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Physical Review A

Classical and quantum mechanics of an integrable limit of the hydrogen atom in combined circularly polarized microwave and magnetic fields

M. J. Raković, T. Uzer, David Farrelly

Published Apr 1, 1998• 12 citations
Publisher

Abstract

The hydrogen atom interacting with a circularly polarized microwave field and a magnetic field perpendicular to the plane of polarization is, in general, a nonintegrable system that may exhibit strongly chaotic motion. However, two integrable limits are known, one of which is separable in elliptical coordinates. A detailed investigation of the classical and quantum mechanics of the separable limit is presented.

DOI
10.1103/physreva.57.2814

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The hydrogen atom interacting with a circularly polarized microwave field and a magnetic field perpendicular to the plane of polarization is, in general, a nonintegrable system that may exhibit strongly chaotic motion. However, two integrable limits are known, one of which is separable in elliptical coordinates. A detailed investigation of the classical and quantum mechanics of the separable limit is presented.
The interaction of a hydrogen or Rydberg atom with a circularly polarized microwave field leads to the creation of global equilibrium points that may be stable or unstable depending on the particulars of the applied field. The additional application of a magnetic field, perpendicular to the plane of polarization, can be used to manipulate both the nature and the stability of these points. We show that stable three-dimensional motion can be maintained that is localized at either a maximum or a minimum in the corresponding surface of zero velocity. At these equilibria, the zero-velocity surface may be locally quadratic in coordinates and stable harmonic-oscillator-like, nondispersive, coherent states can be supported. As the fields are varied, repeated order-chaos transitions may be observed, including various passages through rigorously integrable limits. Classical simulations are presented for a range of field strengths, and the effects of deviations of the microwave field from exactly circular polarization (i.e., elliptical polarization) are examined using a pulsating zero-velocity surface.
The dynamics of Rydberg states of atomic hydrogen perturbed simultaneously by a static electric field and a resonant microwave field of elliptical polarization is analyzed in the quantum perturbative limit of small amplitudes. For some configurations, the secular motion (i.e., evolution in time of the elliptical electronic trajectory) may be classically predominantly chaotic. By changing the orientation of the static field with respect to the polarization of the microwave field, one can modify the global symmetries of the system and break any generalized time-reversal invariance. This has a dramatic effect on the statistical properties of the energy levels.
We present the quantal dynamics of electronic wave packets in the hydrogen atom in the presence of circularly polarized microwave and magnetic fields in regions of classical stability. Whereas wave packets may disperse without the stabilizing influence of a magnetic field, in its presence stable motion can be maintained for wave packets that are localized either at global equilibria that are either a maximum or minimum in a zero-velocity surface. Because these extrema are locally harmonic their vacuum states are truly coherent states in the original sense of Schr\"odinger.
A classical study of the dynamics of atomic Rydberg states interacting with both elliptically and circularly polarized microwave fields is presented. In recent experiments, enhanced ionization rates were observed in elliptically as opposed to circularly polarized fields. A classical phase-space simulation provides good qualitative agreement with experimental findings. These results are explained in terms of the breakdown of almost conserved quantities as the polarization of the field is changed from circular to elliptical.
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]