1 INTRODUCTION

Multi-electron dynamics induced in an atomic system by an intense infrared (IR) field causes several noticeable effects: the broadband enhancement (or the giant resonance) in the spectra of high-order harmonics generation (HHG) in xenon [1–4], the resonant enhancement of individual harmonics in HHG spectra of transition-metal ions [5–7], the multi-electron polarization screening [8], etc. The description of these effects on the qualitative level is possible within existing analytical parameterization of the HHG yield given by the product of the electron wave packet and the photorecombination cross-section [9–11]. The electron wave packet describes the tunneling ionization and single-electron dynamics of the valence electron in an intense IR field, while the photorecombination cross-section, related to the photoionization cross-section by the principle of detailed balance, determines the contribution of multi-electron effects at the recombination stage [12, 13]. The quantitative description of the multi-electron contribution to HHG spectra can be performed within numerical integration of the time-dependent Hartree–Fock method [3, 14–16], the density functional theory [4, 8, 17, 18], and the R-matrix method [19].

The electron-correlated dynamics in an intense IR field can also be induced by the additional interaction of an IR-dressed atomic system with an attosecond pulse having a carrier frequency in the extreme-ultraviolet (XUV) range [20–22]. For instance, the interaction of the XUV pulse with an atomic system can lead to the liberation of electrons from low-lying shells with subsequent Auger decay of the core hole [23]. The joint interaction of the IR field and attosecond pulse with an atomic system opens up the possibility of extracting from experimental data unprecedentedly accurate values for Auger decay times of tens attosecond.

The joint interaction of an intense IR field and attosecond XUV pulse with an atom induces new HHG channels. Recently, the corresponding single-electron HHG channels were studied [11, 24–27]. The theoretical description of XUV-induced channels is based on the perturbative nature of the attosecond pulse interaction with an atomic system (even for intensities \( \sim {\kern 1pt} {{10}^{{16}}}\) W/cm2 comparable to the intra-atomic one) additionally subjected to intense IR field, whose nonlinear IR-induced effects can be treated within the adiabatic theory [11, 28, 29]. The dynamics in one of these XUV-induced HHG channels can be described as follows: an atomic electron is liberated from an atom by IR-induced tunneling, then it propagates semi-classically in the IR-dressed continuum, and finally recombines into the initial state with simultaneous absorption of the XUV photon [11, 24, 25]. This XUV-assisted HHG channel caused the interest for several reasons. First, it makes possible to generate photons with energy exceeding the plateau cutoff in the IR-induced HHG spectrum [30] by the energy of the XUV photon [24, 31]. Second, this channel is of interest for attosecond pulse metrology [25, 26] and for the study of time-frequency dynamics of HHG in an IR field [27].

In this work, we show that multi-electron effects may significantly change the HHG dynamics in the IR field and attosecond XUV pulse. Indeed, let us consider the XUV pulse, whose carrier frequency may induce a resonance between valence and inner shells electrons. An intense IR field ionizes the atomic system, i.e., a valence electron from the outer shell tunnels into the continuum and generates a vacancy in the valence shell. This vacancy can be occupied by an electron from the inner shell due to XUV-induced resonant transition, while the formed vacancy in the inner shell is filled by the electron in the IR-dressed continuum through further photorecombination (see Fig. 1). The energy of the generated photon within this two-electron HHG scenario coincides with the photon energy in a single-electron XUV-assisted HHG channel [24]. However, the efficiency of HHG can be significantly enhanced due to XUV-induced atomic resonance. We study this two-electron HHG scenario by considering the xenon (Xe) atom interacting with IR and XUV pulses within the density functional theory, i.e., in the framework of the time-dependent Kohn–Sham equations (TDKSE) [4]. The numerical results demonstrate that the HHG yield beyond the IR-induced plateau in the HHG spectrum increases due to the XUV-induced resonant two-electron dynamics. Moreover, we show the significant difference in the dependences of HHG yield in single- and two-electron HHG scenarios on the time delay between the IR and XUV pulses and the duration of the XUV pulse. In particular, for a short attosecond pulse, the single-electron XUV-assisted HHG channel is most pronounced for the delay time between the IR and XUV pulses, which is close to the recombination time of an electron with the corresponding energy [25]. In contrast, for the two-electron HHG scenario, the time delay determines the moment of transition from the low-lying shell to the valence state, and for high HHG efficiency, it is sufficient that before this moment, an electron leaves the valence level through the interaction with the IR field.

Fig. 1.
figure 1

(Color online) The sketch of HHG with the participation of two electrons during the resonant excitation of the internal transition by an XUV pulse in xenon atom. Here, \(\hbar \Omega = {{E}_{{5p}}} - {{E}_{{4d}}}\) is the energy of the absorbed photon of the XUV pulse, \({{I}_{p}}\) is the ionization potential, \({{E}_{k}}\) is the energy of the returning electron, \(\hbar \omega = {{E}_{k}} + {{I}_{p}} + \hbar \Omega \) is the energy of the emitted photon, \(\hbar \) is the reduced Planck constant.

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2 TIME-DEPENDENT KOHN–SHAM EQUATIONS

To describe the multi-electron dynamics, we use the TDKSE [32] (the atomic units are used in this work, unless otherwise stated):

$$i\frac{{\partial {{\psi }_{j}}({\mathbf{r}},t)}}{{\partial t}} = \hat {H}{{\psi }_{j}}({\mathbf{r}},t),\quad j = 1...N{\text{/}}2,$$
(1)
$$\hat {H} = - \frac{{{{\nabla }^{2}}}}{2} - \frac{N}{r} + {{\hat {V}}_{{{\text{ee}}}}} + {{\hat {V}}_{{\text{L}}}},$$
(2)

where \({{\psi }_{j}}({\mathbf{r}},t)\) is the time-dependent wavefunction of the jth Kohn–Sham orbital, N = 54 is the total number of electrons in Xe atom coinciding with the charge of the nucleus, \({{\hat {V}}_{{{\text{ee}}}}}\) is the potential describing electron-electron interaction, \({{\hat {V}}_{{\text{L}}}}\) is the operator of the interaction of an atomic electron with IR and XUV pulses. The electron–electron interaction is represented as the sum of the Hartree potential and the exchange-correlation potential in the LB94 approximation [33]:

$${{\hat {V}}_{{{\text{ee}}}}} = {{V}_{{\text{H}}}} + {{V}_{{{\text{xc}}}}},$$
(3)
$${{V}_{{\text{H}}}} = \int \frac{{\rho ({\mathbf{r}}',t)}}{{{\text{|}}{\mathbf{r}} - {\mathbf{r}}'{\text{|}}}}d{\mathbf{r}}',\quad \rho ({\mathbf{r}},t) = 2\sum\limits_{n = 1}^{N/2} {\text{|}}{{\psi }_{n}}({\mathbf{r}},t){{{\text{|}}}^{2}},$$
(4)
$$\begin{gathered} {{V}_{{{\text{xc}}}}} = {{V}_{{{\text{LDA}}}}} - \frac{{{{2}^{{1/3}}}\,\beta {{\chi }^{2}}({\mathbf{r}},t)\,{{\rho }^{{1/3}}}({\mathbf{r}},t)}}{{1 + {{2}^{{1/3}}}3\,\beta \chi ({\mathbf{r}},t)\,{{{\sinh }}^{{ - 1}}}{{{[2}}^{{1/3}}}\chi ({\mathbf{r}},t)]}}, \\ \chi ({\mathbf{r}},t) = \frac{{{\text{|}}\nabla \rho ({\mathbf{r}},t){\text{|}}}}{{{{\rho }^{{4/3}}}({\mathbf{r}},t)}}, \\ \end{gathered} $$
(5)

where \({{V}_{{{\text{LDA}}}}}\) is the exchange-correlation potential in the local density approximation [32], β = 0.05. The index j enumerates Kohn–Sham orbitals, which are characterized by its own principal quantum number (\(n\)), angular momentum (\(l\)) and its projection (\(m\)) onto the quantization axis in accordance with the electronic configuration of the Xe atom, [Kr] \(4{{d}^{{10}}}5{{s}^{2}}5{{p}^{6}}\), where [Kr] is the electronic configuration of the krypton atom. The initial conditions for \({{\psi }_{j}}\) are the eigenfunctions \(\psi _{j}^{{(0)}}\) of the Hamiltonian in the absence of IR and XUV pulses. The wavefunctions \(\psi _{j}^{{(0)}}\) can be written as \(\psi _{j}^{{(0)}} = {{r}^{{ - 1}}}{{R}_{{nl}}}(r){{Y}_{{lm}}}(\theta ,\varphi )\), where \({{R}_{{nl}}}(r)\) is the radial part of the wave function, \({{Y}_{{lm}}}(\theta ,\varphi )\) is the spherical harmonic, \(\theta \) and \(\varphi \) are the polar and azimuthal angles of the spherical coordinate system, respectively. The angle \(\theta \) is measured from the axis \(z\), which coincides with the direction of the electric field.

The interaction with the field is considered in the dipole approximation and the length gauge:

$${{\hat {V}}_{{\text{L}}}} = z\mathcal{F}(t),\quad \mathcal{F}(t) = {{\mathcal{F}}_{{{\text{IR}}}}}(t) + {{\mathcal{F}}_{{{\text{XUV}}}}}(t - \tau ),$$
(6)
$${{\mathcal{F}}_{\alpha }}(t) = {{F}_{\alpha }}{{f}_{\alpha }}(t)\cos ({{\omega }_{\alpha }}t),\quad \alpha = \{ {\text{IR}}{\text{,XUV}}\} ,$$
(7)

where \({{f}_{\alpha }}(t)\), \({{F}_{\alpha }}\), \({{\omega }_{\alpha }}\) are the envelope, peak amplitude, and carrier frequency of IR and XUV pulses, \(\tau \) is the time delay between IR and XUV pulses (\(\tau = 0\) corresponds to the coincidence of envelope maxima of XUV and IR pulses).

In the density functional theory, all physically observable quantities are expressed in terms of electron density \(\rho ({\mathbf{r}},t)\). For example, the dipole acceleration a(t), which determines the probability of generating a harmonic [\(R(\omega )\)] at frequency \(\omega \), is given by the expression

$$a(t) = \frac{{{{d}^{2}}}}{{d{{t}^{2}}}}\int\limits_{ - \infty }^\infty {z\rho ({\mathbf{r}},t)d{\mathbf{r}} = - N\mathcal{F}(t) - N\int \frac{z}{{{{r}^{3}}}}\rho ({\mathbf{r}},t)d{\mathbf{r}}} ,$$
$$R(\omega ) = \frac{{{\text{|}}a(\omega ){{{\text{|}}}^{2}}}}{{2\pi {{c}^{3}}}},\quad a(\omega ) = \int\limits_{ - \infty }^\infty {a(t){{e}^{{i\omega t}}}dt} ,$$

where c is the speed of light.

The results of the TDKSE solution are compared with the results given by the single-active electron (SAE) approximation. SAE results are obtained from the numerical solution of the time-dependent Schrödinger equation for specified form of the effective potential \({{V}_{{{\text{eff}}}}}(r)\) [34] given by a smoothed Coulomb potential [35]

$${{V}_{{{\text{eff}}}}}(r) = - \frac{1}{r}\left[ {\tanh \left( {\frac{r}{a}} \right) + \frac{r}{b}{\text{sec}}{{{\text{h}}}^{2}}\left( {\frac{r}{a}} \right)} \right].$$
(8)

For the initial state we use the state 2p0 represented as \(\psi ({\mathbf{r}},0) = {{R}_{{2p}}}(r){{Y}_{{10}}}(\theta ,\varphi )\). This choice of potential and initial state makes possible to avoid numerical artifacts associated with using 5p0 as the initial state, for which unphysical transitions to a free low-lying level 4d0 occur due to the interaction with the resonant external electric field. We use following constants \(a = 0.564\), b = 12, for which the energy of the state in potential (8) is equal to the ionization potential of the Xe atom (in absolute value). For this case, the low-lying \(1s\) state is located 97.3 eV below the 2p0 level, which is much higher than the considered photon energies of XUV pulse. The energies of the first three excited states \(3s\), \(3p\), \(3d\) are –2.33, –2.59, ‒1.52 eV, respectively, while in the multi-electron system described by TDKSE, the energies of the lower three stationary unoccupied states \(5d\), \(6s\), \(6p\) are –2.33, –3.9, ‒2.22 eV, respectively. Note that the structure of the levels of high-energy states has little effect on the developing of the HHG spectrum, which is well described within the product of electron wave packet and the photorecombination cross section [9–11]. Figure 2 shows the radial part \({{R}_{{2p}}}(r)\) of the initial wave function in SAE approximation, and Kohn–Sham radial parts \({{R}_{{nl}}}(r)\) of the valence orbital \(5{{p}_{0}}\) of the Xe atom and the orbital \(4{{d}_{0}}\) involved in the discussed two-electron HHG mechanism. In SAE approximation, the dipole acceleration is expressed in terms of the wavefunction [8, 35, 36]:

$$a(t) = - \mathcal{F}(t) - \int {\text{|}}\psi ({\mathbf{r}},t){{{\text{|}}}^{2}}\frac{{\partial {{V}_{{{\text{eff}}}}}}}{{\partial z}}d{\mathbf{r}}.$$
(9)
Fig. 2.
figure 2

(Color online) Radial parts \({{R}_{{nl}}}(r)\) of the wave functions of stationary orbitals \(4d\) (thick blue line) and \(5p\) (thin red line) within the density functional theory, and the radial part \({{R}_{{2p}}}\) of the initial \(2p\) state in the SAE approximation (black dashed line).

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3 NUMERICAL RESULTS AND THEIR DISCUSSION

The numerical solution of the system of Eqs. (1) and (2) was performed for the Xe atom interacting with the IR field and attosecond XUV pulse (see Eq. (6)). The IR field was specified in the form of a pulse with a carrier frequency \({{\omega }_{{{\text{IR}}}}} = 1\) eV (\({{\lambda }_{{{\text{IR}}}}} = \) 1.2 μm), envelope fIR(t) = \({\text{co}}{{{\text{s}}}^{2}}(\pi t{\text{/}}{{\mathcal{T}}_{{{\text{IR}}}}})\) (for \(t \in ( - {{\mathcal{T}}_{{{\text{IR}}}}}{\text{/}}2,{{\mathcal{T}}_{{{\text{IR}}}}}{\text{/}}2)\)), where \({{\mathcal{T}}_{{{\text{IR}}}}} = 20.7\) fs (five full periods of the IR field) is the duration of the IR pulse, and peak intensity IIR = \(2 \times {{10}^{{14}}}\) W/cm2. The envelope of the attosecond pulse was parameterized by a Gaussian function fXUV(t) = \({\text{exp}}[ - 2{\text{ln}}(2){{t}^{2}}{\text{/}}\mathcal{T}_{{{\text{XUV}}}}^{2}]\), where \({{\mathcal{T}}_{{{\text{XUV}}}}}\) is the full-width at half-maximum of intensity. For all calculations presented in Figs. 3 and 4, the peak intensity of the XUV pulse is \(5 \times {{10}^{{13}}}\) W/cm2, except Fig. 3c, which also considers the intensity 5 × 1012 W/cm2. The numerical method for solving TDKSE (see Eq. (1)) is described in [4]. The numerically calculated binding energies of Kohn–Sham orbitals in the absence of laser fields (see Table 1 in [4]) are in good agreement with experimental data and indicate the possibility of dipole (resonance) transition between \(5p\) and \(4d\) shells at frequency \(\Omega = {{E}_{{5p}}} - {{E}_{{4d}}} \approx 57.2\) eV (see Fig. 1). Therefore, in the numerical calculations discussed below, the carrier frequency of the attosecond pulse is varied in the range 50–60 eV.

Fig. 3.
figure 3

(Color online) HHG spectra for xenon atom interacting with the IR field and attosecond pulse having frequencies \({{\omega }_{{{\text{XUV}}}}} = {\text{50}}\) eV (a); \({{\omega }_{{{\text{XUV}}}}} = {\text{54}}{\text{.2}}\) eV (b); \({{\omega }_{{{\text{XUV}}}}}\) = 60 eV (c). The duration of the attosecond pulse is \({{\mathcal{T}}_{{{\text{XUV}}}}}\) = 0.6 fs, the time delay between the IR and XUV pulses is \(\tau = - 1.2\) fs, and the peak intensity of the IR pulse is \({{I}_{{{\text{IR}}}}} = 2 \times {{10}^{{14}}}\) W/cm2. Thick black and thick blue lines are the TDKSE solution for the intensity of the XUV pulse \({{I}_{{{\text{XUV}}}}} = 0\) and \(5 \times {{10}^{{13}}}\) W/cm2, respectively; thin red dotted lines are the calculations in the SAE approximation for \({{I}_{{{\text{XUV}}}}} = 5 \times {{10}^{{13}}}\) W/cm2. The dashed cyan line on the panel (c) is the TDKSE solution for \({{I}_{{{\text{XUV}}}}} = 5 \times \) 1012 W/cm2.

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Fig. 4.
figure 4

(Color online) Time-frequency HHG spectrograms for XUV pulse intensity \(5 \times {{10}^{{13}}}\) W/cm2, carrier frequency \({{\omega }_{{{\text{XUV}}}}} = 50\) eV, and three durations of 0.6 (a), 0.4  (c), and 0.2 fs (e) and the corresponding HHG spectra (b, d, and f). The remaining parameters are the same as in Fig. 3. Thin gray solid lines in (a, c, e) denote the dependence of the accumulated energy of a free electron in the IR field on the return time (see equation (11)), thin gray dashed lines are the same as thin solid lines, but shifted by the XUV carrier frequency (a, c) and by the resonance energy \(\Omega \) (e). The horizontal solid line in (b, d, f) shows the position of the \(5p{-} 4d\) resonance; the dotted dashed lines show the cutoff energy in the XUV-induced plateau given by Eq. (7) in (a, c) and Eq. (10) in (e).

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Figure 3 shows HHG spectra for different carrier frequencies of the attosecond pulse. A sharp peak in the IR-induced plateau region is associated with the Rayleigh scattering of the attopulse on the atomic target, and, accordingly, its position is determined by the carrier frequency of the XUV pulse [37]. The HHG spectrum contains two plateau-like structures. The first plateau is developed within well-known three-step HHG scenario in intense IR field [1–4, 30, 38] (see black solid lines in Fig. 3). The second (additional) plateau is developed due to interaction of an attosecond pulse with IR-dressed system (see blue solid lines in Fig. 3). Depending on the laser parameters, the developing of the additional plateau can be determined by both single- and two-electron dynamics. Indeed, the attopulse with carrier frequency \({{\omega }_{{{\text{XUV}}}}}\) = 50 eV and the duration 0.6 fs (frequency bandwidth \( \approx {\kern 1pt} 7\) eV) cannot excite a resonance between 5p and \(4d\) shells, and for this case a single-electron three-step scenario is realized, consisting of tunneling of the valence electron from the 5p subshell in IR field, propagation in the IR-modified continuum with further XUV-assisted recombination into the 5p subshell [24, 25]. For the non-resonant case (see Fig. 3a), the TDKSE solution is in qualitative agreement with SAE approximation results. In contrast to this case, for XUV pulses with carrier frequencies \({{\omega }_{{{\text{XUV}}}}} = {\text{54}}{\text{.2}}\) eV and \({{\omega }_{{{\text{XUV}}}}} = 60\) eV, the spectral width of the pulse is sufficient to excite \(5p\)–\(4d\) resonance, which leads to new effects in the region of the XUV-induced plateau: a significant plateau extension and enhancement of the harmonic’s yield by 1–2 orders of magnitude compared to the non-resonant case (see Figs. 2b and 2c). Note that the HHG spectral density depends linearly on the intensity of the XUV pulse, which can be seen in Fig. 3c: decreasing the intensity of the XUV pulse by an order of magnitude leads to the same decrease in the harmonic yield.

For a more detailed study of the above features, let us consider the dependence of the HHG spectral density on the duration of the XUV pulse, whose carrier frequency is detuned from the \(5p\)–\(4d\) resonance, for the frequency range corresponding to the additional XUV-induced plateau. For relatively long XUV pulses, detuning from the atomic resonance allows the realization of only the single-electron scenario. However, \(5p\)–\(4d\) resonance can be excited by decreasing the duration of the XUV pulse due to the spectral broadening of the pulse. Figure 4 shows the time-frequency spectrograms of the dipole acceleration (panels (a), (c), (e)) and the corresponding HHG spectra (panels (b), (d), (f)) for the fixed frequency of the XUV pulse \({{\omega }_{{{\text{XUV}}}}}\) = 50 eV and different XUV pulse durations (from 0.2 to 0.6 fs). The excitation of resonance can be identified based on the appearance of the characteristic horizontal band in the spectrogram near the resonance energy (see Fig. 4e), indicating the formation of a superposition of \(5{{p}_{0}}\) and \(4{{d}_{0}}\) states.

As follows from Fig. 4a, the resonant transition \(5p \to 4d\) is not excited for \({{\omega }_{{{\text{XUV}}}}} = {\text{50}}\) eV and \({{\mathcal{T}}_{{{\text{XUV}}}}}\) = 0.6 fs, and HHG beyond the cutoff of the main plateau is determined by the single-electron scenario. In particular, the maximum generated frequency in a given channel is determined by the well-known relation [25]:

$$E_{{\text{c}}}^{{({\text{XUV}})}} = {{\mathcal{E}}_{{{\text{IR}}}}}({{t}_{i}},{{t}_{f}} = \tau ) + {{\omega }_{{{\text{XUV}}}}},$$
(10)
$${{\mathcal{E}}_{{{\text{IR}}}}}({{t}_{i}},{{t}_{f}}) = {\text{|}}{{E}_{0}}{\text{|}} + \frac{1}{2}{{\left[ {{{A}_{{{\text{IR}}}}}({{t}_{i}}) - {{A}_{{{\text{IR}}}}}({{t}_{f}})} \right]}^{2}},$$
(11)

where \({{\mathcal{E}}_{{{\text{IR}}}}}({{t}_{i}},{{t}_{f}})\) is the kinetic energy of the photoelectron recombining at the time moment \({{t}_{f}}\), \({{t}_{i}}\) is the ionization moment, \({{E}_{0}}\)is the binding energy of \(5{{p}_{0}}\)orbital. The ionization moment \({{t}_{i}}\) is found from the equation

$${{A}_{{{\text{IR}}}}}({{t}_{i}}) = \frac{1}{{{{t}_{f}} - {{t}_{i}}}}\int\limits_{{{t}_{i}}}^{{{t}_{f}}} {{{A}_{{{\text{IR}}}}}(\xi )d\xi ,} $$
(12)

where \({{A}_{{{\text{IR}}}}}(t)\) is the vector potential of the IR field. For \({{\mathcal{T}}_{{{\text{XUV}}}}} = 0.4\) fs, the single-electron HHG channel is still dominant, but the developing of a plateau associated with the resonant two-electron channel is observed (see Fig. 4d). It can be detected by observing the extension of the HHG spectrum in the frequency range 160–170 eV, which is located beyond \(E_{{\text{c}}}^{{({\text{XUV}})}}\) energy. For \({{\mathcal{T}}_{{{\text{XUV}}}}} = 0.2\) fs (see Figs. 4e, 4f), a smooth additional plateau is developed at frequencies 120–160 eV, at which the peak associated with the single-electron HHG channel no longer appears.

The two-electron scenario for the development of the XUV-induced plateau is realized when two conditions are simultaneously met: (a) the frequency bandwidth of the XUV pulse must cover the resonant frequency and has the intensity sufficient to excite resonance and create a superposition of resonating states, (b) the presence of vacancy in the \(5p\) subshell (created by the IR field) at the moment of interaction of XUV pulse with an atomic system. The specified conditions are met for the attopulse with duration \({{\mathcal{T}}_{{{\text{XUV}}}}} = 0.2\) fs. Within the two-electron scenario (see the description of Fig. 1 above), an electron in the IR-modified co-ntinuum can recombine at any instant after the resonant transition. For this reason, the extent of the XUV‑induced plateau in the resonant case is determined by the maximal energy of the electron in the IR field:

$$E_{{\text{c}}}^{{({\text{XUV}})}} = \max ({{\mathcal{E}}_{{{\text{IR}}}}}({{t}_{i}},{{t}_{f}})) + \Omega ,$$
(13)

and does not depend on the delay time between IR and XUV pulses.

It should be noted that the Xe atom considered in this work has a feature associated with the presence of a giant resonance in the transition from \(4d\) state to the continuum f-state. This resonance leads to a broadband enhancement in the photorecombination cross-section for \(5{{p}_{0}}\) orbital and thus causes enhancement in the HHG yield near 100 eV. The giant enhancement near 100 eV is also clearly visible in HHG spectra in Figs. 3 and 4. However, due to the large width of the giant resonance, the discussed in this work HHG channels beyond the cutoff of the main plateau are also enhanced: in both two-electron and single-electron HHG scenarios, radiative recombination is enhanced due to the presence of a giant resonance of the \(4d\) subshell.

4 CONCLUSIONS

Based on the numerical solution of the time-dependent Kohn–Sham equations, it is shown that the developing of an XUV-induced high-energy plateau in the HHG spectrum of an IR-dressed atom is possible as a result of both single-electron [24, 25] and two-electron scenarios. The two-electron scenario consists of a valence electron tunneling in an IR field, the transition of a second electron from a deeper shell to the valence shell by resonant XUV pulse, and the propagation of the freed electron in the continuum followed by the recombination into a vacancy in a low-lying state. It is shown that in this case, in contrast to the single-electron scenario, the extent of the XUV‑induced plateau is determined by the maximum energy accumulated by the electron in the IR field and does not depend on the time delay between the IR and   XUV pulses. The harmonics yield on the XUV‑induced plateau in the HHG spectrum is 1–2 orders of magnitude higher than for the single-electron scenario, which greatly facilitates the registration of XUV-induced harmonics for low XUV pulse intensities. The use of xenon has an advantage over other noble gases due to the presence of a giant resonance in the matrix element of the transition from continuum to \(4d\) subshell with the maximum near 100 eV, which leads to the further enhancement of the harmonic yield at the XUV-induced plateau.