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Journal of the Optical Society of America B

Theory of electron detection and photon–photoelectron correlations in two-photon ionization

James D. Cresser

Published Aug 1, 1989• 4 citations
Publisher

Abstract

In the theoretical study of photoionization, certain properties of the photoelectrons are not defined in terms of any (even idealized) model for the electron-detection process, in contrast to the well-known Glauber theory of photon detection. Thus, for instance, the energy spectrum of the electron produced in atomic ionization is usually defined simply as the long-time probability for the electron to have a given energy. In this paper a formal model for an electron detector localized to a small region in space, similar to the DeWitt monopole detector, is set up in terms of which results analogous to those obtained for photon detection in the Glauber theory can be obtained. In addition, it is possible to define correlation functions describing the joint probability of detecting an electron and a photon. This correlation function is calculated here in the case of two-photon ionization of an atom modeled as having two discrete energy levels and an ionization continuum. When the atom is subjected to an incident driving field, both a photoelectron and scattered photons arising from the spontaneous decay of the upper of the two discrete levels are produced. Correlations between these photons and the photoelectron are studied here by considering the joint probability of detection of a photoelectron and a scattered photon for a monochromatic driving field.

DOI
10.1364/josab.6.001492

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In the theoretical study of photoionization, certain properties of the photoelectrons are not defined in terms of any (even idealized) model for the electron-detection process, in contrast to the well-known Glauber theory of photon detection. Thus, for instance, the energy spectrum of the electron produced in atomic ionization is usually defined simply as the long-time probability for the electron to have a given energy. In this paper a formal model for an electron detector localized to a small region in space, similar to the DeWitt monopole detector, is set up in terms of which results analogous to those obtained for photon detection in the Glauber theory can be obtained. In addition, it is possible to define correlation functions describing the joint probability of detecting an electron and a photon. This correlation function is calculated here in the case of two-photon ionization of an atom modeled as having two discrete energy levels and an ionization continuum. When the atom is subjected to an incident driving field, both a photoelectron and scattered photons arising from the spontaneous decay of the upper of the two discrete levels are produced. Correlations between these photons and the photoelectron are studied here by considering the joint probability of detection of a photoelectron and a scattered photon for a monochromatic driving field.
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 theory of electromagnetic field measurement by means of photoionization is developed and applied to photoelectron counting. A probability theory involving multitime joint probability functions for a sequence of photoionizations is formulated. A general quantum-theory definition is proposed for the nonexclusive probability function which occurs in the probability theory. Approximations are then introduced to derive expressions for this probability function which involve correlation functions of the photoionization detector and the electromagnetic field-plus-source. The general theory is used to derive quantities of interest in photo-ionization counting experiments. Expressions are derived for (1) the probability Pn(t,t+T){P}_{n}(t, t+T)Pn​(t,t+T) that nnn photo-ionizations are observed in the time interval ttt to t+Tt+Tt+T, and (2) quantities related to Pn(t,t+T){P}_{n}(t, t+T)Pn​(t,t+T), such as its generating function and various moments. Pn(t,t+T){P}_{n}(t, t+T)Pn​(t,t+T) is found to be a compound Poisson distribution determined by the density operator of the field when the latter is expressed in Glauber's PPP representation. Using this result, the character of Pn(t,t+T){P}_{n}(t, t+T)Pn​(t,t+T) is examined for several specific density operators. These correspond to a coherent state, various fields with the mode phases distributed independently of the mode amplitudes, and a "spread-out" coherent state.
Double ionization is studied by direct numerical integration of the time-dependent Schrödinger equation for a model two-electron system in the field of an electromagnetic wave in the case when the photon energy exceeds the energy required for the removal of both electrons of the atom. The probabilities of single-electron and double ionization are obtained as functions of the radiation intensity. The energy spectra and double pulse distributions of photoelectrons are analyzed. It is shown that single-photon ionization plays a significant role only in the limiting cases of weak and strong fields. The obtained results are used in an analysis of the contributions from different channels to the double-ionization process (in particular, for clarifying the role of the electron-electron interaction). The results of numerical calculations are compared with the analytic model of the phenomenon.
Until recently, the nonlinear interaction between light and matter has been restricted to only low photon energies produced by optical lasers. However, about a decade ago, the rise of free-electron laser facilities revolutionized the field of nonlinear light-matter interaction by delivering intense high-energy light pulses. Today, such lasers are used for research in materials science, chemical technology, biophysical science, solid state physics as well as fundamental research. It is the new experimental possibilities provided by free-electron lasers that motivated the work presented in this thesis. Two-photon ionization process is one of the simplest nonlinear interactions in which absorption of two photons by an atom (or a molecule) leads to promoting one of its bound electrons to continuum. This work presents studies of two-photon ionization of neutral atoms. After a brief historical introduction to the topic of nonlinear light-matter interaction, the density matrix describing the state of an atom and a photoelectron following two-photon ionization is derived. In each successive chapter, part of this density matrix is used to obtain characteristic quantities such as total two-photon ionization cross section, photoelectron angular distributions, ion polarization or degree of polarization of fluorescence photon produced by subsequent decay of the photoion. Physical properties of these quantities are studied and intriguing phenomena, such as elliptical dichroism, polarization transfer as well as relativistic and screening effects are investigated.
Angular correlations between two emitted photoelectrons in the sequential two-photon double ionization (TPDI) of atoms are theoretically considered. The general expression for the angular correlation function is analysed and some particular geometries of possible experiments are discussed. The concept of true angular correlations is introduced in contrast to the angular dependence of the two-electron emission from two uncorrelated ionization events. The case of a pure LS coupling description of the TPDI is considered in detail and compared with the general case. As an example, calculations of the angular correlation functions are performed and discussed for sequential TPDI from the 4p valence shell of krypton.
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].