Josephson current and noise at a superconductor/quantum-spin-Hall-insulator/superconductor junction

We study junctions between superconductors mediated by the edge states of a quantum-spin-Hall insulator. We show that such junctions exhibit a fractional Josephson effect, in which the current phase relation has a $4\ensuremath{\pi}$ rather than a $2\ensuremath{\pi}$ periodicity. This effect is a consequence of the conservation of fermion parity---the number of electron $\text{mod}\text{ }2$---in a superconducting junction and is closely related to the ${Z}_{2}$ topological structure of the quantum-spin-Hall insulator. Inelastic processes, which violate the conservation of fermion parity, lead to telegraph noise in the equilibrium supercurrent. We predict that the low-frequency noise due these processes…