Biomedical Engineering Reference
In-Depth Information
is the exciton-photon coupling constant. Here
d
is the inter-band dipole moment
(equal for both dots),
e
λ
(
)
is the unit polarization vector of the photon mode with
k
polarization
ε
r
is the relative dielectric
constant of the semiconductor, and
v
is the normalization volume.
λ
,
ε
0
is the vacuum dielectric constant,
9.2.2
Simulation of the Evolution
The dynamics of the system has been studied using various versions of the density
matrix formalism, ranging from Fermi golden rule description of the occupation
dynamics to non-Markovian simulations based on correlation expansion or time-
convolutionless (TCL) methods.
9.2.2.1
Non-Markovian Master Equation for Carrier-Phonon Dynamics
The
dynamic
induced
by
the
lattice
vibrations
is
governed
by
the
time-
convolutionless equation in the interaction picture
˙
ρ
(
t
)=
L
ph
[
ρ
(
t
)]
,
where
ρ
(
t
)
is a reduced density operator for the carrier subsystem and
t
Tr
ph
H
DQD
−
ph
)
,
H
DQD
−
ph
ph
,
L
[
ρ
(
)] =
−
(
(
τ
)
,
ρ
(
)
⊗
ρ
t
d
τ
t
t
(9.8)
ph
0
with
the
carrier-phonon
interaction
Hamiltonian
in
the
interaction
picture
(
)
H
DQD
−
ph
ph
is the phonon density matrix at thermal equilibrium and
Tr
ph
denotes a partial trace with respect to the phonon degrees of freedom. In
the limit of long times and short reservoir memory, this equation reduces to the
Markovian Lindblad equation with the transition rate from the initial state
n
to the
final state
m
given by the Fermi golden rule,
t
. Here,
ρ
=
[(
−
)
/
]
Γ
2
π
R
nm
E
n
E
m
h
,where
ph
,
n
→
m
1
h
2
q
|
F
nm
(
q
)
|
2
R
nm
(
ω
)=
[
δ
(
ω
−
w
q
)(
n
q
+
1
)+
δ
(
ω
+
w
q
)
n
q
]
is the appropriate spectral density of the phonon reservoir (for a given pair of states).
The details are given in [
52
].
9.2.2.2
The Weyl Operator Method for the Exact Diagonalization
of the Electron-Phonon Coupling Term
If the system under study consists only of the DQD without tunneling between the
dots and off-diagonal terms in the carrier-phonon interaction (which is typical, since
