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 $4\pi$ rather than a $2\pi$ periodicity. This effect is a consequence of the conservation of fermion parity—the number of electron $\text{mod}2$—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 diverges exponentially with temperature $T$ as $T\to 0$. Possible experiments on HgCdTe quantum wells will be discussed.