Josephson current in a four-terminal superconductor/exciton-condensate/superconductor system

We investigate the transport properties of a bilayer exciton condensate that is contacted by four superconducting leads. We focus on the equilibrium regime and investigate how the Josephson currents induced in the bilayer by phase biases applied to the superconducting electrodes are affected by the presence of an exciton condensate in the bulk of the system. As long as the distance between the superconducting electrodes is much larger than the exciton coherence length, the Josephson current depends only on the difference between the phase biases in the two layers. This result holds true in both short- and long-junction limits. We relate it to a new correlated four-particle Andreev process, which occurs at the superconductor/exciton-condensate interface. The system we investigate provides an implementation of the supercurrent mirror proposed by Kitaev as a viable way to realize topologically protected qubits.