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