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Candidate
Andrew Stasiuk | Applied Mathematics, University of À¶Ý®ÊÓƵ
Title
Computational and Theoretical Insights into Multi-Body Quantum Systems
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In generality, perfect predictions of the structure and dynamics of multi-body quantum systems are few and far between.ÌýAs experimental design advances and becomes more refined, experimentally probing the interactions of multiple quantum systemsÌýhas 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, cleverÌýapproximations and assumptions. As the state-of-the-art pushes the quantum frontier to new experimental regimes, many of theÌýold 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. TheÌýtransport of quantum information in a room-temperature dipolar spin network is veritably diffusive in nature, but much lessÌýis known about the transport properties of such a sample at low temperatures. This work presupposes that diffusion is still aÌýgood model for incoherent transport at low temperatures, and proposes a new method to calculate its diffusion coefficient.ÌýThe diffusion coefficient is reported as a function of the temperature of the ensemble. Further, the interaction of an i.i.d.Ìýspin ensemble with a quantized electromagnetic field has long been analyzed via restriction to the Dicke subspace implicit inÌýthe Holstein--Primakoff approximation, as well as other within other approximations. This work reanalyzes the conditions underÌýwhich such a restriction is valid. In regimes where it is shownt that restricting to the Dicke subspace would be invalid, theÌýHamiltonian structure is thoroughly analyzed. Various predictions can be made by appealing to a reduction in effective dimensionalityÌývia a direct sum decomposition.
The main theme of the techniques utilized throughout this work is to appeal to a reduction in difficulty via various theoreticalÌýtools in order to prepare for an otherwise intractable computational analysis. Computational insights due to this techniqueÌýhave then gone on to motivate directly provable theoretical results, which might otherwise have remained hidden behind theÌýcomplexity of the structure and dynamics of a multi-body quantum system.