Biomedical Engineering Reference
In-Depth Information
10.7
Coherent Exciton Dynamics and Level Anticrossing
Population Maps
A level anticrossing map can be generated by calculating the average population
of all exciton levels, as function of the laser excitation energy and applied electric
field. A given coordinate on these maps,
, represents the long time dynamics
of the QDM. The contribution of a pumped exciton is given by an amplitude
p
i
.
Two or more excitons can contribute with their own amplitude to the coordinate
(
(
F
,
h
ω
)
, only when having non-vanishing components in a dressed state [
27
-
29
].
Using the maps corresponding to each exciton state, it is possible to reassemble the
entire spectrum of the system. On the other hand, mapping the population of the
vacuum state
F
,
h
ω
)
00
00
X
=
|
0
, one can obtain the complete LACS spectra, such that all
coordinates
on this map, represents an exciton that is being
depopulated
into the vacuum, or redistributing its weight among all remaining states [
30
].
The first step of this procedure consists in the calculation of the eigenvalue
spectrum of the Hamiltonian in (
10.13
) and the calculation of the unitary dynamics
of the system. To justify unitarity we calculate the typical Rabi period,
T
R
, associated
with a molecular resonance, which depends on all couplings of the Hamiltonian. To
work in the coherent dynamics regime, this period should be of the order of a few
ps [
22
], so that
T
R
τ
X
∼
(
F
,
h
ω
)
1
ns
, the radiative exciton recombination time found in
experiments [
31
,
32
]. An additional requirement is the strong coupling limit for the
resonant transfer processes,
V
F
Δ
X
TB
,
h
/
τ
X
, so that the dynamics of the system
is coherent for times
t
, such that,
T
R
<
t
τ
X
. Therefore if the time evolution is
(
−
iHt
h
unitary, the propagator is given by
U
(
t
)=
exp
)
, so that the population of an
exciton state
|
i
is given at time
t
by
2
P
i
(
t
)=
|
i
|
U
(
t
)
|
0
|
.
(10.14)
If the initial condition is full occupation of the vacuum
(i.e., an “empty” QDM
state), then the averaged population for long integration times is given by
|
0
t
∞
p
i
=(
1
/
t
∞
)
P
i
(
t
)
dt
,
(10.15)
0
where
t
∞
is the pulse duration for a broadband square pulse, which is long enough to
accommodate several Rabi oscillations of the excitonic populations,
T
R
X
.
Only a few Rabi oscillations are enough to compute well-converged long-time
population averages. A lower bound for the initial condition is the time for which
transient effects have elapsed, and an upper bound for the time to end the simulation
would correspond to a value such that damping starts becoming important.
t
∞
<
τ
