{'fraction_non_alphanumeric': 0.03609862416564501, 'fraction_numerical': 0.012169095945148254, 'mean_word_length': 4.90455764075067, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 3, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Analytical model for attosecond time delays and Fano's propensity rules in the continuum

Extracting single photoionization time delays associated with atomic (or molecular) species from attosecond time scale two-photon experiments usually relies on the theoretical description of continuum-continuum transitions. The available models for those processes predict a universal phase contribution, independent of the angular quantum numbers of final states. However, a recent experimental-theoretical study [Fuchs, \emph{et al.} Optica 7, 154 (2020)] determined a sizable time delay dependence on the angular momentum of near-threshold photoelectrons. In this study, we present an analytical model for the two-photon two-color transition matrix amplitudes that reproduces the phase dependence on the angular quantum number of final states. Finally, we show that our analytical model can also describe the generalized Fano's propensity rules [Busto, \emph{et al.} Phys. Rev. Lett. 123, 133201 (2019)] for laser-assisted photoionization.


I. INTRODUCTION
The pioneering work by Eisenbud, Wigner, and Smith (EWS) on interpreting the energy derivative of the phase of the wavefunction resulted in the concept of time delays associated with quantum processes [1][2][3]. For many years, that concept remained a theoretical curiosity because its experimental determination was not feasible: no measurement protocol allowed to extract time shifts on the sub-femtosecond scale. However, that barrier was removed in 2001 when attosecond science became a reality [4,5]. Since then, techniques such as streaking [4] or the reconstruction of attosecond beating by two-photon transitions (RABBITT) [5,6] are the methods of choice for triggering and monitoring electron dynamics in its intrinsic attosecond time scale [7]. Indeed, in 2010 Schultze et al. [8] applied the streaking protocol to determine the single photoionization (relative) time delay of electrons emerging from the 2s and 2p shells in neon atoms. Later on, Klunder et al. [9] obtained the relative time delays for electrons ionized from the 3s and 3p shells in argon atoms, employing the RABBITT scheme.
The experimental realization of a longstanding theoretical prediction prompted a new and still active research area within the ultrafast community that ultimately led to a deeper understanding of the role played by time in quantum mechanics. In fact, as the streaking and RAB-BITT schemes belong to the set of pump-and-probe measurement protocols, it was soon understood that information about the EWS time delay is only accessible if the influence of the infrared probe field is considered [10]. Further theoretical developments show that observed time delays in RABBITT experiments comprise two contribu- * boll@ifir-conicet.gov.ar tions: an EWS-like delay associated with the pump stage and the additional delay originated from the probing process [11].
These theoretical developments were crucial for the successful obtention of (relative) EWS time delay from RABBITT traces, corresponding to different atomic [9] and molecular [12] species. However, the asymptotic approximations on which they rely are equivalent to neglecting the effects of the centrifugal potential on intermediate and final states. Consequently, the continuumcontinuum (cc) phase for different final states becomes a single contribution, independent of the angular quantum numbers of those states. This universal character of the measurement-induced phases should hold as long as the pump stage generates rapidly escaping electrons. In such a scenario, the most relevant contributions to one-photon cc transitions come from space regions where short-range potentials, such as the centrifugal one, are masked by the long tail of the Coulomb interaction between the photoelectron and the parent ion.
The above qualitative argument will be no longer valid for near-threshold photoelectron kinetic energies. Indeed, recent measurements for slow photoelectrons [13] confirmed the breakdown of the universal character of the delay induced during the probing stage. The theoretical analysis of these near-threshold processes [11,[13][14][15] can be exclusively carried out through numerical simulations that, despite being extremely useful for an accurate evaluation of the observables, usually provide only partial explanations for the physical origin behind the observed effects. Therefore, the development of a complementary analytical method can be a valuable tool for providing further insights into these phenomena [16].
In this letter, we present an analytical model that quantitatively describes the dependence of two-photon two-color transition amplitudes on the angular quantum numbers of final states. To establish a connection with some previous experimental and theoretical results on time delays, we focus on the RABBITT scheme: atomic targets in the gas phase irradiated with a combination of odd harmonics of a Ti:sapphire laser plus a synchronized weak fraction of the fundamental infrared (IR) laser. In this situation, the photoelectron spectra display two qualitatively distinct sets of lines: a) dressed harmonic (DH) lines mainly populated by the absorption of one photon from the source of odd harmonics and b) sideband (SB) lines, populated by two-photon transitions, on which the time delay gets imprinted (see Fig. 1 for details). Following standard time-dependent second-order perturbation theory, it is possible to show that sideband signal reads [5], where M ± L,M are the two-photon transition matrix amplitudes connecting the initial bound state to a final state in the continuum with azimuthal and magnetic quantum numbers L and M , respectively. In Fig. 1 we schematically show these reaction pathways for a generic sideband of order 2q and an initial s (l i = 0) state. After solving the angular algebra and the time integrals for infinitely long fields, the transition matrix amplitudes above can be factorized as where A L,M is the angular (algebra) factor and exp(±iω 0 τ ) is a phase factor allowing to control the sideband signal by changing the delay τ between the harmonics and the infrared laser with angular frequency ω 0 . As usual, the radial matrix elements in length gauge are given by where the summation (integration) runs over the entire spectrum of unperturbed states of the target, with energy ε κ , angular momentum λ, and radial wavefunctions R εκ,λ . The initial and final states of the system are also described by eigenstates of the field-free Hamiltonian, with radial wavefunctions R εi,li and R ε k ,L , respectively. The transition matrix amplitudes in Eq. (2) can be derived in closed-form only for hydrogenic atoms [17][18][19]. Therefore, to obtain a simple analytical model able to describe a broader range of targets, some approximations must be made. Firstly, for typical infrared laser intensities in RABBITT experiments, we can focus only on those processes where the extreme ultraviolet photon from the source of high-order harmonics is absorbed first. This is equivalent to neglect the radial matrix element T li L,λ (±ω 0 ) in Eq. (2), that accounts for the dressing of the initial state due to the infrared laser field [20]. The remaining radial matrix element can be evaluated by invoking the Dalgarno-Lewis method [21], to obtain with ρ ε±,λ being the solution of the inhomogeneous differential equation [H λ −ε ± ]ρ ε±,λ = −rR εi,li , where H λ is the laser-free radial Hamiltonian for angular momentum λ, and the energy parameter satisfy ε ± = (2q ∓ 1)ω 0 + ε i . From the graphical representation in Fig. 1, it can be seen that ε ± = κ 2 ± /2 is merely the energy for the lower (upper) dressed harmonic line.
The theoretical developments in Ref. [11] follow from approximating ρ ε±,λ and R ε k ,L in Eq. (4) by their asymptotic forms. Such approximation allows to obtain analytical expressions for the radial matrix elements that factorize as is the phase acquired during the ionization step, triggered by the absorption of one photon with energy Ω ∓ . The second term (φ ± cc ) is the measurement-induced phase due to one-photon cc transitions stimulated by the infrared probe laser. One main contribution in [11] is the derivation of an analytical expression for the φ ± cc phases, allowing thus access to the EWS time delay. Briefly, by measuring the time offset in the oscillation of sideband signals, induced by the controlled modification of the delay τ , a finite differences approximation to the EWS time delay can be obtained by subtracting the measurement-induced time delay (φ + cc − φ − cc )/2ω 0 to the total offset [22]. Our derivation closely follows Ref. [11] to model the radial matrix elements from Eq. (4), but we use the exact radial wavefunction R ε k ,L for the final states instead of its asymptotic approximation. In that case, we get where the expression for the final continuum states was taken from Ref. [23], p F q (a 1 , . . . , a p ; b 1 , . . . , b q ; z) is the generalized Hypergeometric function, and the normalization factors for intermediate and final states in the continuum are given by N ε± and N ε k ,L , respectively. The position-dependent phase in Eq. (5) is given by Φ κ,λ (r) = ln(2κr)/κ + σ λ (κ) − πλ/2 with σ l (κ) = arg[Γ(l + 1 − i/κ)] being the Coulomb phase-shift for partial wave l and momentum κ [11]. Operating over the logarithmic phase and rearranging the factors, I L,λ k,κ± can be obtained in closed form by taking a limit for the integral of a closely related family of functions [24] As the other factors entering Eq. (6) are strictly real [11], the bound-continuum phase acquired during the ionization step and the measurement induced phase, for twophoton processes as those depicted in Fig. 1, must be encoded in the expression for I L,λ k,κ above. As a matter of fact, from Eq. (8) we can recognize the φ ± bc phase in the exp [iσ λ (κ)] phase factor and, therefore, our analytical model allows to disentangle the EWS phase contribution while the cc phases depend on the angular momentum of final states.
The model we present can be applied, in principle, to approximate the two-photon transition matrix amplitudes from any initial state of atomic [9] or molecular species [12,25]. However, to facilitate the comparison with previous theoretical and experimental data, we will firstly concentrate on initial atomic s states. In that case, the dipole selection rules indicate that only final states with angular momentum L = 0, 2 can be populated. The results we get for φ ± cc,L are shown in Fig. 2 by the dashed (L = 0) and dash-dotted (L = 2) lines. From extensive testing, we have found that model works slightly better if momentum for intermediate states is approximated as κ ± k ∓ ω 0 /k. Following this procedure, related to the soft-photon approximation [11] for asymptotic intermediate states, does not change any of the conclusions reached below. To assess the accuracy of our analytical model, we additionally perform two sets of ab-initio calculations. Firstly, we numerically solve the time-dependent Schrodinger equation (TDSE) in velocity gauge for hydrogen atoms, employing the Qprop code [26]. Using a single harmonic and the IR, we obtain the transition amplitudes M ± L,M for the absorption and emission pathways separately. Besides, we calculate the same two-photon transition matrix amplitudes in length gauge from second-order perturbation theory (SOPT), based on the analytical results in Ref. [19]. For both approaches, we obtain the cc phase associated to each transition amplitude by subtracting the contributions φ ± bc and ±ω 0 τ to the total phase [27]. As can be clearly seen in Fig. 2, the results we get for the ab-initio calculations [28] are virtually identical to each other, ruling out any gauge and/or convergence systematic errors. Generally speaking, model results in Fig. 2 for the emission pathways are better than those for the absorption ones. This behavior is not surprising and reflects a diminished descriptive power of our model when it comes to representing absorption pathways. From Fig. (1), it is evident that intermediate states for emission pathways will always have a larger kinetic energy when compared to the sideband final state. In that case, description of the intermediate states with asymptotic wavefunctions qualitatively captures the kinematical conditions under which every model is expected to work better: the lower the kinetic energy, the better the quality of wavefunctions describing that state should be. The situation is reversed for the absorption pathways. Now, intermediate states with lower kinetic energy are described through approximate asymptotic states, which may be a poor description for sufficiently low kinetic energies.
Nonetheless, model results for the absorption pathway and L = 2 quantitatively reproduce the accurate data for electron kinetic energies above ε k ∼ 8 eV, being almost identical to those from the long-range amplitude-corrected (LRAC) asymptotic approximation in Ref. [11]. On the other hand, the L = 0 results for the absorption pathway slightly overestimate ab-initio calculations, with the notable exception of results around E k ∼ 5 eV, that are almost perfectly described. For the emission pathway, the quantitative agreement between the ab-initio and model results in Fig. 2 is prominent, considering the simplicity of our approach. For low kinetic energy, our model faithfully reproduce the phase splitting obtained through the exact calculations, whereas at high energy the results for L = 0, 2 converge to the expected limit given by the LRAC approximation.
The above comparison of individual cc phases φ ± cc,L obtained through different methods is only possible by theoretical analysis. From the experimental point of view, phase differences are the most detailed pieces of information that angular distributions of RABBITT traces encode [13,14]. Recently, these phase differences were experimentally obtained for helium targets [13], and the simulations reported therein allow to conclude that only single-active electron dynamics are triggered within this energy range, in agreement with previous calculations for two-photon two-color processes [14,29]. Therefore, we can directly compare the phase differences φ ± cc,0 − φ ± cc,2 obtained from our model with those from accurate calculations and the available experimental data. These comparisons are presented in Fig. 3, for the absorption and emission processes separately. Our model globally captures the salient features displayed by the phase differences obtained from ab-initio and experimental data: they decrease to eventually zero for larger photoelectron kinetic energies, and, for the same kinetic energy, the results for the absorption pathways are slightly larger than those for the emission ones. The convergence to zero can be easily demonstrated by just replacing the Gauss hypergeometric function in Eq. (8) with the asymptotic approximation 2 F 1 (a, b; c; z) ∼ λ 1 (−z) −a , for |z| → ∞ [30]. In that case, we obtain cc phases identical to those in Eq. (22) from Ref. [11] and, therefore, independent of the angular quantum numbers of final states. On the contrary, the comparatively larger values obtained for the absorption pathway cannot be easily demonstrated as this feature heavily relies on fine details of the Gauss hypergeometric function. As before, we observe a better agreement for the emission pathway where the model quantitatively reproduces the experimental data and ab-initio simulations, for electron kinetic energies above ∼ 5 eV. For smaller kinetic energies, the model captures the trend set by the simulations for hydrogen and the experimental data for helium. The results for the absorption pathway are less accurate, as model calculations overestimate both the experimental data and the ab-initio calculations on the entire range of electron kinetic energies under analysis. Again, the model can reproduce the trend showing an increase in the phase difference for smaller electron kinetic energies. The results for electron kinetic energies below ∼ 5 eV are shown in dashed line to emphasize that for an 800 nm infrared laser, the intermediate states of the absorption pathway will have a kinetic energy below ∼ 3.5 eV, a value perhaps too low to admit a description with asymptotic states.
The primary aim of developing this model was to put forward a general analytical expression able to reproduce phase differences observed for partial waves populating the final channel in sidebands. The previous analysis demonstrates that this goal is achieved. In addition, as a by-product, with our model we can also study the branching ratio for partial waves upon cc transitions. This concept can be traced back, at least, to a few experimental and theoretical studies for the half-RABBITT sidebands obtained when atomic or molecular targets are ionized with a combination of a free-electron laser pulse and an infrared probing field [31,32]. More recently [15], this concept was applied to RABBITT experiments, gener- |T li,± λ−1,λ /T li,± λ+1,λ | λ = 1

Emission pathways
Absorption pathways TDSE SOPT He SOPT [31] He Exp. [32] Model Model Model He SOPT [31] He Exp. [31] He Exp. [32] Kr RPAE [15] Ar RPAE [15] Kr RPAE [15] Model alized for different partial waves, and rationalized as a superset of Fano's propensity rules [33]. In Fig.  4, we compare the moduli ratios |T li,± λ−1,λ /T li,± λ+1,λ | obtained from our model with the benchmark data: random-phase approximation with exchange (RPAE) [15], TDSE and SOPT calculations and the scarce experimental data available in the literature [31,32]. For λ = 1, the model almost perfectly describes ab-initio and experimental results for the emission pathway on the entire energy range. Conversely, the results for the absorption pathway fail for electron kinetic energies below ∼ 5 eV. As before, this behavior may be ascribed to a poor description of low kinetic energy intermediate states with asymptotic wavefunctions. The approximately linear dependence on λ of the kinetic energy value below which model results deviate from the accurate calculations emphasizes the previous conclusion. Further understanding of this trend would require a direct comparison of the approximate intermediate states with those obtained from the numerical resolution of the Dalgarno-Lewis method, which is beyond the scope of the present study.
The final remark on these branching ratios relates to its universal character. From Eq. (6), it is clear that twophoton radial matrix elements factorize as the product of a radial matrix element for the pump stage and a singlephoton cc factor. Therefore, the agreement of our model results with those from more accurate calculations and the experimental data for the branching ratios in Fig. 4 is a direct demonstration that the observed variations are dictated by the single-photon cc transitions. In turn, it means that correlation effects, included in the RPAE cal-culations [15] and experimental data [31,32], play little to no role in these branching ratios.
The above analysis of branching ratios for the radial matrix elements can be systematically performed from a theoretical perspective. On the contrary, the empirical verification of these results has some nuances depending on the initial state. On one hand, for atomic s states there will be a single intermediate state with λ = 1, therefore, the experimental and theoretical results can be directly compared. The agreement we observe in Fig.  4 indicates that the branching ratios are independent of the radial distribution for the initial s state. On the other hand, measurements for initial states with angular momentum larger than zero will be hindered by the fact that final states with L = l i are coherently populated by contributions following two different quantum pathways. In such a scenario, a branching ratio dependence on the radial distribution of the initial state cannot be ruled out. This is in agreement with the findings in Ref. [25] (Fig.  5), where changing the radial distribution for a 1s initial state in hydrogen does not modify an observable closely related to radial matrix elements, whereas the same procedure for initial 2p orbitals induces a dramatic change.
To summarize, we develop an analytical model for the two-photon two-color transition matrix amplitudes that quantitatively reproduce state-of-the-art calculations and experimental data at low photoelectron kinetic energies, without resorting to fitting free parameters. We demonstrate that the effects induced by the centrifugal barrier onto the wavefunctions can be almost completely recovered by just using accurate final states, even if intermediate states are approximated by its asymptotic form. Besides, the functional form we obtained for the matrix elements is fully compatible with the procedure to extract the EWS time delays from the sideband signal oscillation in RABBITT experiments. Additionally, our results can be readily adapted to upgrade models describing higher order cc processes [34], as it only requires to change the description of final states. Finally, we demonstrate that the partial waves branching ratio above a λ-dependent photoelectron kinetic energy can also be accurately retrieved by considering the centrifugal potential effects only on final states. In particular, our model may be useful to disentangle the coherent contributions populating final state partial waves with L = l i in two-color experiments from free-electron laser facilities, where measurements can be performed away from threshold.

ACKNOWLEDGMENTS
The authors acknowledge the financial support from the Agencia Nacional de Promoción Científica y Tecnológica, Project PICT No. 01912 (2015-3392), and from the Consejo Nacional de Investigaciones Científicas y Técnicas de la República Argentina, Project PIP No. 0784 (11220090101026). Part of the results presented in this work have been obtained by using the facili-