Insulating regime of an underdamped current-biased Josephson junction supporting Z(3) and Z(4) parafermions

We study analytically a current-biased topological Josephson junction supporting Z(n) parafermions. First, we show that in an infinite-size system a pair of parafermions on the junction can be in n different states; the 2 pi n periodicity of the phase potential of the junction results in a significant suppression of the maximum current I-m 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 delta. We consider two limiting cases: when the phase evolution may be considered adiabatic, which results in the 2 pi periodicity of the effective potential, and the opposite case, when Landau-Zener transitions restore the 2 pi n 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 I-m 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 I-m at different values of the splitting delta.