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Physical Review A

Systematic study of1s-nsand1s-ndtwo-photon transitions of hydrogenlike atoms

Viorica Florescu, Suzana Patrascu, O. Stoican

Published Sep 1, 1987• 26 citations
Publisher

Abstract

We present the results of a numerical study of the invariant amplitudes of 1s-ns and 1s-nd two-photon transitions in hydrogenlike atoms in the nonrelativistic dipole approximation, based on compact analytic formulas. We report also results valid in the limit n\ensuremath{\rightarrow}\ensuremath{\infty}. The 1s-nd transitions with n>6 are studied for the first time. For other particular cases our numerical results are in agreement with previous calculations in the literature. The investigated energy region of the two-photon spectrum is extended. The position of the transparencies in the second half of the spectrum is found to be almost n independent.

DOI
10.1103/physreva.36.2155

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We present the results of a numerical study of the invariant amplitudes of 1s-ns and 1s-nd two-photon transitions in hydrogenlike atoms in the nonrelativistic dipole approximation, based on compact analytic formulas. We report also results valid in the limit n\ensuremath{\rightarrow}\ensuremath{\infty}. The 1s-nd transitions with n>6 are studied for the first time. For other particular cases our numerical results are in agreement with previous calculations in the literature. The investigated energy region of the two-photon spectrum is extended. The position of the transparencies in the second half of the spectrum is found to be almost n independent.
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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.
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).