Fano resonance in Josephson current

We theoretically study the Fano resonance in Josephson current. The system under consideration is a double-path Josephson junction in which one path is through an insulating barrier and the other one is through a quantum dot (QD). To treat the Coulomb interaction inside the QD, we employ the Hubbard-I approximation, in which the correlation δ n ↑ δ n ↓ ) can be taken into account beyond the Hartree-Fock approximation. It is found that the dependence of the Josephson critical current on the QD energy level has an asymmetric resonance, which is similar to the Fano resonance in normal metal systems. Moreover, we find that the sign of the critical current can change around the resonance, and such behavior is not seen without the Coulomb interaction. These results indicate that the cooperation of the Fano effect and the Coulomb interaction induces the 0-π transition.