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 A

Linear response of a nonrelativistic hydrogenlike atom to a single-mode radiation field. II. Electric dipole approximation: General formalism

Tudor A. Marian

Published Nov 1, 1997• 3 citations
Publisher

Abstract

The interaction of a low-intensity laser field with a nonrelativistic one-electron atom is considered to the first order of perturbation theory and in the electric dipole approximation. The radiation field is turned on adiabatically modifying the initial unperturbed atomic state, which is either an angular-momentum eigenstate $|\mathrm{nlm}〉$ or a Stark state $|{\mathrm{nn}}_{e}m〉$. The first-order correction to the wave function is expressed both in the length and velocity gauges in terms of a vector function called the linear-response vector and depending on the field-free energy eigenstate. We derive in a unifying manner the linear-response vectors in the position representation, as closed-form contour integrals, starting from a unique generating function built up with the Coulomb Green function. The linear-response vectors are then evaluated in momentum space via a Fourier transformation: they are obtained as integral representations and also in explicit form, as generalized hypergeometric functions. With reference to the static limit, we complement earlier results and find the reduced linear-response vectors in momentum space, as Fourier transforms of their coordinate-representation counterparts. The low-frequency behavior of the length-gauge first-order correction to the ground-state wave function is established in coordinate as well as in momentum space. We finally point out the high-frequency limit of the linear response in the velocity gauge.

DOI
10.1103/physreva.56.3988

Related papers

First-order perturbed wave functions for the hydrogen atom in a harmonic uniform external electric field

Viorica Florescu, Tudor A. Marian

We investigate the linear response of a nonrelativistic hydrogenlike atom to a harmonic uniform electric field, which is switched on adiabatically. For the electron initially in an arbitrary bound \ensuremath{\Vert}nlm〉 state, the first-order correction to the wave function has been written in terms of a vector function ${\mathrm{w}}_{\mathrm{nlm}}$. We report an exact closed-form expression of the vector ${\mathrm{w}}_{\mathrm{nlm}}$ that we have derived using the Coulomb Green's function. Our general analytic results prove to be useful in a systematic study of two-photon processes involving the hydrogen atom.

Physical Review A•Dec 1, 1986•34 citations

Bound-bound two-photon transition matrix elements for the hydrogen atom in the dipole approximation

Tudor A. Marian

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, $_{1}\mathrm{F}_{\mathrm{E}}$ and $_{2}\mathrm{F}_{\mathrm{E}}$, 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.

Physical Review A•Apr 1, 1989•11 citations

Stimulated radiative corrections in hydrogen in the presence of a strong laser field

Edmond Arnous, J. Bastian, A. Maquet

The stimulated radiative corrections (or ac-Stark shift, or light shift...) of the energy levels of a hydrogenic atom irradiated by an intense, nonresonant, monomode laser field, are evaluated within the framework of the time-dependent perturbation theory, up to and including fourth order. The calculation is performed by using a Sturmian representation of the Coulomb Green's function. Whenever it is possible, comparison is made with other results, and several limiting cases are considered. If the laser frequency $\ensuremath{\omega}\ensuremath{\rightarrow}0$, one recovers the results of the dc-Stark effect. If $2\ensuremath{\omega}$ becomes larger than the ionization energy of the atomic state considered, the fourth-order level shift acquires an imaginary part which may be connected, via an extension of the optical theorem, to the two-photon ionization cross section. We point out also the difficulties encountered, when trying to get sensible estimates from partial summation of the infinite sums entering the perturbative expression of the fourth-order amplitudes. Finally, the order of magnitude of these various corrections is discussed.

Physical Review A•Feb 1, 1983•21 citations

Non-relativistic quantum electrodynamics for strong laser-atom interaction

Marco Frasca

A formulation of quantum electrodynamics is given that applies to atoms in a strong laser field by perturbation theory in a non-relativistic regime. Dipole approximation is assumed. The dual Dyson series, here discussed by referring it to the Birkhoff theorem for singularly perturbed linear differential equations, can be applied and a perturbation series obtained transforming the Hamiltonian by a Pauli-Fierz transformation. But, if just few photons are present high-order harmonics cannot be generated. So, it is proven that odd high-order harmonics only appear when the laser field is intense and one can substitute the creation and annihilation operators by the square root of the mean number of photons taken to be huge, the field retaining its coherency property as observed experimentally for harmonics. In this case, the Hamiltonian for perturbation theory comes to the Kramers-Henneberger form. The theory has a dipolar contribution when the free-electron quiver motion amplitude is larger than the atomic radius. For a Coulomb potential one has that the outer electron is periodically kicked, and so a prove is given that the same should happen to Rydberg atoms in intense microwave fields. The distribution representing the kicking has a Fourier series with just odd terms. Using a modified Rayleigh-Schrödinger perturbation theory, it is shown that under the same condition of validity of the quiver motion amplitude to atomic radius ratio, the atomic wave function is only slightly modified by the laser field due to the way the energy levels rearrange themselves. This gives a prove of stabilization in the limit of laser frequency going to infinity. Then, perturbation theory can be applied when the Keldysh parameter becomes smaller with respect to the shifted distance between the energy levels of the atom.

arXiv:quant-ph/0004002•Apr 1, 2000•0 citations

Radiative decay of hydrogenlike atoms in an electric field

Mark Hilley, Peter J. Mohr

Photon emission by an atom in the metastable $2S$ state in a weak static electric field is examined within the context of bound-state quantum electrodynamics. Low's method is applied to obtain the expansion in terms of Coulomb wave functions of the propagation function for an electron interacting with the nuclear Coulomb field, the applied electric field, and the quantized radiation field. For a weak applied field, and to lowest relative order in $\frac{\ensuremath{\alpha}}{\ensuremath{\pi}}$, the $2S$-state resonant contribution to the propagation function can be factorized to provide the basis for a calculation of the angular distribution of single-photon radiation and lifetime of the excited state. A numerical evaluation is given, which takes into account effects of electric dipole, magnetic dipole, and magnetic quadrupole emission, relativistic corrections, and finite level widths.

Physical Review A•Jan 1, 1980•50 citations

Non Resonant Two Photon Ionization of Atomic Hydrogen: Exact Expressions of Angular Distribution

M Faye, S T Wane, Seydi Ababacar Ndiaye

Within the framework of second order time dependent perturbation theory and non relativistic dipole approximation, the exact expressions of the radial transition amplitude from 3d, 3p and 3s excited states, were obtained from the integral representation of the Green Coulomb function. Below threshold and above one photon ionization threshold, it can be expressed as an analytic combination of three Appell hypergeometric functions of first order, one of which is degenerated; while at threshold, it can be obtained in terms of an expansion of Gauss confluent hypergeometric functions. The angular coefficients are numerically evaluated and analysed for some available laser wavelengths.

Physica Scripta•Jan 1, 2003•2 citations
The interaction of a low-intensity laser field with a nonrelativistic one-electron atom is considered to the first order of perturbation theory and in the electric dipole approximation. The radiation field is turned on adiabatically modifying the initial unperturbed atomic state, which is either an angular-momentum eigenstate ∣nlm〉|\mathrm{nlm}〉∣nlm〉 or a Stark state ∣nnem〉|{\mathrm{nn}}_{e}m〉∣nne​m〉. The first-order correction to the wave function is expressed both in the length and velocity gauges in terms of a vector function called the linear-response vector and depending on the field-free energy eigenstate. We derive in a unifying manner the linear-response vectors in the position representation, as closed-form contour integrals, starting from a unique generating function built up with the Coulomb Green function. The linear-response vectors are then evaluated in momentum space via a Fourier transformation: they are obtained as integral representations and also in explicit form, as generalized hypergeometric functions. With reference to the static limit, we complement earlier results and find the reduced linear-response vectors in momentum space, as Fourier transforms of their coordinate-representation counterparts. The low-frequency behavior of the length-gauge first-order correction to the ground-state wave function is established in coordinate as well as in momentum space. We finally point out the high-frequency limit of the linear response in the velocity gauge.
We investigate the linear response of a nonrelativistic hydrogenlike atom to a harmonic uniform electric field, which is switched on adiabatically. For the electron initially in an arbitrary bound \ensuremath{\Vert}nlm〉 state, the first-order correction to the wave function has been written in terms of a vector function wnlm{\mathrm{w}}_{\mathrm{nlm}}wnlm​. We report an exact closed-form expression of the vector wnlm{\mathrm{w}}_{\mathrm{nlm}}wnlm​ that we have derived using the Coulomb Green's function. Our general analytic results prove to be useful in a systematic study of two-photon processes involving the hydrogen atom.
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.
The stimulated radiative corrections (or ac-Stark shift, or light shift...) of the energy levels of a hydrogenic atom irradiated by an intense, nonresonant, monomode laser field, are evaluated within the framework of the time-dependent perturbation theory, up to and including fourth order. The calculation is performed by using a Sturmian representation of the Coulomb Green's function. Whenever it is possible, comparison is made with other results, and several limiting cases are considered. If the laser frequency $\ensuremath{\omega}\ensuremath{\rightarrow}0$, one recovers the results of the dc-Stark effect. If $2\ensuremath{\omega}$ becomes larger than the ionization energy of the atomic state considered, the fourth-order level shift acquires an imaginary part which may be connected, via an extension of the optical theorem, to the two-photon ionization cross section. We point out also the difficulties encountered, when trying to get sensible estimates from partial summation of the infinite sums entering the perturbative expression of the fourth-order amplitudes. Finally, the order of magnitude of these various corrections is discussed.
A formulation of quantum electrodynamics is given that applies to atoms in a strong laser field by perturbation theory in a non-relativistic regime. Dipole approximation is assumed. The dual Dyson series, here discussed by referring it to the Birkhoff theorem for singularly perturbed linear differential equations, can be applied and a perturbation series obtained transforming the Hamiltonian by a Pauli-Fierz transformation. But, if just few photons are present high-order harmonics cannot be generated. So, it is proven that odd high-order harmonics only appear when the laser field is intense and one can substitute the creation and annihilation operators by the square root of the mean number of photons taken to be huge, the field retaining its coherency property as observed experimentally for harmonics. In this case, the Hamiltonian for perturbation theory comes to the Kramers-Henneberger form. The theory has a dipolar contribution when the free-electron quiver motion amplitude is larger than the atomic radius. For a Coulomb potential one has that the outer electron is periodically kicked, and so a prove is given that the same should happen to Rydberg atoms in intense microwave fields. The distribution representing the kicking has a Fourier series with just odd terms. Using a modified Rayleigh-Schrödinger perturbation theory, it is shown that under the same condition of validity of the quiver motion amplitude to atomic radius ratio, the atomic wave function is only slightly modified by the laser field due to the way the energy levels rearrange themselves. This gives a prove of stabilization in the limit of laser frequency going to infinity. Then, perturbation theory can be applied when the Keldysh parameter becomes smaller with respect to the shifted distance between the energy levels of the atom.
Photon emission by an atom in the metastable 2S2S2S state in a weak static electric field is examined within the context of bound-state quantum electrodynamics. Low's method is applied to obtain the expansion in terms of Coulomb wave functions of the propagation function for an electron interacting with the nuclear Coulomb field, the applied electric field, and the quantized radiation field. For a weak applied field, and to lowest relative order in $\frac{\ensuremath{\alpha}}{\ensuremath{\pi}}$, the 2S2S2S-state resonant contribution to the propagation function can be factorized to provide the basis for a calculation of the angular distribution of single-photon radiation and lifetime of the excited state. A numerical evaluation is given, which takes into account effects of electric dipole, magnetic dipole, and magnetic quadrupole emission, relativistic corrections, and finite level widths.
Within the framework of second order time dependent perturbation theory and non relativistic dipole approximation, the exact expressions of the radial transition amplitude from 3d, 3p and 3s excited states, were obtained from the integral representation of the Green Coulomb function. Below threshold and above one photon ionization threshold, it can be expressed as an analytic combination of three Appell hypergeometric functions of first order, one of which is degenerated; while at threshold, it can be obtained in terms of an expansion of Gauss confluent hypergeometric functions. The angular coefficients are numerically evaluated and analysed for some available laser wavelengths.