The Fractal Structure In The Auto-ionization Of Atoms In External Fields

Atomic systems played a major role in the birth and growth of quantum mechanics. The dynamics of the highly excited Rydberg atoms in strong external fields has become a challenge topic and received considerable attention over the past years particularly due to their significant importance for studying quantum chaos. During the last 20 years it has been recognized that either the photo-absorption or ionization processes of Rydberg atoms in strong external static fields are intrinsically not separable for variables and unpredictable on the dynamic behaviors in a long time scale although perfectly deterministic. The corresponding chaotic dynamics can be observed only in a non-perturbative regime. Because atoms are simple microscopic low-dimensional systems that have to be described by quantum mechanics and compared with other microscopic complex systems (nuclei, atomic clusters, semiconductors, etc.), the atoms have the great advantage that all the basic components are well understood hence it is possible to write an explicit expression of the Hamiltonian. Thus, they are among the best available prototypes for studying quantum chaos. A major problem is that of understanding how the regular or chaotic behavior of the classical system is manifest in its quantum properties, especially in the semi-classical limit. This is the subject of quantum chaos. More precisely, typical questions at least we would like to answer are as follows:What kind of semiclassical approximations can be used if the system is unperturbative and is not separable?What is the long-term behavior of quantum system?Fortunately the closed-orbit theory based on the Gutzwiller's periodic orbit theory and trace formula of the quantum state density provided a unique bridge connecting the classical mechanics and the quantum world. It can be not only used to approach bounded system but also used suitably to deal with scattering problem in positive energy region. This closed-orbit theory together with recurrence spectroscopy of the scaled variables is actually the best method we know for extracting information from chaotic spectra, where a detailed analysis of the individual lines is not particularly enlightening. In last more than 20 years it achieved great successes in many physics realms. According to the news announced recently by PRA, two literatures of the pioneering work on the closed-orbit theory have been listed in the most cited ten articles from 1988. An extended semi-classical closed- orbit theory will play crucial role in this thesis. From a conceptual point of view, classical chaos can only exist in systems where different degrees of freedom are strongly coupled. This is a consequence of celebrated KAM theorem. Up to now, it is well known that core non- coulomb scattering and external applied magnetic field are the origins creating chaotic dynamics. Chaos or regularity manifests themselves during long-term behavior of the system, whereas atoms are only suitable for studying the transient chaos, that is, chaos on a finite time scale. The best way to clarify this issue is to try to analyze its fractal structure that embedded in the chaotic spectra. Fractals have become a fundamental ingredient of nonlinear dynamics and chaos theory since they were defined 1n the 1970s. Fractal structures appear naturally in dynamical systems, in particular associated with the phase space. The analysis of these structures is especially useful for obtaining information about the future behavior of complex system, since they provide fundamental knowledge about the relation between the behavior of these systems and uncertainty and indeterminism. In order to establish quantum-classical correspondence the temporal-resolved spectra are required, because it provides us with the clues for understanding quantum chaos, very few direct measurements have been done in chaotic regime. Ionization of hydrogen atoms in a microwave field is a first step in this direction, because it contains explicitly time dependence, but can also be reduced to a static problem by use of the Floquet theorem as a consequence of its time periodicity. On the other hand, the exchange of energy between the different degrees of freedom, which is usually difficult to measure in a microscopic multidimensional system, is equivalent here to the number of photons exchanged between the atom and the external field, a much more accessible quantity. A rough estimation of the energy exchange is the ionization probability of the atom, by far the easiest quantity to measure in an experiment. Therefore, studying of the ionization provides a simple tool for analyzing the electron dynamics. Experimentally, we can easily detect whether an atom is ionized after interaction with an applied field, whereas it is much more complicated to measure its internal state, although measuring the electron dynamics poses a difficult technological problem. An important development has been done in 1996 by Noordam, et al. The time dependent auto-ionization rate was measured for alkali-metal rubidium atom in a static electric field by a short pulsed laser. The experimental results demonstrate that ionization occurred via a train of electron pulses rather than by exponential decay as usual. Soon, Robicheaux and Shaw presented quantum calculations which emphasized on explaining the experimental results of Noordam, et al. A complete theoretical analysis on the chaos dynamics in autoionization process was retarded till 2004. Delos et al. predicted that somewhat similar electron pulses trains should occur for a hydrogen atom in parallel electric and magnetic fields. They revived the electron pulsed train entering detector, analyzed its fractal structure and constructed a theoretical skeleton for studying escape dynamics and chaotic properties in microscope atomic or molecular system. For example, as a basic theoretical model it can be used to the transport problem of particles through a microcavity and to the escape process of the optical wave or point particle from potential field. Moreover, the work has encouraged great interests on quantum chaos appear in atoms and molecules.Despite of these successes, however, obvious challenges remain especially for explaining the chaotic origin of the non-hydrogen atoms in external fields. Motivated by preceding developments we investigate the autoionization dynamics of alkali-metal lithium atom in external fields in this thesis. We will pay particularly much more attention on the long-term behavior of quantum system and classical-quantum correspondence. First considering in static electric field situation we calculate quantum autoionization rate using extended semiclassical closed-orbit theory including core short-range model potential and combinable recurrences which display the multiple scattering effect of core, the wave packet state of ejected electron and the electron flux arriving at the detector are gained thus we obtain escape-time plot of ionization rate. This spectrum is temporal domain meanwhile is spatial domain because the arriving time is function of electron initial launched angles. On the other hand, after examining the different spectra via changing the external parameters (external field strength or scaled energy) we found that the autoionization spectra are sensitive to the initial conditions. The analysis of classical chaos is realized by means of Poincaréreturn map, the problem is reduced to a two-dimensional phase space where exists a prominent homoclinic tangle arising from the transverse intersection of the stable and unstable manifold that start from the hyperbolic fixed point. We have testified it is the infinitely repeated transverse intersections of the stable manifold with the line of initial condition that leads to the"fractal"structure in the escape-time plot. Thus a direct connection between the ionization spectrum in general geometric space and the homoclinic tangle manifolds in phase space is found. It allows us to analyze the self-similar fractal structure that embodies approximately the quantum long–term behavior. Our result is valuable to improve understanding of quantum chaos.For further studying the chaotic properties of autoionization of atoms in external fields, we calculate the time- resolved auto-ionization spectrum of Lithium atoms in parallel electric and magnetic fields, and the chaotic behaviors are investigated semiclassically. In the escape-time spectrum of ionized electron, the relation between the fractal structure and classical trajectories concerned resonance is discussed. A general regularity about the geometric shapes of the classical trajectories surrounding the core region is given. We have particularly interested in the emergence of the new trajectories due to the core multiple-scattering effect which already is beyond closed-orbit theoryAs another primary factor creating chaotic dynamics, the influences of magnetic field on the chaotic electron pulsed train of Lithium atoms are undoubtedly very important. We explored the autoionization rate in different strength of the magnetic field in detail and found that as the strength of the magnetic field increases, ionizing pulses become more complicated, chaotic behaviors clearer, which suggests sensitive dependence of escaping trajectories on initial conditions. Indeed, it indicated also the system is intrinsically chaotic.My thesis is summarized as follows:The first chapter is summarization, which briefly introduces the research and development of atomic ionization, the purpose and the significance of the subject we choose and the main work we have done. The second chapter discusses the basic theory we used in this thesis, i.e., the closed orbit theory including core scattering, and we also introduces the model potential and the scaling transformation. In the third chapter, we calculate autoionization rate of Rydberg lithium atoms in a static electric field. Through the Poincarémap found a direct connection between the ionization spectrum in general geometric space and the homoclinic tangle manifolds in phase space is found. It will help us to study the quantum and classical correspondence. The forth chapter calculate the autoionization rate of lithium atoms in the parallel electric and magnetic fields, qualitative describe the shape of the ionizing trajectories from the nesting distinct fractal epistrophes is given. The escape time and the auto-ionization rate that exhibit a series of chaotic pulse train are discussed for different magnetic fields. In the last chapter, we briefly summarize the total subject and give an outlook for the future work.Atomic units are used throughout this paper unless specifically noted.