport can
be seen in the first- and second-order coherences of the emitted microwave
radiation.
