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.