Self-Consistent Dynamics of a Josephson Junction in the Presence of an Arbitrary Environment

We derive microscopically the dynamics associated with the dc Josephson effect in a superconducting tunnel junction interacting with an arbitrary electromagnetic environment. To do so, we extend to superconducting junctions the so-called $P(E)$ theory (see, e.g., Ingold and Nazarov, arXiv:cond-mat/0508728) that accurately describes the interaction of a nonsuperconducting tunnel junction with its environment. We show the dynamics of this system is described by a small set of coupled correlation functions that take into account both Cooper pair and quasiparticle tunneling. When the phase fluctuations are small the problem is fully solved self-consistently, using and providing the exact linear admittance $Y(\omega )$ of the interacting junction.