Biomedical Engineering Reference
In-Depth Information
where the new two-phonon and two-phonon-assisted variables are defined as
B
q
k
=
f
q
f
k
,
B
q
k
=
f
q
f
k
b
q
b
k
f
q
f
k
σ
x
b
q
b
k
,etc.
In the next step, one finds the equation of motion for these new variables, introduc-
ing three-phonon variables. It is clear that the resulting hierarchy of equations is
infinite and has to be truncated at a certain level. For a QD system, this can be done
by setting all the correlated parts of the three-phonon and three-phonon-assisted
variables equal to zero. This amounts to neglecting the correlations involving three
or more phonons or, physically, to neglecting three-phonon processes (that is,
emission or absorption of three or more phonons within the memory time of the
phonon reservoir, which is of the order of 1 ps). The motivation for this procedure is
that higher order correlations should play a decreasing role in the dynamics. From
the equations of motion it is also clear that such higher order correlations develop in
higher orders with respect to the coupling constants
f
k
. On the other hand, truncation
on this level allows one to account for the back-action of non-thermal and coherent
phonons, which is important for a confined system [
57
].
b
q
b
k
,
x
q
k
=
f
q
f
k
σ
x
b
q
b
k
,
x
q
k
=
9.2.2.4
Lindblad Master Equation for Carrier-Photon Dynamics
The decoherence effects induced by radiative environments are described in the
Markov limit by the Master equation of motion in the Lindblad form
˙
ρ
(
t
)=
L
rad
[
ρ
(
t
)]
,
where
1
2
{
Σ
+
(
L
rad
[
ρ
(
t
)] =
Γ
Σ
−
(
t
)
ρ
(
t
)
Σ
+
(
t
)
−
t
)
Σ
−
(
t
)
,
ρ
(
t
)
}
,
(9.15)
†
is the Lindblad dissipator with
Σ
−
(
t
)=(
Σ
+
(
t
))
denoting the operators in the
interaction picture and
E
3
2
|
d
|
Γ
=
(9.16)
πε
0
ε
r
h
4
3
is a spontaneous decay rate for a single QD.
9.2.2.5
Joint Influence of Phonon-Induced Dynamics
and Spontaneous Emission
The evolution of a pair of QDs may be solved exactly only in special cases:
An exact solution is available for an uncoupled system (
V
0) interacting only
with lattice vibrations in the limit of instantaneous state preparation or ultrafast
optical excitation [
58
,
60
] or in the Markov limit for dots coupled only to its
=
