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 pair of parafermions on the can be $n$ different states; $2\pi{n}$ periodicity phase potential results significant suppression maximal current $I_m$ for insulating regime underdamped junction. Second, behaviour realistic finite-size with avoided level crossings characterized by splitting $\delta$. consider two limiting cases: when evolution may considered adiabatic, which decreased effective potential, and opposite case, Landau-Zener transitions restore potential. The resulting is exponentially limits, allows us to propose new detection method establish appearance experimentally, based measuring at values