Applied Mathematics

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This is the collection for the University of Waterloo's Department of Applied Mathematics.

Research outputs are organized by type (eg. Master Thesis, Article, Conference Paper).

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Browsing Applied Mathematics by Author "Cory, David"

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    Analysis of Light-Matter Systems
    (University of Waterloo, 2022-01-18) El Mandouh, Mohamed; Cory, David
    In this thesis we introduce the simplest model of a two–level system coupled to a single mode of an optical cavity, called the Jaynes-Cummings model. This model is then extended to an ensemble of identical two-level systems and is studied in more detail, also known as the Tavis-Cummings model. This model is intractable, but we show that by a clever but simple choice of basis one can reduce the dimensionality of the Tavis-Cumming system. We then demonstrate the effectiveness of this reduction by calculating interesting statistics of the system, and simulating large ensembles of two–level systems which were not practical before. Finally, we examine some dynamics of the Tavis-Cummings model in the presence of photon losses, and introduce a method for population transfer by modulating TLS-cavity interaction strength.
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    Computational and Theoretical Insights into Multi-Body Quantum Systems
    (University of Waterloo, 2021-03-31) Stasiuk, Andrew; Cory, David
    In generality, perfect predictions of the structure and dynamics of multi-body quantum systems are few and far between. As experimental design advances and becomes more refined, experimentally probing the interactions of multiple quantum systems has become commonplace. Predicting this behavior is not a ``one size fits all" problem, and has lead to the inception of a multitude of successful theoretical techniques which have made precise and verifiable predictions through, in many cases, clever approximations and assumptions. As the state-of-the-art pushes the quantum frontier to new experimental regimes, many of the old techniques become invalid, and there is often no tractable methodology to fall back on. This work focuses on expanding the theoretical techniques for making predictions in newly accessible experimental regimes. The transport of quantum information in a room-temperature dipolar spin network is veritably diffusive in nature, but much less is known about the transport properties of such a sample at low temperatures. This work presupposes that diffusion is still a good model for incoherent transport at low temperatures, and proposes a new method to calculate its diffusion coefficient. The diffusion coefficient is reported as a function of the temperature of the ensemble. Further, the interaction of an i.i.d. spin ensemble with a quantized electromagnetic field has long been analyzed via restriction to the Dicke subspace implicit in the Holstein--Primakoff approximation, as well as other within other approximations. This work reanalyzes the conditions under which such a restriction is valid. In regimes where it is shownt that restricting to the Dicke subspace would be invalid, the Hamiltonian structure is thoroughly analyzed. Various predictions can be made by appealing to a reduction in effective dimensionality via a direct sum decomposition. The main theme of the techniques utilized throughout this work is to appeal to a reduction in difficulty via various theoretical tools in order to prepare for an otherwise intractable computational analysis. Computational insights due to this technique have then gone on to motivate directly provable theoretical results, which might otherwise have remained hidden behind the complexity of the structure and dynamics of a multi-body quantum system.
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    Exploring Practical Methodologies for the Characterization and Control of Small Quantum Systems
    (University of Waterloo, 2018-09-04) Hincks, Ian; Cory, David; Emerson, Joseph
    We explore methodologies for characterizing and controlling small quantum systems. We are interested in starting with a description of a quantum system, designing estimators for parameters of the system, developing robust and high-fidelity gates for the system using knowledge of these parameters, and experimentally verifying the performance of these gates. A strong emphasis is placed on using rigorous statistical methods, especially Bayesian ones, to analyze quantum system data. Throughout this thesis, the Nitrogen Vacancy system is used as an experimental testbed. Characterization of system parameters is done using quantum Hamiltonian learning, where we explore the use of adaptive experiment design to speed up learning rates. Gates for the full three-level system are designed with numerical optimal control methods that take into account imperfections of the control hardware. Gate quality is assessed using randomized benchmarking protocols, including standard randomized benchmarking, unitarity benchmarking, and leakage/loss benchmarking.