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 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

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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.

In this thesis, bifurcation phenomena in dynamical systems with cosymmetry and Hamiltonian structure were investigated using numerical methods. Several numerical continuation methods and test functions for detecting bifurcations were presented. The numerical results for various examples are given using a numerical bifurcation analysis toolbox.

This thesis consists of two parts. In the first part, we study stability properties of Hamiltonian systems on the Wasserstein space. Let H be a Hamiltonian satisfying conditions imposed in the work of Ambrosio and Gangbo. We regularize H via Moreau-Yosida approximation to get H[subscript Tau] and denote by μ[subscript Tau] a solution of system with the new Hamiltonian H[subscript Tau] . Suppose H[subscript Tau] converges to H as τ tends to zero. We show μ[subscript Tau] converges to μ and μ is a solution of a Hamiltonian system which is corresponding to the Hamiltonian H. At the end of first part, we give a sufficient condition for the uniqueness of Hamiltonian systems. In the second part, we develop a general theory of differential forms on the Wasserstein space. Our main result is to prove an analogue of Green's theorem for 1-forms and show that every closed 1-form on the Wasserstein space is exact. If the Wasserstein space were a manifold in the classical sense, this result wouldn't be worthy of mention. Hence, the first cohomology group, in the sense of de Rham, vanishes.