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Physical Review Letters

Entanglement and Bell Correlation in Electron-Exchange Collisions

Karl Blum, Birgit Lohmann

Published Jan 21, 2016• 13 citations
Publisher

Abstract

Elastic collisions between initially unpolarized electrons and hydrogenlike atoms are discussed, aiming to analyze the entanglement properties of the correlated final spin system. Explicit spin-dependent interactions are neglected and electron exchange only is taken into account. We show the final spin system to be completely characterized by a single spin correlation parameter depending on scattering angle and energy. Its numerical value identifies the final spins of the collision partners to be either in the separable, entangled, or Bell correlated regions. We emphasize explicit examples for the mixed spin system in order to illustrate the abstract concepts. The analysis of published experimental and numerical data reveals the possibility to create tunable pairs of collision partners with any desired degree of spin entanglement.

DOI
10.1103/physrevlett.116.033201

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Elastic collisions between initially unpolarized electrons and hydrogenlike atoms are discussed aiming to analyze the entanglement properties of the correlated final spin system. Explicit spin-dependent interactions are neglected and electron exchange only is taken into account. We show the final spin system to be completely characterized by a single spin correlation parameter depending on scattering angle and energy. Its numerical value identifies the final spins of the collision partners to be either in the separable, entangled, or Bell correlated regions. The symmetry of the scattering process allows for the construction of explicit examples applying methods of classical communication and local operations for illustrating the concepts of nonlocality versus separability. It is shown that strong correlations can be produced violating Bell's inequalities significantly. Furthermore, the degree of entanglement can be continuously varied simply by changing either the scattering angle and/or energy. This allows for the generation of tunable spin pairs with any desired degree of entanglement. It is suggested to use such nonlocally entangled spin pairs as a resource for further experiments, for example in quantum information processes.

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Abstract In two recent papers, Blum and Lohmann (2016 Phys. Rev. Lett. 116 033201) and Lohmann et al (2016 Phys. Rev. A 94 032331), the possibility of continuously varying the degree of entanglement between an elastically scattered electron and the valence electron of quasi-one electron targets was discussed. Here we present results for elastic electron scattering from atomic hydrogen in the energy regime of 1−10 eV and the full range of scattering angles 0 ° − 180 ° . We confirm previous calculations at very low energies, which predicted that the hydrogen target is not a promising candidate for Bell correlations through electron collisions. This finding remains unchanged in the near-resonance regime of incident electron energies just below 10 eV. In addition to the spin-correlation parameter P , we present the angle-integrated total cross section, as well as the angle-differential cross section at a few representative collision energies.

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Moscow University Physics Bulletin•Jun 1, 2022•0 citations
Elastic collisions between initially unpolarized electrons and hydrogenlike atoms are discussed, aiming to analyze the entanglement properties of the correlated final spin system. Explicit spin-dependent interactions are neglected and electron exchange only is taken into account. We show the final spin system to be completely characterized by a single spin correlation parameter depending on scattering angle and energy. Its numerical value identifies the final spins of the collision partners to be either in the separable, entangled, or Bell correlated regions. We emphasize explicit examples for the mixed spin system in order to illustrate the abstract concepts. The analysis of published experimental and numerical data reveals the possibility to create tunable pairs of collision partners with any desired degree of spin entanglement.
Elastic collisions between initially unpolarized electrons and hydrogenlike atoms are discussed aiming to analyze the entanglement properties of the correlated final spin system. Explicit spin-dependent interactions are neglected and electron exchange only is taken into account. We show the final spin system to be completely characterized by a single spin correlation parameter depending on scattering angle and energy. Its numerical value identifies the final spins of the collision partners to be either in the separable, entangled, or Bell correlated regions. The symmetry of the scattering process allows for the construction of explicit examples applying methods of classical communication and local operations for illustrating the concepts of nonlocality versus separability. It is shown that strong correlations can be produced violating Bell's inequalities significantly. Furthermore, the degree of entanglement can be continuously varied simply by changing either the scattering angle and/or energy. This allows for the generation of tunable spin pairs with any desired degree of entanglement. It is suggested to use such nonlocally entangled spin pairs as a resource for further experiments, for example in quantum information processes.
Abstract In two recent papers, Blum and Lohmann (2016 Phys. Rev. Lett. 116 033201) and Lohmann et al (2016 Phys. Rev. A 94 032331), the possibility of continuously varying the degree of entanglement between an elastically scattered electron and the valence electron of quasi-one electron targets was discussed. Here we present results for elastic electron scattering from atomic hydrogen in the energy regime of 1−10 eV and the full range of scattering angles 0 ° − 180 ° . We confirm previous calculations at very low energies, which predicted that the hydrogen target is not a promising candidate for Bell correlations through electron collisions. This finding remains unchanged in the near-resonance regime of incident electron energies just below 10 eV. In addition to the spin-correlation parameter P , we present the angle-integrated total cross section, as well as the angle-differential cross section at a few representative collision energies.
In two recent papers [Blum and Lohmann, Phys. Rev. Lett. 116, 033201 (2016); Lohmann et al., Phys. Rev. A 94, 032331 (2016)], the possibility of continuously varying the degree of entanglement between an elastically scattered electron and the valence electron of an alkali-metal target was discussed. To estimate how well such a scheme may work in practice, we present results for elastic electron scattering from lithium in the energy regime of 1−−5eV1--5\phantom{\rule{4pt}{0ex}}\mathrm{eV}1−−5eV and the full range of scattering angles ${0}^{\ensuremath{\circ}}--{180}^{\ensuremath{\circ}}$. The most promising regime for Bell correlations in this particular collision system are energies between about 1.5 and 3.0 eV, in an angular range around ${110}^{\ensuremath{\circ}}\ifmmode\pm\else\textpm\fi{}{10}^{\ensuremath{\circ}}$. In addition to the relative exchange asymmetry parameter, we present the differential cross section that is important when estimating the count rate and hence the feasibility of experiments using this system.
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)(e,2e)(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)(e,2e)(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)(e,2e)(e,2e) ionization process of atomic hydrogen show that the entangled electron pairs indeed can be formed in the (e,2e)(e,2e)(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.
In the framework of nonstationary scattering theory, we study the formation of an entangled state of two identical nonrelativistic spin- 1/2 particles as a result of their elastic scattering. The measure of particle entanglement in the final channel is described using pair concurrence. For the indicated quantitative criterion, we obtain general expressions in terms of the direct and exchange scattering amplitudes in the cases of pure and mixed spin states of the pair in the initial channel. We consider the violation of Bell’s inequality in the final channel. We show that as a result of a collision between unpolarized particles, a Werner spin state of the pair forms, which is entangled if the singlet component of the angular differential scattering cross section in the center-of-mass reference frame exceeds the triplet component. We use the process of free electron-electron scattering as an example to illustrate the developed formalism.
Abstract Generation of an entangled spin state as a result of a low-energy elastic collision of two protons is considered. The scattering process is described by a theoretical model that is in good agreement with the available experimental data on the angular differential cross sections of proton–proton collisions at low energies. The entanglement of the spin state of final protons as a function of the scattering angle is analyzed using Bell’s inequality and pair concurrence. The numerical results for different values of the collision energy are compared with the results without considering nuclear forces. It is shown that nuclear interaction significantly enhances the effects of spin entanglement.