Self-consistent dynamics of a Josephson junction in presence of an arbitrary environment

We derive microscopically the dynamics associated with the d.c. 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.