# Signatures of a 4$\pi$-periodic supercurrent in the voltage response of capacitively shunted topological Josephson junctions

## TL;DR

This paper models how a 4π-periodic supercurrent affects the voltage response in topological Josephson junctions, proposing methods to detect Majorana states through Shapiro step analysis and spectral features.

## Contribution

It introduces a semiclassical RCSJ model incorporating capacitance to identify signatures of 4π-periodic supercurrents and Majorana states in Josephson junctions.

## Key findings

- Capacitance causes quenching of odd Shapiro steps in underdamped junctions.
- Presence of 4π-periodic supercurrent alters Fourier spectra and voltage response.
- Period-doubling complicates detection of 4π periodicity in spectral analysis.

## Abstract

We investigate theoretical aspects of the detection of Majorana bound states in Josephson junctions using the semiclassical RCSJ model of junction dynamics. The influence of a 4$\pi$-periodic supercurrent contribution can be detected through its effect on the width of the Shapiro steps and the Fourier spectrum of the voltage signal. We explain how the inclusion of a capacitance term results in a strong quenching of the odd steps when the junction is underdamped, and hence may be used to effectively detect Majorana bound states. Furthermore, in presence of capacitance the first and third steps are quenched to a different degree, as observed experimentally. We examine the emission spectrum of phase-locked solutions, showing that the presence of period-doubling may difficult the measurement of the 4$\pi$-periodic contribution from the Fourier spectrum. Finally, we study the voltage response in the quasiperiodic regime and indicate how the Fourier spectra and the first-return maps in this regime reflect the change of periodicity in the supercurrent.