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Physical Review A

Effect of an intense radiation field on the discrete and continuum spectra of atomic hydrogen

Yves Gontier, M. Trahin

Published Aug 1, 1989• 21 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.40.1351

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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 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.
This paper contains a generalization of results described in previous work. A general formula is given for the total cross section for two-photon ionization of a hydrogenic state. An implicit method for evaluating the second-order radial matrix elements that occur in this expression is described in detail. Numerical results were obtained for the states with principal quantum numbers n = 1 through 5. It is concluded that, when both one-photon and two-photon ionization are energetically possible, the effect of the latter may be expressed as an intensity-dependent correction to the Gaunt factor which may therefore be written as G = g1 + ζIλ3g2. Here g1 is the usual Gaunt factor, I is the intensity of the light in W cm−2, λ is the wavelength in cm, g2 is a dimensionless factor of order unity, and ζ is a constant given by ζ = 0.1504 W−1 cm−1. Graphs of g2 are given as a function of electron energy. These results do not include the effect of three-photon processes, which can also contribute to the first-order intensity-dependent correction to the Gaunt factor as a result of interference between the first- and third-order amplitudes.
We formulate a computationally feasible method for calculating multiphoton ionization rates for atoms exposed to intense fields in the intensity regime where perturbation theory ceases to apply. The method is based on the time-independent picture of ionization, which starts with the Floquet ansatz. The question of the gauge of the radiation field is discussed in some detail. Various expressions for the ionization amplitude are derived from a variational principle, with the radiation field expressed in the velocity gauge. We consider in particular an approximation in which the wave vector developing from the initial state is replaced by a trial vector that is the Floquet expansion truncated just below the threshold for ionization, and in which the wave vector developing into the final state is replaced by a trial vector that is just the wave vector appearing in the Kroll-Watson low-frequency approximation in scattering theory. We have applied this approximation to hydrogen and we present some results for both nonresonant and resonant multiphoton ionization. We argue that the experimentally observed resonance structure in the above-threshold peaks of the ionization signal occurs through the electron jumping from one dressed-state energy-eigenvalue curve to another.
General equations are presented for determining the field-dependent variation of the wavefunctions, matrix elements and intensities of dipole transitions between Stark levels of a hydrogen atom in a field range where the fine-structure effects may be neglected. The Sturm-series resolution for the reduced Coulomb-Green function is used to derive analytical expressions for the first-order corrections in terms of parabolic quantum numbers of initial and final states. Simple forms of these expressions as functions of the upper-state quantum numbers are presented explicitly for the Lyman and Balmer series. The selective action of the dc electric field on the intensity of Stark components of hydrogen lines is revealed in numerical calculations for both and types of emitted radiation. This selectivity may cause, in particular, enhancement of the high-frequency and damping of the low-frequency Stark -lines of the Lyman series.
The partition function of a single hydrogen atom diverges. While the astrophysical solution to this dilemma is the introduction of a density dependent alteration of the partition function, the question of what happens in an arbitrarily rarefied gas remains open. It is argued that in this case blackbody radiation changes the atom’s spectrum. A unified derivation including these effects and the nonrelativistic Lamb shift and spontaneous emission is presented. Then recent numerical calculations performed by workers in the field of Rydberg atom spectroscopy are utilized to estimate the principal quantum number for which blackbody radiation significantly alters the spectrum.