Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review

Sum Rules for Atomic Transition Probabilities

John P. Vinti

Published Aug 15, 1932• 49 citations
Publisher

Abstract

Some new sum rules for atomic transition probabilities are developed; they are similar to the well-known rule of Kuhn, Reiche, and Thomas. They give, if unpolarized light is incident on any gaseous element in any given discrete energy level, the total absorption, excitation plus ionization, with various types of frequency weighting. The results are similar to those of Dirac and Harding, but are more general in that they apply to any discrete state of any atom, rather than to the ground state of hydrogen only. The method differs from that of Dirac and Harding.

DOI
10.1103/physrev.41.432

Related papers

The Thomas-Reiche-Kuhn sum rule for an atom of N equally spaced energy levels

C. Baxter

The Thomas-Reiche-Kuhn sum rule is determined for an atom consisting of an arbitrary number of discrete, equally spaced eigenenergy levels. This generalization is consistent with a two-level atom and with a usually seen form of the sum rule involving the atomic ground state, and resolves an apparent paradox reported elsewhere.

Journal of Physics B Atomic Molecular and Optical Physics•Dec 14, 1992•9 citations

Multiphoton Ionization of Atomic Hydrogen in the Ground State

Yves Gontier, M. Trahin

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{}.

Physical Review•Aug 5, 1968•133 citations

Thomas–Reiche–Kuhn (TRK) sum rule for interacting photons

Salvatore Savasta, Omar Di Stefano, Franco Nori

The Thomas-Reiche-Kuhn (TRK) sum rule is a fundamental consequence of the position-momentum commutation relation for an atomic electron, and it provides an important constraint on the transitionmatrix elements for an atom. Here, we propose a TRK sum rule for electromagnetic fields which is valid even in the presence of very strong light- matter interactions and/or optical nonlinearities. While the standard TRK sum rule involves dipole matrix moments calculated between atomic energy levels (in the absence of interaction with the field), the sum rule here proposed involves expectation values of field operators calculated between general eigenstates of the interacting light-matter system. This sum rule provides constraints and guidance for the analysis of strongly interacting light-matter systems and can be used to test the validity of approximate effective Hamiltonians often used in quantum optics.

De Gruyter eBooks•May 25, 2021•1 citation

Sum rules and expansion formula for Stark radiative transitions in the hydrogen atom

David R. Herrick

An expansion formula for Stark effect transition probabilities is derived in terms of field-free (spherical) transition probabilities. Expansion coefficients are squares of Clebsch-Gordan coefficients from the SO(4) = SU(2) \ifmmode\times\else\texttimes\fi{} SU(2) description of the hydrogen atom. The method provides new sum rules for Stark transitions which explain regularities in previous calculations. Accurate formulas for average Stark transition probabilities obtain from application of the method to an empirical model for field-free transition probabilities. The approximation permits estimation of Stark lifetimes in the large-$n$ regime.

Physical Review A•Nov 1, 1975•24 citations

Calculations of hydrogen atom multiphoton energy level shifts, transition amplitudes and ionization probabilities

Kenneth G. Whitney, C.S. Chang

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).

Journal of Physics B Atomic Molecular and Optical Physics•Jun 23, 2008•1 citation

Multiphoton ionization of the hydrogen atom

S. N. Haque, M. G. Mustafa

In this paper, we have presented the analytical and numerical calculations of the ionization rate (σ/I in units of cm4∙W−1) for the hydrogen atom in the ground state. The summation over infinite intermediate states, discrete plus continuous, has been carried out using our modified technique based on Bebb and Gold's concept of average frequency. Our method improves the results of Bebb and Gold at all frequencies, and the results obtained are in good agreement with the results of Gontier and Trahin, Chan and Tang, and Klarsfeld.

Canadian Journal of Physics•Sep 1, 1987•1 citation
Some new sum rules for atomic transition probabilities are developed; they are similar to the well-known rule of Kuhn, Reiche, and Thomas. They give, if unpolarized light is incident on any gaseous element in any given discrete energy level, the total absorption, excitation plus ionization, with various types of frequency weighting. The results are similar to those of Dirac and Harding, but are more general in that they apply to any discrete state of any atom, rather than to the ground state of hydrogen only. The method differs from that of Dirac and Harding.
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{}.
The Thomas-Reiche-Kuhn (TRK) sum rule is a fundamental consequence of the position-momentum commutation relation for an atomic electron, and it provides an important constraint on the transitionmatrix elements for an atom. Here, we propose a TRK sum rule for electromagnetic fields which is valid even in the presence of very strong light- matter interactions and/or optical nonlinearities. While the standard TRK sum rule involves dipole matrix moments calculated between atomic energy levels (in the absence of interaction with the field), the sum rule here proposed involves expectation values of field operators calculated between general eigenstates of the interacting light-matter system. This sum rule provides constraints and guidance for the analysis of strongly interacting light-matter systems and can be used to test the validity of approximate effective Hamiltonians often used in quantum optics.
An expansion formula for Stark effect transition probabilities is derived in terms of field-free (spherical) transition probabilities. Expansion coefficients are squares of Clebsch-Gordan coefficients from the SO(4) = SU(2) \ifmmode\times\else\texttimes\fi{} SU(2) description of the hydrogen atom. The method provides new sum rules for Stark transitions which explain regularities in previous calculations. Accurate formulas for average Stark transition probabilities obtain from application of the method to an empirical model for field-free transition probabilities. The approximation permits estimation of Stark lifetimes in the large-nnn regime.
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).
In this paper, we have presented the analytical and numerical calculations of the ionization rate (σ/I in units of cm4∙W−1) for the hydrogen atom in the ground state. The summation over infinite intermediate states, discrete plus continuous, has been carried out using our modified technique based on Bebb and Gold's concept of average frequency. Our method improves the results of Bebb and Gold at all frequencies, and the results obtained are in good agreement with the results of Gontier and Trahin, Chan and Tang, and Klarsfeld.