Dynamics of Majorana states in a topological Josephson junction

Topological Josephson junctions carry 4π-periodic bound states. A finite bias applied to the junction limits the lifetime of the bound state by dynamically coupling it to the continuum. Another characteristic time scale, the phase adjustment time, is determined by the resistance of the circuit "seen" by the junction. We show that the 4π periodicity manifests itself by an even-odd effect in Shapiro steps only if the phase adjustment time is shorter than the lifetime of the bound state. The presence of a peak in the current noise spectrum at half the Josephson frequency is a more robust manifestation of the 4π periodicity, as it persists for an arbitrarily long phase adjustment time. We specify, in terms of the circuit parameters, the conditions necessary for observing the manifestations of 4π periodicity in the noise spectrum and Shapiro step measurements.