Josephson current transport through a Quantum Dot in an Aharonov-Bohm Ring

The Josephson current through an Aharonov-Bohm (AB) interferometer, in which a quantum dot (QD) is situated on one arm and a magnetic flux $\Phi$ threads through the ring, has been investigated. With the existence of the magnetic flux, the relation of the Josephson current and the superconductor phase is complex, and the system can be adjusted to $\pi$ junction by either modulating the magnetic flux or the QD's energy level $\varepsilon_d$. Due to the electron-hole symmetry, the Josephson current $I$ has the property $I(\varepsilon_d,\Phi)=I(-\varepsilon_d,\Phi+\pi)$. The Josephson current exhibits a jump when a pair of Andreev bound states aligns with the Fermi energy. The condition for the current jump is given. In particularly, we find that the position of the current jump and the position of the maximum value of the critical current $I_c$ are identical. Due to the interference between the two paths, the critical current $I_c$ versus the QD's level $\varepsilon_d$ shows a typical Fano shape, which is similar to the Fano effect in the corresponding normal device. But they also show some differences. For example, the critical current never reaches zero for any parameters, while the current in the normal device can reach zero at the destruction point.