1. Introduction

The complexity of the world and the processes taking place in it manifests itself through correlations that make it impossible to reduce even such a relatively simple system as an atom with two electrons to independent, non-interacting elements. Electrons interact - that sounds almost trivial. However, if one thinks about multiple ionization, it even almost involuntarily assumes that the electrons leave the atom one by one - independently of each other. In this context, interaction of strong laser light with matter still brings new surprises for more than half century as reviewed by [1]. It was realized quite early that the impact of short but strong laser radiation may lead to multiple ionization of atoms [2–7] and subsequently discussion on collective, correlated vs. uncorrelated, i.e. non-sequential or sequential character of electrons emission has begun [3,8–16]. It soon became apparent that the physics involved seems to be more interesting at intermediate intensities, when the existence of the characteristic knee in the ionization yield, i.e. the measured ionization yield was orders of magnitude larger than expected from the independent electron escape picture, triggered a lively discussion about the importance of the so-called re-scattering mechanism on the non-sequential double ionization [17–19]. The theoretical picture came with the simple-man three-step model [20], that provided a transparent explanation of above-threshold ionization (with electrons leaving the atom with energy far exceeding the threshold [21]) as well as high harmonic generation (HHG) [22] and re-scattering mechanism leading to enhanced double ionization [17–19]. The cold-target recoil-ion-momentum spectroscopy (COLTRIMS) [23–27] allowed one to detect the momenta of an ion and ejected electrons during the ionization event opening the path for a detailed study of both the electron and ion laser-induced dynamics. The measured momentum distributions of the two escaping electrons, in particular the high-resolution ones, representing the famous finger-like structure, finally sealed that we are dealing with one of the most amazing manifestations of the existence of a correlation in nature. Further in-depth studies of these electron-electron correlation during strong-field evolution led to a deeper understanding of ongoing processes and the development of more sophisticated models [28–30]. Spin related effects have also been studied [31–33]. For an additional resolution of ionization delays, coincidence schemes were assisted by a streaking technique [27,34,35]. An alternative path for accessing real-time dynamics implies a combination of coincidence spectroscopy with an attosecond interferometry [36,37]. All these discussed results where limited to simplest case in which correlations reveal itself, namely to two electrons. When it comes to next “nearest neighbor”, three electron system, natural difficulties arise – how to relatively easily visualize results in order to immediately see the correlations? To this end we apply Dalitz plots [38], a graphical tool extensively used in the particle physics community. They were introduced into atomic physics for the description of ion impact ionisation [39–41] and routinely used for the visualization of Coulomb explosion of molecules driven by strong laser fields [42–45]. Here we present their usefulness for extracting qualitative information about the strong laser field triple ionization, based on the numerical solution of the Schrödinger equation in reduced dimensions.

Importantly, recent coincidence experimental schemes [36,37,46–49] are kinematically complete, that is, momenta, position and detection time are measured for each charged particle explicitly. With increased energy resolution [36,37] one is able to catch fine interference structures that form the quantum fingerprint of the laser-driven electron dynamics. Those recent advances removed the limitations on the number of particles detected: there is no need to use ionic momenta for recalculating electronic ones.

The experimental arena is thus ready for three-electron coincidence experiments. In ion-impact induced ionization, the pioneering experiments were demonstrated already [39]. Such an experiment for strong field ionization would definitely open a new page in the understanding of electronic correlations, e.g. in recollisional dynamics induced by IR femtosecond laser pulses. In particular, different double ionization channels of multi-electron atoms (affecting different electrons) cannot be resolved in a two-electron coincidence experiment [50]; the three-electron coincidence scheme, on the other hand, potentially could allow one to track the relative contributions of different channels [51].

The theoretical support needed for these prospective experiments is, however, not yet available partly because numerical simulations for many electron ionization are extremely challenging. A full quantum-mechanical simulations are, at best, possible for two electrons [52,53]. Models for three electrons are at their infancy being limited to either classical Monte Carlo methods [45,54–58] or to time-dependent orbital approaches which, while successful for description of HHG [59], can hardly provide detailed information on the ionization dynamics and momenta distributions.

Recently, we presented introductory studies of strong-field triple ionization, however, due to the intrinsic complexity of the problem, inevitably we restricted ourselves to only investigate the dependence of ionization yields on the laser pulse amplitude [50,51,60,61]. In the present work, we raise the analysis to a significantly higher level, addressing three electrons momenta distributions with use of Dalitz plots for presentation of data. Our method is based on the Eckhardt-Sacha model of reduced dimensionality, derived from the analysis of saddles in the effective adiabatic potential for electrons in an instantaneous electric field [62]. Let us emphasize, that this is a distinctly different approach from the standard Rochester reduction, which limits the motion of electrons to the laser polarization axis [63,64]. In the Eckhardt-Sacha model, electrons moving far from the nucleus along effective directions, even being at similar distance from it, interact negligibly - observe an additional $r_ir_j$ term in the denominator of Eq. (2) below. In this way the overestimation of the Coulomb repulsion, characteristic to the Rochester model, is eliminated.

The use of upgraded software for the quantum-mechanical calculation involving three active electrons combined with Dalitz plots technique used for visualization allows us to study strong-field triple ionization with unprecedented insight. First, we show what kind of patterns are visible in the triple momenta distributions as markers of electron-electron correlations. Second, in order to demonstrate the versatility and possibilities of the presented approach, we examine the influence of the symmetry of the initial state wave function on the observed momenta distribution. This shows how the Pauli principle inevitably influences the dynamics of multi-electron processes.

2. Model

The Hamiltonian of the model studied reads [60]:

Additionally, and novel for three electrons, a twist comes from the fact that for three electrons one has to be careful about the spatial symmetries of the wavefunction. As is known, they are directly related to the electronic spin and Pauli exclusion principle [67]. This has been already noticed in the pioneering studies of Li atom ionization [68–70]. In its $1s^22s^1$ initial state the corresponding Slater wavefunction is a sum of 3 terms (each corresponding to electron permutations) with two electrons with spin up (U) and one down (D). Since the Hamiltonian is spin-independent (neglecting spin-orbit coupling) for non-relativistic strong laser-atom interactions all three components evolve identically and a single combination may be time evolved only. The same would occur for species with electronic $ns^2np^1$ configuration, e.g. for Boron or Aluminum, however, in these cases a real multi-photon regime falls within very low frequencies. As a model system with electronic $ns^2np^1$ configuration we consider an artificial three-electron atom with the first three ionization thresholds corresponding to Neon [50]. Thus, we will refer to that model as Neon. The ground state wavefunction in the model atom is spatially partially antisymmetric (assuming the spin configuration is (UUD) meaning Up-Up-Down for X, Y and Z electrons): $\Psi (X,Y,Z)=\Psi (X,Z,Y)=-\Psi (Y,X,Z)=-\Psi (Z,Y,X)$ [60]. For comparison, we will consider a model atom that has $p^3$ configuration with all spins oriented in the same direction (say UUU) and, thus, a totally antisymmetric spatial wavefunction : $\Psi (X,Y,Z)=-\Psi (X,Z,Y)=-\Psi (Y,X,Z)=-\Psi (Z,Y,X)$. The same spin orientation is dictated by the Hund rule for the $p^3$ configuration of the ground state in a multielectron atom. The corresponding ionization potentials are set to match the first three ionization potentials of Nitrogen, i.e. 0.52 a.u., 1.61 a.u. and 3.92 a.u. with $\epsilon =\sqrt {1.02}$ and $q_{ee}=\sqrt {0.5}$. We will refer, therefore, to that model as Nitrogen.

3. Data representation

The time evolution of the TDSE (for details see Supplement 1) leads to events in which the three electrons are ionized, i.e. three electrons are in the “outer” region after the pulse has ceased. Can we extract useful information on the dynamics from momenta distribution in such a case? The 3D wavefunction allows us to learn about the momentum of outgoing electrons employing the technique of Dalitz plots. This representation is instrumental in particle physics and has been proved to be very useful to disentangle electron-electron and electron-ion correlations in ion-atom collisions [39–41] as well as for the understanding of Coulomb explosion of molecules under strong laser pulses [42–45]. The time-evolved 3D wavefunction is projected onto the surrounding sphere by integrating over the radial coordinate, and then each octant of this sphere is projected onto a plane, forming an equilateral triangle. Each projection plane is chosen perpendicular to the diagonal line belonging to the particular octant; the type of projection is gnomonic with a tangent equal to the radius of the sphere. The borders between octants are formed by the $XY$, $XZ$ and $YZ$ planes, thus the vertices of the Dalitz plots correspond to an exclusive motion of the $X$, $Y$ or $Z$ electron with the other two electrons at rest. We refer to these vertices as $X$, $Y$ and $Z$, respectively. For each Dalitz plot, an internal position is defined by a set of distances $\{ \pi _X,\pi _Y,\pi _Z \}$ to the sides of the triangle opposite to the $X,Y,Z$ vertices, correspondingly (see an illustration in Fig. 1):

 

Fig. 1. Structure of a Dalitz plot. The three-electron ($X,Y,Z$) momentum distribution is mapped into a triangle. Each point inside the triangle is defined by coordinates $\{ \pi _X,\pi _Y,\pi _Z \}$. For each electron $i$, the small $\pi _i$ the small is the ratio of the momentum $p_i$ to the triple-ion momentum value. See the text for more details.

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Let us now see the connection between the simulated data and the data of a prospective experiment. While an all-angles experiment allowing the detection of all ejected electrons independently of their direction in 3D space is a standard tool these days, the restricted-dimensionality code we use is not capable of resolving such an information. The numerical data do allow one to determine the positive or negative direction of electronic motion with respect to the laser polarization axis only. On the other hand, in our calculation, we are able to extract spin-resolved information - a knowledge inaccessible in real experiments because the strong field phenomena in the non-relativistic regime are insensitive to the electron spin. Thus, in order to have a compatible set of data, we provide a Dalitz plots transformation to wash out any spin-resolved fingerprint. Such spin-averaged plots would correspond to angular-integrated (with respect to forth- “$+$” and back- “$-$” propagation) momentum-resolved 3-electron coincidence experiment.

The corresponding transformation that averages over spin directions is simple: for each experimentally-resolved combination of electrons [($+++$), ($++-$), ($--+$) and ($---$)] one collects all three possible orientation of electrons (vertices $XYZ, YZX, ZXY$) and then sums them up. As a result, one is left with four plots per one simulation, compare Fig. 2. Such a procedure is necessary for a system consisting of different spin directions. On the contrary, in the $p^3$ case no averaging is needed (see Supplement 1).

 

Fig. 2. Experimental-type (spin direction averaged) Dalitz plots for triply-ionized state of Neon depict the relative momentum distribution of electrons released in different spatial directions with respect to the external field vector (“$+$” for positive, “$-$” for negative). In the notation $(***)$ the first, second and third signs correspond to $X,Y$ and $Z$ electrons. The interpretation of the position in the plots is presented in Fig. 1.

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4. Results

In order to show the capabilities of an anticipated coincidence experiment on triple strong-field ionization, we simulated the dynamics within three-active-electron models of Nitrogen and Neon atoms – the examples of systems with spatially fully antisymmetric wavefunction and with partially antisymmetric wavefunction, correspondingly. These two models were previously studied in the context of the ionization yield dependence on the laser field amplitude [50,51,60,61]. In essence the results are as follows. At intermediate laser intensities, when the characteristic knee in the ionization yield is observed, there exist several channels leading to a triple ion, i.e. a sequential ionization, when each electron is liberated independently of others; a direct ionization, when all three electrons are set free at once; and a mixed ionization, when two electrons escape from the atom simultaneously and the third one is ejected separately (either single ionization follows the double one or vice versa). For atoms with fully antisymmetric wavefunction in their ground state the channels involving any kind of simultaneous escape (either the double ionization in the mixed triple ionization or the direct triple ionization) are suppressed in comparison to the sequential ionization channel. For atoms with a partially antisymmetric wavefunction, however, the mixed ionization channel is not suppressed [51]. Basing on the fact that in the case of two-electron events the non-sequential ionization manifests its correlated character on the electron’s momentum distributions in a form of the famous finger-like structures [71,72], we expect to recognize indications of correlated motion in the Dalitz plots as well. Furthermore, as the significance of diverse ionization channels in the case of Nitrogen and Neon is different, the correlated motion traces in the Dalitz plots for Nitrogen and Neon, respectively, are expected to differ.

Figure 2 shows a set of four Daliz plots obtained for Neon exposed to a pulse with an electric field amplitude $F_0=0.12$ a.u. One recognizes in panels (a) and (d), which collect momenta of electrons escaping along the polarization axis, a structure centered at the triangle centroid that exhibits a tri-fold symmetry. The structure from panel (a) has three equivalent maxima corresponding to the motion of two electrons with approximately equal momenta and the third one with significantly lower momentum - these maxima point to a mixed ionization: a direct double ionization accompanied by a single ionization event. There is also a maximum right at the triangle centroid that corresponds to the direct triple escape. One could expect a similar structure in panel (d), as this panel collects momenta of electrons escaping in opposite directions. It is not the case, because we limit ourselves to few-cycle laser pulses. On the contrary, for longer pulses, this should be observed. Panels (b) and (c) collect events in which two electrons are moving in the same direction and the third one in the opposite one - again the correlated motion of the former two electrons manifests itself in a structure that is aligned with and symmetric about the bisector falling from the vertex corresponding to the third electron moving in the opposite direction.

Let us now compare the Dalitz plots for Neon and Nitrogen. We limit our discussion to the octants representing electrons moving in the same direction with respect to the polarization axis. Such plots are shown in Fig. 3. Differences in the observed structures are evident. Although in both panels the structures have tri-fold symmetry, in the case of Neon one observes clear maxima on three bisectors at the triangle centroid (see panel (a)), in striking contrast to the zeros in the distributions along each of the bisectors that are visible for Nitrogen (see panel (b)). The spin symmetry clearly manifests itself in the Dalitz plots. For Nitrogen, a spatial totally antisymmetric wavefunction has $\Psi = 0$ for $r_i=r_j, \, i,j=\{1,2,3\}$; this property is conserved during the transformation to the momentum space, leaving the diagonals empty. Such diagonals map to bisectors during the Dalitz plot creation. In the case of Neon, however, there is a single “zero” bisector in the Dalitz plots of the 8-plot set, thus it does not survive rotations and additions provided to obtain the “experimental” spin-averaged plot. The observed difference in the Dalitz plots for Neon and Nitrogen is in line with the recalled suppression of correlated escapes for the case of Nitrogen [51]. At the same time both the N and Ne plots share a common feature: a qualitative change of the Dalitz plot structures around a field amplitude $F_0=0.1$ a.u. (see Fig. 4) - distributions become narrower, centered at, aligned with and symmetric about the bisector falling from the vertex corresponding to the electron moving in the opposite direction to the motion of the other two electrons. Comparison with ionization yields curves [51,60], directly implies a clear correlation between such a change and entering into the “knee” regime, where nonsequential ionization processes are instrumental.

 

Fig. 3. Dalitz plots for Neon (a) do not exhibit “empty bisectors” – in contrast to those of Nitrogen (b). The plots are spin-averaged.

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Fig. 4. Dalitz plots for both Neon (a,b) and Nitrogen (c,d) experience qualitative change when moving from lower field values $F_0=0.08$ a.u. (a,c) to $F_0=0.12$ a.u. (b,d). The plots are spin-averaged.

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Observe, additionally, that plots for higher laser intensities reveal apparent interference fringes (c.f. Fig. 2– Fig. 4). Those are absent for the low laser intensity case, Fig. 4, where only the field at the peak of the pulse contributes to the triple ionization yield. For stronger fields electrons can come back and rescatter, leading to fringes manifesting that now different quantum paths account for triple ionization.

There is one more dissimilarity between the Dalitz plots for N and Ne. In the intensity region below the knee, that is, about $F_0<0.1$ a.u. in our case, a noticeable difference is seen in the general shapes of the structures in $(--+)$ octants (see Fig. 5). The maxima for N are all concentrated below the triangle centroid $O$ – within the triangle $XOY$; the maxima for Ne, on the contrary, are located in the upper part of the plot, within triangles $ZOX$ and $ZOY$. The effect is robust and repeats for a number of intensities. Importantly, the absence of similar differences in $(++-)$ octants (see Fig. 4 panels (a) and (c)) suggests a significant dependence of the signal on the CEP, as should be expected for a few cycle pulse - study of such dependencies are in progress. Indeed, for the low field amplitude triple ionization would almost necessarily imply at least two cycles of ionization with subsequent recollision of an electron or two electrons. Efficiency of these sub-processes is inherently CEP-dependent. From the present data, for Ne there is significant amount of partially correlated escapes - see the maximum on the altitude close to the $Z$ vertex (Fig. 5(a)). The $Z$ electron carries the most of momentum and the remaining two electrons, moving in the opposite direction both carry approximately the same amount of momenta. The extended features parallel to either $XZ$ or $YZ$ edges both correspond to rather uncorrelated motion, i.e. each electron carries different amount of momenta. The situation for N is completely different (Fig. 5(b)): observed features are along the altitudes beginning either at the $X$ or $Y$ vertex. This means that two electrons escaping into opposite directions (back to back motion) have similar momenta as if they were anti-correlated. But most of the momenta is carried by the third electron - therefore these features are on the region altitude that is between the centroid and given vertex, i.e. $XO$ or $YO$.

 

Fig. 5. For low amplitude $F_0=0.06$ a.u. the $(--+)$ plots for Neon (a) and Nitrogen (b) show different shapes: the main structures are concentrated below the center of a Dalitz plot for the former case, and above this center in the latter case. The plots are spin-averaged.

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The discussed numerical scheme is compatible with a prospective experiment, in the sense that both may utilize forth- and back- detectors. A natural question, however, still arises: are these similar effects seen in the real 3D world, at least qualitatively? First note that the Eckhardt-Sacha model was developed based on a semi-adiabatic approximation, in which electrons are moving along the lines of instantaneous saddles [62]. Escape through these saddles gives rise to correlated motion of electrons, i.e. a non-sequential triple ionization. In full space, the correlated electrons will move in the vicinity of those lines. Thus, comparing the 3D situation with the 1D model, blurred features in the Daltiz plots are anticipated and one should expect, at least, partial conservation of the “empty bisectors” effect seen in Fig. 3. The field amplitude dependence visualized in Fig. 4 is likely to remain as well, as it has a clear connection with the existence of a multiple ionization knee, which is a well established effect for both the 3D and 1D models. The behavior represented in Fig. 5 is a curious finding of our particular model, and it not possible to predict its vitality in the 3D case.

5. Summary

The three-electron coincidence scheme appears to be an important tool for studying multiple electron ionization dynamics. We have proposed a possible experimental scheme for detecting electronic momenta after triple ionization of atoms by strong femtosecond laser pulses and have shown how the obtained data can be processed, visualized and analyzed using Dalitz plots. We have demonstrated how differences in the ionization of atoms with different spin symmetry of electronic valence shell can manifest themselves in such plots. The work in progress will analyze in detail the dependence of the Dalitz plots on the carrier-envelope phase which would provide a more detailed understanding of the ionization dynamics involving three active electrons - our preliminary studies indicate a significant sensitivity of Dalitz plots to the choice of the carrier-envelope phase. Identifying different regions of Dalitz plots with different processes (as in the discussion of Fig. 2) may allow direct estimates of the different processes yields. Also, possible resonant processes in multielectron excitations, particularly with short wavelength radiation should directly affect the Dalitz plot representations. Our simulations indicate a need for a theoretical, semiclassical model allowing one for a detailed interpretation of the Dalitz plots for three electron strong-field ionization.