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Physical Review A

Density-matrix formalism for the photoion-electron entanglement in atomic photoionization

Thomas Radtke, S. Fritzsche, A. Surzhykov

Published Sep 27, 2006• 10 citations
Publisher

Abstract

The density-matrix theory, based on Dirac's relativistic equation, is applied for studying the entanglement between the photoelectron and residual ion in the course of the photoionization of atoms and ions. In particular, emphasis is placed on deriving the final-state density matrix of the overall system ``$\text{photoion}+\text{electron}$,'' including interelectronic effects and the higher multipoles of the radiation field. This final-state density matrix enables one immediately to analyze the change of entanglement as a function of the energy, angle and the polarization of the incoming light. Detailed computations have been carried out for the $5s$ photoionization of neutral strontium, leading to a photoion in a $5s\phantom{\rule{0.2em}{0ex}}^{2}S$ ${J}_{f}=1∕2$ level. It is found that the photoion-electron entanglement decreases significantly near the ionization threshold and that, in general, it depends on both the photon energy and angle. The possibility to extract photoion-electron pairs with a well-defined degree of entanglement may have far-reaching consequences for quantum information and elsewhere.

DOI
10.1103/physreva.74.032709

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The density-matrix theory, based on Dirac's relativistic equation, is applied for studying the entanglement between the photoelectron and residual ion in the course of the photoionization of atoms and ions. In particular, emphasis is placed on deriving the final-state density matrix of the overall system ``photoion+electron\text{photoion}+\text{electron}photoion+electron,'' including interelectronic effects and the higher multipoles of the radiation field. This final-state density matrix enables one immediately to analyze the change of entanglement as a function of the energy, angle and the polarization of the incoming light. Detailed computations have been carried out for the 5s5s5s photoionization of neutral strontium, leading to a photoion in a 5s2S5s\phantom{\rule{0.2em}{0ex}}^{2}S5s2S Jf=1∕2{J}_{f}=1∕2Jf​=1∕2 level. It is found that the photoion-electron entanglement decreases significantly near the ionization threshold and that, in general, it depends on both the photon energy and angle. The possibility to extract photoion-electron pairs with a well-defined degree of entanglement may have far-reaching consequences for quantum information and elsewhere.
The theory of multiphoton ionization for an atomic system of arbitrary complexity is developed using a density matrix formalism. An expression is obtained which determines the differential N-photon ionization cross section as a function of the polarization states of the target atom and the incident radiation. The parameters which characterize the photo-electron angular distribution are related to the general reduced matrix elements for the N-photon transition. Two-photon ionization of unpolarized atoms is treated as an illustration of the use of the theory. The dependence of the multiphoton ionization cross section on the polarization state of the incident radiation, which has been observed in two- and three-photon ionization of Cs, is accounted for by the theory. Finally, the photoelectron spin polarization produced by the multiphoton ionization of unpolarized atoms, like the analogous polarization resulting from single-photon ionization, is found to depend on the circular polarization of the incident radiation.
Photoionization is one of the basic processes of light-matter interaction, which may be used to transfer at least part of quantum information from photons to electrons in the continuum in such a way that the final state electron carries composite information of the initial electron and of the incident photon. We examine how photoionization may change entanglement of electrons. Our result shows that entanglement in one degree of freedom is affected by a transition in another degree of freedom through the coupling of the two degrees of freedom.
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.
In this paper, we study the entanglement properties of a pair of electrons ejected simultaneously from an atom following the absorption of a single photon. We show that, in the absence of spin-orbit interaction (SOI), the non-separability of the state of two photoelectrons in a pair depends upon the spins of the atom and of its residual dication. The two electrons produced in a double photoionization experiment may always be entangled, on the other hand, if the SOI is taken into account. In the latter case, entanglement will be affected by, among other things, the photoionization dynamics and the state of polarization of the absorbed radiation.
Quantitative control and measurement of quantum entanglement are essential for advancing quantum technologies. Photoionization induced by ultrashort laser pulses provides a unique platform for studying entanglement between photoelectrons and residual ions, representing one of the most intriguing quantum phenomena in attosecond physics. Although extensive studies have focused on the coherence properties within either the emitted electrons or the ions individually, the electron-ion entanglement has remained largely unexplored. In this work, we bridge this gap by investigating the resonance-enhanced multiphoton ionization of argon atoms driven by two time-delayed ultrashort ultraviolet pulses. Employing state-of-the-art first-principles multi-electron simulations, we demonstrate the ability to reconstruct and precisely manipulate the purity of electron quantum states through detailed analysis of the photoelectron angular distributions. Our results reveal distinct scattering-phase differences among various electron configurations within the same partial wave channel, providing unequivocal evidence of electron-ion correlation and entanglement. With the fast development of free-electron lasers, this study establishes an experimentally feasible framework for directly controlling quantum entanglement in ultrafast ionization processes, offering new insights and powerful methodologies for exploring complex electron dynamics in many-electron systems.