Thesis (Ph. D.)--Ohio State University, 2005. / Title from first page of PDF file. Includes bibliographical references (p. 177-185).

Not available / text

Thesis (Ph. D.)--University of Texas at Austin, 2003. / Vita. Includes bibliographical references. Available also from UMI Company.

Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 94-95). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 94-95). Also available on the Internet.

Thesis (Ph. D.)--Oregon State University, 2007. / Printout. Includes bibliographical references (leaves 151-157). Also available on the World Wide Web.

Cette thèse a pour but de développer et d'utiliser un cadre cohérent permettant de caractériser les phases topologiques dans des milieux spatialement périodiques induites par une perturbation dépendant périodiquement du temps ("phases topologiques de Floquet" ou "isolants topologiques de Floquet"), en présence de symétries. Ces phases sont des généralisation des isolants topologiques apparues lors de l'étude d'isolants topologiques induits par la lumière ainsi que d'analogues ondulatoires des isolants topologiques (en acoustique, mécanique et optique). De nouveaux invariants topologiques caractérisant ces systèmes sont définis, en particulier en présence d'un renversement du temps fermionique. Les cas, déjà connus dans des situations particulières, des classes complexes A et AIII de Cartan-Altland-Zirnbauer sont généralisés à toutes les dimensions, et leur survivance dans les classes réelles est discutée. Les conséquences physiques potentielles dans des systèmes électroniques sont explorées par des simulations de transport résolues en temps, qui concluent à l'existence de conductances différentielles moyennes quantifiées en présence d'un état de bord topologique. / This thesis aims at developing and using a coherent framework to characterize topological states in spatially periodic media stemming from a time-periodic drive (« topological Floquet states » or « Floquet topological insulators »), when symmetries are present. Such states are a generalization of topological insulators, which appeared from the study of the control by light of topological insulators, and from the study of the wave-physics versions of topological insulators (in acoustics, mechanics and optics). New invariants characterizing such systems are defined, in particular when fermionic time-reversal is present. The cases of complex classes A and AIII in the Cartan-Altland-Zirnbauer classification, which are already known in particular cases, are generalized to any space dimension, and their survival in real classes is discussed. Potential physical consequences in electronic systems are explored by time-resolved numerical simulation of transport properties, which show evidence of quantized average differential conductances when a topological edge state is present.

This thesis is concerned with a variety of topics grouped together under the general theme of periodically driven atomic systems. Periodic driving is ubiquitous in most techniques used in atomic physics, be it laser cooling, ion trapping or AC magnetic fields. An in-depth understanding of the behaviour of such systems can be provided through Floquet theory which will develop as a central theme in the following chapters. The thesis is divided in two parts: neutral atoms, and ions and biomolecules. In the first part I discuss a new ⁴¹K-⁸⁷Rb mixture experiment, built during the first year of my DPhil. This species combination has some very broad and low-loss interspecies Feshbach resonances that are instrumental for carrying out the experiments discussed in the first chapter. Unfortunately, the mixture experiment had to be put aside and our attention was shifted to Time-Averaged Adiabatic Potentials (TAAPs) and how these can be extended using multiple Radio-Frequency (RF) fields. This technique opens up the way for precise interferometric measurements. Lastly, the peculiar behaviour of Modulation Transfer Spectroscopy (MTS) of ³⁹K is investigated and a linearising transformation for four-wave mixing processes is presented. In the second part we turn our attention to charged ions and biomolecules and the techniques of ion trapping. We propose a novel technique for co-trapping charged particles with vastly different mass-to-charge ratios and thoroughly explore its consequences. The behaviour of the trap and the stability of equations with periodic coefficients in general is studied using Floquet theory. The normal modes and symmetries of the system also need to be considered in relation to the effectiveness of the sympathetic cooling of the ions. Small systems were simulated using a Molecular Dynamics (MD) approach in order to capture the effect of micromotion and other heating processes.

This thesis covers my work in the field of closed, repeatedly driven, Hamiltonian systems. These systems do not exchange particles with the surrounding environment and their time-evolution is described by Hamilton's equations of motion (in the classical framework) or the Schroedinger equation (in the quantum framework). Their interaction with the environment is encoded into the time-dependence of the system's Hamiltonian. Chapter 1 is an "Overview" in which the status of the field, my contributions and future prospective are outlined. Chapters 2 to 4 provide the theoretical background which is used in Chapters 5 to 7 to derive some original results. These results show that in Hamiltonian systems, after many driving events, universal properties emerge. In particular, using the framework of the linear Boltzmann equation, I have studied the dynamics of a mobile, light impurity in a gas of heavy particles. The impurity's kinetic energy increases and, in the long time limit, approaches a non-thermal asymptotic distribution. The significance of this work is to show explicitly the emergence of a non-thermal distribution in a closed, driven system. Moreover, using the work-fluctuation theorems, I have studied the character of the energy distribution of a generic isolated system driven according a generic protocol. Both thermal and non-thermal distributions can be realized for the same system by changing the characteristics of the driving protocol. These two different regimes are separated by a dynamical phase transition. Finally, I have used the Floquet Theory and the Magnus Expansion to analyze the behavior of a generic interacting system which is driven periodically in time. For fast driving the system is unable to absorb energy and remains localized in the low energy part of the Hilbert space while for slow driving the system absorbs energy and, in the long time limit, it is delocalized in the entire Hilbert space. These two qualitatively different behaviors are separated by a many-body localization transition which is related to the break down of the Magnus expansion at the critical value of the driving frequency.

In this thesis we study Floquet theory on a Banach space. We are concerned about the linear differential equation of the form: y'(t) = A(t)y(t), where t ∈ R, y(t) is a function with values in a Banach space X, and A(t) are linear, bounded operators on X. If the system is periodic, meaning A(t+ω) = A(t) for some period ω, then it is called a Floquet system. We will investigate the existence and uniqueness of the periodic solution, as well as the stability of a Floquet system. This thesis will be presented in five main chapters. In the first chapter, we review the system of linear differential equations on Rn: y'= A(t)y(t) + f(t), where A(t) is an n x n matrix-valued function, y(t) are vectors and f(t) are functions with values in Rn. We establish the general form of the all solutions by using the fundamental matrix, consisting of n independent solutions. Also, we can find the stability of solutions directly by using the eigenvalues of A. In the second chapter, we study the Floquet system on Rn, where A(t+ω) = A(t). We establish the Floquet theorem, in which we show that the Floquet system is closely related to a linear system with constant coefficients, so the properties of those systems can be applied. In particular, we can answer the questions about the stability of the Floquet system. Then we generalize the Floquet theory to a linear system on Banach spaces. So we introduce to the readers Banach spaces in chapter three and the linear operators on Banach spaces in chapter four. In the fifth chapter we study the asymptotic properties of solutions of the system: y'(t) = A(t)y(t), where y(t) is a function with values in a Banach space X and A(t) are linear, bounded operators on X with A (t+ω) = A(t). For that system, we can show the evolution family U(t,s) representing the solutions is periodic, i.e. U(t+ω, s+ω) = U(t,s). Next we study the monodromy of the system V := U(ω,0). We point out that the spectrum set of V actually determines the stability of the Floquet system. Moreover, we show that the existence and uniqueness of the periodic solution of the nonhomogeneous equation in a Floquet system is equivalent to the fact that 1 belongs to the resolvent set of V.