Temporal characterization of electron dynamics in attosecond XUV and infrared laser fields ⁰⁰footnotetext: ^∗ [email protected] ⁰⁰footnotetext: ^‡ chen$\_$[email protected]

We first investigate the dynamic behaviors of electrons emitted only by a single XUV pulse with $t_{C}=\frac{T_{XUV}}{2}$. Figures 1(a) and 1(b) show the time-energy distributions (TEDs) of electrons along the positive x axis (namely $\Theta=$0, called as ”positive direction” below) in a four-cycle XUV pulse with $\varphi_{XUV}$=$0.5\pi$ and 0, respectively, which are calculated by Eq. (1). Noted that in order to intuitively show the physical meaning of the WDL function, the time-energy distribution is used to describe the result obtained by Eq. (1). The energy spectra (white dashed lines) calculated by the SFA theory are also given in Figs. 1(a) and 1(b). One can find that both TEDs look the same and exhibit a broad distribution without fine structures presented in the low-frequency case (See Fig. 2 of Ref. pra-guo ). The maxima of the TEDs are at times close to the time of the center of the XUV pulse and at energies corresponding to the peaks of the energy spectra. In addition, the TEDs in the negative direction ($\Theta=\pi$) also look like those shown in Figs. 1(a) and 1(b), which are plotted in Figs. 2(e) and 2(f). Further, we integrate the TED over the energy to obtain the ITD in Figs. 1(c) and 1(d). As shown in Figs. 1(c) and 1(d), all ITDs are in the shape of the XUV envelope, which are consistent with the previous understanding that the temporal profile of the XUV-induced photoemission rate follows the intensity profile of the incident XUV pulse mod . It is important to point out that in the TEDs there are many interference stripes located in the low-energy region and their positions vary with the CEP of the XUV pulse and the electron’s emission direction.

Give a closer glance at the peaks of the ITDs which are zoomed into in the insets of Figs. 1(c) and 1(d). One can find that minor differences exist among these ITDs. For $\varphi_{XUV}=0.5\pi$, the peak positions of the ITDs in both negative and positive directions slightly deviate from the center of the XUV pulse. There is a left shift in the positive direction with respect to the center of the XUV pulse while a right shift in the negative direction. The absolute values of left and right shifts are equal and estimated to be less than 1.68 as. As shown in the inset of Fig. 1(c), we can find that the ITDs in the positive and negative directions are mirror symmetric with respect to the central axis of the XUV pulse. For $\varphi_{XUV}$=0, the ITD is symmetric with respect to the center of the XUV pulse, in regardless of the emission direction of the electron. However, the maximal value of the ITD in the positive direction is a bit higher than that in the negative direction.

All these fine differences shown in Figs. 1(c) and 1(d) indicate that for a single-photon ionization event induced by an XUV pulse, the ionization characteristic of the electron still has a weak dependence on the emission direction of the electron and the CEP of the XUV pulse. According to our calculations, it is the interference appearing in the low-energy region of the TED that is responsible for the dependence mentioned above.

Next, we investigate the origin of these interference structures. We can approximately write the XUV electric field $\mathbf{E_{XUV}}(t)$ as $(\mathbf{E_{-}}+\mathbf{E_{+}})/2$, where $\mathbf{E_{\mp}}=E_{XUV}f_{0}(t)e^{\mp i(\Omega t+\varphi_{XUV}-\frac{\pi}{2})}\textbf{e}_{x}$ and the slowly varying pulse envelope is assumed, i.e. $f_{0}(t)=\sin^{2}\left[\frac{\pi t}{T_{XUV}}\right]$. When $\mathbf{E_{-}}$ is used, the term $e^{i(\frac{p^{2}}{2}+I_{p}-\Omega)t}$ is present in the transition matrix element, which corresponds to the rotating wave term (RWT). This transition process satisfies energy conservation, in which an electron transitions from the bound state to the continuum state by absorbing an XUV photon. When $\mathbf{E_{+}}$ is adopted, the term $e^{i(\frac{p^{2}}{2}+I_{p}+\Omega)t}$ appears in the transition matrix element, which corresponds to the counter-rotating term (CRT). This ionization process does not satisfy energy conservation where an XUV photon is emitted during the electron transition.

The TEDs of the electrons calculated by the RWT and the CRT components are given in Figs. 2(a) and 2(b), respectively, which show no dependence on the electron’s emission direction or the CEP of the pulse. The white lines in Fig. 2 denote the ITD. As shown in Fig. 2(a), the RWT component dominates the distribution and no interference exists in the TED or ITD. In contrast, for the results from the CRT component shown in Fig. 2(b), there are many strong interference structures in the TED and the corresponding ITD (white line) oscillates. The contribution from the CRT component is much smaller than that from the RWT component (See the color bars of Figs. 2(a) and 2(b)). When both two components are considered (namely, $\mathbf{E_{XUV}}(t)=-\frac{\partial\mathbf{A_{XUV}}(t)}{\partial t}$ is used), a very weak interference structure located in the low-energy region shows up in the TEDs which are displayed in Fig. 2(c) for $\varphi_{XUV}=0.5\pi$ and in Fig. 2(d) for $\varphi_{XUV}=0$. The weak interference structure shows a dependence on $\varphi_{XUV}$ by comparison of the interference patterns shown in Figs. 2(c) and 2(d). This indicates that the CRT component not only survives but also interferes with the RWT component although its contribution is relatively small. It is worth noting that the interference between the RWT and CRT components can also be found in Ref. Lupx , which leads to a weak oscillation in the time-dependent population of the bound state in the attosecond streaking field. Up to now, the structures in Figs. 2(a)-(d) have a common feature that they are symmetric with the center of the XUV pulse and are independent of the electron’s emission direction. It should be noted that we neglect the dressing of the XUV field (namely $A_{XUV}(t)=0$) in the calculations of Figs. 2(a)-2(d). This is done for one reason that we can check the effect of the variable $A_{XUV}(t)$ on the electron dynamics because this variable is usually neglected in the case of an XUV (XUV train) field combined with an IR field due to its small value, e.g., $A_{XUV}(t)\sim 0.01$ a.u. in our paper.

When the variable $A_{XUV}$ is considered, the interference stripes located in the low-energy region which are shown in Figs. 2(e) and 2(f) become clearer compared with those shown in Figs. 2(c) and 2(d) and exhibit the dependence on the electron’s emission direction. This demonstrates that although the value of $A_{XUV}(t)$ is very small, it can still affect the dynamics of the electron. Figure. 2(e) (Figure. 2(f)) shows the TED of the electron emitted along the negative direction in an XUV field with the same parameters as in Fig. 1(a) (Fig. 1(b)), except for $\Theta=\pi$. For $\varphi_{XUV}=0$, the constructive (destructive) interference occurs at the central time of the XUV pulse in the positive (negative) direction, which leads to a sharp (flat) peak of the ITD shown in Fig. 1(d). For $\varphi_{XUV}=0.5\pi$, the interference structure is no longer symmetric with respect to the center of the XUV envelope. The position of the constructive interference stripe closest to the peak of the envelope in the positive (negative) direction that is shown in Fig. 1(a) (Fig. 2(e)) is at the right (left) side of the peak of the XUV envelope, which makes a right (left) shift of the peak of the ITD in positive (negative) direction.

Based on the above calculations, it is concluded that the interference structure presenting in the low-energy region comes from the interference between the RWT and CRT components, which is responsible for the dependence of the ITD on $\varphi_{XUV}$ and the electron’s emission direction. It is worth mentioning that the ratio of the transition amplitude of the electrons from the RWT component to that from the CRT components determines the strength of the interference structure in the TED and thus affects the ITD. According to our analysis, for a fixed XUV frequency, the larger the value of $I_{p}$, the weaker the effect of the interference on the ITD is.

In this section, we add an IR field to investigate its effect on the dynamical characteristics of electrons compared with the case of only a single XUV pulse. In this paper, we choose the IR pulse with fixed $\varphi_{IR}$=0 and a duration of 4 o.c. All times in the following text are in units of optical cycle (o.c.) of the involved IR field. Note that we use the arrival time (namely $t_{C}$) of the center of the XUV pulse in the presence of the IR field instead of the relative phase between the two fields to describe one specific two-color field.

First, we take the case of attosecond streaking with $t_{C}=2$ o.c. and $\varphi_{XUV}=0$ as an example. Figure 3 shows the TEDs of attosecond streaking in the positive (left column) and negative (right column) directions for three IR intensities of $I_{IR}=1\times 10^{11}$ W/cm² (Figs. 3(a) and 3(b)), $1\times 10^{12}$ W/cm² (Figs. 3(c) and 3(d)), and $1\times 10^{13}$ W/cm² (Figs. 3(e) and 3(f)). As shown in Fig. 3, for the lowest intensity of $I_{IR}=1\times 10^{11}$ W/cm², the TEDs in both the positive and negative directions are almost identical, which look the same as those of Fig. 1(b). However, when the IR intensity increases, the TEDs in the positive direction gradually shift to the higher energy region and the widths of the distributions become broader. In contrast, the TEDs in the negative direction have an opposite tendency. One can find that the difference of the TEDs between positive and negative directions is less pronounced for the lowest IR intensity ($I_{IR}=1\times 10^{11}$ W/cm²) and becomes more obvious for the higher IR intensities ($I_{IR}=1\times 10^{12}$ W/cm² and $1\times 10^{13}$ W/cm²).

These features can be approximately explained by a semiclassical picture as follows mod ; 2012jpb ; pengliang ; review3 : An electron in the bound state is ionized by absorbing an XUV photon to one continuum state with momentum $\mathbf{p}_{0}$ at time $t_{0}$, and then it moves only in the IR field. The final kinetic energy detected by the detector is given by

For the ionization time range shown in Fig. 3, the direction of the electron’s drift momentum $-\mathbf{A_{IR}}(t_{0})$ points to the positive direction, which leads to a high-energy (low-energy) shift of the photoelectron spectrum in the positive (negative) direction because of the positive (negative) value of $-\mathbf{p}_{0}\cdot\mathbf{A_{IR}}(t_{0})$. Meanwhile, as shown in Fig. 1(a), for a given ionization time $t_{0}$, the initial momentum $p_{0}$ has a distribution with the value of the center slightly less than $\sqrt{2(\Omega-I_{p})}$. According to Eq. (7), the value of the energy shift is proportional to $p_{0}$. This, together with the distribution of $p_{0}$, results in the change of the width of the TED. It is easy to understand that the discrepancy of the TEDs between these two directions is more obvious for the case of a high IR intensity because of the larger absolute value of the vector potential $\mathbf{A_{IR}}(t_{0})$.

In order to show in detail the influence of the IR intensity on the ITD, we calculate the ITDs for three cases of $t_{C}=1.75$ o.c., 2.0 o.c., and 2.25 o.c. As shown in Figs. 4(a)-4(f), the shapes of the ITDs in the same direction vary with different two-color fields. For the cases of $t_{C}=1.75$ o.c. and 2.25 o.c., the ITDs are no longer symmetric with respect to the center of the XUV pulse, the peaks of which obviously deviate from the center of the XUV pulse. However, for the case of $t_{C}=2.0$ o.c., the ITDs still show symmetric shapes with maxima at $t=t_{C}$. For clarification, the time offsets between the peak of the ITD and the center of the XUV pulse are plotted in Fig. 4(g) as a function of the IR intensity, which are extracted from Figs. 4(a)-4(f). Here, the positive (negative) value of the time offset means time delay (advance). Several characteristics in the cases of $t_{C}=1.75$ o.c. and $t_{C}=2.25$ o.c. can be found in Fig. 4(g): (i) There are time delays in the case of $t_{C}=2.25$ o.c. while time advances in the case of $t_{C}=1.75$ o.c. (ii) The offset value becomes larger with increasing IR field for one specific case. (iii) For the electrons emitted along the same direction, the absolute value of the time offset in the case of $t_{C}=1.75$ o.c. is equal to that in the case of $t_{C}=2.25$ o.c. for the same IR intensity. (iiii) The time offset value of the ITDs in both positive and negative directions are unequal for a given case, which indicates that the weak dependence of the electron behavior on the emission direction.

These features can be ascribed to the change of the electric field. Figure 5 shows the electric fields of the XUV and the IR pulses for three different cases. The whole two-color pulse is also given at the top right of each panel. The presence of the IR field is equivalent to the increase or the decrease in the original XUV field at times marked by the magenta dotted lines for the former and by the green dotted lines for the latter. Here, we mainly focus on the central one optical cycle of the XUV pulse, because the change of the ITD induced by the IR field is mainly confined to this cycle. For instance, the XUV field is enhanced at times $t<t_{C}$ and is suppressed at times $t>t_{C}$ in the case of $t_{C}=$1.75 o.c. This causes $\mathbf{E}(t)\neq-\mathbf{E}(2t_{C}-t)$ in the time range during the XUV pulse and thus leads to the unsymmetric shape of the ITD. Meanwhile, the peak of the ITD shifts to the side of the electric field enhancement. The similar analyses can be used in the case of $t_{C}=2.25$ o.c. One can find that the peaks of the ITDs in the case of $t_{C}=2.25$ o.c. also shift to the right side of the center of the XUV pulse where the electric field is enhanced.

It is worth noting that there is a relationship of $\mathbf{E_{1}}(t_{C1}-\delta t)=-\mathbf{E_{2}}(t_{C2}+\delta t)$, where $\mathbf{E_{1}}$ ($\mathbf{E_{2}}$ ) and $t_{C1}$ ($t_{C2}$) represent the two-color field and the central time of the XUV pulse in the case of $t_{C}=1.75$ o.c. (2.25 o.c.), namely $t_{C1}=1.75$ o.c. ($t_{C2}=2.25$ o.c.), as shown in Fig. 5(a) (Fig. 5(c)), respectively, and $\delta t$ is the time difference between any time $t$ and the central time of the XUV pulse, ranging from -$\frac{\omega}{\Omega}$ o.c. to $\frac{\omega}{\Omega}$ o.c. in the present work. This characteristic of the electric fields results in the equal absolute values of the time offsets for these two cases under the same parameters. However, for the case of $t_{C}=2.0$ o.c., although the electric fields of the central one cycle of the XUV pulse are all increased (They are not visually obvious), they still satisfy the relation of $\mathbf{E}(t)=-\mathbf{E}(2t_{C}-t)$ (See Fig. 5(b)), leading to the symmetric shape of the ITDs.

Based on the characteristics shown in Figs. 3 and 4, we can find that the IR field not only causes the energy shift of the electron explained approximately by the semiclassical picture but also has a obvious effect on the emission time of the electron. This effect becomes greater with increasing IR intensity and more importantly, the time offsets induced by the IR field are different for different two-color fields.

Further, we study the effect of the IR field on the ITDs of electrons emitted from atoms with the higher ionization energy of $I_{p}=2.5$ a.u. Figure 6 displays the ITDs of attosecond streaking for three different cases of $t_{C}=1.75$ o.c. (Figs. 6(a) and 6(b)), 2.0 o.c. (Figs. 6(c) and 6(d)), and 2.25 o.c. (Figs. 6(e) and 6(f)). The parameters are the same as in Fig. 4, except for $\varphi_{XUV}=0.5\pi$ and $I_{p}=2.5$ a.u. As can be clearly shown in Fig. 6, all ITDs are smooth curves and look almost identical. These curves still resemble the envelope of the XUV pulse. By comparing with those shown in Fig. 4, it is found that the impact of the IR field on the ITD is smaller for the case of $I_{p}=2.5$ a.u. Meanwhile, we zoom into the peak of the ITD in the inset of each panel. One can find that some tiny differences still exist among these ITDs. For each case, the ITDs in the positive and negative directions are also mirror symmetric with respect to the axis $t=t_{C}$, which is consistent with the case of only one XUV pulse with $\varphi_{XUV}=0.5\pi$ shown in Fig. 1(c). Another feature is that the ionization rates at ionization times in vicinity of the center of the XUV pulse, in regardless of the emission directions, increase in the case of $t_{C}=1.75$ o.c. while they decrease in the case of $t_{C}=2.25$ o.c., with increasing IR intensity. These are also the results of the change of the electric field (See Fig. 7). Figure 7 displays the two-color pulse with the same parameter as in Fig. 5, except for $\varphi_{XUV}=0.5\pi$. As shown in Fig. 7, the pattern of the XUV field with $\varphi_{XUV}=0.5\pi$ changes with respect to that with $\varphi_{XUV}=0$ shown in Fig. 5. In the case of $t_{C}=1.75$ o.c., the electric field of the central half optical cycle of the XUV pulse is decreased and thus causes the suppression of the yield of the electron. However, the corresponding electric field is increased in the case of $t_{C}=2.25$ o.c., leading to the enhancement of the ionization rate.

For the case of $t_{C}=2$ o.c., the situation is different. As can be clearly shown in insets of Figs. 6(c) and 6(d), with increasing IR intensity the ionization rates gradually increase for the electrons along the positive direction while they decrease for the electrons along the negative direction. This can be understood as follows: According to the semiclassical picture, for the electrons emitted along the negative direction, the direction of the initial momentum $\mathbf{p_{0}}$ after ionized by absorbing an XUV-photon is opposite to the direction of $-\mathbf{A_{IR}}(t_{0})$ under consideration. For the electrons with the small value of $p_{0}$, the original emission direction of the electron can be reversed by the IR field because of the larger value of $-\mathbf{A_{IR}}(t_{0})$, namely $[\mathbf{p}_{0}-\mathbf{A_{IR}}(t_{0})]<0$. More electrons meet above condition compared with the case of the low ionization energy $I_{p}=2$ a.u. This results in the lower yield of electron in the negative direction and the higher yield of electron in the positive direction, which is more obvious for the case of $I_{IR}=1\times 10^{13}$ W/cm².