Entanglement balance of quantum $(e,2e)$ scattering processes

The theory of quantum information constitutes the functional value of the quantum entanglement, i.e., quantum entanglement is essential for high fidelity of quantum protocols, while fundamental physical processes behind the formation of quantum entanglement are less relevant for practical purposes. In the present work, we explore physical mechanisms leading to the emergence of quantum entanglement in the initially disentangled system. In particular, we analyze spin entanglement of outgoing electrons in a nonrelativistic quantum $(e,2e)$ collision on a target with one active electron. Our description exploits the time-dependent scattering formalism for typical conditions of scattering experiments, and contrary to the customary stationary formalism operates with realistic scattering states. We quantify the spin entanglement in the final scattering channel through the pair concurrence and express it in terms of the experimentally measurable spin-resolved $(e,2e)$ triple differential cross sections. Besides, we consider Bell's inequality and inspect the regimes of its violation in the final channel. We address both the pure and the mixed initial spin state cases and uncover kinematical conditions of the maximal entanglement of the outgoing electron pair. The numerical results for the pair concurrence, entanglement of formation, and violation of Bell's inequality obtained for the $(e,2e)$ ionization process of atomic hydrogen show that the entangled electron pairs indeed can be formed in the $(e,2e)$ collisions even with spin-unpolarized projectile and target electrons in the initial channel. The positive entanglement balance—the difference between entanglements of the initial and final electron pairs—can be measured in the experiment.

Figure 1

Spin-unresolved TDCS (a), pair concurrence (b), and entanglement of formation (c) as functions of the electron emission angles ${\theta }_{A,B}$ in the scattering plane when $P_{1,2}=1$ with ${\mathbf{P}}_1\perp {\mathbf{P}}_2$. The nonzero values are shown only in the areas where $\mathrm{TDCS}\ge 0.05\times \max$(TDCS).

Figure 2

Same as in Fig. 1, but for ${\mathbf{P}}_1=-{\mathbf{P}}_2$.

Figure 3

Same as in Fig. 1, but for $P_{1,2}=0$.

Figure 4

Spin-unresolved TDCS (a) and the left-hand side of Bell's inequality (55) [panels (b) and (c)] as functions of the electron emission angles ${\theta }_{A,B}$ in the scattering plane when $P_{1,2}=1$ with ${\mathbf{P}}_1\perp {\mathbf{P}}_2$. The nonzero values are shown only in the angular regions where $\mathrm{TDCS}\ge 0.05\times \max$(TDCS).

Figure 5

Same as in Fig. 4, but for ${\mathbf{P}}_1=-{\mathbf{P}}_2$.

Figure 6

Spin-unresolved TDCS (a) and the spin asymmetry $\mathcal{A}$ (b) as functions of the electron emission angles ${\theta }_{A,B}$ in the scattering plane for $P_{1,2}=0$. The results for the case of $P_{1\left(2\right)}=1$ with $P_{2\left(1\right)}=0$ are identical. The light areas in panel (b) show the angular regions where the spin asymmetry violates Bell's inequality $\mathcal{A}\le 1/\sqrt{2}$ and is “measurable.”