Biomedical Engineering Reference
In-Depth Information
increases, but its amplitude decreases, so in the limit of large energy mismatches the
transitions become very frequent, their amplitude becomes small and, as expected
for systems interacting with disjoint frequency ranges of the photon reservoir, the
decay of fidelity closely follows the exponential function which is characteristic for
uncorrelated systems.
The coupling between the single-exciton states of the QDs becomes effective
when the distance between the systems is close enough. Contrary to the previous
case, where the localized states
were the eigenstates of the system, for
coupled dots the eigenstates are linear combinations of the states
|
10
and
|
01
|
10
and
|
01
, i.e.,
cos
2
|
sin
2
|
|
Ψ
+
=
10
+
01
,
(9.30)
sin
2
|
cos
2
|
|
Ψ
−
=
−
10
+
01
,
(9.31)
where the mixing angle
.Ifthe
coupling exceeds the energy splitting, then these states become relatively close to
the sub- and superradiant states (they become the sub- and superradiant states only
in the ideal case of identical dots, where they correspond to eigenvalues
θ
=
arctg
(
V
/
Δ
)
takes values in the range
[
−
π
,
π
)
V
). In this
case, the transition to the superradiant regime is suppressed and, as can be seen in
Fig.
9.11
b, the superradiant character of the evolution is recovered to a great extent.
Even if the coupling is not sufficiently strong to rebuild the stability, it is clear from
Fig.
9.11
b that its existence reduces the decay rate.
If a pair of uncoupled QDs with different transition energies is prepared in a
superradiant state (red line in Fig.
9.11
c), then the system maintains its superradiant
character and decays with a decay rate 2
±
andthenenters
the subradiant regime. As previously, the fidelity oscillates around the exponential
decay and the collective character of the evolution, in this case this means a decay
with the rate 2
Γ
until
t
∼
π
h
/
(
2
Δ
)
, may be recovered for a sufficiently strong coupling between the
dot. Apart from the limiting case of vanishing and strong coupling, the decay is non-
exponential and its modulation yields information about the origin of the energy
splitting in the system. It is clearly seen in Fig.
9.11
c, where we compare
a decay
of the fidelity for two systems with the same energy splitting
Γ
2
√
V
2
,
originating only from the difference between the two dots (red curve) or from the
system properties and coupling between the single-exciton states (black curve).
As discussed above, the collective character of evolution is destroyed, if the
sub- and superradiant states are spanned in inhomogeneous ensembles with energy
mismatches of the order of
2
E
=
Δ
+
eV. In realistic DQDs the transition energy splitting
is rather on the order of meV than
μ
eV; therefore, once again we will consider the
dynamics of the rapidly decaying state
μ
, but this
time in terms of the exciton occupation of the system and for the technologically
achievable
|
+
and optically inactive state
|−
,
the decay of the exciton occupation shows no oscillations. In the limit of weak
coupling,
V
Δ
=
1 meV. As can be seen in Fig.
9.12
, due to the large value of
Δ
, regardless of whether the system was prepared initially in the
sub- or superradiant state, the exciton occupation decays exponentially with a decay
Δ
