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Physical Review A

Exact summation of higher-order terms in multiphoton processes

Yves Gontier, N. K. Rahman, M. Trahin

Published Dec 1, 1976• 90 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.14.2109

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Physical Review A•Jan 1, 1977•27 citations
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.
It is shown how certain classes of problems dealing with the interaction of radiation with atoms can be formulated in terms of four-operator perturbation series. The case of an isolated atom coupled to a single-mode field is discussed in detail within the resolvent formalism. The process where no photon is effectively absorbed by the atom is described, to all orders, in terms of a series involving four operators which simultaneously increase and/or decrease the photon number and the orbital quantum number. This series is exactly summed and the summation technique is generalized to single-atom, two-field and to two-atom, single-field systems exhibiting zero, one, and two net photon absorptions or emissions. The theory is applied to induced emission of a single photon from a pair of two-level atoms coupled to a monomode radiation field. The energy diagrams, the time dependence of the probability, and the resonance curves are discussed for systems where (i) both atoms are in the upper state and (ii) atom 1 is in the upper state and atom 2 is in the lower state at t=0. In comparing these results to the ones obtained with a single atom, we find that the important intensity-dependent correlations give rise to trapping effects even when the atoms are initially in the upper state.
The problem of calculating transition probabilities for an atom interacting with an intense radiation field is set up within the context of an effective Hamiltonian formalism. This has the virtue that resonance effects are properly accounted for at the outset; more generally, it provides a convenient framework for approximations. Through the restriction of the class of intermediate states to be included in the construction of the effective potential, a unitary (nonperturbative) model is obtained. A successive approximation procedure for determining the effective potential, which is based on continued-fraction representations, is described. While very likely intractable for the true problem, this successive approximation procedure can be implemented, for the model effective potential, with the aid of a generalized Rayleigh-Ritz procedure described here. The level-shift matrix is constructed from the effective potential; it determines the transition probabilities, linewidths, and energy-level displacements induced by the field. We derive variational expressions of the Kohn type for each of the elements of the model level-shift matrix.
Analyses of the resonant multiphoton ionization of atoms require knowledge of ac Stark energy shifts and of multiphoton, bound-to-bound state, transition amplitudes. In this paper, we consider the three-photon photoionization of hydrogen atoms at frequencies that are at and surrounding the two-photon 1s to 2s resonance. AC energy shift sums of both the 1s and 2s states are calculated as a function of the laser frequency along with two-photon 1s ? 2s resonant transition amplitude sums. These quantities are calculated using an extended version of a method, which has often been employed in a variety of ways, of calculating these sums by expressing them in terms of solutions to a variety of differential equations that are derived from the different sums being evaluated. We demonstrate how exact solutions are obtained to these differential equations, which lead to exact evaluations of the corresponding sums. A variety of different cases are analysed, some involving analytic continuation, some involving real number analysis and some involving complex number analysis. A dc Stark sum calculation of the 2s state is carried out to illustrate the case where analytic continuation, pole isolation and pole subtraction are required and where the calculation can be carried out analytically; the 2s state, ac Stark shift sum calculations involve a case where no analytic continuation is required, but where the solution to the differential equation produces complex numbers owing to the finite photoionization lifetime of the 2s state. Results from these calculations are then used to calculate three-photon ionization probabilities of relevance to an analysis of the multiphoton ionization data published by Kyrala and Nichols (1991 Phys. Rev. A 44, R1450).
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.
Excitation of an atomic system subjected to a slowly varying external electromagnetic field is studied using a two-level model. Time evolution of the system is found using an approach which is nonperturbative in the field strength. There is no constraint to small values of the applied field, that is, the field (in appropriate energy units) need not be small compared to the difference in energies of the two levels. Rather, we use the fact that the situation of interest to us is where the frequency of the exciting field is small compared to the frequency associated with the level difference. Transition probabilities and resonance conditions are found which circumscribe both the large- and small-field limits. In the weak-field limit the previous results of high-order perturbation theory are readily recovered. For a monochromatic field the characteristic features of resonance excitation at high harmonic number of the applied field are (a) extremely narrow resonance widths and (b) shifts in resonance positions which are strong functions of field intensity. Because of this sensitivity, we are able to demonstrate that when slow temporal evolution of the field amplitude is taken into account (e.g., due to finite pulse duration) the appropriate mean excitation rate is that due to the uncorrelated contribution of many resonances. The results of this analysis are used to estimate excitation rates in a specific atomic system, Cd12+{\mathrm{Cd}}^{12+}Cd12+, which are then compared to multiphoton-ionization rates. Our calculations suggest that the ionization rate exceeds the excitation rate by several orders of magnitude.
Expressions for energy-level shifts and transition amplitudes of atoms interacting with an incoming radiation field are developed in a completely quantum theoretic way and the results are compared with others. The continued-fraction expressions reproduce both those of Swain and of Gontier et al., and the numerical results for the particular case of Bloch-Siegert shifts in mercury vapor agree with Stenholm's but go further in giving the shape of the resonances. These results are also obtained by a method of direct diagonalization of the Hamiltonian of the coupled system.