Thesis (Ph. D.)--University of Washington, 2001. / Vita. Includes bibliographical references (leaves 233-234).

Thesis (Ph.D)--Mathematics, Georgia Institute of Technology, 2009. / Committee Chair: Bunimovich, Leonid; Committee Member: Bonetto, Federico; Committee Member: Chow, Shui-Nee; Committee Member: Cvitanovic, Predrag; Committee Member: Weiss, Howard. Part of the SMARTech Electronic Thesis and Dissertation Collection.

The Born-Oppenheimer approximation is a mainstay in molecular physics and chemistry and can be considered a two step process. The first step is to solve the electronic problem with nuclei fixed in space while the second step is to then determine the nuclear dynamics on a given electronic potential energy surface. This first-step calculation of the wavefunction and electronic energies for fixed nuclei has been at the center of modern quantum chemistry for decades. While the majority of chemical processes can be investigated by considering these single electronic surface dynamics, there exist problems in which the dynamics are not constrained to a single electronic surface. One such problem that justifies going beyond the typical adiabatic approximation is the determination of energy levels in systems with strongly coupled electronic states. While some work has been done using diabatic or quasidiabatic Hamiltonians to describe such systems, the work has historically been of qualitative accuracy. Model Hamiltonians have been constructed using experimental data to help calibrate the model parameters aided by the use of lower level adiabatic calculations to help inform the model. It is only within the last few years that theorists have been able to attempt parameterization of such models using only ab initio methods. The goal of this work is to develop a computational framework for the parameterization of quantitatively accurate quasidiabatic Hamiltonians based purely on ab initio information and apply it to a notoriously difficult problem that has plagued the theoretical community for decades -- high accuracy treatment of the energy levels of the NO₃ radical. In this dissertation, high-level ab initio calculations that employ the equation-of-motion coupled-cluster method in the single, doubles and triples (EOMIP-CCSDT) have been used in conjunction with a quasidiabatic ab initio approximation to construct a vibronic Hamiltonian for the strongly coupled X²A'₂ and B²E' states of the NO₃ radical. A quartic vibronic coupling model potential of the form advocated by Köppel et al. has been used to determine the energy levels of this system to quantitative accuracy when compared to experimental data. In order to obtain sufficiently accurate potential energy surfaces necessary to parameterize a quantitatively accurate model Hamiltonian, thousands of large calculations had to be run that do not fit in memory on even the largest HPC systems. The resulting large, out-of-core solves do not map to traditional systems in a way to enable any reasonable parallelization. As a result, a new MPI-based utility has been developed to support out-of-core methods on distributed memory systems. This and other advances in scientific computing form the basis of the developed computational framework. / text

The dynamics of mesoscopic two-boson systems that model an interacting pair of ultracold alkali atoms in the presence of electromagnetic potentials are considered. The translational degrees of freedom of such a system can be described by a simple reduced atom Hamiltonian. Introducing time modulations in the laser fields causes parametric variations of the Hamiltonian's Floquet eigenvalue spectrum. Broken symmetries cause level repulsion and avoided crossings in this spectrum that are quantum manifestations of the chaos in the underlying classical dynamics of the systems. We investigate the effects of this phenomenon in the coherent control of excitations in these systems. These systems can be coherently excited from their ground states to higher energy states via a Stimulated Raman Adiabatic Passage (STIRAP). The presence of avoided crossings alter the outcome of STIRAP. First, the classical dynamics of such two-boson systems in double wells is described and manifestations of the same to the quantum mechanical system are discussed. Second, the quantum dynamics of coherent control in the manner discussed above is detailed for a select choice(s) of system parameters. Finally, the same chaos-assisted adiabatic passage is demonstrated for optical lattice systems based on experiments on the same done with noninteracting atoms. / text

Not available / text

Not available / text

Not available / text

Not available

published_or_final_version / Civil and Structural Engineering / Doctoral / Doctor of Philosophy

This thesis primarily discusses the results of two papers, [Hu] and [HaHu]. The first is an overview of algebraic-geometric techniques for integrable systems in which the AKS theorem is proven. Under certain conditions, this theorem asserts the commutatvity and (potential) non-triviality of the Hamiltonian flow of Ad*-invariant functions once they're restricted to subalgebras. This theorem is applied to the case of coadjoint orbits on loop algebras, identifying the flow with a spectral curve and a line bundle via the Lax equation. These results play an important role in the discussion of [HaHu], wherein we consider three levels of spaces, each possessing a linear family of Poisson spaces. It is shown that there exist Poisson mappings between these levels. We consider the two cases where the underlying Riemann surface is an elliptic curve, as well as its degeneration to a Riemann sphere with two points identified (the trigonometric case). Background in necessary areas is provided.