Josephson Current in a Quantum Dot in the Kondo Regime Connected to Two Superconductors

We apply a Gutzwiller-like variational technique to study Josephson conduction across a quantum dot with an odd number of electrons connected to two superconducting leads. We show that, for small values of the superconducting gap, Kondo correlations and superconductivity cooperate to enhance the Josephson current. As the superconducting gap increases, the current changes sign and the system becomes a $\pi$-junction. The $\pi$-junction behavior sets in much before antiferromagnetic correlations at the dot can be treated perturbatively.