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Le Journal de Physique Colloques

HIGHER ORDER STIMULATED RADIATIVE CORRECTIONS IN HYDROGEN

A. Maquet

Published Nov 1, 1982• 0 citations
Publisher

Abstract

We present a calculation of the radiative shifts and broadenings of the energy levels of a hydrogenic atom, irradiated by a strong, non-resonant and monomode laser field. The calculation is performed within the framework of the timedependent perturbation theory, up to fourth-order included, using the sturmian representation of the Coulomb Green's function. The order of magnitude of these corrections as well as several interesting limiting cases are considered and discussed.

DOI
10.1051/jphyscol:1982239

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We present a calculation of the radiative shifts and broadenings of the energy levels of a hydrogenic atom, irradiated by a strong, non-resonant and monomode laser field. The calculation is performed within the framework of the timedependent perturbation theory, up to fourth-order included, using the sturmian representation of the Coulomb Green's function. The order of magnitude of these corrections as well as several interesting limiting cases are considered and discussed.
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.
Nous présentons un calcul des déplacements et élargissements radiatifs des niveaux d'énergies d'un atome d'hydrogène irradié par le rayonnement non-résonnant d'un laser intense monomode. Le calcul est effectué dans le cadre de la théorie des perturbations, jusqu'au quatrième ordre inclus, en utilisant une représentation sturmienne de la fonction de Green coulombienne. Les ordres de grandeurs de ces corrections ainsi que plusieurs cas limites intéressants sont considérés et discutés.
Second-order and fourth-order intensity-dependent corrections to the energy levels of the hydrogen atom are calculated. By comparing the results obtained for these two orders and for various values of the principal quantum number, it is seen that, at a given intensity, the second-order correction becomes more and more questionable as one approaches the ionization limit. It is shown how the divergences, which appear in calculating the matrix elements of the shift operator in a discrete basis of continuum functions, can be avoided by performing the computation in the velocity gauge. The relevant second-order numerical results indicate serious modifications of the continuum spectrum. Intrinsic limitations of the second-order calculations are discussed.
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.
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.
The dependence on the ellipticity of the incident laser field of the harmonics generated by atoms of hydrogen in the multiphoton regime is examined for two fundamental wavelengths, 532 and 1064 nm. The single atom response is calculated nonperturbatively using the Floquet method in conjunction with basis sets of complex Sturmian functions. The results are compared with those obtained using lowest order perturbation theory. Nonperturbative effects modify the strength and polarization of the harmonics at the highest intensities investigated, particularly, for the threshold harmonics and in the vicinity of Stark-shift-induced resonances. No anomalous dependence of the rate of harmonic generation on the ellipticity of the fundamental field is found.