Chemistry Reference
In-Depth Information
In the rotating wave approximation [33], the time evolution of
ρ
(
t
) satisfies the
following Redfield form of the Kossakowski-Lindblad equation
⎛
⎝
⎞
⎠
=
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
u
∗
e
−
iωt
ue
iωt
ρ
11
ρ
12
ρ
21
ρ
22
−
iγ
12
−
iγ
21
ρ
11
ρ
12
ρ
21
ρ
22
ue
iωt
ue
iωt
−
−
ω
−
i
0
i
∂
∂t
u
∗
e
−
iωt
u
∗
e
−
iωt
0
ω
−
i
−
u
∗
e
−
iωt
ue
iωt
iγ
12
−
−
iγ
21
where
u
is the complex Rabi frequency of the laser field (the real and imaginary
parts are the amplitudes of the real fields
E
x
and
E
y
) and
γ
12
,
γ
21
, and
are positive
real constants describing the interaction with the environment. In this equation,
ω
is the energy difference between the ground and excited states and the frequency
of the laser field. In the interaction representation, this equation becomes
⎛
⎞
⎛
⎞
⎛
⎞
u
∗
ρ
11
ρ
12
ρ
21
ρ
22
−
iγ
12
−
ui
21
ρ
11
ρ
12
ρ
21
ρ
22
⎝
⎠
=
⎝
⎠
⎝
⎠
−
u
−
i
u
i
∂
∂t
0
(40)
u
∗
u
∗
0
−
i
−
u
∗
iγ
12
−
u
−
iγ
21
Here, the interaction representation means that we have performed the unitary
transformation
U
to the mixed-state
ρ
:
⎛
⎝
⎞
⎠
10 0 0
0
e
iωt
00
00
e
−
iωt
U
=
0
00
0 1
Since Tr[
ρ
]
1, the density matrix
ρ
depends on three real parameters that can
be given by the coordinates of the Bloch ball:
x
=
=
[
ρ
12
],
y
=
[
ρ
12
], and
2
2
z
ρ
11
. From (40), one deduces that these coordinates satisfy the following
system of inhomogeneous linear differential equations:
⎧
⎨
=
ρ
22
−
x
=−
x
+
u
2
z
y
=−
y
−
u
1
z
(41)
⎩
z
=
(
γ
12
−
γ
21
)
−
(
γ
12
+
γ
21
)
z
+
u
1
y
−
u
2
x
u
1
and
u
2
being two real functions such that
u
=
u
1
+
iu
2
. The dynamics is called
either unital if
γ
12
=
γ
21
, that is, the fixed point of the free dynamics is the center
of the Bloch ball or affine otherwise [21]. Equations (41) can be written in a more
compact form
x
=
F
0
+
u
1
F
1
+
u
2
F
2
