Insulating regime of an underdamped current-biased Josephson junction supporting ${\mathbb{Z}}_3$ and ${\mathbb{Z}}_4$ parafermions

Abstract

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 $2\pi n$ periodicity of the phase potential of the junction results in a significant suppression of the maximal current $I_m$ for an insulating regime of the underdamped junction. Second, we study the behaviour 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 decreased periodicity of the effective potential, and the opposite case, when Landau-Zener transitions restore the $2\pi n$ periodicity of the phase potential. The resulting current $I_m$ is exponentially different in the opposite limits, which allows us to propose a new detection method to establish the appearance of parafermions in the system experimentally, based on measuring $I_m$ at different values of the splitting $\delta$.