<p align="justify">Les transformations de supersymétrie (ou de Darboux) sont appliquées à l'étude du problème inverse, c'est à dire à la construction d'un potentiel d'interaction à partir de données de collisions, en mécanique quantique. En effet, ces transformations permettent de construire de nouveaux potentiels à partir d'un potentiel donné. Leur formalisme est étudié en détail, ainsi que celui correspondant à l'itération de deux telles transformations (paires de transformations).</p><p><p align="justify">La présence d'états liés rend le problème inverse ambigu :plusieurs potentiels ayant des spectres liés différents peuvent avoir les mêmes propriétés pour la description des collisions; de tels potentiels sont dits équivalents en phase. Une décomposition originale du problème inverse est proposée pour gérer efficacement cette ambiguïté :dans un premier temps, un potentiel est construit à partir des données de collision (ce qui constitue le problème inverse proprement dit); dans un second temps, tous les potentiels équivalents en phase au potentiel ainsi obtenu sont construits. Avant ce travail, il était connu que ces deux aspects du problème inverse pouvaient être traités à l'aide de paires de transformations de supersymétrie.</p><p><p align="justify">En ce qui concerne la construction de potentiels équivalents, nous étendons les méthodes existantes à des catégories de potentiels très utilisées en physique nucléaire, à savoir les potentiels optiques (ou complexes), les potentiels en voies couplées et les potentiels dépendant linéairement de l'énergie. En utilisant une paire de transformations permettant d'enlever un état lié, nous comparons les propriétés physiques des potentiels nucléaires profonds (c'est à dire possédant des états liés interdits par le principe de Pauli) et peu profonds. Des calculs dans des modèles à trois corps du noyau à halo d'6He et de la collision 16O+17O à basse énergie n'ont pas révélé d'importantes différences entre ces familles de potentiels. D'autres types de transformations permettent d'ajouter des états liés à énergie et normalisation arbitraires. Cependant, dans le cas à plusieurs voies, leur utilisation est compliquée par la possibilité d'avoir des états liés dégénérés et non dégénérés. Une étude préliminaire à deux voies montre que ces deux types d'états peuvent être traités par supersymétrie.</p><p><p align="justify">En ce qui concerne le problème inverse proprement dit, nous montrons que l'utilisation de transformations simples (plutôt que de paires) permet une meilleure compréhension des méthodes existantes, tant pour l'inversion à moment cinétique orbital fixe que pour l'inversion à énergie fixe. De plus, l'utilisation de transformations simples mène dans certains cas à de nouvelles catégories de potentiels. Ainsi, nous construisons un nouveau potentiel d'interaction nucléon nucléon pour l'onde 1S; ce potentiel possède une singularité en r 2 à l'origine. La possibilité de construire des potentiels profonds par inversion est brièvement discutée. Pour les voies couplées, une étude bibliographique révèle certaines propriétés contradictoires des méthodes existantes, mais une analyse complète reste à faire.</p><p> / Doctorat en sciences appliquées / info:eu-repo/semantics/nonPublished

Quantum mechanics became a foundation for incessant development of versatile computational methods for analysis of chemical and physical properties of molecules and crystals. A huge progress has been made in the fifield of density functional theory, since nowadays this theory offers the best compromise between precision of results and efficiency fiof computation. The chemical bonding analysis can be easily performed with real space methods based on chemical concepts introduced via partitioning of real space into chemically meaningful domains, since the orbital based approach is not well applicable due to the delocalized nature of plane waves. However the practical usage of those methods often requires a signifificant amount of computational resources. Some methods require the evaluation of so called domain overlap matrices, that is a formidable task for complex and low-symmetry systems. In the present research the author enables the investigation of complex solid compounds with real space chemical bonding indicators by introducing the derivation of the expression for the evaluation of the domain overlap matrix elements from the projected-augmented wave method. The corresponding program module was developed, which is capable to perform the real space chemical bonding analysis with a number of methods, like electron localizability indicators, electron localization function, localization/delocalization indices and domain averaged Fermi hole orbitals. The efficiency and the accuracy of the developed implementation is demonstrated by the comparison with the domain overlap matrix elements evaluation from the full-potential linearized augmented plane wave method on a set of simple compounds with three atoms per primitive cell at most. A set of complex periodic structures is analyzed and the capability of the present implementation to unravel intricate chemical bonding patterns is demonstrated.

Quantum simulations, where controllable quantum systems are used to simulate other quantum systems, originally proposed by Richard Feynman, are one of the most remarkable applications of quantum information science. Compared to computation, quantum simulations require much less number of qubits for the m to be practical. In the work described in this thesis, we have performed a few quantum simulations of small quantum systems using Nuclear Magnetic Resonance(NMR) techniques. These simulations have been used to experimentally demonstrate the underlying interesting quantum protocols. All the experiments presented have been carried out using liquid-state or liquid crystal NMR. Numerical pulse optimization techniques have been utilized in some of the experiments, to achieve better control over the spin systems. The first chapter contains “Introduction” to quantum information processing, NMR, and numerical pulse optimization techniques. In chapter 2, we describe quantum simulation of a 3-spin Heisenberg-XY spin chain having only nearest neighbour interactions. Recently, spin chains having pre-engineered short-range interactions have been proposed to efficiently transfer quantum information between different parts of a quantum information processor. Other important proposals involving these spin chains include generating entangled states and universal quantum computation. However, such engineered interactions do not occur naturally in any system. In such a scenario, the experimental viability of these proposals can be tested by simulating the spin chains in other controllable quantum systems. In this work, we first theoretically study the time evolution of bipartite and tripartite entanglement measures for a 3-spin open ended XY spin chain. Then, by simulating the XY interactions in a 3-spin nuclear spin system, we experimentally generate, (i)a bipartite maximally(pseudo-)entangled state(Bell state) between end qubits, and(ii) multipartite(pseudo-)entangled states(Wand GHZ states),starting from separable pseudo-pure states. Bell state has been generated by using only the natural unitary evolution of the XY spin chain. W-state and GHZ-state have been generated by applying a single-qubit rotation to the second qubit, and a global rotation of all the three qubits respectively after the unitary evolution of the spin chain. In chapter 3, we simulate a 3-spin quantum transverse Ising spin system in a triangular configuration, and show that multipartite quantum correlations can be used to distinguish between the frustrated and non-frustrated regimes in the ground state of this spin system. The ground state of the spin system has been prepared by using adiabatic state preparation method. Gradient ascent pulse engineering technique has been utilized to efficiently realize the adiabatic evolution of the spin system. To analyse the experimental ground state of the system, we employ two different multipartite quantum correlation measures, generated from monogamy studies of bipartite quantum correlations. Chapter 4 contains a digital quantum simulation of the mirror inversion propagator corresponding to the time evolution of an XY spin chain. This simulation has been used to experimentally demonstrate the mirror inversion of quantum states, proposed by Albanese et al.[Phys.Rev.Lett.93,230502(2004)], by which entangled states can be transferred from one end of the chain to the other end. The experiments have been performed in a 5-qubit dipolar coupled nuclear spin system. For simulation, we make use of the recently proposed unitary operator decomposition algorithm along with the numerical pulse optimization techniques, which assisted in achieving high experimental fidelities. Chapter 5 contains a digital quantum simulation of the unitary propagator of a transverse Ising spin chain, which has been used to experimentally demonstrate the perfect state transfer protocol of Di Franco et al. [Phys.Rev.Lett.101,230502(2008)]. The importance of this protocol arises due to the fact that it achieves perfect state transfer from one end of the chain to the other end without the necessity of initializing the intermediate spins of the chain, whereas most of the previously proposed protocols require initialization. The experiments have been performed in a 3-spin nuclear spin system. The simulation has also been used to demonstrate the generation of a GHZ state.

Quantum mechanics became a foundation for incessant development of versatile computational methods for analysis of chemical and physical properties of molecules and crystals. A huge progress has been made in the fifield of density functional theory, since nowadays this theory offers the best compromise between precision of results and efficiency fiof computation. The chemical bonding analysis can be easily performed with real space methods based on chemical concepts introduced via partitioning of real space into chemically meaningful domains, since the orbital based approach is not well applicable due to the delocalized nature of plane waves. However the practical usage of those methods often requires a signifificant amount of computational resources. Some methods require the evaluation of so called domain overlap matrices, that is a formidable task for complex and low-symmetry systems. In the present research the author enables the investigation of complex solid compounds with real space chemical bonding indicators by introducing the derivation of the expression for the evaluation of the domain overlap matrix elements from the projected-augmented wave method. The corresponding program module was developed, which is capable to perform the real space chemical bonding analysis with a number of methods, like electron localizability indicators, electron localization function, localization/delocalization indices and domain averaged Fermi hole orbitals. The efficiency and the accuracy of the developed implementation is demonstrated by the comparison with the domain overlap matrix elements evaluation from the full-potential linearized augmented plane wave method on a set of simple compounds with three atoms per primitive cell at most. A set of complex periodic structures is analyzed and the capability of the present implementation to unravel intricate chemical bonding patterns is demonstrated.

Indiana University-Purdue University Indianapolis (IUPUI) / The tight-binding model that has been used for many years in condensed matter physics, due to its analytic and numerical tractability, has recently been used to describe light propagating through an array of evanescently coupled waveguides. This dissertation presents analytic and numerical simulation results of light propagating in a waveguide array. The first result presented is that photonic transport can be achieved in an array where the propagation constant is linearly increasing across the array. For an input at the center waveguide, the breathing modes of the system are observed, while for a phase displaced, asymmetric input, phase-controlled photonic transport is predicted. For an array with a waveguide-dependent, parity-symmetric coupling constant, the wave packet dynamics are predicted to be tunable. In addition to modifying the propagation constant, the coupling between waveguides can also be modified, and the quantum correlations are sensitive to the form of the tunneling function. In addition to modifying the waveguide array parameters in a structured manner, they can be randomized as to mimic the insertion of impurities during the fabrication process. When the refractive indices are randomized and real, the amount of light that localizes to the initial waveguide is found to be dependent on the initial waveguide when the waveguide coupling is non-uniform. In addition, when the variance of the refractive indices is small, light localizes in the initial waveguide as well as the parity-symmetric waveguide. In addition to real valued disorder, complex valued disorder can be introduced into the array through the imaginary component of the refractive index. It is shown that the two-particle correlation function is qualitatively similar to the case when the waveguide coupling is real and random, as both cases preserve the symmetry of the eigenvalues. Lastly, different input fields have been used to investigate the quantum statistical aspects of Anderson localization. It is found that the fluctuations in the output intensity are enhanced and the entropy of the system is reduced when disorder is present in the waveguides.

Indiana University-Purdue University Indianapolis (IUPUI) / More than a decade ago, it was shown that non-Hermitian Hamiltonians with combined parity (P) and time-reversal (T ) symmetry exhibit real eigenvalues over a range of parameters. Since then, the field of PT symmetry has seen rapid progress on both the theoretical and experimental fronts. These effective Hamiltonians are excellent candidates for describing open quantum systems with balanced gain and loss. Nature seems to be replete with examples of PT -symmetric systems; in fact, recent experimental investigations have observed the effects of PT symmetry breaking in systems as diverse as coupled mechanical pendula, coupled optical waveguides, and coupled electrical circuits. Recently, PT -symmetric Hamiltonians for tight-binding lattice models have been extensively investigated. Lattice models, in general, have been widely used in physics due to their analytical and numerical tractability. Perhaps one of the best systems for experimentally observing the effects of PT symmetry breaking in a one-dimensional lattice with tunable hopping is an array of evanescently-coupled optical waveguides. The tunneling between adjacent waveguides is tuned by adjusting the width of the barrier between them, and the imaginary part of the local refractive index provides the loss or gain in the respective waveguide. Calculating the time evolution of a wave packet on a lattice is relatively straightforward in the tight-binding model, allowing us to make predictions about the behavior of light propagating down an array of PT -symmetric waveguides. In this thesis, I investigate the the strength of the PT -symmetric phase (the region over which the eigenvalues are purely real) in lattices with a variety of PT - symmetric potentials. In Chapter 1, I begin with a brief review of the postulates of quantum mechanics, followed by an outline of the fundamental principles of PT - symmetric systems. Chapter 2 focuses on one-dimensional uniform lattices with a pair of PT -symmetric impurities in the case of open boundary conditions. I find that the PT phase is algebraically fragile except in the case of closest impurities, where the PT phase remains nonzero. In Chapter 3, I examine the case of periodic boundary conditions in uniform lattices, finding that the PT phase is not only nonzero, but also independent of the impurity spacing on the lattice. In addition, I explore the time evolution of a single-particle wave packet initially localized at a site. I find that in the case of periodic boundary conditions, the wave packet undergoes a preferential clockwise or counterclockwise motion around the ring. This behavior is quantified by a discrete momentum operator which assumes a maximum value at the PT -symmetry- breaking threshold. In Chapter 4, I investigate nonuniform lattices where the parity-symmetric hop- ping between neighboring sites can be tuned. I find that the PT phase remains strong in the case of closest impurities and fragile elsewhere. Chapter 5 explores the effects of the competition between localized and extended PT potentials on a lattice. I show that when the short-range impurities are maximally separated on the lattice, the PT phase is strengthened by adding short-range loss in the broad-loss region. Consequently, I predict that a broken PT symmetry can be restored by increasing the strength of the short-range impurities. Lastly, Chapter 6 summarizes my salient results and discusses areas which can be further developed in future research.

Nous naviguons présentement sur la vague de la deuxième révolution quantique qui nous dirige vers un océan de possibilités. L’approche tant attendue de l’ordinateur quantique affecte notre société, notamment la sécurité mondiale actuelle. C’est la course pour mettre à jour nos réseaux de communication pour maintenir le droit à la vie privée. En cryptographie, bien que le chiffrement de message soit crucial pour des échanges privés, la sécurité générale de toute communication repose majoritairement sur la sécurité d’une clé. C’est pourquoi l’établissement quantique de clé ou QKD (de quantum key distribution en anglais) est une importante tâche cryptographique qui se doit d’être résistante aux adversaires quantiques. Beaucoup d’avancées ont déjà été faites dans le domaine, en l’occurrence l’usage de la fibre optique qui a mené à l’implémentation réelle de protocoles QKD. Par contre, l’obstacle qui continue de limiter tout progrès est la distance. Celle-ci hausse exponentiellement les erreurs introduites dans l’échange dépassant facilement les taux maximum tolérés actuels après quelques centaines de kilomètres seulement. De ce fait, bien que la théorie semble prometteuse, la mise en pratique de protocoles quantiques demeure un défi. Pour viser l’application mondiale, nous nous devons de prioriser l’efficacité. Ce mémoire présente une variation du fameux protocole BB84 pour maximiser la perfor- mance des applications de QKD en augmentant le taux d’erreurs toléré et, en l’occurrence, la distance entre les partis. Un satellite sera introduit comme troisième parti. Il aidera Alice et Bob à partager une chaine secrète. Celle-ci leur permettra de rouler le protocole BB84 sans dévoiler les bases. De plus, deux techniques seront définies, soient le filtrage et la concentration. Ces dernières serviront lors de la communication classique interactive pour diminuer l’erreur entre nos deux individus tout en limitant le gain d’information de leur ad- versaire. Les bénéfices de cette modification sont la possibilité de recycler les bases secrètes du protocole ainsi que la possibilité d’étendre d’avantage la longueur du canal atteignant ainsi l’objectif de pousser les limites pratiques de QKD. / We are currently sailing on the second quantum revolution wave towards an ocean of pos- sibilities. The long awaited quantum computer is near and it will affect global security as we know it. It is a race against the clock to update our entire communication network to maintain the right to personal privacy. An important cryptographic task is key establish- ment. While communicating privately, the entire security lies mainly in the security of the key used. Therefore, it is crucial that future protocols for key establishment be resistant against quantum adversaries. Over the years, there has been great progress in the field like the practical use of optical fibre leading to quantum key distribution (QKD) protocols implemented in real life. Despite this, a specific obstacle still remains. Distance poses a serious problem as it increases ex- ponentially the amount of errors introduced in the protocol, meaning we easily exceed the maximum rate that we can currently tolerate after only a few hundred kilometers. Hence, what we do in theory may sound promising, but the actual application in reality remains a challenge. To aim for global use, we need to prioritize efficiency. This thesis suggests an alternative to the renowned BB84 protocol to help maximize applications of quantum key distribution by increasing the tolerated error rate and thus, the distance between two parties. A satellite will be introduced as a third party to help Alice and Bob share a secret bit sequence. This bit string will allow them to run a BB84 protocol without revealing the bases. Then, two techniques will be defined: filtering and concentration. They will serve in the classical communication phase to help lower the error rate between our two parties while also limiting the amount of information gained by the adversary. Benefits from this approach are the recycling of the secret bases of the protocol as well as the possible extension of the length of the channel, thus achieving the end goal of pushing the limits of practical implementation of QKD.

Not only is Quantum Mechanics (QM) conceptually rich, it is also a theory that physics students have found abstract and technically formidable. Nevertheless, compared to other classical topics of physics, university students’ understanding of QM has received minimal attention in the physics education literature. The principal purpose of this study was to characterize the variation in the ways that undergraduate physics students depict the basic concepts of QM and to extrapolate the results to scaffold possible changes to instructional practices at the university that provided the context for the study. In so doing, an adaptation of a developmental phenomenographic perspective was chosen. Empirically, the study was approached through in-depth interviews with 35 physics students from two Ethiopian governmental universities after they had been exposed to the traditional QM course for one-third of a semester. Interview responses were analyzed using phenomenographic approach where a picture of students’ depictions was established for each quantum concept by expounding the given responses. For each basic quantum concept addressed, the structure of the description categories was separately constructed, and overall, it was found that naive, quasi-classical ontology and/or variants of classical ways of visualization are dominant in students’ responses. For example, it was found that students’ depictions of the photon concept could be described with three distinct categories of description, which are (a) classical intuitive description, (b) mixed model description and (c) quasi-quantum model description. Similarly, the findings revealed that it is possible to establish three qualitatively different categories of description to picture students’ depictions of matter waves, namely, (a) classical and trajectory-based description, (b) an intricate blend of classical and quantum description and (c) incipient quantum model description. Likewise, it was found that students’ depictions of uncertainty principle can be described as: (a) uncertainty as classical ignorance, (b) uncertainty as measurement disturbance and (c) uncertainty as a quasi-quantum principle. With regard to learning QM, the categories of description made clear several issues: most students did not have enough knowledge to depict the basic concepts of QM properly; they were influenced by the perspective of classical physics and their perceptions in making explanations about QM; and they also applied mixed ideas, one based on their classical model and the other from newly introduced QM. These results are also supported by the findings of previous studies in similar domains. Findings from the study were used to guide the design of multiple representations-based instructions and interactive learning tutorials on the conceptual aspects of QM that has been shown to address specific difficulties identified in the study. Theoretical and practical implications of the study, as well as potential future considerations are drawn. / Mathematics, Science and Technology Education / D. Phil. (Mathematics, Science and Technology Education)

Not only is Quantum Mechanics (QM) conceptually rich, it is also a theory that physics students have found abstract and technically formidable. Nevertheless, compared to other classical topics of physics, university students’ understanding of QM has received minimal attention in the physics education literature. The principal purpose of this study was to characterize the variation in the ways that undergraduate physics students depict the basic concepts of QM and to extrapolate the results to scaffold possible changes to instructional practices at the university that provided the context for the study. In so doing, an adaptation of a developmental phenomenographic perspective was chosen. Empirically, the study was approached through in-depth interviews with 35 physics students from two Ethiopian governmental universities after they had been exposed to the traditional QM course for one-third of a semester. Interview responses were analyzed using phenomenographic approach where a picture of students’ depictions was established for each quantum concept by expounding the given responses. For each basic quantum concept addressed, the structure of the description categories was separately constructed, and overall, it was found that naive, quasi-classical ontology and/or variants of classical ways of visualization are dominant in students’ responses. For example, it was found that students’ depictions of the photon concept could be described with three distinct categories of description, which are (a) classical intuitive description, (b) mixed model description and (c) quasi-quantum model description. Similarly, the findings revealed that it is possible to establish three qualitatively different categories of description to picture students’ depictions of matter waves, namely, (a) classical and trajectory-based description, (b) an intricate blend of classical and quantum description and (c) incipient quantum model description. Likewise, it was found that students’ depictions of uncertainty principle can be described as: (a) uncertainty as classical ignorance, (b) uncertainty as measurement disturbance and (c) uncertainty as a quasi-quantum principle. With regard to learning QM, the categories of description made clear several issues: most students did not have enough knowledge to depict the basic concepts of QM properly; they were influenced by the perspective of classical physics and their perceptions in making explanations about QM; and they also applied mixed ideas, one based on their classical model and the other from newly introduced QM. These results are also supported by the findings of previous studies in similar domains. Findings from the study were used to guide the design of multiple representations-based instructions and interactive learning tutorials on the conceptual aspects of QM that has been shown to address specific difficulties identified in the study. Theoretical and practical implications of the study, as well as potential future considerations are drawn. / Mathematics, Science and Technology Education / D. Phil. (Mathematics, Science and Technology Education)

Les travaux récents en fondements de la théorie quantique (des champs) et en information quantique relativiste tentent de mieux comprendre les effets des contraintes de causalité imposées aux opérations physiques sur la structure des corrélations quantiques. Le premier chapitre de cette thèse est consacré à l'étude des implications conceptuelles de la non-localité quantique, notion qui englobe celle d'intrication dans un sens précis. Nous détaillons comment les récentes approches informationnelles tentent de saisir la structure des corrélations non-locales, ainsi que les questions que ces dernières soulèvent concernant la capacité d'un observateur localisé à isoler un système de son environnement. Le second chapitre détaille les effets de l'invariance de Poincaré sur la détection et la quantification de l'intrication. Cette invariance impose que tous les systèmes soient modélisés en dernière instance dans le cadre de la théorie des champs, ce qui implique qu'aucun système à énergie finie ne puisse être localisé, ainsi que la divergence de toute mesure d'intrication pour des observateurs localisés. Nous fournissons une solution à ces deux problèmes en démontrant l'équivalence générique qui existe entre une résolution spatiale finie des appareils de mesure et l'exclusion des degrés de liberté de haute énergie de la définition du système observé. Cette équivalence permet une interprétation épistémique du formalisme quantique standard décrivant les systèmes localisés non-relativistes et leurs corrélations, clarifiant ainsi l'origine des mesures finies d'intrication pour de tels systèmes. Le dernier chapitre explore un cadre théorique récemment introduit qui prédit l'existence de corrélations quantiques sans ordre causal défini. Procédant par analogie avec le cas des corrélations non-locales, nous présentons quelques principes informationnels contraignant la structure de ces corrélations dans le but de mieux en comprendre l'origine physique. / Recent works in foundations of quantum (field) theory and relativistic quantum information try to better grasp the interplay between the structure of quantum correlations and the constraints imposed by causality on physical operations. Chapter 1 is dedicated to the study of the conceptual implications of quantum nonlocality, a concept that subsumes that of entanglement in a certain way. We detail the recent information-theoretic approaches to understanding the structure of nonlocal correlations, and the issues the latter raise concerning the ability of local observers to isolate a system from its environment. Chapter 2 reviews in what sense imposing Poincaré invariance affects entanglement detection and quantification procedures. This invariance ultimately forces a description of all quantum systems within the framework of quantum field theory, which leads to the impossibility of localized finite-energy states and to the divergence of all entanglement measures for local observers. We provide a solution to these two problems by showing that there exists a generic equivalence between a finite spatial resolution of the measurement apparatus and the exclusion of high-energy degrees of freedom from the definition of the observed system. This equivalence allows for an epistemic interpretation of the standard quantum formalism describing nonrelativistic localized systems and their correlations, hence a clarification of the origin of the finite measures of entanglement between such systems. Chapter 3 presents a recent theoretical framework that predicts the existence of correlations with indefinite causal order. In analogy to the information-theoretic approaches to nonlocal correlations, we introduce some principles that constrain the structure of such correlations, which is a first step toward a clear understanding of their physical origin.