Browsing by Subject "Hamiltonian systems"
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Item Coherent control of cold atoms in a[n] optical lattice(2007) Holder, Benjamin Peirce, 1976-; Reichl, L. E.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.Item Curvilinear coordinate formulation for vibration-rotation-large amplitude internal motion interactions(Texas Tech University, 1986-08) Guan, YuhuaA theory for vibration-rotation-large amplitude internal motion interactions is developed using curvilinear coordinates for the vibrational degrees of freedom. An essential feature of the theory is our coordination of two transformations for the separation of vibration from rotation and vibration from the LAM, in zeroth order. Series expansion in the vibrational coordinates is used to obtain the full vibration-rotation-LAM Hamiltonian. A Van Vleck perturbation approach is used to obtain the effective rotation-LAM Hamiltonian for the molecule in the nth vibrational state. Reduction of the effective Hamiltonian has been made to (1) the zero angular momentum state of the molecule, (2) the zeroth order rotation-LAM Hamiltonian, and (3) the usual vibration-rotation Hamiltonian when the LAM takes on a small amplitude. The theory is applied to the water molecule treating the bending mode as the LAM. Fourier sine functions are used as the basis for the bending mode, harmonic oscillation functions for the two stretching modes, and Wang functions for the rotational motion. Using Hoy-Mills-Strey and Hoy-Bunker force constants and molecular geometry, the vibration-rotation- LAM energy levels for the water molecule have been calculated. The HMS constant yields better vibration-bending results and the HE constants better rotational results.Item Dynamics of quantum control in cold-atom systems(2009-05) Roy, Analabha, 1978-; Reichl, L. E.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.Item Hamilton's equations with Euler parameters for hybrid particle-finite element simulation of hypervelocity impact(2002) Shivarama, Ravishankar Ajjanagadde; Fahrenthold, Eric P.Item Numerical studies of the standard nontwist map and a renormalization group framework for breakup of invariant tori(2004) Apte, Amit Shriram; Morrison, Philip J.Item Plasma turbulence in the equatorial electrojet observations, theories, models, and simulations(2015-12) Hassan, Ehab Mohamed Ali Hussein; Morrison, Philip J.; Horton, Wendell; Fitzpatrick, Richard; Bengtson, Roger; Humphreys, ToddThe plasma turbulence in the equatorial electrojet due to the presence of two different plasma instability mechanisms has been observed and studied for more than seven decades. The sharp density-gradient and large conductivity give rise to gradient-drift and Farley-Buneman instabilities, respectively, of different scale-lengths. A new 2-D fluid model is derived by modifying the standard two-stream fluid model with the ion viscosity tensor and electron polarization drift, and is capable of describing both instabilities in a unified system. Numerical solution of the model in the linear regime demonstrates the capacity of the model to capture the salient characteristics of the two instabilities. Nonlinear simulations of the unified model of the equatorial electrojet instabilities reproduce many of the features that are found in radar observations and sounding rocket measurements under multiple solar and ionospheric conditions. The linear and nonlinear numerical results of the 2-D unified fluid model are found to be comparable to the fully kinetic and hybrid models which have high computational cost and small coverage area of the ionosphere. This gives the unified fluid model a superiority over those models. The distribution of the energy content in the system is studied and the rate of change of the energy content in the evolving fields obeys the law of energy conservation. The dynamics of the ions were found to have the largest portion of energy in their kinetic and internal thermal energy components. The redistribution of energy is characterized by a forward cascade generating small-scale structures. The bracket of the system dynamics in the nonlinear partial differential equation was proved to be a non-canonical Hamiltonian system as that bracket satisfies the Jacobi identity. The penetration of the variations in the interplanetary magnetic and electric fields in the solar winds to the dip equator is observed as a perfect match with the variations in the horizontal components of the geomagnetic and electric fields at the magnetic equator. Three years of concurrent measurements of the solar wind parameters at Advanced Composition Explorer (ACE) and Interplanetary Monitoring Platform (IMP) space missions used to establish a Kernel Density Estimation (KDE) functions for these parameters at the IMP-8 location. The KDE functions can be used to generate an ensemble of the solar wind parameters which has many applications in space weather forecasting and data-driven simulations. Also, categorized KDE functions ware established for the solar wind categories that have different origin from the Sun.Item Receding Horizon Covariance Control(2012-10-19) Wendel, EricCovariance assignment theory, introduced in the late 1980s, provided the only means to directly control the steady-state error properties of a linear system subject to Gaussian white noise and parameter uncertainty. This theory, however, does not extend to control of the transient uncertainties and to date there exist no practical engineering solutions to the problem of directly and optimally controlling the uncertainty in a linear system from one Gaussian distribution to another. In this thesis I design a dual-mode Receding Horizon Controller (RHC) that takes a controllable, deterministic linear system from an arbitrary initial covariance to near a desired stationary covariance in finite time. The RHC solves a sequence of free-time Optimal Control Problems (OCP) that directly controls the fundamental solution matrices of the linear system; each problem is a right-invariant OCP on the matrix Lie group GLn of invertible matrices. A terminal constraint ensures that each OCP takes the system to the desired covariance. I show that, by reducing the Hamiltonian system of each OCP from T?GLn to gln? x GLn, the transversality condition corresponding to the terminal constraint simplifies the two-point Boundary Value Problem (BVP) to a single unknown in the initial or final value of the costate in gln?. These results are applied in the design of a dual-mode RHC. The first mode repeatedly solves the OCPs until the optimal time for the system to reach the de- sired covariance is less than the RHC update time. This triggers the second mode, which applies covariance assignment theory to stabilize the system near the desired covariance. The dual-mode controller is illustrated on a planar system. The BVPs are solved using an indirect shooting method that numerically integrates the fundamental solutions on R4 using an adaptive Runge-Kutta method. I contend that extension of the results of this thesis to higher-dimensional systems using either in- direct or direct methods will require numerical integrators that account for the Lie group structure. I conclude with some remarks on the possible extension of a classic result called Lie?s method of reduction to receding horizon control.Item Renormalization of continuous-time dynamical systems with KAM applications(2006) Kocić, Saša; Koch, HansItem Renormalization of isoenergetically degenerate Hamiltonian flows, and instability of solitons in shear hydrodynamic flows(2003) Gaidashev, Denis Gennad'yevich; Koch, Hans A.; Bona, J. L.Part I of this Thesis presents a study of the renormalization group transformation acting on an appropriate space of Hamiltonian functions in two angle and two action variables. In particular, we study the existence of real invariant tori, on which the flow is conjugate to a rotation with the rotation number equal to the golden mean (ω-tori). We demonstrate that the stable manifold of the renormalization operator at the “simple” fixed point contains isoenergetically degenerate Hamiltonians possessing shearless ω-tori. We also show that one-parameter families of Hamiltonians transverse to the stable manifold undergo a bifurcation: for a certain range of the parameter values the members of these families posses two distinct ω-tori, the members of such families lying on the stable manifold posses one shearless ω-torus, while the members corresponding to other parameter values do not posses any. We also present some numerical evidence for universality associated with the breakup of shearless invariant tori, and compute the relevant critical renormalization and scaling eigenvalues. Part II of the Thesis presents a stability analysis of plane solitonsin hydrodynamic shear flows obeying a (2+1) analogue of the Benjamin–Ono equation. The instability region and the short-wave instability threshold for plane solitons are found numerically. We also determine the dependence of the growth rate on the propagation angle in the longwave limit and demonstrate the existence of a critical angle which separates two types of behaviour of the growth rate.Item Renormalization, invariant tori, and periodic orbits for Hamiltonian flows(2001-05) Abad, Juan José, 1967-; Koch, Hans A.Item Resonance overlap, secular effects and non-integrability: an approach from ensemble theory(2003) Li, Chun Biu; Prigogine, I. (Ilya); Petrosky, Tomio Y.Item The role of the Van Hove singularity in the time evolution of electronic states in a low-dimensional superlattice semiconductor(2007-05) Garmon, Kenneth Sterling, 1978-; Petrosky, Tomio Y.; Reichl, L. E.Over the last three decades, the rapid development of efficient synthetic routes for the preparation of expanded porphyrin macrocycles has allowed the exploration of a new frontier involving “porphyrin-like” coordination chemistry. This doctoral dissertation describes the author’s exploratory journey into the area of transition metal cation complexation using oligopyrrolic macrocycles. The reported synthetic findings were used to gain new insights into the factors affecting the observed coordination modes and to probe the emerging roles of counter-anion effects, tautomeric equilibria and hydrogenbonding interactions in regulating the metalation chemistry of expanded porphyrins. The first chapter provides an updated overview of this relatively young coordination chemistry subfield and introduces the idea of expanded porphyrins as a diverse family of ligands for metalation studies. Chapter 2 details the synthesis of a series of binuclear complexes and illustrates the importance of metal oxidation state, macrocycle protonation and counter-anion effects in terms of defining the final structure of the observed metal complexes. The binding study reported in Chapter 3 demonstrates a strong positive allosteric effect for the coordination of silver(I) cations in a Schiff base expanded porphyrin. Chapter 4 introduces the use of oligopyrrolic macrocycles for the stabilization of early transition metal cations. Specifically, the preparation of a series of vanadium complexes illustrates the bimodal (i.e., covalent and noncovalent) recognition of the non-spherical dioxovanadium(V) species within the macrocyclic cavities. Experimental procedures and characterization data are reported in Chapter 5.Item The role of the Van Hove singularity in the time evolution of electronic states in a low-dimensional superlattice semiconductor(2007) Garmon, Kenneth Sterling; Petrosky, Tomio; Reichl, LindaIn this dissertation we will study a wide range of phenomena from atomic, molecular, and optical to solid-state physics. We will find a common theme in problems from these different branches of physics in that they can all be modeled by some variation of a simple bi-linear Hamiltonian. Each of these models will also share a key feature in that they all contain one or more singularities (called a Van Hove singularity in the context of solid-state) in the density of allowed states associated with a branch point that results near the edge of a continuous energy spectrum. In addition, the fact that each of these models is one-dimensional will maximize the effect of the singularity on the system. We will show that when a discrete state is coupled with the continuum that in the vicinity of the singularity Fermi’s golden rule breaks down; the golden rule normally predicts that the de-excitation rate of the discrete state should be proportional to g 2 where g is the dimensionless coupling constant between the discrete state and the continuum. Relying on a non-perturbative approach, we will show that the de-excitation rate is actually proportional to g 4/3 in the vicinity of the singularity. This results in a dramatic amplification of the decay rate. In the main topic of the dissertation, we will consider a nano-scale semiconductor superlattice with either a single impurity site or multiple impurities (which behave as electron donors or acceptors) in which there are two Van Hove singularities in the density of electron states which occur at the two edges of the conduction band. These singularities result in the non-analytic g 4/3 amplification of the charge transfer rate from the discrete impurity site into the electronic conduction band where g is the coupling constant between the impurity state and the conduction band. We will demonstrate other results including an asymmetry in the optical absorption profile for monochromatic light incident on a core electron state in the single impurity system and bound states in continuum (BIC) for the superlattice system with two impurities.Item Spatially-homogeneous Vlasov-Einstein dynamics(2008-12) Okabe, Takahide; Morrison, Philip J.The influence of matter described by the Vlasov equation, on the evolution of anisotropy in the spatially-homogeneous universes, called the Bianchi cosmologies, is studied. Due to the spatial-homogeneity, the Einstein equations for each Bianchi Type are reduced to a set of coupled ordinary differential equations, which has Hamiltonian form with the metric components being the canonical coordinates. In the vacuum Bianchi cosmologies, it is known that a curvature potential, which comes from the symmetries of the three-dimensional Lie groups, determines the basic properties of the evolution of anisotropy. In this work, matter potentials are constructed for Vlasov matter. They are obtained by first introducing a new matter action principle for the Vlasov equation, in terms of a conjugate pair of functions, and then enforcing the symmetry to obtain a reduction. This yields an expression for the matter potential in terms of the phase space density, which is further reduced by assuming cold streaming matter. Some vacuum Bianchi cosmologies and Type I with Vlasov matter are compared. It is shown that the Vlasov-matter potential for cold streaming matter results in qualitatively distinct dynamics from the well-known vacuum Bianchi cosmologies.Item Topics in Lagrangian and Hamiltonian fluid dynamics : relabeling symmetry and ion-acoustic wave stability(1998) Padhye, Nikhil Subhash, 1970-; Morrison, Philip J.Relabeling symmetries of the Lagrangian action are found for the ideal, compressible fluid and magnetohydrodynamics (MHD). These give rise to conservation laws of potential vorticity (Ertel's theorem) and helicity in the ideal fluid, cross helicity in MHD, and a conservation law for an ideal fluid with three thermodynamic variables. The symmetry that gives rise to Ertel's theorem is generated by an infinite parameter group, and leads to a generalized Bianchi identity. The existence of a more general symmetry is also shown, with dependence on time and space derivatives of the fields, and corresponds to a family of conservation laws associated with the potential vorticity. In the Hamiltonian formalism, Casimir invariants of the noncanonical formulation are directly constructed from the symmetries of the reduction map from Lagrangian to Eulerian variables. Casimir invariants of MHD include a gauge-dependent family of invariants that incorporates magnetic helicity as a special case. Novel examples of finite dimensional, noncanonical Hamiltonian dynamics are also presented: the equations for a magnetic field line flow with a symmetry direction, and Frenet formulas that describe a curve in 3-space. In the study of Lyapunov stability of ion-acoustic waves, existence of negative energy perturbations is found at short wavelengths. The effect of adiabatic, ionic pressure on ion-acoustic waves is investigated, leading to explicit solitary and nonlinear periodic wave solutions for the adiabatic exponent r = 3. In particular, solitary waves are found to exist at any wave speed above Mach number one, without an upper cutoff speed. Negative energy perturbations are found to exist despite the addition of pressure, which prevents the establishment of Lyapunov stability; however the stability of ion-acoustic waves is established in the KdV limit, in a manner far simpler than the proof of KdV soliton stability. It is also shown that the KdV free energy (Benjamin, 1972) is recovered upon evaluating (the negative of) the ion-acoustic free energy at the critical point, in the KdV approximation. Numerical study of an ion-acoustic solitary wave with a negative energy perturbation shows transients with increased perturbation amplitude. The localized perturbation moves to the left in the wave-frame, leaving the solitary wave peak intact, thus indicating that the wave may be stable.