Decoherence plays an important role in the evolution of quantum systems. In this thesis, we theoretically investigate the decoherence dynamics of two-dimensional spin- 1/2 structures coupled to an electron bath, with a focus on the influence of exchange and dipolar coupling. Using a combination of analytical derivations and numerical simulations, we explore the impact of interatomic distance, lattice symmetry, and external magnetic fields on quantum coherence. The study extends previous one-dimensional investigations to two-dimensional square, triangular, and hexagonal Ti-spin structures on an MgO substrate, where spin interactions can be manipulated at the atomic scale using scanning tunneling microscopy (STM). We analyse the eigenenergies and eigenstates of these systems and employ the Lindblad master equation to model decoherence processes. Our results show a competition between the Heisenberg exchange and the dipole-dipole coupling, with special attention to the intermediate regime. Additionally, we characterise spin relaxation and flipping dynamics, highlighting their dependence on structural variations. These insights contribute to the broader understanding of spin coherence in engineered quantum systems and provide a foundation for future experimental and theoretical studies in quantum nanoscience.

In this report, we have taken a look at three different types of trebuchets: the seesaw, the traction trebuchet or mangonel, and the counterweight trebuchet. We have created models to describe their motion by making use of Lagrangian mechanics and have compared them based on their maximum ranges. Moreover, we have also calculated a lower bound for the dimensions of themain beam needed to prevent it from breaking based on a static state. We have seen that the mangonel and trebuchet improved on the seesaw by increasing the range roughly sixfold which was expected. The maximum range of the trebuchet, however, slightly decreased compared to the mangonel. This was something we did not expect but think is due to the fact that the dimensions of the catapults were not optimised. Lastly, we found that the beam needed to be at least 18.7 cm to prevent it from breaking. However, for the real catapults, the beam would most likely need to be thicker to prevent it from breaking.

The aim of this thesis is to simulate quantum spin chains at a finite temperature. This has been achieved by using purification to write the incoherent thermal states of the spin chain as pure states, after which the matrix product state (MPS) formalism to simulate the spin chain. The influence of temperature on the spin chain has been tested by comparing chain magnetization to the strength of the external magnetic field at different temperatures. Real time evolution has also been applied under the assumption that the time evolution is adiabatic. The results have been compared to the analytical solution found by directly calculating the density matrix. The model is very accurate for imaginary time evolution, which is required to evolve the system to the desired temperature. During real time evolution the results oscillate slightly around the analytical solution, which remains constant. This oscillation is a consequence of model truncation and the Suzuki-Trotter approximation. The effect of allocating more numerical resources to limit this phenomenon is explored, and a method to increase reachable timescales by decreasing entanglement growth in the chain over time is tested and verified.

This thesis describes atomic spin chains subject to magnetic anisotropy. Such chains, assembled through means of scanning tunneling microscopy (STM), can be home to a plethora of magnetic states and spin physics. This includes quantum tunneling of magnetization, the injection and delocalization of magnons and, for single atoms, the application of electron spin resonance (ESR). Each phenomenon happens at their own timescale which is dependent on the isolation of the spin chain from the environment, ranging from seconds to picoseconds.

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.

This thesis investigates intrinsic frustrated magnetic systems on an insulating square lattice. The frustrated systems were successfully engineered from magnetic iron atoms on top of an insulating square c(2x2) reconstruction of nitrogen on Cu3Au(100) (copper-gold) surface. Vertical atom manipulation was used to position the magnetic atoms atomically precise on the surface and methods were found to increase the success rate of this. For the investigation of frustration, a numerical simulator was made in Python for modelling frustrated spin structures based on already existing spin models and the following experimental measurements were taken: topography, spectroscopy and current-time traces. The simulations of a frustrated spin system show low energy excited states and the measurements show switching of the system between states.

Physics at the level of an atom is dominated by laws of quantum mechanics. Often, this is entangled with a high complexity in behavior of the systems at that length scale. Unravelling the properties of a material at the atomic level is, therefore, a challenging task that easily supersedes current computational capabilities. A route to circumvent this problem is found in physical realization of simpler quantum systems that are representative of the complex quantum systems one is interested in. These simpler physical systems, unlike their more complex counterparts, can actually be measured and information about the complex system, otherwise inaccessible, gained. This thesis describes experimental work focusing mainly on the property of magnetism in spin chains. To mimic these complex systems, we employ a scanning tunneling microscope (STM) to build atomic chains on solid state surfaces and probe their magnetic properties. The intrinsic strength of STM in building and testing structures with single atom precision makes STM a great candidate for simulation of complex quantum systems. In addition to STM having a role of a quantum simulator, I present work supporting STM as a control device determining the very existence of the magnetic excitations of the atom it measures. Finally, I present experimental findings that suggest we are able to probe the magnetic excitations of the atom with subatomic resolution. In summary, this thesis work presents STM as a powerful probing and control tool for studies on quantum magnetism at the level of a single atom.