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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, aR 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  iG
  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 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.
^ ^
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.