Josephson current through a correlated quantum level: Andreev states and $\pi$ junction behavior

The Josephson transport and the electronic properties of a quantum dot characterized by a single level coupled to superconducting leads is analyzed. Different approximations are used and compared: the mean-field approximation, the second-order perturbation theory in the Coulomb interaction, and the exact diagonalization in the zero bandwidth limit. The system exhibits a rich behavior as a function of the relevant parameters. We discuss in detail the conditions for the observation of $\pi$ junction behavior and the effect of Coulomb interactions on the Andreev states.