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In-Depth Information
The dynamics of the quantum system depicted in Fig. 1 is described in more
detail in [3] where it is analyzed with the coupling constant of the resonant cavity
with the environment (denoted by
k
) and the external radiation field (denoted
by
I
). Both QDs are coupled to the cavity and the coupling is represented by
g
.
Since the cavity and external fields interact with the QDs, the excitonic states
are affected by the radiative and non-radiative decays (
ʳ
r
and
ʳ
nr
respectively).
From this model, the Hamiltonian is given by:
H
=
H
0
+
H
1
+
H
2
,where
H
0
represents the free energy of the QD, the cavity, the phonons and the electro-
magnetic bath;
H
1
represents the internal QD-cavity and QD-QD interactions;
and
H
2
is the interaction between QD and cavity with the electromagnetic bath
and phonons. The master equation for the evolution of the systems dynamics is
given by:
2
dˁ
dt
=
−i
[
H,ˁ
]+
ʳ
r
(
t
)
[(
˃
+
ˁ˃
p
˃
p
−
˃
+
ˁ
)+
h.c
]
2
n
phot,we
−
−
p
=1
2
+
ʳ
r
(
t
)
2
[(
˃
p
−
ˁ˃
+
−
˃
+
˃
p
(
n
phot,we
+1)
ˁ
)+
h.c
]
−
p
=1
+
k
(
t
)
(1)
(
a
†
ˁa
aa
†
ˁ
+
h.c
)
2
n
phot,wc
−
+
k
(
t
)
2
+1)(
aˁa
†
−
a
†
aˁ
+
h.c
)
(
n
phot,wc
2
+
ʳ
nr
(
t
)
2
[(
˃
z
ˁ˃
z
−
˃
z
˃
z
ˁ
)+
h.c
]
.
(2
n
phon,we
+1)
p
=1
Where
H
is the Hamiltonian of the quantum system,
ʳ
r
and
ʳ
nr
are the ra-
diative and non-radiative exciton decay rates,
is the average number
of photons with excitonic Bohr frequency,
ˉ
e
is the Bohr frequency related to
the energy difference among “1” and “0” levels,
a
and
a
†
are the creation and
annihilation operators,
k
(
t
) is the cavity decay rate,
ˉ
c
is the cavity radiation
frequency,
n
phot
,ˉ
e
n
phot
,ˉ
c
is the average number of photons with cavity frequency and
is the average number of phonons with excitonic Bohr frequency,
˃
+
and
˃
−
are the excitation and de-excitation operators (excitonic operator) re-
spectively. Finally,
h.c
is the hermitian conjugate of the preceding term. Equation
(1) is time dependent but it is normalized regarding the spontaneous emission
rate for excitons in the empty electromagnetic field.
Projecting (1) on the Fock states, the density matrix is obtained. From it the
dynamics of the base states of the QD can be extracted to compute the quantum
dynamics of Wootters concurrence [4], which is defined by:
C
(
t
)=
max
0
,
ʻ
1
(
t
)
−
n
phon
,ˉ
e
ʻ
4
(
t
)
ʻ
2
(
t
)
−
ʻ
3
(
t
)
−
.
(2)
Where
ʻ
i
are the eigenvalues of the matrix
R
defined by:
ˁ
∗
·
R
=
ˁ
·
˃
y
·
˃
y
.
(3)
