Biomedical Engineering Reference
In-Depth Information
a
b
D = 6 nm
8 nm
10 nm
Δ = 2 meV
4 meV
6 meV
12
8
4
0
0
2
4
6
4
6
8
10
12
D [meV]
D
[nm]
Fig. 9.7 The rate of phonon-assisted excitation transfer as a function of the energy mismatch for
a few values of the QD separation D ( a ) and as a function of D for a few values of
Δ
( b )at
T
=
4K[ 62 ]
andtothe
Schr odinger picture one obtains a complex evolution. However, in the case of
In general, upon transforming back to the original basis
|
01
,|
10
|
V
|
| Δ |
, which is of particular practical importance, the eigenstates of H DQD are very
close to
|
10
and
|
01
. Moreo ver, in this limit the energy difference between the
2 V 2
eigenstates, h
Ω =
+ Δ
2
is nearly equal to 2
Δ
. In this limit, one obtains an
exponential excitation transfer with the rate
V
Δ
2
γ =
4
π
[
R
( Ω )+
R
( Ω )] ,
with the spectral density
2
h 2 |
sin 2 k z D
| k
2
R
( ω )=
n B ( ω )+
1
2 |
f k |
δ ( | ω |−
w k ) .
Thus, the Markovian equations are particularly useful in the limit of weak coupling,
where the Markovian dephasing rate may be identified with the rate of irreversible
excitation transfer between the dots.
The rate for the phonon-assisted process is governed, on the one hand, by the
amplitude of the Forster coupling which decreases roughly as 1
D 3 . On the other
/
hand, it is strongly influenced by the structure of R
( Ω )
. The latter oscillates as
a function of both
and D [ 63 ] due to the interplay between the wavelength
of the emitted phonon and the QD separation in the molecule (phonons are
preferentially emitted along the strongest confinement limit, i.e., along the DQD
axis). In particular, R
Ω
( Ω )
has a pronounced minimum whenever
Ω
is a multiple of
2
π
hc l /
D which explains the oscillating dependence on both
Δ
and D (see Fig. 9.7 ).
3ns 1 , which is about twice
The transfer rate for
Δ =
4meV, D
=
4
.
5nmis
γ =
2
.
25 ns 1 deduced from fitting to the photon correlation
data [ 9 ]. It should be noted, however, that our modeling is based on certain choices
of parameters that cannot be uniquely determined. First of all, the magnitude of the
Forster coupling for a given inter-dot distance can only be roughly estimated, since
lower than the value of
Γ =
5
.
 
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