Biomedical Engineering Reference
In-Depth Information
9.4.4
Four-Wave Mixing
Effects resulting from collective coupling to electromagnetic field are also visible
in the nonlinear response of DQD samples. Below we will show how this kind
of coupling affects the four-wave mixing response of an ensemble of DQDs. The
modeled experiment consists of exciting the system of many DQDs with two
ultrafast laser pulses delayed in time by
. To obtain the optical response of the
whole ensemble, we first calculate the polarization induced by a single DQD. As in
the previous case, we assume that the laser pulses act symmetrically on both dots
in the pair. After the first excitation, arriving at time
t
τ
, the system evolves
according to the Lindblad equation discussed in Sect.
9.2.2.4
. The second laser
pulse arrives at time
t
=
−
τ
0. Later, the system again evolves according to the Lindblad
equation. In order to extract the signal measured in the four-wave mixing experiment
we pick out only the terms containing the phase factor exp
=
[
i
(
2
φ
2
−
φ
1
)]
,where
the
φ
1
,
2
are the phases of the exciting pulses. The signals emitted by ensembles
of DQDs are usually very weak and thus their detection is based on a heterodyne
technique [
125
], where the signal emitted by the system of QDs is superimposed
on the reference signal. Therefore, in the next step, we calculate the overlap of
the signal induced by quantum dots and the reference signal which arrives at time
t
t
0
and has a Gaussian envelope. Since we want to calculate the signal emitted
by an ensemble of DQDs we average the polarization induced by a single pair
with a Gaussian distribution for the fundamental transition energies
E
1
and
E
2
which, expressed in terms of the average transition energy and the energy mismatch,
becomes a product of two Gaussian functions,
g
=
(
E
1
,
E
2
)=
g
(
E
,
Δ
)=
g
(
E
)
g
(
Δ
)
,
where
g
are the Gaussian distributions of the mean transition energy
and the energy mismatch,
(
E
)
and
g
(
Δ
)
exp
exp
¯
E
2
2
1
√
2
−
(
E
−
)
1
−
(
Δ
−
Δ
)
(
)=
(
Δ
)=
√
2
,
g
E
and
g
E
2
Δ
πσ
2
σ
πσ
Δ
2
σ
E
respectively, with the corresponding variances
σ
Δ
. Following this procedure
and performing some approximations, which are discussed in detail in [
126
], we
find the signal in the form
σ
E
and
3
n
=
1
F
n
(
t
0
,
τ
)
,
F
(
t
0
,
τ
)=
where
d
cos
2
α
1
2
sin
2
i
8
sin
F
n
(
t
0
,
τ
)=
(
α
1
)
(
α
2
)
Δ
g
Δ
(
Δ
)[
F
n
+
(
Δ
)+
F
n
−
(
Δ
)]
.
(9.32)
