Theses and Dissertations from UMD
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New submissions to the thesis/dissertation collections are added automatically as they are received from the Graduate School. Currently, the Graduate School deposits all theses and dissertations from a given semester after the official graduation date. This means that there may be up to a 4 month delay in the appearance of a give thesis/dissertation in DRUM
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Item Constructing an ergodic theory of quantum information dynamics(2024) Anand, Amit Vikram; Galitski, Victor; Physics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)The ergodic theory of classical dynamical systems, originating in Boltzmann's ergodic hypothesis, provides an idealized description of how the flow of information within energy surfaces of a classical phase space justifies the use of equilibrium statistical mechanics. While it is an extremely successful mathematical theory that establishes rigorous foundations for classical chaos and thermalization, its basic assumptions do not directly generalize to quantum mechanics. Consequently, previous approaches to quantum ergodicity have generally been limited to model-specific studies of thermalization, or well-motivated but imprecise general conjectures. In this Dissertation, we develop a general theoretical framework for understanding how the energy levels of a quantum system drive the flow of quantum information and constrain the applicability of statistical mechanics, guided by two prominent conjectures. The first of these, the Quantum Chaos Conjecture (QCC), aims to characterize which quantum systems may thermalize, by postulating a connection between ergodicity or chaos and the statistical properties of random matrices. The second, the Fast Scrambling Conjecture (FSC), is concerned with how fast a quantum system may thermalize, and posits a maximum speed of thermalization in a sufficiently “local” many-body system. This Dissertation is divided into three main parts. In the first part, Theory of Quantum Dynamics and the Energy Spectrum, we tackle these conjectures for a general isolated quantum system through results that may be understood as new formulations of the energy-time uncertainty principle. For QCC, we introduce precise quantum dynamical concepts of ergodicity and quantitatively establish their connections to the statistics of energy levels, deriving random matrix statistics as a special consequence of these dynamical notions. We subsequently build on one of these connections to derive an energy-time uncertainty principle that accounts for the full structure of the spectrum, introducing sufficient sensitivity for many-body systems. The resulting quantum speed limit allows us to prove a precise formulation of FSC from the mathematical properties of the energy spectrum. In doing so, we generalize QCC beyond the statistics of random matrices alone, and FSC beyond requirements of locality, establishing precise versions of these statements for the most general quantum mechanical Hamiltonian. In the second part, Quantum Systems Beyond the Chaotic-Integrable Dichotomy, we demonstrate the need for the aforementioned precise formulations of these conjectures, by showing that looser formulations can be readily violated in “maximally” chaotic or integrable systems that would be most expected to satisfy them. Finally, in the third part, Experimental Probes of Many-Body Quantum Ergodicity, we develop tools to experimentally probe the structure of energy levels associated with ergodic dynamics, and demonstrate a generalization of these probes to open systems in an experiment with trapped ions.Item Classical and quantum dynamics of Bose-Einstein condensates(2017) Mathew, Ranchu; Tiesinga, Eite; Sau, Jay D; Physics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)After the first experimental realization of a Bose-Einstein condensate (BEC) in 1995, BECs have become a subject of intense experimental and theoretical study. In this dissertation, I present our results on the classical and quantum dynamics of BECs at zero temperature under different scenarios. First, I consider the analog of slow light in the collision of two BECs near a Feshbach resonance. The scattering length then becomes a function of the collision energy. I derive a generalization of the Gross-Pitaevskii equation for incorporating this energy dependence. In certain parameter regimes, the group velocity of a BEC traveling through another BEC decreases. I also study the feasibility of an experimental realization of this phenomena. Second, I analyze an experiment in which a BEC in a ring-shaped trap is stirred by a rotating barrier. The phase drop across and current flow through the barrier is measured from spiral-shaped density profiles created by interfering the BEC in the ring-shaped trap and a concentric reference BEC after release from all trapping potentials. I show that a free-particle expansion is sufficient to explain the origin of the spiral pattern and relate the phase drop to the geometry of a spiral. I also bound the expansion times for which the phase drop can be accurately determined and study the effect of inter-atomic interactions on the expansion time scales. Third, I study the dynamics of few-mode BECs when they become dynamically unstable after preparing an initial state at a saddle point of the Hamiltonian. I study the dynamics within the truncated Wigner approximation (TWA) and find that, due to phase-space mixing, the expectation value of an observable relaxes to a steady-state value. Using the action-angle formalism, we derive analytical expressions for the steady-state value and the time evolution towards this value. I apply these general results to two systems: a condensate in a double-well potential and a spin-1 (spinor) condensate. Finally, I study quantum corrections beyond the TWA in the semiclassical limit. I derive general expressions for the dynamics of an observable by using the van Vleck-Gutzwiller propagator and find that the interference of classical paths leads to non-perturbative corrections. As a case study, I consider a single-mode nonlinear oscillator; this system displays collapse and revival of observables. I find that the interference of classical paths, which is absent in the TWA, leads to revivals.