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Abstract
We report on a theoretical study of nonsequential double ionization (NSDI) of magnesium atoms by using combined linearly and circularly polarized fields. By employing a concise model including the dynamic ionic dipole potential, we show that the polarization effects can be controlled by tuning the subcycle waveform of the electric field of the two-color pulses. We demonstrate that the influence of the dipole potential on NSDI depends on the symmetry of two-color laser fields by tracing back the electron trajectories. Furthermore, we propose a method allowing for manipulating the returning trajectories with the initial direction of the tunneled electrons almost unchanged.
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1. Introduction
Electron correlation is ubiquitous in many fundamental processes in light-driven atomic and molecular systems, and has thus been an intriguing topic from both experimental and theoretical sides. A typical example is nonsequential double ionization (NSDI) of atoms and molecules in strong laser fields, which contains significant information about electron correlation and catches much attention in recent years since its discovery [1,2]. This phenomenon shows a strong enhancement in the doubly charged ion yield or double ionization probability versus intensity curve, i.e., a distinct “knee” structure, over predictions by the sequential tunneling theory assuming an independent ionization of the two electrons [3]. The physical picture of NSDI can be understood by the quasiclassical recollision model [4]. In this model, after tunneling through the distorted atomic potential barrier, the electron obtains energy from the oscillating laser field and can be driven back to the parent ion, leading to the release of the second electron by inelastic scattering. This insightful physical picture is also employed to account for other strong-field processes such as high-order harmonic generation (HHG) [5] and high-order above-threshold ionization [6].
In recent years, numerous efforts have been made to successfully control ionization and recollision by employing well-designed laser fields. In Ref [7], it has been shown that the two-electron emission dynamics in NSDI can be controlled by changing the subcycle shape of orthogonally polarized two-color (OTC) laser fields. By tuning the relative intensities of the two-color circularly polarized laser pulses, one can modify distinct pathways to ionization, allowing for unique control of strong-field processes [8]. Another transparent way that has been proposed is simply using an elliptical light waveform to manipulate the recollision dynamics [9]. Recently, Han $et$ $al.$ presents a novel attoclock configuration, where a perturbative linearly polarized light field at 400 nm is used to mark the ionization time of the electrons ionized in circularly polarized (CP) laser fields at 800 nm [10].
In these previous works, tailored laser fields have been used to control the electron dynamics during tunneling and recollision. However, how to control the remaining electrons’ motion after tunneling remains largely unexplored. It has been realized that the remaining electrons’ motion and the relevant multielectron effect, which are closely related to the laser-induced polarization of the ion, have a significant influence on strong-field ionization. Such effects can be found in the photoelectron momentum and angular distributions, low-energy and very low-energy structures [11–14] of the photoelectron spectra from single ionization. Recently, it has been shown that, the induced dipole potential of the ion enhances the recollision probability in NSDI with CP fields [15]. These studies uncover the multielectron effect on single- and double-ionization dynamics. Motivated by these works, one may speculate the control of the remaining electrons’ motion by manipulating the ionic polarization effect. Because Mg is a nearly ideal two electron system displaying a high degree of electron correlation, and its single ion has a large static polarizability so that the remaining electrons can be influenced by the laser light easily, it can be regarded as an appropriate system designed to study the ionic core polarization effect in strong-field double ionization. A key question thus arises: is it possible to control the remaining electrons’ motion in NSDI of such system with tailored laser fields?
In this paper, with a fully three-dimensional (3D) Monte Carlo model including the dynamic dipole potential, we present a detailed analysis of the effects of the laser-induced dipole potential on NSDI of Mg in the two-color linearly and circularly polarized fields. Our calculations cover a wide range of the relative intensities and the relative phase of the two colors, allowing for a deep understanding of the dynamic interaction between the remaining electrons’ dynamic motion and the rescattered electron. The calculations demonstrate that the dipole potential effects can be finely controlled, which is closely related to the symmetry of the combined laser fields by tracing back the electron trajectories. Additionally, we propose a method allowing us to control the rescattering process with the initial conditions of the photoelectrons unchanged.
2. Theoretical model
In this paper, NSDI occurs mainly via recollision excitation to the ionic excited state with subsequent ionization (RESI) pathway [15,16]. In the calculations a 3D Monte Carlo model is employed, which is developed on the basis of recollision impact excitation cross sections and proved to be in good agreement with experiments in many works [15–18]. Considering the diffusion of the returning electron wave packet, the recollision-induced excitation probability depends on the impact parameter $b$. By using an effective, energy-averaged cross section, the NSDI ratio can be expressed as
We here consider the following two-color electric fields in the $(x, z)$ plane
The outermost electron is assumed to be ejected into the continuum by tunneling through the suppressed atomic potential barrier along the combined-field direction with a rate given by the ADK formula [23]. The subsequent evolution of the tunneling electron is determined by Newton’s equation of motion
The returning and directly ionized electron trajectories are picked up to obtain $W_{ret}(E_{ret})$ and $W_{d}(E_{si})$, respectively.
The total potential including the ionic polarization effect has the following form
with $\alpha _{I}$ the static polarizability of the singly charged ion. The second term denotes the laser-induced dipole potential, representing the multielectron effect [12]. We set a cutoff point $r_{c}= \alpha _{I}^{1/3}$ for the range of the ionic polarization. As for $r \leq r_{c}$, the polarization effects will have no contributions because of the shielding of the ionic system [24].3. Results and discussions
In Fig. 1 we show the ratios of Mg$^{2+}$/Mg$^{+}$ as functions of $\Delta \varphi$ with and without the dipole potential included for various $I_{l}/I_{c}$ ratios (a ratio of the intensity of the fundamental beam to that of the second harmonic beam) but keep the total laser intensity fixed, i.e., $6\times 10^{13}$ W/cm$^{2}$. This intensity is selected as it corresponds to the middle of the knee region of the NSDI yield [16,17,25]. It is clear that at $I_{l}/I_{c}$= 5 [Fig. 1(a)], the dipole potential suppresses the ratios of Mg$^{2+}$/Mg$^{+}$; at $I_{l}/I_{c}$= 0.2 [Fig. 1(b)], the dipole potential shows little influence on the ratios of Mg$^{2+}$/Mg$^{+}$; at $I_{IR}/I_{R}$= 0.02 [Fig. 1(c)], the dipole potential significantly enhance the NSDI ratio, which is contrary to the case of $I_{l}/I_{c}$= 5. For the case of $I_{l}/I_{c}$=5, the difference between the calculations with and without the dipole potential included shows a strong dependence on $\Delta \varphi$, reaching a minimum near $\Delta \varphi = 0.5 \pi$. This feature becomes much less pronounced for $I_{l}/I_{c}$ = 0.2 and 0.02. Thus, our calculations demonstrate that one can tune the dipole potential effects by using the $I_{l}/I_{c}$ and $\Delta \varphi$ as control parameters in two-color linearly-circularly polarized fields.
Fig. 1. The ratio of Mg$^{2+}$/Mg$^{+}$ as a function of relative phase $\Delta \varphi$ for (a) $I_{l}/I_{c}$= 5, (b) ${I_{l}/I_{c}}$= 0.2, and (c) ${I_{l}/I_{c}}$= 0.02.
To see more details of the influence of the polarization effect, we present the simulated ratios of Mg$^{2+}$/Mg$^{+}$ from two-color fields with and without the dipole potential included for various $I_{l}/I_{c}$ ratios in Fig. 2. It is obvious that the influence of the dipole potential on NSDI ratio strongly depends on the $I_{l}/I_{c}$ ratio, which is consistent with Fig. 1. This is because that the shape of the combined fields is changed for each $I_{l}/I_{c}$ ratio, which will be discussed in the following.
Fig. 2. The calculated ratio of Mg$^{2+}$/Mg$^{+}$ versus the total intensity ($I_{l} + I_{c}$) for (a) $I_{l}/I_{c}$= 5, (b) $I_{l}/I_{c}$= 0.2, and (c) ${I_{l}/I_{c}}$= 0.02. In each calculation, $\Delta \varphi$ is randomly distributed in the interval [0, $2\pi$].
In order to explore how the dipole potential suppresses or enhances the ratios of Mg$^{2+}$/Mg$^{+}$, we first depict the evolution of characteristic electron trajectories in Fig. 3. In these typical trajectories, the polarization effect makes the electrons miss [Figs. 3(a) and (c)] or return [Figs. 3(b) and (d)] to the core, comparing to the cases without the dipole potential considered with the same initial conditions. To highlight the difference between the electron trajectories in Fig. 3, we mark the cut z = 0 (the green dashed lines in Fig. 3) to clarify the features of them for the different parameters. As shown in Figs. 3(a) and (c), the electron trajectories do not cross the cut z = 0 before returning. After tunneling, the electron is directly driven away from the core with the dipole potential considered, suppressing the recollision probability and NSDI. Since the induced dipole force decreases very rapidly with increasing $r$, we can mainly consider the contributions of the induced dipole forces around the tunneling exit, which pushes the electron away from the core. While in Figs. 3(b) and (d), the tunneled electrons will spiral around the core and go through the cut z = 0 before coming back. In this case, due to the attraction of the dynamic dipole potential, which makes a great contribution when the electron approaches the core, the electron can come back to the core more easily [15]. From above discussion, it is obvious that for different electron trajectories, the dynamic dipole potential has a focusing or defoucsing effect on the electrons. Therefore, in the following we distinguish these trajectories by determining if the electron travels though the cut z = 0.
Fig. 3. Characteristic electron trajectories with or without considering the dipole potential in the laser polarization plane. For comparison purpose, the electron trajectories with (red lines) and without (black lines) the ionic polarization effect evolve with the same initial conditions. The left column including panels (a) and (c), shows the cases of $I_{l}/I_{c}$ = 5 with $\Delta \varphi = 0.2 \pi$ (a) and $\Delta \varphi = 0.5 \pi$ (b). The right column [panels (b) and (d)] presents the results of $I_{l}/I_{c}$ = 5 with $\Delta \varphi = 0.2 \pi$ (b) and $\Delta \varphi = 0.5 \pi$ (d). The blue globe is the core and the dots indicate the instants around 0T, 0.25T, 0.5T and 0.8T after tunneling. The arrows represent the induced dipole forces, with the scale and direction showing the magnitude (relatively) and direction of the force at different times, respectively. The green dashed lines in the figures represent the cut z = 0. The purple dashed curves indicate the distance, i.e., R $=$ 5 a.u.
Note that whether the dynamic dipole force suppresses or enhances the ratio of Mg$^{2+}$/Mg$^{+}$ depends on the $I_{l}/I_{c}$ ratio sensitively, and for different trajectories it will focus or defocus the returning electrons. To reveal the relation between the ratio of $I_{l}/I_{c}$ and the electron trajectories, we calculate the times the returning electrons across the cut z = 0 for various conditions in Fig. 4. The first peak in Fig. 4 corresponds to the statistical distribution of the trajectories shown in Figs. 3(a) and (c), while the other peaks indicate the distribution of the reversed trajectories that across the cut z = 0 for several times. In Fig. 4(a) with $I_{l}/I_{c}$ = 5, the weight of the first peak is dominant, correspondingly the ratio of Mg$^{2+}$/Mg$^{+}$ with inclusion of the dipole potential will be smaller than the ones without the dipole potential considered, as shown in Fig. 1(a). This is due to the suppression of the recollision trajectories caused by the dipole force, as seen in Figs. 3(a) and (c). On the contrary, for the case of $I_{l}/I_{c}$ = 0.02 the reversed electron trajectories dominate [Fig. 4(c)]. Therefore, the dynamic dipole potential tends to enhance the recollision probability and thus NSDI yield, which is similar to the mechanism with a pure CP light [15]. In addition, as for the case of $I_{l}/I_{c}$ = 0.2 shown in Fig. 4(b), the weight of the first peak is comparable to that of the other peaks. As a result, the dipole potential shows little influence on the ratio of Mg$^{2+}$/Mg$^{+}$ [Fig. 1(b)]. When comparing the distributions of the peaks for $\Delta \varphi = 0.5 \pi$ and $0.2 \pi$ for $I_{l}/I_{c}$ = 5, the weight of the reversed trajectories of $\Delta \varphi = 0.5 \pi$ is about two times higher than that of $\varphi = 0.2 \pi$. It matches the features of Fig. 1(a) showing a strong dependence on $\Delta \varphi$, where the difference between the calculations with and without the dipole potential included reaches a minimum around $\Delta \varphi = 0.5 \pi$. While for the other $I_{l}/I_{c}$ ratios, the distributions of the peaks do not vary distinctly, consistent with the results in Figs. 1(b) and (c). Such intriguing characteristic is closely related to the symmetry of the combined fields, as we will show in the following.
Fig. 4. The distributions of the times the returning electrons cross the cut z = 0 for (a) $I_{l}/I_{c}$= 5, (b) $I_{l}/I_{c}$= 0.2, and (c) ${I_{l}/I_{c}}$= 0.02. The counts have been normalized for comparison purposes.
To understand how the two-color fields drive the electron trajectories, we display temporal evolution of the electric field amplitudes for different $I_{l}/I_{c}$ ratios and $\Delta \varphi$ in Fig. 5. When reducing the $I_{l}/I_{c}$ ratio, the shape of the combined electric field will be changed, where the amplitude of the field component along the minor axis (the $x$-axis) increases obviously [see the projection of the laser fields on the $x-z$ plane in Fig. 5]. Therefore, the important contribution to the reversed electron trajectories arises from the electric force on the electrons along the transverse dimension, i.e., the $x$-axis. For the high $I_{l}/I_{c}$ ratio, shown in Figs. 5(a) and (d), the proportion of the intensity of the linearly polarized laser is large, the symmetry of the two-color fields depends heavily on $\Delta \varphi$. As the proportion of the intensity of the circularly polarized laser increases, the influence of $\Delta \varphi$ on the symmetry becomes less, comparing Figs. 5(b) and (e) or Figs. 5(c) and (f). Thus, there are more differences between the distributions of the peaks of $\Delta \varphi = 0.2 \pi$ and $\Delta \varphi = 0.5 \pi$ in Fig. 4 (a), comparing to the cases of Figs. 4(b) and (c). The results also match the $\Delta \varphi$ dependence of the difference between the calculations with and without the dipole potential considered in Fig. 1.
Fig. 5. The time evolution of the two-color electric fields. These parameters are $I_{l}/I_{c}$ = 5 in (a) and (d), $I_{l}/I_{c}$ = 0.2 in (b) and (e), and $I_{l}/I_{c}$ = 0.02 in (c) and (f). For the upper panels and the lower panels, $\Delta \varphi$ are $0.2 \pi$ and $0.5 \pi$, respectively. The electric amplitudes have been multiplied by a factor of 1000 for a more intuitive comparison between the field components along the $x$-axis and $z$-axis.
Furthermore, we can also control the recollision trajectories while maintaining the tunneling direction via this two-color electric fields for a high $I_{l}/I_{c}$ ratio. As shown in Figs. 6(a)-(d), the recollision electron distributions in the space domain at the returning time are correspondingly changed when tuning $\Delta \varphi$, while the electrons are released mainly along the $z$-axis (the little offset along the $x$-axis comes from the influence of the transverse electric field component of CP fields) and ionized near the peak electric field of the fundamental light according to the ADK formula [23]. It is clear that most of the recollision time distribution peaks are at nT + 0.4T and nT + 0.9T for $\Delta \varphi = 0$ and $\pi$, shown in Figs. 6(e) and (f), respectively. The corresponding illustrations of the laser-induced displacement of the electron density of the ionic core at nT + 0.4T and nT + 0.9T are depicted in Figs. 6(g) and (h), respectively. Thus, as $\Delta \varphi$ changes, the returning electron will interact with the bound electrons populated at different angles around the nucleus influenced by the laser field, with the initial tunneling direction unchanged. This is particularly interesting for detecting the dynamic dipole potential of molecules. Over the past decades, a self-imaging method based on coherent electron driven by the intense laser field has emerged and developed, which is based on the strong-filed recollision scenario [4,26,27]. When considering the multielectron polarization, the theory and experiment have a better agreement for odd-even high harmonic generation in CO [28]. And it also affects the angular distributions of the total ionization yields for molecules such as CO$_{2}$ and CS$_{2}$, in particular at high intensities [29]. In order to accurately retrieve the information of the remaining electron wave packet, which plays an instrumental role in many strong-field processes, one should avoid the distortion of the initial conditions of the tunneled electron. Because the tunneling rate depends on the alignment angle and the electron orbital shape for molecules [30], it is important to fix the tunneling direction when detecting the influence of the remaining electrons on the photoelectrons in the space domain.
Fig. 6. The top row including (a) and (b) shows the position of the returning electron at the recollision moment. The second row are the corresponding distributions of the tunneling exit, respectively. The red lines represent the $z$-axis. The third row displays the normalized distributions of recollision time for (e) with the peaks at nT + 0.4T and (f) with the peaks at nT + 0.9T. The lower row consisting of (g) and (h) are the illustrations, where the blue lines are the electric field evolution, the arrows display the instantaneous directions of the fields, the red spots represent the recollision instants, and the rest are the laser-induced displacements of electron density of the ionic core when the electrons return. The $I_{l}/I_{c}$ ratio is 5 with $\Delta \varphi = 0$ and $\Delta \varphi = \pi$ in the left and right columns, respectively.
It is feasible that the laser-driven dynamic electron redistribution in the ionic core can be experimentally detected. A laser-induced inelastic diffraction scheme, based on the intense field-driven atomic NSDI process, has been proposed recently [26]. In principle this method can be used to extract the doubly differential cross sections of the target ions, e.g., Mg$^{+}$, from the two-dimensional photoelectron momentum distributions in the NSDI process. For the same $I_{l}/I_{c}$ ratio, as $\Delta \varphi$ changes, the extracted doubly differential cross sections should be different and sensitive to the laser-driven dynamics of the remaining electron. This is due to the dependence of the recollision electron distributions on $\Delta \varphi$ in both the space and time domains, as shown in Fig. 6. In this way, one can measure the laser-driven transient redistributions of the remaining electrons on the subcycle timescale.
4. Conclusion
In conclusion, we have theoretically investigated the effects of the dipole potential on the double ionization probability of Mg by using the two-color pulses. We employ a 3D Monte Carlo simulation fully considering the laser-induced dipole potential. The current studies clearly identify that the ionic polarization effects can be well controlled by using the $I_{l}/I_{c}$ and $\Delta \varphi$ as control parameters. Our analysis of the electron trajectories reveals that the influence of the dynamic dipole potential on NSDI depends on the symmetry of the two-color laser fields. In addition, this kind of precisely shaping electric fields can also be used to control the rescattering process while keeping the tunneling direction of the first electron almost unchanged.
Funding
National Natural Science Foundation of China (11974380); CAS Project for Young Scientists in Basic Research (YSBR-091).
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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