We consider a quantum wire, containing two Majorana bound states (MBS) at its ends that are coupled to a current lead on one side and to a quantum dot (QD) on the other side. Using the method of full counting statistics we calculate the conductance and the zero-frequency noise. Using an effective low-energy model, we analyze in detail the Andreev reflection probability as a function of the various system parameters and show that it exhibits a Fano resonance (FR) line shape in the case of a weakly coupled QD as a function of the QD energy level when the two MBS overlap. The asymmetry parameter changes sign as the bias voltage is tuned through the MBS overlap energy. The FR is mirrored as a function of the QD level energy as long as tunneling to the more distant MBS is negligible. However, if both MBS are coupled to the lead and the QD, the height as well as the asymmetry of the line shapes cease to respect this symmetry. These two exclusive cases uniquely distinguish the coupling to a MBS from the coupling to a fermionic bound state that is shared between the two MBS. We complement the analysis by employing a discretized one-dimensional p-wave superconductor (Kitaev chain) for the quantum wire and show that the features of the effective low-energy model are robust towards a more complete Hamiltonian and also persist at finite temperature.

We study the equilibrium Josephson current between an s-wave superconductor and a topological superconducting nanowire with Majorana bound states (MBS) at its ends. Within a low-energy model we show analytically that the non-locality of the MBS allows for a finite supercurrent to flow that otherwise would vanish. In particular, we find the critical current to be a function of the difference in the spin canting angles of the Majorana wave functions at the location of the tunnel contact. We complement our analytical calculations by numerically solving the full tight binding model and show how to extract the main features of the low-energy model from the critical current using available experimental techniques.

In this minireview, we outline the recent experimental and theoretical progress in the creation, characterization and manipulation of Majorana bound states (MBSs) in semiconductor-superconductor (SC) hybrid structures. After an introductory overview of the broader field we specifically focus on four of our recent projects in this direction. We show that the emergence of Fano resonances in the differential conductance in a normal lead-Majorana nanowire-quantum dot setup can be exploited to determine if a single MBS is contacted by the normal lead and the quantum dot providing an experimental test of the nonlocality of MBSs. In the second project, the tunnel-coupling to two MBSs in an s-wave SC-Majorana nanowire Josephson junction (JJ) leads to a finite contribution of the MBSs to the equilibrium Josephson current probing directly the local spin-singlet contribution of the Majorana pair. We then shift our focus from MBSs forming in nanowire systems to MBSs forming in topological JJs. In a single sheet of buckled silicene with proximity induced superconductivity two local electric fields can be used to tune the junction between a topologically trivial and topologically non-trivial regime. In a Corbino geometry topological Josephson junction two MBSs harbored in Josephson vortices can rotate along the JJ and, in the course of this, will be exchanged periodically in the phase difference of the JJ. The tunneling current in a metal tip coupled to the JJ is shown to exhibit signs of the anyonic braiding phase of two MBSs.

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