Introduction

Strong-field ionization of atoms in circularly polarized (CP) or close-to-CP intense laser fields is of fundamental interest and has widespread applications in attoclock technique, in which the periodically spinning carrier of a CP field acts as the minute hand while the slowly varying pulse envelope can also serve as the hour hand, providing an ultrafast chronoscope for the ionization dynamics1. Therefore, it is crucial to fully explore and understand the underlying physics which, however, is to some extent still puzzling. At high laser intensities where the electrons are expected to be deprived one by one, i.e, via the sequential ionization, the ionization time was found to deviate from the prediction of the widely accepted single-active-electron approximation2. Even more appealing is the nonsequential double ionization (NSDI) which proceeds in three steps: a preliminarily released electron, subsequently driven back by the oscillating laser field, finally dislodges another electron3,4. With the help of multi-particle coincidence detection techniques, exotic signatures of the field-intervened electron-electron (e-e) correlation5,6,7 or anti-correlation8,9 during the NSDI process have been revealed in experiment and the underlying mechanisms are subsequently explored by classical10, semiclassical11,12,13, and quantum simulations14,15,16,17.

Traditionally, it is believed that NSDI is more favorable in linearly polarized (LP) fields but would be significantly suppressed in CP fields because the released electron might spiral away from the parent ion due to a considerable drift momentum. This scenario was verified in some early experiments18 but not in others19, and thus had aroused a surge of debate during the past years20,21. It was later realized that the initial tunneling momentum might compensate the drift momentum for the alkaline-earth atoms, based on which a phase diagram was predicted, indicating what kind of atoms under which laser parameters can NSDI take place22,23.

Stepping forward to the nonsequential multiple ionization (NSMI), while most of the early experiments were performed with the rare-gas atoms (see, e.g., refs. 24,25,26,27,28,29), the combination of the laser cooling technique and the strong-field reaction microscope has offered the opportunity to extend the discussions to the family of alkali atoms recently. Up to now, however, only the single30,31, double32, and sequential multiple ionizations33 have been explored. In this paper, we focus on the NSMI of alkali atoms under CP radiations and find an shepherd electron effect34. We demonstrate our theory with the rubidium (Rb) atom that has versatile applications in precision measurements35 and Bose-Einstein condensation36, while the physical picture is general and can be extended to other species of atoms. The essential idea is that a CP field can be applied to successively strip several electrons, among which the rescattering of the inner-shell electrons are strongly suppressed while the outermost one (referred to as the shepherd electron thereafter) is stabilized on an elliptical orbit. This effectively produces a vacancy in the atom, therefore with the return of the shepherd electron some intriguing effects can be induced. First, the shepherd electron may further dislodge the inner-shell electrons by recollisions. Second, if no recollision occurs, it can be recaptured after some other inner-shell electrons are ionized as such the Coulomb focusing effect of the parent ion becomes stronger. As a result, we predict a double-knee structure on the ion yield curve that is universal from double to quintuple ionizations. The knee structure in the regime below the saturation (BS) intensity is related to recollision while the above-threshold (AS) one is attributed to frustrated multiple ionization. The latter also leaves footprints with a long tail (or a halo structure) on the recoiled-ion momentum spectrum and can be detected by the attoclock technique.

Results and discussion

Double-knee structure

Our study is facilitated by a classical trajectory Monte Carlo approach with Heisenberg potential (CTMC-H, see Methods), which well describes the multi-shell structure of alkali-metal atoms33,37,38. The CTMC-H model has achieved great success in solving the ionization time puzzle in sequential double ionization of Ar39, interpretation of the kinetic energy release (KER) spectrum of laser-driven dissociation of D\({}_{3}^{+}\) molecule40, as well as quantitatively reproducing the ionization cross section in traditional atomic collisions41 and thus explaining the Wannier threshold law42 and spin polarization43, to name only a few. As demonstrated in Fig. 1a, our classical atom highly resembles the shell structure of Rb, i.e., the outermost 5s electron is much further from the nucleus than the other six 4p electrons which arrange pairwise around the nucleus and occupy the six corners of an octahedron. We exploit the CTMC-H model to simulate the multi-electron dynamics of the laser-driven Rb system. The laser electric field takes the form

$${{{{{{{\boldsymbol{{{{{{{{\mathcal{E}}}}}}}}}}}}}}}}(t)=\frac{{{{{{{{{\mathcal{E}}}}}}}}}_{0}f(t)}{\sqrt{1+{\epsilon }^{2}}}\left[\epsilon \sin (\omega t){\hat{{{{{{{{\boldsymbol{e}}}}}}}}}}_{x}-\cos (\omega t){\hat{{{{{{{{\boldsymbol{e}}}}}}}}}}_{y}\right],$$
(1)

with \({{{{{{{{\mathcal{E}}}}}}}}}_{0}\), ω and ϵ denoting its amplitude, frequency, and ellipticity, respectively. The envelope function is given by \(f(t)={\sin }^{2}\left(\pi t/T\right)\) and T represents the full pulse duration. In our simulations, we consider a maximum of seven active electrons because the laser intensities are not strong enough to ionize the 4s-shell and other even more deeply bound electrons.

Fig. 1: Double-knee structure on the ion yield curve and the associated physical mechanisms.
figure 1

a Schematic illustration of a model Rb atom with seven active electrons irradiated by a circularly polarized laser field, manifesting the shepherd electron effect. b Ion yield of Rbn+(n = 1, 2, 3, …) as a function of the peak laser intensity at three different laser wavelengths, i.e., λ = 3200, 800, and 337 nm, respectively. Here, the results obtained by simulations with seven (4p65s1) active electrons (solid curves) are compared to that with only six (4p6, dashed curves) or five (4p5, dotted curves) active electrons. Such comparison disentangles the decisive role of the inter-shell and intra-shell e-e correlations. c A typical trajectory of quintuple ionization, in which the outermost shepherd (5s, black) electron is driven back and forth at the vicinity of the inner-shell electrons, dislodging the herd (4p electrons) either by soft [e2 (red) and e3 (green)] or hard [e4 (blue) and e5 (violet)] recollisions. The spacial configuration at the instant of the recollision [shaded area of (c)] is shown in (d), where the contour plot shows the effective potential Ω. e and f Demonstrate a typical trajectory of the frustrated quadruple ionization, in which, after the striping of three 4p electrons, the 5s shepherd electron is recaptured by the parent ion during the ramp-off of the external laser field. The gray dashed line in (e) represents the adiabatic solution \({r}_{0}=\sqrt{2}{{{{{{{{\mathcal{E}}}}}}}}}_{0}f(t)/{\omega }^{2}\).

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Figure 1b shows the ion yield of Rbn+(n = 1, 2, 3, . . . ) as a function of the peak laser intensity when the atom is irradiated by a CP field with a full pulse duration of twenty optical cycles. The results are compared at three different wavelengths. In the long-wavelength limit (λ = 3200 nm), it can be seen that each ion yield curve has the same tendency: a sharp ratchet-up followed by a plateau (saturation regime) and again a sharp drop-off. If the wavelength is decreased to λ = 800 nm, a universal double-knee structure appears for all the ionization channels, see the solid curves. The double-knee structure retains at λ = 337 nm and seems more asymmetric.

To reveal the possible physical mechanisms behind the sensitive wavelength-dependent ionization dynamics, we carry out two additional groups of simulations with only six (6e, 4p6) or five (5e, 4p5) active electrons and compare them with the full simulations with seven active electrons (7e, 4p65s1). In the 6e model, we intentionally remove the outermost 5s electron and only consider the laser interaction with the remained six electrons. The 5e model is similar except that another 4p electron is also removed. As shown in Fig. 1b, the characteristic double-knee structure at 800 nm totally disappears in the 6e simulations (dashed curves). We thus infer that the outermost electron is responsible for the NSMI of all the inner-shell electrons, which is different from the rare-gas atoms, in which other channels such as the cascade NSMI are expected to dominate the ionization processes, i.e., the double, triple, and quadruple ionizations respectively induced by recollision of the firstly, secondly, and thirdly released electrons and so on44. The situation becomes different at λ = 337 nm, for which the removal of the 5s electron strongly suppresses the knee-structure in the BS regime but does not change much of the reversed knee-structure in the AS regime. With the further removal of a 4p electron, the double-knee structure again totally disappears. This indicates that both the inter-shell and intra-shell electron correlations play indispensable roles in this case, therefore the correlated electron dynamics is more complicated.

Taking full advantage of the classical trajectory back-analysis22, we can achieve more insight into the underlying mechanisms, please refer to Fig. 1c–f for two types of typical trajectories. We first convert the system into the frame that rotates with the time-dependent CP field45, following the transformation \(u=x\sin \omega t-y\cos \omega t\), \(v=x\cos \omega t+y\sin \omega t\), and w = z. In the rotating frame (u, v, w), the laser field turns out to be an quasi-static electric field. Consequently, the converted system exhibits a conserved quantity \(J={\dot{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}^{2}/2+{{\Omega }}\), which is known as Jacobi’s integral. Here, \(\dot{{{{{{{{\boldsymbol{\sigma }}}}}}}}}=(\dot{u},\ \dot{v},\ \dot{w})\) represents the momentum of the electron in the rotating frame,

$${{\Omega }}=-\frac{1}{r}-\frac{{\omega }^{2}{r}^{2}}{2}+\frac{{{{{{{{{\mathcal{E}}}}}}}}}_{0}uf(t)}{\sqrt{2}}$$
(2)

is the effective potential energy, and J imposes a constraint on the permissible phase space for electron motion. By setting \(\dot{{{{{{{{\boldsymbol{\sigma }}}}}}}}}=0\), an implicit definition of a surface is obtained, wherein the velocity is equal to zero. This surface is referred to as the zero-velocity surface or Hill surface. Figure 1 demonstrates the projection of the Hill surface onto the (u, v) plane through surface or contour plots, in which the parent ion is situated at the origin, creating a deep well around it and is encompassed by a region of higher potential. A maximum of the potential energy landscape can be found approximately at the position \((u={{{{{{{{\mathcal{E}}}}}}}}}_{0}/{\omega }^{2},v=0)\), situated to the right of the parent ion. While at the left side, there exists a saddle point where electrons can cross over the Coulomb barrier.

In the rotating frame, we find that the shepherd electron adiabatically follows the Ω = 0 surface. This gives rise to an estimation of the electron trajectory r0 (gray dashed curve in Fig. 1e)

$${r}_{0}\approx \frac{\sqrt{2}{{{{{{{{\mathcal{E}}}}}}}}}_{0}f(t)}{{\omega }^{2}}.$$
(3)

With the adiabatic ramp on and off of the laser field, the Hill surface firstly expands and then shrinks, and the shepherd electron is gradually pulled out and sent back again to the vicinity of the nucleus. The return of the shepherd electron can trigger two possible processes. First, it might recollide with the inner-shell electrons3 and lead to the NSMI as shown in Fig. 1c, d, which is responsible for the strong enhancement of the ion yield of Rbn+(n = 2, 3, 4, …) as evidenced by the characteristic knee structures in the BS regime46. Despite the shepherd electron has a negligible drift momentum since it is released during the onset of the laser pulse and therefore experiences a small vector potential, it is subsequently driven back and forth at the vicinity of the parent ion by the oscillating laser field and its kinetic energy upon recollision can reach up to \({U}_{p}={{{{{{{{\mathcal{E}}}}}}}}}_{0}^{2}/4{\omega }^{2}\), the ponderomotive energy. Taking again the trajectory shown in Fig. 1c for example, the energy of the returning electron is about 200 eV (at t ≈ 8.5 o.c., see Fig. 2a, b for the corresponding energy evolution), which provides a sufficient amount of energy for the ionization of several inner-shell electrons. Second, after the exfoliation of some inner-shell electrons, the Coulomb focusing effect47 of the parent ion becomes much stronger than what the electron can experience at the very beginning of its leave. So the electron has a large probability to be recaptured by the parent ion, leading to the frustrated MI, see Fig. 1e, f. This is the dominant mechanism for the reversed knee structure in the AS regime. We have conducted a statistical analysis on the return time of the shepherd electron for the reversed knee. As illustrated in Fig. 2c, most of the shepherd electrons are recaptured during the last two or three laser cycles when the laser field is turned off. Here, we would like to note that the shepherd electron effect seems similar to but not a simple variation of the conventional frustrated tunnel ionization (FTI) process48,49: For FTI, it has been shown in prior works that the recapture process takes place predominantly in linearly polarized light but vanishes for circularly polarized light; in contrast, the shepherd electron effect happens in circularly polarized light. It is due to ionization of the outermost electron in the rising edge of the pulse similar to the mechanism demonstrated in ref. 23, whereas the shepherd electron effect on NSMI was not revealed therein.

Fig. 2: Some temporal information of the shepherd electron.
figure 2

a Energy evolution of the electrons corresponding to the typical trajectory shown in Fig. 1c. b The compensated energy66 as a function of time, which is negative for t ≤ 8.5 o.c. so that one can say the shepherd electron remains in the excited state. c Time distribution of the shepherd electron when it is recaptured by the Rb4+ ion. The laser intensity is 5 × 1015 W/cm2.

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Hallow atom and the Attoclock signature

In addition to the double-knee structure on the ion-yield curve, the shepherd electron effect also leaves footprints on the ion momentum spectrum. Taking Rb3+ at I = 5 × 1015 W/cm2 as an example, we plot the momentum distribution (Px, Py) in the polarization plane, as shown in Fig. 3. The momentum distribution displays a donut shape mainly consisting of two rings (Fig. 3a): one colored red and centered close to the zero plus another concentric ring colored yellow with higher momentum33. Besides, there also exist some scattered events beyond \({P}_{r}=\sqrt{{P}_{x}^{2}+{P}_{y}^{2}} \, > \, 6\) a.u. This halo structure is not observed in the 6e simulations (Fig. 3b), again pointing to the pivotal role of the shepherd electron in producing the high-energy ion, which can be more clearly seen in the momentum distribution of Pr, as shown in Fig. 3c. Here, a double-hump structure, with one hump at Pr ≈ 1 a.u. and the other one at Pr ≈ 5 a.u. [corresponding to the inner and outer rings in the (Px, Py) distribution, respectively] can be found both in the 6e and 7e simulations. However, the momentum cuts off at Pr ≈ 6 a.u. for the 6e simulations, but extends up to Pr ≈ 10 a.u. for the 7e simulations.

Fig. 3: Halo structure on the recoiled-ion momentum spectrum and the real-time observation with attoclock technique.
figure 3

The momentum distributions of Rb3+ in the polarization plane at the laser intensity of 5 × 1015W/cm2, as predicted by the (a) seven- (4p65s1) and (b) six- (4p6 only) active-electron models, respectively. The laser wavelength λ is 800 nm. The dashed white circles correspond to Pr = 6 a.u. to guide the eyes. c Shows the distribution of \({P}_{r}=\sqrt{{P}_{x}^{2}+{P}_{y}^{2}}\), where a long tail can be seen from the 7e simulation but absent in the 6e case. This is due to the frustrated multiple ionization, see text for details. d The ionization time distributions of the firstly (black), secondly (red), and thirdly (green) released electrons for the whole ensemble of Rb3+ (scatters), in comparison with those events that end up with high-ion-momentum Pr > 6 a.u. (shaded areas). e, f The same as (c), but for different laser wavelengths.

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The above results can be explained by the simple-man model associated with the over-barrier-ionization (SMOBI), in which the electrons are assumed to be released when the laser field reaches the OBI threshold. Therefore, the absolute value of their momenta can be estimated as \({p}_{en}={{{{{{{{\mathcal{E}}}}}}}}}_{n}/\omega\), where \({{{{{{{{\mathcal{E}}}}}}}}}_{n}={I}_{pn}^{2}/4n\) is the critical field strength leading to the OBI of the n-th electron, and Ipn is the n-th ionization energy of the Rb atom50. Because the outermost electron can be easily ionized during the rising edge of the laser pulse, its momentum is extremely small and thus can be ignored. Therefore, the ion momentum of triple ionization mainly depends on the second and third ionized electrons: The inner ring structure results from the electrons ionized in the opposite direction (Pr ≈ pe3 − pe2), while the outer ring structure is caused by the electrons ionized in the same direction (Pr ≈ pe3 + pe2)51. This explains the first two humps in Fig. 3c, which are the common features of the 7e and 6e simulations. In contrast, we find that the long tail of Rb3+ momentum spectrum that is only observed in the 7e simulations seems to be contributed by three 4p ionized electrons, i.e., Pr ≈ pe2 + pe3 + pe4.

The remained question is how a triply ionized Rb atom acquires such a large energy and is related to the fourth electron? The answer lies in the typical trajectory of Fig. 1e, where after the subsequent ionization of the e2, e3, and e4, the shepherd electron is driven back to the origin when the laser field is ramped off and is recaptured by the Rb4+ to form an excited Rb3+* ion, therefore the recoil ion momentum is equal to the sum of three 4p electrons’ momenta. Due to the trapping of the outermost electron into the Rydberg states, the Rb3+* ion has the same charge mass ratio with the Rb3+ ion and thus can not be distinguished from them, which accounts for the relatively slow decrease of the Rb3+ yields as compared with the results of the 6e model. This consolidates our findings on the pivotal role of the shepherd electron as the dominant mechanism underlying the reversed knee structure.

Since the shepherd electron is pumped into and stabilized on an elliptical orbit even after the deprive of some inner-shell electrons (recall the dashed curve in Fig. 1e at t > 10 o.c.), this effectively produces a transient vacancy in the atom, similar to the hollow atom triggered by X-rays52. In the XUV case, a hollow atom can be produced only when the photoionization rate is comparable to or faster than the Auger decay rate. In our case, the efficiency is ensured by the long life time of the highly excited shepherd electron. Furthermore, for our hollow atom is produced by the absorption of multiple NIR photons, it might have applications in coherent XUV light amplification by means of high-order harmonic emission.

Real-time observation of the shepherd electron effect can be realized with the attoclock technique, in which the field envelope provides a hour hand for the coarse timing of the ionization, with a direct mapping of the momentum to the ionization time of the electrons2. In Fig. 3d, we show the ionization time distributions of the triple ionization, which is constituted of three separated peaks representing the ionization sequence of the three ionized electrons. One might also notice that there exists a sub-peak (shaded areas) following each main peak. These side-band structures are related to those ions with high recoil momentum Pr > 6 a.u., for which the 5s electron is recaptured and the three sub-peaks indeed correspond to the ionization times of three 4p electrons.

As a double-check of the proposed mechanism, we have also calculated the ion momentum spectra at the other two different wavelengths, as shown in Fig. 3e, f, respectively. In the long-wavelength limit, the momentum spectra calculated with the 6e and 7e models are almost coincident and the corresponding halo structure totally vanishes, signifying the disappearance of the shepherd electron effect. On the other side, i.e., in the short-wavelength limit, the 7e calculation yields a momentum spectrum that can even extend to Pr ≈ pe3 + pe4 + pe5, which is significantly broader than the 6e calculation. Back-analysis of the classical trajectories shows that there is a small probability that both the e1 and e2 electrons are recaptured.

Phase diagram

After identifying the dominant mechanisms, below we make some analytical discussions to obtain a more general and intuitive picture, based on which a phase diagram will be presented, showing the parametric space that our predictions could be observable. For this purpose, we first assume that the field-driven ionizations most probably take place at the rising edge of the pulse envelope when the field amplitude reaches the OBI threshold and the corresponding electrons are released at the local maximum (along the major axis) of an elliptically polarized (EP) laser field, as demonstrated in Fig. 4a. At this time, the field amplitudes along the major and minor axes are \({{{{{{{{\mathcal{E}}}}}}}}}_{y0}={I}_{pn}^{2}/4n\) and \({{{{{{{{\mathcal{E}}}}}}}}}_{x0}=\epsilon {{{{{{{{\mathcal{E}}}}}}}}}_{y0}\), respectively. We want to examine if these field-released electrons can be driven back to the parent ion to induce the shepherd electron effects. In the classical scenario, the ionized electron has a random initial transverse (perpendicular to the major axis) momentum that obeys the Gaussian distribution \(W({p}_{\perp })\propto \exp [-{\scriptstyle{\sqrt{2{I}_{pn}}}}{p}_{\perp }^{2}/{{{{{{{{\mathcal{E}}}}}}}}}_{y0}]\)53; if the initial transverse momentum is able to compensate the drift momentum imposed by the laser field, i.e., \({p}_{{{{{{{\mathrm{drift}}}}}}}}={{{{{{{{\mathcal{E}}}}}}}}}_{x0}/\omega\), then the ionized electron has a considerable probability to return to the vicinity of parent ion. In our following calculations, we take \({p}_{\perp }^{c}=3{\scriptstyle{\sqrt{{{{{{{{{\mathcal{E}}}}}}}}}_{y0}}}}{I}_{pn}^{-1/4}\) as the maximal transverse momentum of the tunneled electron (other fast electrons with momentum larger than \({p}_{\perp }^{c}\), only constituting ≈ 0.00004% of the whole electron ensemble, are too rare to be considered), therefore if \({p}_{\perp }^{c} < {p}_{{{\mbox{drift}}}}\) then one can assert that all the electrons can not return and no shepherd electron effect would be observed. With the above consideration, we can obtain a criterion

$${I}_{pn}={n}^{2/5}{(\epsilon /6\omega )}^{-4/5},$$
(4)

from which one can estimate the critical ellipticity or frequency of the laser field that the ionized electron can return.

Fig. 4: Phase diagram.
figure 4

a Sketch of the ionization geometry. b Phase diagram for observing the shepherd electron effects, where the regions I, II, and III indicate no, one, and two (or more) electrons can return, respectively. The laser parameters are ϵ = 1 and λ = 800 nm. This phase diagram is the multi-ionization extension of our previous Fig. 4 of ref. 22, in which, the recollsion phase diagram only depends on the atomic first ionization potentials (and the laser parameters of course). However, in the present phase diagram, taking Li and Sr for example, even though they have almost identical first ionization potentials, they fall into different regimes and thus are expected to demonstrate quite different strong-field multiple ionization behaviors.

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The phase diagram is presented in the parameter plane of the scaled ionization energy \({\tilde{I}}_{pn}={n}^{-2/5}{I}_{pn}\) and the energy gap \({{\Delta }}{I}_{p}={\tilde{I}}_{p(n+1)}-{\tilde{I}}_{pn}\) as shown in Fig. 4b, which is divided into three regions, corresponding to no (region I), one (region II), and two (or more, region III) electrons can return. In region II the double-knee structure can be most clearly observed and unambiguously attributed to the inter-shell electron correlation, while in region III, because two or more electrons can revisit the parent ion, the situation becomes even more complicated. We put the alkali-metal, alkaline-earth-metal, and rare-gas elements onto the phase diagram. Clearly, they are located in different regions and, in particular, the alkali-metal elements are deep in region II and thus are good candidates to observe the shepherd electron effects. The phase diagram can be applied to explain the wavelength-dependence of the ion yield curves in Fig. 1: In the long-wavelength limit, the boundary of region II shifts leftward so that the Rb atom enters region I and thus the double-knee structure can not be observed; the results in the short-wave limit can be explained similarly as the competition between two electrons that can return. For Rb, this implies that both the inter-shell and intra-shell electron correlations play indispensable roles. Furthermore, since in our above derivation, no assumptions on the ellipticity of the laser field were made, we expect the theory can be extended to the more general EP case54 to explore the ellipticity-dependent shepherd electron effects in the future.

Finally, we attempt to compare our theory with the existing experiments. We first notice that a similar reversed knee structure in the AS regime was observed in strong-field multiple ionization of Ar in CP fields55. However, the results there were explained within the shake-up scenario, namely, tunnel ionization of one electron results in simultaneous excitation of the other one or more remaining electrons. In our case, we can safely exclude this scenario because the shepherd electron is initially located far away from the other inner-shell electrons, therefore its departure from the atom has negligible impact on the other electrons, as can be clearly seen in Fig. 1c, e at the very beginning of the laser pulse. More recently, the momentum distributions of the Rb ions were measured at a laser intensity of 3.3 × 1015W/cm 2 33, in which the halo structure can be only seen vaguely. Due to the present limited experimental resolution, more MI events need to be collected and the influence of the laser focal geometry should be avoided, similar to ref. 55, in order to identify the structure more unambiguously.

In summary, we have proposed and achieved insight into the shepherd electron effect in multiple ionization of atoms by a circularly polarized intense laser field, in contrast to the conventional cascade nonsequential multiple ionization. The shepherd electron effect above the ionization saturation regime is rather similar to the hollow atom triggered by X-rays, in the sense that after the release of some inner-shell electrons, the outermost electron is still bound or recaptured by the parent ion. This phenomenon might have applications in coherent XUV light amplification. In experiments, the shepherd electron effects can be traced in real-time with the attoclock technique. Moreover, in recent years, the magneto-optical trap recoil ion momentum spectroscopy has been proposed and equipped in some state-of-the-art experiments, exhibiting the ability to measure the full-dimensional momenta of multiple reaction products56. We hope that our theoretical predictions can also be observed in this kind of experiment.

Methods

Ab initio study of NSMI is currently a formidable task even with high-performance supercomputers, actually only available for two- or three-electron systems such as He and Li under limiting conditions, e.g., interaction with an ultrashort XUV laser pulse57 or in a reduced-dimensional Hilbert space58. We therefore resort to classical trajectory simulations similar to that pioneered in ref. 59. The difference is that we have adopted a Heisenberg potential that allows to extend the simulation to a larger system with seven or more active electrons33,60,61. The Heisenberg potential is adopted for the electron-nucleus interaction, which takes the form \({V}_{{{{{{{{\rm{H}}}}}}}}}({{{{{{{{\boldsymbol{r}}}}}}}}}_{i},{{{{{{{{\boldsymbol{p}}}}}}}}}_{i},{\xi }_{i})={\xi }_{i}^{2}/(4\alpha {{{{{{{{\boldsymbol{r}}}}}}}}}_{i}^{2})\cdot \exp \{\alpha [1-{(\vert {{{{{{{{\boldsymbol{r}}}}}}}}}_{i}\parallel {{{{{{{{\boldsymbol{p}}}}}}}}}_{i}\vert /{\xi }_{i})}^{4}]\}\)37,38. Consequently the total Hamiltonian of the atomic system can be written as

$${H}_{0}=\mathop{\sum }\limits_{i=1}^{N}\left[\frac{{{{{{{{{\boldsymbol{p}}}}}}}}}_{i}^{2}}{2}-\frac{Z}{| {{{{{{{{\boldsymbol{r}}}}}}}}}_{i}| }+{V}_{{{{{{{{\rm{H}}}}}}}}}({{{{{{{{\boldsymbol{r}}}}}}}}}_{i},{{{{{{{{\boldsymbol{p}}}}}}}}}_{i},{\xi }_{i})\right]+\mathop{\sum }\limits_{i,j=1;i < j}^{N}\frac{1}{| {{{{{{{{\boldsymbol{r}}}}}}}}}_{i}-{{{{{{{{\boldsymbol{r}}}}}}}}}_{j}| }.$$
(5)

Here, we have restricted our simulations to seven active electrons and set N = Z = 7 because the laser intensities explored in this work are not strong enough to ionize the 4s-shell and other even more deeply bound electrons. The variables ri and pi respectively denote the position and momentum of the electron i (i = 1 ⋯ ⋯ N), and α serves as a parameter to modulate the stiffness of the potential. Meanwhile, the optimal potential parameters ξi are determined by fitting the ionization potentials of the Rb atom33, i.e., Ip1 = 4.18 eV, Ip2 = 27.29 eV, Ip3 = 39.25 eV, Ip4 = 50.20 eV, Ip5 = 68.44 eV, Ip6 = 82.90 eV, and Ip7 = 98.67 eV. To this end, we add the electrons one by one, and the system energy is minimized at each step and fitted to the experimental values. This gives rise to the atomic parameters ξ1 = 2.5053, ξ2 = 2.4784, ξ3 = 2.4413, ξ4 = 2.4216, ξ5 = 2.3205, ξ6 = 2.2707, and ξ7 = 2.4520. Correspondingly, the distances from the electrons to the nucleus are r1 = 5.77, r2 = 1.43, r3 = 1.29, r4 = 1.24, r5 = 1.02, r6 = 0.94, and r7 = 1.33. The electron configuration is demonstrated in Fig. 1a, which highly resembles the shell structure of a Rb atom. We further apply random Euler rotations

$${{{{{{{\mathcal{A}}}}}}}}=\left[\begin{array}{lll}-{{{{{{{\mathcal{S}}}}}}}}\varphi {{{{{{{\mathcal{S}}}}}}}}\eta +{{{{{{{\mathcal{C}}}}}}}}\varphi {{{{{{{\mathcal{C}}}}}}}}\theta {{{{{{{\mathcal{C}}}}}}}}\eta &-{{{{{{{\mathcal{S}}}}}}}}\varphi {{{{{{{\mathcal{C}}}}}}}}\eta -{{{{{{{\mathcal{C}}}}}}}}\varphi {{{{{{{\mathcal{C}}}}}}}}\theta {{{{{{{\mathcal{S}}}}}}}}\eta &{{{{{{{\mathcal{C}}}}}}}}\varphi {{{{{{{\mathcal{S}}}}}}}}\theta \\ {{{{{{{\mathcal{C}}}}}}}}\varphi {{{{{{{\mathcal{S}}}}}}}}\eta +{{{{{{{\mathcal{S}}}}}}}}\varphi {{{{{{{\mathcal{C}}}}}}}}\theta {{{{{{{\mathcal{C}}}}}}}}\eta &{{{{{{{\mathcal{C}}}}}}}}\varphi {{{{{{{\mathcal{C}}}}}}}}\eta -{{{{{{{\mathcal{S}}}}}}}}\varphi {{{{{{{\mathcal{C}}}}}}}}\theta {{{{{{{\mathcal{S}}}}}}}}\eta &{{{{{{{\mathcal{S}}}}}}}}\varphi {{{{{{{\mathcal{S}}}}}}}}\theta \\ -{{{{{{{\mathcal{S}}}}}}}}\theta {{{{{{{\mathcal{C}}}}}}}}\eta &{{{{{{{\mathcal{S}}}}}}}}\theta {{{{{{{\mathcal{S}}}}}}}}\eta &{{{{{{{\mathcal{C}}}}}}}}\theta \end{array}\right]$$
(6)

to the above configuration to obtain an ensemble of electrons with varying initial positions and momenta, which is implemented by the Monte Carlo sampling technique. Here, φ, η, and θ\((0\le \varphi \le 2\pi ,0\le \eta \le 2\pi ,-1\le \cos \theta \le 1)\) are the Euler angles while \({{{{{{{\mathcal{S}}}}}}}}\) and \({{{{{{{\mathcal{C}}}}}}}}\) represent the sine and cosine functions, respectively. In practice, considering the symmetry of the Hamiltonian H0, all electrons’ position vectors are rotated with the same Euler angles and the momentum of each electron is rotated independently.

Having created a micro-canonical ensemble of classical atoms, we then exploit the CTMC-H model to simulate the dynamics of the laser-driven Rb system \(H={H}_{0}+\mathop{\sum }\nolimits_{i = 1}^{N}{{{{{{{\boldsymbol{{{{{{{{\mathcal{E}}}}}}}}}}}}}}}}(t)\cdot {{{{{{{{\boldsymbol{r}}}}}}}}}_{i}\) by solving the canonical equations dri/dt = ∂H/∂pi, dpi/dt = − ∂H/∂ri numerically with the standard 4–5th Runge-Kutta algorithm. The ionization events are identified according to the final energies of the electrons. For a given laser parameter, more than 106 classical trajectories are traced to ensure the convergence of the statistical results. After running a large ensemble of trajectories, ionization probabilities \({{{{{{{{\mathcal{P}}}}}}}}}_{n}\) are calculated as \({{{{{{{{\mathcal{P}}}}}}}}}_{n}={{{\Xi }}}_{n}/{{{\Xi }}}_{{{{{{{{\rm{tot}}}}}}}}}\), where Ξn is the number of ionization events ending up with the final charge state Rbn+ and Ξtot is the total number of trajectories simulated62. The ion momentum spectra shown in Fig. 3 are obtained from the final momentum vectors of the ionized electrons according to the momentum conservation law.

Before closing this section, we would like to make some additional remarks regarding the CTMC-H model utilized in our simulations. Firstly, there are various procedures for preparing the classical trajectory ensembles (see, e.g., refs. 63,64), in which the distances from the electrons to the core are not fixed and can vary. However, the procedure by Abrines and Percival63 is more suitable for a single-electron system but might suffer from unphysical autoionization (ionization even without the laser field) when extended to multi-electron situations. On the other hand, the pilot model64 can partially alleviate the unphysical autoionization problem due to the soft-core parameter adopted for the core potential. Yet it is restricted to two- or three-electron systems. Therefore neither of these methods can be applied to the current studies on NSMI of Rb, which involves up to seven electrons. In the current CTMC-H model, the radii of the electrons are fixed which correspond to the stable points of the phase space, and thus can effectively prevent the unphysical autoionization without a laser field. Furthermore, the Heisenberg potential is more flexible in optimizing the ionization energies of the electrons, which is crucial for obtaining the expected atomic shell structure and reasonable ionization cross-sections61.

Secondly, this paper aims at demonstrating some intriguing phenomena due to the concept of shepherd electron. We note that a recent paper has shown that the specific form of the potential may quantitatively influence both the ionization yield and the ion momentum distribution65. Nevertheless, in the section of Phase Diagram, our analytical theory that does not rely on the specific form of the ionic potential seems to explain the numerical results very well, which suggests that the general pictures of shepherd electron are not sensitive to the exact form of the core potential. Looking forward, it would be interesting to quantitatively compare the results predicted by different models in a future study.