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Physical Review A

Exact solution of a realistic model for two-photon ionization

Brian Lee Beers, Lloyd Armstrong

Published Dec 1, 1975• 151 citations
Publisher

Abstract

A realistic model for resonant two-photon ionization of an atom is developed which can be solved exactly to give the ionization probability as a function of time, photon intensity, and a few atomic parameters. The exact solutions are compared to various commonly used perturbation results in order to ascertain the regions of validity of the perturbation calculations. It is shown that there are combinations of the parameters for which ionization rates cannot accurately describe the correct results, and that one commonly used perturbation development is not generally applicable to resonant two-photon ionization. Extension of the model to $(n+m)$-photon ionization having an $m$-photon resonance is discussed. The model is applied to two- and three-photon ionization in Cs.

DOI
10.1103/physreva.12.2447

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When an atom is driven by a laser field with a photon energy that is nearly resonant with an atomic transition, it can undergo periodic population transfers between the ground and excited state, known as Rabi oscillations [1], [2]. The aim of this work is to study photoionization from atoms for parameters accessible to a free-electron laser (FEL) [3]. Using time dependent perturbation theory to model photoionization from a Rabi oscillating atom, we have shown that in addition to the expected pathway of ionization from the excited state, non-resonant two-photon ionization will have a comparable amplitude at intensities on the order of 1013W/cm2[3].

Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)•Jun 26, 2023•0 citations
A realistic model for resonant two-photon ionization of an atom is developed which can be solved exactly to give the ionization probability as a function of time, photon intensity, and a few atomic parameters. The exact solutions are compared to various commonly used perturbation results in order to ascertain the regions of validity of the perturbation calculations. It is shown that there are combinations of the parameters for which ionization rates cannot accurately describe the correct results, and that one commonly used perturbation development is not generally applicable to resonant two-photon ionization. Extension of the model to (n+m)(n+m)(n+m)-photon ionization having an mmm-photon resonance is discussed. The model is applied to two- and three-photon ionization in Cs.
I present a perturbation theory of multiphoton ionization based on the resolvent operator. The theory is applied to a detailed study of two-photon ionization under near-resonance conditions. It is shown that a distinction must be made between linear level shifts and widths, and resonance shifts and widths which appear in the transition probability. The resonance shifts and widths exhibit saturation effects and their values tend to be much smaller than those of the linear level shifts and widths. Using numerical examples corresponding to tunable dye lasers and alkali-metal atoms, we show that the above shifts can have a dramatic effect on the ionization rate even when the laser is not in exact resonance with the intermediate level. The question of "one-step" vs "two-step" two-photon ionization is also discussed. It is shown that exact resonance does not necessarily imply a two-step process. In fact, in most cases it is quite the opposite. A quantitative criterion for this distinction is established. Finally, a distinction is made between instrumental saturation and saturation of the transition probability. The two are shown to be of different nature. Further extensions of the work are also discussed.
Detailed descriptions of one- and two-photon ionization are presented. We go beyond conventional perturbation theory to investigate the dynamics of both processes for a broad range of field intensities. The general theory of unstable decaying states is applied to the problem of one-photon ionization (the photoeffect). It is found that a non-trivial dependence of the transition dipole moment on the continuum state energy can lead to corrections to exponential decay, and even oscillations in the bound state population. As examples we consider the photoionization of hydrogen and atoms with autoionizing states. Only in the absence of autoionizing states corrections to exponential decay remain inconsequential for any experimentally relevant intensity. Our description of two-photon ionization is formulated in the Heisenberg picture. Besides the usual two-level atom projection operators we introduce explicitly the operators referring to the atomic continuum. By eliminating from the equations of motion the terms related to the electromagnetic self field as well as the terms referring to the continuum, transverse and longitudinal damping corresponding to spontaneous emission and photoionization is obtained. The use of field correlation functions allows us to account for the partial coherence of the applied field. We develop a technique to extract from the equations of motion a unique two-photon ionization rate. Such a characterization of the ionization process is not only instructive because of its simplicity, it is also in many cases completely satisfactory. The resulting general expression for the two-photon ionization rate is the central result of our treatment. From it we recover several familiar results in different limiting cases. In addition, our general rate expression provides the first description of the complete hierarchy of saturation effects that one should expect. We present a number of examples where we calculate the two-photon ionization rate as a function of the detuning, and where we emphasize the saturation and field correlation effects.
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).
We study three ways in which two-photon ionization, at high light intensities and near an intermediate-state resonance, may be described. We calculate the instantaneous ionization rate, the total ion count, and a saturable rate constant. We find that the total ion count, which is the experimentally interesting quantity in most cases, is very closely modelled by our saturable rate constant. The instantaneous ionization rate bears no useful relation to the total ion count in many cases. In comparing these three descriptions of ionization we have included the effects of detuning from resonance, finite light bandwidth, finite intermediate-state lifetime, and light intensity; we have varied all of these parameters independently over several orders of magnitude. Our conclusions are in good agreement with, and extend the findings of, Ackerhalt and Shore
We develop a time-dependent perturbation theory (PT) beyond the dipole approximation to analyze the two-photon ionization dynamics of atoms exposed to short laser pulses with an arbitrary polarization. In a wide range of laser parameters, the good performance of the PT method is validated by comparing the results of a number of physical quantities with those calculated by numerically solving the time-dependent Schr\"odinger equation beyond the dipole approximation. Subsequent applications of the PT method in the nonresonant regime allow us to unveil the important role of the interferences between the dipole and nondipole transition pathways to the photoelectron momentum shift along the laser-propagation direction. In particular, we find that the ratio of the probability of each dipole transition channel to the total ionization probability oscillates with the increase of the photon energy. The interferences among different pathways and the oscillatory behavior of the ratios jointly lead to a series of minima in the linear momentum transfer for the case of the linear polarization.
When an atom is driven by a laser field with a photon energy that is nearly resonant with an atomic transition, it can undergo periodic population transfers between the ground and excited state, known as Rabi oscillations [1], [2]. The aim of this work is to study photoionization from atoms for parameters accessible to a free-electron laser (FEL) [3]. Using time dependent perturbation theory to model photoionization from a Rabi oscillating atom, we have shown that in addition to the expected pathway of ionization from the excited state, non-resonant two-photon ionization will have a comparable amplitude at intensities on the order of 1013W/cm2[3].