Completed theses projects

Group of Fabian Hassler

Master’s theses

Grigorios Makris: Discretizing the sphere with an irregular lattice: Application to the transverse-field Ising model​

The thesis investigates the discretization of the sphere with irregular lattices, on which physical systems, and in particular the transverse-field Ising model, can be studied. More specifically, irregular lattices are necessary since, for arbitrary number of sites, the sphere cannot be covered with a regular grid. We solve the free particle problem on lattices that are easy to scale up and investigate if the expected SO(3) symmetry is recovered with increasing lattice size. The Fibonacci lattice is shown to be a good candidate for this. Having obtained the nearest neighbor hoppings for the free particle problem, we directly utilize them as interaction coefficients in the transverse-field Ising model placed on the Fibonacci lattice. With this choice of coefficients, the spin spectrum showcases better (approximate) degeneracies of the SO(3) compared to the case with constant coefficients. The search for the critical point of the system is halted due to pronounced finite-size effects. Thesis (PDF)

Steven Kim: Lindblad Equation, Symmetries, and Instabilities in Driven-Dissipative Systems

In this thesis, we study driven-dissipative systems that can be de- scribed by a Lindblad master equation. In particular, we analyze parametrically driven oscillators that exhibit an instability when the driving strength exceeds the damping. In this critical regime, both fluctuations and nonlinearities have to be included for an accurate description of the system. For the degenerate parametric oscilla- tor, it has been shown that the long-time dynamics can be described by a universal Liouvillian. This makes the efficient calculation of observables, such as the photon current, possible. To determine the long-time dynamics, we provide a method to solve the Lindblad equation, which makes the separation of timescales possible. This is used to derive the effective model for the non-degenerate parametric os- cillator, and we analyze the resulting statistics of radiation. Additionally, we discuss how symmetries of the Lindblad equation are described. Thesis (PDF)

Alexander Ziesen: Lattice Gauge Theoretic Analysis of Topological Ordering in the Majorana Toric Code

In this thesis, we consider a two dimensional square lattice of mesoscopic superconducting islands carrying four Majorana zero modes (MZM) each. We consider the islands to form an array of Josephson junctions, where the MZM allow for additional inter-island single electron transfer. This model is known to realize topological order in the form of Kitaev’s toric code. With a calculation of the effective Hamiltonian, we prove that the presence of charge-2e Josephson tunneling has a stabilizing effect on the toric code gap. The goal of this thesis is to investigate the signatures of topological ordering. To this end, we introduce a duality mapping of the MZM to spins yielding a lattice gauge theory and shift the focus from the matter to the gauge perspective. In this form, we can utilize the equal time formulation of the Fredenhagen-Marcu order parameter, a generalized Wilson loop, to detect the topological signatures and give an intuitive picture of (open) loop condensation to understand the topological phase
​transition. Thesis (PDF)

Lisa Arndt: Dual Shapiro steps of a phase-slip junction in the presence of a parasitic capacitance

This thesis investigates the emergence of dual Shapiro steps in the IV-curve of a single Josephson junction in the phase-slip regime. In particular, we analyze the detrimental effect of the parasitic capacitance between the biasing lines and how it can be remedied by an on-chip superinductance. We obtain an explicit analytical expression for the height of dual Shapiro steps as a function of the ratio of the parasitic capacitance to the superinductance. Using this result, we provide a quantitative estimate of the dual Shapiro step height. Our calculations reveal that even in the presence of a parasitic capacitance it should be possible to observe dual Shapiro steps with realistic experimental parameters. Thesis (PDF)

Florian Venn: Algebraic Methods for the 1D Schrödinger Equation

We discuss algebraic methods to obtain exact results about the eigenvalue spectrum of the one dimensional Schrödinger equation. We focus on degeneracies in the spectrum and eigenvalues that are exactly solvable. The central object in our discussion is supersymmetry. First, we give an introduction to the description of problems using superpotentials. We present insights that can be obtained from the supersymmetric structure, in particular the degeneracies between bosonic and fermionic states and that the groundstate admits an algebraic solution. We use these insights to study the double sine potential. By tuning the double sine potential away from the point where it can be described using supersymmetry, we find an extension of supersymmetry that is related to a class of problems for which multiple eigenstates admit algebraic solutions. Next, we prove, for a subclass of these problems, that a degenerate partner state exists for almost all eigenstates. Finally, we highlight that the class of periodic problems with more than one exactly solvable eigenvalue is not limited to the double sine potential. Thesis (PDF)

Elias Walter: Influence of chiral symmetry on electron scattering at disordered graphene boundaries

In this thesis, we study scattering processes at disordered graphene boundaries. In particular, we investigate how the chiral symmetry of quasiparticles in graphene influences diffusive scattering. We find that a boundary that breaks chiral symmetry behaves like a mirror, in the sense that in the long Fermi wavelength limit diffusive scattering is suppressed and incoming electrons are reflected specularly. However, if the disorder on average preserves chiral symmetry, diffusive scattering increases significantly, leading to a breakdown of the mirror-like behavior. Thesis (PDF)

Daniel Otten: Second-Order Coherence of Microwave Photons Emitted by a Quantum Point Contact

In this work we present a diagrammatic approach to calculate current cumulants for the electron transport through a quantum point contact. We provide compact expressions for cumulants up to and including the third order. Furthermore, fluctuations in the electronic current lead to emitted radiation in the microwave regime. In this context the current cumulants are linked to the photon counting statistics of the microwave field. For this setup, we calculate the Fano factor F and the second-order coherence function g(2)(τ). Thesis (PDF)

Dominique Dresen: Quantum Transport of Non-Interacting Electrons in 2D Systems of Arbitrary Geometries

The scattering formalism for describing the trans port properties of systems is discussed. We apply the formalism to GaAs and graphene with different geometries and types of disorder. For example, we discuss impedance matching of graphene to outside leads, Aharonov-Bohm effect in a ring geometry, and how strain-fluctuations in graphene manifest themselves in the transport properties. Thesis (PDF)

Sebastian Rubbert: Tuneable Long Range Interactions in an Array of coupled Cooper Pair Boxes

The Kitaev chain emulating the transverse Ising model can be implemented in an array of superconducting islands with semiconducting nanowires. In this thesis, we will show that adding additional capacitances to the system implements a long-range interaction between the Ising degrees of freedom. Thesis (PDF)

Bachelor’s theses