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 thesis we provide an elementary introduction in finite dimensional representation theory of the Lie groups SU(2) and SU(3) for undergraduate students in physics and mathematics. We will also give two application of representation theory of these two groups in physics: the spin and quark models. We begin with first discussing representation theory for finite groups to create intuition for representations. We will explain notions such as intertwining maps and complete reducibility and we will mention some application of representation theory of finite groups inquantum mechanics. Hereafter, we begin with representation theory for Lie groups and Lie algebras, especially the groups SO(3) and SU(2), as these groups will play an important role in the description of spin. One of the main results is that SU(2) is the universal cover of SO(3). Furthermore, we give a description of spin by means of representation theory of SO(3) and its Lie algebra so(3). We will show that half integer representations of the Lie algebra so(3) cannot be exponentiated to representations of the Lie group SO(3), but it can be exponentiated to its universal cover SU(2). Moreover, we study the irreducible representations of SO(3) inside the Hilbert space L2(R3). We will argue that one of the simplest quantum Hilbert spaces of a particle L2(R3), can be modified to the completion of the tensor product L2(R3) ⊗V, where, V is a finite dimensional Hilbert space that incorporates the internal degrees of freedom: spin. V carries an irreducible projective representation of SO(3). We will also discuss the addition of angular momentum of two particles in quantum mechanics. For this, we show how the tensor product of irreducible representations V and W of so(3) decomposes into SO(3) invariant subspaces of L2(R3). Hereafter, we will turn to representation theory of the Lie group SU(3) for setting up the mathematical framework for analysing the quark model. We will proof that there is a one-to-one correspondence between the irreducible representations of sl(3;C)and SU(3). We will also proof the theorem of the highest weight by which we can classify all the irreducible representations of SU(3) and sl(3;C) by their highest weight. We will also introduce the notion of the Weyl group and show that the Weyl group is a symmetry of weights of the finite dimensional representation of sl(3;C). Other properties of these representation, such as the dimension of the irreducible representations of sl(3;C) will be provided. Lastly, the quark model is discussed by means of representation theory of SU(3). We will show how this model can be used to classify two type of particles which also interact by means of the strong force: baryons and mesons. We show that we can classify the lightest mesons and baryons in so-called multiplets by the irreducible representations of SU(3). However, we will also introduce a modification of the strong force which further refines this model. A topic for further study would be how the symmetry group SU(3) describing Quantum Chromo Dynamics (QCD) can be used for the description of mesons and baryons.

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

Topological insulators and topological superconductors are novel states of matter. One of the most characteristic properties of topological insulators are the topologically protected edge states. While the bulk of the material stays insulating, the edge-state conductance is quantized and topologically protected from backscattering. In topological superconductors the edge states manifest themselves in the form of Majorana bound states: zero energy states inside the superconducting gap that are located at the end of a one-dimensional topological superconductor. Chapter 2 of this thesis contains a detailed review and a discussion of k.p-theory. The k.p-theory allows one to go beyond commonly used effective models and obtain much more detailed description of a semiconductor's band structure around its gap. Topological insulators are often semiconductor-based and topological superconductors can be realized in a hybrid structure that consists of a semiconductor and a conventional superconductor. Chapter 3 covers implementation details of the numerical methods used in this thesis. Quantum spin Hall effect is one example of a topological insulator. Band inversion in HgTe/CdTe or InAs/GaSb two-dimensional system leads to a topological phase that is characterized by topologically protected helical edge states which carry electric current with a quantized conductance. It was believed that in-plane magnetic field would break time reversal symmetry, suppress the conductance and open an energy gap in the edge-state dispersion. However, the experiment conducted by Du et al. reported robust helical edge transport in InAs/GaSb persisting up to a magnetic fields of 12 T. In Chapter 4 of this thesis we show that the burying of a Dirac point in the valence band, a feature of the system dispersion revealed only by the detailed k.p-simulation, explains this unexpected observation. Experimental group of L.P. Kouwenhoven investigated experimentally the details of spin-orbit interaction in InAs/GaSb system in both topological and trivial phases. In Chapter 5 we connect the results of this experiment with our band structure calculations: in the topological phase, a quenching of the spin-splitting is observed and attributed to a crossing of spin bands, whereas in the trivial regime, the Rashba coefficient changes linearly with electric field and the linear Dresselhaus coefficient is constant. In Chapter 6 we take a look into the spin texture of the inverted InAs/GaSb system close to the hybridization gap. Transport measurements conducted by the experimental group of C.M. Marcus in Copenhagen revealed a giant spin-orbit splitting inherent to this system. This leads to a unique situation in which the Fermi energy in InAs/GaSb crosses a single spin-resolved band, resulting in a full spin-orbit polarization. In the last chapter of this thesis we focus on semiconducting nanowires with induced superconductivity that are considered to be a promising platform for hosting Majorana bound states. In this theoretical research conducted together with physcists from ETH Zurich we show that the orbital contribution to the electron g-factor in higher subbands of small-effective-mass semiconducting nanowires can lead to the g-factors that are larger by an order of magnitude or more than a bulk value.@en

Topological band theory has contributed to some of the most astonishing developments in solid-state physics. The unique attributes that arise from topological effects are at the focus of modern experimental and theoretical research. Weyl point, a topological defect at the Fermi surface, enables topological transitions and transport phenomena. Its existence is considerably restricted in natural materials due to the tuning and dimension constraint. Recently, The Weyl points have been predicted to accommodate within superconducting nanostructures in the spectrum of Andreev bound states. Theoretically, one can easily manipulate the dimensionality and the tuning process through elementary approaches with specially designed structures. This opens up a new window for explorations in a higher dimension, high-order topological effects,Majorana states, and other complications even though it may be still experimentally challenging. One realization of such structures is the multi-terminal Josephson junction. The parameters are the superconducting phase differences of the terminals and theWeyl points reside at low energies within the superconducting gap. Chapter 2 of this thesis investigates the topological effect in the quantized transconductance of such a structure considering the presence of the continuous spectrum that is intrinsic to superconductors. This research is based on scattering formalism and relates the Landauer conductance to the continuous spectrumas a background field in the regular topological charge picture. Chapter 3 is based on a very generic superconducting nanostructure setup so long as it hosts Weyl points in it. The research proposes a unit that tunnel-couples such a setup with a quantum dot. The distinct feature of the spectrum, especially the distinction between its spin-singlet and spin-doublet due to spin-orbit coupling, leads to an exploration of the state manipulation. Eventually, through adiabatic and diabatic approaches, one can feasibly realize a full unitary transformation of the spectrum. Because of this, the unit could easily find its promising application in entangled qubits. Chapter 4 also relies on the generic low-energy Weyl point setup in the superconducting nanostructure, but instead, it is weakly tunnel-coupled to regularmetallic leads. We know that spintronics explores the intrinsic spin degree of freedom. It is usually realized on magnetic materials. In the setup of this research, the energy spectrum contains a natural spin-orbit that creates a minimalistic magnetic state in the vicinity of theWeyl point. The spin structure of the spectrum allows fine-controls over the spin and switch between magnetic/non-magnetic state. Hence this chapter’s research focuses on the possible spintronics features based on master equations. Chapter 5 furthers the research of chapter 4. It considers a universal energy scale sets up by the tunnel coupling strength. In the language of the Green’s function, this chapter studies the topological effect through the response function. This set up is a suitable example of low energy Weyl points situated in the presence of a low-energy continuous spectrum brought by electrons in the leads. We have seen in Chapter 1 how the continuous spectrum above the gap modifies the topology leading to a non-quantized contribution to the transconductance. The peculiarity of couplingWeyl points to a low energy continuous spectrum is that the dissipation gives rise to a redefinition of the Berry curvature, whichmanifests as a continuous density of topological charge instead of a pointlike one. This unusual characteristic can be captured by the tunnel current and thus can assist the detection ofWeyl points experimentally. @en

The microscopic theory of superconductivity proposed by John Bardeen, Leon Cooper, and John Robert Schrieffer has been a vital milestone of condensed matter physics and the basis of development of new quantum technologies. It explained the superconductivity as an emergent phenomenon arising from weak phonon-mediated attraction of individual electrons and giving raise to what we know as the superconducting condensate characterized by a complex order parameter that has a modulus and a phase. In many superconductors, the modulus defines a spectral gap. Because of the gap, the superconductor cannot host low-energy single-electron excitations. When an electron excitation comes from a normal metal to a superconductor, it is reflected as a hole, and an incoming hole is reflected as an electron. This process is known as Andreev reflection. In the last 60 years, the superconducting heterostructures have been extensively studied, Josephson junction being the most prominent example. The electrons and holes in such structures perform never-ending roundtrips between the super-conductor interfaces. This gives raise to Andreev bound state spectrum, that determines the supercurrent-phase relation of the nanostructure. There is a recent upheaval of interest in nanostructures with multiple superconducting terminals. One of the subjects of interest is Weyl points: for four or more terminals, the Andreev bands can cross zero energy at a point in the space of superconducting phases space. This is a direct analogy to band crossings in the Brillion zone of topological materials. Another subject is the peculiarities of the spectrum under conditions of the superconducting proximity effect, where the gapped-gapless transition in the space of superconducting phase has been observed. It has been long known that the superconducting phase is, in fact, a quantum variable that is canonically conjugated to charge. It is the basis of all applications for the development of superconducting quantum computing. The 2π periodicity of the superconducting phase also manifests itself in a non-trivial way. It enables events known as quantum phase slips, those promise metrological applications, for instance, a metrological standard of cur-rent. Thus it is of great interest to understand the quantum effects of the superconducting phase in novel and more complex setups. This is the main theme of this thesis.@en