Effective action theory of Andreev level spectroscopy

With the aid of the Keldysh effective action technique we develop a microscopic theory describing Andreev level spectroscopy experiments in non-tunnel superconducting contacts. We derive an effective impedance of such contacts which accounts for the presence of Andreev levels in the system. At subgap bias voltages and low temperatures inelastic Cooper pair tunneling is accompanied by transitions between these levels resulting in a set of sharp current peaks. We evaluate the intensities of such peaks, establish their dependence on the external magnetic flux piercing the structure and estimate thermal broadening of these peaks. We also specifically address the effect of capacitance renormalization in a non-tunnel superconducting contact and its impact on both the positions and heights of the current peaks. At overgap bias voltages the $I-V$ curve is determined by quasiparticle tunneling and contains current steps related to the presence of discrete Andreev states in our system.