  We study scattering of propagating microwave fields by a DC-voltage biased
Josephson junction. At sub-gap voltages, a small Josephson junction works
merely as a non-linear boundary that can absorb, amplify, and diversely convert
propagating microwaves. In the leading-order perturbation theory of the
Josephson coupling energy, the spectral density and quadrature fluctuations of
scattered thermal and coherent radiation can be described in terms of the
well-known $P(E)$ function. Applying this, we study how thermal and coherent
radiation is absorbed and amplified in an Ohmic transmission line and in a
circuit with a resonance frequency. We show when a coherent input can create a
two-mode squeezed output. In addition, we evaluate scattering amplitudes
between arbitrary photon-number (Fock) states, characterizing individual photon
multiplication and absorption processes occuring at the junction.

Superconductivity;Mesoscale and Nanoscale Physics;Quantum Physics