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The fractional Josephson effect remains one of the decisive hallmarks oftopologically protected Majorana zero modes. We analyze the effects of parityviolating quasiparticle poisoning onto the current voltage characteristics oftopological Josephson junctions. We include poisoning events directly withinthe resistively shunted junction (RSJ) model in the overdamped limit both inthe short- and long-junction regime. We calculate the current voltagecharacteristics numerically where poisoning is modeled either via additionalrates in the Fokker-Planck equations or by a time dependent parity and comparethem to the limits of no and strong poisoning rates which we obtainanalytically. Combining the tilted washboard potential with poisoning events,we show that the critical current of the long junction limit can be used as aprobe of the junction topology even in the high temperature poisoning casewhere relaxation- and excitation processes are equally likely. Using the tiltedwashboard potential model we develop three different schemes to measure thepoisoning rate thereby also extending the consideration to two pairs of helicaledge states containing a constriction that allows for tunneling between the twopairs of edge states.

Quasiparticle poisoning effects on the dynamics of topological Josephson junctions