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

TL;DR

The paper investigates how excited core-levels influence spin–polarization entanglement between a photoelectron and an emitted X-ray photon in SPR-XEPECS, focusing on Ti2O3-type and CeF3-type systems. It develops a cluster-model framework with a TiO6 site and a Ce3+ ionic model, and uses a density-matrix formalism to quantify entanglement via fidelity and tangle in second-order XEPECS/Spectroscopic processes. The key finding is that entanglement arises from two distinct mechanisms: core $2p$ spin–orbit coupling and a mechanism involving outer $4f$ spin–orbit coupling with strong $4f$–$4d$ exchange, with strong core-level dependence observed (e.g., $3d\rightarrow3p$ in Ti2O3 shows almost no entanglement due to crystal-field effects, while $4f\rightarrow4d$ in CeF3 yields near-maximal entanglement). This work clarifies how core-level selectivity in X-ray inner-shell excitations governs quantum entanglement, offering guidance for future XEPEPS experiments and quantum information applications at X-ray energies.

Abstract

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 3$d\rightarrow\ $2$p$ and 3$d\rightarrow\ $3$p$ SPR-XEPECS (spin- and polarization-resolved XEPECS) processes for $\rm Ti_{2}O_{3}$-type system, and the 4$f\rightarrow\ $4$d$ SPR-XEPECS process for $\rm CeF_{3}$-type system. In the calculation for $\rm Ti_{2}O_{3}$-type system, we used $\rm TiO_{6}$ 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 $\rm CeF_{3}$-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 3$d\rightarrow\ $2$p$ and 4$f\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 3$d\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.

Figures (6)

• Figure 1: Schematic illustration of the $2p$ core-level excitation process ($\Circled{1}\rightarrow\Circled{2}$) and the $3p$ excitation process ($\Circled{1}^{'}\rightarrow\Circled{2}^{'}$) in $\mathrm{Ti_{2}O_{3}}$. While the $2p^{5}$ state is split by $2p$ spin-orbit interaction, the $3p^{5}3d^{1}$ state is split by 3$d-$3$p$ exchange interaction.
• Figure 2: (a) The spin-resolved 3$p$ XPS, (b) the 3$d\rightarrow\ $3$p$ NXES and XEPECS spectra of $\mathrm{Ti_{2}O_{3}}$-type system. The blue curve in (b) represents the binding energy-integrated NXES spectrum, while the other spectra show the 3$d\rightarrow\ $3$p$ XEPECS at specific binding energies $E_{B} = -3.74$ eV (orange) and $6.32$ eV (green). Each binding energy corresponds to the peak position indicated by the arrow of the same color in (a).
• Figure 3: (a) 3$d\rightarrow\ $2$p$ SPR-XEPECS at $E_{B}$ = 2.64 eV, for 2$p$ XPS in Fig. 2(a) of ref. [17]. (b) 3$d\rightarrow\ $3$p$ SPR-XEPECS at $E_{B}$ = 6.32 eV in the 3$p$ XPS of Fig. \ref{['Fig.2']}(a)
• Figure 4: Real (Re) and imaginary (Im) parts of the density matrix calculated at the peak “B” in Fig. \ref{['Fig.3']}(b). The basis states are defined as products of the photoelectron spin (U: up, D: down) and the emitted photon's linear polarization (1: $\lambda_1$, 2: $\lambda_2$).
• Figure 5: (a) The spin-resolved 4$d$ XPS of $\rm CeF_3$. (b) The 4$f \rightarrow\ $ 4$d$ SPR-XEPECS of $\rm CeF_3$ at the binding energy $E_B = 19.05$ eV indicated by the blue arrow in the inset of (a).