Title: Quantum computing algorithms for quantum chemistry Author: Jakub Višňák Abstract: The topic of this study is the simulation of the quantum algorithm for the diagonalization of the matrix representation of the all-electron Dirac-Coulomb hamiltonian of the SbH molecule. Two different limited CI expansions were used to describe both the ground state (X 0+ ) and the first excited doublet (A 1) by simulating the Iterative Phase Estinamtion Algorith (IPEA). In the simulations numerically performed in this work, the "compact mapping" has been employed for the representation of the evolution operator exp(i Hˆ t); in the theoretical part of the work, the "direct mapping" is described as well. The influence of the metodics for choosing the initial eigenvector estimate is studied in both IPEA A and IPEA B variants. For those variants, the success probabilities pm are computed for different single-points on the SbH dissociation curves. The initial eigenvector estimates based on the "CISD(2)" method are found to be sufficient for both studied LCI-expansions up to internuclear distance R  6 a0. The pm dependence on the overlap between the eigenvector in question and its inital estimate - 2 0  is studied the for IPEA B method. The usability of the both variants of the IPEA in possible later calculations is...

The thesis presents a numerical study of magnetocrystalline anisotropies in dilute ferromagnetic semiconductors and transition metal systems intended to advance the current understanding of the microscopic origins of this relativistic effect and to contribute to the development of spintronic devices with new functionalities. The major part of the work surveys magnetocrystalline anisotropies in (Ga,Mn)As epilayers and compares the calculations to available experimental data. Our model is based on an envelope function description of the valence band holes and a spin representation for their kinetic-exchange interaction with localised electrons on Mn2+ ions, treated in the mean-field approximation. For epilayers with growth induced lattice-matching strains we study in-plane to out-of-plane easy axis reorientations as a function of Mn local-moment concentration, hole concentration, and temperature. Next we focus on the competition of in-plane cubic and uniaxial anisotropies. We add an in-plane shear strain to the effective Hamiltonian in order to capture measured data in bare, unpatterned epilayers, and we provide microscopic justification for this approach. The model is then extended by an in-plane uniaxial strain and used to directly describe experiments with magnetisation direction controlled by...

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.

In the present work, we introduce a never before studied soliton equation called the intermediate mixed Manakov (IMM) equation. Through a pole ansatz, we prove that the equation has N-soliton solutions with pole parameters governed by the hyperbolic Calogero-Moser system. We also show that there are spatially periodic N-soliton solutions with poles obeying elliptic Calogero-Moser dynamics. A Lax pair is given in the form of a Riemann-Hilbert problem on a cylinder. A similar Lax pair is shown to imply a novel spin generalization of the intermediate nonlinear Schrödinger equation. Some conservation laws for the IMM are proven. We demonstrate that the IMM can be written as a Hamiltonian system, with one of these conserved quantities as the Hamiltonian. Finally, a physical interpretation is given by showing that the IMM can be rewritten to describe a system of two nonlocally coupled fluids, with nonlinear self-interactions. / Vi presenterar en aldrig tidigare studerad solitonekvation som vi döper till ‘the intermediate mixed Manakov equation’ (ungefär ‘den mellanliggande kopplade Manakovekvationen’. Kortform: IMM). Genom en polansats bevisar vi att ekvationen har N-solitonlösningar där polparametrarna utgör ett hyperboliskt Calogero-Mosersystem. Vi visar också att det finns rumsligt periodiska N-solitonlösningar vars poler följer elliptisk Calogero-Moserdynamik. Ett Laxpar ges i form av ett Riemann-Hilbertproblem på en cylinder. Vi demonstrerar att ett liknande Laxpar leder till en ny spinngeneralisering av den s.k. INLS-ekvationen. Några bevarandelagar för IMM bevisas. Vi visar att IMM-ekvationen kan skrivas som ett Hamiltonskt system, där Hamiltonianen är en av våra tidigare bevarade storheter. Till sist ger vi en fysikalisk tolkning av vår ekvation genom att demonstrera hur den beskriver ett system av ickelokalt interagerande vätskor, med ickelinjära självinteraktioner.

Experiments for the development of fusion power that are based on magnetic confinement deal with plasmas that inevitably contain energetic (non-thermal) particles. These particles come e.g. from fusion reactions or from external heating of the plasma. Ensembles of energetic ions can excite plasma waves in the Alfvén frequency range to such an extent that the resulting wave fields redistribute the energetic ions, and potentially eject them from the plasma. The redistribution of ions may cause a substantial reduction heating efficiency, and it may damage the inner walls and other components of the vessel. Understanding the dynamics of such instabilities is necessary to optimise the operation of fusion experiments and of future fusion power plants. A Monte Carlo model that describes the nonlinear wave-particle dynamics in a toroidal plasma has been developed to study the excitation of the abovementioned instabilities. A decorrelation of the wave-particle phase is added in order to model stochasticity of the system (e.g. due to collisions between particles). Based on the nonlinear description with added phase decorrelation, a quasilinear version of the model has been developed, where the phase decorrelation has been replaced by a quasilinear diffusion coefficient in particle energy. When the characteristic time scale for macroscopic phase decorrelation becomes similar to or shorter than the time scales of nonlinear wave-particle dynamics, the two descriptions quantitatively agree on a macroscopic level. The quasilinear model is typically less computationally demanding than the nonlinear model, since it has a lower dimensionality of phase space. In the presented studies, several effects on the macroscopic wave-particle dynamics by the presence of phase decorrelation have been theoretically and numerically analysed, e.g. effects on the growth and saturation of the wave amplitude, and on the so called frequency chirping events with associated hole-clump pair formation in particle phase space. Several effects coming from structures of the energy distribution of particles around the wave-particle resonance has also been studied. / <p>QC 20150330</p>

Le groupe de difféomorphismes hamiltoniens à support compact d’une variété symplectique admet une distance naturelle bi-invariante, d’après les travaux de Viterbo, Schwarz, Oh, Frauenfelder et Schlenk, construite à partir des invariants spectraux en homologie de Floer Hamiltonienne. Cette distance, appelée la norme spectrale, s’est révélée être un outil fort utile en topologie symplectique. Par contre, son diamètre reste inconnu en général. En fait, pour les variétés symplectiques fermées, il n’existe même pas de critère pour déterminer si la norme spectrale a un diamètre fini ou infini. Il a été conjecturé que, pour les variétés symplectiquement asphériques, le diamètre de la norme spectrale est infini. Dans cette thèse, nous démontrons que pour tout domaine de Liouville, la norme spectrale a un diamètre infini si et seulement si la cohomologie symplectique du domaine de Liouville en question est non nulle. Ceci généralise un résultat de Monzner-Vichery-Zapolsky et admet plusieurs applications dans le cadre des variétés symplectiques fermées. En particulier, nous démontrons que le produit de deux variétés symplectiquement asphériques a un diamètre spectral infini. Plus généralement, nous démontrons que toute variété symplectiquement asphérique contenant un domaine de Liouville incompressible de codimension zéro avec cohomologie symplectique non nulle doit avoir un diamètre spectral infini. / The group of compactly supported Hamiltonian diffeomorphisms of a symplectic manifold is endowed with a natural bi-invariant distance, due to Viterbo, Schwarz, Oh, Frauenfelder and Schlenk, coming from spectral invariants in Hamiltonian Floer homology. This distance, called the spectral norm, has found numerous applications in symplectic topology. However, its diameter is still unknown in general. In fact, for closed symplectic manifolds there is no unifying criterion for the diameter to be finite or infinite. It has been conjectured that for closed symplectically aspherical manifolds, the spectral norm has infinite diameter. In this thesis, we prove that for any Liouville domain the spectral norm has infinite diameter if and only if its symplectic cohomology does not vanish. This generalizes a result of Monzner-Vichery-Zapolsky and has applications in the setting of closed symplectic manifolds. For instance, we show that the product of two closed symplectically aspherical manifold has an infinite spectral diameter . More generally, we prove that any symplectically aspherical manifold which contains an incompressible Liouville domain of codimension zero with non-vanishing symplectic cohomology must have infinite spectral diameter.

Les travaux de la présente thèse porteront sur le raffinement du modèle du potentiel source-puits (SSP) proposé par Ernzerhof en 2006. Cette méthode permet de calculer la conductance qualitative de dispositifs électroniques moléculaires (MEDs). Dans la première partie de ce travail, le modèle SSP sera amélioré en y intégrant la description de l’interaction d’un champ électromagnétique fort avec le MED. Des expériences récentes ont démontré que des molécules pouvaient interagir fortement avec des plasmons de polaritons de surface (SPP). Ces interactions créent des états liés électron-SPP qui seront exploités pour contrôler la conductance de MEDs. Des formules analytiques expliqueront l’impact des paramètres physiques de ces circuits optoélectroniques sur la conductance de ceux-ci. Dans le même esprit, la seconde partie de cette thèse inclura les interactions électron-noyau au modèle SSP afin de décrire entre autres le courant décohérent d’un MED. Dans ce modèle les interactions noyau-électron seront décrites à partir de l’approximation harmonique et intégrées à l’hamiltonien de façon non-pertubative. Des formules analytiques seront dérivées afin de décrire la conductance de tels MEDs. Finalement, les conséquences du bris de la symétrie de la parité et du temps de la matrice hamiltonienne de la méthode SSP seront découvertes dans la densité spectrale et les fonctions d’ondes des MEDs. / The purpose of this thesis is to expand the scope of the source-sink potential (SSP) method originally proposed by Ernzerhof in 2006. The SSP model allows the computation of the qualitative conductance of molecular electronics devices (MEDs). In the first part of this work, the SSP model will be improved by including the description of interaction between the strong electromagnetic field and the MED. Recent experiments have shown that molecules could strongly interact with surface plasmon of polaritons (SPPs). These interactions will create so-called dressed states that can be used to control the conductance of MEDs. In the second part of this work, the SSP model will be augmented by including electron-nucleus interactions to describe the inelastic current. In this model, the electron-nuclueus interaction will be account for with the help of the harmonic approximation and incorporated into the hamiltonian non-pertubatively. Analytical formulas will be derived that will allow one to understand the impact of physical parameters on the conductance of MEDs. Lastly, the impact of the parity and time symmetry breaking of the SSP matrix hamiltonian will be studied and related to change in the spectral density and in the eigenfunctions of the MEDs.

No description available.

Chaotic transport in Hamiltonian systems is often restricted due to the presence of partial barriers, leading to a limited flux between different regions in phase space. Typically, the most restrictive partial barrier in a 2D symplectic map is based on a cantorus, the Cantor set remnants of a broken 1D torus. For a 4D symplectic map, we establish a partial barrier based on what we call a cantorus-NHIM—a normally hyperbolic invariant manifold with the structure of a cantorus. Using a flux formula, we determine the global 4D flux across a partial barrier based on a cantorus-NHIM by approximating it with high-order periodic NHIMs. In addition, we introduce a local 3D flux depending on the position along a resonance channel, which is relevant in the presence of slow Arnold diffusion. Moreover, for a partial barrier composed of stable and unstable manifolds of a NHIM, we utilize periodic NHIMs to quantify the corresponding flux.

This thesis deals with cohomologies on the symplectic quotient of a Frölicher space which is locally diffeomorphic to a Euclidean Frölicher subspace of Rn of constant dimension equal to n. The symplectic reduction under consideration in this thesis is an extension of the Marsden-Weinstein quotient (also called, the reduced space) well-known from the finite-dimensional smooth manifold case. That is, starting with a proper and free action of a Frölicher-Lie-group on a locally Euclidean Frölicher space of finite constant dimension, we study the smooth structure and the topology induced on a small subspace of the orbit space. It is on this topological space that we will construct selected cohomologies such as : sheaf cohomology, Alexander-Spanier cohomology, singular cohomology, ~Cech cohomology and de Rham cohomology. Some natural questions that will be investigated are for instance: the impact of the symplectic structure on these di erent cohomologies; the cohomology that will give a good description of the topology on the objects of category of Frölicher spaces; the extension of the de Rham cohomology theorem in order to establish an isomorphism between the five cohomologies. Beside the algebraic, topological and geometric study of these new objects, the thesis contains a modern formalism of Hamiltonian mechanics on the reduced space under symplectic and Poisson structures. / Mathematical Sciences / D. Phil. (Mathematics)