Biomedical Engineering Reference
In-Depth Information
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
0
1
2
3
4
t [ps]
t [ps]
Fig. 9.8
Evolution of entanglement of the two-qubit system at various temperatures (0 K—
red
curve
,40K—
green curve
,80K—
blue curve
, 120 K—
violet curve
) with
D
6nm.
Left panel
corresponds to the initial state (
9.28
)andthe
right panel
corresponds to the initial state (
9.29
). The
green, oscillating curve in the left panel corresponds to
V
B
=
6ps
−
1
(at
T
=
=
40 K), the other curves
in the left panel correspond to
V
B
=
0
where
g
q
is defined in Sect.
9.2.2.2
. For long times, the factors cos
w
q
t
and sin
w
q
t
become quickly oscillating functions of
q
and their contribution averages to 0.
Consequently, the phase damping factors
B
ij
decrease form their initial value of 0 to
a certain asymptotic value depending on the material parameters, system geometry
and temperature, while the phase shift factors
A
ij
affect the system evolution at
small times and then average out to zero. As a result, the off-diagonal elements of the
density matrix are reduced and the phase information is partly erased. The following
calculations were performed for two identical, self-assembled GaAs/InGaAs QDs,
for which the wave function widths were taken equal to 4
.
4 nm for electrons and 3
.
6
nm for holes in the in-plane direction, and 1 nm in the growth direction for both.
The calculations of the evolution of the EOF are performed for two initial fully
entangled pure states
=
|
00
+
|
01
+
|
10
−|
11
(
1
)
0
|
ψ
(9.28)
2
=
|
01
−|
10
(
2
)
0
|
ψ
√
2
.
(9.29)
The evolution of the EOF of the qubit pair is shown in Figs.
9.8
and
9.9
.In
the absence of energy shift
V
B
, entanglement decays on a time scale of a few
picoseconds, corresponding to the timescale of the phonon-induced dephasing
process. At low temperatures or for overlapping systems, this process resembles the
decay of coherences in a single system under the same environmental influence [
15
].
However, for a sufficiently large separation between the systems and at sufficiently
high temperatures the initially maximal entanglement present in the state (
9.28
)
decays completely after a finite time even though the phonon-induced dephasing
is always only partial (see left panels of Figs.
9.8
and
9.9
). On the other hand, for
the singlet initial state [Eq. (
9.29
)], the destruction of entanglement is always only
partial mimicking the pure dephasing process (see right panels of Figs.
9.8
and
9.9
).
