Physical Review A

Two-photon transitions in atomic hydrogen: An alternative approach

R. B. Thayyullathil, K. J. Prasanna

Published Apr 1, 1994• 7 citations

Abstract

We present an efficient analytic approach for the evaluation of two-photon transition rates in a hydrogen atom. The method is a variation of the Dalgarno and Lewis method [Proc. R. Soc. London Ser. A 233, 70 (1955)] for the evaluation of the second-order Stark effect in a hydrogen atom.

DOI: 10.1103/physreva.49.2432

Related papers

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

R. B. Thayyullathil, Rajesh Radhakrishnan, M Seema

Journal of Physics A Mathematical and GeneralJul 22, 20039 citations

Studies on Non-Resonant Multiphoton Processes in Atomic Hydrogen

Rajesh Radhakrishnan, Thiyagarajan babu

Jul 1, 20080 citations

Two-photon decay of hydrogenic atoms

J. H. Tung, X. M. Salamo, F. T. Chan

Physical Review ASep 1, 198472 citations

Two-photon absorption of atomic hydrogen from two light beams

J. H. Tung, A. Z. Tang, Greg J. Salamo +1 more

Journal of the Optical Society of America BJun 1, 198637 citations

Nonresonant multiphoton ionization in atomic hydrogen

Rajesh Radhakrishnan, R. B. Thayyullathil

Physical Review AMar 22, 200412 citations

Comment on analytical calculations of two-photon transition amplitudes for the hydrogen atom

Крыловецкий Александр Абрамович, Nikolai L. Manakov, S. I. Marmo

It is shown that analytical results for amplitudes of two-photon transitions from an arbitrary bound state of the hydrogen atom published in two recent papers in this journal reduce to an essentially simpler form that considerably simplifies both analysis and numerical evaluations.

Journal of Physics B Atomic Molecular and Optical PhysicsJan 25, 20056 citations

Physical Review A

Two-photon transitions in atomic hydrogen: An alternative approach

R. B. Thayyullathil, K. J. Prasanna

Published Apr 1, 1994• 7 citations

Publisher

Abstract

DOI: 10.1103/physreva.49.2432

Related papers

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

R. B. Thayyullathil, Rajesh Radhakrishnan, M Seema

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.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.

Journal of Physics A Mathematical and GeneralJul 22, 20039 citations

Studies on Non-Resonant Multiphoton Processes in Atomic Hydrogen

Rajesh Radhakrishnan, Thiyagarajan babu

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.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.

Jul 1, 20080 citations

Two-photon decay of hydrogenic atoms

J. H. Tung, X. M. Salamo, F. T. Chan

The two-photon decay mode of hydrogenic atoms from an arbitrary state ( ${n}_{1}$ , ${l}_{1}$ , ${m}_{1}$ ) to an arbitrary state ( ${n}_{2}$ , ${l}_{2}$ , ${m}_{2}$ ) is studied within the framework of nonrelativistic quantum mechanics. In the dipole approximation, these decay rates, which involve infinite summation over intermediate states, are derived exactly via a general second-order matrix element obtained by Kelsey and Macek and an implicit technique introduced by Dalgarno and Lewis. The results are expressed in terms of hypergeometric functions. For transitions ${n}_{1}s\ensuremath{\rightarrow}{n}_{2}s$, our results reduce to those of Klarsfeld whose starting point is the Coulomb Green's function. For transitions to the ground state, an alternative expression involving a simple one-dimensional integral is presented. The decay rate of the $2s$ metastable state of atomic hydrogen is calculated as an illustration of the method. The result, $\frac{1}{\ensuremath{\tau}}=8.2284$ ${\mathrm{sec}}^{\ensuremath{-}1}$, agrees with Klarsfeld. For transitions of ${n}_{1}s\ensuremath{\rightarrow}1s$ and ${n}_{1}d\ensuremath{\rightarrow}1s ({n}_{1}\ensuremath{\ge}3)$, the transition rates exhibit interesting and unexpected structures. In particular, "zeros" are found in the two-photon emission spectrum indicating that two-photon emission is not possible at certain frequencies. Physically, these "zeros" are the result of destructive interference between the radiating dipole terms associated with the sum over intermediate states. In addition to the emission spectrum the expected coincidence signal between two detectors monitoring the two photons simultaneously emitted during a two-photon transition is calculated as a function of the angle between the detectors. The angular distribution for the ${n}_{1}d\ensuremath{\rightarrow}1s$ transitions is shown to be significantly different from the ${n}_{1}s\ensuremath{\rightarrow}1s$ transitions. Finally, a possible experiment is suggested to test the results presented in this paper.The two-photon decay mode of hydrogenic atoms from an arbitrary state (${n}_{1}$, ${l}_{1}$, ${m}_{1}$) to an arbitrary state (${n}_{2}$, ${l}_{2}$, ${m}_{2}$) is studied within the framework of nonrelativistic quantum mechanics. In the dipole approximation, these decay rates, which involve infinite summation over intermediate states, are derived exactly via a general second-order matrix element obtained by Kelsey and Macek and an implicit technique introduced by Dalgarno and Lewis. The results are expressed in terms of hypergeometric functions. For transitions ${n}_{1}s\ensuremath{\rightarrow}{n}_{2}s$, our results reduce to those of Klarsfeld whose starting point is the Coulomb Green's function. For transitions to the ground state, an alternative expression involving a simple one-dimensional integral is presented. The decay rate of the $2s$ metastable state of atomic hydrogen is calculated as an illustration of the method. The result, $\frac{1}{\ensuremath{\tau}}=8.2284$ ${\mathrm{sec}}^{\ensuremath{-}1}$, agrees with Klarsfeld. For transitions of ${n}_{1}s\ensuremath{\rightarrow}1s$ and ${n}_{1}d\ensuremath{\rightarrow}1s ({n}_{1}\ensuremath{\ge}3)$, the transition rates exhibit interesting and unexpected structures. In particular, "zeros" are found in the two-photon emission spectrum indicating that two-photon emission is not possible at certain frequencies. Physically, these "zeros" are the result of destructive interference between the radiating dipole terms associated with the sum over intermediate states. In addition to the emission spectrum the expected coincidence signal between two detectors monitoring the two photons simultaneously emitted during a two-photon transition is calculated as a function of the angle between the detectors. The angular distribution for the ${n}_{1}d\ensuremath{\rightarrow}1s$ transitions is shown to be significantly different from the ${n}_{1}s\ensuremath{\rightarrow}1s$ transitions. Finally, a possible experiment is suggested to test the results presented in this paper.

Physical Review ASep 1, 198472 citations

Two-photon absorption of atomic hydrogen from two light beams

J. H. Tung, A. Z. Tang, Greg J. Salamo +1 more

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.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.

Journal of the Optical Society of America BJun 1, 198637 citations

Nonresonant multiphoton ionization in atomic hydrogen

Rajesh Radhakrishnan, R. B. Thayyullathil

The infinite summations over the complete set of unperturbed atomic states appearing in the $N$ th-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.The infinite summations over the complete set of unperturbed atomic states appearing in the $N$th-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.

Physical Review AMar 22, 200412 citations

Comment on analytical calculations of two-photon transition amplitudes for the hydrogen atom

Крыловецкий Александр Абрамович, Nikolai L. Manakov, S. I. Marmo

Journal of Physics B Atomic Molecular and Optical PhysicsJan 25, 20056 citations

The two-photon decay mode of hydrogenic atoms from an arbitrary state (

{n}_{1}

{l}_{1}

{m}_{1}

) to an arbitrary state (

{n}_{2}

{l}_{2}

{m}_{2}

) is studied within the framework of nonrelativistic quantum mechanics. In the dipole approximation, these decay rates, which involve infinite summation over intermediate states, are derived exactly via a general second-order matrix element obtained by Kelsey and Macek and an implicit technique introduced by Dalgarno and Lewis. The results are expressed in terms of hypergeometric functions. For transitions ${n}_{1}s\ensuremath{\rightarrow}{n}_{2}s$, our results reduce to those of Klarsfeld whose starting point is the Coulomb Green's function. For transitions to the ground state, an alternative expression involving a simple one-dimensional integral is presented. The decay rate of the

2s

metastable state of atomic hydrogen is calculated as an illustration of the method. The result, $\frac{1}{\ensuremath{\tau}}=8.2284$ ${\mathrm{sec}}^{\ensuremath{-}1}$, agrees with Klarsfeld. For transitions of ${n}_{1}s\ensuremath{\rightarrow}1s$ and ${n}_{1}d\ensuremath{\rightarrow}1s ({n}_{1}\ensuremath{\ge}3)$, the transition rates exhibit interesting and unexpected structures. In particular, "zeros" are found in the two-photon emission spectrum indicating that two-photon emission is not possible at certain frequencies. Physically, these "zeros" are the result of destructive interference between the radiating dipole terms associated with the sum over intermediate states. In addition to the emission spectrum the expected coincidence signal between two detectors monitoring the two photons simultaneously emitted during a two-photon transition is calculated as a function of the angle between the detectors. The angular distribution for the ${n}_{1}d\ensuremath{\rightarrow}1s$ transitions is shown to be significantly different from the ${n}_{1}s\ensuremath{\rightarrow}1s$ transitions. Finally, a possible experiment is suggested to test the results presented in this paper.

The infinite summations over the complete set of unperturbed atomic states appearing in the

N

th-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.