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

www.nature.com/scientificreports OPEN Breakdown of the electric dipole approximation at Cooper minima in direct two-photon ionisation J. Hofbrucker1,2,3*, A. V. Volotka1,2 & S. Fritzsche1,2,3 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 sources1,2. Nowadays, it is also possible to probe these systems in nonlinear regime with free-electron lasers3,4. That is why, in the last decade, much of experimental efforts5–15 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 Zernik16, 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 role17–25. With the possibility of producing intense high-photon energy beams, the multiphoton inner-shell ionisation came in focus as well26–29. 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 ionisation30. In single-photon ionisation, effects beyond the electric dipole approximation have been explored at Cooper minima31. 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 distributions32. It has been shown theoretically before33,34, that strong anisotropic effects can be observed near Cooper minima due to relativistic and correlation effects, which are necessary to explain experimental measurements32,35,36. 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 minimum37. It has been observed, that a similar Cooper minimum is also present in total cross sections of multi-photon ionisation processes38–40, where it appears in a form of a local minimum. In our recent work41, 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) and (n′ + 1)(l + 1) virtual intermediate resonances, 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 ml < l − 1 get ionised. The photoion in this case appears to be in an aligned state, which, in the 1 Helmholtz-Institut Jena, Fröbelstieg 3, D-07743, Jena, Germany. 2GSI Helmholtzzentrum für Schwerionenforschung GmbH, Planckstrasse 1, D-64291, Darmstadt, Germany. 3Theoretisch-Physikalisches Institut, Friedrich-SchillerUniversität Jena, Max-Wien-Platz 1, D-07743, Jena, Germany. *email: Scientific Reports | (2020) 10:3617 | https://doi.org/10.1038/s41598-020-60206-z 1 www.nature.com/scientificreports www.nature.com/scientificreports/ case of inner-shell ionisation, is imprinted in the polarisation degree of subsequent fluorescence light41. 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 photoelectron42,43 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 s1/2 → p1/2 → εd3/2 in the electric dipole approximation, while both electrons can be ionised over an intermediate p3/2 state. Consequently, passing through a Cooper minimum between a pair of n′p3/2 and (n′ + 1)p3/2 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 ion5–8, or fluorescence10–14 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 theore (...truncated)