4π -periodic anomalous Josephson effect between Majorana zero modes

Abstract

Usually the Josephson current is driven by the superconducting phase difference $\ensuremath{\varphi}$ between two superconducting leads. Here we propose an anomalous Josephson current between two Majorana nanowires, which can be driven by both the superconducting phase difference $\ensuremath{\varphi}$ and the orientation difference $\ensuremath{\phi}$ between the Zeeman fields in two Majorana wires. We show that the orientation of the Zeeman field acts as an effective superconducting phase under appropriate conditions, which provides a new signature of Majorana zero modes. Due to the presence of Majorana zero modes, the anomalous Josephson current is $4\ensuremath{\pi}$ periodic in both $\ensuremath{\varphi}$ and $\ensuremath{\phi}$. When both $\ensuremath{\varphi}$ and $\ensuremath{\phi}$ are periodically varying, we predict the existence of peaks in the average Josephson current, which are the analog to Shapiro steps in the ac Josephson effect. Similarly, the existence of Majorana zero modes can be confirmed by the missing of odd peaks. The findings provide a scheme to experimentally verify the $4\ensuremath{\pi}$-periodic Josephson effect, as well as a platform to realize the anomalous Josephson junction.