The Single-Active-Electron Approximation with Angular-Momentum-Dependent Potentials: Application to the Helium Atom

Abstract

We discuss an extension of the Single-Active-Electron (SAE) approximation in atoms by allowing the model potential to depend on the angular-momentum quantum number ℓ. We refer to this extension as the ℓ-SAE approximation. The main ideas behind ℓ-SAE are illustrated using the helium atom as a benchmark system. We show that introducing ℓ-dependent potentials improves the accuracy of key quantities in atomic structure computed from the Time-Independent Schrödinger Equation (TISE), including energies, oscillator strengths, and static and dynamic polarizabilities, compared to the standard SAE approach. Additionally, we demonstrate that the ℓ-SAE approximation is suitable for quantum simulations of light−atom interactions described by the Time-Dependent Schrödinger Equation (TDSE). As an illustration, we simulate High-order Harmonic Generation (HHG) and the three-sideband (3SB) version of the Reconstruction of Attosecond Beating by Interference of Two-photon Transitions (RABBITT) technique, achieving enhanced accuracy comparable to that obtained in all-electron calculations. One of the main advantages of the ℓ-SAE approach is that existing SAE codes can be easily adapted to handle ℓ-dependent potentials without any additional computational cost.

1. Introduction

The Single-Active-Electron (SAE) approximation is a major tool for simulating photoinduced processes in the nonlinear regime of light−atom interactions.1 By providing a simplified description of the atomic system, it offers a framework for efficiently solving the Time-Dependent Schrödinger Equation (TDSE) with high accuracy. According to the principle behind the SAE approximation, the atom of interest is treated as a core plus one active electron, which is typically weakly bound. The core, made up of electrons and the nucleus, is assumed to be frozen, with its electric charge spherically distributed. Thus, the quantum dynamics of the system under the influence of a laser field are governed by the active electron. This simplified picture of the atomic system provided by the SAE approximation can describe the physics of multiphoton single-ionization and, more generally, multiphoton processes involving one-electron transitions, as long as correlation effects are small. In practice, the SAE approximation stands out due to its low computational cost and its ability to capture both the qualitative and quantitative aspects of the underlying physical processes. That is why it has been extensively employed to establish benchmark calculations before running multi-electron codes [1,2,3,4,5,6].

The interaction between the core and the active electron is described by a radial/central model potential. In general, there are two approaches for constructing such models: (i) interpolating the potential directly from ab initio calculations, such as Kohn–Sham density functional theory [7] or Hartree–Slater considerations [8,9,10], and (ii) fitting the potential to reproduce a selected set of experimental, or reference, bound-state energies, e.g., [11,12]. Once the model potential is defined, the accuracy of the SAE energy levels becomes crucial in quantum simulations. This accuracy plays a major role when multiphoton resonant transitions between bound states occur during light−atom interactions. For example as noted in [13], an error of a few percent in the SAE atom’s ground-state energy can result in an error of several hundred percent in multiphoton ionization rates.

To improve the agreement between SAE energy levels and experimental/reference values, we can take advantage of the radial character of the model potential by making it ℓ-dependent.2 We call this the ℓ-SAE approximation. Within the framework of the Time-Independent Schrödinger Equation (TISE), this approach has already been used successfully for the description of the bound states of alkali atoms [12,15,16,17,18], as well as in subsequent collision calculations, e.g., [19,20,21,22]. A general scheme to tackle, in principle, any atom was presented in [8] in the context of the Hartree–Fock–Slater method. In contrast, to the best of our knowledge, ℓ-dependent potentials have rarely been used in TDSE-based simulations. Although the idea behind the ℓ-SAE approximation is hinted at [23], it has only been applied to cesium to study High-order Harmonic Generation near and below the threshold [24]. For this atom, the ℓ-dependent potential was constructed based on the energies, but the accuracy of other TISE-based physical properties was not investigated.

The principal goal of the present work is to demonstrate that the ℓ-SAE approximation not only improves the agreement between experimental energies and the ℓ-SAE values but also enhances the accuracy of relevant properties/observables in the context of light−atom interactions. For concreteness, and due to its experimental relevance, we use the helium atom as a prime example. Interestingly, several ℓ-independent SAE model potentials for helium can be found in the literature [7,13,25,26,27,28,29,30]. However, to the best of our knowledge, no ℓ-dependent model for helium has been reported. While our focus is on a particular atom, it is important to note that the ℓ-SAE approximation is particularly effective for the alkali targets mentioned above, but it can reasonably be expected to also be appropriate for light quasi-two-electron atoms such as beryllium and the other alkaline-earth elements, whose structure is well described within the $LS$-coupling scheme.

This manuscript is organized as follows. We begin by constructing an ℓ-dependent model potential for helium in Section 2. Then, we investigate the accuracy of the model by calculating various atomic properties based on the TISE, with the corresponding results presented in Section 3. In Section 4, we use the ℓ-SAE approximation in time-dependent simulations to obtain results on HHG and 3SB-RABBITT. Whenever possible, we compare our results with reference values from the literature, including standard SAE approaches and other methods that account for correlation effects.

Unless stated otherwise, atomic units (a.u.) are used throughout this manuscript.

2. Potentials for Helium in the ℓ-SAE Approximation

The ℓ-SAE approximation applied to the helium atom offers a suitable framework for a proof of principle beyond quasi-one-electron alkali targets, since helium is well described within the $LS$-coupling scheme with a tightly-bound inner electron (see Figure 1). In particular, the excited bound states can be labeled by ${\left(1sn\mathsf{\ell }\right)}^{1,3}L$ in a non-relativistic treatment.3 Therefore, the energies can be grouped according to $\mathsf{\ell }=L$ and the total spin S. In this work, we focus on the spin-singlet states, $S=0$. However, the approach to tackle triplet states is similar.

In the ℓ-SAE approximation, the radial part of the bound-state wavefunction of the active electron, labeled by $\left(n\mathsf{\ell }\right)$, is determined from the reduced radial Schödinger equation

$$-{\displaystyle \frac{1}{2}}{\displaystyle \frac{d^2{R}_{n\mathsf{\ell }}\left(r\right)}{d{r}^2}}+\left[V_{\mathsf{\ell }}\left(r\right)+{\displaystyle \frac{\mathsf{\ell }(\mathsf{\ell }+1)}{2r^2}}\right]R_{n\mathsf{\ell }}\left(r\right)=E_{n\mathsf{\ell }}{R}_{n\mathsf{\ell }}\left(r\right);{\int }_0^{\infty }{R}_{n\mathsf{\ell }}^2\left(r\right)dr=1.$$

(1)

Here, $V_{\mathsf{\ell }}\left(r\right)$ is the model potential for a given ℓ. In particular, we consider

$$V_{\mathsf{\ell }}\left(r\right)=-{\displaystyle \frac{1}{r}}-\left({\displaystyle \frac{1}{r}}+a_{\mathsf{\ell }}+b_{\mathsf{\ell }}r\right)e^{-c_{\mathsf{\ell }}r},$$

(2)

where $\{a_{\mathsf{\ell }},b_{\mathsf{\ell }},c_{\mathsf{\ell }}\}$ is a set of free ℓ-dependent parameters to be determined. This model potential exhibits the required asymptotic behavior at both small and large r, namely,

$$V_{\mathsf{\ell }}\left(r\right)\to -{\displaystyle \frac{2}{r}},r\to 0,$$

(3a)

and

$$V_{\mathsf{\ell }}\left(r\right)\to -{\displaystyle \frac{1}{r}},r\to \infty .$$

(3b)

To ensure that $V_{\mathsf{\ell }}\left(r\right)$ has infinitely many bound states, the conditions $c_{\mathsf{\ell }}>0$ and $V_{\mathsf{\ell }}^{\prime }\left(r\right)>0$ are imposed. To optimize the value of the parameters $\{a_{\mathsf{\ell }},b_{\mathsf{\ell }},c_{\mathsf{\ell }}\}$ for fixed ℓ, we minimize the weighted sum

$$W(a_{\mathsf{\ell }},b_{\mathsf{\ell }},c_{\mathsf{\ell }})=\sum _n{\omega }_{n\mathsf{\ell }}\left|{\displaystyle \frac{E_{n\mathsf{\ell }}-E_{n\mathsf{\ell }}^{\left(exact\right)}(a_{\mathsf{\ell }},b_{\mathsf{\ell }},c_{\mathsf{\ell }})}{E_{n\mathsf{\ell }}^{\left(exact\right)}}}\right|,$$

(4)

where ${\omega }_{n\mathsf{\ell }}>0$ is a weight factor for the state ($n\mathsf{\ell }$). Additionally, $E_{n\mathsf{\ell }}$ denotes the energy obtained from the numerical solution of Equation (1), with $V_{\mathsf{\ell }}\left(r\right)$ as given in (2). In turn, $E_{nl}^{\left(exact\right)}$ denotes the recommended values of the spin-singlet energy taken from the NIST database [31]. Therefore, W in (4) represents the weighted sum of the relative errors over a selected number of bound states, and its minimization determines the optimal configuration of the free parameters. Specifically, we considered the lowest five bound states from the series $\left(1sn\mathsf{\ell }\right)$ for each value of ℓ. The weights are chosen manually to uniformly maximize the number of exact digits in energies $E_{n\mathsf{\ell }}$.

An accurate determination and minimization of (4) requires accurate values for $E_{nl}$. In our FORTRAN implementation, they were obtained by means of the Lagrange Mesh Method (LMM), as described in [32], which is a highly accurate approximate variational method that discretizes the equation on a non-uniform grid. This ultimately leads to the diagonalization of the discretized Hamiltonian matrix. The LMM used in this work, whose implementation is based on [33], was then coupled to the MINUIT subroutine [34] to carry out the minimization of $W(a_{\mathsf{\ell }},b_{\mathsf{\ell }},c_{\mathsf{\ell }})$. Due to the small number of grid points that LMM requires (∼200), combined with the high efficiency of MINUIT, the minimization procedure is fast and computationally inexpensive. Increasing the grid size beyond 200 points did not significantly change the results within double precision.

Table 1. Optimal parameters for the potentials $V_{\mathsf{\ell }}\left(r\right)$ of Equation (2), designed for the singlet states ${\left(1sn\mathsf{\ell }\right)}^{1}L$ of helium.

ℓ	${\mathit{a}}_{\mathsf{\ell }}$	${\mathit{b}}_{\mathsf{\ell }}$	${\mathit{c}}_{\mathsf{\ell }}$
0	$-0.39353$	0.49574	1.19128
1	$-1.27450$	0.29052	1.19128
$\mathsf{\ell }\ge 2$	$-0.62884$	0.10516	1.19128

shows the optimal parameters. The coefficients for $\mathsf{\ell }\ge 2$ are assumed to be the same since the centrifugal barrier, together with the Coulomb term $-1/r$, mainly determines the energies $E_{n\mathsf{\ell }}$ with high accuracy. For this reason, the coefficient $c_{\mathsf{\ell }}$ was optimized for ${\left(1s^2\right)}^{1}S$ and kept fixed for non-S states. A plot of the potentials $V_{\mathsf{\ell }}\left(r\right)$, multiplied by r, is shown in Figure 2 for $\mathsf{\ell }=0,1,2$. Noticeable differences can be observed, especially at intermediate distances $1\lesssim r\lesssim 5$. Of particular interest is the fact that $r{V}_{\mathsf{\ell }}\left(r\right)$ is clearly larger than $-1$ for the S and P Rydberg series. This would be impossible for purely screening one nuclear charge by the inner $1s$ electron. However, the potentials also account for the exchange integral, which represents a repulsive effect for the singlet series.

In our model, the ionization potential is $I_p=0.903571\mathrm{a}.\mathrm{u}.$ according to $V_0\left(r\right)$. Meanwhile, the recommended value is $I_p^{\left(\mathrm{NIST}\right)}=0.903569\mathrm{a}.\mathrm{u}.$ [31]. Thus, there is a discrepancy of $2\times {10}^{-6}$ a.u. In

Table 2. Helium energy levels for selected low-lying spin-singlet states ${\left(1sn\mathsf{\ell }\right)}^{1}L$ with $n\le 4$. Both reference (NIST) and ℓ-SAE energies are rounded to the number of digits shown.

State	Energy (a.u.)
	NIST	ℓ-SAE
${\left(1s^2\right)}^{1}S$	−2.9036	−2.9036
${\left(1s2s\right)}^{1}S$	−2.1460	−2.1427
${\left(1s2p\right)}^{1}P$	−2.1238	−2.1238
${\left(1s3s\right)}^{1}S$	−2.0613	−2.0613
${\left(1s3d\right)}^{1}D$	−2.0556	−2.0557
${\left(1s3p\right)}^{1}P$	−2.0551	−2.0551
${\left(1s4s\right)}^{1}S$	−2.0336	−2.0337
${\left(1s4d\right)}^{1}D$	−2.0313	−2.0313
${\left(1s4f\right)}^{1}F$	−2.0313	−2.0313
${\left(1s4p\right)}^{1}P$	−2.0311	−2.0311

, we present the energies for the first ten low-lying states of helium. For these levels, the maximum deviation is $0.003$ a.u., which is found for the ${\left(1s2s\right)}^{1}S$ state and represents a 0.002% error. It is well known that this state is notoriously difficult to describe also by ab initio methods [35].

We also investigated the accuracy of the ℓ-SAE energies for the Rydberg S, P, D, and F series up to $n=35$, the largest value for which reference values are available in the NIST database. As seen in Figure 3, the percent error smoothly decreases for $n\ge 5$ for each series.

For the purposes of the present work, the accuracy in energies provided by the potentials $V_{\mathsf{\ell }}\left(r\right)$ is sufficient to show the scope and improvement that the ℓ-SAE approximation offers. Moreover, we observed that increasing the number of free parameters4 in $V_{\mathsf{\ell }}\left(r\right)$ does not lead to a significant improvement in the accuracy of the energy levels.

By construction, the ℓ-SAE energy levels are more accurate than those from standard SAE models. For example, the deviations from the reference NIST ionization energy, based on the SAE potentials presented in [7,27,28,36], are 4.5%, 0.03%, 1.5%, and 2.4%, respectively. However, a detailed validation of the ℓ-SAE approach requires the study of other observables and relevant physical quantities. In the next section, based on the solution of the reduced radial TISE (1), we calculate some of them, especially those that are relevant in the context of light−atom interactions.

3. TISE-Based Atomic Properties

Once the functions $R_{n\mathsf{\ell }}$ are determined by solving (1), the wave functions of the spin-singlet bound states ${\left(1sn\mathsf{\ell }\right)}^{1}L$ are given by

$${\psi }_{1sn\mathsf{\ell }}^{\left(He\right)}({\mathit{r}}_1,{\mathit{r}}_2)={\displaystyle \frac{1}{\sqrt{2}}}\left[{\psi }_{1s}^{\left(H{e}^{+}\right)}\left({\mathit{r}}_1\right)·{\psi }_{n\mathsf{\ell }m}\left({\mathit{r}}_2\right)+{\psi }_{1s}^{\left(H{e}^{+}\right)}\left({\mathit{r}}_2\right)·{\psi }_{n\mathsf{\ell }m}\left({\mathit{r}}_1\right)\right];$$

(5a)

$${\psi }_{n\mathsf{\ell }m}\left({\mathit{r}}_i\right)=R_{n\mathsf{\ell }}\left(r_i\right)Y_{\mathsf{\ell }m}({\theta }_i,{\phi }_i);i=1,2.$$

(5b)

Here, $Y_{\mathsf{\ell }m}(\theta ,\phi )$ denotes the familiar spherical harmonic of degree ℓ and order m. These wave functions can be straightforwardly extended to continuum states, specifically when the active electron has been detached from the core. They are used below to calculate observables. However, as will be shown, only the functions $R_{n\mathsf{\ell }}\left(r\right)$ are relevant in practice.

3.1. Oscillator Strengths

Given two bound states, say ${\left(1sn\mathsf{\ell }\right)}^{1}L$ and ${\left(1s{n}^{\prime }{\mathsf{\ell }}^{\prime }\right)}^1{L}^{\prime }$, we computed the averaged oscillator strength between them, which is defined by5

$$f_{n^{\prime }{\mathsf{\ell }}^{\prime }n\mathsf{\ell }}={\displaystyle \frac{2}{3}}{\displaystyle \frac{\max \{{\mathsf{\ell }}^{\prime },\mathsf{\ell }\}}{2\mathsf{\ell }+1}}(E_{n^{\prime }{l}^{\prime }}-E_{nl}){\left(R_{n^{\prime }{\mathsf{\ell }}^{\prime }n\mathsf{\ell }}\right)}^2.$$

(6)

Here, the dipole integral is given by

$$R_{n^{\prime }{\mathsf{\ell }}^{\prime }n\mathsf{\ell }}={\int }_0^{\infty }r{R}_{n^{\prime }{\mathsf{\ell }}^{\prime }}\left(r\right)R_{n\mathsf{\ell }}\left(r\right)dr.$$

(7)

See [37] for details. The selection rule, $\mathsf{\ell }={\mathsf{\ell }}^{\prime }\pm 1$, must be considered in (6) to define the dipole-allowed transitions. For some of these transitions, the corresponding oscillator strengths are shown in

Table 3. Averaged oscillator strengths $f_{n^{\prime }{\mathsf{\ell }}^{\prime }n\mathsf{\ell }}$ for some selected dipole-allowed transitions between spin-singlet helium bound states. For each state in the first column, the first row corresponds to the results obtained in the framework of the ℓ-SAE approximation. The reference values [38] appear in the second row. All numbers are rounded to the digits displayed.

	$1^{1}S$	$2^{1}S$	$3^{1}S$	$4^{1}S$
$2^{1}P$	0.241254	0.328273	$-0.139731$	$-0.025059$
	0.276165	0.376440	$-0.145470$	$-0.025870$
$3^{1}P$	0.066093	0.156859	0.625162	$-0.313562$
	0.073435	0.151342	0.626193	$-0.307507$
$4^{1}P$	0.027180	0.047910	0.141081	0.880303
	0.029863	0.049155	0.143889	0.858021
	$2^{1}P$	$3^{1}P$	$4^{1}P$
$3^{1}D$	0.714401	$-0.021810$	$-0.015310$
	0.710164	$-0.021140$	$-0.015303$
$4^{1}D$	0.121109	0.649095	$-0.040219$
	0.120270	0.648104	$-0.040062$
	$3^{1}D$	$4^{1}D$
$4^{1}F$	1.015314	0.002622
	1.015083	0.002492

and compared with reference values from [38]. The maximum deviation reaches 13% for the $2^{1}S\to 2^{1}P$ transition, mostly due to the inaccuracy of the $2^{1}S$ state. Overall, however, the remaining deviations are significantly smaller than those obtained with the widely used SAE potential proposed by Tong and Lin [27], which reaches almost a factor of 3 for the $3^{1}P\to 3^{1}D$ transition.

3.2. Static and Dynamic Polarizabilities

We now investigate the ℓ-SAE predictions for the so-called static and dynamic polarizabilities defined by

$$\alpha \left(0\right)={\displaystyle \frac{4}{3}}\sum _n{\displaystyle \frac{|R_{np1s}{|}^2}{E_{np}-E_{1s}}}$$

(8a)

and

$$\alpha \left(\omega \right)={\displaystyle \frac{4}{3}}\sum _n{\left|R_{np1s}\right|}^2{\displaystyle \frac{E_{np}-E_{1s}}{{(E_{np}-E_{1s})}^2-{\omega }^2}},$$

(8b)

respectively. The summations in (8a) and (8b) are carried out over both bound and continuum states. In (8b), $\omega$ is the frequency that characterizes a monochromatic (thus infinitely long) laser field interacting with the atom.

Our ℓ-SAE model generally yields polarizabilities smaller than the reference values presented in [39], which account for correlation effects, see Figure 4. For example, the static polarizability comes out as $\alpha \left(0\right)=1.296921\mathrm{a}.\mathrm{u}.$, while the reference value is $\alpha \left(0\right)=1.383193\mathrm{a}.\mathrm{u}.$, corresponding to a difference of 6%. For the dynamic case, we observed qualitative agreement for all frequencies considered, namely, $\omega \in [0,0.89]$. The error for the dynamic polarizability does not exceed 8% in the region $\omega \in [0,0.77]$. In the vicinity of $\omega \approx E_{np}-E_{1s}$, the dynamic polarizability exhibits sharp antisymmetric peaks, known as resonances. Thanks to the accurate ℓ-SAE bound-state energies (cf. Table 2), the resonance positions are well reproduced. The standard SAE approximation, due to its limited energy accuracy, is unable to accurately locate these resonances.

3.3. Photoionization Cross Section

Even though the ℓ-SAE approach yields improved bound-state energies, one might assume that observables without explicit dependence on them will not necessarily improve in accuracy. Calculating the photoionization cross section provides a testing ground, as it encodes the transition rate from an initial bound state to a final continuum state due to the absorption of a photon with frequency $\omega$. Therefore, the energy part depends solely on the ionization potential provided by the model.

In the electric dipole approximation, the photoionization cross section is given by [40]

$$\sigma ={\displaystyle \frac{4{\pi }^2\omega }{3c}}|\langle R_{1s}\left|r\right|R_{ϵp}{\rangle |}^2;(ϵ=\omega -I_p),$$

(9)

where $ϵ$ denotes the energy of the released p-electron. In (9), we assume that the helium atom is initially in its ground state.

Figure 5 shows $\sigma$ from the ℓ-SAE approximation and two SAE models for comparison. In general, the ℓ-SAE provides cross sections closer to the benchmark values than the standard SAE models. One of the SAE models we used, SAE (I), yields an accurate ionization potential $I_p=0.9038$ a.u. [27], with an accuracy comparable to that of the ℓ-SAE model. Therefore, we conclude that our ℓ-SAE approach also improves the bound-continuum dipole matrix elements corresponding to the extension of (7), due to the enhanced accuracy of the continuum wave function $R_{ϵp}$ that also enters in (9).

4. Solution of the TDSE in the ℓ-SAE Approximation

In this Section, we investigate the dynamics of the active electron within the ℓ-SAE approximation for simulations of pulsed laser−atom interactions. We limit our models to the dipole approximation and use the length gauge.

4.1. Generalities

The ℓ-SAE approach is particularly suitable when the wave function is expressed in terms of partial waves written in spherical coordinates $\{r,\theta ,\phi \}$, i.e.,

$$\psi (t,r,\theta ,\phi )=\sum _{\mathsf{\ell }=0}^{L_{max}}\sum _{m=-\mathsf{\ell }}^{\mathsf{\ell }}{u}_{\mathsf{\ell }m}(t,r)Y_{\mathsf{\ell }m}(\theta ,\phi ).$$

(10)

The equations governing the time evolution6 of the components $u_{\mathsf{\ell }m}(t,r)$ are barely modified when ℓ-dependent potentials are considered: the only required modification is the replacement $V\left(r\right)\to V_{\mathsf{\ell }}\left(r\right)$. Thus, standard SAE codes require minimal adjustments to implement the ℓ-SAE approximation, thereby making it particularly convenient for computational applications. In this context, we updated the SAE code described in [42] to handle ℓ-dependent potentials. Essentially, the code uses the celebrated second-order Strang splitting to propagate the initial-state wave function, typically the ground state, in time according to

$$\psi (t,r,\theta ,\phi )=e^{-i{\displaystyle \frac{t}{2}}{H}_0}{e}^{-it{H}_I\left({\displaystyle \frac{t}{2}}\right)}{e}^{-i{\displaystyle \frac{t}{2}}{H}_0}\psi (0,r,\theta ,\phi )+\mathcal{O}\left(t^3\right),$$

(11)

where $V_{\mathsf{\ell }}\left(r\right)$ only appears in $H_0$. Further technical details on the computational implementation are provided in [43]. In what follows, we consider a linearly polarized laser field. As a result, the functions $u_{\mathsf{\ell }0}\left(r\right)$, $\mathsf{\ell }=0,1,\dots$ are the only contributing components in (10).

4.2. High-Order Harmonic Generation (HHG)

We consider a helium atom interacting with an intense short-pulse laser, leading to the generation of high-order harmonics. In particular, we consider the linearly-polarized laser field

$$\mathit{E}\left(t\right)=\sqrt{{\displaystyle \frac{2I_0}{{ϵ}_{0}c}}}{sin}^2\left({\displaystyle \frac{\omega t}{2N}}\right)cos\left(\omega t\right)\widehat{\mathit{z}},$$

(12)

where c is the speed of light, ${ϵ}_0$ the vacuum permittivity, $I_0$ the laser peak intensity, N the number of optical cycles, and $\omega$ the (central) frequency of the driving laser. The electric field is assumed to vanish outside the time interval $0<t<2\pi N/\omega$. We considered two sets of field parameters: (i) $\omega =0.183280$ a.u. (248.6 nm), $N=30$, $I_0=1\times {10}^{14}$ W/cm² and (ii) $\omega =0.042822$ a.u. (1064 nm), $N=60$, $I_0=5\times {10}^{13}$ W/cm². For these sets, benchmark results were recently established in the context of HHG [44].

As soon as the pulse is over, we calculate the spectral density, which represents the dipole-radiated energy per frequency $\omega$. It is given by [45]

$$S\left(\omega \right)={\displaystyle \frac{2}{3\pi c^3}}{\left|\tilde{a}\left(\omega \right)\right|}^2;\tilde{a}\left(\omega \right)={\int }_{-\infty }^{\infty }a\left(t\right)e^{i\omega t}dt.$$

(13)

Here, $a\left(t\right)$ is the second derivative with respect to the time of the induced dipole moment $d\left(t\right)=-\langle \psi \left(t\right)\left|z\right|\psi \left(t\right)\rangle$. Below, we compare our induced dipole moment and its spectral density with the results reported in [44], especially those coming from a standard SAE model and the RMT code [2], which accounts for both multi-electron and multi-channel dynamics.

4.2.1. Induced Dipole Moment

For both sets of field parameters, (i) and (ii), the corresponding dipole moment is shown in Figure 6 and Figure 7, respectively. The reference results from RMT and the SAE approximation are also included. In general, we observe that the induced dipole moment predicted by our ℓ-SAE model is smaller than the reference obtained via RMT, which presumably yields the most accurate results [44]. However, this is expected, since the static polarizability in our ℓ-SAE model is 8% smaller than the actual value. In contrast, the SAE approach used in [44] predicts a slightly larger dipole moment. Despite the differences between the dipole moments, they are quite similar. In fact, the differences become more noticeable when looking at their spectral content, as discussed below.

4.2.2. Spectral Density

Figure 8 exhibits spectral density compared to the RMT and SAE predictions for both sets of field parameters (i) and (ii). We observe that the ℓ-SAE model captures the main features of the HHG spectra. In contrast to the SAE for set (i), the ℓ-SAE results are similar to the RMT predictions from a 6-state model. For set (ii), the ℓ-SAE results are again close to the 3-state RMT data, especially in the plateau region.

4.3. 3SB-RABBITT

As a final investigation, we simulated 3SB-RABBITT, a variant of 1SB-RABBITT in which the XUV comb consists of multiple odd harmonics of the frequency-doubled IR field. Consequently, the photoelectron spectrum in 3SB-RABBITT exhibits three sidebands between the main peaks, denoted as low, center, and high. For simplicity, we provide only a brief discussion of this interferometry technique, focusing on the results obtained for the observable of our interest, the RABBITT phase ${\Phi }_R\left(\theta \right)$.

Motivated by a recent experimental realization of 3SB-RABBITT in helium [46], quantum simulations [47] were performed using the RMT code. In this approach, four different models were used in RMT: 1-, 3-, 6-, and 10-state models, representing the number of states in the close-coupling expansion. Thus, we can compare the performance of the ℓ-SAE approximation with each RMT model. In our ℓ-SAE simulations, we used the same experimentally based pulse employed in [47], which consists of a time-delayed combination of linearly polarized IR and XUV pulses with femtosecond duration. Additionally, we applied the same data-processing method as in [47].

When the time delay $\tau$, given in fractions of the IR period, between the two fields is varied, the photoelectron signal at the detection angle $\theta$, denoted by $S(\tau ,\theta )$, oscillates according to

$$S(\tau ,\theta )=A\left(\theta \right)+B\left(\theta \right)cos[4\omega \tau -{\Phi }_R\left(\theta \right)],$$

(14)

where A is the average signal and B is the amplitude of the oscillation. These parameters depend on the energy of the ejected electron (i.e., the specific sideband of interest) and are determined by fitting the signal to (14). To calculate $S(\tau ,\theta )$, we project the wave function at the end of the pulse onto the continuum states, which are obtained for each ℓ by solving the TISE with the potential $V_{\mathsf{\ell }}\left(r\right)$.

Here, we present results for the SB 14 group with an IR intensity of $7\times {10}^{11}$ W/cm². This group corresponds to three sidebands located between the energy peaks corresponding to absorption at the 13th and 15th harmonics. Similar quality in the ℓ-SAE results is observed for the other sidebands reported in [47]. Figure 9 shows the ${\Phi }_R\left(\theta \right)$ phase for the low, center, and high SBs from the ℓ-SAE approach and three different RMT models. There is overall qualitative agreement between the phases, especially at angles $\theta \le {55}^{\circ }$. For these angles, in the case of the center and high SBs, the quality of the ℓ-SAE results is comparable to that of the 6- and 10-state RMT models. Beyond ${55}^{\circ }$, except for the lower SB, significant deviations appear. Specifically, the phase jump over a narrow angular range (the full jump is shown in the right panels of Figure 9 for better visibility) occurs in opposite directions. This could be related to (i) the very small photoelectron signal, making the fitting procedure potentially unreliable; (ii) the difference between the ℓ-SAE and RMT bound-state energy levels; or (iii) an effect of channel coupling. Further investigations regarding the validity of the ℓ-SAE approach in a 3SB-RABBITT setup are currently in progress [48].

It is worth noting that running the ℓ-SAE code for a single time delay requires approximately two hours of computing time on a modern personal computer with about 10 cores. In contrast, the same simulation using the 10-state RMT model takes an entire day and nearly 800 processors, thus requiring the use of a supercomputer.

5. Conclusions

In this work, we developed and described the ℓ-SAE approach, which incorporates model potentials that depend on the angular momentum of the active electron. As demonstrated with the helium target, these potentials yield more accurate bound-state energies than those obtainable with the standard SAE approach. The ℓ-SAE method also improves the accuracy of the calculated oscillator strengths, the static and dynamic polarizabilities, and the photoionization cross section.

In the context of TDSE simulations, the inclusion of ℓ-dependent potentials does not require significant modifications to existing SAE computer codes, and the computational resources needed for its implementation remain largely unchanged. As a result, we were able to simulate HHG and 3SB-RABBITT processes efficiently and found that, for the observables of interest here, i.e., single-electron processes, the ℓ-SAE results are of similar accuracy to those from coupled-state all-electron models such as RMT, except for the RABBITT phase for large detection angles.

Finally, we note that the ℓ-SAE approximation can also be extended to study molecular systems, as demonstrated for the diatomic molecule CO₂ in [49].

6. A Few Words on Barry Irwin Schneider by J.C.d.V.

I had the chance to talk to Barry a few times, and I was always impressed by his extensive knowledge. In particular, we discussed the LMM, one of the methods used in this work. During our conversations, he encouraged me to develop a user-friendly computational implementation of LMM. Although the idea had already been in my mind for some time, it was Barry’s suggestion that ultimately convinced me to develop such a code. Today, my user-friendly implementation is not only used by myself but also by colleagues and educators, thus furthering the impact of the method.