In recent years, there has been significant research in fabricating semiconductor quantum dot-superconductor hybrid devices, which are promising candidates for creating Kitaev chains capable of hosting Majorana bound states (MBSs). These MBSs are expected to be able to store quantum information in a topologically protected manner scaling exponentially with the Kitaev chain length, making them robust against local perturbations and decoherence. However, the fabrication of such devices is complex and requires many design decisions, whilst the resulting devices require extensive characterisation to determine their operating parameters. Currently, there is a lack of theoretical tools to guide the design and characterisation of these devices, and although some systemic protocols and tools have been developed, much of the design process is still based on heuristics and trial-and-error. This thesis aims to address this gap by developing a theoretical framework and computational tools to model and simulate quantum dot Kitaev chain devices at the microscopic level. A pipeline is proposed and demonstrated to model such devices, starting from solving the Schrödinger-Poisson electrostatics problem to determine the potential landscape, then using this potential to solve the quantum transport problem and extract the relevant parameters from a generalised Kitaev chain model. This tool could be used to guide the design of future devices and to characterise existing devices, providing insights into their behaviour and performance. ... Read more

This thesis explores the quantum Zeno effect, a phenomenon in which frequent measurements can inhibit the evolution of a quantum system. The aim of the thesis is to make this statement mathematically precise. To this end, the first chapter introduces the mathematical language needed throughout the thesis. The following chapter then develops the basic formalism of quantum theory based on [2], explaining how states describe the possible condition of a system and how observables describe the possible outcomes of measurements. This leads to chapter 4, where quantum measurements are discussed as mathematical procedures rather than merely informal acts of observation. In particular, we consider idealized properties that a measurement may satisfy, such as repeatability, and focus on the sharp measurements relevant for the Zeno effect. With these ingredients in place, the thesis culminates in a rigorous formulation of the idea expressed in the opening sentence. The main result concerns the limiting behavior of a quantum system subjected to increasingly frequent measurements of a particularly ideal type, namely L¨uders measurements. Between successive measurements, the system is allowed to evolve according to its usual time evolution. Under suitable assumptions, we prove that in the limit of infinitely frequent measurements the time evolution of the system can freeze.
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The recent quest for large resonator photon numbers in circuit quantum electrodynamics (cQED) has led to the discovery of ionization in Transmon-resonator systems. Ionization compromises the quantum non-demolition nature of Transmon readout. Furthermore, it can lead to dephasing in elements coupled to the Transmon, which is detrimental to optomechanical schemes using an auxiliary qubit to create quantum states. Since the transverse nature of the usual dipolar capacitive coupling lies at the origin of ionization, our team engineered a new parametric coupling scheme that can potentially suppress ionization by being inherently more longitudinal.

Using a large detuning in combination with parameters chosen based on branch analysis, ionization in the system was largely suppressed. This suppression enabled the observation of the collapse and revival of the Transmon potential under parametric coupling at high resonator photon numbers. The measured Transmon Stark shift indicates a collapse of the potential at 12,300 photons and a subsequent revival, in agreement with the derived theoretical model. Resonator phase-space measurements further confirmed the collapse and revival. The newly identified revived regime potentially enables coherent Transmon operation at high photon numbers. ... Read more

This thesis characterises the behaviour of the nonequilibrium heat capacity C_N(β) on random tournament graphs T_N without an energy landscape for large N. In this context, the heat capacity C_N(β) is completely determined by the geometry of the underlying network and is derived from the quasipotential. The quasipotential is obtained by solving a discrete Poisson equation involving the random generator L_N of a non-reversible Markov chain over the tournament.

Numerical results suggest a self-averaging property, leading to our main conjecture:

N · C_N(β) = 4C₂(β) (1 + N⁻¹/² · ξ_C)

where C₂(β) is the heat capacity of a fundamental two-level system and the random variable ξ_C → N(0, τ_C) in distribution.

To prove the conjecture, we provide three main contributions:

1. We derive a coarse-grained generator Ḡ_N in score-space that represents the system in an idealised situation where the score bins B_s (which count the number of vertices with a certain score) attain their expected value. We solve its corresponding Poisson problem and prove that it captures the ensemble-averaged 4C₂(β) limit.
2. We shift focus to the statistics of the score bins, which are globally dependent random variables. Using a change of measure, we establish a concentration result and a local Central Limit Theorem (CLT) for the sizes of the score bins N_s.
3. We introduce an auxiliary generator G_N that bridges the vertex and score spaces, linking Ḡ_N and L_N. We demonstrate that G_N satisfies the conjectured Gaussian scaling by deriving the pseudoinverse Ḡ_N† and solving the Poisson problem for G_N perturbatively.

We conclude by providing a viable route to proving that C_N(β) is close to the heat capacity of G_N. ... Read more

his thesis investigates the conserved Runge–Lenz vector in systems governed by inverse-square central forces. By analyzing its associated symmetries through the framework of Lie groups and Lie algebras, we explore its role in both classical and quantum mechanical settings. In each case a hidden so(4) symmetry is revealed. In the classical regime, this is mapped to a SO(4) group action. Whilst in the quantum regime, this symmetry is used to calculate the energy levels of the hydrogen atom. This thesis was written as part of the Bachelor’s programs in Applied Physics and Applied Mathematics at Delft University of Technology ... Read more

Understanding the ground-state properties of many-body systems is a computational challenge in condensed-matter physics. MeanFi is a Python package that performs self-consistent Hartree-Fock calculations on non-superconducting tight-binding models and aims to find the ground state solution of a Hamiltonian with density-density interactions. This thesis presents how this package is generalized to also perform these calculations for superconducting tight-binding models. First, a complete derivation of the mean-field expansion is given by applying Wick’s contractions and the mean-field approximation. This expansion is then transformed into the Bogoliubov-de Gennes basis to explicitly include superconducting terms in the Hamiltonian. Second, the self-consistency criterion is adapted by constraining the solution space by enforcing symmetries on the solution by using Qsymm. Third, finite-temperature calculations are added to the algorithm and the total charge of the system replaces the electron filling-factor that was used in MeanFi, introducing a minimization problem to the algorithm. Last, the updated algorithm is applied to a 1D-Hubbard model with attractive interactions and the resulting superconducting gap as a function of temperature matches theoretical predictions from BCS-theory. ... Read more

The rotational velocities of stars in galaxies indicate that either Newtonian gravity breaks down on this scale, or that there is extra mass in the universe that has not been observed. The former is known as Modified Newtonian Dynamics (MOND), whereas the latter option is known as the dark matter paradigm. In this thesis a version of MOND is considered where the field equation for gravity is modified and becomes nonlinear. To simulate MOND, new N-body codes are needed due to the field equation not being linear anymore and in this thesis a new particle mesh method is developed. This new particle mesh method follows the same steps as the normal particle mesh method, except at the step where the acceleration field on the grid is calculated. This step is replaced by an iterative method to find the MONDian acceleration field on the grid. The code was then tested in the deep MOND regime using four cases of which analytical solutions of the MOND Poisson equation are known. These cases are: linear motion, the two-body problem, a ring system and an isothermal sphere. Using these test cases, numerous conclusions were drawn. From the linear motion test it was found that in the code bodies can interact with themselves, resulting in non-physical accelerations. The two-body test showed that the magnitude of the relative error in the force as calculated by the particle mesh code is roughly on the same order as the same error when calculating the Newtonian force using a particle mesh code. Furthermore, all tests showed that energy is conserved quite well, even though momentum need not be conserved due to the self interactions, and that generally the conservation of angular momentum is violated by the code. The tests also showed that the analytical formulae that were derived for these systems gave largely the same predictions as the algorithm, hence verifying both the formulae as the algorithm. The code could be sped up by writing it in C, the accuracy can be improved by reducing self-interactions and it can be extended by adding a method to handle close encounters. Some applications of the code are given, including: simulating the solar system, wide binary stars, tidal dwarf galaxies and galaxy clusters.
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Entanglement is an essential resource for a variety of applications, such as distributed quantum computing and quantum cryptography. However, long-distance entanglement generation is challenging because of two reasons: photon loss occurs as an exponential function of distance through an optical fiber and the no-cloning theorem prevent us from directly amplifying the photons. Therefore, quantum repeaters that enable long-distance communication to realize quantum information-based protocols are desired. One way to achieve a higher entanglement generation rate is to make many attempts of generating entangled states in parallel, a process known as time-multiplexing. There has been previous work investigating the performance of time-multiplexed entanglement generation using processing nodes, but only at the elementary link level. Furthermore, this analysis was restricted to the rate of entanglement generation, with no concern for the fidelity. In this work, we go further by investigating both the fidelity and the rate of entanglement generation of time-multiplexed protocols by analyzing the secret key rate (SKR) of quantum key distribution. Moreover, we also study setups with one and two repeaters. Specifically, we investigate the impact of different hardware parameters on the SKR. Among other results, we conclude that swap gate time is a key factor for achieving higher SKR. We also examine what effect different repeater-chain protocols have on the performance of repeater chains with limited hardware resources. We find that having the repeater send photons in alternating fashion towards both end nodes results in a higher SKR than generating entanglement sequentially. Besides, we investigate what is the most efficient distribution of communication qubit (CQ) in a protocol with multiple repeaters. We ascertain that the repeater chain setup in which the number of CQs in a node is equal to that node's number of neighbors makes best use of its resources. ... Read more

Spin qubit in semiconductor quantum dot arrays offers a promising platform for future scalable quantum computing with its small size and compatibility with modern semiconductor industry. To scale up the quantum dot arrays, one of the major challenges is the wiring bottleneck, as a high density of control lines might need to be integrated into a small chip. A proposal to solve this problem is using the shared control protocol, in which multiple qubits could be controlled by a shared line. For this, the most critical requirement is to realize uniformity across the quantum dot array, such that a single control signal could lead to an identical response in all dots involved. However, such uniformity is hard to achieve due to the variation of the device fabrication, and tackling this problem via materials and fabrication optimization only appears to be a daunting challenge.

In this thesis, we propose a potential solution to achieve uniformity of threshold voltage in such share-controlled systems. This solution is based on the hysteresis behavior of turn-on voltage in the heterostructure field-effect transistor (HFET) devices hosting the quantum dot array. In the Ge/SiGe HFET devices with hole as carrier, we found that the drift of turn-on voltage can be caused by population of 2DHG under negative gate voltage and reversed by applying positive gate voltage. We attribute this effect to trapping and detrapping processes on the dielectric surface of the device. Following this discovery, an automatic feedback control program was designed, in which gate voltage pulses are applied to control the trap filling level such that potential landscape in the device corresponds to the desired turn-on voltage. Using this program, we performed deeper investigations of the turn-on voltage shift including its relaxation and history-dependent stability. A hypothetical physical model for observations in these experiments is followed. For practical application of this effect, the feasibility to locally define and control the turn-on voltage is also demonstrated. Based on these results, we present a proposal for addressable manipulation of potential landscape in share-controlled quantum dot array, which might potentially realize the threshold voltage uniformity for scalable quantum dot array in the future. ... Read more

Level-set percolation on the Discrete Gaussian Free Field (DGFF) turned out to be a hot topic within mathematical physics over the last couple of years. In particular, the DGFF on Z^d , with homogeneously weighted nearest-neighbour interactions, i.e. all conductances equal to 1, has been studied in detail. These models can be simulated with great efficiency. In this research, we abandon the homogeneity requirement and look at three-dimensional DGFFs with arbitrary conductances. Our goal is to find a quick and reliable method to simulate such DGFFs on a finite lattice. Since this is, in essence, a high-dimensional Gaussian sampling problem, we investigated this problem using the Conjugate Gradients (CG) linear solver as a Gaussian sampler. To see how it performed, we compared our implementation of the CG sampler with known methods for DGFFs in the unit conductance case. Finally, as a showcase of our implementation, we studied level-set percolation on a DGFF with a simple checkerboard conductance pattern. Our main conclusion is that the CG algorithm is very suitable for simulating Discrete Gaussian Free Fields. Since it does not make any assumptions on the conductances, it can be used to generate DGFFs with arbitrary conductances. However, there are still a number of issues with our implementation. The biggest one is concerning the stopping tolerance of the CG sampler. Once the tolerance is set smaller than some lattice size-dependent threshold, the percolative behaviour of the resulting sample changes drastically. We have not been able to explain this. Moreover, we would recommend making the implementation usable for parallel computing. We have been limited to relatively small lattice sizes during this project. Consequently, the use of certain finite-size scaling arguments when analysing level-set percolation might not always have been as justified. Finally, based on our study of the DGFF on a lattice with checkerboard conductances a and b (a < b), we conjectured that, in its percolative behaviour, this DGFF resembles a DGFF defined on a lattice with constant conductance c, where c is a weighted average of a and b. The weight of a is expected to be larger than the weight of b. ... Read more

context: The long-term evolution of a self-gravitating astrophysical disk can be modeled using secular perturbation theory. Recently, Batygin published a paper where he claims that such a disk with a special density can be described by a Schrödinger equation by using this method. aims: In this thesis, we will study the secular perturbation theory applied to an astrophysical disk with the same density as Batygin, using the Laplace-Lagrange equations. We will take the continuum limit of those equations, and try to find a wave equation like Batygin. methods: We first apply the Laplace-Lagrange equations to a disk with a large number of planets. Then we take the continuum limit of an infinite number of planets. We then compare the numeric results of the discrete disk to the analytic results of the continuum limit. results: The eigenmodes of the system are well approximated by damped sinusoids. Mode number n changes sign n times. The eigenvalues are linear in the mode number. conclusions: The eigenmodes do not satisfy a wave equation. ... Read more

In this thesis, networks of coupled quantum harmonic oscillators are studied. The dynamics of these networks are determined by single-frequency vibrations of the entire network called normal modes. We study the behavior of the nor- mal modes when the network is coupled to a thermodynamical heat bath by looking at the Lindblad Master Equation of the system. From this equation, we determine the rate at which the normal modes decay. Certain normal modes decay very slowly, and some do not decay at all. These normal modes are called quasi-noiseless and noiseless clusters respectively. We determine what happens to the noiseless clusters when the network pa- rameters are very slightly perturbed. We have found that two distinct types of noiseless clusters can be identified. The first type disappears with even the slightest perturbation, making it useless in practice. The second type instead be- comes quasi-noiseless, making it a viable candidate for applications. We show how to determine the degree to which these noiseless clusters become quasi- noiseless by looking at the other normal modes of the network. We also explain how a network of oscillators, including an optional heat bath, can be simulated with an optical setup as described in [3]. We suggest this setup can be used to verify our findings. ... Read more

Superconducting-normalconducting-superconducting (SNS) transmons with 2-facet Al-shell nanowires are qubits compatible with magnetic fields above 10 mT. There are important correlations of the room temperature nanowire resistance with the chance of the qubit being measurable: at a resistance of 2−3 kΩ, the qubit is almost guaranteed to work. The chance of success halves every 2−3 kΩ increase. This information can be used to increase the yield. The flux noise power spectral density (PSD) of a model spin-1/2 fluctuator has been investigated as a function of the magnetic field using the Zeeman interaction. Not only the fluctuations parallel to the magnetic field contribute, but also the fluctuations perpendicular to the magnetic field. Cross-terms cancel out. The flux noise PSD of the SQUID is a linear combination of these spin PSDs when the spins are spatially uncorrelated. The magnetic field suppresses the parallel spin-axis noise PSD contribution as cosh^(-2)(μB/kBT). The magnetic field changes the perpendicular spin-axis PSD contribution due to the Larmor precession frequency peak 2fZ~μB, but does not influence the PSD contribution at frequencies higher that the Larmor precession frequency. When rotational asymmetry in the SQUID geometry is present, the PSD contributions of the perpendicular and parallel components can be separated. In our setup, a perpendicular coil is used to align the magnetic field with the transmon plane. The alignment procedure of maximizing the resonator frequency vs. the perpendicular coil field has been verified. To measure the flux noise, the perpendicular coil is first used to change the flux bias by large amounts. Then a dedicated flux bias is used to make a fine-grained sweep over the flux without flux-jumps, to calibrate the magnetic field at the SQUID. We have found a signal of the flux noise at zero field and at field. A flux noise amplitude of A~1000 μΦ_0 has been found at zero magnetic field. ... Read more

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 ... Read more

In this study, ultrathin films of the itinerant 4d ferromagnet SrRuO3 were epitaxially deposited on SrTiO3 and capped with a thin LaAlO3 layer. Top gates and a dielectric layer were patterned onto contacted films and magnetotransport properties were characterized at low temperatures as a function of top gate voltage. A particular focus was placed on the anomalous Hall resistivity. The magnitude of the Hall signal and the sheet resistance were shown to vary with top gate voltage. In particular, the anomalous Hall loops were compared to numerical tight-binding models. The model is proposed as an alternate explanation to the skyrmion picture and as a complement to the two-channel phenomenological model put forth to explain the unusual low-temperature anomalous signal of ultrathin SrRuO3 . Model predictions were found to be valid at
low temperatures in semiconducting Ru-deficient SrRuO3 films. ... Read more

The study of the dynamics of large, complex networks is generally very hard. Analytical solutions are rarely available and numerical solutions require immense computation times. Recently, Gao et al. [1]have proposed a new theoretical approach to analyse the average behaviour of complex networks. In this research we discuss the derivation of the mean field approximation and the resulting one dimensional, so called, ”effective” equation. We offer an alternative derivation. We also give an expression for the error in the form of a differential equation. Applying noise to a model of plants and pollinators uncovers a weak point in the given formalism, the effective equation does not correctly predict the average behaviour of the network. For lower dimensional systems, projecting the synchronization manifold on a phase plot reveals why the mean field approximation works well in the specific case. Finally, the theory is applied to some existing models. First, the Generalized Lotka-Volterra model, where it struggles in specific cases where the network has no stable fixed points but the theory does predict one. Second, the Kuramoto model, where a synchronization state of the oscillators is correctly predicted by the theory, and third, a one dimensional array of Josephson junctions where synchronization is also correctly predicted along with the correct stability criteria. ... Read more

We have explored the concept of mobile computational fluid dynamics (CFD) solving. Using the computational
power of a smartphone we tried to generate real time simulations of relatively simple transient natural
convection problems in the laminar regime. The focus lies on finding a compromise between accuracy, speed
and stability: we want to make a real timemobile CFD solver that is as accurate as possible.
A JAVA application has been created that runs on Android. The SIMPLE algorithm has been implemented
in order to solve for the flow and heat transfer. The SIMPLE algorithm on the application was tested for
runtime performance and accuracy.
For a test problem, a real time simulation on a Nexus 5X has been realized with an accuracy of 5.4% on a
21x21 grid. Running the algorithm on a desktop gave a simulation 10 times faster than real time. Comparing
this algorithm to a version converted to MATLAB, gave similar solving speed. We concluded that the bottleneck
in the algorithm, solving matrix equations, could not be easily improved, because MATLAB solvers
perform quite optimal: a significantly faster CFD solver implementing the SIMPLE algorithm probably does
not exist.
Nevertheless, when trying to use the obtained results on water and air, we could not obtain any real time
solutions. It is up to further research to define restictions that can guarantee real time solutions for fluids. ... Read more