Chemistry Reference
In-Depth Information
with the vector
x
of coordinates (
x, y, z
) and the three vector fields
F
0
,
F
1
, and
F
2
defined by
⎛
⎞
⎛
⎞
⎛
⎞
−
x
0
−
z
0
⎝
⎠
⎝
⎠
⎝
⎠
F
0
=
−
y
F
1
=
z
y
F
2
=
(
γ
12
−
γ
21
)
−
(
γ
12
+
γ
21
)
z
−
x
C. Geometric Analysis of Lindblad Equation
The objective of this section is to present control techniques that can be used to
analyze finite-dimensional quantum systems whose dynamics is governed by the
Lindblad equation. In particular, our goal is to solve the control problem associated
with the two-level dissipative quantum system modeled in Section IV.B [Eq. (41)].
We consider the time-optimal problem with constraint
u
1
+
u
2
≤
1, which is not
restrictive up to a rescaling of the dissipative parameters. The energy minimization
control problem can be analyzed along the same lines and shares similar properties.
First, we give some general results to present the geometric framework.
1.
Symmetry of Revolution
We consider a rotation of angle
θ
with axis (
Oz
):
X
=
x
cos
θ
+
y
sin
θ
Y
=−
x
sin
θ
+
y
cos
θ
Z
=
z
and a similar rotation on the controls:
v
1
=
u
1
cos
θ
+
u
2
sin
θ
v
2
=−
u
1
sin
θ
+
u
2
cos
θ
We obtain the system
X
=−
X
+
v
2
Z
Y
=−
Y
−
v
1
Z
Z
=
γ
−
−
γ
Z
+
v
1
Y
−
v
2
X
+
This defines a one-dimensional symmetry group. By construction,
v
1
+
v
2
=
u
1
+
u
2
