Breakdown of the electric dipole approximation at Cooper minima in direct two-photon ionisation

We predict breakdown of the electric dipole approximation at nonlinear Cooper minimum in direct two-photon K–shell atomic ionisation by circularly polarised light. According to predictions based on the electric dipole approximation, we expect that tuning the incident photon energy to the Cooper minimum in two-photon ionisation results in pure depletion of one spin projection of the initially bound 1s electrons, and hence, leaves the ionised atom in a fully oriented state. We show that by inclusion of electric quadrupole interaction, dramatic drop of orientation purity is obtained. The low degree of the remaining ion orientation provides a direct access to contributions of the electron-photon interaction beyond the electric dipole approximation in the two-photon ionisation of atoms and molecules. The orientation of the photoions can be experimentally detected either directly by a Stern-Gerlach analyzer, or by means of subsequent Kα fluorescence emission, which has the information about the ion orientation imprinted in the polarisation of the emitted photons.

The interest in the inner-shell dynamics of atoms and molecules has been rising ever since excitation and ionisation of the strongly bound electrons became accessible by the first XUV and x-ray light sources 1,2 . Nowadays, it is also possible to probe these systems in nonlinear regime with free-electron lasers 3,4 . That is why, in the last decade, much of experimental efforts [5][6][7][8][9][10][11][12][13][14][15] have been paid to studying the fundamental properties of nonlinear light-matter interaction, and finding use in applied fields such as nonlinear spectroscopy. The theoretical developments were in many respects following the experimental trail. Starting with the pioneering work of Zernik 16 , who calculated the nonresonant two-photon ionisation cross section for hydrogen, and continuing with further theoretical development into direction of many-photon absorption and ionisation of complex atoms. In particular, significant progress has been made in the description of the two-photon ionisation of outer-shell electrons of noble gas atoms, where electron correlations play a significant role [17][18][19][20][21][22][23][24][25] . With the possibility of producing intense high-photon energy beams, the multiphoton inner-shell ionisation came in focus as well [26][27][28][29] . In these relativistic regimes, the wavefunction contraction has been found to play an important role, while contributions of higher multipoles remained to be generally less important, similarly as in the outer-shell ionisation 30 .
In single-photon ionisation, effects beyond the electric dipole approximation have been explored at Cooper minima 31 . Such a Cooper minimum arises at a incident photon energy, where the dominant dipole element passes through a local minimum. This minimum does not only influence the shape of the total cross section, but more significantly, can strongly affect the photoelectron angular distributions 32 . It has been shown theoretically before 33,34 , that strong anisotropic effects can be observed near Cooper minima due to relativistic and correlation effects, which are necessary to explain experimental measurements 32,35,36 . Moreover, large nondipole contribution has been predicted in the XUV + IR two-photon above-threshold ionisation of neon 1s electron, when XUV photon energy matches the Cooper minimum 37 .
It has been observed, that a similar Cooper minimum is also present in total cross sections of multi-photon ionisation processes [38][39][40] , where it appears in a form of a local minimum. In our recent work 41 , we showed that for the two-photon ionisation of an nl (l > 0) electron, these nonlinear Cooper minima can be always found between any pair of n l ( 1) where the dominant channel l → l + 1 → ε(l + 2) vanishes. Two-photon ionisation at this minimum is then described by the channel(s) l → l − 1 → εl, and in the case of ionisation by circularly polarised laser, only the electrons with orbital momentum projections m l < l − 1 get ionised. The photoion in this case appears to be in an aligned state, which, in the www.nature.com/scientificreports www.nature.com/scientificreports/ case of inner-shell ionisation, is imprinted in the polarisation degree of subsequent fluorescence light 41 . As the nonlinear Cooper minima are a property of the fundamental two-(or generally multi-)photon ionisation process itself, they are expected to strongly influence many other observables such as photoelectron 42,43 or Auger electron polarisation, as well as their angular distributions.
In this paper, we consider the case of atomic K-shell ionisation (l = 0). In this case, the ionisation by circularly polarised light proceeds only via the single nonrelativistic channel s → p → εd, and therefore, all relative characteristics such as photoion polarisation and photoelectron angular distribution stay always the same. However, when we account for the spin orbit interaction, only the s electron with one of the initial spin projections can be ionised via the single channel s 1/2 → p 1/2 → εd 3/2 in the electric dipole approximation, while both electrons can be ionised over an intermediate p 3/2 state. Consequently, passing through a Cooper minimum between a pair of ′ n p 3/2 and + ′ n p ( 1) 3/2 resonances, one might expect a pure depletion of one of the spin projections of the initial s electron. However, this conclusion holds true only in the electric dipole approximation. Here, we will show that in the case of two-photon ionisation of K-shell at nonlinear Cooper minima, accounting for beyond-dipole contributions becomes inevitable. In other words, the fragile spin nature of the nonlinear Cooper minimum of a fine-structure channel gives us the opportunity to access multipole contributions in nonlinear light-matter interaction processes. To demonstrate the breakdown of the electron-dipole approximation on examples, we propose similar conditions as considered in the recent experiments, where either ion 5-8 , or fluorescence 10-14 yields were detected as a signature of two-photon K shell ionisation. However, instead of solely detecting the yields, we suggest to additionally carry out measurements of degree of polarisation of photoions or fluorescence photons.
To explain the suggested scenario in detail, we start with theoretical description of two-photon ionisation of s-electrons with the use of density matrix theory. We use this theoretical approach to demonstrate the breakdown of the electric dipole approximation at nonlinear Cooper minimum on examples of nonsequential two-photon ionisation of neutral germanium atoms and helium-like neons ion by right-circularly polarised light.

Methods
In what follows, we will consistently use the typical many-electron notation in which J represents the total angular momentum, M its projection and α all further numbers that are needed for unique characterisation of an atomic state. Corresponding lower case notation is used for one-electron notation. We consider K-shell ionisation of an atom in an initial many-electron state α J M i i i by two right-circularly polarised photons γ with energy ω. After the interaction, a photoelectron p m e e , with asymptotic momentum p e and spin projection m e , is emitted from the atom, leaving it in an excited state J M We can schematically represent this process as with the notation [J] = (2J + 1). The two photons are assumed to come from the same source, and hence, have equivalent wavevector k, as well as polarisation, which can be conveniently described in helicity (λ) representation. Since we consider right-circularly polarised photons, their polarisation state can be described by a single Stokes parameter P 3 denoting their degree of polarisation 44 The two-photon ionisation process can be analysed by measuring the properties of the emitted photoelectrons 9 , or the photoions [5][6][7][8] . In most experiments, however, the two-photon K-shell ionisation of atoms has been detected by measuring subsequent fluorescence light [10][11][12][13][14] . We will, therefore, also examine this second process, in which the excited atom J M f f f α | 〉 relaxes into a lower energy state α J M 0 0 0 by emission of fluorescent photon γ 0 with energy ω Kα In the next subsection, we will first provide a description for the simultaneous interaction of the atom with two photons. Then, we will derive the density matrices of the produced ion as well as of the fluorescence photon and analyse their properties. The results in this paper have been obtained within independent particle approximation, however, recently published many-electron code based on multi-configuration Dirac-Fock method promises easy-to-use tools for calculations of multi-photon absorption or emission processes 45 . Relativistic units (ℏ = c = m = 1) are used throughout the paper, unless otherwise stated.

Two-photon transition amplitude.
In second-order perturbation theory, the absorption of two photons is described by the many-electron transition amplitude n n n n 2 1 where n 1 and n 2 are single-electron states, and † a n , a n represent the corresponding creation and annihilation operators, respectively. Moreover, α μ is the four-vector Dirac matrices and A ( ) ω λ μ is the photon wavefunction. According to the particle-hole formalism, a state with a hole in a substate s m 1 a 1/2 has the same angular momentum properties as an electron with total angular momentum projection −m a . Within the independent particle approximation, the final many-electron state can be obtained using the creation and annihilation operators as † a a e e a 1/2 with typical Clebsch-Gordan notation .. . . . . , . For future analysis, it is convenient to perform the decomposition of the photon field into its electric (p = 1) and magnetic (p = 0) components, with a multipole order J and its projection M The term J = p = 1 represents the electric dipole approximation, while other values of J and p will be refered to as contributions beyond the electric dipole approximation. Further simplification of the transition amplitude can be achieved by expanding the photoelectron wavefunction into its partial waves 47 In order to calculate the above transition amplitudes, we solve the Dirac equation with a screening potential, which partially accounts for the inter-electronic interaction. We use the core-Hartree potential, which corresponds to a potential created by all bound electrons except of the active electron.
There are specific incident photon energies ω between two intermediate level resonances E E n j n j n n n n ω < < ′ for which one of the amplitudes ( ) U 0 l j j n = , due to balance of the its spectral contributions 41 . The photon energy, at which the dominant ionisation channels drops to zero, we call nonlinear Cooper minimum, and we will demonstrate that at this energy all possible observables of the two-photon ionisation process are strongly influenced. Hence, studying the response of atoms at this energy is the key to understanding nonlinear light-matter interaction.
Spin population of the produced ion. Upon the simultaneous absorption of the two photons by the atom, one of the initially bound 1s electrons is promoted into continuum. The state of the system after ionisation α p J M m , f f f e e therefore consists of a singly charged ion in an excited state and a photoelectron. The density matrix of this state can be written as (2020) 10:3617 | https://doi.org/10.1038/s41598-020-60206-z www.nature.com/scientificreports www.nature.com/scientificreports/ Using Eqs. (2) and (3) as well as the fact that we do not consider any detection of the photoelectron, we trace out the photoelectron degrees of freedom from the density matrix (12) and obtain density matrix corresponding to the excited photoion only To get a better feeling about the properties of this observable, let us provide the expression for ion orientation after two-photon ionisation of a 1s of a closed-shell atom in the electric dipole (J 1 = J 2 = 1 and p 1 = p 2 = 1) approximation  10 whenever only one of the spin projections of the K-shell electrons can be ionised. This happens only if the incident beam is fully polarised (P 3 = ±1) and when the ionisation channel U d 3/2 strongly dominates the process. One important example, where this channel dominates is at the nonlinear Cooper minimum. A detailed explanation of this phenomenon will shown together with demonstration of its effects on examples in the results section. (13) fully describes all properties of the excited photoion. However, in the case of inner-shell ionisation, where the created hole is surrounded by electrons in higher orbitals, it is often more convenient to detect the photoion state indirectly via subsequent fluorescent emission. We, therefore, need to describe the radiative decay of the excited ion into its lower energy state J M 0 0 0 α via Kα emission. The photon density matrix describing the fluorescent photon with momentum k 0 in the helicity representation λ 0 has the general form

Properties of subsequent fluorescence photons. Equation
This general expression can be simplified with the use of the photon field of Eq. (8) and the Wigner-Eckart theorem 48 .
We obtain with the polar and azimuthal angles θ and φ with respect to the  z axis, which was chosen to be along the incoming photon direction  k. In the case of two-photon ionisation of an s electron by circularly polarised incident light the dependence on φ vanishes due to symmetry reasons and the degree of circular polarisation of the subsequent fluorescence photon P J 3 ( ) 0 , resulting in a final ion state with total angular momentum J 0 = 1/2 and J 0 = 3/2 corresponding to α K 2 and K 1 α can be written in the electric dipole approximation as

(3/2) 10
The simple expressions above clearly demonstrate, that the fluorescence photon is fully determined by the orientation of the ion and carries the spin polarisation information. Since the Kα 1 and Kα 2 fluorescent photons have different energies, they are experimentally distinguishable 11 . This means that the degrees of circular polarisation corresponding to each radiative decay can be measured and analysed separately. From the above expression, it is clear that detecting the fluorescence light emitted by the decay of the p 1/2 electron will yield larger signal in measurements of the degree of polarisation than the one emitted by the decay of the p 3/2 electron.

Results
In the following section, we study numerically the polarisation of neutral germanium and helium-like neon following the non-sequential two-photon K shell ionisation by right-circularly polarised light. Within the electric dipole approximation, there are only few possible ionisation channels which the active electron can undertake (see Fig. 1). While electrons with both spin-up and spin-down can be ionised from the K shell via the intermediate state with p 3/2 symmetry into a final d 5/2 partial wave, only spin-down electron can be ionised via the p 1/2 or p 3/2 intermediate states to the d 3/2 partial wave. Therefore, under conditions where the channel with final d 3/2 symmetry strongly dominates the process, only the spin-down electron will ionised and the produced ion will be strongly oriented.  (11). Due to the cancellation, the channel involving the final d 5/2 partial wave vanishes and only the spin-down electron will be ionised. Therefore, based on the prediction of the electric dipole approximation, the ion orientation parameter at nonlinear Cooper minimum (as well as at intermediate p 1/2 resonances) should be equal to unity. However, in the examples below, we show that numerical calculation carried out beyond the dipole approximation reveals breakdown of this prediction in the case of Kshell ionisation. The numerical results are obtained in the independent particle approximation with the core-Hartree screening potential. In order to estimate the accuracy, we also perform the calculations with the Perdew-Zunger, Kohn-Sham, and Dirac-Slater potentials [49][50][51] . In all the potentials, the nonlinear Cooper minima, degrees of polarisation as well as the cross sections vary not more than by 10%.

Polarisation of fluorescent light after two-photon ionisation of 1s electron of germanium.
In the following example, we consider ionisation of the K shell electron of neutral germanium (E 1s = 11.1 keV) by two-right-circularly polarised photons, with subsequent detection of Kα fluorescent light. A similar scenario has been already realised with linearly polarised beam 11 at the SACLA free-electron laser. The experimentally determined total ionisation cross section for ω = 5.6 keV is 0 64 10 exp 59 σ ≈ . × − cm 4 s. The uncertainty associated with the experimental value is not mentioned, however, we can assume that it is not better than 50% (typical for cross section data). It is, therefore, in a reasonable agreement with our theoretical result σ theo ≈ 2 × 10 −59 cm 4 s 29 . In our calculations, we consider ionisation of the zero-spin isotopes of germanium (which accounts for about 92% of the naturally occurring germanium) by two photons with energies in the range between 9.5−11 keV. This range covers photon energies matching the 1s → 2p and 1s → 3p intermediate resonances, as well as a nonlinear Cooper minimum between the resonances. The corresponding results are presented in Fig. 2. To guide the eye, the upper plot shows the well-known quantity of total cross section as a function of the incident photon energy, calculated within (dashed, yellow) and beyond (full, black) electric dipole approximation. The lower plot shows the degree of circular polarisation of the subsequent fluorescent photon measured along the axis of the incoming beam, with the same colour notation as in the upper plot. The level resonances in the upper plot describe the sequential ionisation process when the electron from a given shell is ionised by one photon, and simultaneously, the 1s electron is promoted into the hole by the second photon. The logarithmic scale of the figure also reveals the nonlinear Cooper minimum in the total cross section in a form of a local minimum, which lies around ω = 10.35 keV. The position of the nonlinear Cooper minimum could be, therefore, determined from measurements of the total cross section. However, it is also clear from the figure, that apart from dipole forbidden transitions at the 1s → 2s and 1s → 3s resonances, the calculations within the electric dipole approximation are in a very good agreement with the multipole calculations. Hence, it would be very challenging to access information about the multipole transitions from measurements of the total cross section, instead, other observables need to be inspected.
The lower plot of Fig. 2, reveals hitherto elusive information. Specifically, clear deviations in the degree of polarisation appear for either for photon energies matching an intermediate p 1/2,3/2 level (sequential ionisation) resonances, or at nonlinear Cooper minimum. While the degree of circular polarisation at level resonances agrees between electric dipole and multipole calculations, a breakdown of the dipole approximation is clearly visible at the nonlinear Cooper minimum. The high degree of polarisation of the fluorescent photons at this point drops strongly due to the contributions of the generally weak multipole ionisation channels. This breakdown of the electric dipole approximation can be better seen in Fig. 3, which shows the degree of circular polarisation only in the vicinity of the nonlinear Cooper minimum. The left subfigure shows the degree of polarisation of the fluorescence photon within the electric dipole approximation as well as with inclusion of higher multipoles. The exact position of the Cooper minimum of each channel is marked with a dashed vertical lines. The right side of the figure shows the two electric dipole and one (of many) multipole multipole channels, and demonstrates that the multipole contributions dominate the process at the nonlinear Cooper minimum. It is due to the contributions of electric quadrupole transitions that both initial electrons can be ionised, and the produced ion is no longer strongly oriented. As a consequence, the subsequent fluorescent photons also posses low degree of polarisation.
ion orientation after two-photon ionisation of ne 8+ . Since, measurements of fluorescence are not always possible, other methods of measuring the ion orientation need to be carried out. The polarisation state of the ion can be detected directly with a Stern-Gerlach analyser 52 . Here, we consider the ion detection Fig. 1. Possible electric dipole ionisation channels in nonsequential two-photon ionisation of an s 1/2 state by two right-circularly polarised photons. While both spin projections of the initial state can be promoted to a final d 5/2 partial wave, selection rules dictate that only spin-down electron can be ionised into a partial wave with d 3/2 symmetry. This figure has been generated using Mathematica 11.0.0.0 (https://www.wolfram.com/ mathematica/) and Inkscape 0.92 (https://inkscape.org/). (2020) 10:3617 | https://doi.org/10.1038/s41598-020-60206-z www.nature.com/scientificreports www.nature.com/scientificreports/ technique to study a similar scenario as in the experiments by Doumy et al. 7 , who reported the total cross section of two-photon ionisation of Ne 8+ . The experimentally determined cross section was orders of magnitude greater than theoretically predicted values 26,29,53 . We here suggest, that carrying out a similar experiment (but with a measurement of the ion orientation) could allow to test the theoretical agreement with the experiment at higher accuracy, and thus, elucidate the reason of the discrepancy. Figure 4 shows the total cross section as well as the ion orientation of ionisation of Ne 8+ by two right-circularly polarised photons. As seen from this figure, a nonlinear Cooper minimum appears for incident photon energy about 20 eV lower than the one used in ref. 7 . A measurement of this Cooper minimum with circularly polarised beam at this point, would allow one to accurately test theoretical models. The ion orientation posses the same properties as the degree of circular polarisation of fluorescence (see Eq. (19)). It is therefore clear, that also ion orientation will be sensitive to higher multipoles at Cooper minimum. Instead of reviewing the breakdown of the electric dipole approximation, in this example, we demonstrate the sensitivity of the ion orientation to the polarisation purity of the incident beam. The right side of Fig. 4 shows that de-tuning the purity of incident polarisation by mere 0.1% results in a drop of the ion orientation at nonlinear Cooper minimum by around 30%. If the incident beam is only 99% circularly polarised, the ion polarisation decreases to about 20% of the value of pure polarised beam. This high sensitivity of the ion orientation appears uniquely at the nonlinear Cooper minimum, as the influence of other channels increases quickly with polarisation de-tuning. At photon energies matching an intermediate level resonances, the process is effectively determined by one of the channels only, hence, all other channels are negligible. Due to this fact, the ion orientation is no longer extremely sensitive to the de-tuning the polarisation purity of the incident light. The purity of ion orientation at these resonant photon energies could be, however, influenced by the level widths of the fine-structure levels.    7,11 . In contrast to the experiments, we suggest performing the experiments with circularly polarised beams, which are already available at Linac Coherent Light Source (LCLS) at Standford 54 as well as FERMI at Trieste 4 . Moreover, number of other free-electron facilities include polarisation control in their upgrade plans 55,56 . With these experimental possibilities, the nonlinear Cooper minimum can play a key role in detailed understanding of nonlinear light-matter interaction. Moreover, the degree of ion orientation (or degree of circular polarisation of fluorescent light) has been found to be extremely sensitive to the polarisation purity of ionising light and hence could be used for measuring the polarisation purity of free-electron laser beams.

Conclusion
The concept of Cooper minimum was generalised to two-photon ionisation of inner-shell electrons, where it originates from fine-structure splitting of the 2p shell into the 2p 1/2 and 2p 3/2 subshells. The exact positions of the minima can be experimentally detected by measuring the ion, electron or fluorescent yields, and can serve as a sensitive tool to test the agreement between theory and experiment. It is worth noting that the polarisation of the incident light does not play any role in the position of the nonlinear Cooper minimum. The Cooper minima will appear in the total cross sections (and electron, photon or ion yields) for ionisation by beam of arbitrary polarisation.
It has been shown that the nonlinear Cooper minima can be utilised to access the multipole contributions from measurements of polarisation properties of the produced photoion. According to predictions based on the electric dipole approximation, it is expect that tuning the incident photon energy to the Cooper minimum in two-photon ionisation by circularly polarised light results in pure depletion of one spin projection of the initially bound 1s electrons, and hence, leaves the ionised atom in an oriented state. We showed that by inclusion of electric quadrupole interaction, dramatic drop of orientation purity is obtained. This breakdown of the electric dipole approximation can be readily experimentally detected either directly by Stern-Gerlach analyzer, or by means of subsequent Kα-fluorescence, which has the information about the ion orientation imprinted in its polarisation degree. Fig. 4. Ion orientation after nonsequential two-photon ionisation of Ne 8+ by two right-circularly polarised photons. Left: the total cross section as well as the orientation parameter  (1/2) 10 as a function of incident beam energy. The experimentally determined cross section 7 is marked with a red cross. Right: Sensitivity of the ion orientation to the purity of polarisation of the incident photons is shown in detail.