Introduction

Sequential multi-photon processes have attracted attention with the recent advent of X-ray free-electron lasers (XFELs)1,2,3,4,5. XFELs provide ultraintense and ultrashort X-ray pulses6 with high degrees of typically linear polarisation7,8. Exposed to such an intense X-ray pulse, an atom absorbs more than one photon predominantly via sequences of one-photon ionisation events9,10 accompanied by Auger-Meitner decay or X-ray fluorescence11,12. As a consequence of these so-called X-ray multi-photon ionisation dynamics9, high atomic charge states are formed within a single X-ray pulse10,11,12,13,14,15. Because this unavoidably damages the electronic structure of the irradiated sample16,17,18,19,20, applications of XFELs such as biomolecular imaging21,22,23,24,25,26,27,28,29 are affected. Therefore, understanding X-ray multi-photon ionisation dynamics is critical. For atoms, they have been widely explored theoretically and/or experimentally based on ion10,11,12,13,14,15,30,31,32,33,34,35,36, electron12,35,36,37, and photon12,14,33,36 spectra. Complementary information can be obtained by studying the electron cloud alignment of atomic ions, but this requires computationally expensive quantum state-resolved descriptions of atomic ions and X-ray-induced transitions36,38,39.

An alignment of the electron cloud of atomic ions (hereinafter: ion alignment) with nonzero orbital-angular momentum is induced by photoionisation with different ionisation probabilities for different orbital-angular-momentum projection states40,41,42,43,44,45,46. As a consequence of this ion alignment, subsequently emitted Auger-Meitner electrons and fluorescence photons are anisotropically distributed46,47,48,49,50,51 and fluorescence photons are also polarised47,52,53. Angular distributions of photoelectrons emitted from the aligned ions are likewise affected54 supplementary to their fundamental anisotropy55,56,57. On the one hand, all these make ion alignment experimentally accessible42,43,44,51,53. On the other hand, angle-resolved spectroscopy experiments58,59,60,61 may profit from its theoretical study. However, how ion alignment is affected by X-ray multi-photon ionisation dynamics driven by an intense X-ray pulse is so far unknown.

Comparably strong alignments have been encountered in strong-field ionised62,63 and resonantly-excited64 atoms. Further, sequential double and triple ionisations driven by extreme ultraviolet (XUV) pulses have been investigated via photoelectron angular distributions54,60,61,65,66. Recently, alignment in XUV-ionised39 and X-ray-ionised38 atoms up to Kr3+39 and Ar3+38, respectively, has been theoretically explored with a focus on a single ionisation step. In all these studies, multiple competing sequences of photoionisation and accompanying decay processes were not systematically involved, either because they are absent or to simplify computations. But, they matter for the interaction with intense X-ray pulses.

Extending a former study38, we here investigate the time evolution of ion alignment during an intense linearly polarised X-ray pulse. To this end, we performed state-resolved X-ray multi-photon ionisation dynamics calculations36,38 for atomic argon (Ar), which simulate the time evolution of individual quantum-state populations. An individual quantum state is defined by an electronic configuration together with quantum numbers (L, S, ML, κ). Here, L is the total orbital-angular momentum with projection ML, S is the total spin-angular momentum (whose projection is irrelevant), and κ guarantees uniqueness. Individual quantum states differing only in ML form an energy level, defined by an electronic configuration together with a term 2S+1L(κ). Enabled by these calculations, not only time-resolved charge-state distributions and electron and photon spectra but also ion orbital- and spin-angular momentum and alignment are explored. We demonstrate that ion alignment (averaged over all individual quantum states populated at a given time) shows a highly non-linear behaviour and is suppressed by X-ray multi-photon ionisation dynamics. However, we find that the degree of alignment reached is not negligible for the individual atomic charge states populated.

Results

State-resolved X-ray multi-photon ionisation dynamics for Ar are calculated for an X-ray pulse having a temporal Gaussian pulse envelope of 10 fs FWHM (full width at half maximum) and a fluence (not volume-integrated) of 1012 photons per μm2 (unless noted otherwise). These are typical XFEL pulse parameters1,2,4, in particular for atomic, molecular, and optical physics instrumentations67,68,69,70. Such a high fluence has been realised not only in a series of gas-phase XFEL experiments on atoms, molecules, and clusters10,13,31,37,71,72, but also in many serial crystallography experiments27,28,29. Even higher fluences are required for single-particle imaging at XFELs16,19,20,29. We consider X-rays being linearly polarised along the z axis (quantisation axis) and intentionally chose Ar at a photon energy of 1.5 keV to increase the possibility of ion alignment induced by non-trivial ionisation dynamics. At this photon energy, there is no K-shell (1s) ionisation, which cannot create ion alignment without electron correlation effects39. Instead, L-shell (2s and 2p) photoionisation in Ar initiates non-trivial X-ray multi-photon multiple ionisation, including M-shell (3s and 3p) electrons. The target system can be any atomic species, including open-shell systems, when the above conditions are fulfilled. We chose neutral Ar atoms because corresponding gas-phase samples are straightforward to produce for experiments, and it is beneficial to have zero alignment at the beginning.

Ion, electron, and photon spectra

In order to examine the X-ray multi-photon ionisation dynamics of Ar driven by an intense linearly polarised X-ray pulse, we present in Fig. 1 calculated electron and photon spectra and charge-state distribution.

Fig. 1: Overview of X-ray multi-photon ionisation dynamics of Ar at a photon energy of 1.5 keV, a fixed fluence of 1012 photons per μm2, and a pulse duration of 10 fs FWHM.
figure 1

a Time-resolved photoelectron spectrum. The temporal shape of the X-ray pulse is depicted with a grey-shaded area. b Time-resolved Auger-Meitner electron spectrum. c Time evolution of the charge-state distribution. d Time-integrated fluorescence spectrum. In panels a, b, and d, double arrows indicate the subspectra, defined by the involved subshells, i.e., L1 (2s), L2,3 (2p), M1 (3s), and/or M2,3 (3p). In b and d, the first subshell index refers to the initial vacancy, whereas the latter refer to the subshells from which the electrons are removed. Single arrows indicate ions involved in the transition via initial charge state, involved electronic configuration(s) (holes indicated), and/or terms (2S+1L). For all panels, the statistical errors obtained via bootstrap estimate are too small to be shown.

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During the intense X-ray pulse, several photons per atom are absorbed sequentially. Consequently, a wealth of spectral lines characterises the time-resolved photoelectron spectrum in Fig. 1a. Here, the vertical and horizontal axes are the time and energy, using 0.4-fs and 1-eV bins, respectively. The photoelectron spectrum is grouped into large L-shell and minor M-shell spectra. Each subspectrum is dominated by lines for the ionisation of neutral Ar, occurring early in time. These lines are followed in time by a flat quasi-continuum, moving toward lower energy and weaker lines with time. The quasi-continua are generated by photoionisation of ions produced by preceding ionisation and/or decay processes. These intermediate ions, generally, exhibit a manifold of relatively weak state-to-state transitions between different energy levels.

L-shell ionisation is mostly immediately accompanied by Auger-Meitner decay as demonstrated by the time-resolved Auger-Meitner electron spectrum in Fig. 1b. Similar to the photoelectron spectrum, the Auger-Meitner electron spectrum exhibits two energetically-separated quasi-continua, the L–LM and L–MM spectra. Here, an Auger-Meitner channel L–MM means that an initial L-shell vacancy is filled by an electron from the M-shell, accompanied by the ejection of another electron from the M-shell. In the L–LM channel the initial L-shell vacancy decays such that in the final state, one electron has been removed from a higher-lying L-shell orbital and the other electron has been removed from the M-shell. The L–LM and L–MM spectra are generated by a plethora of energy levels involved in the given channels. The L–LM spectrum dominates at early times up to around 5 fs, but is energetically forbidden later. In contrast to the L–LM spectrum, the L–MM spectrum extends to times beyond the X-ray pulse due to the long lifetimes of some intermediate ion states. Nonetheless, most Auger-Meitner decays ( ≈ 95%) take place within 40 fs after the peak of the X-ray pulse.

As a result of sequences of one-photon ionisation accompanied by Auger-Meitner decay, charged ions are formed as shown by the time-resolved charge-state distribution in Fig. 1c. Here, the sum of probabilities Pq of all charge states q (q = 0, +1, ⋯ , +16) is unity at each time. As can be seen, charged ions are formed sequentially as time goes by, with most changes before and around the peak of the X-ray pulse. When the X-ray pulse is over, most ions ( ≈ 78% at 40 fs) have reached a charge of +8 to +12.

The highly-charged ions relax via fluorescence into stable states when the X-ray pulse is over and Auger-Meitner decay is suppressed by a lack of M-shell electrons. This is depicted in Fig. 1d by the time-integrated fluorescence spectrum, generated by Ar7+ to Ar13+. Fluorescence is a very weak process, especially without K-shell holes. The associated lifetimes range from 674 fs (Ar15+ 1s22s02p03s03p1) to 8 × 105 fs (Ar15+1s22s02p13s03p0) (state-resolved fluorescence rate calculations38). Therefore, fluorescence predominantly occurs for highly-charged ions after the end of the X-ray multi-photon ionisation dynamics. Only about 1% of fluorescence decays take place within 40 fs after the peak of the X-ray pulse.

Orbital- and spin-angular-momentum distributions

The time-resolved orbital-angular-momentum distribution is of interest because alignment parameters consider ions with definite orbital-angular momentum L. In Fig. 2a, we plot the time-resolved probability PL of ions with definite L (L = 0, …, 4), which is given by all individual quantum states with quantum number L populated at a given time. Initially, neutral Ar is in an S-state (L = 0) and is included in PL=0. Induced by the X-ray pulse, neutral Ar turns into ions in P-states (L = 1; up to  ≈ 43%). In subsequent sequences of one-photon ionisation accompanied by Auger-Meitner decay, also D-states (L = 2; up to 38%), a few F-states (L = 3;  < 16%), and little G-states (L = 4;  < 3%) are formed at the expense of S- and P-states. F- and G-states are rare because unstable electronic configurations with several holes simultaneously in the 2p- and 3p-shells are required. As a consequence of their instability, they (stepwise) decay into stable S- and P-states. Note that these decays occur on time scales much longer than the time window in Fig. 2a. In the end (when all decay processes are completed) many P-states, some S- and D-states, but neither F- nor G-states are left over (Table 1). In Fig. 2b, the mean of the orbital-angular-momentum distribution \(\bar{L}={\sum }_{L}L{P}_{L}\) and its width \(\scriptstyle{\Delta }_{L}=\sqrt{{\sum }_{L}{L}^{2}{P}_{L}-{\bar{L}}^{2}}\) are shown. The maximum mean and width are reached shortly after the peak of the X-ray pulse (at about 4 fs), which stresses that at this time the maximum number of ions with L > 1 is achieved.

Fig. 2: Time evolution of ion orbital-angular momentum L and spin-angular momentum S of Ar induced by an intense XFEL pulse with a fixed fluence of 1012 photons per μm2.
figure 2

a Probability PL of individual L. b Mean of the orbital-angular-momentum distribution together with its width (area around mean). c, d The spin-angular-momentum distribution is presented likewise (with S = 5/2, … ,4 collected in one line). The temporal pulse shape is depicted by the grey-shaded areas in a and c. For all panels, the statistical errors obtained via bootstrap estimate are too small to be shown.

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Table 1 Orbital- and spin-angular-momentum distributions when the pulse is over and all decay processes are completed
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The time-resolved spin-angular-momentum distribution, shown in Fig. 2c, d, scrutinises the spin S (S = 0, 1/2, …, 4) instead of L. Similar to the orbital-angular-momentum distribution, the majority of ions is found in low non-zero spin states (S = 1/2, 1, 3/2) during and after the X-ray pulse. The maximum mean spin is also reached shortly after the peak of the X-ray pulse. The difference is that the spin-angular-momentum distribution barely evolves after the X-ray pulse is over (compare Fig. 2c and Table 1). This is because Auger-Meitner and fluorescence decays change S only by 1/2 or not at all, respectively (non-relativistic selection rules). Consequently, decays from some very high spin states, generally having low L, are spin-forbidden.

Ion alignment dynamics

Before defining the time-dependent alignment parameter, let us explore in Fig. 3a–d the time-resolved distribution of orbital-angular-momentum projections ML for the given L. For ions with definite L, the probabilities \({P}_{{M}_{L}}\) of ML = −L, …, +L are obtained by summing up probabilities of all individual quantum states with ML for the given L and at a given time. Note that \({P}_{{M}_{L}}={P}_{-{M}_{L}}\) because no orientation38,45 is created by linearly polarised X-rays. For the dominating P-state (L = 1), we observe in Fig. 3a that at the beginning of the X-ray pulse and the X-ray multi-photon ionisation dynamics, P-states are aligned along the X-ray polarisation direction (\({P}_{{M}_{L} = 0} > {P}_{{M}_{L} = \pm 1}\)). In the course of X-ray multi-photon ionisation dynamics, their alignment is reduced (\({P}_{{M}_{L} = 0}\) decreases, whereas \({P}_{{M}_{L} = \pm 1}\) increases), until an almost uniform distribution (\({P}_{{M}_{L} = 0}={P}_{{M}_{L} = \pm 1}=1/3\)) is reached shortly after the peak of the X-ray pulse (at about 3 fs). For higher L (Fig. 3b–d), no uniform distribution (the dashed line) is reached after the X-ray pulse and, consequently, their alignment is less reduced (which is further discussed later). The statistical errors are larger than those for L = 1 due to smaller numbers of realisations.

Fig. 3: Alignment dynamics of Ar ions induced by an intense XFEL pulse.
figure 3

a–d The distribution of orbital-angular-momentum projections ML of ions with definite orbital-angular momentum L is shown by their probabilities \({P}_{{M}_{L}}\) as a function of time at a fixed fluence of 1012 photons per μm2. The dotted grey lines at 1/(2L + 1) indicate a uniform distribution. e–h From the probabilities \({P}_{{M}_{L}}\) the alignment parameter A20 for a definite L is calculated and compared for three different fluences. i The mean alignment parameter \({\bar{A}}_{20}\), averaged over all L, is shown for the same fluences. In all panels, the error bars every 2 fs indicate the statistical error obtained via bootstrap estimate. Insufficient statistics are avoided by the restriction PL≥0.005 at each time step (a–h). The temporal pulse shape is sketched by grey shading in d and i.

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The degree and direction of alignment is described by the alignment parameter A20, which is also applicable to ion ensembles. Based on the probabilities \({P}_{{M}_{L}},{A}_{20}\) for ions with definite L is defined as38,45,73,74

$${A}_{20}(L)=\sqrt{\frac{5}{{f}_{L}}}\times {\sum}_{{M}_{L}}\left[3{M}_{L}^{2}-L(L+1)\right]{P}_{{M}_{L}},$$
(1)

where fL = (2L + 3)(L + 1)L(2L − 1). The alignment parameter is negative (A20 < 0), when ions with smaller ∣ML∣ are more populated than others, and positive (A20 > 0), when ions with larger ∣ML∣ are more populated. A20 = 0 indicates a uniform distribution. The meaning of A20 is further elaborated on in the next section, and extreme values of A20 are listed in Table 2. The A20 calculated with the time-resolved probabilities \({P}_{{M}_{L}}\) in Fig. 3a–d is shown for the given L in Fig. 3e–h (blue line for a fixed fluence of 1012 photons per μm2). For P-states, A20 is initially about  − 0.16, which is already not an extremely strong alignment compared to the perfect alignment of \({A}_{20}=-\sqrt{2}\) (Table 2). With time ∣A20∣ decreases almost to zero. For higher L, A20 is also negative, but with only a weakly declining trend in time.

Table 2 Perfect alignment (A20 < 0) and anti-alignment (A20 > 0) where all ions have either ML = 0 or ML = ∣L∣, respectively
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Figures 3 e–h also show how the alignment of each L depends on the X-ray fluence applied (number of incoming X-rays per unit area9). Fluence-dependencies are important in volume integration75, necessary for quantitative comparisons with experiments. The fluence of 1012 photons per μm2, considered so far, is compared with two lower fluences: 5 × 1011 photons per μm2 and 1011 photons per μm2. No strong fluence-dependence of A20 is observed because ionisation dynamics are mainly only reduced and delayed with lower fluence. For lower fluences, therefore, the evolution of A20 is delayed, so that the alignment is less reduced with time. Thus, the saturation value reached after the X-ray pulse is sensitive to the X-ray fluence applied, and its absolute magnitude decreases as the fluence increases. We observe that the absolute magnitude of this saturation value is enhanced with increasing L, which is to be further analysed later. As a result, for L > 1, alignment is not negligible for all fluences, in contrast to P-states.

Figure 3i demonstrates what happens to the ion alignment when averaging over L. It shows the mean alignment parameter \({\bar{A}}_{20}={\sum }_{L}{P}_{L}\times {A}_{20}(L)\) (unaligned neutral Ar included). As can be seen, \({\bar{A}}_{20}\) exhibits a highly non-linear behaviour. Before the X-ray pulse, neutral Ar is unaligned (\({\bar{A}}_{20}\)=0). At the beginning of the X-ray pulse, p-shell ionisation of neutral Ar creates a negative ion alignment (\({\bar{A}}_{20} < 0\)) and \(| {\bar{A}}_{20}|\) increases as more ions are produced. Around the peak of the X-ray pulse, \(| {\bar{A}}_{20}|\) is maximised (\({\bar{A}}_{20}\approx -0.047\) for 1012 photons per μm2). At this time, most neutral Ar is already ionised (see Pq=1 in Fig. 1c). If there were only single photoionisation, \(| {\bar{A}}_{20}|\) would continue increasing up to \({\bar{A}}_{20}\approx -0.12\) (ensemble of Ar1+ produced in 59% by p-shell ionisation). However, due to progressing photoionisation accompanied by Auger-Meitner decay involving mainly P- and D-states, \(| {\bar{A}}_{20}|\) noticeably decreases. A saturation value is reached at the end of the X-ray pulse, which is clearly sensitive to the X-ray fluence applied. On the one hand, the higher the fluence, the more \(| {\bar{A}}_{20}|\) is reduced compared to its maximum because of enhanced X-ray multi-photon ionisation dynamics. On the other hand, for low fluence (1011 photons per μm2), only around 50% of the atoms are ionised at all and, thus, the maximum of \(| {\bar{A}}_{20}|\) is very small. As a consequence, to see a maximum alignment, the fluence should be large enough to ionise most atoms ( > 1011 photons per μm2), but should not induce too much X-ray multi-photon ionisation dynamics ( < 1012 photons per μm2).

To analyse the ion alignment further, we present in Fig. 4a–d charge-resolved alignment parameters \({A}_{20}^{q}\) for different L. Note that \({A}_{20}(L)={\sum }_{q = 1}^{16}{P}_{q}^{L}{A}_{20}^{q}(L)\) with the charge-state probabilities \({P}_{q}^{L}\) for definite L (\(\mathop{\sum }_{q = 1}^{16}{P}_{q}^{L}=1\)). \({P}_{q}^{L}\) evolves sequentially in time and charge similar to Pq (Fig. 1c) though lack of initially neutral Ar causes much larger \({P}_{q}^{L}\) for low q at early times. \({A}_{20}^{q}\) is clearly non-zero and exhibits very different values and signs for different q and L. For P-states, intermediately-charged ions are weakly aligned (\({A}_{20}^{q}\, < \,0\)) or weakly anti-aligned (\({A}_{20}^{q}\, > \,0\)) along the X-ray polarisation (Fig. 4a). But Ar1+ produced early by single 2p or 3p ionisation38 is clearly aligned along the X-ray polarisation (\({A}_{20}^{q = 1}\approx -0.21\)) and Ar14+ and Ar15+ produced less and later are clearly anti-aligned (\({A}_{20}^{q = 14}\approx +0.05\)). For P-states, only \({P}_{{M}_{L} = 0}\) and \({P}_{{M}_{L} = \pm 1}\) determined the A20 value. \({P}_{{M}_{L} = 0}\, > \,{P}_{{M}_{L} = \pm 1}\) results in a negative A20, while \({P}_{{M}_{L} = 0}\, < \,{P}_{{M}_{L} = \pm 1}\) gives a positive A20. Typically, p-shell ionisation preferentially increases \({P}_{{M}_{L} = 0}\)38. On the other hand, the ML = 0 → ML = 0 transition is prohibited when the sum of ion and involved bound electron angular momenta is odd, which effectively increases \({P}_{{M}_{L} = \pm 1}\). The interplay of these two tendencies combined with the manifold of parallel and competing state-to-state transitions reduces the degree of alignment and changes its direction, already for Ar2+. For comparison, a single (unaveraged) sequence of 2p ionisation and Auger-Meitner decay leads to \({A}_{20}^{q = 2}\approx 0.1\)38. In contrast to P-states, \(| {A}_{20}^{q}|\) for L > 1 is larger and becomes clearly positive only for very highly charged ions (q ≥ 12). This is attributed to a reduced effect of prohibited transitions on the alignment when the ions involved in a transition have higher L. It also explains why in Fig. 3 a non-vanishing alignment is observed for L > 1 at long times in contrast to L = 1. In Fig. 4e, the charge-resolved alignment parameter \({\bar{A}}_{20}^{q}\) averaged over all L is presented for selected q. From Figs. 3i and 4e, we conclude that the reduction of ion alignment is mostly the result of averaging over all L and q. Our results suggest that L-resolved or q-resolved quantities are beneficial for detecting ion alignment dynamics.

Fig. 4: Differently charged Ar ions demonstrate different degrees and directions of alignment.
figure 4

a–d The charge-resolved alignment parameter \({A}_{20}^{q}\) for different orbital-angular momenta L. Red and blue indicate \({A}_{20}^{q} > 0\) and \({A}_{20}^{q} < 0\), respectively. e The average over all L for selected charge states q. Grey shading indicates the temporal pulse shape. Error bars obtained via bootstrap estimate are shown only in e. In a–d, they are not shown, but are slightly larger than those in e. Insufficient statistics are avoided by the restriction \({P}_{q}^{L}\ge 0.005\) at each time step. A fluence of 1012 photons per μm2 is used.

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Spatial shapes of aligned electron clouds

To develop some intuition about ion alignment, we show in Fig. 5 spatial shapes of electron clouds for selected ions produced at some exemplary times t. At the times selected, the corresponding ion yield becomes the largest during the time evolution; the corresponding alignment parameters as a function of time are given in Figs. 3 and 4. In Fig. 5a, we consider ions with different L, but average over all q. The spatial shape of the electron cloud, averaged over all projections ML, is determined by using the quantity

$$| {\bar{Y}}_{L}(\theta ){| }^{2}=\mathop{\sum }_{{M}_{L}=-L}^{L}{P}_{{M}_{L}}{\left\vert {Y}_{L}^{{M}_{L}}(\theta ,\phi )\right\vert }^{2}.$$
(2)

Here, \({Y}_{L}^{{M}_{L}}\) is a spherical harmonic (squaring eliminates the ϕ-dependence) and \({P}_{{M}_{L}}\) is the time-resolved probability for definite L at time t (Fig. 3a–d). In Fig. 5b, we consider ions in a few selected charge states q, but average over all L. In this case, the spatial shape of the electron cloud, \(| {\bar{Y}}_{q}(\theta ){| }^{2}\), is the L-average over all \(| {\bar{Y}}_{L}(\theta ){| }^{2}\) [Eq. (2)] after replacing the \({P}_{{M}_{L}}\) with the \({P}_{{M}_{L}}^{q}\) underlying \({A}_{20}^{q}\) (Fig. 4). Taking the vertical axis as the X-ray polarisation (z) axis, we observe for most selected electron clouds an oval, prolate shape (corresponding to A20 < 0). Only highly-charged ions (q ≥ 12) exhibit an oblate shape (A20 > 0). For a few selected electron clouds (L = 4 in Fig. 5a and q = 1, 15 in Fig. 5b) the deviations from a uniform distribution (circle; dotted grey lines in Fig. 5) are quite pronounced and clearly visible in the polar plots.

Fig. 5: Spatial shape of the electron cloud of Ar ions produced by an intense XFEL pulse at specific points t in time.
figure 5

a Electron-cloud shape \(| {\bar{Y}}_{L}{| }^{2}\) for ions with definite orbital-angular momentum L (charge-averaged) at time t with maximum PL (Fig. 2a). b Electron-cloud shape \(| {\bar{Y}}_{q}{| }^{2}\) for ions in a few selected charge states q (L-averaged) at time t with maximum Pq (Fig. 1c). The insets in both panels show polar plots of a few selected electron-cloud shapes. The vertical axis (θ = 0 rad and θ = π rad) corresponds to the X-ray polarisation axis. A fluence of 1012 photons per μm2 is used.

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Discussion

The present paper presents a complete state-resolved description of X-ray multi-photon ionisation dynamics, including electron-cloud alignment in an XFEL pulse. The calculated alignment parameter for Ar1+ (\({A}_{20}^{q = 1}\approx -0.21\)) acceptably agrees with previous works38,44,54, taking into account the photon-energy dependence. Due to X-ray multi-photon ionisation, the magnitude of the alignment parameter after averaging over L and q becomes relatively small ( < 0.05) as depicted in Fig. 3i. However, when we analyse the alignment parameter for individual L and q (Fig. 4a–d), individual L (Fig. 3e–h), and individual q (Fig. 4e), its magnitude is larger than the averaged value. Non-trivial ion alignment dynamics for individual quantum states are predicted. We expect that this result can be generalised to other atomic species (ionised under suitable conditions), photon energies (below the respective K-shell threshold), and (femtosecond) pulse durations. Observing electron-cloud alignment dynamics induced by XFEL pulses requires few-femtosecond- or even attosecond-resolved measurements sensitive to ML quantum-state distributions.

Desirable are time-resolved measurements of Auger-Meitner electron angular distributions42,44,48,49,76 generated in XFEL experiments. Best suited for this propose are transitions involving only a single continuum wave for the Auger-Meitner electron48,49, e.g., final ions with zero L. Then, the anisotropy parameter47,50,77,78,79,80 is directly proportional to the alignment parameter for the initial ion. Future developments in this direction will enable deeper insights into electron-cloud alignment in an XFEL pulse, including the impact of effects so far neglected, such as non-dipole effects54,66,81 or spin-orbit coupling62,63,64,82. A further possibility is to apply circularly polarised X-ray pulses8,83,84,85,86. Even though the direction of alignment changes40, based on the present results, we expect similar electron-cloud alignment dynamics during a circularly polarised X-ray pulse. However, how the orientation induced by the circularly polarised X-ray pulse behaves is an open question.

Lastly, we would like to comment on electric alignment (orientation) in molecular targets induced by XFEL pulses. Here, we demonstrate that the polarisation of the X-ray beam is imprinted on the atomic electron-cloud dynamics on ultrafast time scales. We anticipate a similar impact on the electron-cloud dynamics in molecules. However, this depends in detail on the strength of interatomic interactions and remains to be investigated in a future study.

Methods

For X-ray multi-photon ionisation dynamics calculations, we employ XATOM20,87,88, which has been successfully applied for interpreting XFEL experiments11,12,13,14,15,31,32,34,35,89. XATOM has recently been extended to state-resolved ionisation dynamics with a Monte Carlo implementation36, based on a non-relativistic quantum-state-resolved electronic-structure framework38 for isolated atoms. For any given electronic configuration, it provides zeroth-order states with quantum numbers (L, S, ML, κ) and term-specific first-order-corrected energies ELSκ. On this basis, state-to-state transition parameters for photoionisation, Auger-Meitner decay, and fluorescence are calculated from first principles. Since they are independent of the spin projection MS, MS is neglected in this work.

A Monte Carlo on-the-fly rate-equation method11,12 is employed for describing the time evolution of individual quantum-state populations. The number of individual quantum states involved is already very large for argon36, i.e., 216, which would be challenging without a Monte Carlo on-the-fly approach. Next to the Monte Carlo method a couple of simplifications are critical to efficiently perform calculations. We made several approximations: non-sequential two-photon absorption34,90, higher-order many-body processes such as double photoionisation91 and double Auger-Meitner decay92, coherent effects93,94,95,96, inter-channel coupling97,98, relativistic effects32,99, finite-nuclear-size effects99, and non-dipole effects100,101 are not included in this work.

Due to the Monte Carlo method all presented results contain statistical errors. Statistical errors are obtained via bootstrap estimate102 using 50 bootstrap samples of 200,000 Monte Carlo trajectories. For the results in Figs. 1 and 2, errors are two orders of magnitude smaller than the dominant quantities themselves and are, consequently, omitted. The calculated errors become quite large, particularly for propagation at early times and for large L in Fig. 3, because the number of ions of interest is too small. To avoid such insufficient statistics, results in Figs. 3 and 4 are only plotted when the underlying ion probability is ≥0.005. We expect that the overall trend shown in Figs. 3 and 4 will remain unchanged when more Monte Carlo trajectories are used.