The fractional Josephson effect remains one of the decisive hallmarks of topologically protected Majorana zero modes. We analyze the effects of parity violating quasiparticle poisoning onto the current voltage characteristics of topological Josephson junctions. We include poisoning events directly within the resistively shunted junction (RSJ) model in the overdamped limit both in the short-and long-junction regime. We calculate the current voltage characteristics numerically where poisoning is modeled either via additional rates in the Fokker-Planck equations or by a time dependent parity and compare them to the limits of no and strong poisoning rates which we obtain analytically. Combining the tilted washboard potential with poisoning events, we show that the critical current of the long junction limit can be used as a probe of the junction topology even in the high temperature poisoning case where relaxation-and excitation processes are equally likely. Using the tilted washboard potential model we develop three different schemes to measure the poisoning rate thereby also extending the consideration to two pairs of helical edge states containing a constriction that allows for tunneling between the two pairs of edge states.

We propose a tunable topological Josephson junction in silicene where electrostatic gates could switch between a trivial and a topological junction. These aspects are a consequence of a tunable phase transition of the topologically confined valley-chiral states from a spin-degenerate to a spin-helical regime. We calculate the Andreev bound states in such a junction analytically using a low-energy approximation to the tight-binding model of silicene in proximity to s-wave superconductors as well as numerically in the short-and long-junction regime and in the presence of intervalley scattering. Combining topologically trivial and non-trivial regions, we show how intervalley scattering can be effectively switched on and off within the Josephson junction. This constitutes a topological Josephson junction with an electrically tunable quasiparticle poisoning source. arXiv:1808.02809v1 [cond-mat.mes-hall]

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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