In this thesis the stored energy and its fluctuations of a central spin battery with nearest-
neighbour interactions between the battery spins are investigated. Using analytical ex-
pressions, it is shown that for 2 battery spins and equal strength in the flip-flop interaction
g and nearest-neighbour interaction J, the fluctuations are minimal whenever the bat-
tery is maximally charged when taking at least four charge spins. Similarly, whenever
the formed envelopes of the energy have a zero, the fluctuations have a global maximum.
In the same limit, it could also be seen that an increase of the charge spins Nc and
spin-ups m, resulted in a higher global maximum of the stored energy. Furthermore for
2 battery spins, taking the limit J ≫ g results in a situation where the battery cannot
be charged at all, whereas taking the limit g ≫ J results in a central spin battery where
no nearest-neighbour interactions are present; its stored energy as a function of time is
a single cosine function, that is always able to reach its theoretical maximum.
Similar results were found for systems with more than 2 battery spins. Increasing J with
constant g resulted in a decrease of the global maximum of the energy, dropping from its
theoretical maximum to its minimum. Opposite behaviour could be seen when increasing
g with constant J. Whenever the global maximum of the energy crossed the line E = 0,
the fluctuations at the same moment in time formed a peak.

Random walks have been used in a number of different fields for a long time. With the rise of of quantum random walks a lot of these applications have been made more efficient. Recently there have been a lot of improvements in the realization of these quantum random walks in real world systems.
In this research, the effect of uncertainty in the measurement time and measurement frequency are studied on three problems that make use of a quantum random walk. These problems are the network centrality problem, the graph isomorphism problem and the spatial search problem.
These effects are studied by first looking at the theory behind these problems after which the theoretical results of these three problems are calculated. These theoretical results will then be compared to simulated results that have a certain level of uncertainty in the measurement time or frequency.
For both the network centrality problem and the spatial search problem we concluded that only a small error is made when uncertainties are introduced in the system. This implies that these problems are solvable in realizations of the quantum random walk in real world systems. For the Graph isomorphism problem we saw that the error that is made when uncertainties are introduced was large which implies that this problem would only be solvable for small uncertainties.

This work is a bachelor thesis featuring a collection of explorations of the Kuramoto model in the presence of noise in one and two dimensions. The model was explored analytically and numerically. It contains a study of literature and new analytical derivations. An attempt to link the fraction of oscillators to the order parameter resulted in experience through the process of deriving the result, but a plot of the first few terms of this expansion showed it not being close. The assumption on which the expansion was based however was shown to be valid in simulations. More successfully an upper bound on the probability of synchronization was found. This probability was found on a ring consisting of N oscillators, and verified of a ring of 3 oscillators. The results was a bound better than one that was already calculated, though the method of verification used was not ideal. An analysis of the stability of solutions on this 3-ring resulted a visual representation of entrainment and an amount of stable solutions possible, which was 1. This result was in accordance with the more general result following from the investigation of a system of coupled rings with arbitrary sizes of either ring. We then expanded this system to conclude that the stability of the entire system could be split into the stability of either ring and the possibility of having a solution for the linking phase difference. We elaborated on general practices when numerically analyzing SDEs and integrations methods. The Kuramoto model proved to be kind in the sense that the Euler-forward and Milstein scheme coincided. In a more physics-centered approach we studied the Ising and XY model. This prompted literature study on the Kosterlitz Thouless transition, through analysing the spin-wave Hamiltonian and the energy of free vortex of a pair of vortices. We also then explored conservation of the vortex-pair behaviour in a noise including Kuramoto model, which verified a duality in partition functions as described in ref. [1].

In this project, we have done a calibrated measurement on a previously designed and fabricated current pumped Josephson Parametric amplifier. We have installed a microwave switch into our crygenic fridge tobe able to get a calibrated response measurement using Short-Open-Load calibration, which we used withmeasurements of the gain and flux tunability of the device. This showed that our previous measurements onthe gain deviated significantly, with about 10dB. We also did a thermal calibration on the amplifier by heatingthe plate it was attached to, and found that the input noise approximates the limit of a half quantum of noise.

In 2019 werd er voor het eerst een foto van een zwart gat gepubliceerd. Het betreffende super zware zwarte gat ligt in het midden van het sterrenstelsel Messier 87. In dit onderzoek willen we simulaties maken van hoe een foto van een zwart gat eruit ziet. Zowel voor een achtergrond (zoals de sterrenhemel) als de accretieschijf die eromheen zit. De vraag is vervolgens welke informatie uit de foto gehaald kan worden. Om de foto te maken, werd een ray-tracer gemaakt. Hierbij kijk je waar een lichtstraal, die op het scherm komt (een oog of camera), vandaan komt. De bewegingsvergelijkingen zijn vereenvoudigd door constantes van de beweging te zoeken om de geodeetvergelijkingen te versimpelen. Vervolgens zijn de bewegingsvergelijkingen numeriek opgelost. Er is gevonden dat lichtstralen van veraf niet in een zwart gat vallen bij $\frac{\sqrt{27}}{2}R_s$. $R_s$ is de Schwarzschild straal. Dit is de straal waarvandaan licht niet meer kan ontsnappen van een zwart gat. De gesimuleerde plaatjes zijn 128x128 pixels. In de gesimuleerde plaatjes had het zwart gat ongeveer $2.5R_s$. Dit komt nauw overeen maar er wordt een aanzienlijke fout gemaakt door de resolutie van de plaatjes. Verder is kwalitatief de oriëntatie en rotatieparameter $a$ goed uit de plaatjes te halen.

The defining challenge of connecting NISQ quantum computers over large distances is efficient microwave to optical transduction. In this work, we demonstrate a new platform for piezo-optomechanical conversion by combining a crystalline silicon nanobeam photonic crystal cavity and a suspended thin-film lithium niobate acoustic resonator. The goal is to combine the excellent optomechanical properties of the silicon nanobeam cavity with coherent excitations in a piezoelectric resonator in order to achieve large effective coupling between the microwave excitation and the optical cavity. In order to fabricate this hybrid silicon-on-lithium niobate device, we explore a technique termed ‘slapping’ where a loosely connected suspended nanostructure is patterned such that a tapered optical fiber can be used to rip it away and place it in an arbitrary location on another chip. We report a single photon optomechanical coupling rate g_0 = 5.1 kHz and a single photon microwave to optical efficiency of η_{μ→o} = 6.1 × 10−6 at 25μW of input optical power in a device limited by a suboptimal optical interface.

In this report the goals is to theoretically construct a shift resistant encoding in a rotor system. The report starts with an overview of the rotor space. In the context of which the difference between PVM and POVM is highlighted. After this overview, the shift resistant encodings are discussed. These two concepts are then combined and a shift resistant encoding in the rotor space is presented. Lastly a system which hints at an implementation is presented.. Of this system the Hamiltonian is discussed.

This thesis will focus on verifying the area-law for the entanglement entropy SN for spin-1 2 lattice systems in 2 dimensions, with ferromagnetic interactions defined by the nearest neighbour XX-model. The area-law implies that the SN of a subsystem is expected to scale proportionally to the size of the boundary through which that subsystem interacts with the rest of the system. By writing a numerical model in MATLAB, which uses computational devices to circumvent the huge memory requirement that quantum simulations usually demand, it is possible to analyse 2 dimensional systems up to a lattice size of 5 × 5 for the first time. These finite systems have been assigned periodic boundary conditions, as if it were a torus. It doesn’t seem possible to evaluate such systems analytically, unlike 1 dimensional systems, as we will show. The area-law has been shown in a direct way by looking at entanglement entropy, as well as the indirect way by analysing the connected spin correlation.

In this bachelor thesis, the paper ”Classical synchronization indicates persistent entanglement in isolated quantum systems” by Dirk Witthaut et al. is looked into and discussed. First, synchronization in a classic sense is explained. The Kuramoto model is introduced, and a few properties of this model are defined. In the next part of the theoretical background, the creation and annihilation operators are defined. An example is given on how to derive these operators, and how to write the Hamiltonian in the form of creation and annihilation operators. It becomes clear that the Hamiltonian of a vector potential in vacuum does not have any coupling terms. This is why no synchronization will occur here. Assuming the Hamiltonian has a different form with two-body interactions and a coupling factor, then it will have some sort of interaction between modes. Dirk Witthaut published in his paper a way to derive the Kuramoto equation from this Hamiltonian. First, the time derivative of the expecta- tion value of the â n operator is evaluated by using the Ehrenfest theorem. This returns multiple three point functions. These can also be evaluated by the Ehrenfest theorem, but that will only result in more coupled equations and five point functions. To solve it, a first order mean field approximation is used. According to Witthaut, this results in a series of coupled complex differential equations which can be rewritten into the Kuramoto equations. Witthaut then further elaborates this result in the remainder of his paper. My calculations point towards a different conclusion. Witthaut made a mistake when calculating the commutation relations that were used in the Ehrenfest theorem. This resulted in a different system of coupled equations, which I couldn’t elaborate into the Kuramoto model. This is the conclusion of this bachelor thesis.

This thesis analyses a lumped element circuit proposed for an analogue quantum simulation of opto-mechanics. The circuit consists of two resonators, a simple LC-resonator, and a similar resonator in which the inductor is replaced by a SQUID as a flux tuneable inductance. The interaction between the two resonators is established by mutual inductance between the inductor in the LC-resonator and the SQUID loop part of the other resonator. This makes the resonance frequency of the SQUID-resonator a function of the flux in the LC-resonator. It is shown that mutual and self inductances in the SQUID loop give rise to two transcendental constraints, for the magnetic flux in the SQUID loop, and the generalised flux across the SQUID. By an approximate solution for the constraints and under the assumption that the loop inductance is small compared to the SQUID inductance, we derive an approximate description for the circuit dynamics. We demonstrate that the circuit Hamiltonian contains the asymmetric opto-mechanical interaction, but in addition also a self-Kerr non-linearity in the analogue optical cavity, as well as a weak cross-Kerr interaction between the two resonators.

The NV-center in diamond is a promising system for quantum information processing. The typical qubits are the electron spin in the NV-center and the surrounding 13C nuclear spins. Fast and precise control of many 13C spins is desirable but challenging due to crosstalk. Crosstalk is the unwanted effect of a control pulse on the other qubits. The coupled NV-13C system is explored, and numerical techniques to optimize the control of multiple and single qubits are used. In this report the effect of the Bloch-Siegert shift in the weak driving regime is investigated first. This is done by studying the effect of a π-pulse when two carbon qubits are driven at the same time on resonance. It turns out that in this weak driving regime the effect of the off resonance is very small, and changes the fidelity not more than 0.3%. Furthermore the effect of a square πpulse on 4 nearby qubits is studied. If the pulse is not optimized (thus square), the pulse cannot be done in less than 100µs while retaining at least 99% fidelity. By optimizing the envelope of the driving field using constrained optimization the pulse can be done in 22µs while retaining at least 99% fidelity.