Biomedical Engineering Reference
In-Depth Information
phonon energies are usually too small to enable transitions between the dots without
the mediation of other interactions), the Hamiltonian can be diagonalized exactly
[
49
,
53
]. The Hamiltonian is now restricted to the DQD Hamiltonian of Eq. (
9.3
)
without the tunnel coupling (
V
0), the free phonon Hamiltonian [Eq. (
9.4
)], and
the exciton-phonon interaction given by Eq. (
9.5
), where
n
=
=
m
and
n
correspond to
states of charge carriers localized in single QDs.
If we further restrict the system to the lowest energy excitonic states, then the
unitary transformation that diagonalizes this Hamiltonian is as follows
H
†
H
DQD
+
=W
H
W
=
H
ph
,
with the operator
1
i
=
0
1
j
=
0
|
ij
ij
|
W
ij
.
W=
(9.9)
W
ij
are the Weyl shift operators and are given by
[
q
(
g
(
1
)
∗
g
(
2
)
∗
W
00
=
0
,
W
11
=
exp
−
)
b
q
−
H
.
c
.
]
q
q
g
(
2
)
∗
g
(
1
)
∗
[
q
[
q
W
01
=
exp
b
q
−
H
.
c
.
]
,
W
10
=
exp
b
q
−
H
.
c
.
]
,
q
q
with
g
(
i
)
f
(
i
)
w
q
,where
f
(
i
)
=
/
are phononic coupling constants corresponding to the
q
q
q
two dots,
f
(
1
)
,
f
(
2
)
=
F
10
,
10
(
q
)
=
F
01
,
01
(
q
)
(cf. Eq. (
9.5
)). The DQD Hamiltonian is
q
q
now equal to
H
DQD
=
E
1
(
|
1
1
|⊗
I)+
E
2
(I
⊗|
1
1
|
)+
Δ
E
(
|
1
1
|⊗|
1
1
|
)
,
(9.10)
while the free phonon Hamiltonian remains unchanged. The DQD energies are
shifted and are equal to
E
i
=
ε
i
−
∑
q
w
q
|
q
g
(
2
)
q
.
The evolution operator generated by the full Hamiltonian can be written in terms
of the Weyl operators and the diagonalized Hamiltonian
g
(
i
)
∑
q
w
q
g
(
1
)
2
,and
|
Δ
E
=
Δε
−
2Re
q
†
U
t
,
U
t
=W
W
t
(9.11)
U
t
.Since
H
is diagonal, the evolution
described by
U
t
is trivial and the evolution of the reduced DQD density matrix can
be found in the form
U
t
=
i Ht
U
t
W
where
exp
(
−
)
and
W
t
=
Tr
ph
W
t
e
−
i Ht
σ
0
e
i Ht
†
†
t
ρ
(
˜
t
)=
W
W
W
,
(9.12)
where
σ
0
denotes the combined initial state of the DQD and phonon system, and the
trace is taken over phononic degrees of freedom.
