Measuring ionization time lag of polar molecules with a calibrated attoclock

Electrons in atoms and molecules can not respond immediately to the action of intense laser field. There is a time lag (about 100 attoseconds) between instants of the field maximum and the ionization-rate maximum. This lag characterizes the response time of the electronic wave function to a strong-field ionization event and has important effects on dynamics of the ionized electron. For polar molecules with a large permanent dipole, the direct measurement or calculation of the absolute time lag is difficult. Here, a calibrated attoclock procedure, which is related to a simple Coulomb-induced temporal correction to electron trajectories, is proposed to measure the relative time lag of two different ionization events. Using this procedure, the relative lag of polar molecules in two consecutive half laser cycles can be probed with high time resolution.


Introduction
Whether there is a finite response time in the interaction between light and matter is a basic conceptual problem in quantum physics [1][2][3][4]. Answering this problem is important for understanding light-induced physical processes and for precisely measuring and controlling these processes in ultrafast science. Although experimental and theoretical studies on this issue have made a lot of progress in recent years [4], there are still many questions to be addressed. For example, theoretically, because there is no time operator, the definition of time is difficult in quantum mechanics [3]. Experimentally, whether the measurement itself will influence the measured process is also a question [2]. However, the definition and measurement of time is unambiguous in classical mechanics.
For the system of an atom or a molecule interacting with a strong infrared (IR) laser pulse, the bound electron can escape from the laser-Coulomb-formed barrier through tunneling [5,6]. This tunneling event triggers rich physical processes such as above-threshold ionization (ATI) [7][8][9][10][11] and high-order harmonic generation (HHG) [12][13][14][15]. These processes have important applications in attosecond science [16][17][18]. When the laser electric field is strong enough, the behavior of the tunneling electron becomes classical after it exits the barrier [14,15]. Therefore, in essence, the strong-field induced ultrafast processes contain the transition from quantum process (related to tunneling ionization) to classical-like process (related to the electron motion after tunneling). So, this strong-field system also provides an ideal platform to investigate the response time. Specifically, if there is a finite response time in the tunneling-ionization process of this system, the response time can be identified through its influence on the subsequent laser-driven classical motion of the ionized electron.
Recent research showed that [19] in the interaction of a strong IR laser pulse with the He atom, the time when the laser intensity reaches the peak is inconsistent with the time when the atom has the maximal instantaneous ionization rate. There is a time lag between these two times. This time lag is related to the Figure 1. Sketch of the Coulomb-induced ionization time lag for the polar molecule HeH + with a PD in one laser cycle. When the electron exits the laser-Coulomb-formed barrier (red-solid curve) at a time t0, due to the Coulomb effect, it can not be free immediately. It is accelerated by the laser field for a period of ∆t. At the time t ′ 0 = ∆t + t0, the instantaneous energy of the electron begins to be larger than zero and the electron becomes free. For atoms and symmetric molecules, this time lag ∆t is the same when the tunneling event occurs in the first (with Ex(t) > 0) or the second (with Ex(t) < 0) half cycle of one laser cycle. For polar molecules, due to the effect of the PD which is directing from He to H, the situation is different. When the laser polarization is antiparallel (parallel) to the PD in (a) [(b)], the energy of the ground state |0⟩ of HeH + is dressed up (down) and the exit position x(t0) is nearer to (farther away from) the nuclei. Accordingly, the Coulomb effect is stronger (weaker) and the time lag ∆t is larger (smaller) in the first (second) half cycle. The ionization of HeH + provides a platform for identifying the difference of ∆t (i.e. the relative lag) between diverse laser-dressed electronic states. Embedded graph is the ground state wave function of HeH + . classical behavior of the long-range Coulomb potential which prevents the tunneling electron from moving away from the nucleus instantaneously. It has a profound influence on the subsequent ultrafast dynamics (such as ATI and HHG) of the electron after tunneling. This lag concept can be used to explain complex strong-field physical phenomena, e.g. the up-down asymmetry of photoelectron momentum distribution (PMD) of atoms in an orthogonally-polarized two-color laser filed [19], the left-right asymmetry of PMD of asymmetric molecules in a linearly-polarized laser field [20] and the remarkable shift of HHG emission time [21]. Therefore, it is considered that this lag reflects an essential response time in strong-laser-matter interaction.
This time lag has also shown its significance on attosecond measurement. According to the electron-trajectory theory [15,22] based on strong-field approximation (SFA) [23,24] where the Coulomb effect is neglected, both ATI and HHG can be described with complex electron trajectories, which are represented by the final (for ATI) or instantaneous (for HHG) momentum of the escaped electron and the timing of the electron when it leaves or returns to the nuclei. These trajectories build a bridge between the experimental observables and the temporal information of relevant dynamical processes, with providing a theoretical tool for attosecond measurement. The Coulomb-modified SFA (MSFA) which considers the effect of long-range Coulomb potential [25][26][27], however, showed that the Coulomb-induced ionization time lag leads to an important temporal correction to ATI and HHG electron trajectories [19].
Experimentally, this attosecond-scale lag can not be measured directly at present. For indirect measurement, it needs to develop accurate theories to deduce this time lag from observables. Theoretically, different atomic and molecular targets may have different effects in strong laser field, so it is also not easy to develop theories which include these effects to directly calculate this lag. In [19], with numerical solution of the time-dependent Schrödinger equation (TDSE) and MSFA, this time lag is approximately evaluated through analyzing the time-dependent ionization probability and comparing results of long-range Coulomb potential to short-range ones. But there is uncertainty about how to define the instantaneous ionization of a laser-driven system. However, it is possible to measure the lag difference between the timing of ionization from distinct electronic states in a single laser pulse [3]. The strong-field ionization dynamics of polar molecules with a large permanent dipole (PD) differ remarkably in the two half cycles of a laser cycle [20,28]. It is interesting to study the time lags of an asymmetric system in these two half cycles. In particular, such an asymmetric system with a large PD also provides a chance for identifying this lag difference between diverse laser-dressed electronic states (see figure 1).
The focus of the paper is whether this lag difference can be identified with a simple approach and high time resolution without the need of solving TDSE or MSFA. Such approaches are highly desired in experimental studies, especially for complicated atomic and molecular targets for which TDSE and MSFA simulations are not easy to achieve. In the following, we will show that a calibrated attosecond-clock procedure based on the use of PMD in a strong elliptical laser field provides such possibilities.

Methods
We choose the polar molecule HeH + as the study objective, which is the simplest heteronuclear molecule with a stable ground state. For the asymmetric system, the TDSE simulations of strong-field ionization dynamics can be more easily executed. Moreover, it can also be manipulated in present strong-filed experiments [29]. Due to the effect of PD, the HeH + system in strong laser fields stretches rapidly toward larger internuclear distances R and the ionization of the molecule also mainly occurs at larger R [30]. We therefore study the ionization dynamics of HeH + in elliptical laser fields at a stretched distance of R = 2 a.u. in the Born-Oppenheimer approximation. We assume that the molecular axis of HeH + is located in the xy plane. The TDSE of the polar molecule in two-dimensional cases is solved with the spectral method [31]. Relevant numerical details are introduced in [32]. Analytically, we use a MSFA model which also includes the PD effect [33]. For convenience, we call this model MSFA-PD to differentiate it from the general MSFA without PD. The details for these strong-field models can be found in [20,28]. At R = 2 a.u., the HeH + system has the ionization potential of I p = 1.44 a.u. and the value of PD calculated is D = −0.36 a.u.
The elliptical electric field E(t) has the form of Here, E 0 is the laser amplitude corresponding to the peak intensity I and ϵ is the ellipticity. ω is the laser frequency and f (t) is the envelope function. ⃗ e x (⃗ e y ) is the unit vector along the x(y) axis. Here, the value of ϵ = 0.87 is used, implying that the component E x (t) dominates in ionization. We assume that the molecular axis is oriented parallel to ⃗ e x and the heavier (lighter) nucleus is located on the right (left) side. We use trapezoidally shaped laser pulses with a total duration of 15 cycles, which are linearly turned on and off for three optical cycles, and then kept at a constant intensity for nine additional cycles.

Time lags of HeH +
In figure 1, we show the dynamics difference between tunneling ionization of HeH + in the two half cycles of a laser cycle in the plateau part of the pulse, described with the MSFA-PD. For the first half laser cycle with E x (t) > 0 in figure 1(a), the ground state |0⟩ is dressed up and the electron exits the barrier along the H side. At the exit time t 0 , the exit position x(t 0 ) of the tunneling electron is nearer to the nuclei and the electron is subject to a stronger Coulomb force which precludes the electron to escape. The electron with zero average momentum is accelerated by the laser field from the tunnel exit x(t 0 ) to the position x(t ′ 0 ), then the instantaneous energy of the electron including both the kinetic energy and the potential energy begins to be larger than zero and the electron becomes free. Due to the stronger Coulomb force at the exit position in this case, the time difference of ∆t = t ′ 0 − t 0 which is defined as the ionization time lag is also larger. We denote this lag along the H side ∆t H . The situation reverses for the case of the second half laser cycle in figure 1 where the Coulomb force is weaker at the exit position and the time lag ∆t is also smaller. We denote this lag along the He side ∆t He . When the absolute value of ∆t may be influenced by the methods used in calculations, the difference between ∆t H and ∆t He , which reflects the essential dynamics difference for polar molecules with a large PD in two consecutive half laser cycles, is insensitive to the methods. We mention that from the perspective of the tunneling time delay, the barrier formed in figure 1(a) with smaller Stark-shifted ionization potential I p is narrower than in figure 1(b) with larger I p . Intuitively, tunneling will spend more time for the case of figure 1(b) with a wider barrier than for figure 1(a) with a narrower barrier. These are different from TDSE and model predictions, as to be shown below. More discussions related to figure 1 also see section 3.6.

Offset angles of HeH +
In figure 2, we present PMDs of HeH + in elliptical laser fields obtained with different methods. For shorter laser wavelengthes, some electrons which are located in the ground state are pumped into then ionized from the first excited state, and the excited-state ionization channel therefore has an important role in strong-field dynamics of HeH + [32]. Here, we focus on longer wavelengthes (longer than 800 nm) for which the Stark-shifted ground-state channel, as introduced in figure 1, dominates in ionization [28,29].
For TDSE results in figure 2(a), the offset angle θ (θ, the angle between the y axis and the axis which goes through the origin and the part of the PMD with the maximal amplitude) in the upper half plane is θ u = 6.8 • and that in the lower half plane is θ l = 5.1 • , with a difference of ∆θ = 1.7 • . This difference is also reproduced by the MSFA-PD with θ u = 4.8 • and θ l = 4 • , as seen in figure 2(b). This difference disappears in the MSFA simulations which consider the asymmetric Coulomb potential of HeH + but neglect the PD effect,

Coulomb-calibrated attoclock
Without the Coulomb effect, the PMD in elliptical laser fields is symmetric with respect to the y axis and there is a one-to-one mapping between ionization time and PMD, that is, the PMD acts like an 'attosecond clock' [34]. One can read the ionization time t 0 from this clock with atomic-scale resolution, by virtue of the simple-man (SM) mapping p = −A(t 0 ) given by the classical model [14]. Here, p is the photoelectron momentum and A(t) is the vector potential of the electric field E(t). The Coulomb effect destroys this symmetry and leads to the nonzero offset angle θ. According to the general attoclock procedure [35], for the elliptical laser field with a high ellipticity ϵ ≈ 1, the offset angle θ can be written as θ = θ delay + θ ref . Here, the term θ delay ≈ ωτ is considered to be related to the possible tunneling-induced time delay τ , while the term θ ref is a calculated zero-time reference by semiclassical simulations. This reference term θ ref is considered to arise from the Coulomb scattering as the escaped electron travels and to be unrelated to the concerned time delay τ .
Based on the discussions on the ionization time lag in the introduction part, we consider that the whole offset angle is induced by this lag (i.e. the response time) ∆t = t ′ 0 − t 0 . That is θ ≈ ω∆t (for ϵ ≈ 1). Then one can retrieve the time lag information ∆t directly through the offset angle in PMD. By doing so, we in fact introduce the lag ∆t into the SM mapping. That is p = −A(t ′ 0 = t 0 + ∆t), where this time lag manifests itself as a simple temporal correction to the classical prediction of the SM. Below, we will call the attoclock procedure with the new mapping p = −A(t ′ 0 = t 0 + ∆t) the Coulomb-calibrated attosecond clock (CCAC). In the following, we will show that the predictions of CCAC, TDSE and MSFA for the relative lag are near to each other.

Absolute lags by different methods
In figure 2(d), we plot the function curve of the time t versus the angle θ with θ = arctan[A x (t)/A y (t)], which is defined by the new mapping according to CCAC and is applicable for the value of ϵ being somewhat smaller than 1. Here, for simplicity, we have replaced t ′ 0 by t. When the angle is located in the first (I) and the third (III) (the second (II) and the fourth (IV)) quadrants, we define that it is plus (minus). With the offset angle obtained in TDSE or experiments, one can deduce the ionization time information t = t 0 + ∆t through the function curve.
Next, we perform comparisons between the predictions of TDSE, MSFA and CCAC for this time lag. Relevant results are shown in figures 3 and 4. In figure 3(b), the TDSE instantaneous ionization probability is obtained with excluding the components of the first seven bound eigenstates from the norm and the MSFA-PD one is obtained with evaluating the weight of these electron trajectories having the instantaneous energy larger than zero at the time t [28]. The time derivative of the instantaneous ionization probability gives the ionization rate in figure 3(c). The laser electric fields of E x (t) and E y (t) and the minus vector potentials of −A x (t) = p x and −A y (t) = p y in one laser cycle of 6 T-7 T are plotted in the first row of figure 3, with dividing the time region into four parts of I to IV corresponding to PMDs with momenta (p x , p y ) in quadrant 1 to quadrant 4.
The time-dependent ionization probability of TDSE in figure 3(b) shows a remarkable asymmetry in the first (6 T-6.5 T) versus the second (6.5 T-7 T) half laser cycles. This asymmetry is well reproduced by the MSFA-PD. The corresponding ionization rates of TDSE and MSFA-PD in figure 3(c) show the maxima in each half laser cycle, as indicated by the vertical arrows. These maxima correspond to the brightest parts of the PMD in the upper and the lower half planes. Relevant times t related to these maxima are also presented here, which deviate from the instants of t = 6.25T and t = 6.75T at which the electric field E x (t) reaches its peak. These times can be understood as the ionization times of photoelectrons with the maximal amplitudes in the first and the following half laser cycles, and this deviation of these times from the peak time t 0 of the electric field E x (t) denotes the Coulomb-induced ionization time lag ∆t = t − t 0 . In figure 3(d), we show the predictions of CCAC for the ionization time t with the offset angles obtained from TDSE. These ionization times obtained with TDSE, MSFA-PD and CCAC in each half laser cycle differ from each other, indicating that the value of the corresponding ionization time lag relative to the peak time of E x (t) depends on the approach used in evaluations. However, the situation is different for relative lags, as shown in figure 4.

Relative lags by different methods
In figure 4, we plot the time lags (left column) and the lag differences (right) in these two half laser cycles for varied laser wavelengthes and intensities, obtained with different methods. One can observe that when the time lags for different methods and laser parameters differ remarkably, the lag differences are located in a time region of 10-25 attoseconds. In particular, the predictions of CCAC for this lag difference are near to the TDSE ones. By comparison, the MSFA-PD results deviate somewhat from the TDSE predictions, with an upper limit of about 10 attoseconds for this deviation. This deviation is easily understood, since in MSFA-PD simulations, the approximate descriptions of the Coulomb and PD effects both can induce disagreement with the TDSE. The similarity between TDSE, MSFA-PD and CCAC predictions for this lag difference suggests that one can distill the relative time information from PMDs measured in experiments through CCAC with a high time resolution.

Discussions
Recent experiments [36] have reported the measurement of PMD for an asymmetric dimer Ar-Kr + with a large internuclear distance R in elliptical laser fields. The measured PMD also shows a remarkable asymmetric structure with the distribution in the lower half plane with larger amplitudes splitting into two parts. The offset angles related to these two parts are shown to encode time information of direct ionization and resonance ionization, respectively. By using a proposed self-referenced molecular attoclock scheme which uses the offset angle of direct ionization as a reference to remove the Coulomb-related angle component θ ref , the lifetime of the resonance states can then be probed with high accuracy. By comparison, phenomena in the present work are related to the difference between the angles in the upper and lower half planes corresponding to electrons ionized from the HeH + system in two consecutive half laser cycles. This angle difference is attributed to the effect of the permanent dipole which results in the ground-state energy dressed up and down in these two neighboring half laser cycles, respectively. In addition, we consider that the whole offset angle is related to the Coulomb-induced ionization time lag. As the accurate value of this lag in one half laser cycle is not easy to determine, we show that the relative lag in these two half laser cycles can be accessed with high accuracy. Therefore, both results in [36] and in the present work suggest that through using a self-calibrated attoclock procedure to remove the uncertain in measurements, the relative time lag associated with different aspects of tunnel ionization of polar molecules including Stark shift and resonance excitation can be probed with high time resolution.
In the general attoclock procedure, as discussed in section 3.3, the offset angle θ is considered to include the tunneling-time-related part θ delay and Coulomb-related part θ ref . The latter is generally considered to arise from the Coulomb-induced deflection when the tunneling electron moves far away from the nucleus and therefore be unrelated to tunneling time. In our work, we analytically study strong-field ionization of HeH + with a developed MSFA model that considers the PD effect. Specifically, we describe tunneling with PD-included SFA where only imaginary time is needed for tunneling and we further solve the Coulomb-included Newton equation for every SFA electron trajectory to consider the Coulomb effect. As the PMD obtained by the general Coulomb-free SFA does not show the offset angle, this angle appearing in our model predictions is completely induced by the Coulomb potential. We note that for strong-field ionization of a delta-potential atom, the calculated PMD also does not show the offset angle. Because we attribute the whole offset angle to Coulomb-induced ionization time lag (i.e. the response time), and the imaginary time required for tunneling can entropically give cause to a real time delay [37], this 'real tunneling time' could also be the one included in the Coulomb-induced ionization time lag. Indeed, recent experimental and theoretical studies in [38] also reveal a remarkable probabilistic tunneling time at a time scale similar to this lag discussed in the paper. Relevant studies use a Feynman path integral (FPI) approach where the total transmitted wavefunction can be expressed as a sum of all possible paths with each path relating to a deterministic tunneling time. The calculated FPI tunneling time shows a broad distribution and is comparable with that extracted from experimental measurement over a wide laser-intensity range, suggesting a significant uncertain about the beginning time of valence-electron wave packet evolution.
The reasons for attributing the whole angle to the response time are as follows. The Coulomb effect is more remarkable for the tunneling electron being near the nucleus than being far away from the nucleus. In other words, the main Coulomb effect concentrates on the tunneling process when the electron passes through the Coulomb-laser-formed barrier. Intuitively, the Coulomb force will slow down the electron, making the ionization time of the tunneling electron to shift from the Coulomb-free prediction t 0 to a later Coulomb-included time t ′ 0 = t 0 + τ . This simple shift picture is supported by the instantaneous ionization rate of TDSE, which also shows a distinct shift of the ionization peak from t 0 to t ′ 0 . In particular, with the simple shift picture or the lag concept, we find that many complex strong-field phenomena can be understood easily, including the origin of the offset angle in PMD. Because a quantitative description for this lag which needs to develop new Coulomb-included strong-field theories is not accessible, we qualitatively attribute the whole angle to this time lag here. On the other hand, the MSFA-PD approach used in the paper considers the Coulomb effect after the electron exits the barrier and therefore underestimates the Coulomb force. As a result, the MSFA predictions of the offset angle are smaller than the TDSE ones on the whole.
Very recently, a semiclassical theory termed as TRCM (tunneling-response-classic-motion) has been proposed to describe the response time of the electron inside an atom to light in strong-field tunneling ionization [39]. This ionization time lag discussed above corresponds to the response time and can be strictly defined in the theory. The main ideas in the theory are as follows. When the tunneling electron exits the barrier at the time t 0 , due to the exit position x(t 0 ) (which is about 10 a.u. for generally laser and target parameters used in experiments) is not far away from the nucleus, the tunneling electron is still located at a quasi-bound state (characterized by a bound wave packet consisted of high-energy bound eigenstates of the field-free Hamiltonian of the studied system). This state approximately agrees with the virial theorem and contains the basic symmetry requirement of the Coulomb potential on the electronic state. A small period of time τ is therefore needed for the tunneling electron to evolve from the quasi-bound state into an ionized state. Then at the time t ′ 0 = t 0 + τ (see figure 1), the electron can be considered to be free and the Coulomb effect can be neglected. The evolving process is called as the response process. It has a small time scale of about 100 attoseconds and a small space ∼0.33 a.u. for the H atom), forming a narrow time-space boundary between quantum and classic from which the response time can be clearly defined and determined. The quantity of the time lag τ depends on the exit position x(t 0 ) ≈ I p /E 0 . Due to the Stark effect related to PD for HeH + , the laser-dressed ionization potential and accordingly the exit position differ for the active electron inside HeH + exiting the barrier in the first and the second half laser cycle (also see figure 1). This corresponding time lag therefore also differs for these two half cycles and this difference can be probed with the CCAC procedure as discussed in the paper.
With the TRCM theory, the absolute time lag of atoms can be well evaluated by a simple formula which is related to some basic laser and atomic parameters and is independent of the observable. The offset angle deduced from this evaluated lag is in quantitative agreement with a series of recent attoclock experiments. In particular, it has also been shown that the CCAC procedure proposed here corresponds to the adiabatic version of the TRCM theory which holds for small Keldysh parameters [5]. Specifically, the adiabatic TRCM gives similar results to the full TRCM theory for cases of relatively high laser intensity and long laser wavelength. It can also be used to explain experimental results with adiabatic laser intensity calibrating procedure [40]. For molecules with a relatively large internuclear distance R and especially for polar molecules with a large PD, the situation is complex, and many new effects such as multi-center-induced interference effects and PD-induced excited-state effects need to be further considered in this semiclassical theory. The extension of this theory to molecules is one of the main aims in the following work.
It should also be mentioned that (1) in the present work, we use laser wavelengths longer than 800 nm. For shorter wavelengths such as λ = 400 nm, the PD-induced excited-state effect is remarkable with the PMD in elliptical laser fields showing a distinct tail. The analysis on the instantaneous ionization probability showed that this tail can be attributed to the excited-state contributions and encodes time information of sub-cycle excited-state dynamics [32]. In particular, for short wavelength of 400 nm, the experimental and theoretical studies also showed that the PMD of Ar in elliptical laser fields presents clear ATI rings (corresponding to ATI peaks in energy spectrum) and the deflection angles related to each ring differ from each other. This energy(ring)-dependent angle shift has been attributed to different influences of Coulomb potential on the ionization time of electron trajectories related to different photoelectron energy [41]. For our present cases, the laser wavelength is long and the ground-state energy is dressed due to PD. Therefore the ring and the ring-dependent lag are unresolved. The influence of PD on energy-dependent angle shift at short laser wavelengths deserves a detailed study in the future. (2) The interference effect arising from two-center characteristic of the molecular potential can also play a role in the offset angle. For molecules with a small R, the interference effect is characterized by the 'interference' of a tunneling-induced imaginary electronic momentum between these two atomic centers, and mainly affects the offset angle when the molecular axis is aligned not parallel to the major axis of laser polarization [42]. Moreover, the orientation degree of the asymmetric samples also has an important influence on PMD. For a small orientation degree, the polar molecule will behave similarly to a symmetric molecule. In the paper, we have presumed the parallel alignment with perfect orientation. (3) For molecules with large R, the interference effect can manifest itself as the interference of the real electronic momentum between these two atomic centers [43], resulting in some complex interference patterns which will influence the identification of the offset angle. In addition, for large R, the energy gap between the ground state and the first excited state becomes smaller and the resonance between these two lowest states can induce resonance-enhanced ionization, which will also remarkably influence the PMD and therefore influence the deduction of this lag. Moreover, the asymmetry of the molecular orbital will also play a role in the present procedure. For polar molecules with a small asymmetry, the PD effect is small and the lag difference discussed in the paper is also difficult to resolve. Considering these complex effects related to molecular properties, we conclude that our procedure discussed in the paper is more applicable for polar molecules with a large PD, a relatively small R, a high orientation degree and the parallel alignment at long laser wavelengths.

Conclusion
In summary, we have studied ionization dynamics of HeH + in strong elliptical IR laser fields. We have shown that due to the PD effect, the Coulomb-induced ionization time lags differ for ionization events occurring in two consecutive half laser cycles. This lag difference (about 20 attoseconds) is well mapped in the PMD. Using a procedure termed as CCAC which considers the Coulomb-induced temporal correction to classical predictions, we are able to distill this difference from PMD with a high time resolution. The Coulomb-induced ionization time lag characterizes the response time of the electronic wave function to a strong-field ionization event. This response time is general and has important effects on tunneling-triggered ultrafast electron dynamics. The present work provides a feasible manner for evaluating the response time, especially for probing the differences of response time between diverse targets and diverse electronic states.

Data availability statement
The data that support the findings of this study are available from the corresponding author on reasonable request. The data generated and/or analysed during the current study are not publicly available for legal/ethical reasons but are available from the corresponding author on reasonable request.