Introduction

Post-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 picture5,6, 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 resolution8,9,10,11,12,13,14,15. 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 (e1) firstly experiences a recollision with the parent univalent ion and delivers energy to the bounded electron (e2). In the below-threshold regime, the maximal kinetic energy of e1 upon recollision is smaller than the ionization potential of e2, so e2 can be only pumped to an excited state, as illustrated in Fig. 1. Then e2 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 e2 will travel directly to the detector16,17,18,19, i.e., the post-tunneling process of e2 has been largely ignored. In fact, after tunneling ionization, e2 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 expectation20,21.

Fig. 1: Sketch map to illustrate the below-threshold nonsequential double ionization process.
figure 1

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 at time t2r, or it may also be recaptured into a Rydberg state of ion.

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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-shaped22,23 and anti-correlated24 patterns of correlated electron momentum distribution (CEMD), and also the transition between them25, 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 × 1014 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 anti-correlation. This transition of CEMD from cross-shaped 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.

Fig. 2: Simulated results for below-threshold nonsequential double ionization of Ar.
figure 2

a–d Simulated correlated electron momentum distributions (CEMDs) of Ar for different laser pulse durations at the intensity of 1.25 × 1014 W cm−2. The carrier-envelope phases are averaged. Each CEMD is normalized to itself. e Simulated yield ratio Y2&4/Y1&3 for different pulse durations and different intensities. Y1&3 (Y2&4) denotes the integrated yield in the first and third (the second and fourth) quadrants in the CEMD. The numbers given in the legends denote peak laser intensities with units of 1014 W cm−2. The open circles are calculated by only considering direct trajectories of e2 (see text for details). f Measured results extracted from Ref. 25. The black short-dashed lines in e and f serve as indications of the intensity dependence.

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To quantitatively characterize the CEMD, in Fig. 2e we plot the ratio Y2&4/Y1&3 for different pulse durations and different intensities. Y1&3 (Y2&4) denotes the integrated yield in the first and third (the second and fourth) quadrants. We also present the measured results25 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 e2 is directly knocked out by e1, 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 e2 at 4 fs and 30 fs, respectively. Here, we define it as the recolliding trajectory if the minimal distance of e2 from the residual ion is less than the tunnel exit. Otherwise, it is the direct trajectory. Since momenta of direct trajectories of e2 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 e2 have dominant contributions for all pulse durations as depicted by the ratio Yres/Ydir (Yres and Ydir 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.

Fig. 3: Distributions corresponding to recolliding trajectories and direct trajectories of e2.
figure 3

a, b Correlated electron momentum distributions (CEMDs) corresponding to direct trajectories of e2. c, d CEMDs corresponding to recolliding trajectories of e2. e Pulse-duration dependence of Yrec/Ydir, the ratio between the integrated yields of recolliding and direct trajectories for e2 for all events or only double ionization (DI) events. f Pulse-duration dependence of YFDI/YDI, the ratio between the probabilities of frustrated double ionization (FDI) and DI when e2 is confined to direct trajectories. The laser intensity is 1.25 × 1014 W cm−2. Each CEMD is normalized to itself. The carrier-envelope phases are averaged.

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Intuitively, the Coulomb focusing effect imposed on e2 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 Yres/Ydir 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. In the recapture process, e2 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 out26,27,28,29. In our calculation, more than two-thirds of direct e2 are recaptured into the Rydberg states of Ar+ at 4 fs, and the probability of FDI for direct e2 decreases quickly with increasing pulse duration, as shown in Fig. 3f. Compared with recolliding trajectory of e2, direct e2 cannot move far away from Ar2+ 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 e2 being recaptured means fewer of them contribute to DI, resulting in larger relative contribution of recolliding trajectories of e2 to DI. In brief, the enhanced FDI probability significantly enlarges the relative contribution of recolliding trajectories of e2 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 e2 are considered, the calculated Y2&4/Y1&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 e1 which is determined by the microscopic dynamics of the recollision process for e1. According to our calculations, the first- and third-return recolliding trajectories of e1 are dominant for the laser parameters interested here. For other returns, either the return energy is too small to excite e2, or the collision probability is negligible due to the spreading of the wave packet30. In Fig. 4a, b, we present the CEMDs corresponding to the first- and third-return trajectories of e1, respectively, in 1.25 × 1014 W cm−2, 30 fs laser pulse. Note that all trajectories of e2 (direct and recolliding trajectories) are included. The CEMD for the first-return trajectories of e1 (Fig. 4a) shows a cross-shaped pattern, whereas that for the third-return 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 e1. 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 process31.

Fig. 4: Distributions corresponding to the first- and third-return trajectories of e1.
figure 4

a Correlated electron momentum distributions (CEMDs) corresponding to the first return. b CEMDs corresponding to the third return. c Pulse-duration dependence of Y3rd/Y1st, the ratio of the integrated yield of the third-return to that of the first-return trajectories of e1. The laser intensity is 1.25 × 1014 W cm−2. Each CEMD is normalized to itself. The carrier-envelope phases are averaged.

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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 e1 will show a band-like distribution along the p1z = 0 axis with the maxima away from the origin, i. e., vanishing momentum of e1 but much higher momentum of e2 (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 anti-correlation (Fig. 5b). These band-like distributions can be understood as follows. The final momentum of e1 is determined by the residual momentum after exciting e2 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 × 1014W cm−2), the magnitudes of them for the first-return trajectories of e1 are nearly equal, resulting in a vanishing momentum of e1. When the laser intensity increases, the band will become tilted towards the main diagonal23 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 e1. Since electrons ionized at times separated by a half optical cycle will have opposite momenta, there is one band on each side of p1z = 0 axis. Actually, there are also two bands for the first return, but they merge together.

Fig. 5: Effect of Coulomb field of Ar2+ on correlated electron momentum distributions (CEMDs).
figure 5

a, b CEMDs without performing electron indistinguishability symmetrization, and only recolliding trajectories of e2 with ionization time nearest the collision time of e1 are included. c Schematic representation of the laser electric field E(t) and the corresponding vector potential A(t) for pulse duration of 30 fs. e1 collides with the ion most probably at the crossing of E(t) at t1r or \(t_{1r}^\prime\). Upon the collision, e2 is excited, and then is ionized most probably at the peak of the laser field at t2i or \(t_{2i}^\prime\). The subsequent evolution of the canonical momentum p2z–A(t) for the recolliding trajectories of e2, denoted as I and II, are presented to illustrate the Coulomb-field effect of Ar2+. d, e CEMDs calculated by replacing the ionization amplitude \(M_{{{\tilde{{{{{\bf{P}}}}}}}_{2}}}^{(3)}\) in Eq. 1 with the standard SFA. Trajectories of e1 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.

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The anti-correlation between the two electrons for the third-return trajectories of e1 is illustrated in Fig. 5c. The recollision of e1 most probably occurs around the crossing of the electric field at t1r or \(t_{1r}^\prime\). Since the magnitude of the drift momentum after recollision, which is equal to –A(t1r) (vector potential at the recollision time), is larger than the residual momentum for the third-return recolliding trajectories of e1, its final momentum is in the direction of the drift momentum. If the recollision of e1 occurs at t1r, the final momentum of e1 will be positive, corresponding to the right band in Fig. 5b. Upon recollision, e2 is pumped to the first excited state, then it is most probably ionized at the subsequent electric field peak at t2i. If the Coulomb attraction of the ion is not considered and no recollision occurs, e2 will have vanishing final momentum. This can be seen clearly in Fig. 5d, e, in which the CEMDs are obtained by calculating \(M_{{{\tilde{{{{{\mathbf{P}}}}}}}_{2}}}^{(3)}\) in Eq. 1 (see the Methods) with the standard SFA. But if the ionic Coulomb potential is taken into account, momenta of e2 for recolliding trajectories (trajectory I) shift to the negative direction, opposite to the direction of the final momentum of e1 (see Fig. 5c). This is exactly the situation of the right-band distribution in Fig. 5b. The left band corresponds to the situation that e1 recollides with the ion at \(t_{1r}^\prime\) and e2 is ionized at \(t_{2i}^\prime\). 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 e1 occurs at t1r while e2 is ionized at \(t_{2i}^\prime\), 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 recollision probability of e2 in the presence of the Coulomb field of Ar3+ is higher than that for Ar2+ 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 × 1014 W cm−2 (Fig. 6b)—the arms get thinner and longer. When the laser intensity increases to 8 × 1014 W cm−2 which is still lower than the threshold intensity of 8.6 × 1014 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 e1 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 conditions32.

Fig. 6: Simulated results for below-threshold nonsequential double ionization of Ar+ which finally becomes Ar3+ in 400 nm laser pulse.
figure 6

a Ratio between the integrated yields of recolliding and direct trajectories for e2. b–d Normalized correlated electron momentum distributions. The carrier-envelope phases are averaged.

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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 multi-return trajectories of the first ionized electron e1, the recollision and recapture processes of the second ionized electron e2. Especially, recollision process of e2, which is enhanced relatively by the recapture process of e2, is found to play an important role in electron–electron correlation. We expect that the recollision process of e2 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 single-electron dynamics. The transition magnitude is expressed as (atomic units m = ħ = e = 1 are used)

$$M\left( {\mathop {{{{{{{{\mathbf{P}}}}}}}}}\limits^\sim _1,\mathop {{{{{{{{\mathbf{P}}}}}}}}}\limits^\sim _2} \right) = \mathop {\sum}\limits_s {M_{\mathop {{{{{{{{\mathbf{P}}}}}}}}}\limits^\sim _2}^{\left( 3 \right)}\left( {t_{2i}^s,t_{1r}^s} \right)M_{\mathop {{{{{{{{\mathbf{P}}}}}}}}}\limits^\sim _1}^{\left( 2 \right)}\left( {t_{1r}^s} \right)M_{\mathop {{{{{{{{\mathbf{P}}}}}}}}}\limits^\sim _1}^{\left( 1 \right)}\left( {t_{1r}^s,t_{1i}^s} \right)}$$
(1)

in which different trajectories labeled with s are summed coherently. \(M_{\mathop {{{{{{{{\mathbf{P}}}}}}}}}\limits^\sim _1}^{\left( 1 \right)}\left( {t_{1r}^s,t_{1i}^s} \right)\), describing the tunneling ionization of e1 at \(t_{1i}^s\) and its subsequent propagation in the laser field until colliding with the parent ion at time \(t_{1r}^s\), is calculated using the quantum-trajectory Monte Carlo (QTMC) method33,34 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, e1 will excite e2 and then move to the detector. This excitation process is described by \(M_{\mathop {{{{{{{{\mathbf{P}}}}}}}}}\limits^\sim _1}^{\left( 2 \right)}\left( {t_{1r}^s} \right)\) which is calculated with conventional S-matrix theory. Finally, e2 is ionized through tunneling at \(t_{2i}^s\) from the excited state, and then propagates in the laser field until the end of the pulse, which is described by \(M_{\mathop {{{{{{{{\mathbf{P}}}}}}}}}\limits^\sim _2}^{\left( 3 \right)}\left( {t_{2i}^s,t_{1r}^s} \right)\) calculated with the Coulomb-corrected strong field approximation (CCSFA) method35. The sin-squared pulse shape is employed in our calculation. A model potential36 is applied to mimic the Coulomb field of Ar2+ felt by e2 in its propagation. Only the first excited state 3s3p6 with zero magnetic quantum number37 is included in the present calculations. The depletion of the excited state is also taken into account in calculating \(M_{\mathop {{{{{{{{\mathbf{P}}}}}}}}}\limits^\sim _2}^{\left( 3 \right)}\left( {t_{2i}^s,t_{1r}^s} \right)\)18. In our model, the different-return trajectories of e1 can be distinguished according to the travel time tt defined as the interval between the ionization time \(t_{1i}^s\) and the recollision time \(t_{1r}^s\). For trajectories with tt 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 end of the laser pulse, say, electrons with negative energy are believed to be recaptured (for more details of the method see Supplementary Methods).