Theory of microwave spectroscopy of Andreev bound states with a Josephson junction

We present a microscopic theory for the current through a tunnel Josephson junction coupled to a nonlinear 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)NATUAS0028-083610.1038/nature12315] 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 multiphoton processes. The calculated spectra are in fair agreement with the experimental data.