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arXiv:cond-mat/0112094 · Physical review. B, Condensed matter

Bell inequalities and entanglement in solid-state devices

N. M. Chtchelkatchev, G. Blatter, Gordey B. Lesovik, Thierry Martin

Published Oct 31, 2002• 236 citations
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Abstract

Bell-inequality checks constitute a probe of entanglement---given a source of entangled particles, their violation is a signature of the nonlocal nature of quantum mechanics. Here, we study a solid-state device producing pairs of entangled electrons, a superconductor emitting Cooper pairs properly split into the two arms of a normal-metallic fork with the help of appropriate filters. We formulate Bell-type inequalities in terms of current-current cross correlators, the natural quantities measured in mesoscopic physics; their violation provides evidence that this device indeed is a source of entangled electrons.

DOI
10.1103/physrevb.66.161320

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Bell-inequality checks constitute a probe of entanglement---given a source of entangled particles, their violation is a signature of the nonlocal nature of quantum mechanics. Here, we study a solid-state device producing pairs of entangled electrons, a superconductor emitting Cooper pairs properly split into the two arms of a normal-metallic fork with the help of appropriate filters. We formulate Bell-type inequalities in terms of current-current cross correlators, the natural quantities measured in mesoscopic physics; their violation provides evidence that this device indeed is a source of entangled electrons.
The possibility for detecting energy entanglement in normal metal−−--−−superconductor junctions is examined. We prove that two electrons in a NS structure originating from the same Cooper pair are entangled in the energy subspace. This work follows previous works where spin entanglement was studied in similar circuits. The device consists of a superconducting beam splitter connected to two electronic Mach-Zehnder interferometers. In each arm of the interferometers, energies are filtered with coherent quantum dots. In contrast to previous studies of zero-frequency cross-correlations of electrical currents for this system, attention is drawn to finite-time measurements. This allows one to observe two-particle interference for particles with different energies above and below Fermi level. Entanglement is first characterized via the concurrence for the two-particle spatial density matrix. Next, we formulate the Bell inequality, which is written in terms of finite-time noise correlators, and thereby we find a specific set of parameters for which entanglement can be detected.
We propose a spin-independent scheme to generate and detect two-particle entanglement in a mesoscopic normal-superconductor system. A superconductor, weakly coupled to the normal conductor, generates an orbitally entangled state by injecting pairs of electrons into different leads of the normal conductor. The entanglement is detected via violation of a Bell inequality, formulated in terms of zero-frequency current cross correlators. It is shown that the Bell inequality can be violated for arbitrary strong dephasing in the normal conductor.
Electron entanglement: splitting pairs One of the most counterintuitive aspects of quantum mechanics is non-locality, manifesting as spatially separated objects influencing each other in a direct way. Experimental verifications of such concepts have been conducted successfully using entangled pairs of photons to test so-called Bell inequalities, but similar demonstrations using electrons have remained elusive owing to the inherent difficulty of creating and splitting pairs of entangled electrons. Lukas Hofstetter and colleagues now demonstrate that this is possible in a Y-shaped electron entangler consisting of a superconductor at its stem, coupled via tunnelling barriers to two separate quantum dots along the fork-like branches of the device. The entangled pairs of electrons are naturally formed in the superconductor, which in its ground state consists of Cooper pairs of electrons, and split by Coulomb interactions and careful tuning of the energy levels of the quantum dots. The study paves the way for the study of electronic entanglement and experimental tests of Bell inequalities in the solid state.
We study the time-dependent electron-electron and electron-hole correlations in a mesoscopic device which is splitting an incident current of free fermions into two spatially separated particle streams. We analyze the appearance of entanglement as manifested in a Bell inequality test and discuss its origin in terms of local spin-singlet correlations already present in the initial channel and the action of post-selection during the Bell-type measurement. The time window over which the Bell inequality is violated is determined in the tunneling limit and for the general situation with arbitrary transparencies. We compare our results with alternative Bell inequality tests based on coincidence probabilities.
New strategies to generate and to detect spin-entangled pairs of electrons in mesoscopic solid-state heterostructures are proposed. A bilayer-graphene device is designed in which Cooper pairs from an s-wave superconductor are injected into an electrostatically-defined topological channel. Due to a particular band structure, two electrons of a Cooper pair propagate in opposite directions and are hence spatially separated if atomic-scale defects are sparse. Neither energy filtering nor Coulomb repulsion are required. The device can be interpreted as a normal/superconducting/normal junction, in which emission of nonlocal singlets is equivalent to local Andreev reflection, in contrast to the widespread identification of Cooper pair splitting with crossed Andreev reflection. To detect entanglement, the Josephson current through two parallel single-level quantum dots is investigated at all occupations. Combining exact diagonalization and perturbation theory, signatures of nonlocal Cooper pair transport are identified in the critical current, a macroscopic quantity. The model reproduces recent experimental observations and predicts a nonlocal triplet ground state on the quantum dots due to a tunable superconductor-mediated exchange coupling, which, too, is visible in the critical current exactly if nonlocal Cooper pairs are transported. Entanglement detection is pursued, too, by converting spin-entangled Cooper pairs into polarization-entangled photons, which are subsequently probed by a Bell-type measurement. A closed emission cycle is constructed, such that no correlations between the photons and the electrons remain, which are detrimental for entanglement transfer. Inevitable side channels and imperfections are identified and accounted for by a particular measurement protocol. Furthermore, it is discussed how spin-charge separation can be used to extract signatures of nonlocal spin singlets from the average current through an electronic beam splitter. This complements a known mechanism based on exchange statistics by which only the more difficult-to-access noise is sensitive for entanglement. The device can be realized by two crossed nanowires or by quantum-Hall edge states in Corbino geometry. The thesis also contains a review on entanglement and on recent theoretical and experimental progress in solid-state systems towards entanglement detection and generation, focusing on Cooper pair splitting.