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We study analytically an underdamped current-biased topological Josephsonjunction. First, we consider a simplified model at zero temperature, where theparity of the non-local fermionic state formed by Majorana bound states (MBSs)localized on the junction is fixed, and show that a transition from insulatingto conducting state in this case is governed by single-quasiparticle tunnelingrather than by Cooper pair tunneling in contrast to a non-topological Josephsonjunction. This results in a significantly lower critical current for thetransition from insulating to conducting state. We propose that, if the lengthof the system is finite, the transition from insulating to conducting stateoccurs at exponentially higher bias current due to hybridization of the stateswith different parities as a result of the overlap of MBSs localized on thejunction and at the edges of the topological nanowire forming the junction.Finally, we discuss how the appearance of MBSs can be establishedexperimentally by measuring the critical current for an insulating regime atdifferent values of the applied magnetic field.

Critical current for an insulating regime of an underdamped current-biased topological Josephson junction