Abstract
Recollision is the most important posttunneling process in strongfield 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 belowthreshold nonsequential double ionization process by introducing a Coulombcorrected quantumtrajectories method. We will reproduce the experimentally observed crossshaped and anticorrelated patterns in correlated twoelectron momentum distributions, and the transition between them. Both the crossshaped and anticorrelated 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.
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Introduction
Postionization process has been the focus of strongfield 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 strongfield phenomena, such as highorder abovethreshold 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 laserinduced electron diffraction (LIED)^{7} and laserinduced electron inelastic diffraction (LIID)^{8}, are proposed and successfully applied in imaging of atomic and molecular ultrafast dynamics and structure with spatialtemporal resolution^{8,9,10,11,12,13,14,15}. However, the recollision in the abovementioned strongfield 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 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}, i.e., the posttunneling 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 posttunneling 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 Coulombcorrected quantumtrajectories (CCQT) method, we identify the key role played by the recollision between the second ionized electron and the divalent ion in the belowthreshold NSDI process. We find that, only when this recollision is included, the experimentally observed crossshaped^{22,23} and anticorrelated^{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 crossshaped 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 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 anticorrelated 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 anticorrelated 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 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 twothirds 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 crossshaped and anticorrelated 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 first and thirdreturn 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 first and thirdreturn 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 firstreturn trajectories of e_{1} (Fig. 4a) shows a crossshaped pattern, whereas that for the thirdreturn trajectories (Fig. 4b) exhibits an anticorrelated pattern. As shown in Fig. 4c, the ratio of the integrated yield of the thirdreturn trajectories to that of the firstreturn increases quickly with increasing pulse duration. Correspondingly, the CEMD changes from a crossshaped to an anticorrelated pattern. Therefore, the transition between the two patterns of CEMD with increasing pulse duration is the result of increasing contribution of the thirdreturn trajectories of e_{1}. The significant contribution of the thirdreturn trajectories can be attributed to the Coulomb focusing effect from the univalent cation. A similar effect has also been reported for highorder ATI process^{31}.
Crossshaped and anticorrelated CEMDs
Next, we will explain how the crossshaped and anticorrelated patterns of CEMDs are formed by the recolliding trajectories of the two electrons. Without indistinguishability symmetrization, the firstreturn trajectories of e_{1} will show a bandlike 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 thirdreturn 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 bandlike 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 firstreturn 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 fasterincreasing residual momentum. For the thirdreturn, its return energy is smaller than that of the firstreturn, so the residual momentum is not enough to compensate for 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 anticorrelation 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 \(t_{1r}^\prime\). 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 thirdreturn 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 \(M_{{{\tilde{{{{{\mathbf{P}}}}}}}_{2}}}^{(3)}\) in Eq. 1 (see the Methods) with the standard SFA. But if the ionic Coulomb potential is taken into account, momenta of 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 rightband distribution in Fig. 5b. The left band corresponds to the situation that e_{1} recollides with the ion at \(t_{1r}^\prime\) and e_{2} is ionized at \(t_{2i}^\prime\). As a consequence, the two electrons are emitted backto back and the CEMD exhibits an anticorrelated pattern. In addition, it is also possible that the recollision of e_{1} occurs at t_{1r} while e_{2} is ionized at \(t_{2i}^\prime\), 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 anticorrelated 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 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 thirdreturn 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 belowthreshold 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 Coulombcorrected quantumtrajectories (CCQT) method to describe the belowthreshold 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 belowthreshold 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 Coulombcorrected quantumtrajectories (CCQT) method by taking advantage of the welldeveloped Coulombcorrected 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. \(M_{\mathop {{{{{{{{\mathbf{P}}}}}}}}}\limits^\sim _1}^{\left( 1 \right)}\left( {t_{1r}^s,t_{1i}^s} \right)\), describing the tunneling ionization of e_{1} 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 quantumtrajectory Monte Carlo (QTMC) method^{33,34} which is efficient to obtain large number of hardcollision 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 \(M_{\mathop {{{{{{{{\mathbf{P}}}}}}}}}\limits^\sim _1}^{\left( 2 \right)}\left( {t_{1r}^s} \right)\) which is calculated with conventional Smatrix theory. Finally, e_{2} 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 Coulombcorrected strong field approximation (CCSFA) method^{35}. The sinsquared 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 \(M_{\mathop {{{{{{{{\mathbf{P}}}}}}}}}\limits^\sim _2}^{\left( 3 \right)}\left( {t_{2i}^s,t_{1r}^s} \right)\)^{18}. In our model, the differentreturn trajectories of e_{1} can be distinguished according to the travel time t_{t} defined as the interval between the ionization time \(t_{1i}^s\) and the recollision time \(t_{1r}^s\). 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 nthreturn 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).
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.
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Acknowledgements
The calculations were performed on High Performance Computing System of Shanxi University. This work was partially supported by the National Key Program for S&T Research and Development (No. 2019YFA0307700), the National Natural Science Foundation of China (Grants Nos. 11874246, 91950101, 11774215, 11874247, 12147215).
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X.H., W.Y., and J.C. designed the research; X.H. and Y.B. performed all the simulations; X.H., Y.B., C.L., M.L., and J.Z. analyzed data; X.H., W.L., W.Y., and J.C. discussed the results; X.H., W.L., W.Y., and J.C. wrote the paper.
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Hao, X., Bai, Y., Li, C. et al. Recollision of excited electron in belowthreshold nonsequential double ionization. Commun Phys 5, 31 (2022). https://doi.org/10.1038/s42005022008092
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DOI: https://doi.org/10.1038/s42005022008092
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