Josephson currents in quantum Hall devices

We consider a simple model for a superconductor/normal-metal/superconductor Josephson junction in which the "normal metal" is a section of a filling-factor nu = 2 integer quantum Hall edge. We provide analytic expressions for the current/phase relations to all orders in the coupling between the superconductor and the quantum Hall edge modes, and for all temperatures. Our conclusions are consistent with the earlier perturbative study by Ma and Zyuzin [Europhys. Lett. 21, 941 (1993)]: The Josephson current is independent of the distance between the superconducting leads, and the upper bound on the maximum Josephson current is inversely proportional to the perimeter of the Hall device.