Electron Entanglement via interactions in a quantum dot Gladys León 1, Otto Rendon 2, Horacio Pastawski 3, Ernesto Medina 1 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 Description of the device [1] 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 non- degenerate 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). 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. 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 . The effect of broadening and non-locality due to coupling to semi-infinite leads on the resulting transmission. Computational method. The T-matrix formalism is used to compute the transition amplitude between the initial state  i  and final state  f  : ; The above expression is recursive The last expression is exact, using the Greens function and unperturbed part Equation 2.b.: Equation 2.a.: Results All graphs were built with the following conditions: V R =V L =V E L =-1 E d =0 (figure 3.a. and 3.b.) Self-energy  of one dimensional lead  R =0.5 FIG. 2: The diagram is built from the tight-binding Hamiltonian, equation 1, the initial energy of the electrons is the same, and each box represents 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. R1 R2  Colored boxReal states Black boxVirtual states References: [1] W. D. Oliver, F. Yamaguchi, Y. Yamamoto, Phys. Rev. Lett. 88, (2002). 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. 3c. Singlet transition amplitude versus dot energy E d, with U=10 and semi-infinite leads. 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. Equation 1.: Hamiltonian of the System Leads Dot 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. VLVL VRVR R1R1  d + U dd L FIG. 1: Energy level diagram. The external leads are coupled to the dot with a coupling strength V L,R. Electron-electron interactions are only considered within the dot. The initial and final states are: R2R2 Instituto Venezolano de Investigaciones Científicas Centro de Física Apartado Caracas 1020A, Venezuela