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

TL;DR

This paper analyzes the insulating regime of underdamped current-biased Josephson junctions supporting $ ext{Z}_n$ parafermions, revealing how their unique periodicity affects maximal current and proposing an experimental detection method.

Contribution

It provides an analytical study of $ ext{Z}_n$ parafermion-supported Josephson junctions, highlighting the impact of finite size effects and Landau-Zener transitions on the current and periodicity.

Findings

Maximal current $I_m$ is suppressed in the insulating regime due to $2 ext{pi}n$ periodicity.

Finite-size effects cause avoided level crossings characterized by splitting $ extdelta$.

Different regimes of phase evolution lead to exponentially different $I_m$, enabling experimental detection of 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 $\delta$.