Andreev multiple reflection and current in a junction with a quantum dot embedded in a superconducting ring

We investigate the properties of Andreev multiple reflection and supercurrent in a normal-metal junction with a quantum dot embedded in a mesoscopic superconducting ring. The quantum dot is in the Coulomb blockade regime and contains several energy levels. The supercurrent through the junction is created by an external magnetic flux threaded through the ring. The supercurrent is related to the multiple Andreev reflection at interfaces between the superconductor and the normal metal, which is influenced by the Coulomb blockade on the quantum dot. Using a tight-binding model with Bogoliubov-de Gennes pairing potential, the supercurrent versus the external flux is calculated. By switching on the gate voltage applied on the dot, the supercurrent shows a series of peaks and valleys for various values of the flux through the ring. The results suggest the strong effect of the quantum dot on the supercurrent in such a structure, and the possible methods of controlling the supercurrent.