Signatures of Majorana bound states in the electronic transport through finite-size topological superconductors

Abstract

Majorana bound states (MBSs) offer a promising route to fault-tolerant quantum computation, because of their non-Abelian anyonic exchange statistics. They emerge as protected boundary modes of one dimensional topological superconductors (TSCs). Due to the finite size of these TSCs the wave functions of the two MBSs can spread across the whole TSC which leads to the possibility to access both MBSs at the same end of the TSC. First, we consider a spinless metallic lead-TSC-quantum dot setup in which the Majorana system is described with a Kitaev chain. Here, we show that a pair of Fano resonances arises as a function of dot level energy in the differential conductance. In an analytical low-energy description, we show that in the case of isolated MBS, i.e. only one MBS is contacted by the lead and the second MBS is only contacted by the quantum dot, these Fano resonances are invariant under a sign change of the dot level energy. This symmetry, however, is broken as soon as we allow the quantum dot to not only couple to one but also to the second MBS. Next, we consider a spinful model, in which the MBS system is given by a semiconducting nanowire with Rashba spin orbit interaction, proximity induced s-wave superconductivity and an applied Zeeman field. In this scenario, we find that even without a coupling to the dot the transport properties can be used to determine the different couplings to both MBSs. Furthermore, we find that the spin canting angles of the MBSs have a profound influence on the low-energy transport properties. We underline our analytical findings with a numerical treatment of the proposed transport setup where we apply the mean field approximation for the Coulomb energy selfconsistently. For the Josephson junction we use the quasidegenerate perturbation theory to obtain an effective low-energy Hamiltonian. Our calculations show that the MBSs only contribute to the equilibrium Josephson current if both of them can be adressed by electron tunneling from the lead. Moreover, we find that the critical current is oscillating as a function of applied Zeeman field and exhibits a sign change at parity crossings. A numerical analysis reveals the contributions of higher energy states due to a residual s-wave pairing in the topologically non-trivial regime which shadow the signatures related to the MBSs. We therefore suggest an experimental scheme that uses quasiparticle poisoning to unveil the Majorana contributions.