Chemistry Reference
In-Depth Information
Let
be the six-dimensional vector of coordinates
x
i
. Equation (13) reads in a
more compact form as:
x
x
=
u
1
F
1
+
u
2
F
2
(15)
where
F
1
and
F
2
are the vector fields
⎛
⎞
⎛
⎞
x
4
−
x
3
x
4
x
5
−
⎝
⎠
⎝
⎠
x
3
x
2
+
x
6
x
1
F
1
=
F
2
=
.
(16)
−
x
1
−
x
5
x
6
−
x
2
x
4
−
−
x
3
−
x
3
x
4
since
i
=
1
x
i
The dynamics takes place on a five-dimensional sphere
S
5
1.
The interaction of the system with the laser field consists of a dipolar interaction
with constant dipolar terms coupling only neighboring states. It is known that
this type of system is not completely controllable by unitary evolution [32]. The
noncomplete controllability of the system can be understood geometrically as
follows: We define new coordinates
X
i
by
⎧
⎨
=
1
X
1
=
√
2
(
x
1
−
x
5
)
1
X
2
=
√
2
(
x
1
+
x
5
)
X
3
=
x
3
.
⎩
X
4
=
x
4
1
X
5
=
√
2
(
x
2
+
x
6
)
1
X
6
=
√
2
(
x
2
−
x
6
)
For any choice of the controls
u
1
and
u
2
in Eq. (15), only the coordinates
{
X
1
,X
3
,X
5
}
and
{
X
2
,X
4
,X
6
}
are coupled between each other. In addition, they
fulfill the following relations
X
1
+
X
3
+
X
5
=
R
i
X
2
+
X
4
+
X
6
=
R
f
where
R
i
and
R
f
are two real constants, such that
R
i
R
f
1. The system thus
evolves on two spheres that we denote
S
i
and
S
f
. The radii
R
i
and
R
f
of the
two spheres are determined from the initial state of the system and are constant
for unitary evolution. To simplify the geometrical description of the control, we
consider in this chapter that one of the radius is equal to one and the other to zero.
+
=
