Insulating regime of an underdamped current-biased Josephson junction supporting Z3 and Z4 parafermions

We study analytically a current-biased topological Josephson junction supporting Zn parafermions. First, we show that in an infinite-size system a pair of parafermions on the junction can be in n different states; the 2πn periodicity of the phase potential of the junction results in a significant suppression of the maximum current Im for an insulating regime of the underdamped junction. Second, we study the behavior of a realistic finite-size system with avoided level crossings characterized by splitting δ. We consider two limiting cases: when the phase evolution may be considered adiabatic, which results in the 2π periodicity of the effective potential, and the opposite case, when Landau-Zener transitions restore the 2πn periodicity of the phase potential. We also study the case with time-reversal symmetry and show that breaking this symmetry gives different phase periodicity reductions. resulting current Im is exponentially different in the opposite limits, which allows us to propose another detection method to establish the appearance of parafermions in the system experimentally, based on measuring Im at different values of the splitting δ.