Generation And Detection Of Electronic Entanglement In Solid State Systems

Quantum entanglement, a kind of nonlocal correlation, is a fundamental property of quantum physics. It is the crucial point to understand the quantum theory, and moreover, it plays an important role in modern quantum information and computation processes. Recently, creating and detecting spin entangled electron pairs in solid state physics has attracted much interest, for electron spin has a long coherence time, and is easy to manipulate and integrate. Creation and detection of electronic entanglement are the precondition for the quantum information and computation implementations, so that the first part of our work are focused on the two aspects. Conventional s-wave superconductor is considered as a natural source of spin entangled electrons, for the Cooper pairs in the superconductor are in the spin singlet state. Through crossed Andreev reflection (CAR), two electrons in the Cooper pair can be split into different terminals, keeping their spins still entangled. In recent years, CAR has been observed experimentally, while the efficiency is still very low, and whether the entanglement still survives is need to be proven. In this thesis, we first study CAR in a narrow quantum spin Hall system (QSH); then we propose that the helical edge states in QSH can be utilized to test Bell inequality (BI) for a spin entangled electron pair; at last, we design a spintronic quantum eraser to quantitatively probe the two-electron entanglement, which is a new method for the detection of entanglement.Specifically, In chapter two, we investigate the CAR process in a narrow QSH, with one of the edges in contact with a bulk of s-wave superconductor. Without the inter-edge coupling, CAR is totally suppressed by the helicity of the edge state; when the QSH is narrow enough inducing an inter-edge coupling, CAR can happen. We find that CAR probability can be greatly enhanced by the proper tuning of the gate voltages. In this case, both the normal Andreev reflection and the elastic cotunneling processes quench in certain energy window. In the edge apart from the superconductor, where the superconducting gap is negligibly small, quasiparticles can transport freely, so that CAR can occur even when the width of the superconductor region is much larger than the superconducting coherence length. The interference by the free propagating waves of quasiparticle between two normal metal-superconductor interfaces results in a resonant CAR, i.e., a perfect splitting of entangled electrons. The resonant CAR can be observed through the conductance and current-current correlation.CAR has been realized in experiment, though one question still remains, that is whether the separated electrons can keep their spin entanglement as the prediction of the theory. This requires further experimental certification. In chapter three, we pro-pose to test BI on solid state spins by utilizing the helical edge states in QSH. Bell test remains a challenge in solid state physics at present, for the environment induced decoherence effect and the difficulty on the detection of spin with high efficiency. The edge states of QSH is robust and protected by the bulk topology. Without time re-versal breaking, the back scattering is forbidden, which means a long spin relaxation and coherence length. The contact between the edge states and the detecting termi-nal does not lead to any scattering either, which guarantees a high efficiency of spin detection. Moreover, due to spin polarization direction and the moving direction are bounded for helical particles, the manipulation on electron spins can be achieved in an all-electrical-controlled manner. Therefore, the spin correlation along different polar-ization direction, which is required by the Bell test, can be realized by the convenient gate voltage tuning. In experiment, the BI can be expressed by the current correla-tion function. We predict that the violation of BI exists in a large parametric range under proper choice on the parameters, and the maximal violation of BI is likely to be achieved.The test on BI is a standard method for entanglement detection, and it is easy to judge whether a state is entangled or not in this way. However, the Bell test is difficult to quantitatively probe the degree of entanglement, for a maximal violation of BI is required by sweeping all possible spin polarization directions. In chapter four, we develop a new method to quantitatively probe the degree of entanglement, taking advantage of the principle named quantum eraser effect. The main idea is that, for a pair of spin entangled electrons in a general EPR state, we can design a quantum eraser with the correlated two-spin state, and we find that the efficiency of the quantum eraser is determined by the degree of spin entanglement. We find the concurrence of two spin-entangled electrons is directly given by the Aharonov-Bohm (AB) oscillation amplitude of the current correlation (Fano factor). For a more general case with an unpolarized or a mixed entangled state, the amplitude of the AB oscillation is still a linear function of the concurrence. It provides a new way to quantitatively probe the electron entanglement. The concurrence can be measured through the oscillation of the Fano factor, which is simpler than the Bell test.In the second part of our work, we investigate the scheme for detecting the su-perconductors with finite momentum pairing. Just after the celebrated BCS theory of superconductivity, Fulde-Ferrell (FF) and Larkin-Ovchmnikov (LO) predicted inde-pendently that finite momentum pairing may occur in some type-Ⅱ superconductors at strong magnetic field, which is called FFLO (or LOFF) state. In the past five decades, great endeavors have been paid trying to unveil this novel phase, unfortunately, only indirect evidences have been reported. In chapter five, we propose an Andreev interfer-ometer composed of a branched Y-junction and an FF superconductor to directly detect the Cooper pair momentum in the FF superconductor, which provides the most con-vincing evidence for finite momentum pairing. In this device, the subgap conductance oscillation is determined only by the phase modulation of the FF superconductor, with-out any intervention of other uncontrollable phases during the multiple scatterings at the Y-junction. The Cooper pair momentum can be measured through the conductance spectrum. This interferometer can have intriguing applications in the identification of the possible finite momentum pairing in non-centrosymmetric superconductors.