Thesis (Ph. D.)--University of Texas at Austin, 2001. / Vita. Includes bibliographical references (leaves 118-125). Available also in a digital version from Dissertation Abstracts.

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The dynamics of non-interacting, ultracold alkali atoms in the presence of counter-propagating lasers (optical lattice systems) is considered theoretically. The center of mass motion of an atom is such a system can be described by an effective Hamiltonian of a relatively simple form. Modulation of the laser fields implies a parametric variation of the effective Hamiltonian's eigenvalue spectrum, under which avoided crossings may occur. We investigate two dynamical processes arising from these near-degeneracies, which can be manipulated to coherently control atomic motion. First, we demonstrate the mechanism for the chaos-assisted, or multiple-state, tunneling observed in recent optical lattice experiments. Second, we propose a new method for the coherent acceleration of lattice atoms using the techniques of stimulated Raman adiabatic passage (STIRAP). In each case we use perturbation analysis to show the existence of a small, few level, subsystem of the full effective Schrödinger equation that determines the dynamics. / text

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The main principle of this thesis is to employ the geometric representation of Hamiltonian dynamics: in a broad sense, we study how to construct, in phase space, geometric structures that are related to a dynamical system. More specifically, we study the problem of constructing phase-space tori that are approximate invariant tori of a given Hamiltonian; also, using the constructed tori, we define an integrable Hamiltonian closely approximating the original one. The methods are generally applicable; as examples, we use gravitational potentials that are of interest in stellar dynamics. First, we construct tori for box and loop orbits in planar, barred potentials, thus demonstrating the applicability of the scheme to potentials that have more than one major orbit family. Also, we show that, in general, the construction scheme needs two types of canonical transformations together: point transformations as well as those expressed by generating functions. To complete the construction scheme, we show how to furnish the tori with consistent coordinate systems, i.e., how to recover the angle variables of a torus labelled by its actions. Next, the developed methods are employed in creating invariant phase-space tori in nonintegrable potentials supporting minor-orbit families. These tori are used to define an integrable Hamiltonian H₀, and a modified form of the standard Hamiltonian perturbation theory is then used to demonstrate that a minor-orbit family can be treated as one made up of orbits trapped by a resonance of H₀. Finally, we generalize the scheme further by constructing tori in time-reversal asymmetric Hamiltonians (by considering the motion in a rotating frame of reference), and study the transition from locally contained stochasticity to global chaos. Using both near-integrable 'laboratory' Hamiltonians and those for which we construct tori, we investigate the transition in the light of the resonance overlap criterion.

Thesis (Ph. D.)--University of Texas at Austin, 2007. / Vita. Includes bibliographical references.

Thesis (Ph. D.)--University of Texas at Austin, 2007. / Vita. Includes bibliographical references.

The goal of this dissertation has been to develop a method that enables one to calculate accurate, rigorous lower bounds to the eigenvalues of the standard nonrelativistic spin-free Hamiltonian for an atom with N electrons. Lower bounds are necessary in order to complement upper bounds obtained from the Hartree-Fock and Rayleigh-Ritz techniques. Without accurate lower bounds, it is impossible to estimate the error of the approximate values of the energies. By combining two heretofore distinct methods and using the symmetry properties of the Hamiltonian, this goal has been achieved. The first of the two methods is the method of intermediate problems. By beginning with an appropriately chosen "base operator" H⁰, one forms a sequence of intermediate Hamiltonians Hᵏ, k = 1,2,..., whose corresponding eigenvalues form a sequence of lower bounds to the eigenvalues of the original Hamiltonian H. Complications which occurred in this method due to the stability of essential spectra under compact perturbations were later surmounted with the use of abstract separation of variables by D. W. Fox. The second technique, the effective field method, provides a lower bound operator to the interelectron repulsion term in H that is of the form of a sum of separable potentials. This latter technique reduces the eigenvalue problem for H⁰ to a sum of single particle operators. With the use of a special potential, the Hulthen potential, one may construct an explicitly solvable base problem from the effective field method, if one uses the method of intermediate problems to calculate lower bounds to non-S states. This base problem is then suitable as a starting point for the method of intermediate problems with the Fox modifications. The eigenvalues of the new base problem are already comparable to the celebrated Thomas-Fermi energies. The final part of the dissertation provides a practical procedure for determining the physically realizable spectra of the intermediate operators. This is accomplished by restricting the Hamiltonian to subspaces of proper physical symmetry so that the resulting lower bounds will be to eigenvalues of physical significance.

Numerical methods are usually necessary in solving Hamiltonian systems since there is often no closed-form solution. By utilizing a general property of Hamiltonians, namely the symplectic property, all of the qualities of the system may be preserved for indefinitely long integration times because all of the integral (Poincare) invariants are conserved. This allows for more reliable results and frequently leads to significantly shorter execution times as compared to conventional methods. The resonant triad Hamiltonian with one degree of freedom will be focused upon for most of the numerical tests because of its difficult nature and, moreover, analytical results exist whereby useful comparisons can be made.