Dynamics of Majorana States in a Topological Josephson Junction

TL;DR

This paper investigates how the 4pi periodicity of topological Josephson junctions manifests through Shapiro steps and noise spectrum, emphasizing the importance of circuit parameters in observing these effects.

Contribution

It provides a detailed analysis of the conditions under which 4pi periodicity effects can be observed in Josephson junctions, linking circuit parameters to experimental signatures.

Findings

Even-odd effect in Shapiro steps depends on phase adjustment time being shorter than bound state lifetime.

A peak at half the Josephson frequency in noise spectrum is a robust indicator of 4pi periodicity.

Specific circuit conditions are identified for observing 4pi effects in experiments.

Abstract

Topological Josephson junctions carry 4pi-periodic bound states. A finite bias applied to the junction limits the lifetime of the bound state by dynamically coupling it to the continuum. Another characteristic time scale, the phase adjustment time, is determined by the resistance of the circuit "seen" by the junction. We show that the 4pi periodicity manifests itself by an even-odd effect in Shapiro steps only if the phase adjustment time is shorter than the lifetime of the bound state. The presence of a peak in the current noise spectrum at half the Josephson frequency is a more robust manifestation of the 4pi periodicity, as it persists for an arbitrarily long phase adjustment time. We specify, in terms of the circuit parameters, the conditions necessary for observing the manifestations of 4pi periodicity in the noise spectrum and Shapiro step measurements.