Classical interpretation for the influence of XUV pulse width on the streaking time delay and the oscillation amplitude of the momentum shift

We numerically investigate both the streaking time delay and the oscillation amplitude of the momentum shift of the photoelectron and justify them physically by developing a classical model based on the weak field approximation. The streaking time delay is insensitive to the extreme ultraviolet (XUV) pulse duration, while the oscillation amplitude obviously reduces as the XUV duration increases. This XUV duration dependence is attributed to the ionization probability of electron at initial times other than the peak of the XUV pulse. We propagate the classical electron trajectories originating at different initial times in the coupled Coulomb-laser (IR) potential and average the momentum shift for each trajectory over the width of the XUV pulse. By extracting the streaking time delay and the oscillation amplitude from this averaged momentum shift, the classical model results and the time-dependent Schrödinger equation results are found to be in good agreement. Both the insensitivity of the streaking time delay and the sensitivity of the oscillation amplitude on the XUV pulse width are well explained by our classical model considering initial ionization time average. Analytical estimation for the oscillation amplitude is obtained from the model of initial ionization time average.

In usual attosecond streaking schemes, the bound-state electron is ionized by absorbing one XUV photon in presence of a weak collinearly polarized infrared (IR) laser pulse. The energy of the photoelectron ejected forward along the laser polarization is streaked by the external IR pulse and, thus, depends on the phase of the IR pulse at the instant of ionization. According to the simpleman's picture or the strong-field approximation (SFA) where the influence of the Coulomb potential of the ion on the photoelectron is neglected, the photoelectron with energy E = ω X − I p (ω X is the XUV photon energy and I p is the ionization potential) is immediately exposed to the IR field with vector potential A IR (τ ) when the XUV pulse arrives at time τ . Propagating the classical electron in the IR field, it may get an additional momentum δp, once the IR pulse concludes, as a function of its ionization time τ (atomic units are taken throughout this paper unless otherwise stated), The momentum shift δp(τ ) as a function of arriving time τ of the XUV pulse can be detected in the attosecond streaking experiments. However, the breakdown of the oversimplified SFA model equation (1) is common in both experimental measurements and strict theoretical simulations. In turn, another formula similar to equation (1) is frequently employed to describe the streaking spectrum [28][29][30] δp(τ ) = −κA IR (τ + τ s ), with κ and τ s two fitting parameters. Though κ ≈ 1 is frequently assumed, the inclusion of κ produces a better fit for the streaking oscillation from accurate simulations. We refer to the apparent time-shift τ s defined in equation (2) as the streaking time delay and κ as the oscillation amplitude of the momentum shift.
In the streaking experiments, τ is the time shift between the peaks of the XUV and IR laser pulses which can be systemically varied. The streaking time delay is related to a large number of effects, such as Eisenbud-Wigner-Smith (EWS) time delay [31,32], Coulomb-laser coupling (CLC) [28,33], electron correlation [21,32,34] and the initial-state polarization or dipole-laser coupling [35]. The accurate modeling for the streaking time delay in multielectron systems is often challenging. As an example, the pioneering experiment on neon atom by Schultze et al [16] triggered considerable theoretical investigations [27,34,[36][37][38][39][40]. Only recently, the discrepancy between experiment and theory has been partially resolved and attributed to the influence of the shake up process [41] which was not accounted for in the experiment [16].
The streaking time delay of hydrogenic atoms and ions can be numerically simulated with high precision by solving the time-dependent Schrödinger equation (TDSE), and the origin of the delay appears well understood. The streaking time delay of the ground-state hydrogenic atoms is often attributed to two contributions, the EWS delay from the short-range behavior of the Coulomb potential and the CLC from the combined influence of the IR pulse and the long-range Coulomb potential. The time shifts from streaking simulation of hydrogenic atoms well coincide with those extracted from RABBIT (reconstruction of attosecond harmonic beating by interference of two-photon transitions) [42,43]. The classical-quantum correspondence has been well explored in studying the streaking time delay [28-30, 33, 42, 44]. Simulations based on the classical trajectory Monte Carlo (CTMC) method yield delays in close agreement with the TDSE results [29,33,44]. Rather accurate delays can be extracted from the most probable classical trajectory with the initial position of the electron carefully chosen. More recently, the relation of streaking time delay to the EWS delay is analytically derived by Saalmann and Rost through analyzing the most probable classical trajectory in the Krammer-Henneberger (KH) frame [45].
While exploring the streaking dynamics most of the aforementioned reports [29,30,45,46] have focused mainly on the streaking time delay and have not discussed the oscillation amplitude of the momentum shift of the electron, which relates to the strength of the IR field [2] and the characteristics of the x-ray pulse [1,3]. However, to explore the complete dynamics it is useful to investigate the oscillation amplitude κ beside the streaking time delay. The oscillation amplitude κ has been demonstrated to be larger than 1 for smaller kinetic energies of the photoelectrons, approaching to SFA limit i.e. κ → 1 for large enough photoelectron kinetic energy [28]. In the framework of SFA, it is shown that the XUV pulse duration can have a significant influence on the oscillation amplitude of momentum shift [47]. In analyzing the streaking time delay through the Eikonal-approximation (EA) [28], the classical numerical (CN) calculations [46], the classical analytical calculations in KH frame [45] and the improved-EA (IEA) [30], the most probable trajectory of the electron originated at the peak of the XUV pulse is considered while neglecting the effect of the XUV width. However, the models only considering the most probable trajectory are not able to explain the effects of XUV pulse width.
In the present paper, we study the streaking time delay and amplitude in photoionization of hydrogen atom with a moderate intensity XUV pulse assisted by an IR pulse by numerically solving both the 1D and 3D TDSE. The streaking dynamics can depend on the intensity of the XUV pulse in the nonperturbative regime [57]. In consistence with most previous theoretical studies, here we focus on the perturbative photoionization regime, where the XUV intensity dependence is not important. The streaking time delay and the oscillation amplitude κ are extracted by fitting the momentum shift of the ionized electron, in comparison with the momentum of the electron ionized by XUV pulse only, to equation (2). We develop a classical model based on the weak field approximation (WFA) for physical understanding of the effect of the temporal width of the XUV pulse on the streaking time delay and the oscillation amplitude κ. An expression of momentum shift, in close relation to previously reported EA and IEA formulas, is derived. The comparison of TDSE results with the EA, IEA, WFA and the CN results are presented and the advantage of the present WFA model is demonstrated in comparison with the EA and the IEA. To include the contribution from the trajectories originated at initial times other than the peak of the XUV pulse, the calculated momentum shift of the electron is averaged over the width of the XUV pulse. The streaking time delay and the κ from the averaged momentum shift are compared with numerically calculated TDSE results and we find a satisfactory agreement. We further derive a simple analytical expression for the dependence of κ on the XUV pulse duration and the streaking field frequency. The insensibility of the steaking time delay τ s on the XUV pulse duration is also a natural deduction in our derivations of initial time average.
The remaining of this article is organized as follows: In section 2, we briefly describe the numerical methods to solve the 1D TDSE for the model potential and the 3D TDSE for the real Coulomb potential. We present the classical analysis in section 3. The classical accurate expression for the energy shift is given in section 3(A). We develop the WFA in section 3(B), showing its relation to EA and IEA in section 3(C), and describe the ensemble average of the momentum shift over the width of the XUV pulse in section 3(D). The extension of the model to the 3D space is described in section 3(E). In section 4, the results and the corresponding discussions are presented. In section 5 we summarize our findings.

Quantum simulations
Our scheme to solve the 3D TDSE is presented in [48] in detail and has been successfully applied in a large number of physical simulations [49][50][51][52]. The method is based on the finite-element discrete variable representation method [53] and split-Lanczos algorithm [48]. Our method to solve the 1D TDSE was recently presented in reference [54]. In the following, we introduce the 1D TDSE in more details and explain how the momentum shifts in streaking scheme are extracted from the calculated photoelectron momentum spectra.
The 1D TDSE writes as where is the soft-core Coulomb potential which eliminates the singularity of the Coulomb potential at the origin, but keeps the Coulomb tail at the far end and therefore supports the Rydberg series in the eigen spectrum [54]. The ionization potential of hydrogen atom can be modeled by considering the soft parameter a = 2 [28]. The electron-laser interaction Hamiltonian is given in the length gauge by or in the velocity gauge by The electric field E(t) is related to the vector potential A(t) through E(t) = −dA(t)/dt. We use a cosine-squared function to describe the envelope of the IR pulse, whose vector potential is given by with the IR photon energy ω IR = 1.55 eV corresponding to a wavelength of 800 nm, the pulse duration T IR of 5 laser cycles, and the amplitude A 0 IR corresponding to a peak intensity of I IR = 1 × 10 12 W cm −2 . The time-delayed XUV pulse is assumed to have a Gaussian envelope, where τ is the time delay between the XUV and the IR laser pulse. The variables with subscript X indicate the parameters of the XUV laser pulse. The 1D TDSE, equation (3), is numerically solved with the finite-difference method discretizing the spatial coordinate x and the Lanczos propagator [48] for the short-time propagation of the wave function. Typically, the spatial grid with |x| = 1000 with grid spacing ∆x = 0.1 a.u. and the temporal grid with time step ∆t = 0.01 a.u is considered. We obtain the initial ground state of hydrogen atom with the imaginary-time-propagation method. In order to avoid reflection from the boundaries of the simulation box, we applied the wave-function-splitting method [55], in which the wave function is split into inner and outer parts periodically and the propagation of the outer part is performed analytically through the Volkov propagator. The smoothed splitting function used for the splitting of the wave function is F s (x) = 1/[1 + exp (x − |x c |/∆)], where ∆ = 5 a.u and x c = 400 a.u. is the width of the crossover region and the dimension of the splitting box, respectively. The wave function is freely propagated for t = 500 a.u. after the laser pulses conclude.
The momentum spectrum P(p, τ ) is obtained by collecting the photoelectron at the end of propagation from the projection of the wave function to the following plane wave with p being the photoelectron momentum. The delay-dependent momentum shift δp(τ ) is extracted from the peak shift in the momentum spectra i.e.
where p 0 max is the momentum at which the ionization probability P(p) reaches the largest value in the region p > 0 when only the XUV pulse is applied, and similarly p max (τ ) represents the peak position in the momentum spectrum when the IR pulse exists. Except for using the peak positions, the momentum shift can be, alternatively, extracted from the expectation value of the momentum [28], The streaking time delay τ s is then obtained by fitting the numerical momentum shifts to equation (2) applying the method of least squares. We notice that a high-precision result of τ s with error smaller than 2 × 10 −3 as can be obtained with a relative large step of delay time about 65 a.s.

Classical expressions of the photoelectron energy shifts
In the absence of the IR laser pulse, the final asymptotic momentum p 0 of the classical electron is related to its initial position x i and initial momentum p i through the conservation law of mechanical energy, with V(x i ) being the Coulomb potential at initial position given in equation (4). Introducing the IR laser pulse, a new asymptotic momentum p f (τ ) differing from p 0 will be derived from the initial momentum p i and the initial position x i . The asymptotic momentum p f (τ ) can be obtained by solving the Newton's equation numerically. In the following, we try to explicitly express the asymptotic momentum shift δp = p f − p 0 as an integral and derive various approximate expressions. We analyze the classical trajectory of the electron in the 1D space firstly. Such an analysis can be extended to the dominant trajectory along laser polarization in the realistic 3D space.
It is convenient to firstly study the energy shift δE of the asymptotic photoelectron by adding the IR pulse. The momentum shift δp is related to the energy shift δE by when δp/p 0 ≪ 1, which is usually satisfied in the routine streaking experiments where A IR /p 0 ≪ 1. Since the Coulomb force is conservative, the energy variation δE equals exactly the work of the IR pulse when the electron moves from x i to infinity, with p(t) be the momentum of the electron and −F IR (t) the electric force of the IR pulse along the trajectory.
To make the influence of the Coulomb potential obvious, we estimate the time integration in equation (14) with the method of integration by parts, where we have applied the boundary condition A IR (∞) = 0 and the Newton's second law, Corresponding to equation (4), the Coulomb force F C = −dV/dx is given by The first time integration in equation (16) can be analytically estimated by converting it to an integration with respect to A IR . Denoting the first two terms in equation (16) as δE 1 and the third term in equation (16) as δE 2 , we express the energy shift as with and In equation (21) we have changed the time integration in equation (16) to the x integration. p(x) is the position-dependent momentum of the photoelectron as the electron is moving in the CLC potential, while t(x) represents the time the electron arrives at the position x under the action of Coulomb force and laser field.

The weak field approximation
Up to now, the expressions (19)-(21) of the energy shift are accurate in classical mechanics without involving any approximations. Next we introduce the WFA by assuming that The condition equation (22) is usually well satisfied in the streaking experiments. Considering the condition equation (22) and the fact that p i > p 0 the first term in equation (20) is neglected, and therefore The explicit expressions of p(x) and t(x) are required for the calculation of the spatial integration in equation (21). In condition (22), the influence of the IR pulse may be neglected as zeroth-order approximation and p(x) and t(x) are then given as (see equation (12)) and Inserting equations (24) and (25) into equation (21), we obtain the δE 2 under WFA, 3.3. Relation of the WFA to the EA and IEA Next, we compare the present WFA with the EA initially developed for the strong-IR field ionization in the tunneling regime [56], but also utilized to analyze the streaking experiments where the IR field is typically weak [28]. The energy shifts of the present WFA model have extremely similar expressions with the previous EA model, which was derived from a quite different theoretical frame. In the EA model, the term comparable to δE 1 (equation (23)) in WFA is given by [28][29][30], We note that the present equation (23) can be converted into the above EA expression by assuming and approximating p i with The condition (28) holds only for the initial positions x i far from the core, and it indicates that the EA results would approach to the WFA results at large XUV photon energies when the photoelectron energies are larger than the absolute value of the Coulomb potential at the classical orbit of the ground state. The first term in equation (27) is exactly the expression of SFA.
The remaining term in EA model is also similar to equation (26) in the WFA, and the difference is on the time t(x) the photoelectron arrives at point x. In EA model, the electron arrives at the point x with a constant velocity. Therefore, Inserting equations (30) and (31) into equation (21), δE 2 in EA is obtained as It is known that EA does not work well in addressing the streaking time delay. Recently, Wang et al [30] suggested the IEA to give a much better description of the streaking time delay. In IEA, δE 1 is given by This expression is completely identical to the one (equation (23)) in WFA. In the second term of equation (33) the asymptotic momentum p 0 in EA has been replaced by the average of the asymptotic (p 0 ) and initial (p i ) momentum. δE 2 in IEA is given by In comparison with the present WFA equation (26), the difference is in the expression of t(x), In equation (35), the averaged momentum when the electron moves from x i to x is assumed to be p i /2 + p WFA (x)/2, which is only accurate in the assumption of uniform acceleration.

Ensemble average of the classical trajectories
In most of the classical calculations [28,30,45], the most probable trajectory of the electron originated at the peak of the XUV pulse τ and at the center of the Coulomb potential x i = 0 is propagated in the coupled IR and Coulomb potential. However, the momentum shift of the classical electron strongly depends on the initial position through the CLC. Therefore, as in the CTMC calculations [33] the ensemble average, of the initial positions x i from where the classical electron starts its motion, is taken with ψ 0 (x i ) being the initial ground-state wave function while δp(τ, x i ) is the momentum shift defined in equation (13). τ is the initial time and x i is the initial position from where the electron starts its motion in the coupled Coulomb-laser potential. By taking the average the results will be independent of the choice of initial position around the center of the Coulomb potential.
In general, the most probable trajectory originated at the peak of the XUV pulse is considered and the influence of the temporal width of the XUV pulse is neglected. However, the electron has a probability to be ionized at initial times other than the peak of the XUV pulse. Therefore, we emphasize that for complete understanding of the streaking dynamics, the ensemble average of the initial times representing the ionization time of the electron should be taken. The constant C = π 4 ln 2 . To calculate the time average without the initial position average, δp x (τ ′ ) in equation (37) will be replaced by δp(τ ′ ).
To make the time-average effect obvious, we reformulate the time integral in equation (37) by adjusting the integration variable With the help of equation (2), the momentum shift δp x (τ ′ + τ ) is explicitly expressed as where κ′ and τ ′ s are the classical amplitude and streaking time delay without considering the initial time average. Due to the contribution of the Gaussian envelope in equation (38), only the values of δp x (τ ′ + τ ) near τ ′ = 0 play significant roles in the integral. Expanding δp x (τ ′ + τ ) into Taylor series around τ ′ = 0 and neglecting terms with higher orders than τ ′ 3 , we obtain Inserting equation (40) into equation (38), we find that the first term of equation (40) produces the momentum shift without initial time average, the contribution of the second term in equation (40) is zero, and the third term contributes a modulation of the original streaking spectrogram. In fact, all terms in the series of equation (40) ∝ τ ′ 2n−1 (here n > 0 is an integer) contribute zero in the time integral of equation (38), while the modulations from terms ∝ τ ′ 2n can be easily seen by assuming that the derivatives of the streaking pulse envelope is negligible. The neglect of the IR pulse envelope is frequently taken in the analysis of streaking spectrogram [45].
Employing the slowly-varying envelope approximation (SVEA) approximation, we can neglect the derivatives of the streaking pulse envelope. Then the derivative part of the third term in equation (40) can be approximated by Similar to the term ∝ τ ′ 2 , all those terms ∝ τ ′ 2n are ∝ A IR (τ + τ ′ s ), which immediately leads to the conclusion that the streaking time delay τ s = τ ′ s , i.e. the delay remains the same even after the initial time average, but the amplitude κ of momentum shift will be significantly modulated. Inserting equation (41) into equation (40) firstly and then inserting the reformulated equation (40) into equation (38), we find the modulated κ from initial time average to be It is obvious from equation (42) that the amplitude κ will decrease with the increase in XUV duration and IR photon energy. In the following, we will show that the amplitude κ from the numerical calculations of equation (38) can be well reproduced by equation (42). We note that similar pulse duration dependence as equation (42) is reported in [47] under the framework of SFA, where the influence of the Coulomb potential is not included. In the SFA model, κ is always smaller than 1 for all photon energies and durations of the XUV pulses, while this prediction can be violated as shown in the followings.

The application of WFA model in the 3D space
Though the above derivations are developed for 1D system, the extension to the 3D system is straightforward. Depending on the initial position and momentum, the trajectory of the photoelectron in the 3D space can be a curve instead of a straight line as in the 1D space. In principle, one may take the initial position of photoelectron as an ensemble representing the quantum ground-state wave function [33]. However, it has been noted that accurate streaking time delays can be obtained by analyzing the most probable trajectory along the laser polarization. Earlier, Ivanov and Smirnova took the initial position to be velocity-dependent to match the phase shift in the scattering state [42]. More recently, Wang et al [30] took the initial position to coincide with the peak position of the radial density of the ground state. For the present 3D hydrogen atom with the Coulomb potential given by V(x) = − 1 x , we obtain accurate streaking time delays and amplitudes of momentum shift by assuming that the photoelectrons are produced at the position x i = 1 a.u. with the direction of the initial momentum p i coinciding with the asymptotic momentum along the laser polarization.

Typical streaking spectrogram
The resulting streaking spectrogram obtained by solving the 1D TDSE, equation (3), for the photoionization of the ground-state hydrogen atom in the forward direction is shown in figure 1(a). The XUV intensity is chosen to be I X = 1 × 10 15 W cm −2 with the central energy of the XUV photon ω X = 31 eV. The figure shows the oscillation of momentum shift with time delay τ between the XUV and the IR laser pulses. The momentum shift δp(τ ) extracted from the streaking traces is shown in figure 1(b), where the prediction equation (1) from the simpleman's picture is also given. By fitting the momentum shift δp(τ ) to equation (2), we observe a time delay between the momentum shift and IR vector potential, shown in the inset of figure 1(b).

Comparison of TDSE, WFA, EA and IEA results
Next, we solve the 1D TDSE for a range of XUV photon energies ω X and compare the extracted streaking time delay τ s and the oscillation amplitude κ with the results from CN calculations (solving equation (17) numerically), WFA model (equations (23) and (26)), EA model (equations (27) and (32)) and IEA model (equations (33) and (34)). Firstly, we do not consider the initial time average effects in the CN calculations and various models. In figure 2(a), we show the streaking time delay from TDSE calculations and models. The figure depicts that the magnitude of streaking time delay reduces and approaches to the SFA result with the increase of the central XUV photon energies as the Coulomb-laser interaction becomes weaker for larger kinetic energies of the photoelectron [28]. Moreover, it illustrates that the streaking time delay from the classical approximations are in good agreement with the TDSE results. The absolute values of the streaking time delay are comparable to the results reported elsewhere [30,33]. Rather, the results from the IEA, WFA and CN calculations are more accurate than the results from the EA in comparison with the TDSE results. We find results from the present WFA model are closer to the CN results than the IEA model, although the differences are tiny.
In figure 2(b), we plot the κ extracted from the quantum mechanical streaking traces and the classical analytical and numerical calculations. One can observe that there is a global shift between the results from TDSE and the classical calculations. The oscillation amplitude κ from the classical calculations is larger than 1 for smaller energies of the photoelectron and approaches to 1 for high kinetic energies. However, the  oscillation amplitude κ from the TDSE is always less than 1 beside the global shift between the oscillation amplitude κ from the expectation value and the peak value of the momentum. Yet, the overall trend of κ from all the calculations is the same i.e. decreasing with the increase in the XUV photon or photoelectron energy. The apparent difference of κ from TDSE and various models can be understood as the consequence of neglecting the variation of initial ionization time in those models.

Effect of initial time and initial position average
The obvious difference in TDSE results and various models clearly indicate the break down of the models based on the propagation of the most probable trajectory. The effect of the ensemble average over the initial position and initial time becomes indispensable to investigate. Thus, we analyze the effect of the initial position x i on the streaking time delay and the oscillation amplitude κ using different analytical approximations i.e. WFA, IEA, and EA in figure 3. It can be seen that the streaking time delay extracted from the momentum shift without taking the initial time average (lines in figure 3(a)) is the same with the streaking time delay extracted from the initial time-averaged momentum shift (markers in figure 3(a)). But, the oscillation amplitude κ extracted from the initial time-averaged momentum shift is shifted globally  downward, see figure 3(b). Beside the downward global shifting of the κ it can be seen that κ from the averaged momentum shift may be less than 1. Moreover, one can observe that the streaking time delay and κ from different approximations coincide for large positive x i . In this region the electron does not pass through the deep part of the ionic potential and hence observe a weak CLC. The momentum p(x) approaches to the asymptotic momentum p 0 and thus both IEA and WFA approach to EA.
In figure 4(a), we show the streaking time delay from the numerical TDSE calculations and classical analytical and numerical calculations after taking the initial time average of the momentum shift over the width of the XUV pulse using equation (37). The streaking time delay from TDSE and the averaged classical analytical and numerical calculations are in good agreement and the absolute values of the streaking delay from the averaged momentum shift are the same as the streaking delay from the momentum shift without taking its average, see figure 2(a). It illustrates that the streaking time delay dynamics in photoionization of hydrogenic atoms can be sufficiently understood and explained by the propagation of the most probable trajectory of the photoelectron ionized at the peak of the XUV pulse and the center of the ionic potential in the combined IR laser and Coulomb potential.
Furthermore, looking at figure 4(b) one can observe that the oscillation amplitude κ from the TDSE calculations and the classical analytical and numerical calculations averaged over the initial times only through the width of the XUV pulse are in good agreement. It indicates that for complete understanding of the streaking dynamics it is indispensable to take initial times ensemble average of the momentum shift, of the classical electron propagating in the combined IR and Coulomb potential, over the width of the XUV pulse.
We take the ensemble average of the classical momentum shift first over the initial position and then averaged this position-averaged momentum shift over the initial times through the width of the XUV pulse. The results extracted from the initial position-time-averaged momentum shift and only from the initial time-averaged momentum shift are illustrated in figures 5(c1) and (c2). Comparing the results in Figure 5. (a1)-(c1) The streaking time delay, and (a2)-(c2) the oscillation amplitude κ from the momentum spectra by solving TDSE and the WFA after averaging both over the initial position and initial time (markers), and only over the initial time (lines) (c1) and (c2) using equation (37) are calculated for a range of XUV photon energy ωX. The oscillation amplitude κ shifts globally downward with increasing the XUV width and is in agreement with the TDSE results for each width of the XUV pulse. figure 5(c1) and in figure 5(c2) , one can observe that there is no apparent difference there. Consequently, only averaging the classical momentum shift over the initial time is enough to explain the streaking dynamics such that initial position is carefully chosen.

Effect of XUV pulse width
The initial time average effect would lead to strong dependence of the oscillation amplitude κ on the duration of the XUV pulse though the dependence of the streaking time delay on the duration of the XUV is negligible. In figure 5(a1), the streaking time delay from the expectation value of the momentum equation (11) is shown. It depicts that with decreasing the width of the XUV pulse the streaking time delay shifts to larger negative values in smaller energy region of the photoelectron. This shift of the streaking time delay can be explained in terms of its variation with the photoelectron energy. The photoelectron wave packet has a certain bandwidth around the central energy due to the finite width of the XUV pulse. Therefore, the obtained time delay for an ionized wave packet can be considered as an average over contributions by particular energies within the bandwidth. Due to the nonlinear variation of streaking time delay with the kinetic energy, the obtained time delay for a wavepacket will be smaller than the contribution from the central kinetic energy. The difference between the obtained time delay and the time delay for the central kinetic energy decreases, and hence the time delay for a wave packet increases as the energy bandwidth decreases i.e. the XUV pulse width increases [46]. This type of increase in time delay does not occur while extracting the time delay from momentum corresponding to the maximum probability of the ionized electron, see figure 5(b1). Neither this happens for the streaking time delay extracting from classical momentum shift averaged over the initial time and averaged over both the initial position and initial time, figure 5(c1). It can also be observed that taking the ensemble average of the momentum shift through WFA over the initial time only or both over the initial position and initial time does not influence the streaking Figure 6. (a1)-(c1) The streaking time delay, and (a2)-(c2) the oscillation amplitude κ from the momentum spectra by solving 3D TDSE and the WFA after averaging over the initial time (c1) and (c2) using equation (37) are calculated for a range of XUV photon energy ωX. The intensity of XUV pulse IX = 1 × 10 12 W cm −2 and IR pulse IIR = 4 × 10 11 W cm −2 pulse is considered. The time delay and the oscillation amplitude κ inherits the same characteristics as in 1D case. The stars, triangles, squares and circles are calculated by using equation (42) for TX = 0, 200 as, 300 as, and 400 as, respectively. delay, figure 5(c1). This confirms that analyzing the streaking time delay in ionization of hydrogenic atoms is enough to propagate the most probable trajectory originated at the peak of the XUV pulse and at the center of the ionic potential in coupled Coulomb-laser potential and extract the time delay without taking the ensemble average.
Turning to the oscillation amplitude κ in figures 5(a2)-(c2), one can observe that κ shifts globally downward with increasing the width of the XUV pulse pertaining the decreasing trend with the XUV photon energy. The most important feature is the absolute value of κ which was predicted to be greater than 1 for lower energies and approach to the SFA results for larger energies of the XUV photon or photoelectron [28]. However, it can be seen from figures 5(a2) and (b2), that κ strongly depends on the temporal width of the XUV pulse. For smaller width, i.e. T X = 200 as, κ is greater than 1 for smaller energies and approaches to smaller values for larger energies. But for larger widths this is not the case. This discrepancy is resolved by averaging the momentum shift of the classical electron over the initial times through the width of the XUV pulse using equation (37) and extracted κ from this averaged momentum shift, see figure 5(c2). It can be clearly seen that the averaged classical results for each width of the XUV pulse, figure 5(c2), are in good agreement with the quantum mechanical numerical results from TDSE, figures 5(a2) and (b2). This clarifies that the propagation of the most probable classical trajectory in the coupled Coulomb-laser potential is not enough to fully understand the streaking dynamics. Hence, we take the average of the momentum shift over the initial time only and both over initial position and initial times, figures 5(c1) and (c2). It can be observed that the initial position average does not influence the streaking delay and the oscillation amplitude κ considerably.
Up to now, our discussions of results are focused on the calculations in the reduced 1D space. Next, we will show that all the observations in the 1D space can be extended to the real 3D space. As we see from 1D case that the initial position average does not have considerable influence on the streaking delay and the oscillation amplitude κ, see figure 5(c2). In the 3D WFA model, we only include the initial time average of the characteristic trajectories. We show the 3D TDSE and WFA results in figure 6. It illustrates that the 3D system inherits the same characteristics. The streaking time delay in the small energy region, extracted from the expectation value of momentum of TDSE calculation, slightly shifts to larger negative values with decreasing the width of the XUV, see figure 6(a1). Meanwhile, the streaking time delays from the peak  the oscillation amplitude κ from the momentum spectra calculated by solving 3D and 1D TDSE and WFA against the XUV photon energy with XUV width 300 a.s. The time delay and the κ from 1D, 3D TDSE and WFA calculations are in good agreement. The black squares and the red filled hexagrams represent the data extracted from [30] and [33], respectively. momentum of TDSE calculation (figure 6(b1)) and the WFA model (figure 6(c1)) are almost the same for different XUV duration. We observe downward global shift of the oscillation amplitude κ with the increase of the XUV width figures 6(a2)-(c2) as in 1D case figures 5(a2)-(c2). We also extract κ by exploiting equation (42) for different widths of the XUV and different energies of the XUV photon, shown as markers (stars, triangles, squares and circles), in figure 6(c2). The lines from the numerical calculation of equation (37) and the markers (triangles, squares and circles) from equation (42) are in consistence with each other for the corresponding width of the XUV pulse, justifying the approximation to neglect the higher-order derivatives in equation (40) and the SVEA in (41).
The more rigorous check for the reliability of the analytical equation (42) is given in figure 7, where κ is shown as a function of T X . Equation (42) reproduces the numerical result of (38) nearly perfectly for small T X , while the difference becomes larger as T X increases. The fact that figure 7 underestimates the value of κ at large T X can be attributed to the neglect of the term ∝ τ ′ 4 in the expansion of equation (40). The term ∝ τ ′ 4 has opposite sign to the term ∝ τ ′ 2 , leading to a larger κ in comparison with equation (42). Rather surprisingly, the agreement between present simple model and numerical TDSE results is also good for a wide range of pulse duration T X .
Finally, we rigorously compare the 1D TDSE and WFA (3D) results with those of 3D TDSE in figure 8. The choice of different dimensions only lead to a slight difference in both the streaking time delay and κ. Our WFA model including initial time average well reproduces the κ from expectation values through TDSE in a wide range of XUV photon energies. The global shift between the κ calculated from the expectation value (the circles) of momentum and from the peak momentum (the dotted line) exists as in 1D TDSE case (the black triangles and the blue dashed line), figure 8(b).

Summary
In conclusion, we have developed a classical model based on WFA to understand the XUV pulse duration dependence of the streaking time delay τ s and the oscillation amplitude of the momentum shift κ in the streaking spectrogram from TDSE. We formulate the momentum shift of WFA in analogy to those of EA and IEA. Higher precision of the present WFA formula is demonstrated. Though τ s is not sensitive to the variation of the XUV pulse duration, the amplitude κ does. This observation in numerical TDSE calculation can be well explained by our present model, in which the photoelectron is not only ionized at the peak of the XUV pulse but also ejected at time that the XUV field is not negligible. By doing the initial ionization time average, we obtain an analytical estimation for the XUV duration dependence of the amplitude κ, see equation (42). The reduction of κ as XUV duration increases, observed in TDSE calculations, is well reproduced by the simple and analytical equation (42). We demonstrate that these physical dynamics stay the same in both reduced 1D space and the realistic 3D space. The difference of τ s and κ in the 1D calculation and the 3D calculation is not significant. Our study emphases the important role of the initial ionization average in understanding the streaking spectrogram in classical simulations and may be useful in charactersing the XUV pulse and the light fields through streaking configuration. The quantitative dependence of the κ factor on the XUV pulse duration, discovered from the hydrogen atoms in the present study, can be expected in a broad range of systems including molecules and solids.

Data availability statement
The data of the present investigations may be provided upon reasonable request from the authors. The data cannot be made publicly available upon publication because they are not available in a format that is sufficiently accessible or reusable by other researchers. The data that support the findings of this study are available upon reasonable request from the authors.