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Journal of Physics B Atomic Molecular and Optical Physics

Quasiclassical dipole matrix elements for high atomic states and stochastic dynamics of hydrogen atoms in microwave fields

B. Kaulakys

Published Feb 14, 1991• 13 citations
Publisher

Abstract

Analytical expressions for dipole matrix elements between high atomic states of nonhydrogenic and perturbed hydrogenic atoms are derived. The theoretical investigation is fulfilled for discrete and continuum atomic states and for the one-dimensional atom. It is shown that the quantal transition probabilities between perturbed atomic states depend on the nature of perturbation, e.g. matrix elements for the hydrogenic atom in an electromagnetic field differ from those of the nonhydrogenic atom. The results obtained generalize and increase the accuracy of some known expressions for the nonhydrogenic atom and extend them for the perturbed hydrogenic atom. Further, the analysis of classical dynamics and the derivation of mapping equations of motion for the one-dimensional atom in a harmonic field is fulfilled. The relationship between the energy change of the classical atom in a harmonic field and expressions for dipole matrix elements is revealed and analysed.

DOI
10.1088/0953-4075/24/3/013

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Analytical expressions for dipole matrix elements between high atomic states of nonhydrogenic and perturbed hydrogenic atoms are derived. The theoretical investigation is fulfilled for discrete and continuum atomic states and for the one-dimensional atom. It is shown that the quantal transition probabilities between perturbed atomic states depend on the nature of perturbation, e.g. matrix elements for the hydrogenic atom in an electromagnetic field differ from those of the nonhydrogenic atom. The results obtained generalize and increase the accuracy of some known expressions for the nonhydrogenic atom and extend them for the perturbed hydrogenic atom. Further, the analysis of classical dynamics and the derivation of mapping equations of motion for the one-dimensional atom in a harmonic field is fulfilled. The relationship between the energy change of the classical atom in a harmonic field and expressions for dipole matrix elements is revealed and analysed.
Quantum transition amplitudes between photonic states for the one-dimensional model of a hydrogenic Rydbcrg atom in a microwave field arc obtained. For this purpose the reduced Hamiltonian and the correspondence principle for transitions between strongly coupled states are used. The result reveals the resonance nature of the atom-har-monic-field interaction. The recently discovered phenomenon of quantum limitation of chaos in classically chaotic systems [1-3] has aroused an additional interest in experimental and theoretical investigations of a hydrogen atom in a microwave field. Much attention has been focused upon the high frequency , regime, when the microwave field frequency to is greater than the classical orbital frequenc [4—8]y . It is just in this case that a quantum limitation of the classical diffusion is observed. The new experimental results [4,5] support the idea that quantum effects suppress classical chaos, however until now there is no evidence for quantum localization by interference effects. The detailed quantum description is complicated and, therefore, is not quite suitable for the interpretation of experimental results. On the other hand, widely used model descriptions and considerable simplifications [1-3] in explaining the phenomena of quantum interference and chaos localization often lack some important features of the system. As a result, there exist various additional explanations on classical chaos limitation by quantum effects [6-8]. From our point of view an adequate quasi-classical theory containing the main quantum effects is needed. Such a theory would be of help in interpreting the experiments and in explaining the existing discrepancies between classical and quantum descriptions. In this paper the analytic quasi-classical expression for amplitudes of transitions between photonic states of the one-dimensional atom in a harmonic field has been obtained. It is believed that this result will be helpful for those engaged in the investigation of quantum chaos. Our consideration is based on a strong coupling correspon-1991 by Allcrton Press, Inc. dence principle [9] which gives the semi-classical transi-tion amplitude. A well-known transformation from an autonomous Hamiltonian in a -dimensional phase space to a new Hamiltonian in a restricted phas e space with dimensions depending periodically on a new effec-tive time parameter [10] has also been used. The Hamil-tonian of the one-dimensional model of a hydrogen atom in an external microwave field of amplitude F and fre-quency is given by [10]
The ionization of hydrogen in strong microwave fields is a fundamental problem in atomic physics and nonlinear dynamics. Transitions between dressed states take place simultaneously at several resonance frequencies when a strong static field is added parallel to a linearly polarized microwave field of comparable strength. We present a quantum-mechanical treatment of this intricate problem which reproduces its distinctive features very well. A clear picture of the roles of the fields emerges through an innovative mapping of the perturbation Hamiltonian. This mapping also has the computational advantage of collapsing the infinite sums over off-shell matrix elements into simpler on-shell expressions.
This chapter begins with a discussion of radiation by a microscopic classical oscillator with radiative damping. It then turns to the Schrödinger equation for the hydrogen atom, and an extended analysis of the solutions to the Schrödinger equation for the hydrogen atom in polar coordinates is then presented. Real orbitals are constructed from linear combinations of the complex Schrödinger equation solutions. The significance of real orbitals for molecular structure and the consequent introduction of rotating and vibrating dipoles are then introduced. A semiclassical model (quantal atom and classical E-M field) for the absorption and emission of radiation is developed and compared to the Einstein A and B coefficients. It is shown how the time-dependence of the radiatively coupled non-stationary states of the hydrogen atom are analogous to the classical harmonic oscillator. The essentials of atomic transitions, as well as molecular rotational, vibrational, and electronic spectroscopy complete the chapter.
We develop practical formulas for the calculation of the matrix elements of the interaction of the electromagnetic field with an atomic state, beyond the long-wavelength approximation. The atom-plus-field Hamiltonian is chosen to have the multipolar form, containing the electric, paramagnetic, and diamagnetic operators. The final workable expressions include the interactions to all orders and are derived by first expanding the fields in partial waves. The electric-field operator reaches a constant value as the radial variable becomes large, contrary to the result of the electric-dipole approximation (EDA) where the value of the corresponding operator increases indefinitely. Applications are given for Rydberg states of hydrogen up to n=50n=50n=50 and for free-free transitions in a Coulomb potential. Such matrix elements are relevant to a number of real and virtual processes occurring during laser-atom interactions. The computation is done numerically, using a combination of analytic with numerical techniques. By comparing the results of the EDA with those of the exact treatment, it is shown that the former is inadequate in such cases. This finding has repercussions on the theory and understanding of the physics of quantum systems in high-lying Rydberg levels and wave packets or in scattering states.
The quantum manifestations of chaos in the hydrogen atom in a generalized van der Waals potential, which includes the celebrated quadratic-Zeeman-effect problem under an appropriate limiting condition, are studied in detail. Using group-theoretical methods, we derive the matrix elements in an algebraic form. As the Hamiltonian is scale invariant, we introduce an appropriate scaling parameter. By considering the unperturbed hydrogen-atom problem, we demonstrate that suitable changes in the scaling parameter stabilize different parts of the spectrum depending upon the parameter's value. For the present generalized van der Waals potential problem, we utilize this property of the scaling parameter effectively to improve the convergence of eigenvalues while diagonalizing the matrices for various parametric values. Then, we vary one of the system parameters in the range [0,3], study the level statistics, and observe a GOE-Poisson-Brody-Poisson-Brody-Poisson-GOE-type (where GOE means Gaussian orthogonal ensemble) of transition regime hitherto unidentified in any of the perturbed hydrogen-atom problems. Our results are not only in agreement with random-matrix-theory predictions but also justify classical and semiclassical investigations.