Switching the sign of Josephson current through Aharonov-Bohm interferometry

We investigate the DC Josephson effect in a superconductor-normal metal-superconductor junction where the normal region consists of a ballistic ring. We show that a fully controllable $\pi$-junction can be realized through the electro-magnetostatic Aharonov-Bohm effect in the ring. The sign and the magnitude of the supercurrent can be tuned by varying the magnetic flux and the gate voltage applied to one arm, around suitable values. The implementation in a realistic set-up is discussed.