Quantum Non-integrability

The following sections are included:

• PREFACE

• CONTENTS

The behaviors of energy levels and eigenfunctions for parametrized Hamiltonians are studied. We review the codimensions for degeneracies for general parameter spaces, thereby establishing the degree to which specific degeneracies are generic. We then present a scheme for systematically investigating the dependences of energy eigenvalues and eigenfunctions on the topological structure of the parameter space, and examine in detail how this structure affects degeneracies in the system. The relationship of the degeneracies to the twistings of eigenfunctions and their topological characterization is demonstrated; in particular, we show how topological indices can knot energy bands together. We also discuss the metamorphoses of the knotting patterns and the corresponding twists in the eigenfunctions, and obtain general rules which govern these changes. Some necessary mathematical concepts and results in algebraic topology are reviewed.

In this article, a theory which we have recently developed is reviewed. This theory addresses the long standing problem of quantum-classical correspondence. It begins with the axiomatic structure of quantum mechanics, from which the basic concept of dynamical degrees of freedom emerges and the associated physical geometry of an arbitrary quantum system is constructed. Such a geometrical space possesses both complex and symplectic structures, the two basic ingredients to guarantee the rigorous correspondence between quantum and classical kinematics. Also, it is remarkable that the quantum-classical correspondence of dynamics can be mediated by a quenched index obtained from the geometry. The intermediate stage from quantum to classical dynamics is described by the semiquantal dynamics. The main property of the semiquantal dynamics is that it manifests the effect of quantum fluctuation on classical trajectories, and quantum fluctuation depends explicitly on the quenched index. This offers a way to study “chaos in quantum systems”. We also prove a theorem which links dynamical symmetry and quantum integrability, and conclude that “quantum chaos” is related to dynamical symmetry breaking. The generic rule of quantum chaos is explored. Throughout the article, many basic quantum systems are used to illustrate these important points.

The analysis is done for the quantum properties of systems that possess dynamical chaos in classical limit. Two main topics are considered: (i) the problem of quantum macroscopical description of the system and the Ehrenfest-Einstein problem of the validity of the classical approximation; and (ii) the problem of levels spacing distribution for the nointegrable case. For the first topic the method of projecting on the coherent states base is considered and the ℓn1/ħ time for the quasiclassical approximation breaking is described. For the second topic the discussion of GOE and non-GOE distributions is done and estimations and simulations for the non-GOE case are reviewed.

We review the ionization of highly excited (Rydberg) atoms by microwave fields, emphasizing the hydrogen and helium atom experiments performed at Stony Brook during the past few years. We discuss some of the hydrogen data that have covered the wide range of scaled frequencies between 0.02-2.8 with principal quantum numbers n₀ in the range 24-90 and a number of discrete frequencies ω/2π between 7.58-36.02 GHz. Hydrogen experiments with two-frequency driving, as well as single-frequency helium experiments, have focused thus far on low scaled frequencies. Comparisons of experimental results for 3D hydrogen atoms with classical and quantal theoretical calculations, even when carried out only for a 1D model atom, have shown how the dynarnical behavior of the strongly driven hydrogen atom may be classified into six regimes of the scaled frequency. In several of these regimes we make detailed comparisons with theoretical calculations and with some experimental results from other laboratories. Specifically, we evaulate the validity or lack thereof of various scaling laws that have appeared in the literature. Much of the interest in the periodically driven hydrogen atom stems from ionization being associated classically in most of these regimes with the onset of irregular or chaotic trajectories. Though the strongly driven atom is certainly a quantal system, perturbative quantal treatments are usually inapplicable. Understanding its nonlinear classical dynamics has given powerful insight into the complicated, rich behavior of a strongly-driven, simple atomic system whose low-dimensional classical counterpart is chaotic.

The classical dynamics of two electrons moving in the presence of a single, fixed, nucleus is largely chaotic. What is the effect of this on the spectra of two-electron atoms? Regular sequences of bound states and resonances are seen experimentally to quite high quantum numbers. Scattering cross sections are smooth, with well understood threshold effects in both theory and experiment. Where are the chaotic levels and/or chaotic scattering regimes? Or, are atomic systems too quantum-like to show any manifestations of the underlying classical dynamics? To investigate these questions, and the corresponding problem of semiclassical quantization, we have investigated the classical and quantum dynamics of a model two-electron atom wherein each electron is confined to one-dimensional motion on opposite sides of the nucleus. Periodic orbits have been enumerated, and a simple coding found (as has been independently done by Kim and Ezra, Richter and Wintgen). This allows application of the Gutzwiller trace formula to relate classical and quantum spectra. Fully quantum computations have been carried out using the method of complex coordinates to allow variational computation of resonances. A regime is found, beginning at approximately the n = 10 threshold, where the resonance eigenvalues behave like a two-dimensional gas, and a Pechukas-type theory of their motion is developed. This is one analogue of classical chaos. Further, near n = 30, estimates indicate that the typical spacing of the real parts and the average magnitude of the imaginary parts of the “gas” of resonances are approximately equal. In this regime we predict the onset of quantum chaotic scattering.

The driven Morse oscillator is a common model used in the studies of many molecular dynamical processes, including photodissociation, vibrational overtone excitation, dissociation of van der Walls molecules, and surface desorption of admolecules. We review here the classical and quantum dynamics of this system, of which the essential features are universal to many driven anharmonic oscillators. Our emphasis here is on the role of phase space structures on threshold dissociation dynamics, sensitive dependence of dissociation on field parameters, chaotic scattering, and the possibility of stabilizing collisional complexes by external fields. Symmetry is shown to play a key role in the organization of periodic orbits, resonance islands, and homoclinic orbits. Quantum studies are reviewed in light of the classical results. Experimental relevance of these results is also closely addressed.

Quantum chaotic scattering is a new subfield of quantum chaos which focusses on the common features underlying such diverse subjects as, e.g., Ericson fluctuations in nuclear physics and conductance fluctuations in solid state physics. With the help of a physical example (the “Dipole Model”) quantum chaotic scattering is introduced and its pertinent features are discussed. Examples for chaotic scattering taken from various areas of physics and physical chemistry demonstrate importance and universality of quantum chaotic scattering. The connection to random matrix theory is made. It is shown that the quantum scattering matrix in the classically chaotic parameter regime resembles a random matrix drawn from Dyson’s circular ensemble in the limit ћ → 0. Physical scattering potentials are studied in detail and a transition from regular to chaotic and back to regular quantum scattering as a function of the incident energy is presented. Several experimental setups are proposed which may provide an experimental test for the theoretical predictions.

The present status and future problems in both the classical-level theory and full quantum theory of nuclear collective dynamics are discussed by putting special emphasis on their relation to the classical and quantum order-to-chaos transition dynamics, respectively. The nonlinear dynamics between the collective and single-particle excitation modes of motion specific for the finite, self-sustained and self-organizing system as the nucleus is discussed within the time-dependent Hartree-Fock (TDHF) theory, the basic equation of which is shown to be formally equivalent to the Hamilton’s canonical equations of motion in the classical nonlinear dynamical system. An importance to relate the structure of the TDHF symplectic manifold with an inexhaustible rich structure of the classical phase space in the nonlinear system is stressed. A full quantum theory of nuclear collective dynamics is proposed under a dictation of what has been developed in the classical-level TDHF theory. It is shown that the proposed quantum theory enables us to explore exceeding complexity of the Hilbert space. It is discussed that a resonant denominator known as a source of the extraordinary rich structure of the phase space trajectories, also plays a decisive role in generating a rich structure of the quantum Hilbert space.