Under-threshold RABBITT in argon

The process of reconstruction of attosecond bursts by beating of two-photon transitions (RABBITT) can involve a transition from under the ionization threshold. Such an under-threshold RABBITT (or uRABBITT) was demonstrated experimentally and analyzed theoretically in He and Ne. In the present work, we explore an analogous process in the argon atom. The uRABBITT in Ar reveals the familiar physical effects: a phase transition across the threshold and the symmetry modification of the photoelectron momentum distribution. It can also be used for mapping the electronic structure of the target atom bound states.


Introduction
Resolving photoemission in time was one of the most significant achievements of atomic and molecular physics in the past decade. The process of reconstruction of attosecond bursts by beating of two-photon transitions (RABBITT) (Paul et al 2001, Mairesse et al 2003 has been credited with this achievement. Starting with the pioneering work by Klünder et al (2011), RABBITT has been widely used for time resolution of photoemission from atoms, molecules and condensed matter (see Pazourek et al 2015, Borrego-Varillas et al 2022and Kheifets 2023 for reviews). The RABBITT process involves two ionizing transitions driven by odd harmonics of the fundamental laser frequency Ω 2q±1 = (2q ± 1)ω supplemented by absorption or emission of one ω photon. This two-photon ionization process creates an even order sideband SB 2q in the photoelectron spectrum. The population of this sideband oscillates as the time delay between the Ω and ω laser pulses varies: Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Here A and B are the magnitude parameters whereas C is the RABBITT phase which can be converted to the atomic time delay τ a = C/(2ω) (Véniard et al 1996, Dahlström et al 2012. In a conventional RABBITT, both the ±ω ionization processes involve transitions in the continuum. However, when Ω 2q−1 < I p , the +ω transition should proceed from under the threshold. This discrete-to-continuum transition is strongly promoted by an overlap with a bound state of the target atom. Such an under-threshold RABBITT (or uRABBITT for brevity) was observed experimentally in He (Swoboda et al 2010) and Ne (Villeneuve et al 2017). A detailed theoretical study (Kheifets and Bray 2021) revealed the uRABBITT phase transition across the ionization threshold. By sharpening the energy resolution, the uRABBITT process was shown to map faithfully the target atom bound states (Kheifets 2021). Both the experiment (Villeneuve et al 2017) and theory (Kheifets 2022) revealed a strong modification of the photoelectron momentum distribution across the ionization threshold. Very recently, the rainbow RABBITT technique was used to study the uRABBITT process in He (Autuori et al 2022, Drescher et al 2022, Neoriĉić et al 2022. In these works, the base frequency of a wideband infrared (IR) pulse was fixed while narrow bandpass filters were used to scan the photoelectron spectrum over the resonance.
In the present work, we uncover the uRABBITT process in the argon atom which was not reported previously. Unlike the two other atoms, He and Ne, the fundamental laser frequency should be doubled changing the wavelength from the 800 nm to 400 nm range 1 . By doing so, the transition from the ground 3p to the excited 3d bound state can be facilitated with the fifth harmonic of the fundamental frequency as illustrated graphically in figure 1. Accordingly, the sideband SB 6 undergoes the uRABBITT phase transition in the photon energy range 2.8 ⩽ ω ⩽ 2.9 eV (442 ⩾ λ ⩾ 427 nm). This transition is accompanied by a noticeable modification of the photoelectron momentum distribution. Both effects can be used as convenient tools for probing and mapping the target atom electronic structure.
The rest of the paper is organized in the following way. In section 2 we give a brief outline of our computational methods. Section 3 contains our main numerical results divided into angular resolved and angular integrated spectra. We conclude in section 4 by outlining further extensions of the present work.

Methods
We follow closely our previous works (Bray et al 2018, Kheifets andBray 2021). In brief, we solve numerically the time-dependent Schrödinger equation (TDSE) in a singleactive electron approximation 2 : Here the radial part of the atomic Hamiltonian contains an effective one-electron potential V(r). This potential is obtained by localizing the non-local Hartree-Fock (HF) potential as prescribed by Wendin and Starace (1978). Thus obtained localized (LHF) potential returns the bound state energies listed in table 1. The interaction HamiltonianĤ int (t) in equation (2) is written in the velocity gaugê The vector potential represents the linearly polarized XUV and IR fields. The extreme ultraviolet (XUV) field is taken as an attosecond pulse train (APT) containing 41 pulselets of the opposite polarities and 30 as duration. The total width of the APT is about 30 fs whereas that of the IR pulse is close to 15 fs. The APT is centered at ω x = 15ω and its spectral width Γ ≃ 0.2 eV. The base frequency ω is incremented with a typical step of 0.02 eV. The IR and XUV intensities are both 1 × 10 10 W cm −2 . Solution of equation (2) is sought with the spherical-coordinate implicit derivatives method implemented in the computer code by Morales et al (2016). The photoelectron momentum distribution (PMD) was obtained by projecting the time-dependent wave function at the end of the propagation on the basis of the plane waves Here the indexes n, l, m denote the initial atomic bound state (the Ar 3p m ground state). The photoelectron momentum k is defined in the Cartesian plane in which theẑ axis is aligned with the joint polarization direction of the XUV and IR pulses. Projection on the k y = 0 plane P(k x , k z ) serves to determine the angular symmetry of the main harmonic peaks and the sidebands. The angular integrated photoelectron spectrum is obtained as The sideband population is determined as The XUV/IR time delay τ is implicit in equations (5)- (7) but not shown for brevity of notations.

Angular resolved photoelectron spectra
The PMD P(k x , k z ) projected on the k y = 0 plane is exhibited in figure 2. The top row of panels shows the primary harmonic peaks H7, 9 and 11 corresponding to the Ar atom prepared in the 3p m initial state (m = 0 -left and m = 1 -center) or summed over m (right). The primary harmonics display clearly the Y 2m angular symmetry as illustrated in the bottom row of panels. This supports the Fano (1985) propensity rule which favors strongly the discrete-continuous transitions with an increase of the angular momentum. The same applies to the discrete-discrete transitions as is seen from the optical oscillator strength at comparable energies (see table 1). When summed over m, the m = 0 symmetry still persists especially in the H7 harmonic peak. In the middle row of panels, the sidebands SB6, 8 and 10 are displayed whereas the primary harmonic peaks are masked for clarity. The sidebands show the Y 1m angular symmetry. This is in sharp contrast to the analogous spectra of the Ne atom in the initial 2p m state which demonstrate the Y 3m symmetry (Villeneuve et al 2017, Kheifets 2022. It was suggested by Busto et al (2019) that the same angular momentum increase propensity holds for the transitions in the continuum. The Ar 3p m case does not satisfy this rule. Summation over m smooths the angular anisotropy of the sidebands except for SB6 which still retains the 3p m=0 angular character.

Angular integrated spectra
The spectra shown in figure 2 correspond to the XUV/IR time delay τ = 0. Explicit τ dependence is shown in figure 3 which displays the RABBITT traces recorded at the two photon energies ω of 2.8 eV (left) and 2.9 eV (right). These traces are formed by vertical stacks of several angular-integrated spectra taken at various XUV/IR time delays. The primary harmonic features in these spectra are masked for better clarity. The SB's oscillate with time delay as prescribed by equation (1). The SB centers are joined by solid lines to show more clearly the SB dispersion. While this dispersion is very weak for SB8 and above, the SB6 falls out from this nearly perfect line up. It deviates visibly to the left at ω = 2.8 eV and to the right at ω = 2.9 eV.
To highlight this deviation more clearly, we plot in figure 4 the RABBITT phase parameters C for the three lowest sidebands SB6, 8 and 10. The left panel displays the C dependence on the photoelectron energy E whereas the right panel shows the photon energy ω dependence. While the C parameters of the SB8 and 10 are essentially flat, SB6 shows a rapid oscillation of the C parameter in the left panel. The nature of this oscillation can be explained by a simple perturbation theory model introduced by Drescher et al (2022). According to this model, the phase of the two-photon transition amplitude from the ground g to the final f states near the resonance behaves as arg a (near)res g→f → π/2 ω < ω ng 3π/2 ω > ω ng Here ω ng is the transition frequency connecting the ground state g and an intermediate discrete state n. To visualize this phase shift by one unit of π, we add it to the RABBITT phase for ω ⩽ 2.9 eV. This way the phase dependence on the photon frequency becomes smooth and shows a very distinct RABBITT-uRABBITT phase transition. Not only does the C parameter oscillate in SB6. The corresponding B parameter also shows an oscillatory structure. Figure 5 lines up the C and B parameters side-by-side for the lowest SB6, 8 and 10. As in figure 4, the corresponding C and B parameters are flat for SB8 and 10 whereas SB6 displays well synchronized oscillations in the C and B parameters. The B parameter peaks very sharply when the submerged harmonic H5 crosses the discrete 3d and 4d levels as can be seen from the energy diagram exhibited in the right panel of figure 1. We could anticipate a similar oscillation of the phase C and magnitude B parameters near the discrete 5d and higher states. But we do not observe it in the present simulation either because our photon energy scan stops at 3.1 eV or we do not have a sufficient energy resolution.     The uRABBITT phase transition in SB6 is also reflected in the angular dependence of the C parameter when the photoelectron emission angle θ is taken away from the polarization direction corresponding to θ = 0. This dependence is shown in figure 6 for the two photon energies ω = 2.8 eV (left) and ω = 3 eV (right). The 'normal' SB8 and 10 show very little angular dependence in their C parameters below the magic angle θ m = 54.7 • corresponding to the kinematic node of Y 20 . At the same time, the C parameter of SB6 starts deviation from the polarization direction at considerably smaller photoelectron emission angles.

Conclusion
We demonstrate the clear signatures of the uRABBITT process in the argon atom which was discovered earlier in the He and Ne atoms. The present demonstration of uRABBITT in Ar is first of its kind. Due to peculiarity of the bound state structure, uRABBITT in Ar occurs at the fundamental frequency in the 400 nm wavelength range. This range can be reached by doubling the photon frequency of the most commonly used 800 nm Chirped pulse amplification lasers. The uRABBITT in Ar is facilitated by the harmonic peak H5 crossing the 3d and 4d energy levels. This is accompanied by a strong oscillation of the RABBITT phase C and magnitude B parameters of SB6 whereas the analogous parameters for higher order sidebands remain essentially flat. The SB symmetry analysis shows a clear violation of the propensity rule suggesting an angular momentum increase in the continuum transitions. This rule holds for the analogous transitions in the Ne atom.
The observations made in the present work suggest an universality of the uRABBITT process. Its direct link with the target atom electronic structure allows to map the binding energies and oscillator strengths for the discrete transitions. The same technique can be applied to more complex targets such as molecules. We hope that our findings will stimulate these studies.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).