**1. Introduction**

The study of attosecond photoionization dynamics has been made possible by coherent light sources in the extreme ultraviolet (XUV) regime based on high-order harmonic generation (HHG) [1]. Experimental techniques that were originally used for pulse characterization, such as the Reconstruction of Attosecond Beating By Interference of Two-photon Transitions (RABBIT) [2] and the attosecond streak-camera [3], have proved useful to gain novel insights into the time it takes for electrons to escape the binding potentials of atoms [4–16], molecules [17–21], and solid-state targets [22–24]. The main observables are delay-dependent modulations in the photoelectron spectra that arise due a phase-locked laser probe field in the infrared (IR) regime [25–30]. For "weak" fields, these modulations can be understood from perturbation theory, where absorption of one XUV photon (Ω) is followed by exchange of one IR photon (±*ω*). It is a rather technical task to evaluate laser-driven continuum–continuum transitions numerically in the presence of the longrange Coulombic potential: *k* → *k* [26,31,32]. Thus, analytical continuum–continuum phase shifts *φ*cc(*k*, *<sup>k</sup>*), have been derived by using the Wentzel–Kramers–Brillouin (WKB) approximation, in order to interpret the RABBIT delays at sufficiently high kinetic energy of the photoelectrons [33]. Asymptotic theories based on the Eikonal Volkov Approximation (EVA) have also been developed [34]. The main result of these asymptotic theories is that delays observed in RABBIT experiments can be separated into two terms: (i) a finite-difference approximation to the Wigner–Smith–Eisenbud delay of the photoelectron

**Citation:** Vinbladh, J.; Dahlström, J.M.; Lindroth, E. Relativistic Two-Photon Matrix Elements for Attosecond Delays. *Atoms* **2022**, *10*, 80. https://doi.org/10.3390/ atoms10030080

Academic Editors: Anatoli Kheifets, Gleb Gribakin and Vadim Ivanov

Received: 20 June 2022 Accepted: 24 July 2022 Published: 2 August 2022

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after absorption of one XUV photon: *τ*W [35–37], and (ii) a universal continuum–continuum delay: *<sup>τ</sup>*cc(*k*; *<sup>ω</sup>*), with an analytical expression that only depends on the final momentum of the photoelectron and the frequency of the IR field. In the case of a single angular momentum channel *λ* = *i* + 1 with *i* being the initial angular momentum, this separation has been successfully implemented to measure the Wigner-like delay of the 2*s*-orbital in neon atoms [12]. In the more general case, where multiple intermediate angular momenta are populated, *λ* = *i* ± 1, the probe process becomes more complicated and care must be taken to account for the weight of all intermediate transitions, which leads to an "effective" Wigner delay [33]. As an example, the RABBIT delay measured close to the 3*p*-Cooper minimum in argon [38] is much reduced in magnitude when photoelectrons are detected over all emission angles, rather than along the polarization axis of the fields [7]. Nonetheless, the asymptotic theory has been extended to interpret delays from molecules, where contributions of multiple partial waves in the initial orbital and the orientation of the target relative to the laser polarization, adds more complexity to the process [21,39]. Although it has been shown that the separability of delays remains valid at high kinetic energies, by using full two-photon matrix elements from time-independent R-matrix theory [40], the target-specific delay in molecules *τ*PI, cannot be interpreted as a Wigner-like delay, due to interference effects of multiple partial waves in the two-photon transitions [39] and various channel coupling effects [21,40]. The use of full two-photon R-matrix theory [40] is undoubtedly an important milestone in the field of photoionization delays from molecules, which has allowed for quantitative analysis of many recent experiments [18–21].

In the case of atoms, full two-photon matrix elements have been used for a decade to compute delays in photoionization at various levels of Many-Body Perturbation Theory (MBPT) [41–44]. Although the importance of the random-phase approximation with exchange (RPAE) for attosecond science was first realized by Kheifets [45,46], numerical simulations of the one-photon ionization process, developed by Amusia [47], are inherently insufficient to interpret RABBIT delays. Thus, a two-photon approach was developed, whereby the many-body response of XUV absorption was computed at the level of RPAE, and the IR exchange in the continuum was computed numerically by using an effective onebody ionic potential [41,42]. This two-photon matrix approach has shown good agreemen<sup>t</sup> with a range of RABBIT experiments [7,8,12,13,48]. Noble gas atoms consist of multiple valence states, which implies experimentally unresolved ionic states with magnetic quantum numbers |*m*| ≤ *i*. However, any problem with incoherent final channels can easily be avoided by detecting photoelectrons along the polarization axis *z*ˆ, where only *m* = 0 contributes. In this configuration, it has been shown that a numerically obtained continuum–continuum delay, *τ*MBPT cc , can be accurately separated from the one-photon Wigner delay *τ*MBPT W , computed for photoelectrons along the polarization axis with the unique ionic state *m* = 0 [41,43]. In this way, a precise separation of delays has been demonstrated down to 5 eV, which is much lower than the high-energy regime predicted by the original asymptotic theory [33]. The two-photon matrix elements have also been used to study effects beyond the asymptotic approximation. Firstly, a strong angle-dependence of RABBIT delays from the isotropic helium atom was evidenced in experiments by Heuser et al. [48]. Secondly, the role of universal asymmetries between absorption and emission processes in the continuum was identified by Busto et al. [49]. Finally, a weak angularmomentum dependence of continuum–continuum phases was measured by Fuchs et al. in helium atoms [50]. The latter discovery was in good agreemen<sup>t</sup> with theoretical predictions based on exact two-photon matrix elements for hydrogen, provided by Taïeb [33], as well as full two-photon matrix elements based on MBPT [13]. Thus, several effects that depend on the exact form of continuum states have been identified by using RABBIT delay measurements in recent years [51].

Due to the energy spacing between the odd harmonics from HHG, ΔΩ = 2*ω*, the temporal resolution in traditional RABBIT experiments is limited to probe processes that are much shorter than 2 *π*/ΔΩ = *Tω*/2 ≈ 1.3 fs (assuming an IR laser system with *h*¯ *ω* = 1.55 eV). At a first glance, this seems to preclude any studies of autoionizing dynamics

in atoms or molecules, which typically unfold on a time scale of a few femtoseconds, or more [29,52]. However, the subject of combined time–frequency non-linear metrology is quite subtle, and it has been found that a high-energy resolution of photoelectrons in RABBIT sidebands can be used to reconstruct autoionizing processes in time [11]. In this case of resonant excitation, via bound Rydberg states or autoionizating states, it is useful to consider the RABBIT scheme as a combination of one "structured" (resonant) path and another "unstructured" (reference) path [10,11,13,16,53,54]. In this case, the phase variation of the resonant path is typically much stronger than any continuum–continuum (or other non-resonant) phase shift, and the phenomena can be understood by expanding Fano's model for autoionization to laser-assisted photoionization, within the strong-field approximation [55], or by using approximate two-photon two-color matrix elements [56,57]. In the latter works, it was shown that finite pulses, in the time domain, can lead to nonperiodic structures in RABBIT experiments due to autoionizing states. The two-photon Fano model has proven essential to disentangle dynamics from multiple autoionizing states measured by the RABBIT technique [14]. Although we find that the theory development for autoionization in RABBIT is another milestone in the field, we will not consider this class of processes in the following work. Rather, we will focus on correlation effects in unstructured continuum, where MBPT is a numerically efficient route to describe correlation effects and RABBIT data can be safely assumed to be periodic.

Despite these many successes, there remained disagreement between experimental and theoretical results for the relative RABBIT delay between the 3*p* and 3*s* orbitals in argon, first measured by Klünder et al. in 2011 [5], which was mostly ascribed to the low signal close to the correlated minimum in the 3*s*-partial photoionization cross section [6,41,42,58–60]. The fact that this exceptionally deep minimum from 3*s* arises due to correlation effects, was first showed by Amusia in 1972 by applying the RPAE method to describe photoionization from inner atomic orbitals [61]. By using two-photon matrix elements, it has now been shown that the position, height, and sign of the associated RABBIT delay from 3*s* is similarly sensitive to correlation effects [41,42], which largely stems from the sensitivity of the onephoton Wigner delay peak from the correlated minimum in the photoionization cross section [62]. In order to solve this long-standing problem, a full two-photon two-color RPAE (2P2C-RPAE) method was developed for RABBIT delays [44]. This new method allowed for detailed examination of correlated IR exchange processes. It was found that, apart from a rather minor discrepancy at the correlated 3*s*-minimum in argon, the universal separability of the MBPT continuum–continuum delay and Wigner delay was achieved. However, this discrepancy was still not enough to reach agreemen<sup>t</sup> with the experimental results [5,6]! It was not until the argon experiment was repeated, with higher photon energies by Alexandridi et al. in 2021 [15], that excellent agreemen<sup>t</sup> with 2P2C-RPAE results was found in a broad energy range above the 3*s*-minimum in argon. It was also concluded that the long-standing 3*p*-3*s* problem was caused by an "accidental" harmonic satellite, namely the <sup>3</sup>*s*23*p*<sup>4</sup>(<sup>1</sup>*D*)<sup>4</sup>*p*(<sup>2</sup>*P*) shake-up process, predicted by Wijesundera and Kelly in 1989 by using MBPT [63], which overlapped with the 3*s*-RABBIT sidebands. Prior to that, the importance of "two-electron-two-hole" excitations in argon had been found by Amusia and Kheifets by considering effects beyond RPAE in 1981 [64].

The 2P2C-RPAE method also opened up for gauge-invariance tests of the RABBIT theory [44]. It was concluded that the so-called length-gauge formulation of light-matter interaction was much favoured, which is in line with the gauge theory of Kobe [65,66]. In the velocity-gauge formulation of RABBIT, it was found that the interaction with the second photon required a more detailed many-body treatment, beyond the one-body ionic potential, with important contributions from both time-orders of the fields XUV+IR and IR+XUV. Although it was shown that only the *complete* 2P2C-RPAE theory leads to gaugeinvariant results, the approximate one-body treatment of the IR-exchange was shown to be an excellent approximation in length gauge. For this reason, we will use the length gauge, with an effective ionic potential to describe IR exchange processes, in our current work, which aims to quantitatively account for relativistic effects in RABBIT experiments.

The study of relativistic effects is quite a recent development in attosecond physics. In our view, Saha et al. have pioneered this field with calculations of relativistic one-photon Wigner delays [62,67,68], based on the relativistic random phase approximation (RRPA). Although RRPA theory was originally developed in the late seventies by Johnson and Cheng to describe one-photon ionization cross sections in heavy elements [69,70], the interest in such phenomena is revived by recent RABBIT experiments that have targeted heavy elements. First, Jordan et al. [71] and Jain et al. [72] have compared photoelectrons from the fine-structure split valance orbitals: <sup>4</sup>*pj* and <sup>5</sup>*pj* with *j* = 1/2 and 3/2 of krypton and xenon atoms, respectively, and secondly, Jain et al. [73] and Zhong et al. [74], have compared photoelectrons from inner orbitals in xenon, down to the 4*d* orbital. The 4*d* orbital is of special interest because it is known to posses a giant collective resonance in the photoionization cross section, as evidenced by MBPT in the early seventies by Amusia and Wendin [75,76]. Thus, it is now possible to study the role of sizable relativistic effects, such as the spin-orbit effect in xenon, in the time domain with RABBIT. This opens a call for time-dependent methods to solve the Dirac equation for heavy many-electron atoms; as an example we mention the recently developed relativistic time-dependent configuration–interaction singles (RTDCIS) method [77], but also extend the computation of two-color, two-photon matrix elements to the relativistic domain. Concerning the lack of such relativistic theories, we mention that in refs. [71,73], the experiments were accompanied by photoionization delay calculations with one-photon matrix elements at the level of RRPA for XUV absorption, whereas various asymptotic formulas from nonrelativistic theory were used to account for IR exchange effects. Our goal here is to treat the whole process within a relativistic framework and below we discuss the different points where the relativistic treatment differs from that of the non-relativistic one with an effective ionic potential for IR exchange [41–43]. We also mention that the method presented here has already been utilized in various projects, such as [49,74], without any detailed description of the theoretical formulation. A full development of the two-photon, two-color relativistic random phase approximation (2P2C-RRPA) is beyond the scope of the present work, but we expect that it would not lead to any major modification of the results presented here, because we base our entire theory on the length gauge formulation of the light–matter interaction, where the one-body ionic potential description of IR exchange processes is a good approximation [44].

In Section 2 below, some basic concepts are introduced, and the relativistic scattering phases, as well as the asymptotic form of the continuum solutions, are discussed in detail. Section 3 discusses photoionization delay in a relativistic framework, and in Section 4 the many-body implementation is outlined, and the technique to calculate the needed two-photon matrix elements is explained. Some results are finally shown in Section 5.
