Josephson junction of finite-size superconductors on a topological insulator under a magnetic field

TL;DR

This paper theoretically investigates a Josephson junction on a topological insulator with finite-size superconductors under a magnetic field, revealing conditions for fractional AC Josephson effects and Majorana fermion braiding signatures.

Contribution

It demonstrates the occurrence of $2n \pi$ fractional AC Josephson effects without fine tuning and proposes experimental methods to observe Majorana braiding operations.

Findings

Fractional AC Josephson effects occur with midgap states.

Rich structure of Shapiro spikes facilitates experimental detection.

Observation of Majorana braiding through Josephson current measurements.

Abstract

We theoretically study a Josephson junction formed by two finite-size $s$-wave SCs on a topological insulator under a magnetic field. At certain conditions, the junction hosts the chiral Majorana modes enclosing the two finite-size SCs. The interplay of the extended chiral Majorana modes and the states inside the junction can results in nontrivial topological effects such as the $2n \pi$ fractional AC Josephson effects predicted in Ref.~\cite{ChoiSim} We show that the $2n \pi$ fractional AC Josephson effects can occur in a realistic situation, such as the presence of the midgap states, without requiring fine tuning of the parameters of the junction. We also find that the Shapiro spikes of the junction show a rich structure in a wide range of the AC voltage bias, facilitating experimental identification of the $2n \pi$ fractional AC Josephson effects. Moreover, we discuss how to observe the non-commutativity of the operations that braid the Majorana fermions of the junction, by measuring the Josephson current. Finally, we study the state evolution of the junction when the junction hosts a different number of Majorana zero modes from the case of the $2n \pi$ fractional AC Josephson effects.