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

We study analytically a current-biased topological Josephson junction supporting ${\mathbb{Z}}_{n}$ parafermions. First, we show that in an infinite-size system a pair of parafermions on the junction can be in $n$ different states; the $2\ensuremath{\pi}n$ 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 $\ensuremath{\delta}$. We consider two limiting cases: when the phase evolution may be considered adiabatic, which results in the $2\ensuremath{\pi}$ periodicity of the effective potential, and the opposite case, when Landau-Zener transitions restore the $2\ensuremath{\pi}n$ 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 $\ensuremath{\delta}$.