Biomedical Engineering Reference
In-Depth Information
9.2.2.3
Correlation Expansion Techniques
The correlation expansion technique is a standard method used for the description of
quantum kinetics of interacting carriers and phonons in semiconductor systems of
any dimensionality [
54
]. It has been successfully applied to carrier-phonon kinetics
in QDs driven by an optical field [
55
-
57
], yielding a reliable description beyond the
instantaneous excitation limit, where an exact solution is available for decoupled
systems [
58
], and beyond the weak perturbation case which allows a perturbative
treatment [
59
]. Compared to higher-dimensional systems, in QDs coherent and non-
equilibrium phonons play a larger role because of the localized polaron effect.
Therefore, a reliable description of the carrier phonon-kinetics in these systems
requires a high enough degree of the correlation-expansion technique [
57
].
Various implementations of this technique differ in notation and in the choice
of dynamical variables. Here, let us start from the Hamiltonian (
9.5
) restricted to
only diagonal (
n
m
) couplings to the two ground exciton states in the two QDs
and to LA phonons. By shifting the phonon modes in
H
(
1
)
DQD
=
H
(
2
)
DQD
ph
(where
the two terms describe the phonon interaction with each dot) according to
b
q
→
b
q
−
[
+
−
−
ph
F
01
,
01
(
q
)+
F
10
,
10
(
q
)]
/
(
2
hw
q
)
one gets the coupling Hamiltonian in the form
b
q
+
q
H
DQD
−
ph
=
q
b
†
−
f
q
(
|
01
01
|−|
10
10
|
)
,
where
f
q
=[
F
01
,
01
(
q
)
−
F
10
,
10
(
q
)]
/
2. In the first step, one defines three dynam-
ical variables
x
,
y
,
z
describing the carrier state,
x
=
σ
x
(
t
)
,...
,where
σ
i
(
t
)=
e
i Ht
/
h
σ
i
e
i Ht
/
h
basis, in the Heisenberg
picture. These three variables are the coordinates of the evolving Bloch vector,
uniquely determining the reduced density matrix of the carrier subsystem.
From the Heisenberg equations of motion one finds the dynamical equations for
these three variables,
are Pauli operators, written in the
|
01
,|
10
k
k
x
=
i
[
H
,
σ
x
]
=
−
Δ
y
−
4
y
Re
B
k
−
4
y
Re
y
k
,
(9.13)
and analogous for
y
and
z
. Obviously, this set of equations is not closed, but involves
the new phonon variables
B
k
=
f
k
b
k
, as well as phonon-assisted variables of the
form
y
k
=
,
denote the correlated part of a product of operators, obtained by subtracting all
possible factorizations of the product.
Next, one writes down the equations of motion for the new variables that
appeared in the previous step, for instance,
f
k
σ
y
b
k
=
f
k
(
σ
y
b
k
−
σ
y
b
k
)
. The double angular brackets,
...
2
y
k
=
i
[
H
,
y
k
]
=
Δ
x
k
−
2
Vz
k
−
iw
k
y
k
+
|
f
k
|
(
iyz
+
x
)
q
(
x
q
k
+
x
q
k
)+
4
x
k
q
q
(
B
q
k
+
B
q
k
)
,
+
2
Re
B
q
+
2
x
(9.14)
