Andreev bound states in hybrid superconductor-semiconductor devices can have near-zero energy in the topologically trivial regime as long as the confinement potential is sufficiently smooth. These quasi-Majorana states show zero-bias conductance features in a topologically trivial phase, mimicking spatially separated topological Majorana states. We show that in addition to the suppressed coupling between the quasi-Majorana states, also the coupling of these states across a tunnel barrier to the outside is exponentially different for increasing magnetic field. As a consequence, quasi-Majorana states mimic most of the proposed Majorana signatures: quantized zero-bias peaks, the 4π Josephson effect, and the tunneling spectrum in presence of a normal quantum dot. We identify a quantized conductance dip instead of a peak in the open regime as a distinguishing feature of true Majorana states in addition to having a bulk topological transition. Because braiding schemes rely only on the ability to couple to individual Majorana states, the exponential control over coupling strengths allows to also use quasi-Majorana states for braiding. Therefore, while the appearance of quasi-Majorana states complicates the observation of topological Majorana states, it opens an alternative route towards braiding of non-Abelian anyons and protected quantum computation.

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The proximity effect in hybrid superconductor-semiconductor structures, crucial for realizing Majorana edge modes, is complicated to control due to its dependence on many unknown microscopic parameters. In addition, defects can spoil the induced superconductivity locally in the proximitized system, which complicates measuring global properties with a local probe. We show how to use the nonlocal conductance between two spatially separated leads to probe three global properties of a proximitized system: the bulk superconducting gap, the induced gap, and the induced coherence length. Unlike local conductance spectroscopy, nonlocal conductance measurements distinguish between nontopological zero-energy modes localized around potential inhomogeneities, and true Majorana edge modes that emerge in the topological phase. In addition, we find that the nonlocal conductance is an odd function of bias at the topological phase transition, acting as a current rectifier in the low-bias limit. More generally, we identify conditions for crossed Andreev reflection to dominate the nonlocal conductance and show how to design a Cooper pair splitter in the open regime.

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Nowadays, quantum computers, a promising direction of computer hardware development, suffer too much from errors caused by disturbance of qubits to compete with the state-of-the-art classical computers. Topological phases in condensed matter physics offer a solution: the topological edge states emerging in such a phase are spatially separated by an insulating bulk, which makes a qubit constructed from such topological states much more resilient to local perturbations. Majorana states are examples of topological edge states, and form the main focus of research on topological quantum computing due to indications of successful creation and detection of Majorana states in one-dimensional superconductor-semiconductor hybrid devices. Quasi-Majorana states share most characteristics of Majorana states, but appear in the topologically trivial phase and appear at the same position, in contrast to topological, spatially separated Majorana states. For this reason, quasi-Majorana states are not protected from noise, and hence appear to be not useful for quantum computing. For a long while, quasi-Majorana states were considered a nuisance, because they mimic the local signatures of topological Majorana states, and therefore can make a false positive signature in the search for Majorana states. In trying to understand how Majorana devices work, I started this thesis with an investigation of electrostatics in Majorana devices. An applied gate voltage sets the chemical potential in a Majorana nanowire, relevant for the creation of Majorana states, but this is influenced by other electrostatic components in the environment. I found that these electrostatic effects introduce non-universal, geometry-dependent behaviour of two Majorana characteristics: the shape of the topological phase boundary and the oscillations of the Majorana splitting energy. In addition to controlling the band structure, gate electrodes alter the transport properties of electrons by creating a tunnel barrier or a constriction in the potential. Studying such constrictions, I have demonstrated that the confinement potential barriers are smooth, allowing to measure the helical gap in the band structure, which agrees with experimental observations. Since quasi-Majorana states appear at the slope of a smooth confinement potential, the results of my simulations of the electrostatics motivated to continue with an investigation of these states. I showed that quasi-Majorana states not only have an exponentially suppressed energy as a function of magnetic field, but also have an exponentially different tunnel coupling across the barrier where they are located. This realization allowed me to strengthen the recent observations of similarity between Majorona states and quasi-Majorana states, and conclude that tunneling measurements can not distinguish Majorana states from quasi-Majorana states as a matter of principle. Because of this extreme similarity, I turned to study a possible alternative strategy to distinguish topological Majorana states from quasi-Majorana states. This strategy ix x SUMMARY focusses on rectifying behaviour in the nonlocal conductance through a Majorana wire connected to two normal leads. This phenomenon measures a global topological phase transition, rather than a local measure of the density of states, and therefore it is not influenced by the presence of quasi-Majorana states. The similarity of quasi-Majorana and topological Majorana states also leads to an unexpected consequence. Braiding (an exchange of two Majorana states), a building block of a topological quantum computer, can also be done with quasi-Majorana states. Although quasi-Majorana states appear next to each other, their couplings are exponentially different, which allows to control them individually. Braiding quasi-Majorana states can even be advantageous, because it requires less precise control over system parameters. I therefore conclude that braiding of quasi-Majorana states is within experimental reach, and opens an alternative route on realizing a quantum computer. @en

The motion of an electron and its spin are generally not coupled. However in a one-dimensional material with strong spin-orbit interaction (SOI) a helical state may emerge at finite magnetic fields, where electrons of opposite spin will have opposite momentum. The existence of this helical state has applications for spin filtering and cooper pair splitter devices and is an essential ingredient for realizing topologically protected quantum computing using Majorana zero modes. Here, we report measurements of a quantum point contact in an indium antimonide nanowire. At magnetic fields exceeding 3 T, the 2 e2/h conductance plateau shows a re-entrant feature toward 1 e2/h which increases linearly in width with magnetic field. Rotating the magnetic field clearly attributes this experimental signature to SOI and by comparing our observations with a numerical model we extract a spin-orbit energy of approximately 6.5 meV, which is stronger than the spin-orbit energy obtained by other methods.@en

One of the promising platforms for creating Majorana bound states is a hybrid nanostructureconsisting of a semiconducting nanowire covered by a superconductor. We analyze the previouslydisregarded role of electrostatic interaction in these devices. Our main result is that Coulombinteraction causes the chemical potential to respond to an applied magneticfield, while spin–orbitinteraction and screening by the superconducting lead suppress this response. Consequently, theelectrostatic environment influences two properties of Majorana devices: the shape of the topologicalphase boundary and the oscillations of the Majorana splitting energy. We demonstrate that bothproperties show a non-universal behavior, and depend on the details of the electrostatic environment.We show that when the wire only contains a single electron mode, the experimentally accessibleinverse self-capacitance of this mode fully captures the interplay between electrostatics and Zeemanfield. This offers a way to compare theoretical predictions with experiments@en