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Physical Review A

Higher-Order Effects in Resonant Multiphoton Processes

Yves Gontier, M. Trahin

Published Jun 1, 1973• 47 citations
Publisher

Abstract

The multiphoton transitions in the hydrogen atom are investigated by taking into account the higher-order terms arising from the $S$-operator expansion. These higher-order effects originate induced shifts and induced breadths on the virtual-atomic-energy levels. It is shown that these perturbations play an important role for resonant multiphoton processes. In this case, the theory predicts a particular flux dependence of the transition rates. Furthermore, the well-known dispersion curves are strongly modified in the neighborhood of any resonance. Thus the transition rates can exhibit several peaks. Numerical results are given for a three-photon bound-free transition from the $2S$ state of hydrogen.

DOI
10.1103/physreva.7.1899

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The multiphoton transitions in the hydrogen atom are investigated by taking into account the higher-order terms arising from the SSS-operator expansion. These higher-order effects originate induced shifts and induced breadths on the virtual-atomic-energy levels. It is shown that these perturbations play an important role for resonant multiphoton processes. In this case, the theory predicts a particular flux dependence of the transition rates. Furthermore, the well-known dispersion curves are strongly modified in the neighborhood of any resonance. Thus the transition rates can exhibit several peaks. Numerical results are given for a three-photon bound-free transition from the 2S2S2S state of hydrogen.
Within the framework of time-dependent perturbation theory, general expressions are derived for the cross section of multiphoton processes induced by radiation acting on a hydrogen atom. Bound-bound transitions involving emission of a single quantum and bound-free transitions are calculated for a wide range of photon energies. In the case of resonance, the transition probability is also discussed. The procedure used to evaluate the sums over intermediate states is described. The numerical computation allows one to compare the results obtained for various competing processes.
I have obtained a general analytic result for calculating any higher-order correction to the transition amplitude for NNN-photon absorption by writing the time-dependent perturbation theory in the SSS-matrix form. In obtaining this result, I have made use of the concept of average energy as has been defined by Bebb and Gold. A formal expression has been obtained for the upper bound of the higher-order correction and the results are discussed for the case of the hydrogen atom.
The summation of all the higher-order contributions to multiphoton absorption in atoms is presented. The nonperturbative expressions for the transition amplitudes are expressed in terms of continued fractions. They account for the whole perturbation originated by the radiation field on the atom but do not exhibit explicit expression for the level shift. In the case of radiation fields of high intensity the only reliable determination of the level shift involves a careful location of the poles of the resolvent operator. The theory has been tested in considering a two-level atom. An extremely rapid convergence of the continued fractions has been observed, and accurate results concerning high-order resonance shifts have been obtained. The expressions for the transition amplitudes being nonperturbative with regard to the field strength, they can be considered as analytic continuations of the perturbation series. This provides a useful approach to the problem of ultraintense fields.
The present paper applies the method of high-intensity quantum electrodynamics developed earlier to several problems of current experimental interest. In this paper the intensitydependent shifts and widths of atomic levels are calculated and applied to a given transition near resonance. The technique is demonstrated with three examples: the multiphoton ionization of atomic cesium and hydrogen in atomic beams and third-harmonic production in an atomic vapor. It is found that the multiphoton-ionization rate in a relatively weak laser beam (of order 108{10}^{8}108 W/cm2{\mathrm{cm}}^{2}cm2) is determined by the intensity-dependent level width (6f6f6f configuration) as well as the interference of the fine-structure states, and that the rates for hydrogen, in a much more intense beam (of order 1012{10}^{12}1012 W/cm2{\mathrm{cm}}^{2}cm2) are determined by both the intensity-dependent widths and shifts as‘\stackrel{\mathrm{`}}{\mathrm{s}}s‘ well as the wave-function renormalization constants. The agreement of these two examples of calculations with the experiments done recently is satisfactory. For third-harmonic production the role of two-photon resonance is emphasized, and the dependence of the generated third-harmonic intensity on the intensity and frequency of the fundamental is discussed. Near resonance, the ratio of third-harmonic intensity to fundamental intensity saturates.
A successive-approximation procedure for calculating multiphoton-transition amplitudes is presented. The theory, designed for use in the strong-field domain where ordinary perturbative methods are unsuitable, provides an extension of previous work along these lines by inclusion of the effects of free-free transitions, i.e., interactions with the field which take the atomic system from one continuum state to another. This approach is based on the introduction of an effective Hamiltonian which operates only in the subspace of states, degenerate or nearly degenerate, among which real transitions can take place. Contributions from virtual intermediate states are included in the structure of the effective potential. This approach provides the framework for the construction of models, defined by the approximations made for the effective potential, which build in the constraints of unitarity and which properly account for the effects of intermediate-state resonances. The present work is focussed on those new features of the problem which arise from the fact that both the electron and the residual atom to which it was originally bound can interact with the field in asymptotic states. To avoid complications associated with long-ranged Coulomb effects, the residual atom is assumed to be neutral. (The initial state is a negative ion.) An approximation procedure is described for calculating the field-modified asymptotic states appropriate for such a system. By way of an illustrative example a low-frequency approximation is given which leads to considerable simplification of the calculational procedure.