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Physical Review A

Multiphoton Processes in a Hydrogen Atom

Yves Gontier, M. Trahin

Published Nov 1, 1971• 139 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.4.1896

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Physical Review A•Jan 1, 1973•12 citations
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.
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.
The general theory of multiphoton ionization of hydrogen atoms by simultaneous absorption of several photons is discussed. In the nonrelativistic dipole approximation, the calculations lead to a recursion relationship concerning angular functions and involve radial matrix elements containing many summations over intermediate states. A general method for exactly evaluating several sums is described in detail and applied to the photo-ionization of atomic hydrogen in the ground state. Numerical results are presented as dispersion curves of the transition rate $\frac{{\ensuremath{\sigma}}_{n}}{{I}^{N\ensuremath{-}1}}$ for a range of wavelengths from 1000 up to 7200 \AA{}.
We have discussed the problems of non linear interaction between electromagnetic radiations with atoms from semi-classical point of view. Time dependent Schrödinger equation for single electron system is solved by using perturbative technique to obtain transition probability. Higher order perturbation is also discussed which is used in multiple processes, in which two or more quanta are emitted instead of a single photon. The theory is based on assumption that the perturbation is small. From this transition probability ionization rate and absorption cross-section of hydrogen atom is calculated. Its variation with photon energy and field strength is analyzed which agrees very well with experimental observations.The Himalayan PhysicsVol. 3, No. 32012Page : 40-43
In the present thesis we have formulated the Dalgarno-Lewis procedure\nfor two-and three-photon processes and an elegant alternate expressions\nare derived. Starting from a brief review on various multiphoton processes\nwe have discussed the difficulties coming in the perturbative treatment\nof multiphoton processes. A small discussion on various available methods\nfor studying multiphoton processes are presented in chapter 2. These\ntheoretical treatments mainly concentrate on the evaluation of the higher\norder matrix elements coming in the perturbation theory. In chapter 3 we\nhave described the use of Dalgarno-Lewis procedure and its implimentation on second order matrix elements. The analytical expressions for twophoton\ntransition amplitude, two-photon ionization cross section, dipole\ndynamic polarizability and Kramers-Heiseberg are obtained in a unified\nmanner. Fourth chapter is an extension of the implicit summation technique\npresented in chapter 3. We have clearly mentioned the advantage of\nour method, especially the analytical continuation of the relevant expressions\nsuited for various values of radiation frequency which is also used for\nefficient numerical analysis.\nA possible extension of the work is to study various multiphoton processcs\nfrom the stark shifted first excited states of hydrogen atom. We\ncan also extend this procedure for studying multiphoton processes in alkali\natoms as well as Rydberg atoms. Also, instead of going for analytical expressions,\none can try a complete numerical evaluation of the higher order\nmatrix elements using this procedure.
Transition amplitudes corresponding to bound-bound multiphoton transitions with the emission or absorption of one and two Raman photons in the hydrogen atom have been evaluated following a recent method of Reiss. Results show that the transition probability varies smoothly for low values of NNN, where NNN is the number of intense photons absorbed or emitted during the transition. For larger values of NNN the amplitude, however, has oscillatory behavior. A particular feature of the present method of calculation is that we find the "reduced" transition amplitude for induced emission of two Raman photons is enhanced compared with single-Raman-photon emission. Another distinctive feature is the appearance of many more extrema in the transition amplitude for higher transitions for multiphoton processes at high intensity.