Biomedical Engineering Reference
In-Depth Information
Δ
∼
Γ
/
changes with decreasing energy mismatch. In the intermediate range of
2,
the spectrum is non-Lorentzian and difference between quantum dots coupled to
common and separated reservoirs is clearly seen. Then, with decreasing
h
,the
spectrum of collectively coupled systems switches to the sum of two Lorentzians
with different widths and opposite signs centered around the zero frequency (only
in this case the polarization has two exponential components). In the hypothetical
case of quantum dots emitting to separate reservoirs, the spectrum is a sum of two
identical Lorentzians.
Δ
9.4.3.2
Coupled Quantum Dots
Below we analyze the role of the coupling between single-exciton states (
V
=
0,
V
B
=
0) in the optical response to a laser pulse excitation. In this case, the transition
to the collective regime takes place for the energy mismatches of the order of the
coupling between the systems which we assume to be larger than the relaxation
rate,
V
becomes the smallest frequency parameter
of the system and one can simplify the discussion by retaining only terms linear
in
h
Γ
. Thus the decay rate
Γ
0 (for more detailed discussion, see [
124
]). With these
assumptions, a decay of the linear polarization is described by a sum of two
exponential components with decay rates
Γ
or assume
Γ
=
Γ
±
=[
1
∓
2
V
/E
]
Γ
,where
E
is the energy
splitting. One of these rates, subradiant, decays slowly with a rate
Γ
−
/
2
<
Γ
/
2
and its amplitude vanishes in the limit of
Δ
V
while the second, superradiant,
component decays rapidly with a rate
Γ
+
/
2
>
Γ
/
2 and dominates the evolution in
the limit of strongly coupled dots.
In Fig.
9.18
a and c, we show the evolution of the envelopes of the optical beats for
three sets of parameters corresponding to the same energy splitting
6ps
−
1
.One
effect that is seen in the dynamics of the systems coupled to common as well as to
separate reservoirs is the decrease of the beating amplitude. Although the excitation
pulse acts symmetrically on both dots, the occupation of one dot has a larger overlap
with one of the eigenstates and thus the two systems emit radiation with different
amplitudes which leads to the reduction of the beating amplitude of the total signal.
The second effect, i.e., the change in the decay rate, is observed only in systems
collectively coupled to electromagnetic environment (Fig.
9.18
a). For large energy
mismatches (
E
=
Δ
V
) the polarization decays with a characteristic for single dots
decay rate
at the same time)
the polarization decay is transferred from a single exponential decay to a sum of two
exponential factors with different rates (long- and short-living terms). In the limit
of strong coupling (
V
Γ
/
2. With the increasing coupling (and decreasing
Δ
) only the superradiant state is excited (it coincides with
the optically active symmetric superposition) and thus the signal decays with twice
larger rate than in the case of
Δ
V
. On the contrary, the signal emitted by a pair
of dots coupled to separated reservoirs always decays with a single decay rate.
Also in this case, the collective effects are more clearly seen in the absorption
spectrum shown in Fig.
9.18
b, as well as the difference between common and
Δ
