Excited core-level dependence of entanglement between a photoelectron and an emitted X-ray photon in X-ray inner-shell excitation

We theoretically investigated how the quantum entanglement between the spin of the photoelectron and the polarization of the emitted X-ray photon depends on the excited core-level, using the 3d\rightarrow\2$p$ and 3d\rightarrow\3$p$ SPR-XEPECS (spin- and polarization-resolved XEPECS) processes for $\mathrm{T}{\mathrm{i}}_2{\mathrm{O}}_3$-type system, and the 4f\rightarrow\4$d$ SPR-XEPECS process for $\mathrm{C}\mathrm{e}{\mathrm{F}}_3$-type system. In the calculation for $\mathrm{T}{\mathrm{i}}_2{\mathrm{O}}_3$-type system, we used $\mathrm{T}\mathrm{i}{\mathrm{O}}_6$ cluster model with the full-multiplet structure of the Ti ion and the charge-transfer effect between Ti 3$d$ and ligand O 2$p$ orbitals. For $\mathrm{C}\mathrm{e}{\mathrm{F}}_3$-type system, we used ionic model with the full-multiplet structure of the Ce ion. We found two distinct mechanisms for entanglement generation in the 3d\rightarrow\2$p$ and 4f\rightarrow\4$d$ cases. The first is generated by the spin-orbit interaction of the 2$p$ core electron, whereas the second is generated by the spin-orbit interaction of the 4$f$ valence electron and strong exchange interaction between the 4$f$ and 4$d$ electrons. However, in the 3d\rightarrow\3$p$ case with the strong 3$d-$3$p$ exchange interaction, we found that the entanglement is not generated due to the crystal field effect. These results reveal the existence of two distinct mechanisms for entanglement generation in X-ray inner-shell excitation processes.