  We present a microscopic theory for the current through a tunnel Josephson
junction coupled to a non-linear environment, which consists of an Andreev
two-level system coupled to a harmonic oscillator. It models a recent
experiment [Bretheau, Girit, Pothier, Esteve, and Urbina, Nature (London) 499,
312 (2013)] on photon spectroscopy of Andreev bound states in a superconducting
atomic-size contact. We find the eigenenergies and eigenstates of the
environment and derive the current through the junction due to inelastic Cooper
pair tunneling. The current-voltage characteristic reveals the transitions
between the Andreev bound states, the excitation of the harmonic mode that
hybridizes with the Andreev bound states, as well as multi-photon processes.
The calculated spectra are in fair agreement with the experimental data.
