Biomedical Engineering Reference
In-Depth Information
pulse width long enough to capture several Rabi oscillations of the excitonic level
populations. When,
e
B
(
T
)
,
, the resulting Hamiltonian basis contains
a total of 14 neutral exciton states. This corresponds to: the vacuum
0
00
X
; a pair of
monoexcitons
1
10
X
h
B
(
T
)
=
{
0
,
1
,
2
}
01
01
X
; a pair of spatially indirect monoexcitons,
1
01
X
01
,
,
10
X
; direct
biexciton states
2
20
X
02
02
X
; indirect biexciton states
2
02
X
02
,
,
20
X
; a delocalized biexciton
11
11
X
and a remaining set of trion-like states,
0
11
X
20
11
11
20
X
.
Up to single occupancy of holes and electrons,
e
B
(
T
)
,
,
11
X
,
02
X
,
, the per-
turbed Hamiltonian corresponding to the monoexciton manifold in (
10.8
), is
given by
h
B
(
T
)
=
{
0
,
1
}
H
tot
=
H
D
+
H
tun
+
H
V
F
+
H
opt
.
(10.10)
The Hamiltonian in (
10.10
) includes the diagonal contribution of (
10.8
)plusthe
vacuum state
, the tunneling matrix elements of electrons and holes,
H
tun
,
the resonant energy transfer interaction,
V
F
, and the laser excitation or optical
perturbation,
H
opt
. Explicitly,
|
0
|
+
∑
i
H
D
=
|
0
0
E
i
|
i
i
|
t
e
c
t
h
c
01
01
X
01
10
X
01
01
X
01
10
X
H
tun
=
|
|
+
H
.
+
|
|
+
H
.
V
F
c
01
01
X
10
10
X
H
V
F
=
|
|
+
H
.
H
opt
=
Ω
T
e
−
i
ω
t
+
Ω
B
e
−
i
ω
t
01
01
X
e
i
ω
t
01
01
X
10
10
X
e
i
ω
t
10
10
X
|
0
|
+
|
0
|
|
0
|
+
|
0
|
.
(10.11)
In the model, only spatially direct monoexcitons are coupled to the radiation
field by
0
01
X
1
10
X
, for QD(T) and
QD(B), respectively. On the other hand, the spatially indirect excitons have typically
a much weaker oscillator strength; its inclusion in the model is straightforward
but will be ignored in what follows. Here,
Ω
T
(
t
)=
0
|
μ
T
·
E
(
t
)
|
and
Ω
B
(
t
)=
0
|
μ
B
·
E
(
t
)
|
μ
i
are the interband transition dipole
moments and
E
is the electric field component of the radiation pulse amplitude.
In order to simplify the equation of motion, we have performed the rotating
wave approximation (RWA) and a unitary transformation that removes the time-
dependent fast oscillatory terms of the matter-radiation interaction [
24
]. The latter
approximation is carried out by the following transformation,
(
t
)
†
ih
∂Λ
H
†
H
=
Λ
Λ
+
Λ
∂
t
Λ
=
∑
k
e
−
i
ξ
k
t
|
k
k
|,
(10.12)
