# Topological phase diagram of a three-terminal Josephson junction: From the conventional to the Majorana regime

## TL;DR

This paper investigates how the transconductance in three-terminal Josephson junctions changes during a topological phase transition from s-wave to p-wave (Majorana) induced by an in-plane magnetic field, revealing fractional quantization due to Majorana quasiparticles.

## Contribution

It provides a comprehensive analysis of the topological transition in Josephson junctions, highlighting the fractionalization of transconductance quantization caused by Majorana states.

## Key findings

- Transconductance quantization becomes fractional in the Majorana regime.
- The transition is characterized as a function of magnetic field and flux.
- Majorana quasiparticles lead to effective half-Cooper pair charge pumping.

## Abstract

We study the evolution of averaged transconductances in three-terminal Josephson junctions when the superconducting leads are led throughout a topological phase transition from an $s$-wave to a $p$-wave (Majorana) phase by an in-plane magnetic field $B_x$. We provide a complete description of this transition as a function of $B_x$ and a magnetic flux $\Phi$ threading the junction. For that we use a spinful model within a formalism that allows to treat on equal footing the contribution to the transconductance from both the Andreev subgap levels and the continuum spectrum. We unveil a fractionalization in the quantization of the transconductance due to the presence of Majorana quasiparticles, reflecting the effective pumping of half a Cooper pair charge in the $p$-wave regime.