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Journal of Physics B Atomic Molecular and Optical Physics

A study of quantum dynamics in the classically chaotic regime

J G Leopold, D Richards

Published Jun 28, 1988• 23 citations
Publisher

Abstract

The authors examine the effect of the Coulomb singularity on the classical and semiclassical dynamics of the one-dimensional model for atomic hydrogen, sometimes called the surface-state-electron model, in a periodic electric field and determine a necessary condition, in terms of the field strength, frequency and the initial quantum number, for the violation of the uncertainty principle. They show that when this violation occurs the quantal transition probabilities are significantly less than the equivalent classical probabilities. This criterion is tested numerically and shown to provide a good estimate of the boundary to one of the regions in which classical dynamics fails. The authors also show that when the uncertainty principle is not violated the fluctuations of the quantal about the classical probability spectra can be related to structure in classical phase space. These fluctuations are due to different classical and quantal transport through cantori.

DOI
10.1088/0953-4075/21/12/007

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The authors examine the effect of the Coulomb singularity on the classical and semiclassical dynamics of the one-dimensional model for atomic hydrogen, sometimes called the surface-state-electron model, in a periodic electric field and determine a necessary condition, in terms of the field strength, frequency and the initial quantum number, for the violation of the uncertainty principle. They show that when this violation occurs the quantal transition probabilities are significantly less than the equivalent classical probabilities. This criterion is tested numerically and shown to provide a good estimate of the boundary to one of the regions in which classical dynamics fails. The authors also show that when the uncertainty principle is not violated the fluctuations of the quantal about the classical probability spectra can be related to structure in classical phase space. These fluctuations are due to different classical and quantal transport through cantori.
An analysis of the semiclassical regime of the quantum-classical transition is given for open, bounded, one-dimensional chaotic dynamical systems. Environmental fluctuations-characteristic of all realistic dynamical systems-suppress the development of a fine structure in classical phase space and damp nonlocal contributions to the semiclassical Wigner function, which would otherwise invalidate the approximation. This dual regularization of the singular nature of the semiclassical limit is demonstrated by a numerical investigation of the chaotic Duffing oscillator.
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 intrinsic multivaluedness of interaction process, revealed in Part I of this series of papers, is interpreted as the origin of the true dynamical (in particular, quantum) chaos. The latter is causally deduced as unceasing series of transitions, dynamically probabilistic by their origin, between the equally real, but incompatible 'realisations' (modes of interaction) of a system. The obtained set of realisations form the causally derived, intrinsically complete "space of events" providing the crucial extension of the notion of probability and the method of its first-principle calculation. The fundamental dynamic uncertainty thus revealed is specified for Hamiltonian quantum systems and applied to quantum chaos description in periodically perturbed systems. The ordinary semiclassical transition in our quantum-mechanical results leads to exact reproduction of the main features of chaotic behaviour of the system known from classical mechanics, which permits one to "re-establish" the correspondence principle for chaotic systems (inevitably lost in any their conventional, single-valued description). The causal dynamical randomness in the extended quantum mechanics is not restricted, however, to semiclassical conditions and generically occurs also in essentially quantum regimes, even though partial "quantum suppression of chaos" does exist and is specified in our description, as well as other particular types of the quantum (truly) chaotic behaviour.
Based on a quantum dynamical invariant of motion, I, we study the classical limit of a semiclassical Hamiltonian composed by a full quantum harmonic oscillator plus a classical particle plus a "semiclassical" coupling quartic term. The motion-invariant is closely related to the uncertainty principle. The classical limit (CL) is determined by the relationship between I and the total energy of the system, defining an adimensional invariant Er. We find that the CL coincides with the results of a purely classical treatment. Both invariants allow to follow the transit between quantum nonchaotic to the classical chaotic regime. Particularly, with Er we define the threshold above which chaos appears, and the interval during which both regimes co-exist.
How does the existence of chaos in classical dynamics affect the quantum spectrum of the system? The atom in a magnetic field is a prototype of systems where complete classical and quantal calculations are possible. Transition from regular to chaotic trajectories are linked quantally to the progressive breaking of some peculiar symmetries of the system. We propose a method and its experimental realization on lithium atom to clearly see a transition between regularity and chaos on a quantum spectrum by selecting the starting state of the excitation. In addition, experimental and calculated spectra on hydrogen shows a remarkably good agreement.