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Physical Review A

Variational approximation for two-color ionization

Fei Zhou, Leonard Rosenberg

Published Sep 1, 1991• 15 citations
Publisher

Abstract

A variational procedure is developed for the calculation of the multiphoton-ionization rate of an atom interacting with a superposition of two time-dependent electric fields, one weak and the other intense and slowly varying. Trial functions are chosen that take into account the effect of both the atomic binding potential and the external fields. The variational expression contains, as its leading term, an approximation for the transition amplitude that was derived previously. The correction term is carefully analyzed here with regard to its dependence on suitably defined frequency and intensity parameters. The analysis provides firmly based estimates of the order, relative to the leading term, of the error in the approximation. A sum rule is derived which, owing to the cancellation of higher-order correction terms, allows for an accurate representation, in a relatively simple form, of the total transition rate summed over the net number of photons exchanged with the low-frequency field. A modified final-state trial function is presented which may be used, without loss of accuracy, to account for the existence of scattering resonances with energies close to that of the emergent electron. The calculational procedure is tested numerically on a one-dimensional model problem; the results are consistent with the more formal analysis of the relative order of the correction term and the accuracy of the sum rule.

DOI
10.1103/physreva.44.3270

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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
A variational procedure is developed for the calculation of the multiphoton-ionization rate of an atom interacting with a superposition of two time-dependent electric fields, one weak and the other intense and slowly varying. Trial functions are chosen that take into account the effect of both the atomic binding potential and the external fields. The variational expression contains, as its leading term, an approximation for the transition amplitude that was derived previously. The correction term is carefully analyzed here with regard to its dependence on suitably defined frequency and intensity parameters. The analysis provides firmly based estimates of the order, relative to the leading term, of the error in the approximation. A sum rule is derived which, owing to the cancellation of higher-order correction terms, allows for an accurate representation, in a relatively simple form, of the total transition rate summed over the net number of photons exchanged with the low-frequency field. A modified final-state trial function is presented which may be used, without loss of accuracy, to account for the existence of scattering resonances with energies close to that of the emergent electron. The calculational procedure is tested numerically on a one-dimensional model problem; the results are consistent with the more formal analysis of the relative order of the correction term and the accuracy of the sum rule.
In the final state of a multiphoton-ionization process the ejected electron may interact simultaneously with the external laser field and with the residual atomic system. A procedure for calculating transition probabilities for such processes is described which incorporates the effects of resonances in the final continuum states of the atom. By using standard methods of scattering theory, it is shown how these resonant states may be separated out leaving a "background" amplitude which is a smooth function of the energy, and which therefore lends itself more readily to numerical evaluation. Such an evaluation may be based on the generalized Lippmann-Schwinger integral equation presented here. The constraint imposed by unitarity on the background amplitude is derived, and it is pointed out how approximations may be set up which satisfy unitarity automatically. A simplified model is described in order to illustrate this and other features of the theory and to make contact with some earlier work on this subject. Finally, a low-frequency approximation is developed in which final-state interaction effects are accounted for in a relatively simple analytical form.
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.
Excitation of an atomic system subjected to a slowly varying external electromagnetic field is studied using a two-level model. Time evolution of the system is found using an approach which is nonperturbative in the field strength. There is no constraint to small values of the applied field, that is, the field (in appropriate energy units) need not be small compared to the difference in energies of the two levels. Rather, we use the fact that the situation of interest to us is where the frequency of the exciting field is small compared to the frequency associated with the level difference. Transition probabilities and resonance conditions are found which circumscribe both the large- and small-field limits. In the weak-field limit the previous results of high-order perturbation theory are readily recovered. For a monochromatic field the characteristic features of resonance excitation at high harmonic number of the applied field are (a) extremely narrow resonance widths and (b) shifts in resonance positions which are strong functions of field intensity. Because of this sensitivity, we are able to demonstrate that when slow temporal evolution of the field amplitude is taken into account (e.g., due to finite pulse duration) the appropriate mean excitation rate is that due to the uncorrelated contribution of many resonances. The results of this analysis are used to estimate excitation rates in a specific atomic system, Cd12+{\mathrm{Cd}}^{12+}Cd12+, which are then compared to multiphoton-ionization rates. Our calculations suggest that the ionization rate exceeds the excitation rate by several orders of magnitude.
The Keldysh approximation to treat the multiphoton ionization of atoms is reconsidered. It is shown that, if one consistently uses the hypothesis under which the approximation should be valid (essentially, that of a weak, short-range binding potential), a Keldysh-like term results as an approximation to the first term of a uniformly convergent series in powers of the binding potential. No cancellation occurs when higher-order terms are taken into account. This result allows one to consider the Keldysh approximation as a well-defined theoretical model, without implying, however, that it is adequate to describe multiphoton ionization of real atoms.
A successive-approximation procedure for calculating multiphoton-transition amplitudes is presented. The theory, designed for use in the strong-field domain where ordinary perturbative methods are unsuitable, provides an extension of previous work along these lines by inclusion of the effects of free-free transitions, i.e., interactions with the field which take the atomic system from one continuum state to another. This approach is based on the introduction of an effective Hamiltonian which operates only in the subspace of states, degenerate or nearly degenerate, among which real transitions can take place. Contributions from virtual intermediate states are included in the structure of the effective potential. This approach provides the framework for the construction of models, defined by the approximations made for the effective potential, which build in the constraints of unitarity and which properly account for the effects of intermediate-state resonances. The present work is focussed on those new features of the problem which arise from the fact that both the electron and the residual atom to which it was originally bound can interact with the field in asymptotic states. To avoid complications associated with long-ranged Coulomb effects, the residual atom is assumed to be neutral. (The initial state is a negative ion.) An approximation procedure is described for calculating the field-modified asymptotic states appropriate for such a system. By way of an illustrative example a low-frequency approximation is given which leads to considerable simplification of the calculational procedure.
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).