Biomedical Engineering Reference
In-Depth Information
potential barrier. Thus, the interdot barrier experienced by the heavy-holes changes
with the width of the interdot GaAs barrier,
d
, leading to strong modifications of
the exciton transition energies. Our model uses input numerical parameters from
atomistic simulations carried out by Bester et al. [
21
]. In doing this, we explicitly
incorporate the effects of changing
d
, since the exciton energies were obtained by
configuration interaction fitted to tight binding parameters. The
d
-dependent input
parameters include single particle energies, electron-hole Coulomb interactions,
and electron and hole tunneling energies.
We construct the basis for a two-site Hamiltonian representing the QDM, as
products of electron and hole states,
,
where the subindex indicates the QD in which the single particle is localized; all
single particle states are assumed to be the ground state in each QD/band. On this
basis, the non-interacting Hamiltonian is given by
|
e
T
|
h
T
,
|
e
T
|
h
B
,
|
e
B
|
h
T
,and
|
e
B
|
h
B
⎛
⎝
⎞
⎠
.
E
e
T
h
T
t
h
t
e
0
t
h
E
e
T
h
B
0
t
e
=
H
0
(10.8)
t
e
0
E
e
B
h
T
t
h
0
t
e
t
h
E
e
B
h
B
where the diagonal matrix elements of
H
0
are the neutral exciton transition energies,
given in terms of the single particle energies,
ε
i
, and the electron hole Coulomb
interaction,
U
eh
. Explicitly,
U
e
TT
E
e
T
h
T
=
ε
e
T
−
ε
h
T
+
U
e
TB
E
e
T
h
B
=
ε
e
T
−
ε
h
B
+
U
e
BT
E
e
B
h
T
=
ε
e
B
−
ε
h
T
+
U
e
BB
,
E
e
B
h
B
=
ε
e
B
−
ε
h
B
+
(10.9)
For simplicity we have only shown the exciton manifold corresponding to single
neutral excitons. A similar construction can be made when including the biexciton
manifold or charged exciton states.
10.5
Effective Exciton Hamiltonian
In our model exciton pumping is generated by a coherent external laser field, which
results in coherent Rabi oscillations among all different exciton states [
22
,
23
]. Now,
lets us introduce a notation for exciton states derived from the optically perturbed
Hamiltonian,
H
. This corresponds to constructing an optically pumped excitonic ba-
sis
e
B
e
T
h
B
h
T
X
,where
h
B
(
T
)
is the occupation number of holes in the B(T) QDs, and
e
B
(
T
)
the number of electrons, respectively [
4
]. The QDM is pumped by broadband laser
pulse of frequency
ω
, which pumps nearby exciton states. We assume a rectangular
