Influence of an embedded quantum dot on the Josephson effect in the topological superconducting junction with Majorana doublets

TL;DR

This paper explores how an embedded quantum dot influences the Josephson effect in topological superconducting junctions with Majorana doublets, revealing conditions for normal, fractional, and $ ext{pi}$-period currents.

Contribution

It provides a detailed analysis of the impact of quantum dot coupling and magnetic fields on the Josephson effect in Majorana doublet systems, highlighting new conditions for fractional and $ ext{pi}$-period currents.

Findings

Quantum dot coupling affects Josephson current types.

Finite magnetic field induces fractional Josephson effect in odd-parity cases.

Coexistence of direct and indirect couplings suppresses fractional effects.

Abstract

One Majorana doublet can be realized at each end of the time-reversal-invariant Majorana nanowires. We investigate the Josephson effect in the Majorana-doublet-presented junction modified by different inter-doublet coupling manners. It is found that when the Majorana doublets couple indirectly via a non-magnetic quantum dot, only the normal Josephson effects occur, and the fermion parity in the system just affects the current direction and amplitude. However, in the odd-parity case, applying finite magnetic field on the quantum dot can induce the appearance of the fractional Josephson effect. Next, when the direct and indirect couplings between the Majorana doublets coexist, no fractional Josephson effect takes place, regardless of finite magnetic field on the quantum dot. Instead, the $\pi$-period current has an opportunity to appear in some special cases. All the results are clarified by analyzing the influence of the fermion occupation in the quantum dot on the parity conservation in the whole system. We ascertain that this work will be helpful for describing the dot-assisted Josephson effect between the Majorana doublets.