In this work, Clebsch-Gordan coefficients are studied from both a quantum mechanical and a mathematical perspective. In quantum mechanics, Clebsch-Gordan coefficients arise when two quantum systems with a certain angular momentum are combined and the total angular momentum is to be found. We start by discussing the relevant postulates of quantum mechanics. From there we move on to a discussion of quantum angular momentum, the combining of quantum systems, and Clebsch-Gordan coefficients. Finally, an algorithm for calculating the Clebsch-Gordan coefficients is developed. A firm grasp of linear algebra is required to understand this quantum mechanical perspective. The mathematical perspective is rooted in representation theory. The Clebsch-Gordan coefficients show the decomposition of the tensor product of irreducible representations of the three dimensional rotation group into the direct sum of irreducible representations. We explain the necessary theory to understand how this works. This includes a discussion of irreducible unitary representations, spherical harmonics, direct sum representations, and tensor product representations. Some experience with group theory is required to understand the mathematical perspective.

In this text we used a quantum system of two quantum particles, specificallytwo qubits. One qubit acts as a detector in an equal weight superposition ofspin up and spin down. The two qubits in the system are entangled. To domultiple measurements on the system, we want to rotate the system back to theequal weight superposition, which is destroyed after a measurement.We have calculated the time evolution of the density matrix of this quantumsystem using a standard differential equation. We altered this differential equa-tion using the counting fields method. After this we used the density matrix tocalculate the optimal angle of rotation to return the qubit to the equal weightsuperposition. We did this for an ideal system without outside influences andfor a system with outside influences. We derived a relationship between theoutside influences on the system,γ, whereγ= 2 represents an ideal system,and a variable inside the system,z. This relation is2γ=z.We also described an algorithm to simulate a random quantum trajectory.We found that the value of the spin in the z-direction does go to the expectedvalue of 0 when the amount of single trajectories in the average trajectory islarge. This is true for all values ofγ. For the value of the spin in the x-directionwe found that the theoretical curve correctly predicts the behavior of an idealsystem. We found that for a non ideal system, whenγ >2, the theoreticalcurve does not correctly predict the behavior of a trajectory. The theory fails topredict the speed with which the spin in the x-direction goes to 0 in a non idealsystem, where the theory is significantly slower at going to 0. Also the theorypredicts the starting value to be lower than 1 in a non ideal system, which isnot the case in practice

Majorana bound states (MBS) have non-Abelian exchange statistics, which means exchanging the position of two MBS changes the state of the system. This attracted attention for multiple reasons: It is a quantum effect without a classical analogue, it introduces topology to condensed matter physics and the non-Abelian exchange processes allow quantum computing with intrinsic error protection. For a long time research on MBS was purely theoretical, because there was no experimentally accessible idea how to create them. This changed with the appearance of a recipe combining conventional superconductivity, semiconductors and a magnetic field. Now such setups exist and their conductance indicates the successful creation of MBS, but challenges regarding their quality, conclusive detection and control remain. In this thesis I propose different strategies for avoiding open experimental problems when creating MBS, and identify new pathways for detecting them. Most proposals for creating MBS are similar in two regards: They confine electrons to lower dimensional systems, for example a 1-dimensional wire, and they use magnetic field to couple to the electron spin. I start this thesis by developing a system where the magnetic field forces electrons onto cyclotron orbits, thereby spatially confining them without relying on the system’s geometry. This leads to an increased resilience against imperfections. Imperfections in real systems are not the only technical problem. One example is the combination of superconductivity and a magnetic field. Superconductors expel weak magnetic fields, while strong magnetic fields induce vortices in the superconductor which have a nonsuperconducting region in their core. In order to avoid these problems I turn to a completely different regime and develop a system that does not require a magnetic field to create MBS. Instead the role of the magnetic field is taken by a combination of supercurrents and spin-orbit coupling in the semiconductor. Then I turn to the detection of MBS, which is always a competition between two goals: finding a unique signature only caused by MBS and relying on a simple setup. One well-established signature of MBS is single electron transport through a superconductor, because without MBS a superconductor only transports electrons in pairs. Of course single electron transport is not a unique signature at all: it happens in most conductors. That makes it hard to distinguish MBS from undesired side effects. I develop a setup that adds falsifiability to this signature. In addition to detecting single electron transport, my scheme allows to block it if it is due to MBS, therefore making MBS and a normal conduction channel distinguishable. In the last part of my thesis I do not propose a system, but instead I analyze and simulate an unexpected outcome of an experiment. My colleagues were studying a driven superconducting resonator coupled to a Josephson junction (two superconductors separated by a short barrier) with the goal of using it to detect MBS. Completely unexpectedly it turned out to be a high quality microwave laser. In a collaborative effort we explain this previously overlooked phenomenon.