Presentaci

1
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 al1. 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.
Hamiltonian of the System
Equation 1.
VR
VL
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).
?d U
?d
R1
L
Dot
Leads
R2
On-site Coulomb energy
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 coupling term is the off diagonal part of the
Hamiltonian that characterizes the transfer of
electrons between the leads and dot.

• 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 selfenergy ?.
• The effect of broadening and non-locality due to
  coupling to semi-infinite leads on the resulting
  transmission.

Colored box
Real states
Black box
Virtual states
Computational method. The T-matrix formalism is
used to compute the transition amplitude between
the initial state ?i ? and final state ?f ?

?
Equation 2.a.
Equation 2.b.
The above expression is recursive
The last expression is exact, using the Greens
function and unperturbed part
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.

• All graphs were built with the following
  conditions
• VRVLV
• EL-1
• Ed0 (figure 3.a. and 3.b.)
• Self-energy ? of one dimensional lead
• ?R0.5

Results
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.
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 Ed,with U10 and semi-infinite leads. Each
curve corresponds to different coupling V.
References 1 W. D. Oliver, F. Yamaguchi, Y.
Yamamoto, Phys. Rev. Lett. 88, 37901 (2002).