Theses and Dissertations from UMD
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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 Numerical Studies of Quantum Chaos in Various Dynamical Systems(2020) Rozenbaum, Efim; Galitski, Victor; Physics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)We study two classes of quantum phenomena associated with classical chaos in a variety of quantum models: (i) dynamical localization and its extension and generalization to interacting few- and many-body systems and (ii) quantum exponential divergences in high-order correlators and other diagnostics of quantum chaos. Dynamical localization (DL) is a subtle phenomenon related to Anderson localization. It hinges on quantum interference and is typically destroyed in presence of interactions. DL often manifests as a failure of a driven system to heat up, violating the foundations of statistical physics. Kicked rotor (KR) is a prototypical chaotic classical model that exhibits linear energy growth with time. The quantum kicked rotor (QKR) features DL instead: its energy saturates. Multiple attempts of many-body generalizations faced difficulties in preserving DL. Recently, DL was shown in a special integrable many-body model. We study non-integrable models of few- and many-body QKR-like systems and provide direct evidence that DL can persist there. In addition, we show how a novel related concept of localization landscape can be applied to study transport in rippled channels. Out-of-time-ordered correlator (OTOC) was proposed as an indicator of quantum chaos, since in the semiclassical limit, this correlator's possible exponential growth rate (CGR) resembles the classical Lyapunov exponent (LE). We show that the CGR in QKR is related, but distinct from the LE in KR. We also show a singularity in the OTOC at the Ehrenfest time tᴱ due to a delay in the onset of quantum interference. Next, we study scaling of OTOC beyond tᴱ. We then explore how the OTOC-based approach to quantum chaos relates to the random-matrix-theoretical description by introducing an operator we dub the Lyapunovian. Its level statistics is calculated for quantum stadium billiard, a seminal model of quantum chaos, and aligns perfectly with the Wigner-Dyson surmise. In the semiclassical limit, the Lyapunovian reduces to the matrix of uncorrelated finite-time Lyapunov exponents, connecting the CGR at early times, when the quantum effects are weak, to universal level repulsion that hinges on strong quantum interference. Finally, we consider quantum polygonal billiards: their classical counterparts are non-chaotic. We show exponential growth of the OTOCs in these systems, sharply contrasted with the classical behavior even before quantum interference develops.Item Dynamics of wave packets in the quantum Lorentz gas(2005-06-30) Goussev, Arseni; Dorfman, J. Robert; Physics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)This dissertation addresses the dynamics of a quantum particle moving in an array of fixed scatterers. The system is known as the Lorentz gas. The scatterers are taken to be two- or three-dimensional hard-spheres. The quantum Lorentz gas is analyzed in two dynamical regimes: (i) semiclassical regime, and (ii) high-energy diffraction regime. In both regimes the dynamics of the quantum particle is found to be determined by properties characterizing chaotic dynamics of the counterpart classical Lorentz gas. Thus, this dissertation provides an attempt to more deeply understand the role that classical chaos plays in quantum mechanics of nonintegrable systems. In the semiclassical regime, the quantum particle is represented by a small Gaussian wave packet immersed in the array of scatterers. The de Broglie wavelength of the particle is considered to be much smaller than both the scatterer size and the typical separation between scatterers. It is found that for times, during which the wave packet size remains smaller than the scatterer size, the spreading of the quantum wave packet is exponential in time, and the spreading rate is determined by the sum of positive Lyapunov exponents of the corresponding classical system. The high-energy diffraction approximation allows one to analytically describe the dynamics of large wave packets in dilute scattering systems for times far beyond the Ehrenfest time. The latter is defined as the time during which the evolution of the wave packet is predominantly classical-like. The following two conditions are satisfied by the system in the high-energy diffraction regime: (i) the ratio of the particle s de Broglie wavelength to the scatterer size is much smaller than unity, and (ii) this ratio is much larger than the ratio of the scatterer size to the typical separation between scatterers. The time-dependent autocorrelation function is calculated for wave packets in hard-disk and hard-sphere geometrically open billiard systems. The envelope of the autocorrelation function is shown to decay exponentially with time, with the decay rate determined by the mean Lyapunov exponents and the Kolmogorov-Sinai entropy of the counterpart classical system.