1. Introduction

As exposed to strong laser fields, the Coulomb potential of an atom or a molecule is bent by the external field with forming a barrier out of which the valence electron can tunnel [1,2]. Thereafter, the tunneling electron can be considered as a classical particle, which is moving in the external field and can be driven back to and recollide with the parent ion. The laser-assisted tunneling triggers rich ultrafast physical processes such as above-threshold ionization (ATI) [3–8], high-harmonic generation (HHG) [9–13], and non-sequential double ionization (NSDI) [14–16], etc., which have attracted great interest in recent years. These processes can be well understood by strong-field models of the classical one (CM) [3,11] or the quantum one arising from strong-field approximations (SFA) [1,5,6,17,18]. The development of the laser technology further provides possibilities for using these processes to probe the electronic dynamics with unprecedented time resolution of attosecond. In relevant ultrafast probing experiments, one measures the observables such as the photoelectron momentum distribution (PMD) in ATI or the harmonic spectrum in HHG. Then using the CM or the SFA as a benchmark, one can deduce the attosecond dynamical information of the target from these observables [19,20].

The CM or the SFA with the saddle-point method describes the motion of the tunneling electron in terms of electron trajectories. It is well established that there are long and short electron trajectories contributing to ATI [5,6] or HHG [12]. The long and short trajectories are well resolved in the temporal domain. Specifically, for ATI, the long (short) electron trajectory has an ionization time located in the falling (rising) part of the laser field in one laser cycle. In particular, the long trajectory is also associated with a rescattering process. For HHG, both long and short electron trajectories are related to rescattering and have ionization times located in the falling part of the laser field. In comparison with the short one, the long trajectory has a earlier ionization time and a later return time. For ATI, these trajectories are characterized by the tunneling-out time of the electron versus the corresponding photoelectron momentum, with establishing a bridge between the experimental observables (such as the PMD) and the desired temporal information (such as the tunneling-out time). Similar situations also go for HHG.

One of the limitations of these models is the omission of the Coulomb effect on the tunneling electron. Many studies have shown that the Coulomb potential has important influences on strong-field dynamics of the tunneling electron [21,22]. For example, the Coulomb potential will remarkably affect the ATI photoelectron energy spectrum [23], momentum [24] and angular [25] distributions, etc.. Nevertheless, the influences of Coulomb potential on time-resolved strong-field electron dynamics are not clear. Especially, quantitative even semi-quantitative evaluations on these influences are absent. These evaluations are highly desired for more precise attosecond measurements and controls of electrons in atoms, molecules, and solids. They are also important for establishing a full Coulomb-modified physical picture of strong-field electron dynamics.

Here, we make efforts to clarify the Coulomb effect on time-trajectory-resolved electron dynamics for ATI and HHG. We begin our discussions with ATI in an orthogonally polarized two-color (OTC) field, which allows one to resolve the contributions of long and short electron trajectories directly from the photoelectron momentum distributions. We show that the Coulomb potential induces a time difference of the tunneling electron between the moments of exiting the barrier and being free. This time difference (longer than a hundred attoseconds) is well mapped in the momentum distributions, resulting in a remarkable increase of contributions of long trajectory (more than 20$\%$) to ATI. Then we extend our discussions to HHG. We show that the tunneling-out times of Coulomb-modulated HHG trajectories are earlier (about 25 attoseconds) than expected, leading to a marked increase of contributions of short trajectory (about one order of magnitude) to HHG.

2. Numerical and analytical methods

We assume that the fundamental field is along the x axis and the additional second-harmonic field is along the y axis. In the length gauge, the Hamiltonian of the model He atom interacting with the laser field has the form of H$(t)=H_0+\mathbf {r}\cdot \mathbf {E}(t)$ (in atomic units of $\hbar = e = m_{e} = 1$). Here, the term $H_0=\mathbf {p}^2/2+V(\mathbf {r})$ is the field-free Hamiltonian, and $V(\mathbf {r})=-{Z e^{-\rho r^{2}}}/{\sqrt {r^{2}+\xi }}$ with $r^{2}=x^{2}+y^{2}$ is the Coulomb potential. $\rho$ is the screening parameter with $\rho =0$ for the long-range potential and $\rho =0.5$ for the short-range one. $\xi =0.5$ is the smoothing parameter, and $Z$ is the effective charge which is adjusted in such a manner that the ionization potential of the model system reproduced here is $I_p=0.9$ a.u.. The OTC electric field $\mathbf {E}(t)$ used here has the form of [26,27] $\mathbf {E}(t)=\vec {\mathbf {e}}_{x}{E}_x(t)+\vec {\mathbf {e}}_{y}{E}_y(t)$ with ${E}_x(t)=f(t)E_{0}\sin {(\omega _{0}t)}$ and ${E}_y(t)=\mathcal {E} f(t)E_{0}\sin {(2\omega _{0}t+\phi )}$. $\vec {\mathbf {e}}_{x}$ ($\vec {\mathbf {e}}_{y}$) is the unit vector along the $x$ ($y$) axis. $\phi$ is the relative phase between these two colors. ${E}_0$ is the maximal laser amplitude relating to the peak intensity $I$ of the fundamental field ${E}_x(t)$. $\mathcal {E}$ is the ratio of the maximal laser amplitude for the second-harmonic field $E_{y}(t)$ to ${E}_0$. $\omega _{0}$ is the laser frequency of $E_{x}(t)$ and $f(t)$ is the envelope function. We use trapezoidally shaped laser pulses with a total duration of 20 optical cycles and linear ramps of three optical cycles. The details for solving time-dependent Schrödinger equation (TDSE) of $i\dot {\Psi }(t)=$H$(t)\Psi (t)$ with spectral method [28] and obtaining the PMD can be found in [29]. Unless mentioned elsewhere, the laser parameters used are $I=5\times 10^{14}$W/cm$^{2}$, $\omega _0=0.057$ a.u. ($\lambda =800$ nm), $\phi =\pi /2$ and $\mathcal {E}=0.5$.

To analytically study the Coulomb effect on time-resolved dynamics of ATI, we first calculate SFA-based ATI electron trajectories, characterized by the complex time $t_s=t_0+it_x$ and the drift momentum $\mathbf {p}$, with the following saddle-point equation [5,6]:

There are some points associated with our MSFA treatments which should be stressed before beginning further discussions. First, in our MSFA simulations, the initial conditions are as in [31], the Coulomb potential modifies the final momentum of a SFA electron trajectory but does not influence the amplitude of this trajectory, similar to the treatments in [25,31]. Secondly, using the instantaneous electron energy $E_a(t)$ for definition of ionization time within the laser pulse has its limitations, as the instantaneous electron energy can change its sign with time during the laser pulse. This point can be clearly seen in Fig. 3(c). At the end of the laser pulse $E_a(t)$ can indeed be used to understand, whether the trajectory is captured into a Rydberg state (frustrated tunneling ionization) or corresponds to an ionization event. However, in this paper, we are addressing the influence of Coulomb potential on the temporal aspect of electron trajectory. For comparison with SFA predictions where the ionization time $t_i$ is also the tunneling-out time $t_0$ with the positive instantaneous energy $[\dot {\mathbf {r}}(\mathbf {p},t_i=t_0)]^2/2$, in our MSFA simulations, we introduce the definition of ionization time $t_i$ with $E_a(t_i)=[\dot {\mathbf {r}}(\mathbf {p},t_i)]^2/2+V(\mathbf {r})>0$ where the Coulomb potential is included. In addition, a absorbing procedure is also used to “capture" the rescattering electrons which are very near the nucleus. With these treatments, the statistics on relevant trajectories show that most of ionization events, which occur at $t_i$ and are identified with the value of $E_a(t)$ becoming larger than zero for the first time at the time $t=t_i$, correspond to ionization events at the end of the laser pulse. The time-dependent continuum populations of MSFA, relevant to the definition of ionization time, also agree with the TDSE ones (as shown in Fig. 2(b)), giving support to this definition. With this definition, our MSFA simulations show that the Coulomb effect induces the correction to the effective exit time (i.e., the ionization time $t_i$) of the electron, which in contrast to the SFA, is slightly delayed, and this originates in the fact that the presence of the coulomb potential requires the electron to acquire additional kinetic energy from the field before having a total energy greater than zero. This delay can be used as a simple tool to understand the complex influence of the Coulomb potential on ATI (as discussed in Fig. 1 to Fig. 4) and HHG (as discussed in Fig. 5 to Fig. 6).

3. Cases of ATI

Asymmetric PMD.—The calculated PMDs of TDSE for He with long-range and short-range potentials are presented in Figs. 1(a) and (b). The long-range TDSE results in Fig. 1(a) show a butterfly-like structure with a remarkable up-down asymmetry with respective to the axis of $p_y=0$. The distribution in Fig. 1(a) has larger amplitudes for $p_y<0$. By contrast, this up-down asymmetry remarkably decreases for the short-range results in Fig. 1(b). Further simulations with SFA, related to a delta potential, give an up-down symmetric distribution, as shown in Fig. 1(d). Simulations with MSFA indeed reproduce the remarkable up-down asymmetry, as shown in Fig. 1(c), implying that this asymmetry arises from the Coulomb effect. The symmetry degree, which is defined as the ratio of the total amplitudes with $p_y>0$ to those of $p_y<0$, is about 0.51 in Fig. 1(a) and 0.48 in Fig. 1(c). This degree is near the unity in Fig. 1(b) of short-range potentials. For comparison, the prediction of the CM for $p_x$ vs $p_y$ is also plotted in Fig. 1(a), which shows a symmetric structure, similar to the SFA one. Using the symmetry degree, one can more easily judge the ratio of long-trajectory contributions versus short ones to ATI, as to be discussed below. Alternatively, one can evaluate this up-down asymmetry of PMDs with calculating the expectation value of $<p_y>$. This value is zero in Fig. 1(d) and is near zero in Fig. 1(b). By comparison, this value is about -0.16 a.u. in Fig. 1(a) and -0.12 a.u. in Fig. 1(c).

 

Fig. 1. PMDs of He obtained with different methods. (a) TDSE with the long-range potential. (b) TDSE with the short-range potential. (c) MSFA with the long-range potential. (d) SFA without considering the Coulomb effect. The prediction of CM (the white line) is also plotted in (a).

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Ionization time lag.—A sketch of the CM prediction of drift momenta $p_x(t)=-A_x(t)$ and $p_y(t)=-A_y(t)$ for one laser cycle of the fundamental field is presented in Fig. 2(a). It clearly shows that with the steering of the second-harmonic field of OTC [26,27], the contributions of the long (short) trajectory related to the fundamental field are mapped in the third (III) and the fourth (IV) quadrants with $p_y<0$ (the first (I) and the second (II) quadrants with $p_y>0$), providing a manner for resolving the contributions of long versus short electron trajectories directly from PMDs.

 

Fig. 2. Analyses for Coulomb-induced ionization time lag in ATI. (a) The prediction of CM for the drift momenta of $p_x(t)=-A_x(t)$ and $p_y(t)=-A_y(t)$. (b) The comparison of time-dependent continuum populations calculated with TDSE, MSFA and SFA. (c) The ionization velocity predicted by MSFA for all values of $p_y$ and only for $p_y<0$. (d) The distribution as functions of the tunneling-out time $t_0$ and the ionization time $t_{i}$ predicted by MSFA. Results are presented in one laser cycle of $2\pi /\omega _0$. In (b), for comparison, the prediction of MSFA with the short-range potential is also presented and the MSFA and SFA curves are multiplied by a vertical scaling factor to match the TDSE one. The log$_{10}$ scale is used in (d).

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To understand how the Coulomb effect influences the ATI, we further calculate the time-dependent continuum populations during the laser pulse $I(t)$ obtained by different methods. The TDSE ones are obtained with evaluating $I(t)=1-\sum _{m}|\langle m|\Psi (t)\rangle |^2$. Here, $|m\rangle$ is the bound eigenstate of the field-free Hamiltonian H$_0$ obtained through imaginary-time propagation. The model ones are obtained with calculating $I(t)=\sum _{\mathbf {p},t}|F(\mathbf {p},t_0)|^2$ at $E_a(t)>0$. Relevant results are presented in Fig. 2(b). The SFA results in Fig. 2(b) clearly show a remarkable increase just around the time of $t=5.25T$ or $t=5.75T$ with $T=2\pi /\omega _0$, at which the fundamental field arrives at its peak. The TDSE results, however, show a remarkable increase around a time later than $t=5.25T$ or $t=5.75T$. This time-delay phenomenon is reproduced by the MSFA. For comparison, here, we also show the result of MSFA with the short-range potential, which is similar to the SFA one.

The MSFA predictions of the ionization velocity (the time difference of the continuum population) are presented in Fig. 2(c) with considering all contributions and contributions only related to $p_y<0$. Here, the curve of $p_y<0$ shows large amplitudes at times earlier than $t=5.25T$ or $t=5.75T$, implying that electrons born at the rising part of the fundamental field also contribute remarkably to $p_y<0$, different from the predictions of SFA and CM. Further analyses tell that the long trajectory relating to a rescattering process still dominates the contributions to $p_y<0$, and the short one associated with direct ionization dominates the cases of $p_y>0$. The ratio of short-trajectory contributions to long ones is about 0.48, which agrees with the MSFA symmetry degree, indicating that for the present case, the Coulomb effect induces a remarkable increase of long-trajectory contributions (about $30\%$ relative to the SFA predictions) to ATI. Previous TDSE studies have also indicated this remarkable increase [33], which is obtained with performing a suitably chosen momentum-space analysis on the TDSE wave function. However, the potential mechanism is not discussed. Here, with the MSFA, the time-resolved mechanism can be accessed.

In Fig. 2(d), we further present the distributions $F(\mathbf {p},t_0)\equiv F(t_{i},t_0)$. Here $t_{i}$ is the MSFA prediction of the ionization time related to the trajectory ($\mathbf {p},t_0$). One can observe that the distributions deviate upward from the diagonal line, and this upward deviation is more striking for the time $t_0$ later than $5.25T$ or $5.75T$. The results clearly show the Coulomb induced large time lag (longer than 100 attoseconds on average) for the tunneling-out time $t_0$ and the ionization time $t_i$.

Next, we perform analyses on how the Coulomb effect influences the electron trajectory. In Fig. 3, we show two typical time-dependent MSFA electron trajectories with comparing with the SFA ones. The SFA ones are obtained with assuming $V(\mathbf {r})\equiv 0$ in the MSFA treatments. We focus on the displacement of $\mathbf {r}(\mathbf {p},t)$, drift momentum of $\dot {\mathbf {r}}(\mathbf {p},t)-\mathbf {A}(t)$ and instantaneous energy of $E_a(t)$ for the electron trajectory. Results in the left column show a SFA trajectory in the second quadrant with $p_x<0$ and $p_y>0$. This trajectory corresponds to a short trajectory related to the fundamental field. With considering the Coulomb effect, the MSFA predicts that this trajectory will shift to the fourth quadrant with $p_x>0$ and $p_y<0$. The shifted trajectory corresponds to a long trajectory of the fundamental field, which is relating to a rescattering process, as the displacement and the drift momentum of the shifted trajectory in Figs. 3(a) and 3(b) show. According to the SFA, the energy of the trajectory is larger than zero just at the tunneling-out time $t_0$ (i.e., the time origin of the trajectory). By comparison, there is an obvious time difference for the MSFA ionization time $t_i$, at which the energy of the trajectory becomes larger than zero, and the tunneling-out time $t_0$, as the inset in Fig. 3(c) shows. Results in the right column show the case of one electron trajectory for which the Coulomb effect is not remarkable. Here, the MSFA and SFA predictions are similar for time evolution of this electron trajectory. The Coulomb effect does not change the direction of the drift momentum, but modifies the values of displacement, momentum and energy for this trajectory, as seen in Figs. 3(d) to 3(f). One can also observe from the inset in Fig. 3(f), the MSFA ionization time is also somewhat later than the SFA one.

 

Fig. 3. Comparisons for time evolution of two typical electron trajectories predicted by MSFA and SFA. (a) and (d) Displacement. (b) and (e) Drift momentum. (c) and (f) Energy. Results in the left (right) column correspond to the MSFA electron trajectory with (without) changing the directions of its initial drift momenta under the influence of the Coulomb potential. The insets in (c) and (f) show the enlarged results around the time origins of the trajectories in (c) and (f).

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Quantitative comparisons.—We have also performed simulations at various laser parameters. Relevant TDSE and MSFA results for this symmetry degree (the left column) and the corresponding time lag (right) are presented in Fig. 4. Here, the value of this lag is taken as the time difference between the maximum of the electric field and the instant around which the ionization increases remarkably, as the horizontal arrow shows in Fig. 2(b). One can observe from Fig. 4 (a), on the whole, the TDSE symmetry degree increases when increasing laser intensities and wavelengthes. This situation reverses for the corresponding time lag, as shown in Fig. 4(d), implying an anti-correlation between the values of symmetry degree and time lag. These characteristics are basically reproduced by the MSFA, as seen in Figs. 4(b) and 4(e).

 

Fig. 4. Symmetry degree of PMDs (a-c) and ionization time lag (d-f) for He obtained with different methods. (a) and (d): TDSE. (b) and (e): MSFA. (c) and (f): Differences of MSFA predictions minus TDSE ones. Results are calculated at different laser intensities and wavelengthes of the fundamental field, as shown. The relative phase $\phi$ and amplitude $\mathcal {E}$ of the second harmonic field used are as in Fig. 1.

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Although the MSFA gives an applicable description for the dependence of symmetry degree and time lag on laser parameters, there is a remarkable quantitative difference between TDSE and MSFA predictions, as shown in Figs. 4(c) and 4(f). For the symmetry degree, this difference spreads from -0.2 to 0.2, and seems irregular with respect to the laser parameters, implying that the TDSE symmetry degree is easily influenced by other factors beyond the description of SFA, such as multiphoton ionization which is sensitive to the laser parameters. By contrast, for the time lag, this difference, spreading from 8 attoseconds to 45 attoseconds, is smaller for lower laser intensities and shorter laser wavelengthes on the whole. As we change the parameters used in V$(\mathbf {r})$ in MSFA simulations, these differences for the lag at different laser parameters can not be eliminated. The above comparisons suggest that for a more accurate description of this Coulomb induced time lag, the MSFA needs to be further developed.

4. Cases of HHG

In the above discussions, we have shown that the Coulomb effect will remarkably influence ATI electron trajectories. Due to that the ionization is the first step of many strong-field processes, one can expect that the Coulomb effect will also affect the dynamics of relevant processes. As a case, in the following, we show that how the Coulomb effect influences dynamics of the HHG.

In Fig. 5, we show the comparisons of HHG electron trajectories of He in a single-color field (the fundamental $\omega _0$ field), obtained with SFA, MSFA and the modified classical model (MCM) that considers the exit position [34]. The SFA-based HHG electron trajectories, characterized by the complex ionization and return times of $t_s$ and $t_s'$ and the harmonic energy $\Omega$, are obtained with the following saddle-point equations

 

Fig. 5. Comparisons of HHG long (L) and short (S) electron trajectories of He born at half a laser cycle, obtained with MSFA, SFA, and MCM, for the tunneling-out time $t_0$ (a) and the return time $t_r$ (b) versus the return energy $E_p$ scaled with $U_p=E_0^2/(4\omega ^2_0)$, and the return time $t_r$ versus the harmonic amplitude (d). Results in (c) show time-frequency analyses of TDSE simulations where short-trajectory contributions are only considered [36]. The vertical-solid line in (a) indicates the peak time of $5.25T$. Those in (b) and (d) indicate the HHG cutoff positions, which divide the trajectories into long-trajectory (right branches) and short-trajectory (left) parts. The gray curve in (d) indicates the results of MSFA with the short-range potential. The inset in (d) shows the short-trajectory HHG spectra of MSFA, SFA and TDSE, where the curves are shifted for comparison. The HHG results are calculated with a linearly-polarized single-color laser field and the laser parameters used are $I=5\times 10^{14}$W/cm$^{2}$ and $\omega _0=0.057$ a.u. ($\lambda =800$ nm).

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Remarkable differences are also observed for the HHG amplitudes predicted by MSFA and SFA, as seen in Fig. 5(d). Here, the SFA amplitude for one harmonic with energy $\Omega$ is approximately evaluated with the expression $(1/\tau )^{1.5}e^{b'}$. Here, $b'$ is the imaginary part of the HHG quasiclassical action $S'(\textbf {p}_{st},t_s,t_s',\Omega )=\int _{t_s}^{t_s'}dt'[(\textbf {p}_{st}+\textbf {A}(t'))^2/2+\textbf {I}_{p}]-\Omega t_s'$ at relevant complex saddle points ($t_s,t_s'$) of Eq. (6) [37] and only minus values of $b'$ are considered. Note, the SFA-based HHG amplitude $(1/\tau )^{1.5}e^{b'}$ and the MSFA-based one $(1/\tau )^{1.5}e^{b}$ are associated with different actions of $S'$ and $S$ and different saddle-point equations of Eq. (6) and Eq. (1), respectively. When the predictions of SFA and MSFA are similar for long trajectories (right branches), they differ remarkably for short trajectories (left). The short-trajectory MSFA amplitudes are one order of magnitude higher than the SFA ones for lower energy (such as $E_p=U_p$, indicated by the vertical-dotted line). The MSFA short-trajectory results in Fig. 5(d) are nearer to the TDSE ones in Fig. 5(c), with showing comparable amplitudes for lower and higher energy. Our extended simulations show that these remarkable amplitude differences for MSFA and SFA hold for relatively low laser intensities and short laser wavelengthes. As seen in Fig. 5(a), the HHG tunneling-out times predicted by MSFA are nearer to the peak time of the laser field at which the tunneling amplitudes are larger. This effect is expected to be mainly responsible for increased short-trajectory amplitudes of MSFA, in comparison with SFA.

We mention that the HHG amplitudes predicted by MSFA with the short-range potential agree with the SFA ones, as shown by the gray curve in Fig. 5(d), suggesting that even for HHG, our MSFA simulations can return to SFA cases when the Coulomb effect is weak. In addition, a direct comparison of the HHG spectra of TDSE, MSFA (with the long-range potential) and SFA, also shows that the MSFA predictions are nearer to the TDSE ones, as the inset in Fig. 5(d) shows.

In Fig. 6, we further present the comparisons for HHG return times of short trajectory, predicted by MSFA, SFA and TDSE. The TDSE short-trajectory return times are obtained with finding the locally maximal amplitudes of HHG time-frequency distributions associated with short-trajectory TDSE simulations, as shown in Fig. 5(c). The short-trajectory TDSE simulation allows one to differentiate contributions of short trajectory from those of long trajectory and multiple returns. But it is not capable of resolving contributions of long trajectory from multiple-return ones [36]. We therefor choose the TDSE short-trajectory return time as a reference to compare the predictions of MSFA and SFA. In addition, the predictions of MSFA and SFA for long trajectory only differ remarkably from each other at lower electron energy, as seen in Fig. 5(b). The comparison is encouraging for cases with various laser parameters in Fig. 6. On the whole, for electrons with energy higher than $0.5U_p$ beyond the energy region of near-threshold harmonics which have complicated origins [38], the TDSE results are earlier (about 15 attoseconds) than the SFA ones and are very near to the MSFA predictions. This point can be clearly observed from the enlarged results in the insets in Fig. 6. In [39], with the TDSE simulations, it has been shown that the HHG return times for short trajectory are earlier than the SFA predictions. Here, this phenomenon is reproduced with the MSFA, along with the predictions of earlier tunneling-out times of short trajectory relative to the SFA predictions.

 

Fig. 6. HHG short-trajectory return times of He, obtained with MSFA, SFA and TDSE at different laser parameters. The laser field used is a linearly-polarized single-color field as in Fig. 5. The laser wavelengthes used are $\lambda =800$ nm (the left column) and $\lambda =1200$ nm (right). The laser intensities used are as shown in each panel. The insets in each panel show the enlarged results in some energy regions of $1U_p\leq E_p\leq 2U_p$ in the corresponding panels.

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5. Discussions

The Coulomb induced ionization time lag revealed here occurs after the electron tunnels out of the laser-Coulomb-formed barrier and it is different from the tunneling time, which occurs just in the tunneling process. Indeed, in attoclock experiments [24], the comparison between experimental and theoretical PMD results implies the exclusion of this lag before the tunneling time can be accessed. This lag, arising from the interaction of the electron with both of the strong laser field and the Coulomb potential, is also different from the Wigner delay [40], which results only from the interaction of the electron with the Coulomb potential. This time lag discussed here can be characterized by the time delay between the maximum of the electric field and the instant of ionization, as treated in Fig. 4. In this meaning, this lag discussed here is somewhat similar to the time delay revealed in [41], which is obtained with monitoring the TDSE probability current of an one-dimensional system and is attributed to the wave function’s inability to adopt instantaneously to the external field.

It should be stressed that in the paper, we have limited our discussions to atomic cases. For molecules with more degrees of freedom, besides of the Coulomb effect, the molecular orientation relative to the laser polarization and the nuclear motion should be considered in MSFA treatments for exploring time-resolved electron dynamics. Because the energy gap between the ground state and the first excited state of the molecules is usually smaller (especially for molecules with larger internuclear distances) and these two lowest states can be strongly coupled together by the strong laser field, the effect of excited states also needs to be considered in molecular cases. In addition, for polar molecules, the effect of permanent dipole [42,43] should also be included.

6. Conclusions

In summary, we have studied the influence of Coulomb potential on temporal aspects of ATI and HHG. We have shown that the Coulomb effect gives rise to a large time lag (longer than 100 attoseconds) between the tunneling-out time and the ionization time for ATI electron trajectories. This time lag increases the contributions of long trajectory to ATI. Accordingly, it also shifts the HHG tunneling-out time towards the peak time of the laser field, resulting in a remarkable increase of short-trajectory contributions to HHG. As this lag has a striking influence on ATI and HHG electron trajectories which are important in attosecond measurements, our work gives suggestions on relevant experiments. Because this lag is general for atoms and molecules with long-range Coulomb potentials, our work also opens a perspective for understanding more complex strong-field processes, such as ATI and NSDI from oriented polar molecules, where both of the permanent dipole and the Coulomb-induced large time lag are expected to play a nontrivial role.