Recollision of excited electron in below-threshold nonsequential double ionization

Recollision is the most important post-tunneling process in strong-field physics, but so far has been restricted to interaction between the first ionized electron and the residual ion in nonsequential double ionization. Here we identify the role of recollision of the second ionized electron in the below-threshold nonsequential double ionization process by introducing a Coulomb-corrected quantum-trajectories method. We will reproduce the experimentally observed cross-shaped and anti-correlated patterns in correlated two-electron momentum distributions, and the transition between them. Both the cross-shaped and anti-correlated patterns are attributed to recolliding trajectories of the second electron. The effect of recollision of the second electron is significantly enhanced by the stronger Coulomb potential of the higher valence residual ion, and is further strengthened by the recapture process of the second electron. Our work paves a potential way to image ultrafast dynamics of atoms and molecules in intense laser field. The recollision of photoexcited electrons with ions is a fundamental process in light-matter interaction, but so far only the rescattering of a single electron has been considered. Here by considering the recollision of a second electron theoretically, experimentally-observed momentum distributions can be understood.

P ost-ionization process has been the focus of strong-field atomic and molecular physics in the past thirty years. A semiclassical perspective, in which the recollision process plays a key role, is established with great effort. It can well explain many intriguing strong-field 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., Ref. [1][2][3][4] for review 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 laser-induced 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 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 first ionized electron and the residual ion.
In the NSDI process, one electron (e 1 ) firstly experiences a recollision with the parent univalent ion and delivers energy to the bounded electron (e 2 ). In the below-threshold 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] , i.e., the post-tunneling process of e 2 has been largely ignored. In fact, 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 quantumtrajectories (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.

Results
Comparison with experimental results. Figure 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 Supplementary Note 1 for details of the fitting procedure). As shown in Fig. 2, for shorter pulse durations (Fig. 2a,  b), the distributions show a cross shape with the maxima lying at the origin. While for longer pulses (Fig. 2c, d), the electrons are more homogeneously distributed over the four quadrants, actually, prefer the second and fourth quadrants, which indicates an anticorrelation. This transition of CEMD from cross-shaped to anticorrelated 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. 2e 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. 2f 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 located in the first and third quadrants, becomes more significant with increasing intensity, but is not included here.
Recolliding trajectories of e 2 . 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 Fig. 1 Sketch map to illustrate the below-threshold nonsequential double ionization process. At time t 1i , e 1 is first ionized by the laser field, then it is driven back to collide the parent univalent ion and excites e 2 at time t 1r . e 2 is ionized from the excited state by the laser field at a later time t 2i . After that, e 2 may travel directly to the detector, or it may be driven back to recollide with the divalent ion similar to e 1 at time t 2r , or it may also be recaptured into a Rydberg state of ion. 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 Fig. 3a, b. Whereas the CEMD for recolliding trajectories exhibits a cross structure at 4 fs ( Fig. 3c), and exhibits an anti-correlated pattern at 30 fs (Fig. 3d). Meanwhile, recolliding trajectories of e 2 have dominant contributions for all pulse durations as depicted by the ratio Y res /Y dir (Y res and Y dir denote the yields of recolliding and direct trajectories, respectively) for double ionization (DI) events in Fig. 3e, as a consequence, the total CEMDs also shows a cross or an anti-correlated pattern at 4 fs or 30 fs, respectively.
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. 3e in which the ratio Y res /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  corresponding to direct trajectories of e 2 . c, d CEMDs corresponding to recolliding trajectories of e 2 . e Pulse-duration dependence of Y rec /Y dir , the ratio between the integrated yields of recolliding and direct trajectories for e 2 for all events or only double ionization (DI) events. f Pulse-duration dependence of Y FDI /Y DI , the ratio between the probabilities of frustrated double ionization (FDI) and DI when e 2 is confined to direct trajectories. The laser intensity is 1.25 × 10 14 W cm −2 . Each CEMD is normalized to itself. The carrier-envelope phases are averaged. recapture or FDI process. In the recapture process, e 2 first goes out, then evolves in the external field, and finally is captured into Rydberg states (see the typical trajectory in Supplementary Note 2). Rydberg orbits, especially for high Rydberg states, often lie far away from the parent ion, so the tunneling electrons do not necessarily come back to the residual ion to reach the Rydberg states, and recapture actually occurs when electrons are going out [26][27][28][29] . In our calculation, more than two-thirds of direct e 2 are recaptured into the Rydberg states of Ar + at 4 fs, and the probability of FDI for direct e 2 decreases quickly with increasing pulse duration, as shown in Fig. 3f. 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. 2e).
Transition of CEMD with increasing pulse duration. 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 . According to our calculations, the firstand 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 30 . In Fig. 4a, b, we present the CEMDs corresponding to the firstand 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. 4a) shows a cross-shaped pattern, whereas that for the thirdreturn trajectories (Fig. 4b) exhibits an anti-correlated pattern. As shown in Fig. 4c, 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. A similar effect has also been reported for high-order ATI process 31 .
Cross-shaped and anti-correlated CEMDs. Next, we will explain how the cross-shaped and anti-correlated 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. 5a). 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. 5b). These band-like distributions can be understood as follows. The final momentum of e 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 for the drift momentum, resulting in a non-vanishing 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 thirdreturn trajectories of e 1 is illustrated in Fig. 5c. The recollision of e 1 most probably occurs around the crossing of the electric field at t 1r or t 0 1r . Since the magnitude of the drift momentum after recollision, which is equal to -A(t 1r ) (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. 5b. 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 Fig. 5d, e, in which the CEMDs are obtained by calculating M ð3Þ P 2 in Eq. 1 (see the Methods) with the standard SFA. But if the ionic Coulomb potential is taken into account, momenta of e 2 for recolliding trajectories (trajectory I) shift to the negative direction, opposite to the direction of the final momentum of e 1 (see Fig. 5c). This is exactly the situation of the right-band distribution in Fig. 5b. The left band corresponds to the situation that e 1 recollides with the ion at t 0 1r and e 2 is ionized at t 0 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 0 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.
Prediction for higher valence ions. It is expected that the effect of the Coulomb field for higher valence ion will be stronger. This can be demonstrated in the DI process of Ar + . As shown in Fig. 6a, the c Schematic representation of the laser electric field E(t) and the corresponding vector potential A(t) for pulse duration of 30 fs. e 1 collides with the ion most probably at the crossing of E(t) at t 1r or t 0 1r . Upon the collision, e 2 is excited, and then is ionized most probably at the peak of the laser field at t 2i or t 0 2i . The subsequent evolution of the canonical momentum p 2z -A(t) for the recolliding trajectories of e 2 , denoted as I and II, are presented to illustrate the Coulomb-field effect of Ar 2+ . d, e CEMDs calculated by replacing the ionization amplitude M ð3Þ P 2 in Eq. 1 with the standard SFA. Trajectories of e 1 are confined to the first return in a, d, and the third return in b, e. Each CEMD is normalized to itself. The carrier-envelope phases are averaged.
recollision probability of e 2 in the presence of the Coulomb field of Ar 3+ is higher than that for Ar 2+ shown in Fig. 3e. The CEMD also exhibits a strong dependence on the laser intensity and pulse duration. A similar to Fig. 2a but more obvious cross structure appears in the CEMD for laser field of 400 nm with pulse duration of 4 fs and intensity of 4 × 10 14 W cm −2 (Fig. 6b)-the arms get thinner and longer. When the laser intensity increases to 8 × 10 14 W cm −2 which is still lower than the threshold intensity of 8.6 × 10 14 W cm −2 , the CEMD transits to a correlated pattern (Fig. 6c). If increasing the pulse duration to 16 fs, the CEMD then transits back to the cross structure (Fig. 6d). It is the result of increasing contribution of the third-return trajectories of e 1 which just meets the requirements of cross structure. It is noteworthy that, for convenience of experimental observation, we employ 400 nm laser pulses in the above calculations for NSDI of Ar + which enable us to apply higher laser intensity to obtain higher ionization probability but remains in the below-threshold region. The additional complexity in experimental aspect comes from preparing Ar + instead of Ar atoms as targets, but it should not be an impossible task under current experimental conditions 32 .

Conclusions
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 induced by the Coulomb field of univalent, divalent, and higher valence 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.

Methods
To describe the below-threshold NDSI process both coherently and quantitatively, it has to incorporate both 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 Coulomb-corrected quantum-trajectories (CCQT) method by taking advantage of the well-developed Coulomb-corrected methods dealing with singleelectron dynamics. The transition magnitude is expressed as (atomic units m = ħ = e = 1 are used) in which different trajectories labeled with s are summed coherently.
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 33,34 which is efficient to obtain large number 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 2 ð Þ P 1 t s 1r À Á which is calculated with conventional S-matrix 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 3 ð Þ P 2 t s 2i ; t s 1r À Á calculated with the Coulomb-corrected strong field approximation (CCSFA) method 35 . The sin-squared pulse shape is employed in our calculation. A model potential 36 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 37 is included in the present calculations. The depletion of the excited state is also taken into account in calculating M 3 ð Þ P 2 t s 2i ; t s 1r À Á 18 . 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 s 1i and the recollision time t s 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. In addition, e2 may not be ionized eventually but be recaptured into the Rydberg state of ion by the Coulomb potential, which can be distinguished by checking its energy at the Fig. 6 Simulated results for below-threshold nonsequential double ionization of Ar + which finally becomes Ar 3+ in 400 nm laser pulse. a Ratio between the integrated yields of recolliding and direct trajectories for e 2 . b-d Normalized correlated electron momentum distributions. The carrierenvelope phases are averaged.
end of the laser pulse, say, electrons with negative energy are believed to be recaptured (for more details of the method see Supplementary Methods).

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.

Code availability
The codes associated with this manuscript are available from the corresponding author on reasonable request.