Recollision of excited electron in below-threshold nonsequential double ionization

Consensus has been reached that recollision, as the most important post-tunneling process, is responsible for nonsequential double ionization process in intense infrared laser field, however, its effect has been restricted to interaction between the first ionized electron and the residual univalent ion so far. Here we identify the key role of recollision between the second ionized electron and the divalent ion in the below-threshold nonsequential double ionization process by introducing a Coulomb-corrected quantum-trajectories method, which enables us to well reproduce the experimentally observed cross-shaped and anti-correlated patterns in correlated two-electron momentum distributions, and also the transition between these two patterns. Being significantly enhanced relatively by the recapture process, recolliding trajectories of the second electron excited by the first- or third-return recolliding trajectories of the first electron produce the cross-shaped or anti-correlated distributions, respectively. And the transition is induced by the increasing contribution of the third return with increasing pulse duration. Our work provides new insight into atomic ionization dynamics and paves the new way to imaging of ultrafast dynamics of atoms and molecules in intense laser field.

PACS numbers: 133.80.Rv, 33.80.Wz, 42.50.Hz Recollision is responsible for many intriguing strongfield phenomena, such as high-order above-threshold ionization (HATI), high harmonics generation (HHG), and nonsequential double ionization (NSDI), and also serves as the foundation of attosecond physics (see, e.g., Refs. [1][2][3][4] for reviews and references therein). In the recollision picture [5,6], an electron is liberated from the neutral atom or molecule through tunneling, then is driven back by the laser field to collide with the parent ion elastically or inelastically, or recombine with the ion, resulting in HATI, NSDI and HHG, respectively. Since the electron strongly interacts with the ion, the products upon recollision carry information of the parent ion, and can be used to probe its structure and dynamics. Based on the recollision process, different methods, such as laserinduced electron diffraction (LIED) [7] and laser-induced electron inelastic diffraction (LIID) [8], are proposed and successfully applied in imaging of atomic and molecular ultrafast dynamics and structure with unprecedented spatial-temporal resolution [8][9][10][11][12][13][14][15]. However, the recollision in the above-mentioned strong-field processes and ultrafast imaging methods is limited to interaction between the ionized electron and univalent ion.
In the NSDI process, one electron (e 1 ) firstly experi- * Electronic address: wfyang@stu.edu.cn † Electronic address: chen˙jing@iapcm.ac.cn ences a recollision with the parent univalent ion and deliver energy to the bounded electron (e 2 ). In the belowthreshold regime, the maximal kinetic energy of e 1 upon recollision is smaller than the ionization potential of e 2 , so e 2 can be only pumped to an excited state, as illustrated in Fig. 1. Then e 2 is ionized from the excited state by the laser field at a later time, dubbed as recollision excitation with subsequent ionization (RESI) process. Usually, it is believed that e 2 will travel directly to the detector [16][17][18][19]. However, after tunneling ionization, e 2 may be driven back to recollide with the divalent ion or be recaptured into a Rydberg state of ion as illustrated in Fig. 1. Due to the strong Coulomb field of the divalent ion, these post-tunneling dynamics may be prominent. It has been recently reported experimentally and theoretically that the probability of recapture in double ionization, dubbed as frustrated double ionization (FDI), is much higher than expectation [20,21].
In this work, by introducing a Coulomb-corrected quantum-trajectories (CCQT) method, we identify the key role played by the recollision between the second ionized electron and the divalent ion in the below-threshold NSDI process. We find that, only when this recollision is included, the experimentally observed cross-shaped [22,23] and anti-correlated [24] patterns of correlated electron momentum distribution (CEMD), and also the transition between them [25], can be well reproduced.
To describe the below-threshold NDSI process both coherently and quantitatively, it has to incorporate both arXiv:2106.06395v1 [physics.atom-ph] 11 Jun 2021 FIG. 1: Sketch map to illustrate the below-threshold NSDI process. At time t1i, e1 is first ionized by the laser field, then it is driven back to collide the parent univalent ion and excites e2 at time t1r. e2 is ionized from the excited state by the laser field at a later time t2i. After that, e2 may travel directly to the detector, or it may be driven back to recollide with the divalent ion similar to e1, or it may also be recaptured into a Rydberg state of ion.
the quantum effect and the Coulomb interaction between the residual ion and the ionized electrons in a uniform theory. To achieve this, we introduce a Coulombcorrected quantum-trajectories (CCQT) method by taking advantage of the well-developed Coulomb-corrected methods dealing with single-electron dynamics. The transition magnitude is expressed as (atomic units m = = e = 1 are used) in which different trajectories labelled with s are summed coherently. M p1 (t s 1r , t s 1i ), describing the tunneling ionization of e 1 at t s 1i and its subsequent propagation in the laser field until colliding with the parent ion at time t s 1r , is calculated using the quantum-trajectory Monte Carlo (QTMC) method [26,27] which is efficient to obtain large amount of hard-collision trajectories. Trajectories with minimum distance from the ion less than 1 a.u. are selected to consider the hard collision for the subsequent calculation. Upon collision, e 1 will excite e 2 and then move to the detector. This excitation process is described by M p1 (t s 1r ) which is calculated with conventional Smatrix theory. Finally, e 2 is ionized through tunneling at t s 2i from the excited state, and then propagates in the laser field until the end of the pulse, which is described by M p2 (t s 2i , t s 1r ) calculated with the Coulomb-corrected strong field approximation (CCSFA) method [28]. The sin-squared pulse shape is employed in our calculation. A model potential [29] is applied to mimic the Coulomb field of Ar 2+ felt by e 2 in its propagation. Only the first excited state 3s3p 6 with zero magnetic quantum number [30] is included in the present calculations. The depletion of the excited state is also taken into account in calcu- p2 (t s , t s ) [18] (see the method in Supplement 1). Fig. 2 displays the calculated results for Ar under different pulse durations to compare with the experimental results in Ref. [25]. Intensities higher than the measured ones by 0.25 × 10 14 W/cm 2 are used in the present calculations (see Supplement 1 for details of the fitting procedure). As shown in Fig. 2, for shorter pulse durations (4 fs and 8 fs), the distributions show a cross shape with the maxima lying at the origin. While for longer pulses (16 fs and 30 fs), the electrons are more homogeneously distributed over the four quadrants, actually, prefer the second and fourth quadrants, which indicates an anti-correlation. This transition of CEMD from crossshaped to anti-correlated patterns is in agreement with the measured results reported in Ref. [25], although there is some discrepancy in details. In the measurement, the transition occurs when pulse duration increases from 4 fs to 8 fs, whereas in Fig. 2 it occurs when pulse duration increases from 8 fs to 16 fs. This discrepancy may be due to that the pulse shape and duration employed in our calculations are not exactly the same as that in the

measurements.
To quantitatively characterize the CEMD, in Fig. 2(e) we plot the ratio Y 2&4 /Y 1&3 for different pulse durations and different intensities. Y 1&3 (Y 2&4 ) denotes the integrated yield in the first and third (the second and fourth) quadrants. We also present the measured results [25] in Fig. 2(f) for comparison. In general, the simulation reproduces most of the features in the measured results. The ratio increases with pulse duration and becomes saturated at 16 fs when the intensity is fixed, and it decreases with laser intensity both for pulse durations of 8 fs and 16 fs. However, compared with the measured results, the simulation obviously overestimates the ratio for the highest intensity. This discrepancy can be attributed to that the contribution of the process that e 2 is directly knocked out by e 1 , whose distribution mainly locates in the first and third quadrants, becomes more significant with increasing intensity, but is not included here.
In Fig. 3, we present CEMDs corresponding to recolliding trajectories and direct trajectories of e 2 at 4 fs and 30 fs, respectively. Here, we define it as the recolliding trajectory if the minimal distance of e 2 from the residual ion is less than the tunnel exit. Otherwise, it is the direct trajectory. Since momenta of direct trajectories of e 2 are much smaller than that of recolliding trajectories, CEMDs for direct trajectories are localized around the origin for both 4 fs and 30 fs pulses, as shown in Figs. 3(a) and 3(c). Whereas the CEMD for recolliding trajectories exhibits a cross structure at 4 fs [ Fig. 3(b)], and exhibits an anti-correlated pattern at 30 fs [ Fig. 3(d)]. Meanwhile, recolliding trajectories of e 2 have dominant contributions for all pulse durations as depicted by the ratio Y rec /Y dir (Y rec and Y dir denote the yields of recolliding and direct trajectories, respectively) for double ionization (DI) events in Fig. 3(e), as a consequence, the total CEMDs also shows a cross or an anti-correlated pattern at 4 fs or 30 fs, respectively.
But why the relative contribution of the recolliding trajectories of e 2 is so high? Intuitively, the Coulomb focusing effect imposed on e 2 by the divalent cation, which is much stronger than that of the univalent cation in ATI process, will effectively enhance the probability of recollision. We can indeed see this clearly from Fig. 3(e) in which the ratio Y rec /Y dir with all events included is greater than 1. But it is still much smaller than the ratio considering only DI events. This deviation is the result of the important contribution of recapture or FDI process. More than two-thirds of direct e 2 are recaptured into the Rydberg states of Ar + at 4fs, and the probability of FDI for direct e 2 decreases quickly with increasing pulse duration, as shown in Fig. 3(f). Compared with recolliding trajectory of e 2 , direct e 2 cannot move far away from Ar 2+ at the end of the pulse due to its much lower momentum, especially in shorter laser pulse, therefore is easier to be recaptured by the strong Coulomb field of the divalent ion. More direct e 2 being recaptured means fewer of them contribute to DI, resulting in larger relative contribution of recolliding trajectories of e 2 to DI. In brief, the enhanced FDI probability significantly enlarges the relative contribution of recolliding trajectories of e 2 to DI, and eventually induces the experimentally observed cross-shaped and anti-correlated patterns. In addition, this point is strongly supported by the fact that when only the direct trajectories of e 2 are considered, the calculated Y 2&4 /Y 1&3 is significantly different from the experimental result [see Fig. 2 The specific pattern of CEMD also requires the appropriate momentum of e 1 which is determined by the microscopic dynamics of the recollision process for e 1 . In the recollision process of e 1 , it may miss the parent ion at its first return but collide with the ion at the subsequent returns. In our model, the different-return trajectories of e 1 can be distinguished according to the travel time t t defined as the interval between the ionization time t 1i and the recollision time t 1r . For trajectories with t t in the interval [(n/2)T, ((n + 1)/2)T ] (T is the optical cycle), we denote them as the nth-return trajectories [31,32]. According to our calculations, the first-and third-return recolliding trajectories of e 1 are dominant for the laser parameters interested here. For other returns, either the return energy is too small to excite e 2 , or the collision probability is negligible due to the spreading of the wave packet. In Figs. 4(a) and 4(b), we present the CEMDs corresponding to the first-and third-return trajectories of e 1 , respectively, in 1.25 × 10 14 W/cm 2 , 30 fs laser pulse. Note that all trajectories of e 2 (direct and recolliding trajectories) are included. The CEMD for the first-return trajectories of e 1 [ Fig. 4(a)] shows a crossshaped pattern, whereas that for the third-return trajectories [ Fig. 4(b)] exhibits an anti-correlated pattern. As shown in Fig. 4(c), the ratio of the integrated yield of the third-return trajectories to that of the first-return increases quickly with increasing pulse duration. Correspondingly, the CEMD changes from a cross-shaped to an anti-correlated pattern. Therefore, the transition between the two patterns of CEMD with increasing pulse duration is the result of increasing contribution of the third-return trajectories of e 1 . The significant contribution of the third-return trajectories can be attributed to the Coulomb focusing effect from the univalent cation.
The similar effect has also been reported for high-order ATI process [32]. Next, we will explain how the cross-shaped and anticorrelated patterns of CEMDs are formed by the recolliding trajectories of the two electrons. Without indistinguishability symmetrization, the first-return trajectories of e 1 will show a band-like distribution along the p 1z = 0 axis with the maxima away from the origin, i. e., vanishing momentum of e 1 but much higher momentum of e 2 [ Fig. 5(b)]. Whereas the CEMD for the third-return consists of two bands and the maximum of the left (right) band lies in the up (low) part, giving rise to an anticorrelation [ Fig. 5(c)]. These band-like distributions can be understood as follows. The final momentum of   1 is determined by the residual momentum after exciting e 2 and the drift momentum it obtains from the laser field. Since forward scattering is favored in this inelastic scattering process, the residual momentum and the drift momentum are in opposite directions and will cancel with each other. At the present intensity (1.25 × 10 14 W/cm 2 ), the magnitudes of them for the first-return trajectories of e 1 are nearly equal, resulting in a vanishing momentum of e 1 . When the laser intensity increases, the band will become tilted towards the main diagonal [23] due to the faster-increasing residual momentum. For the third-return, its return energy is smaller than that of the first-return, so the residual momentum is not enough to compensate the drift momentum, resulting in a nonvanishing momentum of e 1 . Since electrons ionized at times separated by a half optical cycle will have opposite momenta, there is one band on each side of p 1z = 0 axis. Actually, there are also two bands for the first return, but they merge together.
The anti-correlation between the two electrons for the third-return trajectories of e 1 is illustrated in Fig. 5(a). The recollision of e 1 most probably occurs around the crossing of the electric field at t 1r or t 1r . Since the magnitude of the drift momentum after recollision, which is equal to −A (t r ) (vector potential at the recollision time), is larger than the residual momentum for the third-return recolliding trajectories of e 1 , its final momentum is in the direction of the drift momentum. If the recollision of e 1 occurs at t 1r , the final momentum of e 1 will be positive, corresponding to the right band in Fig. 5(c). Upon recollision, e 2 is pumped to the first excited state, then it is most probably ionized at the subsequent electric field peak at t 2i . If the Coulomb attraction of the ion is not considered and no recollision occurs, e 2 will have vanishing final momentum. This can be seen clearly in Figs . This is exactly the situation of the right-band distribution in Fig. 5(c). The left band corresponds to the situation that e 1 recollides with the ion at t 1r and e 2 is ionized at t 2i . As a consequence, the two electrons are emitted back-to back and the CEMD exhibits an anti-correlated pattern. In addition, it is also possible that the recollision of e 1 occurs at t 1r while e 2 is ionized at t 2i , which will produce a correlated CEMD. But since its contribution is smaller due to the depletion effect of the excited state, the total CEMD will still exhibit an anti-correlated pattern.
In conclusion, we propose a Coulomb-corrected quantum-trajectories (CCQT) method to describe the below-threshold NSDI process both coherently and quantitatively. It enables us to well reproduce different kinds of CEMDs observed in experiments, and uncover the rich underlying physics which is enhanced by the Coulomb field of univalent and divalent ions, including the multireturn trajectories of the first ionized electron e 1 , the recollision and recapture processes of the second ionized electron e 2 . Especially, recollision process of e 2 , which is enhanced relatively by the recapture process of e 2 , is found to play an important role in electron-electron correlation. We expect that the recollision process of e 2 can be applied to develop a new scheme to image the ultrafast evolution of the molecular structure and dynamics induced by the strong laser field.