Biomedical Engineering Reference
In-Depth Information
Here,
α
1
,
2
are the areas of the exciting pulses and
e
iE
(
t
0
−
τ
)
/
h
dtP
(+)
n
1
E
(
−
)
ref
F
n
±
(
Δ
)=
±
(
t
,
Δ
)
(
t
)=
4
1
(9.33)
2
ref
2
E
+
τ
/
σ
h
exp
exp
E
2
σ
(
t
0
−
τ
)
E
(
t
0
−
τ
)
×
−
∓
i
Φ
n
±
(
Δ
)
,
h
2
2
2
ref
E
2
(
h
+
τ
ref
σ
E
/
h
)
2
(
+
τ
σ
)
where
τ
ref
is the variance of the reference signal and
exp
exp
2
2
ref
E
τ
2
±
Φ
1
±
(
Δ
)=
β
−
[
−
β
±
Γτ
]
,
(9.34)
h
2
(
+
τ
2
ref
2
E
)
8
σ
±
(
Δ
)=
β
+
β
−
exp
exp
exp
2
2
ref
E
τ
i
E
τ
h
−
∓
[
−
Γτ
]
,
Φ
(9.35)
2
(
h
2
2
2
8
+
τ
ref
σ
E
)
Φ
3
±
(
Δ
)=
−
β
∓
exp
exp
2
2
ref
−
(
E /
2
±
V
B
)
τ
i
(
V
B
±E
)
τ
h
iV
B
(
t
0
−
τ
)
−
−
h
2
2
2
ref
E
2
(
+
τ
ref
σ
E
)
(
h
+
τ
σ
/
h
)
×
exp
[
−
(
1
+
β
∓
)
Γτ
]
,
(9.36)
with
.
To calculate the nonlinear response from the ensemble of DQDs in the time
domain (FWM signal), we integrate Eq. (
9.33
) numerically with the weight factor
g
β
±
=
1
±
2
V
/E
. Another integration, over the arrival time
t
0
of the reference pulse, gives the
time integrated four-wave mixing signal (TIFWM) as a function of the delay time
(
Δ
)
τ
which is used to characterize phase decoherence.
Due to the Gaussian term in the formula (
9.32
), the signal is restricted to short
ranges of delay times
/
σ
E
, centered around
t
0
and thus the DQD
response shows “photon echo” type behavior. The contribution
F
1
(
τ
, of the width
∼
h
t
0
,
τ
)
[Eqs. (
9.32
)
and (
9.34
)] has a different structure of phase factors exp
(
i
E
τ
/
h
)
which results in
different [compared to
F
2
(
t
0
,
τ
)
and
F
3
(
t
0
,
τ
)
] character of evolution. The energy
mismatch
Δ
and the energy splitting
E
vary across the ensemble and thus the terms
exp
interfere destructively when the signal emitted by different DQDs is
summed. The relevant phase factor appearing in the first component
F
1
depends
only on
t
0
−
τ
(
i
E
τ
/
h
)
/
σ
E
by the Gaussian term in (
9.32
);
thus, the spread of the phase factors is also limited and independent of
which is limited to the width
∼
h
τ
. Therefore,
the phase factor induces only oscillations in Im
F
1
(
(Fig.
9.19
aande)whichare
always in phase with the center of the echo peak and thus the area of the echo pulse
(time-integrated four-wave mixing signal) is constant.
The contributions
F
2
(
t
0
,
τ
)
t
0
,
τ
)
and
F
3
(
t
0
,
τ
)
contain phase terms proportional to
E
τ
/
h
and
(
V
B
±E
)
τ
/
h
which induce variations of the phase of the signal at
t
0
=
τ
,
with the delay time
, and subsequently variations in the shape and magnitude of
the echo signal. The strong dependence on the delay times
τ
is clearly seen, if one
compares Fig.
9.19
b and c with Fig.
9.19
f and g. Consequently, the total measured
τ
