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arXiv:cond-mat/0505579 · Physical Review B

Clauser-Horne inequality and decoherence in mesoscopic conductors

Elsa Prada, Fabio Taddei, Rosario Fazio

Published Sep 21, 2005• 8 citations
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Abstract

We analyze the effect of decoherence on the violation of the Clauser-Horne (CH) inequality for the full electron counting statistics in a mesoscopic multiterminal conductor. Our setup consists of an entangler that emits a flux of entangled electrons into two conductors characterized by a scattering matrix and subject to decoherence. Loss of phase memory is modeled phenomenologically by introducing fictitious extra leads. The outgoing electrons are detected using spin-sensitive electron counters. Given a certain average number of incoming entangled electrons, the CH inequality is evaluated as a function of the numbers of detected particles and on the various quantities characterizing the scattering matrix. When decoherence is turned on, we show that the amount of violation of the CH inequality is effectively reduced. Interestingly we find that, by adjusting the parameters of the system, there exists a protected region of $Q$ values for which violation holds for arbitrary strong decoherence.

DOI
10.1103/physrevb.72.125333

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We analyze the effect of decoherence on the violation of the Clauser-Horne (CH) inequality for the full electron counting statistics in a mesoscopic multiterminal conductor. Our setup consists of an entangler that emits a flux of entangled electrons into two conductors characterized by a scattering matrix and subject to decoherence. Loss of phase memory is modeled phenomenologically by introducing fictitious extra leads. The outgoing electrons are detected using spin-sensitive electron counters. Given a certain average number of incoming entangled electrons, the CH inequality is evaluated as a function of the numbers of detected particles and on the various quantities characterizing the scattering matrix. When decoherence is turned on, we show that the amount of violation of the CH inequality is effectively reduced. Interestingly we find that, by adjusting the parameters of the system, there exists a protected region of QQQ values for which violation holds for arbitrary strong decoherence.
In this paper we derive the Clauser-Horne (CH) inequality for the full electron-counting statistics in a mesoscopic multiterminal conductor and discuss its properties. We first consider the idealized situation in which a flux of entangled electrons is generated by an entangler. Given a certain average number of incoming entangled electrons, the CH inequality can be evaluated for different numbers of transmitted particles. Strong violations occur when the number of transmitted charges on the two terminals is the same (Q1=Q2),{(Q}_{1}{=Q}_{2}),(Q1​=Q2​), whereas no violation is found for ${Q}_{1}\ensuremath{\ne}{Q}_{2}.$ We then consider two actual setups that can be realized experimentally. The first one consists of a three terminal normal beam splitter and the second one of a hybrid superconducting structure. Interestingly, we find that the CH inequality is violated for the three terminal normal device. The maximum violation scales as 1/M1/M1/M and 1/M2{1/M}^{2}1/M2 for the entangler and normal beam splitter, respectively, 2M2M2M being the average number of injected electrons. As expected, we find full violation of the CH inequality in the case of the superconducting system.
We discuss the Clauser–Horne (CH) inequality for electrons in mesoscopic systems formulated in terms of the full counting statistics (FCS). A derivation of such an inequality is provided for a prototype setup consisting of an entangler attached to two conducting wires, that brings pairs of spin-entangled electrons into spatially separated counters, where detection takes place. Violation of the CH inequality is analysed as a function of the various parameters characterizing the system. The effect of dephasing, which can occur in realistic wires, is also addressed. As expected, the extent of violation is monotonically suppressed by increasing dephasing strength. The CH inequality is finally applied to a three-arm normal-metallic beam splitter. Our results show that violation takes place even in the absence of interaction.
We consider a system of two spin-12\frac{1}{2}21​ particles, initially in an entangled Bell state. If one of the particles is interacting with an environment (e.g., a collection of NNN independent spins), the two-particle system undergoes decoherence. Using a simple model of decoherence, we show that this process has two consequences. First, the maximal amount by which the Clauser-Horne-Shimony-Holt inequality is violated decays to zero. Second, the set of directions of measurement for which the inequality is violated is reduced in the course of decoherence. The volume of that set is bounded above by $\text{const}\ifmmode\times\else\texttimes\fi{}{|r|}^{2}$, where rrr is the decoherence factor. We obtain similar results for the case when each of the two particles is in interaction with a separate environment.
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.
In this article, we have discussed about the scattering of charge and spin in spin-resolved mesoscopic systems. We have proposed a method to generate and detect the entanglement of electronic spin and electronic orbital degrees of freedom. It is observed that in a spin-resolved mesoscopic system, all form of entanglement existing between different electronic degrees of freedom can be generated. A method to detect Bell's inequality violation in terms of zero frequency current cross-correlators has been mentioned. Further, we have also studied the effect of quantum interference between spin flip and non-spin flip scattering matrices in determining the current and noise. We have obtained the expression for shot noise of spin and charge current when an arbitrarily polarised charge current is scattered at the spin resolved potential.