Spin-entanglement between two freely propagating electrons: Experiment and theory

Theory predicts that electron pairs, which are emitted from a crystalline surface upon impact of spin-polarized low-energy electrons, can be spin-entangled. We quantify this entanglement by the von Neumann entropy, which we show to be closely related to the spin polarization of the emitted electrons. Measurement of the spin polarization therefore facilitates an experimental study of the entanglement. As target we used a Cu(111) surface, which exhibits an electronic surface state giving rise to a high pair emission intensity. Experimental spin polarization spectra for several orientations of the reaction plane broadly agree with their theoretical counterparts. They are consistent with spin entanglement of the electron pair at a macroscopic distance.

Figure 1

A primary electron ${\mathrm{e}}_1$ interacts with a valence electron ${\mathrm{e}}_2$, this leads to the emission of electrons ${\mathrm{e}}_3$ and ${\mathrm{e}}_4$. The reaction plane is the paper plane and the spin polarization of the primary beam is perpendicular to it. The spin polarizing mirror measures the spin polarization of the emitted electron ${\mathrm{e}}_3$ perpendicular to the scattering plane.

Figure 2

Theoretical results from Cu(111) for normally incident primary electrons with $E_1=24$ eV and spin ${\sigma }_1=+$. The two outgoing electrons have energies $E_3$ and $E_4$ and equal polar angles ${\vartheta }_3={\vartheta }_4={45}^{\circ }$. Surface geometry sketches are in the right-hand column, the $x$ axis is along the $[1,-1,0]$ direction and the $y$ axis along $[-1,-1,2]$. The red line marks the intersection of the reaction plane with the surface for azimuthal angles $\phi =0^{\circ }$, ${30}^{\circ }$, and ${90}^{\circ }$ as indicated. The plane normal to the reaction plane (dashed green line) is a mirror plane for $\phi =0^{\circ }$ and a nonmirror plane in the cases $\phi ={30}^{\circ }$ and ${90}^{\circ }$. The energy distributions in panels (a)–(i) are presented as functions of the energy difference $E_3-E_4$ of the two outgoing electrons and of the valence electron energy $E_2$ minus $E_F$, which by virtue of energy conservation is equivalent to the sum energy $E_3+E_4$. In each panel, the vertical dashed line marks the special case of equal energies of the two electrons. (a)–(c) show results for azimuthal angle $\phi =0^{\circ }$. (a) Total intensity ${\left|f\right|}^2+{\left|g\right|}^2+{|f-g|}^2$ [cf. Eq. (7)]. (b) Entropy of formation $\tilde{S}$ [cf. Eq. (8)]. (c) Intensity difference ${\left|f\right|}^2-{\left|g\right|}^2+{|f-g|}^2$, which is equal to the spin polarization $P_3$ of electrons ${\mathrm{e}}_3$ in Fig. 1 [cf. Eq. (9)] times the total intensity. (d)–(f) Same as (a)–(c) but for $\phi ={30}^{\circ }$. (g)–(i) Same as (a)–(c) but for $\phi ={90}^{\circ }$. The results for $\phi ={60}^{\circ }$ are not shown, because they are identical with those for $\phi =0^{\circ }$. For primary spin ${\sigma }_1=-$, the total intensity and the entropy of formation are the same as for ${\sigma }_1=+$, but the intensity difference has the reversed sign.

Figure 3

Weighted averages $\overline{S}$ of entropy (solid blue lines) and ${\overline{P}}_3$ of spin polarization (solid red lines), which were—for the same conditions as the results in Fig. 2—obtained by integrating the numerators and denominators of Eqs. (8) and (9), respectively, over the valence energy interval from $-1.0$ eV up to the Fermi level. Neglecting the parallel-spin intensity term ${|f-g|}^2$, this procedure yields the corresponding quantities ${\overline{S}}^{\mathrm{ap}}$ (dashed blue lines) and ${\overline{P}}_3^{\mathrm{ap}}$ (dashed red lines) for antiparallel spins only. (a)–(c) show the results for the azimuthal angles $\phi =0^{\circ }$, ${30}^{\circ }$, and ${90}^{\circ }$. Results for $\phi ={60}^{\circ }$ are the same as those for $\phi =0^{\circ }$.

Figure 4

Spin polarization $P_3$ as a function of the energy difference $E_3-E_4$ for different azimuthal angles $\phi$ for primary energy $E_1=24$ eV. The theoretical results (solid red lines) were obtained, for primary spin ${\sigma }_1=+$, from Eq. (9) by averaging over the valence energy interval $E_2$ from $-1$ eV up to $E_F$, which includes the entire Shockley surface state. The experimental results are represented by fat dots inside a rectangle. The horizontal extension indicates the energy integration range. The vertical extension is the statistical uncertainty at each dot. The experimental (theoretical) polarization scale is on the left (right). The constant scaling factor between them takes into account various depolarization effects.