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Electron Entanglement via interactions in a quantum dot Gladys León1, Otto Rendon2, Horacio Pastawski3, Ernesto Medina1 1 Centro de Física, Instituto Venezolano de Investigaciones Científicas, Caracas, Venezuela 2 Departamento de Física-FACYT, Universidad de Carabobo, Valencia, Venezuela 3 Facultad de Matemáticas, Astronomía y Física, Universidad Nacional de Córdoba, 5000 Cordoba, Argentina Instituto Venezolano de Investigaciones Científicas Centro de Física Apartado 21827 Caracas 1020A, Venezuela Abstract. We study a spin-entangler device for electrons, mediated by Coulomb interaction U via a quantum dot proposed by Oliver et al[1]. The main advantage of this model, compared to others in the literature, is that single particle processes are forbidden. Within this model we calculate two electron transmission in terms of the T-matrix formalism to all orders in the tunneling amplitudes V and in the presence of i) external orbitals and ii) semi-infinite leads, to show the appropriate limits of a perturbative treatment. New qualitative results are found when external leads are considered non-perturbatively. In particular we recover Oliver's fourth order results in the `external orbital’ case, in the limit of small coupling of the dot to the external states, and a small imaginary part is added to the eigenergies. When leads are attached, the system effectively filters the singlet portion to all orders of perturbation theory. We discuss the role of the coulomb site interaction in the generation of the entangled state. Equation 1.: Hamiltonian of the System VR VL d + U d i Description of the device [1] f R1 L R R2 FIG. 1: Energy level diagram. The external leads are coupled to the dot with a coupling strength VL,R. Electron-electron interactions are only considered within the dot. The initial and final states are: ^† The model consists of one input and two output leads attached to a quantum dot with no occupied states (figure1 ). The arrangement of levels is such that single or double occupancy of the dot does not conserve energy and thus only virtual states can comply within the energy uncertainty. A virtual double occupancy of the dot incurs in an on-site Coulomb energy U. The external contacts are considered either nondegenerate leads, with a relatively narrow energy bandwidth, or single level localized states . Single electron transmission is avoided by placing the incoming and the two outgoing leads off resonance. However, the lead energies can be arranged so that two-electron co-tunneling events conserve energy (figure1). Hˆ Hˆ 0 Vˆ † † † H 0 , aˆ , , aˆ , , d cˆ , , cˆ , , U nˆ nˆ , , Vˆ f Colored box Real states Black box Virtual states V a ˆ , , cˆ h . c . † , , On-site Coulomb energy The coupling term is the off diagonal part of the Hamiltonian that characterizes the transfer of electrons between the leads and dot. i a , k , a , k , ' 0 ^† ^† ^† triplet t ^† a R1, aR 2, a R1, a R 2, 0 ;singlet s Dot Leads ^† 1 2 Focal issues. • The limits of the perturbation when the coupling strength V between the dot and “leads” is increased: The character of the Leads is considered as: a.- External orbital state. b.- Semi-infinite leads, with self–energy . R1 • The effect of broadening and non-locality due to coupling to semi-infinite leads on the resulting transmission. Computational method. R2 i L,k L,k 1 The T-matrix formalism is used to compute the transition amplitude between the initial state i and final state f : 2 1 f 2 1 E L L,k1 L,k2 2 ^ ^ ^ T V V Equation 2.a.: 1 ^ ^ ^ H0 T ; ^ 1 0 T i V G iG i Equation 2.b.: The above expression is recursive FIG. 2: The diagram is built from the tight-binding Hamiltonian, equation 1, the initial energy of the electrons is the same, and each boxrepresents either real (initial and final) or virtual (intermediate) states, with their spins. When the electrons are in the dot their spins are drawn on the horizontal line within the box. The wavy lines indicate one of the directed path of fourth order in V conceived by Oliver et al.[1], in their perturbative computation. ^ ^ The last expression is exact, using the Greens function Gˆ and unperturbed part Gˆ 0 TS i T s Results Tt i T t 0 All graphs were built with the following conditions: •VR=VL=V •EL=-1 •Ed=0 (figure 3.a. and 3.b.) •Self-energy of one dimensional lead •R=0.5 Fig. 3a. Normalized singlet transition amplitude versus on site Coulomb energy U, with localized external states, in semi-log scale. Each curve corresponds to different coupling V. Fig. 3b. Normalized singlet transition amplitude versus on site Coulomb energy U, with semi-infinite leads. Each curve corresponds to different coupling V to the dot. References: [1] W. D. Oliver, F. Yamaguchi, Y. Yamamoto, Phys. Rev. Lett. 88, 37901 (2002). Fig. 3c. Singlet transition amplitude versus dot energy Ed,with U=10 and semi-infinite leads. Each curve corresponds to different coupling V.