Josephson current in strongly correlated double quantum dots

We study the transport properties of a serial double quantum dot (DQD) coupled to two superconducting leads, focusing on the Josephson current through the DQD and the associated 0-$\pi$ transitions which result from the subtle interplay between the superconductivity, the Kondo physics, and the inter-dot superexchange interaction. We examine the competition between the superconductivity and the Kondo physics by tuning the relative strength $\Delta/T_K$ of the superconducting gap $\Delta$ and the Kondo temperature $T_K$, for different strengths of the superexchange coupling determined by the interdot tunneling $t$ relative to the dot level broadening $\Gamma$. We find strong renormalization of $t$, a significant role of the superexchange coupling $J$, and a rich phase diagram of the 0 and $\pi$-junction regimes. In particular, when both the superconductivity and the exchange interaction are in close competion with the Kondo physics ($\Delta\sim J\sim T_K$), there appears an island of $\pi'$-phase at large values of the superconducting phase difference.