Using the full three-dimensional classical ensemble model, we have investigated nonsequential triple ionization (NSTI) of Ne by intense linearly polarized laser fields systematically. Trajectory back analysis enables us to identify the various NSTI channels at different intensities in an intuitive way. The momentum distributions of the triply ionized ions calculated by this model agree well with the experimental results over a wide range of laser intensities [J. Phys. B 41, 081006 (2008)]. With this classical model we achieve insight into the complex sub-laser-cycle dynamics of the correlated three electrons in NSTI.
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Nonsequential double or multiple ionization (NSDI or NSMI) is a fundamental process in laser-atom interaction. Due to the strongly correlated behavior of the participating electrons, NSDI and NSMI have attracted increasing attention since the observation of the knee structure in the ion yield versus laser intensity curve . Experimental and theoretical studies [2, 3, 4, 5, 6, 7, 8, 9] have provided convincing evidences that the recollision model is responsible for NSDI and NSMI caused by strong electron correlations . According to this model, an electron is first ionized through tunneling, and then is driven back to the parent ion by the oscillating field and shares energy with other electrons, leading to the release of one or more further electrons. The analyses of the doubly charged ion momentum distributions and the coincident photoelectron energy spectra, especially the correlated electron momenta have provided more details of the recollision dynamics in NSDI. For example, the finger-like structure in the correlated electron momentum spectrum from NSDI of helium implies the backscattering at the nucleus upon recollision .
For nonsequential triple ionization (NSTI), the recollision dynamics is more complex because three correlated electrons are involved. Depending on the laser intensity, different ionization channels contribute to strong-field NSTI . For instance, the purely nonsequential (NS) ionization channel (0-3), the combined sequential and nonsequential (S/NS) ionization channel where two electron tunnel sequentially and the third electron is release through NS process (0-1-3). Though the intensity-dependent ion yields [13, 14] as well as the ion momentum spectra [15, 16, 17, 18] have been reported for NSTI, the understanding of NSTI is still fragmentary. A recent experiment  has measured the momentum distributions of Ne 3+ and Ne 4+ by the 800 nm pulses at the intensity range from 1015 to 2 × 1016 W/cm 2. By tracing the intensity-dependent evolution of the recoil-ion momentum distributions, the authors have succeed in identifying the various triple ionization (TI) channels. However, the details of recollision which is the heart of attosecond science , remain obscure.
Theoretically, an accurate description of NSDI or NSMI needs full quantum theory. However, because of the enormous computational demand, full-dimensional solution of the timedependent schrödinger equation has only been performed on NSDI of helium by 400 nm laser pulses , and it seems impossible to extend this quantum-mechanical calculation to NSTI that involves three correlated electrons in the foreseeable future. Fortunately, it has been demonstrated that the classical treatment is a valid approximation in describing strong-field NSDI and it has been successful in unveiling the attosecond correlated electron dynamics [8, 22, 23, 24]. About ten yeas ago, the ion momentum distribution from NSTI has been studied with a classical model starting from a highly excited compound state . However, recollision, the most important step of NS ionization, is not investigated with this model. In another ‘thermalization’ model , it is assumed that the two bound electrons are ionized simultaneously after an ‘attosecond thermalization’, and the combined S/NS (0-1-3) channel is not taken into account. In contrast to these approaches, the classical ensemble model  proposed by Ho et al is able to describes the entire process of NSTI and provides a clear interpretation of NSTI. In this paper, with this full three-dimensional (3D) classical ensemble model [8, 27], we have described the entire process of TI, from the beginning to the end of the laser pulse, and both the purely NS (0-3) and combined S/NS (0-1-3) channels are taken into account in the model. We obtained the characteristic knee structure in the intensity-dependent ion yield curve and reproduced the intensity-dependent ion momentum distributions which have been observed in the recent experiment . Through back analysis of the triply ionized trajectories, the ionization channels at different laser intensities are clearly identified and the sub-laser-cycle correlated electron dynamics underlying NSTI is demonstrated.
2. The classical ensemble model
The classical ensemble model has been widely used in understanding NSDI before [8, 22, 23, 28, 29] and it is extended to studying NSTI in this paper . The evolution of the three-electron system is determined by the Newton’s motion equations (atomic units are used throughout this paper unless stated otherwise):
where the subscript i is the electron label which runs from 1 to 3. and are the ion-electron interaction and electron-electron interaction, respectively. E(t) =ẑf(t)E 0 sin(ωt) is the electric field, where ẑ is the laser polarization direction and f(t) is the pulse shape, which has two-cycle turn on, six cycles at full strength and two-cycle turn off. In our calculation, the wavelength is 800 nm. Similar to , the screening parameter a is set to be 1.0 to avoid autoionization, and b is set to be 0.1. The initial energy of the system is set to be −4.63 a.u., i.e., the sum of the first, the second and the third ionization potentials of Ne. To obtain the initial state, the ensemble is populated starting from a classical allowed region for the energy of -4.63 a.u. Then the available kinetic energy is distributed randomly between the three electrons in the momentum space. Before the laser field is turned on, each atom is allowed to evolve a sufficient long time to obtain stable position and momentum distributions [22, 23].
3. Results and discussions
Figure 1 shows the double and triple ionization (DI and TI) yields versus laser intensity. In consistent with the experimental results  and previous two-dimensional (2D) calculation , the knee structures are clearly seen in both curves, which imply NS ionization channel dominating DI and TI. There is no essential difference between our 3D calculation and previous 2D calculation , meaning the out-of-plane effects are negligible in the strong-field NSDI and NSTI. After the saturation intensity, the DI yield decreases with the increase of the intensity. This is because the sequential TI begins to deplete the DI yield. In this paper, we concentrate on the intensity range where NS dominates TI.
Figure 2 displays the momentum distributions of the triply ionized ions as a function of the laser intensity. The ion momenta are obtained by the sum of the three emitted electrons momentum vectors because the momentum of the absorbed photons is negligibly small. Note that the distributions in fig. 2 are plotted in units of in order to account for the intensity dependence of the drift momentum received by the recoil ion , where Up is the ponderomotive energy. Figs. 2(a), 2(c) and 2(e) illustrate the two-dimensional ion momentum distributions for I = 1.0 × 1015 W/cm 2, 2.0 × 1015 W/cm 2 and 4.0 × 1015 W/cm 2, respectively. The horizontal axis denotes the longitudinal (in the direction parallel to the laser polarization, i.e., z axis) momentum and the vertical axis corresponds to the perpendicular (along y axis) momentum. Clearly, for all intensities, the perpendicular momenta are concentrated around zero while the longitudinal momenta have wide distributions, which indicates NSTI channels dominate TI at these intensities . In figs. 2(b), 2(d) and 2(f), we show the longitudinal momentum distributions of the ions from figs. 2(a), 2(c) and 2(e), respectively. For I = 1.0 × 1015 W/cm 2, the spectrum exhibits a double-hump structure, with two well-defined maxima at . When the intensity increases to 2.0 × 1015 W/cm 2, the two maxima move to , and when the intensity further increases, the double-hump structure evolves into a wide single peak. The shapes of the spectra and the intensity-dependent evolution agree well with the experimental study , where the authors demonstrated that the purely NS (0-3) process is responsible for the double-hump spectrum with maxima at and the combined S/NS (0-1-3) process is responsible for the spectrum with maxima at .
The classical trajectory method allows us to find out the TI trajectories and back analyze their time evolutions, and thus provides an intuitive way to identify the different ionization channels. The typical trajectories are plotted in fig. 3, presented in longitudinal coordinate z versus time (figs. 3(a), 3(c), 3(e) and 3(g)) and energy versus time (figs. 3(b), 3(d), 3(f) and 3(h)). At I = 1.0 × 1015 W/cm 2, as shown in figs. 3(a) and 3(b), one electron gets ionized firstly, and then is driven back by the electric field to recollide with the bound electrons, leading to the three electrons ionized immediately. It is a typical (0-3) process and we find that it is the dominating channel for TI at this intensity. For this example, the recolliding electron misses the core on its first three returns, and the energy transfer occurs on its fourth return. The miss of the first three returns is due to the transverse displacement, which has been detailedly discussed in ref..
For I = 4.0 × 1015 W/cm 2, as shown in figs. 3(c) and 3(d), the first and the second electrons are released sequentially and the third electron is ionized through a recollision with the second electron, which is driven back by the oscillating field. This trajectory represents the (0-1-3) process, which dominates the TI at this intensity. For the modest intensity, both (0-3) and (0-1-3) processes contribute to NSTI, as displayed in figs. 3(e)–3(h). Our statistic reveals that about 65% of the TI occur through the (0-3) channel and about 35% through the (0-1-3) channel at 2.0 × 1015 W/cm 2.
Back analysis of the TI trajectories also allows us easily to determine the recollision and ionization times, which can provide insight into the sub-laser-cycle dynamics of NSTI. We define the recollision time tr to be the instant when one electron comes within the region r = 3 a.u. after its departure from the core, and the TI time tTI to be the instant when all of the three electrons achieve positive energies, where the energy contains the kinetic energy, the ion-electron interaction and half electron-electron interaction. Note that in the (0-3) process, three electrons are involved in the recollision encounter, while only two electrons are involved for the (0-1-3) process.
Figure 4. (a) shows the laser phase tTI (in cycles) at the time of TI versus laser phase tr at recollision for the intensity 1.0 × 1015 W/cm 2 . It is clearly seen that most of recollisions occur in the time range 0.4T–0.5T (or 0.9T–1.0T, where T is the laser cycle), just before the zero crossing of the field. It is consistent with the prediction of the simple-man model  and previous classical studies on NSDI [8, 23]. The absence of the population along the diagonal tTI = tr in fig. 4(a) indicates that there are not impact TI. Recently, a statistic thermalization model has been proposed for NSMI of Ne . Via comparing the experimental distributions to those predicated by the classical model, the authors demonstrated that the thermalization of the recolliding electron with the bound electrons takes several hundreds attoseconds. Obviously, our results is consistent with this thermalization picture: as shown in fig. 4(a), most of the TIs occur slightly before the field maximum (0.25T and 0.75T), which means a subfemtosecond time delay.
For I = 2.0 × 1015 W/cm 2, we have divided the TI trajectories into two parts based on whether the electrons ionize through the (0-3) or (0-1-3) processes. Figure 4(c) shows the phase tTI at time of TI versus the phase tr at recollision for the (0-3) trajectories. For this intensity, except for the largest cluster ionized around the field maxima, there is considerable population along the diagonal tTI = ti, which is a reminder of impact TI. Figs. 4(b) and 4(d) display the TI phase tTI versus recollision phase tr of the (0-1-3) trajectories at the intensities 4.0 × 1015 W/cm 2 and 2.0 × 1015 W/cm 2, respectively. For these trajectories, the population along the diagonal tTI = ti implies that TIs occur immediately after the two-electron recollision at both intensities.
In a previous paper, based on the classical rescattering model, it was predicted that at certain intensities where both (0-3) and (0-1-3) processes contribute, the momentum distribution of the triply ionized ions can exhibit up to four maxima . However, the four-maximum shape is observed neither in the experimental study  nor our calculation. Even at I = 2.0 × 1015 W/cm 2, where both (0-3) and (0-1-3) processes have considerable contribution to TI, the four-maximum structure is not observed. Why does the momentum distribution not exhibit four maxima? We divide the TI trajectories into (0-3) and (0-1-3) trajectories, and the ion momentum distributions for the two classes of trajectories are displayed separately in fig. 5. It is clearly seen that for the (0-3) trajectories the two peaks of the spectrum are around . It is obviously narrower than the momentum spectrum for I = 1.0 × 1015 W/cm 2, where the peaks locate at (see fig. 2(a)). For the (0-1-3) trajectories, the peaks of the ion momentum spectrum are located at (the blue curve in fig. 5). Figure 5 reveals that the peaks of the ion momentum spectra for the (0-3) and (0-1-3) trajectories are very close, and they merge into two wider peaks (see the solid black curve in fig. 5). As a result, the four-maximum structure is unobservable though both (0-3) and (0-1-3) channels have considerable contributions to TI.
In the classical consideration, one electron less released at a laser phase close to a zero crossing will lead to a reduction of in the triply ionized ion momentum . Then the question arises: why does the ion momentum of the (0-1-3) trajectory become smaller by only comparing with (0-3) trajectory? Through carefully examining of fig. 4 we find that the laser phase at recollision changes with the laser intensity: for I = 1.0 × 1015 W/cm 2 recollisions occur between 0.4T–0.5T (0.9T–1.0T), while for I = 2.0 × 1015 W/cm 2 recollisions occur around 0.35T (0.85T). As a consequence, for I = 2.0 × 1015 W/cm 2 one electron less in the recollision encounter will lead to a decrease of in the ion momentum.
The correlated electron momentum spectrum can provide more detailed insight into the electron dynamics, thus in fig. 6 we display the longitudinal momentum spectra of the correlated electrons from NSTI. Figures 6(a) and 6(c) show the correlated momentum of the (0-3) trajectories, where the intensities are 1.0 × 1015 W/cm 2 and 2.0 × 1015 W/cm 2, respectively. Note that in both plots we have only illustrated the longitudinal momenta of two arbitrary electrons from the three-electron recollision. It is clearly seen that the electron pairs are concentrated in circular regions in the first and third quadrants. For 1.0 × 1015 W/cm 2, the centers of the circles are located at ( , ), while for 2.0 × 1015 W/cm 2, they are located at ( , ). The position change of these clusters indicates that the electrons are released at different laser phases for 1.0 × 1015 W/cm 2 and 2.0 × 1015 W/cm 2, as has discussed above. For the (0-1-3) trajectories, only two electrons are involved in recollision, and their orrelated electron momenta along the laser polarization are shown in figs. 6(b) and 6(d), where the intensity are 4.0 × 1015 W/cm 2 and 2.0 × 1015 W/cm 2, respectively. For these trajectories, the correlated electron momentum distributions also exhibit overall maxima in the first and third quadrants. However, in striking contrast to the (0-3) trajectories, a obvious V-shape structure is observed. This is very similar to the correlated momentum distribution of helium NSDI at high intensity (1.5 PW/cm 2) presented in a recent study , where it has been interpreted as a consequence of coulomb repulsion and typical (e,2e) kinematics. The difference between the correlated momentum distributions of the (0-3) and (0-1-3) trajectories implies the different microscopic dynamics in the three-electron and two-electron recollisions, which calls for further studies.
At the relatively low intensity, the V-shape (or finger-shape) structure has been observed for NSDI of helium , where the authors identified the features corresponding to the binary and recoil lobe in field-free (e,2e) recollision. However, this V shape has never been observed for NSDI of Ne [5, 33]. This implies that the V shape in the correlated momentum spectrum depends on the atomic structure at the relatively low laser intensity. At the relatively high intensity, the V shape is both observed in NSDI of helium experimentally  and predicted in NSDI of Ne + by our calculations, meaning that this feature is independent of the species of the targets at the high intensity. In other words, the above analysis implies that the corresponding microscopic dynamics for the V-shape structure in the correlated electron momentum spectrum of NSDI at the high laser intensity is different from that at the relatively low intensity. Further theoretical and experimental studies are needed to confirm this issue.
In summary, strong-field NSTI has been systematically investigated and the sub-laser-cycle dynamics of the three correlated electrons has been demonstrated with the full 3D classical ensemble model. Our calculations have taken into account both the purely NS (0-3) and combined S/NS (0-1-3) channels, and the various channels at different laser intensities are clearly identified by tracing individual TI trajectories. Back analysis shows that the laser phase at recollision moves towards the maximum of the electric field with the increase of the laser intensity. Because of this change, the peaks in the ion momentum spectrum of the (0-3) trajectories are very close to those of the (0-1-3) trajectories at modest laser intensity where both (0-3) and (0-1-3) channels contribute to NSTI. As a result, the four-maximum shape predicted by the classical rescattering model  is unobservable. The ion momentum distributions calculated by this classical model agree well with the experimental results  over a wide range of laser intensities. Additionally, we find that the correlated electron momentum distributions of the (0-3) trajectories are distinctly different from those of the (0-1-3) trajectories. This difference implies the different microscopic dynamics in the three-electron and two-electron recollisions, which calls for further detailed studies.
This work was supported by the National Natural Science Foundation of China under Grant No. 10774054, National Science Fund for Distinguished Young Scholars under Grant No.60925021, and the 973 Program of China under Grant No. 2006CB806006.
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