## Abstract

The multielectron dynamics in nonsequential triple ionization (NSTI) of neon atoms driven by mid-infrared (MIR) laser pulses is investigated with the three-dimensional classical ensemble model. In consistent with the experimental result, our numerical result shows that in the MIR regime, the triply charged ion longitudinal momentum spectrum exhibits a pronounced double-hump structure at low laser intensity. Back analysis reveals that as the intensity increases, the responsible triple ionization channels transform from direct (e, 3e) channel to the various mixed channels. This transformation of the NSTI channels leads to the results that the shape of ion momentum spectra becomes narrow and the distinct maxima shift towards low momenta with the increase of the laser intensity. By tracing the triply ionized trajectories, the various ionization channels at different laser intensities are clearly identified and these results provide an insight into the complex dynamics of the correlated three electrons in NSTI.

© 2013 Optical Society of America

## 1. Introduction

Detailed understanding of the correlated multielectron dynamics driven by the strong laser field is essential to extend our knowledge on laser-matter interaction and the concepts of attosecond physics to many-body microscopic systems [1]. Due to the strongly correlated electron-electron behavior, nonsequential double or multiple ionization (NSDI, NSMI) of atoms by the intense laser pulse have attracted continuously increasing attention [2–16] since the observation of the dramatically enhanced double ionization (DI) yields [17]. The experiments on the ellipticity dependence of the DI yield [18, 19], especially the differential measurements of recoil ion and emitted electron momenta [1, 8, 20], provide strong evidences that the recollision mechanism is dominantly responsible for the NSDI and NSMI process [21]. According to this recollision scenario, an electron is ionized by the laser field, and then is driven back by the oscillating field and interacts with its parent ion, leading to the release of one or more electrons. Further analyses of the recoil ion momentum distributions [20,22], the photoelectron energy distributions [23,24] and the correlated electron spectra [1, 5, 8, 25] provide more detailed information about the ionization process.

For nonsequential triple ionization (NSTI), though intensity-dependent ion yields [26,27] as well as ion momentum distributions [28–30] have been measured, the understanding of strong-field NSTI mechanisms remains restricted. For instance, a recent experiment [30] has measured the momentum distributions of *Ne*^{3+} by the 795-nm pulses at the intensity range from 10^{15} to 2 × 10^{16}*W/cm*^{2}. The various triple ionization (TI) channels at different laser intensities have been identified by tracing the intensity-dependent evolution of the recoil-ion momentum spectra. The authors speculated that the (0–3) and (0-1-3) processes are responsible for spectrum with the maxima at
$\pm 4\sqrt{{U}_{p}}$ and
$\pm 2\sqrt{{U}_{p}}$, respectively. While, at the transitional intensities such as *I* = 3*PW/cm*^{2} and *I* = 6*PW/cm*^{2} [see Figs. 1(e) and 1(h) in [30]], the responsible processes for TI remains 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 time dependent schrödinger equation has only been performed on NSDI of helium by 400 nm laser pulses [31], and in the long-wavelength regime, it seems impossible to extend this quantum-mechanical calculation to NSTI that involves three correlated electrons in the foreseeable future. Thus, the efforts aimed to provide a detailed picture of NSTI have been recently concentrated on the development of classical approaches [15,32–37]. However, due to the variety of the possible pathways, including different combinations of sequential (S) and nonsequential (NS) ionization steps [30, 33, 35, 38] as well as the possibility of recollision excitations in certain intensity regime, up to now, the understanding of strong-field TI mechanisms remains fragmentary.

The wavelength of the laser pulses in those previous studies are mainly in the near-infrared (NIR) region (*λ* ≤ 1 *μm*). With the advance of ultrafast laser technology, the mid-infrared (MIR) laser pulse (1 *μm* <*λ* ≤ 10 *μm*) have become available [39]. Recently, the interaction of atoms and molecules with MIR laser fields has drew much attentions and some new phenomena and applications have been observed and proposed [40–43]. For example, an unexpected spike-like structure at the low energy region of the above-threshold-ionization (ATI) spectra in the MIR regime has been experimentally observed [40, 41]. In the study of high-harmonic generation (HHG), it has shown that the MIR laser pusle not only produces much more energetic harmonic photons but also reduces harmonic chirps [43], which is beneficial for attosecond pulse generation. The strong-field NSDI and NSTI of atoms and molecules by the MIR pulses has also attracted experimental attention [44,45]. In [45], the momentum distributions of *Ne*^{3+} and *Ar*^{3+} by the 1300-nm pulses at the intensity *I* = 0.4*PW/cm*^{2} have been measured. The authors reported that the shapes of longitudinal momentum distributions for the *Ne*^{3+} and *Ar*^{3+} all exhibit a clear double-hump structure that is different from those at NIR wavelength [29,30]. This result indicates that in the MIR regime, the responsible dynamics of correlated multielectron for TI process is different from that in the NIR regime.

In this paper, the correlated multielectron dynamics in strong-field NSTI of neon atoms driven by 1600-nm laser pulses is investigated with a three-dimensional (3D) full classical ensemble model [13, 15]. The numerical result shows that the triply charged ion momentum spectrum exhibits a pronounced double-hump structure at the low laser intensity, which is well consistent with the experimental result [45]. Back analysis reveals that at low laser intensity, the direct (e, 3e) ionization channel is dominantly responsible for NSTI, where one electron ionizes firstly, then it is driven back by the laser field to recollide with the parent nucleus and kicks out the other two electrons immediately. While for moderate intensity, besides the (e, 3e) channel, there is a mixed ionization channel where the first and second electrons are ionized through a recollision-induced (e, 2e) process and the third electron is excited after recollision and then released by laser field near the subsequent field maximum. Furthermore, the two TI channels are contribute significantly to NSTI at moderate intensity. For the high intensity, the combined sequential and nonsequential (S/NS) ionization channel where the two electrons are released sequentially and the third electron is ionized through a recollision with the second electron plays a dominant role in NSTI. Consequently, the shape of the ion momentum spectra becomes narrow and the distinct maxima shift to the low momenta as the intensity increases.

## 2. The full classical ensemble model

The full classical ensemble model has achieved success in understanding of NSDI and NSTI [13–15, 35] and it has been described in detail in [13, 35]. The evolution of the three-electron system is determined by the classical equation of motion (Atomic units are used throughout the paper if not stated otherwise.)

*i*is the electron label which runs from 1 to 3.

**E**(

*t*)=

*ẑf*(

*t*)

*E*

_{0}sin(

*ω*t) is the electric field, where the

*ẑ*is the laser polarization direction and

*f*(

*t*) is the pulse shape which has two cycles turn-on, six cycles at full strength and two cycles turn-off. In our calculation, the wavelength is 1600nm. The nucleus-electron and electron-electron interaction are represented by a 3D soft-Coulomb potential ${V}_{\mathit{ne}}\left({\mathbf{r}}_{i}\right)=-3/\sqrt{{\left({\mathbf{r}}_{i}\right)}^{2}+{a}^{2}}$ and ${V}_{ee}\left({r}_{i},{r}_{j}\right)=1/\sqrt{\left({\mathbf{r}}_{i}-{\mathbf{r}}_{j}\right)+{b}^{2}}$, respectively. The soft parameter

*a*is employed to avoid autoionization, which sets the lower limit of

*a*. There is also an upper limit for

*a*, which is determined by the condition that there is a classically allowed region for the three electrons with the total energy of the ground-state energy of the target. For the targets investigated in this paper, the lower and the upper limits of

*a*are about 0.945 and 1.06 a.u., respectively. Here, similar to [13,14], the screening parameter

*a*is set to be 1.0 a.u. Note that a small change of the parameter

*a*has little impact on the statistical results presented in our paper. The parameter

*b*is included to avoid the coulomb singularity in our calculations. It could be set to equal any other small value (including zero). In our work, we set b=0.1 a.u. To obtain the initial value, the ensemble is populated starting from a classically allowed position for the neon atom with ground state energy of −4.63 a.u., i.e., the sum of the first, the second and the third ionization potential of neon. The available kinetic energy is distributed randomly between the three electrons randomly in momentum space. Then the electrons are allowed to evolve a sufficient long time in the absence of the laser field to obtain stable position and momentum distribution [13, 14]. After the laser pulse is turned off, if three electrons have positive energy, we define triple ionization.

## 3. Results and discussions

In Fig. 1, we present momentum distributions of the triply charged ions as a function of the laser intensity. The data in Fig. 1 are plotted in units of
$\sqrt{{U}_{p}}$ in order to account for the intensity dependence of the drift momentum received by the recoil ion [30], where the *U _{p}* is the ponderomotive energy. Figures 1(a), 1(c) and 1(e) illustrate two-dimensional ion momentum distributions for 0.5

*PW/cm*

^{2}, 1

*PW/cm*

^{2}and 2

*PW/cm*

^{2}, respectively. Here, the horizontal axis shows the longitudinal (parallel to the laser polarization direction, i.e., z axis) ion momentum and the vertical axis corresponds to the transverse (along x axis) momentum. For all intensities, the transverse momenta are concentrated around zero. The longitudinal momenta, instead, exhibit two clear maxima at non-zero values, which indicates NSTI channels dominate TI at present intensities [22]. An important feature of these spectra is that the width of the spectra becomes narrow with the increase of the intensity. The longitudinal momentum distributions, i.e., the projections of data of two-dimensional ion momentum spectra onto the horizonal axes, could illustrate this feature more clearly [see the right column of Fig. 1].

For 0.5*PW/cm*^{2}, Fig. 1(b) displays the longitudinal momentum distributions of triply charged ions. The *Ne*^{3+} spectrum is essentially identical to the distributions reported in [45] for TI of *Ne* at the intensity of 0.4*PW/cm*^{2} with the wavelength of 1300 nm. The spectrum exhibits a clear double-hump structure, with two-defined peaks at
$\pm 4\sqrt{{U}_{p}}$ and almost no ions produced with zero longitudinal momentum are observed. When the intensity increases to 1.0*PW/cm*^{2} and 2.0*PW/cm*^{2}, the two peaks move to
$\pm 2\sqrt{{U}_{p}}$ [see Fig. 1(d)] and
$\pm 1.2\sqrt{{U}_{p}}$ [see Fig. 1(f)], respectively. This indicates that the peaks shift to the low momenta with the increase of the laser intensity. In addition, comparing Figs. 1(b), 1(d) and 1(f), one can find that the valley of the ion longitudinal momentum distributions becomes shallow as the intensity increases. These results indicate that in the MIR regime, the responsible microscopic electron dynamics for NSTI at different intensities are different and complex.

Tracing back the temporal evolution of TI trajectories allows us to unveil the microscopic multielectron dynamics of atomic TI, and thus provides an intuitive way to identify the different ionization channels. In present intensity regime, we find that there are three main NSTI channels. First, direct (e, 3e) ionization channel, i.e. one electron gets ionized firstly, and then it is driven back by the electric field to recollide with the parent nucleus, leading to the three electrons ionized immediately. This TI process is defined as (0–3) channel. Second, a combined S/NS ionization process where 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 TI process is defined as (0-1-3) channel. Third, a mixed TI process is called (0-2-3) channel. In this channel, one electron is ionized by the laser field, and then it is driven back by the electric field to recollide with the parent ion, leading to the second electron emitted immediately and the third electron excited after recollision and released by the laser field near the subsequent field maximum.

Three sample trajectories for (0–3) [the left column], (0-1-3) [the right column] and (0-2-3) [the middle column] TI channels are plotted separately in Fig. 2, presented in longitudinal coordinate z [the upper rows], energy [the middle rows], and longitudinal momentum [the bottom rows] versus the time for each electron, respectively. For the NSTI trajectory in the left column, it is clearly seen that the three electrons are set free immediately after recollision [see Figs. 2(a) and 2(b)] and accumulate almost the same high longitudinal drift momentum after the laser is over [see Fig. 2(c)]. While for the trajectory in the middle column, only two electrons are set free almost immediately after recollision and they acquire the similar high longitudinal drift momentum from the laser field [see the gray solid and red dotted curves in Fig. 2(f)]. The third electron is excited after recollision and released by the laser field [see the blue solid curve in Fig. 2(e)]. After the laser is over, it acquires a very small drift momentum [see the blue solid curve Fig. 2(f)]. For the NSTI trajectory in the right column, the first and the second electrons are emitted sequentially [see the gray solid and red dotted curves in Fig. 2(h)] and the third electron is released through a recollision with the second electron [see the blue solid and the red dotted curves in Fig. 2(h)].

Similar to [46], we further statistically analyze the percentage contributions of different TI channels to NSTI. As shown in table 1, the relative contribution of the (0–3), (0-2-3) and (0-1-3) channels changes with laser intensity. For *I* = 0.5*PW/cm*^{2}, the statistic result reveals that about 76% of the TI occur through the (0–3) channel and it is the dominant channel for TI at low intensity. At *I* = 2.0*PW/cm*^{2}, the contribution of the (0-1-3) channel to the total TI yield is over 70%, which indicates that the (0-1-3) channel is dominantly responsible ionization channel for TI at high intensity. For *I* = 1.0*PW/cm*^{2}, the results show that both (0–3) and (0-2-3) channels contribute significantly to NSTI and the contributions of the (0–3) and (0-2-3) channels to the total TI yield are about 41% and 59%, respectively. Based on these results, we may draw the conclusion that in the MIR regime, the responsible triple ionization channels for NSTI are sensitively dependent on the laser intensity.

According to the different TI channels, we have segregated the trajectories shown in Figs. 1(b), 1(d) and 1(f), respectively, and the ion momentum distributions for the various classes of trajectories are displayed separately in Figs. 3(a), 3(b) and 3(c). At *I* = 0.5*PW/cm*^{2}, it is clearly seen that for (0–3) trajectories the two peaks of the spectrum are around
$\pm 4\sqrt{{U}_{p}}$ [see the red curve in Fig. 3(a)]. For the (0-2-3) trajectories, both *I* = 0.5*PW/cm*^{2} and *I* = 1.0*PW/cm*^{2}, the peaks also locate near
$\pm 2\sqrt{{U}_{p}}$ [see the blue curves in Figs. 3(a) and 3(b)]. At *I* = 2.0*PW/cm*^{2}, Fig. 3(c) shows that for the (0-1-3) trajectories, the peaks locate at
$\pm 1.2\sqrt{{U}_{p}}$ [see the dark green curve in Fig. 3(c)]. The above results reveal that in the MIR regime, the (0-2-3) channel is responsible for the double-hump spectrum with maxima at
$\pm 2\sqrt{{U}_{p}}$. Furthermore, if the (0-1-3) ionization channel is dominantly responsible for NSTI in this regime, the locations of the ion momentum peaks are much less than the values of
$\pm 2\sqrt{{U}_{p}}$. This is different from that at NIR regime [30], where the authors speculated that the (0-1-3) process is responsible for the spectrum with maxima at
$\pm 2\sqrt{{U}_{p}}$. Additionally, Fig. 3 shows that the peaks of the ion momentum spectra for the (0–3) and (0-2-3) [see Figs. 3(a) and 3(b)] or the (0–3) and (0-1-3) [see Fig. 3(c)] trajectories are very close. Thus, the shape of the longitudinal ion momentum spectra all show two wider peaks [see Figs. 1(b), 1(d) and 1(f), respectively] and the predicted four-maximum structure in [33] is unobservable in the MIR regime.

To give an overall understanding of intensity-dependent evolution of the ion momentum spectra for the various trajectories, we analyze the times of TI and recollision for different intensities. Back analysis of the TI trajectories also allows us easily to determine the recollision and ionization times, which can provide insight into the sub-cycle dynamics of NSTI. We define the recollision time *t _{r}* 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

*t*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) and (0-2-3) channels, three electrons are involved in the recollision encounter, while only two electrons are involved for the (0-1-3) channel.

_{TI}Figure 4(a) shows the triple ionization phase *t _{TI}* versus recollision phase

*t*(both in cycle) for the intensity 0.5

_{r}*PW/cm*

^{2}[47]. It is clearly seen that the most of recollisions occur in the range 0.35T–0.40T (or 0.85T–0.90T, where the T is the laser cycle), just before the zero crossing of the laser field. It is consistent with the predication of the simple-man model [21]. As shown in the Fig. 4(a), the dominant part of the population are along the diagonal

*t*=

_{TI}*t*, which indicates that the TIs occur immediately after the three-electron recollision. As a consequence, for (0–3) trajectories, the three electrons from NSTI are more likely to set free with a small initial momentum close to the zero crossing of the field and accumulate the same high longitudinal drift momentum, which means that the three electrons are emitted predominantly to the same direction with similar longitudinal momenta [see Fig. 2(c)].

_{r}For *I* = 1.0*PW/cm*^{2}, we have separated the TI trajectories according to whether they get ionized through the (0–3) or (0-2-3) channel. Figures 4(c) and 4(d) display the phase *t _{TI}* at time of TI versus the phase

*t*at recollision for the (0–3) and (0-2-3) trajectories, respectively. For the (0–3) trajectories, similar to the case at 0.5

_{r}*PW/cm*

^{2}, most of the population are along the diagonal

*t*=

_{TI}*t*[see Fig. 4(c)]. But, for this intensity, the most of recollisions occur in the range 0.30T–0.35T (or 0.80T–0.85T) [see Figs. 4(c) and 4(d)] and it is less than that at 0.5

_{r}*PW/cm*

^{2}. Particularly, Fig. 4(d) shows that for the (0-2-3) trajectories, most of the TIs occur around the field maximum (0.25T and 0.75T), which is a reminder of impact excitation TI because there is a sub-cycle time delay between TI and recollision. Thus, for the (0-2-3) trajectories, only two electrons accumulate the same high longitudinal drift momentum and the third electron acquires a very small drift momentum from the laser field [see Fig. 2(f)].

For the intensity 2.0*PW/cm*^{2}, Fig. 4(b) shows the most of recollisions occur in the range 0.28T–0.33T (or 0.78T–0.83T), just after the maximum of the laser field. As shown in the Fig. 4(b), the dominant part of the population are along the diagonal *t _{TI}* =

*t*implies that TIs occur immediately after the two-electron recollision. From what has been discussed above, we may draw the conclusion that the contribution of the (0–3) channel to the total yield decreases gradually, and both the (0-2-3) and (0-1-3) channels (the partly sequential channel) make more and more contributions to NSTI. That is to say, the contribution of the partly sequential channel increases gradually as the intensity increases. As a result, the width of the ion momentum spectra becomes gradually narrow with the increase of the laser intensity.

_{r}Moreover, through careful examination of the Fig. 4 we find that the laser phase at recollision varies with the laser intensity: for the intensities 0.5*PW/cm*^{2}, 1.0*PW/cm*^{2} and 2.0*PW/cm*^{2}, recollision times *t _{r}* are around 0.375T(or 0.875T), 0.33T(or 0.83T) and 0.30T(or 0.80T), respectively. According to the classical consideration, if an electron is set to an oscillating laser field of frequency

*ω*and strength

*E*(

*t*) =

*E*

_{0}(

*t*)

*sin*(

*ω*t) at a time

*t*

_{0}(where the

*E*

_{0}(

*t*) is a pulse envelope function), it acquires the drift momentum ${P}_{z}^{e}\left({t}_{0}\right)=2\sqrt{{U}_{p}}\mathit{cos}\left(\omega {\text{t}}_{0}\right)$ after the laser is over [28, 30, 33, 34]. For strong-field NSTI of atoms, the ion momentum vector is equal to the negative sum of the three emitted electrons momentum vectors. Note that in the (0–3) channel, the three electrons are emitted by recollision and escape with similar high drift momentum. As a consequence, for 0.5

*PW/cm*

^{2}, the two-defined peaks of the ion momentum spectrum locate near ${P}_{z}^{\mathit{ion}}=\pm 3\times {P}_{z}^{e}\left({t}_{r}\right)=\pm 3\times 2\sqrt{{U}_{p}}\mathit{cos}\left(\omega \times 0.375T\right)=\pm 4.2\sqrt{{U}_{p}}$. While for the (0-1-3) channel, an electron is ionized by the laser field and only acquires a very small longitudinal drift momentum from the laser field. The other two electrons are set free immediately after recollision and accumulate the similar high drift momentum. Thus, for 2.0

*PW/cm*

^{2}, ${P}_{z}^{\mathit{ion}}=\pm 2\times {P}_{z}^{e}\left({t}_{r}\right)=\pm 2\times 2\sqrt{{U}_{p}}\mathit{cos}\left(\omega \times 0.30T\right)=\pm 1.2\sqrt{{U}_{p}}$, respectively. At the intensity 1.0

*PW/cm*

^{2}, the locations of the ion momentum peaks are slightly larger than the values of $\pm 1.9\sqrt{{U}_{p}}$ since the (0-2-3) channel has considerable contributions to TI at this intensity where the third electron is ionized by the laser field near the field maximum. Therefore, the peaks shift to the low momenta with the increase of the laser intensity.

In addition, a recent study by Lötstedt and Midorikawa has demonstrated that inclusion of the laser magnetic field reduces significantly NSDI probability and this reduction is remarkably different for different targets [48]. To explore if the laser magnetic field significantly influences on the NSTI of neon atoms, we have performed a preliminary calculation with the magnetic field for the case of *I* = 2.0*PW/cm*^{2}. Our calculation shows that when the magnetic field is taken into account the triple ionization yield is about 0.044%, which is a little lower than the yield of 0.047% without the magnetic field. For two-dimensional ion momentum distributions, the numerical results show that, regardless of considering the magnetic field or not, the transverse momenta are concentrated around zero and the longitudinal momenta distribute in the range of
$-3\sqrt{{U}_{p}}~+3\sqrt{{U}_{p}}$. The main feature of the two-dimensional ion momentum distributions do not change with or without the magnetic field, i.e., the magnetic field has little impact on the momentum distribution. Furthermore, by back analysis of the classical trajectories, we find that including the magnetic field, the contribution of the (0-1-3) channels to the total triple ionization is about 73%, which is very close to 71% of the case without the magnetic field. By comparing these results with and without the magnetic field above, it can be found that the laser magnetic field has little impact on NSTI probability, the two-dimensional ion momentum spectrum and the dominant ionization channel contributing to NSTI.

## 4. Conclusion

In summary, the correlated three-electron dynamics in strong field NSTI of neon atoms by MIR laser pulses has been systematically investigated with the full 3D classical ensemble model. The various ionization channels at different laser intensities have been clearly identified by tracing TI trajectories. At low laser intensity, the triply charged ions momentum spectrum exhibits a pronounced double-hump structure and it is well consistent with the experimental result [45]. Back analysis reveals that the (0–3) ionization channel is dominantly responsible for NSTI process at low intensity. While for moderate intensity, both (0–3) and (0-2-3) channels contribute significantly to NSTI. At high intensity, the (0-1-3) channel play a dominant role in NSTI. As a result, the shape of ion momentum spectra becomes narrow and the distinct maxima shift towards the low momenta as the intensity increases. With the help of classical trajectory diagnosis, we achieve insight into the complex dynamics of the correlated three electrons in NSTI.

## Acknowledgments

This work was supported by the National Science Fund for Distinguished Young Scholars under Grant No. 60925021, National Natural Science Foundation of China under Grant No. 11004070, and the 973 Program of China under Grant No. 2011CB808103.

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