Josephson coupling through a quantum dot

We derive, via fourth-order perturbation theory, an expression for the Josephson current through a gated interacting quantum dot. We analyze our expression for two different models of the superconductor-dot-superconductor (SDS) system. When the matrix elements connecting dot and leads are featureless constants, we compute the Josephson coupling J_c as a function of the gate voltage and Coulomb interaction. In the limit of a diffusive dot, we compute the probability distribution P(J_c) of Josephson couplings. In both cases, π junction behavior (J_c<0) is possible, and is not simply dependent on the parity of the dot occupancy.