Isolated hybrid normal/superconducting ring in a magnetic flux: From persistent current to Josephson current

We investigate the ground state current of an isolated hybrid normal/superconducting ring (NS ring), threaded by an Aharonov-Bohm magnetic flux. We calculate the excitation spectrum of the ring for any values of the lengths of the normal metal and of the superconductor. We describe the nonlinear flux dependence of the energy levels above and below the gap edge. Using a harmonics expansion for the current, we isolate the contribution due to these nonlinearities and we show that it vanishes for a large normal segment length d(N). The remaining contribution is very easy to evaluate from the linearized low energy spectrum. This decomposition allows us to recover in a controlled way the current-flux relationships for SNS junctions and for NS rings. We also study the crossover from persistent current to Josephson current in a multichannel NS ring at finite temperature.