Evaluation of quantum entanglement state between photoelectron spin and emitted photon polarization in spin and polarization resolved XEPECS of $\rm Ti_{2}O_{3}$

TL;DR

This work addresses quantum entanglement between the spin of photoelectrons and the linear polarization of emitted X-ray photons in the XEPECS process for $Ti_{2}O_{3}$. It develops a realistic $TiO_{6}$ cluster model with full multiplet structure and $3d$–$2p$ charge transfer, and computes the joint spin–photon density matrix $ ho_{ ext{} ho}$ via second-order perturbation using dipole transitions under rotating-wave approximation. The results show finite coherence between spin and polarization, with a fidelity $F oughly 0.69$ and tangle $T oughly 0.14$ for a chosen geometry, and reveal how entanglement diminishes with stronger hybridization ($ riangle$) and crystal-field splitting ($10Dq$), while increasing $ riangle$ drives the system toward a single-configuration Ti$^{3+}$ limit. The findings emphasize the crucial roles of charge transfer and crystal-field effects in real materials and point to emission-angle and geometry as strong levers for controlling X-ray entanglement in quantum-optical spectroscopy.

Abstract

We theoretically investigated the mechanism of quantum entanglement between the spin of photoelectrons and linear polarization of emitted X-ray photons in the 3$d\rightarrow\ $2$p$ XEPECS process for $\rm Ti_{2}O_{3}$. In the calculation, we used a realistic $\rm TiO_{6}$-type cluster model with the full multiplet structure of the Ti ion and the charge-transfer effect between the Ti 3$d$ and ligand O 2$p$ orbitals. We found that quantum entanglement occurs between the spin of photoelectrons and linear polarization of emitted X-ray photons and that it depends on the angular geometry in the XEPECS process. In addition, we found that the degree of spin and polarization entanglement decreases as the Ti 3$d\ $- O 2$p$ hybridization becomes stronger and as the crystal field modifies the electronic states in terms of the tangle, an index for the degree of entanglement. These results highlight the crucial role of the charge transfer and crystal field effects in determining entanglement properties in real material systems.

Figures (4)

• Figure 1: (a) A schematic view of the photo-induced transition between the 2$p$ and 3$d$ levels with emissions of photoelectron and emitted X-ray photon. (b) The geometrical setting of the linearly polarized incident photon, the photoelectron with spin (U for up and D for down), and the emitted photon with polarization (1 for $\lambda_{1}$ and 2 for $\lambda_{2}$).
• Figure 2: (a) The 2$p$ XPS, (b) the 3$d\rightarrow\ $2$p\ $NXES and (c) the 3$d\rightarrow\ $2$p\ $XEPECS of $\rm Ti_{2}O_{3}$. Each color spectrum in (c) corresponds to the specific binding energy indicated by the arrow with the same color in (a). (d) Photoelectron's spin and emitted photon's polarization-resolved 3$d\rightarrow\ $2$p\ $XEPECS at the unresolved red spectrum in (c). The energies of the initial state ($E_{g}$), intermediate state ($E_{i}$), and final state ($E_{f}$) are each defined relative to the configuration averaged energy of the first electronic configuration. Based on these definitions, the horizontal axes are defined according to Eqs. (\ref{['XPS']}) and (\ref{['NXES']}): the binding energy is $E_{B} = E_{i} - E_{g}$, and the emitted photon energy is $\omega = E_{B} + E_{g} - E_{f}$.
• Figure 3: Real (Re) and imaginary (Im) parts of the density matrix for spin and polarization entanglement states calculated at the peak "B" in Fig.\ref{['Fig.2']}(d). The basis used here is the combined states between photoelectron spin (U for up and D for down) and emitted photon polarization (1 for $\lambda_{1}$ and 2 for $\lambda_{2}$).
• Figure 4: Relationship between tangle $T$ and linear entropy $S_{L}$ using the $\rm TiO_{6}$-type cluster model with various numbers of electronic configuration (config.) included in the simulation: one config. ($\ket{d^{1}}$) with crystal field splitting energy $10Dq = 0$ eV, one config. ($\ket{d^{1}}$), two configs. ($\ket{d^{1}}, \ket{d^{2}\underline{L}}$) and three configs ($\ket{d^{1}}, \ket{d^{2}\underline{L}}, \ket{d^{3}\underline{L}^{2}}$) with $10Dq = 0.5$ eV. The solid line indicates the Werner states, and the grey area corresponds to physically impossible states.MUNRO2PETERS