A minimal Kitaev chain can be realized by coupling two quantum dots to a superconductor on both sides. Andreev Bound States inside the superconductor mediate two types of interdot couplings: Cross Andreev Re ection (CAR) and Elastic Co-tunneling (ECT). Spin-orbit interaction (SOI) enables equal strengts of CAR and ECT, where Majorana Bound States with a quadratic protection emerge. By extending the chain to more dots, the protection is predicted to become stronger. In this project, semiconducting-superconducting InSb nanowires provide both SOI and superconductivity. We develop systematic procedures to tune a two-site device to hold Majorana Bound States. Next, quantum transport processes on a three-site device are studied. Sequential processes combining CAR and ECT are observed.

The electrostatics has effect on conductance of nanowire devices. In this model electrostatics are described by nonlinear coupling of the Poisson and the Schr¨odinger equations. In presence of magnetic and electric field and spin-orbit interaction, conductance develops a feature called the helical gap. This gap is characterised by a drop of conductance and is the main focus of the research. The solver is based around an Anderson mixing scheme, and specific class of points has been discovered for which the solver performs poorly. For those points, an temperature annealing subroutine has been put in place to speed up convergence. This subroutine efficiently solves the system for some small finite temperature. The solver has also been expanded to solve systems for magnetic field pointed in any direction of the y, z plane. As a result, it is now possible to perform simulations for different magnitudes and directions of magnetic field, which are a handy tool for understanding the behaviour of conductance as a function of VG in real nanowires. The relation between energy and conductance has been researched. The size of helical gap is found to scale linearly with the Zeeman energy EZ, while other features of conductance scale nonlinearly

In this thesis, the methods of Matrix Product States are examined. We give background in the origin of the methods and explain in detail how the method can be applied to a series of Josephson junctions. Numerical results are given for the ground state energy analysis of these series.

A method (1SBA) is derived and used to calculate disordered band structures. The validity of the method is investigated and the disordered band structures are compared to transport calculations.