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Abstract
Electron correlation in nonsequential double ionization (NSDI) of molecules by counter-rotating two-color circularly polarized (TCCP) fields is investigated with a three-dimensional classical ensemble model. Numerical results indicate that the two electrons from NSDI of molecules in counter-rotating TCCP fields show strong angular correlation and the angular correlation behavior sensitively depends on the internuclear distance. With the internuclear distance increasing, the dominant behavior of electron pairs evolves from correlation to anti-correlation. It leaves a clear imprint on the ion momentum distributions, which exhibit an inverted Y-shape distribution at a small internuclear distance and a triangle-shape distribution at a large internuclear distance. Back analysis indicates that the asymmetric electron energy sharing by soft recollision and longer time delay of double ionization are responsible for more anti-correlated emissions at large internuclear distances.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
When atoms and molecules are irradiated by an intense laser pulse, the potential of the parent ion is distorted and suppressed by the electric field of the laser pulse. The valence electron of atoms and molecules can tunnel through the suppressed barrier by the laser electric field into continuum state. After ionization the free electron is mainly governed by the subsequent laser electric field. The electron ionized at some particular phases of the laser electric field can return and recollide with the parent ion [1,2]. The second electron is released by recollision-induced direct ionization (RII) or recollision-induced excitation with subsequent field ionization (RESI) [3–9]. This phenomenon is called nonsequential double ionization (NSDI) [10–19]. The two electrons involved in NSDI are highly correlated because of the recollision process. In the past three decades much attention has been concentrated on electron correlation behavior and underlying dynamics process of NSDI [20–28].
After ionization the motion of the ionized electron is governed by the laser electric field. If the wave shape of the laser electric field can be flexibly tailored [29,30], one can control the motion trajectory of the ionized electron and its returning and recollision processes. In recent years counter-rotating two-color circularly polarized (TCCP) laser pulses have been proposed to steer the trajectory of the ionized electron. By changing the parameters of two laser pulses the wave shape of the combined electric field can be flexibly controlled in a two-dimensional plane [31,32]. The counter-rotating TCCP fields have driven the ionized electron to collide with the parent ion and induced high-brightness circularly polarized high-order the harmonics [33]. Several types of interesting interference fringes are found and deeply studied in two-dimensional electron momentum distribution of ATI of atoms driven by TCCP fields [34–37]. In 2016 the theoretical work by Chaloupka $et$ $al$. found the counter-rotating TCCP fields can induce occurrence of NSDI [38]. They observed the motion trajectories of the two electrons from NSDI events and identified the second electron still released by recollision in counter-rotating TCCP fields. Subsequently, the ionization enhancement from NSDI by counter-rotating TCCP fields is demonstrated in experiment [39–41]. Further studies have shown that NSDI yield, electron momentum distributions and recollision trajectories strongly depend on the field amplitude ratio and the relative phase in counter-rotating TCCP fields [42–45]. Double-recollision trajectory and its intensity dependence are demonstrated in NSDI of atoms by counter-rotating TCCP laser fields [43,44].
Compared with atoms, for molecules there are more degrees of freedom that effect the electron correlation behavior and underlying dynamics in NSDI of molecules, such as the internuclear distance. In a linear polarization field NSDI yields of diatomic molecules first increase and then decrease with the internuclear distance increasing [46,47]. At a large internuclear distance a considerable part of NSDI events occur through an internal-collision process [48]. It can be expected that in counter-rotating TCCP fields the electron correlation behavior and ultrafast dynamics in NSDI of molecules also strongly depend on the internuclear distance of molecules. But now for counter-rotating TCCP fields only NDSI of molecules with small internulcear distance are reported [49–51]. In this paper, we focus on the internulcear distance dependence of NSDI of molecules by counter-rotating TCCP fields. Numerical results show that the two electrons from NSDI of molecules in counter-rotating TCCP fields have strong angular correlation and the angular correlation behavior sensitively depends on the internuclear distance. With the internuclear distance increasing the dominant behavior of electron pairs evolves from correlation to anti-correlation. Furthermore, based on bake analysis of NSDI trajectories we study the underlying dynamics process of the internuclear distance dependence of angular correlation of electron pairs in NSDI.
2. Classical ensemble model
Due to the huge computational demand of numerically solving the three-dimensional (3D) time-dependent Schröinger equation for two-electron systems in strong laser fields, in the past decades numerous studies on strong-field double ionization have resorted to classical models [52–54] which have been widely recognized as reliable and useful approaches for interpretation and prediction of strong-field double ionization phenomena [22,23,43–45,50,55,56]. In this paper, we employ the 3D fully classical ensemble model proposed by Eberly and coworkers [53,54] to study the electron correlation behaviour and underlying dynamics process in NSDI of molecules with different internuclear distances driven by counter-rotating TCCP fields. In this model, the evolution of the two-electron system is determined by the Newton’s equations of motion (atomic units are used throughout unless stated otherwise):
In this work two circularly polarized laser pulses with the wavelengths of 1600nm and 800nm are simultaneously shined on the target. The 1600-nm field rotates clockwise and the 800-nm field rotates anticlockwise. The electric field of the laser pulse is given by
To obtain the initial conditions for Eq. (1), the ensemble is populated starting from a classically allowed position for the ground state energy of the target molecule. In our work we select H$_2$ as the target molecule with internuclear distances $R$=2 a.u., 4 a.u. and 6 a.u. to study the effect of the internuclear distance on electron correlation behavior and recollision process in NSDI of molecules. The corresponding ground state energies are -1.64 a.u., -1.26 a.u. and -1.17 a.u. The available kinetic energy is distributed between the two electrons randomly, and the directions of the momentum vectors of the two electrons are also randomly assigned. Then the two-electron system is allowed to evolve a sufficient long time (200 a.u.) in the absence of the laser field to obtain stable position and momentum distributions, which are the initial ensemble for Eq. (1). Then the laser field is turned on and all trajectories are evolved in the combined Coulomb and laser fields. The double ionization event is determined if the energies of the two electrons are both larger than zero at the end of the laser pulse.
3. Results and discussions
Figure 1(a) shows the magnitude of the combined laser electric field (dashed) and negative vector potential (solid). The electric filed consists of three segments per cycle, i.e., (k+1/6)T$<$t$<$(k+3/6)T, (k+3/6)T$<$t$<$(k+5/6)T and (k+5/6)T$<$t$<$(k+7/6)T. Each segment corresponds to a electric field maximum. The negative vector potential is minimal as the electric field achieves its maximum. The two-dimensional (2D) combined laser electric field E(t) (dashed) and the corresponding negative vector potential -A(t) (solid) are shown in Fig. 1(b). The arrows show the time evolution direction. The electric field traces out a trefoil pattern. Each lobes are plotted with different colors. A lobe of the electric field corresponds to a side of the negative vector potential. The square blocks and solid circles mark the field maxima and their negative vector potential.
Fig. 1. One cycle of the combined laser field. (a) the magnitude of the laser electric field (dashed) and negative vector potential (solid). (b) The combined laser electric field E(t) (dashed) and the corresponding negative vector potential -A(t) (solid). The arrows show the time evolution direction. The electric field traces out a trefoil pattern. A lobe of the electric field corresponds to a side of the negative vector potential. The square blocks and solid circles mark the field maxima and their negative vector potential.
Similar to the case of linear polarization, in counter-rotating TCCP fields NSDI yield of molecules first increases and then decreases with the internuclear distance and achieves to its maximum at R=4 a.u. The NSDI probabilities for R=2 a.u., 4 a.u. and 6 a.u. are 0.006%, 0.39% and 0.13% respectively. Figure 2 shows the ion momentum distributions in the polarization plane for the nuclear distances $R$=2 a.u., 4 a.u. and 6 a.u. The ion momenta are obtained by $\textbf {p}_{Ar^{2+}}=-(\textbf {p}_{e1}+\textbf {p}_{e2}$). The ion momentum distribution from $R$=2 a.u. exhibits an inverted Y shape. Most ions are far from the origin with a large momentum, which is signature of correlation emissions of the two electrons [4,5], i.e., the two electron evolved in NSDI are emitted to the same hemisphere. By contrary, the ions from $R$=6 a.u. mainly cluster around the origin. It indicates that the two electrons are more likely released to the opposite hemispheres and thus the ions have a small momentum. This implies that the two electrons mainly show anti-correlation behavior. For $R$=4 a.u., the ion momentum distribution show a maximum around the origin. But the distribution is broader than that from $R$=6 a.u. It indicates for $R$=4 a.u. there are more correlated emission events compared with the case of $R$=6 a.u.
The discussion above indicates that the two electrons are more likely emitted to the same hemisphere for the small internuclear distance and to the opposite hemispheres for the large internuclear distance. To more clearly show the correlation between the final emission directions of the two electrons, we present the distributions of the angles between the final emission directions of the two electrons for the three internuclear distances in Fig. 3. The distribution from $R$=2 a.u. show a peak around 20$^{\circ }$ with a long tail extending to 180$^{\circ }$. The two electrons are most likely emitted to the same hemisphere at an angle of 20$^{\circ }$. For $R$=4 a.u., the angular distribution shows a plateau over a wide range of 20$^{\circ }$~160$^{\circ }$, which means correlated emissions and anti-correlated emissions are comparable. Few electron pairs are released at an angle of 0$^{\circ }$ or 180$^{\circ }$. For $R$=6 a.u., the angular distribution is also broad and show a peak near 120$^{\circ }$. It indicates that anti-correlated emissions dominate in NSDI for the internuclear distance $R$=6 a.u. These results show that the angular correlation behavior of the two electrons in NSDI of molecules strongly depends on the internuclear distance. With the internuclear distance increasing the dominant behavior of electron pairs in NSDI of molecules evolves from correlation to anti-correlation.
Fig. 3. Distributions of the angles between the final emission directions of the two electrons at the end of the laser pulse for three different nuclear distances.
Here it is defined as correlated (anti-correlated) emission when the angle between the final emission directions of the two electrons is smaller (larger) than 90$^{\circ }$. Statistical results show that the ratios of the anti-correlated events in NSDI are 25% for $R$=2 a.u., 48% for $R$=4 a.u. and 56% for $R$=6 a.u. In Fig. 4 we separately present the ion momentum distributions for correlated events (left column) and anti-correlated events (right column) at R=2 a.u. (upper row), 4 a.u. (middle row) and 6 a.u. (bottom row). For R=2 a.u., the ions from correlated emissions are distributed in three islands away from the origin with large momentum. For R=4 a.u and 6 a.u., the ions from correlated emissions are distributed along three sides of a triangle. For anti-correlated emission events, the two electrons are emitted to the opposite hemispheres. For all three internuclear distances ion momentum approaches zero and clusters the origin.
Fig. 4. Ion momentum distributions for correlated events (left column) and anti-correlated events (right column) at R=2 a.u. (upper row), 4 a.u. (middle row) and 6 a.u. (bottom row).
To obtain a deep understanding for ultrafast dynamics in NSDI of molecules in counter-rotating TCCP fields and its dependence on the internuclear distance, we trace the classical NSDI trajectories and find that all NSDI trajectories only experience one recollision. Different from linear polarization fields [48], no internal collision is found for molecules with large internuclear distances in counter-rotating TCCP fields. To perform statistical analysis, we find out the single ionization time t$_{SI}$, the recollision time t$_{R}$ and the double ionization time t$_{DI}$. Here, the single ionization time is defined as the instant when one electron achieves positive energy or is outside the nuclear well. The double ionization time is defined as the instant when both electrons achieve positive energies. The recollision time is defined as the instant of the closest approach after the first departure of one electron from the parent ion. After the recollision, one electron quickly escapes from the parent ion and the other electron may quickly ionize or be delayed some time and then released by subsequent laser field. Based on the final ionization order after recollision the two electrons are defined as the first and the second electron.
Previous studies for linear polarization indicate that the correlation behavior of the electron pairs in NSDI is related to the time delay between the recollision and final double ionization [16,54]. Generally the time delay is small and the two electrons are more likely are emitted to the same hemisphere (correlated emissions). The time delay is long and more anti-correlated emissions occur. Figure 5(a) show the distributions of the time delay for three different internuclear distances. One can see that the time delay distribution exhibits a very sharp peak at 0.2T for $R$=2 a.u. It means that NSDI mainly occurs with a short time delay of 0.2T and thus most electron pairs are emitted into the same hemisphere (correlated emissions) for $R$=2 a.u. For $R$=4 a.u. and $R$=6 a.u. the time delay distributions are broader and many NSDI occur with a longer time delay. So there are more anti-correlated emissions for $R$=4 a.u. and $R$=6 a.u. than for $R$=2 a.u.
Fig. 5. Distributions of the time delay between the recollision and final double ionization for all NSDI events (a) and anti-correlated events (b) for three different nuclear distances.
To more clearly show the underlying dynamics process of the increase of the anti-correlated emissions with the internuclear distance, in Fig. 5(b) we present the time delay between the recollision and final double ionization for anti-correlated emission events for three different internuclear distances. Compared with the case of $R$=2 a.u., for $R$=4 a.u. more anti-correlated emissions originate from NSDI events with time delay about 0.1T and longer than T. We examine the traveling time of the free electron from the single ionization to recollision (t$_R$-t$_{SI}$) and the collision distance between the two electrons at recollision, as shown in Fig. 6. One can see that the travel time of the electron for $R$=2 a.u. is longer than that for $R$=4 a.u., as shown in Fig. 6(a). Thus the returning energy of the electron is larger for $R$=2 a.u. Meantime the recollision distance for $R$=2 a.u. is much smaller than that for $R$=4 a.u. Lower returning energy and the soft collision for $R$=4 a.u. result in that more bound electrons obtain a small energy transfer from the returning electron and are ionized with a time delay longer than T. Those excited electrons are emitted by subsequent electric field. Ignoring the initial momentum at ionization instant, the angle of the most probable emission directions from the adjacent field lobes is 120$^{\circ }$ [see those solid circles in Fig. 1(b)]. Correlated emissions occur when those excited electrons are emitted at the same field lobe as the recollision, and anti-correlated emissions occur when those excited electrons are emitted at the other two field lobes. Thus more anti-correlated emissions occur for those events with time delay longer than T. A such sample trajectory with the delay time of 1.28T for $R$=4 a.u. is presented in Figs. 7(a) and 7(b). The recollision occurs at 2.30T approaching a field maximum. After recollision the second electron is excited and then ionized at 3.58T. The excited electron is emitted at a different field lobe from the recollision. Finally the two electons are emitted at an angle of 146.3$^{\circ }$, which corresponds to anti-correlated emission. For those events with time delay about 0.1T, after recollision the two electrons are emitted at the same electric field lobe. They will acquire similar momenta from the electric field. But for soft recollision at $R$=4 a.u., the energy is asymmetrically shared between the two electron during recollision [7]. As shown in Fig. 7(c), after recollision the second electron has near-zero energy and the first electron escapes with a residual momentum. The initial residual momentum reflects the final emission directions of the first electrons. Finally the two electrons are emitted to the opposite hemispheres with an angle of 130.4$^{\circ }$ [see Fig. 7(d)]. Thus there are more anti-correlated emissions for $R$=4 a.u. than for $R$=2 a.u. As shown in Fig. 5(b), for $R$=6 a.u. more anti-correlated emissions originate from NSDI events with time delay about 0.1T and 0.3T. The ionization energy is smaller for $R$=6 a.u. and thus double ionization happens faster than the case of $R$=4 a.u. More double ionization events occur with time delay 0.3T, i.e., the next lobe after recollision. The first electrons and the second electrons are emitted at two adjacent electric field lobe and thus finally is anti-correlated. A such sample trajectory with a delay time of 0.29T is shown in Figs. 7(e) and 7(f). The two electrons from this trajectory are finally emitted at an angle of 120.2$^{\circ }$. Additionally the recollision distance for $R$=6 a.u. is slightly larger than that for $R$=4 a.u. [see Fig. 6(b)]. So more events with asymmetric electron energy sharing by soft recollision. This results in the increase of anti-correlated emissions for those fast NSDI events with time delay about 0.1T.
Fig. 6. Distributions of the traveling time (a) and distributions of the recollision distance for three different nuclear distances.
Fig. 7. Three sample NSDI trajectories with the delay times of 1.28T (upper row), 0.045T (middle row) and 0.29T (bottom row). The left and right columns show the time evolution of electron energies and the electrons’ path in the field plane respectively. The dark arrows indicate the recollision instant.
4. Conclusion
In conclusion, we have investigated NSDI of molecules with different internuclear distances by counter-rotating TCCP fields with a three-dimensional classical ensemble model. The ion momentum distribution shows an inverted Y-shape distribution at small internuclear distance and a triangle-shape distribution at large internuclear distance. It indicates that with the internuclear distance increasing the dominant behavior of electron pairs evolves from correlation to anti-correlation. Statistics for the angles between the final emission directions of the two electrons shows that the two electrons from NSDI of molecules in counter-rotating TCCP fields have strong angular correlation. Back analysis indicates that the asymmetric electron energy sharing by soft recollision and longer time delay of double ionization are responsible for more anti-correlated emissions at large internuclear distances.
Funding
National Natural Science Foundation of China (11504302, 12004323, 12074329, 61475127); National Training Program of Innovation and Entrepreneurship for Undergraduates (202010635066); Key Scientific Research Projects of Higher Education Department of Henan Province of China (20A140025); Nanhu Scholars Program for Young Scholars of Xinyang Normal University.
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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