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Abstract
We present theoretically obtained photoelectron momentum distributions (PMDs) for the strong field ionization of argon in an elliptically polarized laser field at a central wavelength of 400 nm. Three different theoretical approaches, namely, a numerical solution of the time-dependent Schrödinger equation (TDSE), a nonadiabatic model, and a classical-trajectory Monte Carlo (CTMC) model are adopted in our calculations. From the TDSE calculations, it is found that the attoclock offset angle (most probable electron emission angles with respect to the minor axis of the laser’s polarization ellipse) in the PMD increases with rising ATI order. While this result cannot be reproduced by the CTMC model, the nonadiabatic model achieves good agreement with the TDSE result. Analysis shows that the nonadiabatic corrections of the photoelectron initial momentum distribution (in both longitudinal and transverse directions with respect to the tunneling direction) and nonadiabatic correction of the tunneling exit are responsible for the ATI order-dependent angular shift.
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1. Introduction
The ultrafast dynamics of atoms and molecules subject to intense laser fields have been extensively studied in the last a few decades (see, e.g., [1–4]). As a hot issue of strong field atomic, molecular and optical (AMO) physics, above-threshold ionization (ATI), which was first discovered by Agostini et al. [5], has attracted a lot of attention. The ionization dynamics may be comprehended by the physical picture of either multi-photon ionization (MPI) or tunneling ionization (TI) [6–12]. To distinguish these two regimes, a Keldysh parameter $\gamma = \left [ I_{p}/ \left ( 2U_{p} \right ) \right ] ^{1/2}$ [13], where $I_{p}$ is the ionization potential and $U_{p}$ is the ponderomotive energy of the laser field, has been introduced. For $\gamma \ll 1$, the TI dominates and the dynamics can be understood by the semi-classical theory [6–8]. For $\gamma \gg 1$, the MPI becomes important and the photoelectron spectra show a multitude of ATI peaks and resonance structures [9–12]. For $\gamma \sim 1$, the electron dynamics is not adiabatic, i.e., the quasistatic approximation is not valid, and the nonadiabatic effect [14–18] starts to play a significant role in the tunneling process.
Nonlinear processes in the TI regime have attracted a lot of attention. The tunnelling electron in the first step is related to other nonlinear phenomena, for instance, nonsequential double ionization [19–21], high harmonic generation [22–24], and Rydberg state excitation [25–28]. These intriguing strong-field phenomena can be well-described by the semiclassical model [21,24,25], in which, an electron tunnels through a distorted Coulomb barrier nonperturbatively, and then flows away along a classical trajectory [29,30]. Typically, the semiclassical model can be used to explain the dynamics under "adiabatic" tunneling conditions, which is valid in the case of long wavelength and high laser intensity [31,32]. Recently, it has been discovered that, semiclassical models become inaccurate in describing photoionization dynamics if the nonadiabatic effect becomes important [15–17,33,34]. For example, the distortion of the initial momentum distributions can be attributed to nonadiabatic corrections of the tunneling step [18,35–38].
Previous studies indicate that the nonadiabatic effect has a significant impact on photoelectron dynamics under close-to-circularly or circularly polarized laser fields [15–18,33–38]. Regarding the highly suppressed photoelectron rescattering process [39], the instantaneous ionization time is mapped to the final photoelectron angular distribution (PAD) in the polarization plane in a close-to-circular [40–45] or bicircular [46,47] laser field. One can investigate the tunneling time delay and attosecond-resolved electron dynamics with this technique, which is also dubbed attoclock or attosecond angular streaking [40–47]. The attoclock technique relies on the measurement of the photoelectron momentum distributions (PMDs), which is characterized by an angular offset between the most probable photoelectron emission direction and the minor axis of the polarization ellipse. Asymmetry of photoelectron angular distributions in femtosecond laser field, which is important in attoclock experiments, has attracted a lot of attention for a few decades [15,33,48–52]. It has been shown that the nonadiabatic effect contributes significantly to the angular offset [15,33,48]. Similarly, it has been demonstrated that the angular offset depends on the ATI order (or the electron energy) [49–51]. With a dedicated numerical model, it is shown that the nonadiabatic corrections of the initial momentum and the exit position of photoelectrons can be applied to explain this phenomenon [51]. However, the formation of a physical picture of how the nonadiabatic effect would cause such an ATI order dependence of the angular offset is still hindered by the absence of detailed classical-trajectory analyses.
In this work, we calculate the photoelectron momentum distributions of argon ionized by an elliptically polarized (EP) laser field at 400 nm with a numerical solution of the time-dependent Schrödinger equation (TDSE), a nonadiabatic model and a classical-trajectory Monte Carlo (CTMC) model. TDSE calculations indicate that the PMD shows distinct angular shift and the most probable angle with respect to the minor axis of the laser polarization ellipse increases with rising ATI order. This energy-dependent angular shift can be well reproduced by the nonadiabatic model, but failed by the CTMC model. Analysis shows that the nonadiabatic corrections of the photoelectron initial momentum distributions (in both longitudinal and transverse directions) and nonadiabatic correction of the tunneling exit are together responsible for the ATI order dependent angular shift.
2. Theoretical methods
2.1 Numerical solution of the time-dependent Schrödinger equation
In this paper, the ATI process in the EP pulse is simulated by numerically solving the TDSE. We solve the two-dimensional TDSE in length gauge [53–57]:
where $\Psi (\mathbf {r},t)$ is the electron wave-function, $\mathbf {r}$ the electron position in the polarization plane. The Hamiltonian $H(\mathbf {r},t)$ is given by: Here $V(\mathbf {r})=-1/\sqrt {\mathbf {r}^{2}+a}$ is applied to approximate the potential of Ar$^{+}$ and the soft-core parameter $a$ is set to be 0.39 to match the ionization potential. The external laser field is described by $\mathbf {E}(t)$:2.2 Adiabatic model
The adiabatic model employed in this work is coined CTMC model, where the adiabatic approximation (see, e.g., [30,58]) has been applied. In the calculation, we assume that the electron is released from a bound state to continuum through tunneling. In the parabolic coordinates, the Schrödinger equation for a hydrogen-like atom in a uniform field can be expressed as [31]
After tunneling, the electron’s evolution is described by a classical Newtonian equation, i.e., $\frac {\partial ^{2}\mathbf {r}}{\partial t^{2}}=-\mathbf {E}(t)-\frac {\mathbf {r}}{r^{3}}$, where r is the distance from the electron to the nucleus and E(t) the electric field of the laser pulse.
2.3 Nonadiabatic model
To describe the ultrafast dynamics more accurately, the nonadiabatic effect will give rise to corrections of the initial momentum distributions, tunneling exit, and ionization rate of the tunneled electron, if compared with the adiabatic case. Here we briefly describe how to obtain the initial conditions and the weight of an electron orbit with the nonadiabatic effect considered [16,36].
Based on the strong-field approximation (SFA) [60,61], the transition probability from the ground state to a continuum state can be described as $\Gamma =\mathrm {exp}\left \{ -2\mathrm {Im} S\right \}$, where $S=\int _{t_{s}}^{t_{0}}dt\left \{ \frac {1}{2}\left [ \mathbf {P}+\mathbf {A}(t) \right ]^{2}+I_{p}\right \}$ and $t_{0}$ is the ionization time and $t_{s}$ is the complex transition point. The $\mathbf {P}$ is the conserved canonical momentum. Here the $t_{s}$ should satisfy the saddle-point equation $\left [ \mathbf {P}+\mathbf {A}(t_{s}) \right ]^{2}+2I_{p}=0$, where $t_{s}=t_{0}+it_{i}$, the real part $t_{0}$ is the ionization time and the imaginary part $t_{i}$ denotes tunneling time through the barrier. The saddle-point equation can be rewritten by
In the SFA theory, the electron trajectory under the barrier is given by $\mathbf {r}(t)=\int _{t_{s}}^{t}dt'\left [ \mathbf {P}+\mathbf {A}(t') \right ]$, and the tunnel exit is taken as the real part of the sub-barrier trajectory at $t_{0}$, i.e., $\mathbf {r}(t_{0},t_{i})=\mathrm {Re} \int _{t_{0}+it_{i}}^{t_{0}}dt'\left [ \mathbf {P}+\mathbf {A}(t') \right ]$ [61]. Thus, the coordinates of the tunnel exit ($\mathbf {r}_{0}$) can be written as [16,48],
With the initial momentum, tunnel exit and ionization rate given above, the nonadiabatic effect can be well considered and the following evolution of the photoelectron will be achieved by numerically solving the Newtonian equation. In order to consider the photoelectron interference effect, we include the phase of each classical trajectory. In our nonadiabatic model, the phase of each trajectory is given by a procedure similar to that of the semiclassical two-step (SCTS) model [62],
In our calculations, 5$\times$10$^{7}$ electron trajectories are calculated to obtain the PMD for each intensity. The probability of each asymptotic momentum is determined by
3. Results and discussions
In Figs. 1(a) and 1(b), the PMDs for Ar subject to an EP laser field at the intensity of $3\times 10^{14} W/cm^{2}$ and the ellipticity of 0.71 at 400 nm are obtained with TDSE method and nonadiabatic model, respectively. In both panels, distinct ATI rings, which spaced by one photon energy, can be identified and the results obtained by the two methods are qualitatively consistent to each other. To illustrate the nonadiabatic effect, calculations based on the adiabatic model, where the nonadiabatic effect has been totally ignored, have been performed and the results are presented in Fig. 1(c). In contrast to the results in Figs. 1(a) and 1(b), the ATI rings disappear in Fig. 1(c). This can be attributed to absence of the quantum phases (and the interference [63–66]) of classical trajectories in the adiabatic model.
Fig. 1. (a)-(c) show the PMDs of Ar in the polarization plane of the elliptical laser field at ellipticity of 0.71 calculated by TDSE (a), nonadiabatic model (b), and CTMC model (c). (d) and (e) show the photoelectron angular distributions (PADs) for the 1st- (black squares), the 2nd- (red circles), and the 3rd-order (blue triangles) ATIs of (a) and (b), respectively. (f) shows the PADs with same momentum interval as in (e). The colored arrows depict the maxima of the angular distributions. The laser intensity is $3\times 10^{14} W/cm^{2}$, the wavelength is 400 nm.
In an attoclock experiment, the photoelectron angular distribution obtained from PMD is usually employed to extract the most probable photoelectron emission angle with respect to the minor axis of the laser polarization ellipse. With the data shown in Fig. 1, the photoelectron distributions versus the angle between the photoelectron emission direction and the minor axis of the laser polarization ellipse for different ATI orders are extracted and shown in Figs. 1(d)–1(f). As shown in Figs. 1(d) and 1(e), for the TDSE and nonadiabatic model calculations, the angle corresponding to the peak of the distribution, i.e., the most probable angle, increases with rising ATI order. On the other hand, for the CTMC calculations, the most probable angle is almost unchanged with rising ATI order.
To illustrate the origin of the angular offset presented in Fig.1, we further calculated the PMDs with the CTMC model and nonadiabatic model when the Coulomb potential is ignored, as shown in Fig. 2. It is found that the angular offset, which is clear in Fig. 1(e), disappears and all the yield maxima appear in the direction of the minor axis in the polarization ellipse. Thus, it is clear that the influence of the Coulomb potential is the origin of the angular offset. Comparing Fig. 1(e) with Fig. 1(f), it is interesting to note that the influence of Coulomb potential in the CTMC model and the nonadiabatic model is different and the angular offset varying with respect to ATI order is significantly only in calculations of the nonadiabatic model.
Fig. 2. PMDs for Ar calculated with (a) the CTMC model and (b) the nonadiabatic model. The laser parameters are identical to those of Fig.1 and the Coulomb potential has been ignored in the calculations.
To shed more light on the physics behind the results presented above, we depict the photoelectron initial longitudinal and transverse momentum distributions with respect to the laser phases of electron tunneling for different ATI orders in Figs. 3(a)–3(c) and Figs. 3(e)–3(g), respectively. As we can see, both the photoelectron initial longitudinal and transverse momentum distributions are dependent on the laser phase. For the 1st-order ATI, the highest yield appears at $p_{\parallel }\approx 0$. As the ATI order increases, the momentum interval covered by the photoelectron initial longitudinal momentum distributions becomes broader. The $p_{\parallel }$ distributions of photoelectrons, which is extracted from the data inside the white rectangles in Fig. 3(a)-(c), are presented in Fig. 3(d), where the $p_{\parallel }$ corresponding to the peak of the distribution moves to the negative direction with rising ATI order. In contrast, for the data in the white rectangles in Figs. 3(e)–3(g), the $p_{\perp }$ corresponding to the peak of the distribution shifts from $p_{\perp }< 0$ to $p_{\perp }>0$ with rising ATI order.
Fig. 3. (a)-(c) show the initial longitudinal momentum distributions at the tunnel exit with respect to the laser phase for the 1st- (a), the 2nd- (b), and the 3rd-order (c) ATIs of Fig. 1(b). (d) shows the distribution of the half-cycle-averaged longitudinal momentum distribution from the white rectangle in (a,b,c). (e)-(g) show the initial transverse momentum distributions at the tunnel exit with respect to the laser phase for the 1st- (e), the 2nd- (f), and the 3rd-order (g) ATIs of Fig. 1(b). (h) shows the distribution of the half-cycle-averaged transverse momentum distribution from the white rectangle in (e,f,g).
In order to analyze the influence of the photoelectron initial longitudinal momentum distributions, which is induced by the nonadiabatic effect, on the angular offset, the PMDs with the initial longitudinal momentum of $p_{\parallel }= 0$ have been calculated with the nonadiabatic model and the results are shown in Figs. 4(a) and 4(b). In Figs. 4(a) and 4(b), it can be found that the angular offset of the yield maximum is larger for the 1st-order ATI, which is qualitatively different from the results in Figs. 1(b) and 1(e). Because only the initial longitudinal momentum has been changed in the calculations shown in Figs. 4(a) and 4(b), we can understand that the initial longitudinal momentum distributions in the nonadiabatic model will produce a larger angular offset for higher ATI order.
Fig. 4. (a) shows the PMD calculated by nonadiabatic model with $p_{\parallel } = 0$. (b) shows the PADs for the 1st- (black squares), the 2nd- (red circles) ATIs of (a). (c) shows the PMD calculated by the nonadiabatic model with adiabatic initial transverse momentum and $p_{\parallel } = 0$. (d) shows the PADs for the 1st- (black squares), the 2nd-order (red circles) ATIs of (c). (e) and the inset show the initial exit distributions for different ATI orders of Figs. 1(e) and 1(f), respectively. (f) shows the PADs of the 2nd-order ATI with the tunnel exit shifted by 2, 3, and 4a.u., respectively. (g) shows the tunneling exit distribution of photoelectrons in each case of (f). (h) shows the photoelectron distributions with respect to the strength of the Coulomb potential in each case of (f).
To investigate the influence of the nonadiabatic corrections of photoelectron initial transverse momentum distributions on the angular offset, the photoelectron initial transverse momentum distributions in the nonadiabatic model have been replaced by the corresponding distributions with adiabatic approximation and, in the meantime, the initial longitudinal momentum is still been set to be $p_{\parallel }= 0$. The results are presented in Figs. 4(c) and 4(d), where the angular offset of the yield maximum is smaller for the 1st-order ATI. This result is exactly opposite to that in Figs. 4(a) and 4(b). Thus, we can understand that the influence of nonadiabatic corrections of the initial transverse momentum distributions will give rise to a larger angular offset for lower ATI order.
Additionally, the nonadiabatic effect can influence the tunneling exit and may in turn affect the PAD. The tunneling exit distributions of photoelectrons for each ATI order employed in the nonadiabatic model are shown in Fig. 4(e). As we can see, with the nonadiabatic effect included, the electron tunneling exit distribution locates nearer to the core and becomes narrower for higher ATI order. On the other hand, in the inset of Fig. 4(e), for the CTMC model, the tunneling exit distributions are insensitive to the ATI order and appear further from the core.
To show the influence of the tunneling exit on the angular shift, we have manually shifted the tunneling exit distribution for the 2nd-order ATI. The shifted distributions are presented in Fig. 4(g) and the corresponding Coulomb potential felt by the photoelectrons is shown in Fig. 4(h). Apparently, if the distributions are shifted further from the core, the strength of the Coulomb potential will become smaller, which, in turn, may reduce the angular shift. Indeed, as shown in Fig. 4(f), the angular shift are smaller for larger tunneling exit. Moreover, the angular offsets of all the shifted distributions are significantly smaller than the corresponding unshifted angular offset in Fig. 1(e), where the tunnel exit is much closer to the core (see the red line in Fig. 4(e)).
To comprehend the physics intuitively, two typical electron trajectories from the 1st- and 2nd-order ATI in the nonadiabatic model are calculated and presented in Fig. 5(a). Moreover, we have further depicted a third trajectory with identical parameters to the one from the 2nd-order ATI except for a different tunneling exit, which is 4 a.u. further away from the core. In Fig. 5(b) and the inset, we show the portions of the trajectories near the core, where the initial positions of all the three trajectories can be revealed. By comparison of the two trajectories of the 2nd-order ATI in Fig. 5(a), it can be found that, with further tunneling exit, the electron will gain smaller velocity along $z-$axis and larger velocity along $x-$axis. As a result, the emission angle with respect to the minor axis of the laser polarization ellipse will be changed from 56$^{\circ }$ to 29$^{\circ }$, which is identical to that of the most probable angle of the 1st-order ATI. In Figs. 5(c) and 5(d), we present the time evolution of the electron momenta along x and z directions, respectively, for the three trajectories. We can find that the shift of the tunneling exit will lead to a deviation of $p_{x}$ from 0.3 a.u. to 0.51 a.u., and a deviation of $p_{z}$ from -0.51 a.u. to -0.27 a.u. Thus, change of angular offset is closely related to the different amount of photoelectron momentum variation along $x-$ and $z-$ directions, which, in turn, is relevant to the influence of tunneling exit.
Fig. 5. (a)-(b) show the typical orbits of the 1st-, 2nd-ATI and the 2nd ATI with tunneling exit shifting for 4 a.u. (c)(d) show the $p_{x}$ and $p_{z}$ with respect to the evolution time, respectively. All the orbits are calculated using the nonadiabatic model.
4. Conclusion
In conclusion, we calculate the PMDs of argon atom ionized by an EP laser field at a central wavelength of 400 nm with a numerical solution of the TDSE, a nonadiabatic model and a CTMC model. TDSE calculations indicate that the PMD shows distinct angular shift and the most probable angle with respect to the minor axis of the laser polarization ellipse increases with rising ATI order. This energy-dependent angular shift can be well reproduced by the nonadiabatic model, but not by the CTMC model. Further analysis shows that the initial longitudinal momentum, the initial transverse momentum and the tunneling exit induced by the nonadiabatic effect are together responsible for the ATI order-dependent angular shift.
Funding
National Key Research and Development Program of China (Grant No. 2019YFA0307700); National Natural Science Foundation of China (No. 11804374, No. 11834015, No. 11847243, No. 11874392, No. 11974383, No. 12004391, No. 12074106, No. 12074109); China Postdoctoral Science Foundation (Grant No. 2019M662752, Grant No. 2020T130682); Science and Technology Department of Hubei Province (No. 2019CFA035, No. 2020CFA029).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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