Quench Dynamics of Josephson Current in a Topological Josephson Junction

The $4\pi$ Josephson Effect is a distinguishing feature of a topological Josephson junction. However, stringent conditions make it hard to observe in experiments. Here we numerically study the transient transport properties in a topological Josephson junction. We show that the $4\pi$ Josephson current can be sustained for a significant time (around several $\mu s$ with suitable conditions). Furthermore, we compare the behaviors of Josephson current in different conditions, identifying three main regimes: First, when both the superconducting wires of the Josephson junction lie in the topologically nontrivial region, the $4\pi$ Josephson current can appear with a suddenly applied DC voltage. Second, when one superconducting wire lies in the trivial region and the other one lies in the non-trivial region, the Josephson current is $2\pi$ periodic but unstable with the evolving of time. Third, when both wires lie in the trivial region, a stable $2\pi$ Josephson current is observed. These results can facilitate fine-tuning of the experiment parameters in order to finally observe the $4\pi$ Josephson current in a topological Josephson junction.