Tuning the the fundamental periodicity of the current-phase relation in multiterminal diffusive Josephson junctions

Conventional superconductor/insulator/superconductor (SIS) Josephson junctions, devices where two superconductors are separated by a tunnel barrier are technologically important as elements in quantum circuits, particularly with their key role in superconducting qubits. An important characteristic of Josephson junctions is the relation between the supercurrent Is and the phase difference $\phi$ between them. For SIS junctions, the current-phase relation is sinusioidal and 2$\pi$ periodic. Other types of Josephson junctions, where the material between the superconductors is a weak link or a normal metal (N) may have non-sinusoidal current-phase relations that are still 2$\pi$ periodic. We show here that a multi-terminal diffusive SNS Josephson junction with 4 superconducting contacts can show a current phase relation between two of the contacts that is a superposition of 2$\pi$ and 4$\pi$ periodic components whose relative strength is controlled by the phase difference between the other two contacts, becoming 2$\pi$ or 4$\pi$ periodic for certain values of this phase difference. This tunability might have applications in tailoring the Hamiltonians of superconducting quantum circuits.