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arXiv:cond-mat/0605428 · Physical Review B

Energy entanglement in normal metal–superconducting forks

K. V. Bayandin, Gordey B. Lesovik, Thierry Martin

Published Aug 31, 2006• 34 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.74.085326

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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.
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.
We propose a device which implements a solid-state nanostructured electron entangler. It consists of a single-walled carbon nanotube connected at both ends to normal state electrodes and coupled in its middle part to a superconducting nanowire. Such a device acts as an electronic beam splitter for correlated electrons originating from the superconductor. We first show that it can be used to detect positive (bosonic-like) noise correlations in a fermionic system. Furthermore, it provides a source for entangled electrons in the two arms of the splitter. To generate entangled electron states, we propose two kinds of set-up based either on spin or energy filters. They respectively consist of ferromagnetic pads and of a system of electrostatic gates which define quantum dots. The fabrication of this device would require state-of-the-art nanofabrication techniques, carbon nanotube synthesis and integration, as well as atomic force microscopy imaging and manipulation.
Abstract Correlations are fundamental in describing many body systems - not only in natural sciences. However, in experiments, correlations are notoriously difficult to assess on the microscopic scale, especially for electron spins. Here, we demonstrate a direct measurement of the spin cross-correlations between the currents of a Cooper pair splitter, an electronic device that emits electrons originating from Cooper pairs in a superconductor. While it is firmly established theoretically that these electron pairs form maximally spin-entangled singlet states with opposite spin projections, no spin correlation experiments have been demonstrated so far. We use ferromagnetic sidegates, compatible with superconducting electronic structures, to individually spin polarize the transmissions of two quantum dots fabricated in the two electronic paths, which act as tunable spin filters. The signals are detected in standard transport and in highly sensitive transconductance experiments. We find that the spin-cross correlation is negative, compatible with spin singlet emission, and deviates from the ideal value mostly due to a finite overlap of the Zeeman split quantum dot states. Our results demonstrate a new route to perform spin auto- and cross correlation experiments in nanometer scaled electronic devices, especially suitable for those relying on magnetic field sensitive superconducting elements, like unconventional, triplet or topologically non-trivial superconductors, or to perform Bell tests with massive particles, like electrons.
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 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.
Charge and spin current correlations are analyzed in a source of spin-entangled electrons built from a superconductor and two quantum dots in parallel. In addition to the ideal (crossed Andreev) channel, parasitic channels (direct Andreev and cotunneling) and spin flip processes are fully described in a density matrix framework. The way they reduce both the efficiency and the fidelity of the entangler is quantitatively described by analyzing the zero-frequency noise correlations of charge current as well as spin current in the two output branches. Spin current noise is characterized by a spin Fano factor, equal to 0 (total current noise) and $\ensuremath{-}1$ (crossed correlations) for an ideal entangler. The violation of the Bell inequalities, as a test of nonlocality (entanglement) of split pairs, is formulated in terms of the correlations of electron charge and spin numbers counted in a specific time window $\ensuremath{\tau}$. The efficiency of the test is analyzed, comparing $\ensuremath{\tau}$ to the various time scales in the entangler operation.