Quantum technology is evolving faster than ever. Currently, all eyes are on the quantum computer, the promising computer that can solve problems which are unsolvable for regular computers. In order to understand this new technology, it is necessary to understand the qubit, the basic unit of quantum information. This can be done by means of the density matrix: a mathematical representation of the state of a system. The aim of this thesis is to find the density matrix of a single qubit in closed (isolated) and open quantum systems. In the case of a closed system, an alternating sequence of two processes with different Hamiltonians is considered, which both last a fixed amount of time after one another. These systems have been solved using a direct formula for a 2 × 2 matrix with distinct eigenvalues raised to the power n for any n ∈ N. In the case of an open system, dissipation is taken into account compared to a single process in a closed system. These open systems have been solved numerically using the Lindblad master equation and the spin-boson model to model the environment as a bath of bosons. However, the use of multiple approximations and assumptions questions the validity of the results for ’strong’ interactions. Suggestions for further research include investigating quantitatively when the Lindblad equation is valid to use and solve open quantum systems using different models for the environment.

Quantum Markov Semigroups (QMS) describe the evolution of a quantum system by evolving a projection or density operator in time. QMS are generated by a generator obeying the well-known Lindblad equation. However, this is a difficult equation. Therefore, the result that the Lindblad form greatly simplifies in the case of the generator commuting with the modular automorphisms group, is useful. Unfortunately, the proof only works for finite dimensional Hilbert spaces, which is why the aim of this thesis is to generalise this result to countably infinite dimensional Hilbert spaces. To this end, the Lindblad equation is derived from both a mathematical and physical perspective. Where the former relies on rigorous proof and the latter relies on approximations.   In the rigorous case the theory of unital completely positive maps is used. Furthermore, multiple topologies are considered which put less stringent conditions on the operators of interest than the norm topology. Additionally, the Haar measure is used on the unitaries of the bounded linear operators to construct the explicit Lindblad form. To derive the result by employing physical assumptions the interaction picture is used. The physical derivation starts from the Von Neumann equation and uses multiple assumptions to obtain the final Lindblad form. The most important physical assumptions are: the Born approximation, the Markov approximation and the rotating wave approximation.   Furthermore, the main result is the generalisation of the simplified Lindblad form. This simplified form holds for generators commuting with the modular automorphisms group in case the Hilbert spaces are countably infinite dimensional. However, this requires the domain of the generator to be restricted to trace class operators with the identity operator artificially added. Additionally, the generator needs to map strongly convergent sequences to weakly convergent sequences. It also needs to be self-adjoint with respect to the Hilbert-Schmidt inner product. Lastly, the generator is assumed to be self-adjoint with respect to the Gelfand-Naimark-Segal (GNS) inner product <X, Y>=Tr(σ X*Y) for σ a density operator. This last assumption implies that the generator commutes with the modular automorphisms group, which is the symmetry we are considering. Hence, the two previous assumptions are the additional requirements needed to generalise the result, besides the restriction of the domain. Therefore, it is recommended for further research to generalise the result for the domain extended to the bounded operators B(H). It should be noted that the proof heavily relies on the Hilbert space structure induced by the Hilbert-Schmidt inner product. Consequently, the generalisation for the bounded operators would probably require a different approach. Another recommendation is to try and lift the sequence and self-adjoint requirements on the generator. In addition, it is interesting to investigate which physical systems actually have the symmetry of generators commuting with the modular automorphisms group.

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].

Chiral induced spin selectivity (CISS) is an unexplained phenomenon in physics, in which the interaction within chiral molecules brings about a selectivity in the spin of the electrons. The Breit interaction has been suggested as a possible explanation, but has not yet been researched in the context of CISS. In this thesis, first a derivation of the Breit equation is given, which in the second part is being used to construct a simulation of electron transport through a simple chiral molecule. The spin-polarization for the transmission including the Breit interaction was compared to the case excluding the Breit interaction, and the two were found to be substantially different. From that it followed that the influence of the Breit interaction on the transport of an electron through a chiral molecule is significant and it should therefore not be disregarded in calculations on CISS.

While life is present everywhere on earth, each individual species can only grow and proliferate in a specific set of conditions. Moreover due to inherent fluctuation of the environment, individual organisms are often periodically confronted to stressful conditions that prohibit growth, reproduction and increase mortality. One of the most common strategy to cope with these harmful conditions is to shield and quietly wait for the storm to finish. Indeed upon change of the environment, many organisms enter "dormancy", whereby they differentiate into a distinct resting form with additional protection, storage and with greatly reduced internal activity (dormant state). Although relying on widely different molecular basis, specialized dormant stage are found in virtually every group of living organisms, including endospores of bacteria, spores of fungi, seeds of plants, cysts of protists and diapaused eggs of animals. Even after sometimes years of apparent inactivity, dormant organisms can resume growth and reproduction upon sudden improvement of environmental conditions. How such organisms manage to keep the potential to resume growth while having a nearly ceased activity? Over the last 20 years, scientists have uncovered various physiological and molecular mechanisms that control entry and exit of dormancy. However a basic understanding of what exactly happens during dormancy, i.e how to survive while stopping nearly all internal activity, is still missing. Two reasons are likely responsible for that knowledge gap. A first conceptual obstacle is a general view that since by definition nothingmuch happens during dormancy, nothing important happens. A second technical obstacle is the lack of sensitive enough instruments to quantify the "nearly ceased" activity during dormancy. As a result, very few studies have established the concrete link between a vanishing internal activity and the ability to survive during dormancy in a single experimental system. In this thesiswe propose to focus on dormant Saccharomyces cerevisiae yeast spores to specifically investigate the links between gene expression and the ability to survive during week-long dormancy.@en

The reflections composing this thesis examine the usage and necessity of quantum theory, with an emphasis on systems featuring mechanical resonators. The first chapter introduces the quantum formalism, reviews the historical motivation for the quantization of harmonic oscillators, and presents a derivation of the interaction between the electromagnetic field and mechanical motion in several distinct systems. The second chapter examines the nature of physical effects such as state transfer, squeezing, entanglement, and sideband asymmetry, and how they naturally emerge in non-quantum contexts. A dynamical statistical theory is introduced to aid the quantum/classical comparison, and standard measurement models are reviewed due to their strict connection to non-classicality criteria. The third chapter deals uniquely with quantum effects occurring in systems with mechanical elements, such as phonon anti bunching, parametric down conversion in electromechanical systems, creation and interference of macroscopic super positions in spin-cantilever systems, and collapse and revivals of mechanical motion and mechanical state dependent transmission in membrane-in-the-middle geometries. The fourth and last chapter discusses pervading issues with defining the classical limit, the quantum/classical comparison and definitions of non-classicality. @en

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.

We discuss theoretically the coupling of magnetization and infrared photons in whispering gallery mode cavities.@en

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.

Quantum communication has been shown to be vastly superior to classical communication in many problems. However no general statements exist which tells us how much better quantum communication is to its classical counterpart. In this thesis it was studied the minimum amount of classical bits required to exactly simulate a quantum communication process. The quantum communication process specifically studied was a quantum prepare and measurement communication problem. It has been shown that the calculation of the amount of classical bits of communication required for simulation reduces to a minimization-maximization optimization problem. Several results have been presented for for solving this optimization problem and in addition a link was made between classical simulations of quantum communication and a recent debate on the reality of the quantum state.

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.

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.

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.

This thesis analyzes the occurence of the quantum Zeno effect in a qubit in different situations. A system with a particle with spin 1/2, which represents a qubit, and detector is considered. The detector is modeled by a coordinate q, which has a Gaussian distribution with dispersion σ. There is the free evolution of the qubit and the interaction with the detector. Moreover, there are calculated algebraic expressions for the probabilities of the qubit to be in one of its states at certain moments in time by considering the wave function and density matrix at that time and consequently tracing out the detector coordinate q. Furthermore, there is done an analysis using plots of the evolution of these probabilities in time for different situations. In the situation with one continuous measurement and no free evolution, the qubit remains in its initial state, as expected.In the situation where periods of only free evolution and only measurements alternate, the probabilities keep the same value during the measurement, so then the evolution of the system freezes, and the probabilities evolve in a sine form during the free evolution, in line with the expectation.To neglect free evolution during the measurement, tm << tev is assumed, so there are no series of fast subsequent measurements or a continuous measurement. In this case, the quantum Zeno effect does not occur.Furthermore, we consider the situation where periods of only free evolution and periods with a measurement during free evolution alternate. Now the oscillations continue in time, due to the ongoing free evolution. The larger the influence of the interaction between the qubit and detector during the measurement and the smaller the influence of the magnetic field of the free evolution, the higher the equilibirium position of the oscillations of the probability for the qubit to remain in its initial state is in time. Moreover, the amplitude of these oscillations is smaller. However, due to the free evolution the qubit always has a probability to undergo a transition to its other state. The continuous measurement does not freeze the evolution of the system totally. The quantum Zeno effect does not occur.In the situation where the dispersion σ of the detector coordinate q goes to 0, there is a perfect measurement. The larger σ, the larger the measurement error in the detector resulting in dissipation of the system. The oscillations of the probabilities damp in time.Recommendations for further research include plotting the evolution of the probabilities for a longer period in time with another integration tool. Also, it would be interesting to work out the assumptions done in this analysis to more realistic conditions. Furthermore, another distribution of the detector coordinate q and other qubit states might be interesting to work out in follow-up research.

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.

In the last decades there has been an increasing interest in computing the local strain at the atomic scale of materials. By knowing aspects of the local strain in a lattice, one has information about measurements of distortions of lattice parameters concerning shifts, deformations and defects computed with respect to a smooth, defect-free reference region. Multiple methods have been implemented so far in order to map the strain of two-dimensional lattice patterns, which are obtained through means of a High Resolution Electron Microscope (HRTEM). The functioning of a HRTEM is based on the same principles as an optical microscope, but it uses a beam of electrons instead of visible light. One of the computational methods which then processes the obtained two-dimensional images is called the Geometrical Phase Analysis (GPA) and makes use of a very important mathematical tool, the Fourier transform. The GPA method lies at the center of this project and consists of several steps. First, the Fourier transform of the lattice image is plotted and two Bragg peaks corresponding to two linearly independent frequency vectors in the power spectrum are chosen. Next, a mask is applied around these peaks, separately. In my project I have chosen to apply the Hann smoothing filter. Then, the inverse Fourier transform is applied to the masked image and the phase (also called the raw phase) is plotted. The next step is to compute the reduced phase, which is defined at a local pixel as being the raw phase from which the following product is subtracted: 2𝜋 ⃗𝑔 ⋅ ⃗𝑟, where ⃗𝑟 is the vector corresponding to a pixel in the real space and ⃗𝑔 the frequency vector corresponding to the Bragg peak around which one has applied the mask. At this point the reference region is computed by choosing a smooth, homogeneous area in the reduced phase image. In order to obtain the strain, one needs an optimal frequency vector ⃗𝑔 defined at every pixel of lattice. In order to do so, a minimization process defined in the context of a computer algorithm in the programming language Python has been implemented. These computations should lead to obtaining the lattice strain, which is calculated by taking the symmetric part of the derivation of the displacement obtained from the two linearly independent Fourier components, which is in turn called the distortion. The antisymmetric part of the distortion is the rotation component and serves as a check for the correctness of the computational method. The goal of my project is not only to provide a solid theoretical background for the GPA method and to discuss the strain at atomic level in several lattice patterns, but to also provide a rigorous computer algorithm that makes these computations reality. This algorithm, opposed to pre-existing software, facilitates the reader’s process immensely by the large amount of detail which is given at every step, detail which easily motivates and supports the reader in potential side-steps they would want to take in order to make the method their own.