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Journal of Physics A Mathematical and General

Three-photon transitions from ground state to bound states in atomic hydrogen

R. B. Thayyullathil, Rajesh Radhakrishnan, M Seema

Published Jul 22, 2003• 9 citations
Publisher

Abstract

In this paper, we present an efficient alternative method for the evaluation of the three-photon transition matrix element in the dipole approximation from the ground state to bound states in atomic hydrogen. This method is a variation of the Dalgarno–Lewis method for the treatment of the second-order Stark effect in the hydrogen atom. In this approach, the infinite double sum over the complete set of states including the continuum states present in the third-order perturbation theory result is treated exactly. The closed analytical expression obtained for the matrix element, as a function of incident photon energy, clearly displays all singularities present in the original third-order perturbation theory result.

DOI
10.1088/0305-4470/36/31/309

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In this paper, we present an efficient alternative method for the evaluation of the three-photon transition matrix element in the dipole approximation from the ground state to bound states in atomic hydrogen. This method is a variation of the Dalgarno–Lewis method for the treatment of the second-order Stark effect in the hydrogen atom. In this approach, the infinite double sum over the complete set of states including the continuum states present in the third-order perturbation theory result is treated exactly. The closed analytical expression obtained for the matrix element, as a function of incident photon energy, clearly displays all singularities present in the original third-order perturbation theory result.
We describe a unified approach for the determination of ac-Stark correction and Kramers-Heisenberg dispersion formula. In both cases the contribution from infinite intermediate summation appearing in the expression for the corresponding matrix elements are evaluated exactly in the dipole approximation for the ground state of hydrogen atom using a variation of the Dalgarno-Lewis method. The analytical expressions obtained can be efficiently used for the numerical evaluation of matrix element for all values of incident photon energy and comparison is made with results obtained by different methods.
The infinite summations over the complete set of unperturbed atomic states appearing in the NNNth-order perturbation theory for multiphoton ionization are performed using the Dalgarno-Lewis method. The relevant transition matrix elements are written in a closed integral form exhibiting all the analytic properties of the amplitude as a function of incident photon energy. The cross sections for two- and three-photon ionization for atomic hydrogen are calculated for both linearly and circularly polarized light with a wide range of photon energy spectrum including the near resonance, and numerical comparison is made with the values obtained by different methods.
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.
By use of the linear-response-vectors method, the Kramers-Heisenberg matrix elements for arbitrary bound-bound transitions of a nonrelativistic hydrogenlike atom have been evaluated. To illustrate the discussion, Rayleigh and Raman scattering have been selected as relevant examples of second-order processes. We have derived, in a unitary manner, A- and \ensuremath{\varphi}-gauge compact analytic formulas, both for angular momentum and Stark states. Our results are expressed in terms of two new generalized hypergeometric functions with four variables, 1FE_{1}\mathrm{F}_{\mathrm{E}}1​FE​ and 2FE_{2}\mathrm{F}_{\mathrm{E}}2​FE​, which are finite sums of Gauss hypergeometric functions. Special attention is given to the two-photon transitions involving the atomic ground state. For angular momentum states, we mention the connection that we have established between our general results and earlier ones.
Within the framework of nonrelativistic quantum mechanics, the analysis of the two-photon absorption by atomic hydrogen is generalized to the case of two incident beams with arbitrary direction and polarization (including circular polarization). In the dipole approximation, the second-order matrix elements responsible for two-photon absorption are transformed into a finite sum consisting of the product of a radial part and an angular part. Exact calculation of the angular part predicts that, for the s–s transition only, transparency is obtained regardless of the frequencies of the two beams with perpendicular polarizations. In addition, circularly polarized light is found to be more efficient (up to a factor of 1.5) for the s–d two-photon transition. The radial parts of the matrix elements are accurately evaluated by using the Coulomb Green’s function technique and an implicit technique of Dalgarno and Lewis. Their calculation indicates that “zeros” exist in the two-photon absorption spectrum, thereby predicting that absorption is not possible at certain frequencies. It should be noted that in the calculated spectrum, near or at resonance points, the linewidths of the intermediate levels must be included in order to obtain accurate results.