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Physical Review A

Semiclassical Electrodynamics of Bound Systems in Intense Fields

Howard R. Reiss

Published Mar 1, 1970• 140 citations
Publisher

Abstract

A general approximation method is developed for the treatment of bound quantum systems in the presence of an intense external electromagnetic field of long wavelength. The method is based upon a unitary transformation which approximately removes the electromagnetic field from the problem. The accuracy of the technique increases with the number of photons involved in any given process; and in the case of large photon number, the results have a simple, concise form resembling first-order perturbation theory. It is possible to handle intense-field problems with small numbers of photons also, but then the formalism takes on a complexity analogous to the corresponding perturbation expressions for the same number of photons. The theory is extended to the case where two electromagnetic field modes are present, one or both of which may be intense. Bound-bound transitions in a hydrogenlike atom are calculated as an illustration of the application of the method. One of the explicit nonlinear effects to appear is an intensity-dependent deviation of the transition probability behavior from ${I}^{n}$ for $n$-photon transitions, where $I$ is the field intensity. It is demonstrated that intensity effects are much more probable with bound electrons than with free electrons.

DOI
10.1103/physreva.1.803

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Physical Review Letters•Oct 26, 1970•65 citations
A general approximation method is developed for the treatment of bound quantum systems in the presence of an intense external electromagnetic field of long wavelength. The method is based upon a unitary transformation which approximately removes the electromagnetic field from the problem. The accuracy of the technique increases with the number of photons involved in any given process; and in the case of large photon number, the results have a simple, concise form resembling first-order perturbation theory. It is possible to handle intense-field problems with small numbers of photons also, but then the formalism takes on a complexity analogous to the corresponding perturbation expressions for the same number of photons. The theory is extended to the case where two electromagnetic field modes are present, one or both of which may be intense. Bound-bound transitions in a hydrogenlike atom are calculated as an illustration of the application of the method. One of the explicit nonlinear effects to appear is an intensity-dependent deviation of the transition probability behavior from In{I}^{n}In for nnn-photon transitions, where III is the field intensity. It is demonstrated that intensity effects are much more probable with bound electrons than with free electrons.
An approximation method is developed which yields very simple and general analytical results for electromagnetic bound-bound or bound-free transitions for any arbitrarily high order of interaction with the electromagnetic field. The approximation is perturbative in the sense that transition probabilities are proportional to the field intensity raised to the power NNN, where NNN is the order of the interaction. Limiting values of electromagnetic field strength are given within which the approximation is valid. These limits depend upon the binding potential which produces the bound states. A feature of the results is that they show the onset of high-order dominance effects which can occur at high-field intensities.
The quantum-electrodynamic frequency shifts of a two-level system in spontaneous emission and in an intense quantized radiation field are given up to eighth order in the coupling parameter, using the recursive perturbation method developed previously by one of the authors. The first six orders of the latter agree with the semiclassical result of Shirley and thus disagree with Chang and Stehle's calculation. The eighth-order term agrees with a recent semiclassical calculation by Ahmad and Bullough. The transition from the quantum mechanical case (the number of photons small) to the classical case (the number of photons very large) is also exhibited in our approach.
A new approach to the study of the spontaneous emission of the quantum system driven by a high-intensity laser field is developed. This approach is based on the accurate consideration of quantum system interaction with vacuum quantized field modes in the first order of perturbation theory, while the intense laser field is considered classically beyond the perturbation theory which allows to observe any-order stimulated processes governed by classical field. The proposed approach is applied to the study of a number of quantum systems in intense laser field. The obtained data are compared with those obtained in the frames of semiclassical approximation typically used for analyzing of the strong-field dynamic. It is found that the applicability of the semiclassical approach is strictly limited. It is valid for calculation of transitions to the initially populated state only if the population of this state is close to unity during the pulse and in the after-pulse regime. If its population is depleted, the semiclassical approach fails.
We study the general problem of matter-laser interaction within the framework of nonrelativistic quantum electrodynamics, with particular emphasis on strong laser field effects. Consequently, we formulate a well-defined approximation leading in a straightforward manner toward the conventional semiclassical mode of description. Namely, we arrive naturally to two coupled equations of motion: (i) the Schr\"odinger equation which governs the quantum dynamics of an atomic system driven by classically described radiation field (composed of an incoming laser pulse plus radiation emitted from the atom), and (ii) the classical Maxwell wave equation which describes the emission of radiation from the mentioned atomic source. Employing the formalism of adiabatic Floquet theory, we derive a simple criterion of validity of the just described semiclassical approach. It shows that the semiclassical treatment is justified in most situations. On the other hand, it turns out that the semiclassical approximation breaks down completely in certain special but realistic cases, regardless of the fact that the incoming laser pulse contains a huge number of photons. Under such special circumstances, we anticipate new effects arising due to the quantized nature of the radiation field, to be observable, for example, in harmonic generation spectra. Our considerations are illustrated more explicitly using a simple model of a two-level atom strongly driven by a laser. The quantum dynamics of this model problem is resolved within the framework of quantum electrodynamics while adopting well-defined and physically justifiable approximations. As an outcome, analytic formulas are found serving as a quantitative criterion of (non)applicability of the semiclassical approach and demonstrating the breakdown of semiclassical theory under well-defined conditions. An illustrative numerical calculation is provided.
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.
Results valid for arbitrarily high intensity of electromagnetic field are presented for induced emission from an excited state of an atom. Comparison with perturbation theory shows that perturbative calculations fail quantitatively and qualitatively at lower intensity than had been expected. Distinctive intense-field features are discussed such as vanishing of the transition probability for certain intensities and photon multiplicities, and appearance of two extrema of the transition amplitude for multiphoton processes at very high intensities.