Competing time scales and their experimental signatures in dissipative 1D Kitaev chain*

Using the Lindblad master equation formalism, we study the oscillation of Josephson current in coupled Kitaev chains with the a dissipative boundary. In the super operator representation,the ac Josephson current is calculated in both the topological trivial and nontrivial phases. Different periodicities can be observed with different time scales determined by the energy scales of the applied voltage. If the voltage is small compared to the bulk and edge states dissipation rate, we will see a 2pi periodic current in both the topological trivial and nontrivial phase. By increasing the voltage so that so that the relevant time scale is larger than the edge dissipation rate, we can see a 4pi oscillating current in the topological phase. At large voltage when the period of ac current is much smaller than the time scale of correlation propagating from the junction to the boundary, a 4pi current can be seen in both the topological and trivial phase. Interestingly, the 4pi Josephson current in the trivial phase is induced by the chemical potential gradient of the reservoir with no analogy in equilibrium systems.