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Abstract: We demonstrate experimentally a new quantum state tomography protocol for photoelectrons. The measured reduced density matrix, originating from different noble gases, shows that the photoelectron purity decreases due to ion–electron entanglement induced by spin–orbit interaction. Opening up the possibility for directly studying quantum properties of photoelectrons on a ultrafast timescale.
Photoelectron spectroscopy has been instrumental in the studying of the quantum nature of atoms. The discovery of the photoelectric effect [1] during the $20^{th}$ century was crucial for the development of quantum mechanics. The advent of attosecond science have further improved the field by measurements, not only of the modulus of the momentum, but also of its phase. Attoscience has opened up the study of dynamics on the electron timescale, but until recently mostly for pure quantum states.
Recent developments in atto-/femtosecond science can now account for mixed quantum states as well [2,3,4]. Allowing the retrieval for the full retrieval of the quantum state. We have developed a protocol called KRAKEN, where reduced density matrix elements are measured by the use of a bichromatic laser field as a probe of the created electron state. By scanning over the pump-probe delay and wavelength separation, $\delta\omega$, the sparse density matrix is retrieved. We have developed a Bayesian machine-learning algorithm based on a Hamiltonian Monte-Carlo method to extract the full density matrix based on the measured sparse density matrix. The results are then benchmarked against independent ab initio calculations. The measured (and theoretical) purities of He, Ne and Ar are $0.97\pm0.11$ (1.00), $0.83\pm0.15$ (0.79) and $0.59\pm0.04$ (0.61) respectively, showing a clear difference between the noble gases, due to spin-orbit interaction and the subsequent electron-ion entanglement.
Future developments aim at decreasing the measurement-time by using a poly-chromatic probe and pushing towards quantum process tomography, opening up towards photoelectron time-resolved studies of quantum properties.
[1] A. Einstein, ̈Uber einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt, Annalen der Physik, 322, 10.1002/andp.19053220607 (1905).
[2] C. Bourassin-Bouchet et al, Phys. Rev. X, 10, 031048 (2020).
[3] M. J.J. Vrakking, Phys. Rev. Lett., 126, 113203 (2021).
[4] H. Laurell et al, Phys. Rev. Research, 4, 033220 (2022).
