Quench dynamics of the Josephson current in a topological Josephson junction

The $4\ensuremath{\pi}$-periodic Josephson effect is a distinguishing feature of a topological Josephson junction. However, stringent conditions make it hard to observe in experiments. In this work, we study the transient transport properties in a topological Josephson junction numerically. We show that the $4\ensuremath{\pi}$ Josephson current can be sustained under nonequilibrium conditions. The properties of the Josephson current are analyzed for different conditions and three main regimes are identified. First, when both the superconducting wires of the Josephson junction lie in the topologically nontrivial region, a $4\ensuremath{\pi}$ Josephson current can appear upon suddenly applying a dc voltage. Second, when one superconducting wire lies in the trivial region, while the other wire lies in the nontrivial region, the Josephson current is $2\ensuremath{\pi}$ periodic but the component of the higher-order Josephson current increases. Third, when both wires lie in the trivial region, a stable $2\ensuremath{\pi}$ Josephson current is observed. Most importantly, the fractional Josephson effect is fragile in the presence of disorder. Hence experiments should be designed carefully to eliminate the effect of disorder. These results could be helpful to optimize fine-tuning of the experimental parameters to observe the $4\ensuremath{\pi}$-periodic Josephson current in a topological Josephson junction.