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arXiv:physics/0502016 · Physical Review A

Lippmann-Schwinger description of multiphoton ionization

Igor A. Ivanov, Anatoli S. Kheifets

Published Apr 14, 2005• 8 citations
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Abstract

We outline a formalism and develop a computational procedure to treat the process of multiphoton ionization (MPI) of atomic targets in strong laser fields. We treat the MPI process nonperturbatively as a decay phenomenon by solving a coupled set of the integral Lippmann-Schwinger equations. As basic building blocks of the theory, we use a complete set of field-free atomic states, discrete and continuous. This approach should enable us to provide both the total and differential cross sections of MPI of atoms with one and two electrons. As an illustration, we apply the proposed procedure to a simple model of MPI from a square-well potential and to the hydrogen atom.

DOI
10.1103/physreva.71.043405

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We outline a formalism and develop a computational procedure to treat the process of multiphoton ionization (MPI) of atomic targets in strong laser fields. We treat the MPI process nonperturbatively as a decay phenomenon by solving a coupled set of the integral Lippmann-Schwinger equations. As basic building blocks of the theory, we use a complete set of field-free atomic states, discrete and continuous. This approach should enable us to provide both the total and differential cross sections of MPI of atoms with one and two electrons. As an illustration, we apply the proposed procedure to a simple model of MPI from a square-well potential and to the hydrogen atom.
An approximate non-perturbative theory of multiphoton ionization of a hydrogen atom by two strong radiation fields is outlined. The special case is then considered for when one of the fields is of low intensity and high frequency and the other field is of high intensity and low frequency. A set of representative calculations has been performed for this case; that is, (a) angular distributions for given numbers of absorbed and emitted low-frequency photons and different geometries; (b) total cross-sections versus the strong-field intensity; (c) angular distributions summed over all the low-frequency photon exchanges, showing a fourfold symmetry in a geometry in which each ionization channel exhibits, instead, only a twofold symmetry; (d) the average energy gained by the ejected electrons. Calculations are performed for a strong low-frequency field of arbitrary polarization. In many respects, the present calculations are expected to complement those of single-field multiphoton ionization. It is hoped that the results of the calculations reported here may prompt systematic measurements also in this new class of processes, as the predictions are well within the state of the art of present multiphoton-ionization experiments.
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.
A general framework for description of quantum systems in strong external fields is established. The most important application of the theory is concerned with the multiphoton ionization of atoms and molecules. The adiabatic approximation is developed. The Hamiltonian of the system is diagonalized by a time-dependent unitary transformation, which generates an interaction energy vanishing for very slow fields. A general analysis of the photoionization rates is performed. A specific model is considered: one-dimensional motion in the δ-function potential and external harmonic electric field. This model allows the exact solution for time-dependent eigenvalues and eigenfunctions. These solutions are compared with the conventional free electron approach to the ionization states.
A simple, "exactly solvable" atomic model will be presented that helps to discuss some of the interesting features of multiphoton ionization of atoms. The electromagnetic field is treated non-perturbatively, it can be strong. The atomic electrons are assumed to be non-interacting (Hartree model). They are bound in a harmonic oscillator potential (one- or three-dimensional). At the ionization energy, the potential is cut off and the free electron has no Coulomb attraction. The Schrodinger equation has exact analytic solutions valid in the inner or outer regions. If the field is turned on adiabatically, the exact solution can be found by matching these solutions at the boundary; both of their value and the normal derivative at all times. Thus the machine calculation reduces to repeated application of fast Fourier transforms and to the solution of linear equations (unfortunately a lot of them).
The most important results of study of multiphoton ionization of atoms are reviewed. Discussions are presented of specific features of the process, theoretical methods of description of ionization of atoms in a strong light field, the basic principles of design of experiments in which intense laser radiation is used, results of investigation of direct and resonance processes of multiphoton ionization, and the mutual relation between multiphoton processes (multiphoton excitation and ionization) and various phenomena leading to perturbation of the spectrum of bound states of an electron in an atom under the influence of a strong light field.