⇦Chinese Physics Letters, 2022, Vol. 39, No. 9, Article code 093201 Manipulating Nonsequential Double Ionization of Argon Atoms via Orthogonal Two-Color Field Yingbin Li (李盈傧)1†, Lingling Qin (秦玲玲)1†, Aihua Liu (刘爱华)2,7*, Ke Zhang (张可)1, Qingbin Tang (汤清彬)1, Chunyang Zhai (翟春洋)1, Jingkun Xu (许景焜)3*, Shi Chen (陈实)4, Benhai Yu (余本海)1*, and Jing Chen (陈京)5,6 Affiliations 1College of Physics and Electronic Engineering, Xinyang Normal University, Xinyang 464000, China 2Institute of Atomic and Molecular Physics, Jilin University, Changchun 130012, China 3School of Physics and Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan 430074, China 4Center for Applied Physics and Technology, HEDPS, and School of Physics, Peking University, Beijing 100871, China 5Institute of Applied Physics and Computational Mathematics, Beijing 100088, China 6Shenzhen Key Laboratory of Ultraintense Laser and Advanced Material Technology, Center for Advanced Material Diagnostic Technology, and College of Engineering Physics, Shenzhen Technology University, Shenzhen 518118, China 7State Key Laboratory of Transient Optics and Photonics, Chinese Academy of Sciences, Xi'an 710119, China Received 17 June 2022; accepted manuscript online 28 July 2022; published online 6 August 2022 †These authors contributed equally to this work.
*Corresponding authors. Email: aihualiu@jlu.edu.cn; xujingkun@hust.edu.cn; hnyubenhai@163.com Citation Text: Li Y B, Qin L L, Liu A H et al. 2022 Chin. Phys. Lett. 39 093201    Abstract Using a three-dimensional classical ensemble model, we investigate the dependence of relative frequency and relative initial phase for nonsequential double ionization (NSDI) of atoms driven by orthogonal two-color (OTC) fields. Our findings reveal that the NSDI probability is clearly dependent on the relative initial phase of OTC fields at different relative frequencies. The inversion analysis results indicate that adjusting the relative frequency of OTC fields helps control returning probability and flight time of the first electron. Furthermore, manipulating the relative frequency at the same relative initial phases can vary the revisit time of the recolliding electron, leading that the emission direction of Ar$^{2+}$ ions is explicitly dependent on the relative frequency.
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 DOI:10.1088/0256-307X/39/9/093201 © 2022 Chinese Physics Society Article Text Many fascinating nonlinear physical phenomena, such as high-order harmonic generation (HHG),[1-3] high-order above threshold ionization (HATI),[4,5] strong-field photoelectron holography,[6] and nonsequential double ionization (NSDI),[7-13] can be created by the interaction of atoms and/or molecules with intense laser pulses. NSDI is a typical multi-electron phenomenon that gives a way to study and manipulate electron correlation, and it has become one of the research hotspots in the fields of strong-field atomic and molecular photophysics during the last few decades. Previously, both theories and experiments have confirmed that NSDI can be adequately explained by the three-step re-scattering model.[14] In the three-step model, the outermost electron can be emitted via tunneling when the laser field becomes comparable to the binding Coulomb field. The emitted electron is then accelerated by the oscillating laser electric field and recollides with the parent ion core, resulting in recollision-impact-ionization (RII) or recollision-excitation with subsequent ionization (RESI).[15,16] Previous experimental and theoretical studies have revealed that laser parameters such as the wavelength and intensity of a linearly polarized laser field,[17,18] the carrier envelope phase of a few-cycle laser pulse,[19-21] have a significant influence on the electron correlation features of NSDI. A two-color field has more parameters that can be flexibly altered than linearly polarized laser fields,[22-24] such as relative polarization direction, relative initial phase, relative intensity, and relative frequency. As a result, it can be used to effectively control electron microdynamics, and it has been widely used to control a variety of strong field processes,[25-37] including HHG,[25,26] proton directional emission in molecular dissociation,[27] and strong field tunneling ionization,[28,29] and so on. More intriguingly, since the time and space of OTC pulses are mutually connected, the emission and re-collision of electron wave packets on the polarization plane can be established at the attosecond time scales.[30,31] Experimental and theoretical studies show that by regulating the relative initial phase of the OTC field, the return time of the first electron wave packet can be controlled within attosecond accuracy, allowing for effective control of the electron correlation characteristics of NSDI.[32-37] In this work, we use a three-dimensional classical ensemble model to investigate the dependence of recollision dynamics and electron correlation characteristics on relative initial phase for NSDI driven by OTC fields at the relative frequency $\gamma =2$ and $\gamma =3$. Our results show that changing the relative initial phase of the OTC field can control the NSDI probability, and the NSDI probability trend is consistent with the returning probability. By adapting the relative frequency of OTC fields, the flight time and kinetic energy of electrons before recollision can be controlled. Furthermore, the direction of emission of Ar$^{2+}$ ions is explicitly dependent on the relative frequency. For the relative frequency $\gamma = 2$, the emission momentum of Ar$^{2+}$ ions shows an obvious asymmetric distribution near the relative initial phase of $0.6\pi /1.6\pi$, while for the relative frequency $\gamma = 3$, the asymmetric distribution of Ar$^{2+}$ ion momentum disappears near the relative initial phase of $0.6\pi /1.6\pi$, and the momentum distribution of Ar$^{2+}$ ions shows weak asymmetry near the relative initial phase of $0.1\pi /1.1\pi$. Method. The NSDI is widely described using theoretical models such as quantum models for numerical solution of the time-dependent Schrodinger equation, the $S$-matrix model based on strong field approximation, and the semiclassical and classical ensemble models.[38-46] The numerical solution of the time-dependent Schrödinger equation is undoubtedly the most accurate method, but it is extremely computationally demanding, if not impossible.[47,48] The classical ensemble model has been widely used to study the NSDI by strong field. Since it can not only overcome the problem of large amounts of calculation, but also provide the entire NSDI process in a very intuitive way. It should be noted that the NSDI is a quantum process, and using a single classical trajectory to represent the ionization process is obviously unreasonable and meaningless. However, on a statistical level, the experimentally measured ionization signal is a classical quantity, which is the statistical result of the quantum system. As a result, the statistical results obtained using the classical ensemble model are reasonable and crucial for the analysis of quantum processes such as strong field NSDI. In recent years, the classical ensemble model has been instrumental in determining the electron momentum correlation of atomic and molecular strong field NSDI. The evolution of a two-electron system in the classical ensemble model is determined by Newton's equations of motion (atomic units are used unless stated otherwise): \begin{eqnarray} d^{2}{\boldsymbol r}_{i} /dt^{2}=-\nabla [V_{\rm ne} (r_{i})+V_{\rm ee} (r_{1},r_{2})]-{\boldsymbol E}(t), \tag {1} \end{eqnarray} where subscript $i= 1,2$ is the electron index, $r_{i}$ is the position of the $i$-th electron, and $E(t)$ is an orthogonally polarized two-color laser electric field. The term $V_{\rm ne} (r_{i})=-2/\sqrt {r_{i}^{2}+a^{2}}$ represents core-electron potential energy, and $V_{\rm ee} (r_{1},r_{2})=1/\sqrt {(r_{1} -r_{2})^{2}+b^{2}}$ is the electron-electron potential energy. To avoid self-ionization and unphysical numerical singularity, soft core parameters $a$ and $b$ are set to 1.5 a.u. and 0.05 a.u.,[46,49] respectively. The size of the ensemble (i.e., the number of classically modeled atoms) is 10 million for our calculations. The initial positions and momenta of the two electrons are randomly assigned, and they satisfy the energy constraint that the total energy equals the negative sum of the first two ionization potentials of the target atom: \begin{align} E_{\rm tot}=\,&\Big({\frac{p_{1}^{2} }{2}-\frac{2}{\sqrt {r_{1}^{2} +a^{2}}}}\Big)+\Big({\frac{p_{2}^{2} }{2}-\frac{2}{\sqrt {r_{2}^{2} +a^{2}}}}\Big)\notag\\ &+\frac{1}{\sqrt {r_{12}^{2} +b^{2}}}\notag\\ =\,&-(I_{p 1} +I_{p 2}), \tag {2} \end{align} where $P_{i}$ is the momentum of the $i$-th electron, $I_{p1}$ and $I_{p2}$ are the first and second ionization potentials of the target atom, respectively, $r_{12}$ is the relative position of the two electrons. In this study, we use argon as our target atom, whose initial total energy is $-1.59$ a.u. First, the whole system is allowed to evolve a long enough time (200 a.u.) before field-free propagations to obtain a stable momentum and position distribution.[32] In Fig. 1, we show the initial position and momentum distributions of the electrons distribution in phase space. Then the laser pulse is activated. We record the energy evolution of two electrons every 0.01 laser optical cycle and define the double ionization (DI) event if both electrons obtain positive energy at the end of the laser pulse.
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Fig. 1. The initial position and momentum distributions of the electrons in phase space before field-free propagations.
In our current simulations of this work, the OTC electric field is written as $\boldsymbol{E}(t)=f(t)[E_{x} (t){\hat{\boldsymbol x}}+E_{y} (t){\hat{\boldsymbol y}}]$, where $f(t)$ is a trapezoidal pulse envelope with two cycles turning on, four cycles plateau, and two cycles turning off. $E_{x} (t)=E_{x 0} \cos (\omega_{x} t)$ and $E_{y} (t)=E_{y 0} \cos (\omega_{y} t+\Delta \phi)$ are along the $x$ and $y$ axes, respectively. $E_{x 0}$ and $E_{y 0}$ are the amplitudes of polarized electric fields. The terms $\omega_{x}$ and $\omega_{y}$ are the frequencies of polarized electric fields along the $x$ and $y$ axes, respectively. The relative frequency is defined as $\gamma ={\omega_{y} } / {\omega_{x}}$, $\Delta \phi$ is the relative initial phase of the two electric fields. In this study, the intensities of electric fields are $1 \times 10^{14}$ W/cm$^{2}$ along the $x$ and $y$ axes. Numerical Results and Discussions. Figure 2 displays the probabilities of NSDI as a function of relative initial phase of the OTC field for the relative frequency $\gamma =2$ and $\gamma =3$. The wavelengths are 400 nm + 800 nm (blue dots), 400 nm + 1200 nm (red boxes), 800 nm + 1600 nm (blue diamonds) and 800 nm + 2400 nm (red triangles), respectively. For $\gamma = 2$, as shown in the blue curves with 400 nm + 800 nm and 800 nm + 1600 nm, the NSDI probability curves show a pronounced four-peak structure, and the peaks are located around $0.1\pi /1.1\pi$ and $0.6\pi /1.6\pi$, respectively. The peaks near $0.6\pi /1.6\pi$ are significantly higher than those of $0.1\pi /1.1\pi$. Our simulation results are in good agreement with the experimental results of neon.[35] For $\gamma = 3$ with 400 nm + 1200 nm and 800 nm + 2400 nm, the NSDI probability curves show an obvious two-peak structure. The peaks are located near $0.1\pi /1.1\pi$. Interestingly, compared with the $\gamma =2$ case, its NSDI probabilities near $0.6\pi /1.6\pi$ are strongly suppressed, and close to the minima of each curves. Therefore, under different relative frequencies, the dependence of NSDI probability on the relative initial phase of OTC field is significantly different.
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Fig. 2. Variation of NSDI yield with relative initial phase of OTC field at two relative frequencies: 400 nm + 800 nm (blue dots), 400 nm + 1200 nm (red boxes), 800 nm + 1600 nm (blue diamonds) and 800 nm + 2400 nm (red triangles).
According to the three-step recollision model, the returning probability of the first electron is a key factor affecting the NSDI probability. Taking 400 nm + 800 nm and 400 nm + 1200 nm as examples, we calculated the probability of the first electron returning to the parent ion after the single ionization occurs, as shown in Figs. 3(a) and 3(b). Here, the single ionization time is defined when the energy of the electron is greater than zero or the distance between the electron and the parent nuclear ion is larger than 6 a.u. One issue should be noted, the energy of electron does not exclude the electron-correlation energy. We compute the energy of each electron by including the kinetic energy, ion core-electron potential energy, and half of the electron-electron repulsion energy (that is, electron-correlation energy). When the first electron returns to the parent nuclear ion after a single ionization, if the distance between it and the second electron is less than 3 a.u., the return event is defined. Comparing the curves in Figs. 2, 3(a) and 3(b), the dependence of the return probability on the relative initial phase is basically consistent with the dependence of the corresponding NSDI probability on the relative initial phase under the two relative frequencies. This demonstrates that if the relative frequency is fixed, adjusting the relative initial phase of the OTC field can effectively control the first electron's return probability and thus regulate the NSDI probability. To comprehend the changes in the electron's returning probability with relative initial phase in the OTC field at both 400 nm + 800 nm and 400 nm + 1200 nm, we analyze the flight time of the first electron under various relative initial phases, as shown in Figs. 3(c) and 3(d). Flight time is defined as the time elapsed between single ionization and its first recollision. Figure 3(c) shows that for 400 nm + 800 nm, the smaller part of the flight time is mainly concentrated in the vicinity of the relative initial phases $0.6\pi /1.6\pi$. The results of 400 nm + 1200 nm shown in Fig. 3(d) are different, and the smaller part of the flight time is mainly concentrated around the relative initial phase $0.1\pi /1.1\pi$. In the process of recollision, the shorter the flight time is, the weaker the diffusion of the first electron wave packet is, and the greater the probability of electrons returning to the collision nucleus ion is.[35] Therefore, at 400 nm + 800 nm, the probability curve of NSDI shows an indigenous peak at the relative initial phase $0.6\pi /1.6\pi$ and nearby, while at 400 nm + 1200 nm, the probability curve of NSDI shows an indigenous peak at the relative initial phases $0.1\pi /1.1\pi$.
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Fig. 3. [(a), (b)] The recollision probability with the relative initial phase of the OTC field at two relative frequencies, i.e., at the wavelengths 400 nm + 800 nm and 400 nm + 1200 nm, respectively. [(c), (d)] The flight time with the relative initial phase of the orthogonal two-color field at the laser wavelengths 400 nm + 800 nm and 400 nm + 1200 nm, respectively.
To further understand why the peaks of NSDI probability appear at $0.6\pi /1.6\pi$ for 400 nm + 800 nm and at $0.1\pi /1.1\pi$ for 400 nm + 1200 nm. In Fig. 4, we plot amplitudes [(a), (b)] and Lissajous [(c), (j)] curves of the combined fields for 400 nm + 800 nm and 400 nm + 1200 nm laser pulse. In our calculation, we set the phase of 800/1200 nm pulse as 0, and vary the phase of the 400 nm laser pulse. For the two cases, one can see that the 0 and $\pi$ phase differences give maximum electric field [see Figs. 4(a) and 4(b)]. From the ADK theory, the stronger electric field can give birth more tunneling electrons. Thus we may expect peaks at 0 and $\pi$ for both the cases. For the NSDI, the recollision is very important. If the laser fields in both directions are synchronous, the returning electron can impact on the parent ion core directly, and thus enhance the probability of NSDI. We can see from the Lissajous curves [see Figs. 4(c)–4(j)], for the 400 nm + 800 nm case, when phase difference are 0.5$\pi$ and 1.5$\pi$, the laser field in both directions are synchronous. For the 400 nm + 1200 nm case, they are 0 and $\pi$. Therefore, the NSDI peaks are expected for the 400 nm $+$800 nm case at 0.5$\pi$ and 1.5$\pi$, and for the 400 nm $+$1200 nm case at 0 and $\pi$. The above analysis does not consider the effects of envelop of laser pulse and Coulomb potential of atoms. When they are considered, the peaks of NSDI probability have been shifted by 0.1$\pi$.
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Fig. 4. The amplitudes [(a), (b)] and Lissajous curves [(c)–(i)] of the combined fields. The wavelengths are 400 nm + 800 nm [(a), (c)–(f)] and 400 nm + 1200 nm [(b), (g)–(j)], respectively.
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Fig. 5. Momentum distribution of Ar$^{2+}$ of NSDI along the polarization direction of the 400 nm and the relative initial phase of the OTC pulses: (a) 400 nm + 800 nm, (b) 400 nm + 1200 nm. The data is normalized to the maxima of each graph.
Figure 5 shows the momentum distribution of Ar$^{2+}$ along the polarization of 400 nm pulse, as well as its dependence on relative initial phase of OTC pulses. For the 400 nm + 800 nm case, as shown in Fig. 5(a), we observe that argon ions have a very significant asymmetric distribution near the relative initial phase of $0.6\pi /1.6\pi$, which is consistent with the experimental results.[35] Figure 5(b) shows the same results in the OTC field with 400 nm + 1200 nm. Obviously, for the 400 nm + 1200 nm case, as shown in Fig. 5(b), the asymmetric distribution of Ar$^{2+}$ ion momentum near the relative initial phases of $0.6\pi /1.6\pi$ disappears, while the distribution of Ar$^{2+}$ ion momentum near the relative initial phases of $0.1\pi /1.1\pi$ shows less asymmetry. In Fig. 6 we present the joint momentum distribution [(a)–(d)] and recollision time distribution [(e)–(h)] of NSDI with different relative frequencies ($\gamma =2$, 3) and phases ($0.1\pi /0.6\pi$), respectively. The pink dotted lines represent the 400-nm electric field, and the red dotted lines represent the 800-nm electric field [(e), (f)] or 1200-nm electric field [(g), (h)]. Figures 6(a)–6(d) show the joint electron momentum distributions along the $y$ axis (the polarization direction of the 400 nm laser field) for the 400 nm + 800 nm and 400 nm + 1200 nm laser fields, respectively, and the corresponding relative initial phases are $0.1\pi /0.6\pi$. Figures 6(e)–6(h) show the probability distributions of NSDI events with respect to the collision time under various laser parameters in Figs. 6(a)–6(d). We note that the correlated electron momentum distribution in Fig. 6 is consistent with the recoil momentum distribution of divalent ions in Fig. 5. For 400 nm + 800 nm, when the relative initial phase is $0.1\pi$, as shown in Fig. 6(a), the correlated electron momentum spectrum is mainly distributed in the second and fourth quadrants, indicating that the two electrons are mainly emitted in the opposite direction, and the asymmetry in Fig. 5(a) is weak. When the relative initial phase is equal to $0.6\pi$, as displayed in Fig. 6(b), the momentum spectrum of the correlated electron is mainly distributed in the first quadrant, indicating that the two electrons are mainly emitted in the positive direction of the $y$ axis, and correspondingly Fig. 5(a) has a strong asymmetry. To understand the dynamics of electron emission, we further analyze the recollision time where the recollision time is defined as the nearest moment between the two electrons in the process of the first electron returning to the parent ion core after single ionization. As shown in Figs. 6(e) and 6(f), when the relative initial phase equals $0.1\pi$, the recollision time is mainly concentrated at the peak of the 400 nm laser field, and the corresponding vector potential is small. The final state velocity of the electron is mainly determined by the post-collision velocity, so the correlated electron is mainly emitted in the opposite direction. When the relative initial phase is $0.6\pi$, the recollision time is mainly concentrated at the zero value of the descending edge of the 400 nm laser field, and the corresponding vector potential is large. The final state velocity of the electron is mainly determined by the corresponding vector potential, so the associated electron is mainly emitted in the positive direction of the $y$ axis. For 400 nm + 1200 nm, as shown in Figs. 6(g) and 6(h), when the relative initial phase is $0.1\pi$, the recollision time is mainly concentrated at the zero value of the ascending edge of the 400 nm laser field, which leads to the emission of the associated electron in the negative direction of the $y$ axis. Therefore, the momentum spectrum of the associated electron is mainly concentrated in the third quadrant. The recoil ion has a positive momentum as shown in Figs. 6(c) and 5(b). When the relative initial phase is $0.6\pi$, the scattering time is mainly concentrated at the peak of the 400 nm laser pulse, resulting in the emission of the associated electron mainly along the opposite direction. Therefore, the momentum spectrum of the associated electron is mainly distributed in the second and fourth quadrants, and the ion recoil momentum is close to zero as shown in Figs. 6(d) and 5(b).
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Fig. 6. Joint momentum distribution [(a)–(d)] and collision time distribution [(e)–(h)] of NSDI with different relative frequencies and phases. The pink dotted lines represent the 400-nm electric field. The red dotted lines represent the 800-nm electric field in [(e), (f)] and 1200-nm electric field in [(g), (h)].
Furthermore, we present the relationship between the relative initial phase of OTC in NSDI and the probability of RESI at 400 nm + 800 nm and 400 nm + 1200 nm, as shown in Fig. 7. The RESI channel is defined as a difference of more than a quarter optical cycles between the double ionization time and the collision time. The results reveal that the RESI channel in NSDI has a considerable dependence on the relative initial phase at these two relative frequencies. By careful comparison, one can see that for 400 nm + 1200 nm the RESI channel is more sensitive to the relative initial phase than that of 400 nm + 800 nm. More than that for 400 nm + 800 nm, the probability curve displays a bimodal form and the peaks are located around $0.7\pi$ and $1.7\pi$, respectively. The difference is that for 400 nm + 1200 nm the probability curve shows a valley bottom located at $0.6\pi$ and $1.6\pi$, respectively. The dependence of this RESI channel probability curve on OTC relative frequency can be understood as follows. In Fig. 8, we show the NSDI probability with respected to the returning kinetic energy and relative initial phase. One sees that for 400 nm + 800 nm the returning energy of the first electron is small around $0.6\pi /1.6\pi$ [see Fig. 8(a)], so the energy transferred to the second electron during the recollision is not sufficient to ionize it immediately, leading that the second electron enters to the excited state firstly and then is ionized by the latter laser pulses, where the RESI channel dominates; whereas for 400 nm + 1200 nm, the returning energy of the first electron is relatively larger near $0.6\pi /1.6\pi$ [see Fig. 8(b)], so the second electron is easier to be ionized directly during the recollision, resulting in the suppression of RESI channel.
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Fig. 7. Probability of RESI in NSDI as a function of OTC relative initial phase. The laser wavelengths are 400 nm + 800 nm, 400 nm + 1200 nm, respectively.
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Fig. 8. NSDI probability with respected to the returning kinetic energy and relative initial phase. The wavelengths of laser field are (a) 400 nm + 800 nm and (b) 400 nm + 1200 nm.
In summary, we have used the classical ensemble model to study the dependence of the NSDI of argon atoms driven by the OTC field on the relative initial phase under different relative frequencies. The results show that the NSDI probability can be controlled by changing the relative initial phase of the OTC field. The analysis shows that adjusting the relative frequency of the laser field can control the returning probability as well as the flight time of the first electrons. In addition, the momentum direction of Ar$^{2+}$ ions has a strong dependence on the relative frequency of the OTC field. The dependence is related to the relative initial phase of the OTC pulses. The inversion analysis shows that the adjustment of the relative frequency will affect the recollision time distribution of the first electron and therefore to manipulate and control the dual-electron emission dynamics. Finally, the dependence of the probability of the RESI channel in NSDI on the relative frequency of the laser is discussed, and we find that the RESI channel is more sensitive to the relative initial phase for 400 nm + 1200 nm than that of 400 nm + 800 nm. Acknowledgements. This work was supported by the National Natural Science Foundation of China (Grant Nos. 12074329, 12004323, 12104389, 1174131, and 91850114), the Nanhu Scholars Program for Young Scholars of Xinyang Normal University, and the Open Research Fund of State Key Laboratory of Transient Optics and Photonics. 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