Chemistry Reference
In-Depth Information
For the energy, we have
θ
i
T
C
2
cos
2
θ
i
+
C
3
sin
2
θ
i
cos
2
ϕ
i
C
E
=
+
whose minimum can be determined as above.
Case (c)
To simplify the discussion, here we limit the study to the time-optimal control
problem. As already mentioned, we determine the target state and the angle
α
such
that the travel time of the three extremal trajectories on
S
f
is the same. The optimal
trajectories and the optimal fields for the extremal starting from (
θ
=
π/
2
,ϕ
=
π/
2) are displayed in Fig. 15. When changing
p
θ
to
p
θ
with the same value
of
j
, we obtain two trajectories starting, respectively, from (
θ
−
=
π/
4
,ϕ
=
0) and
(
θ
π/
2. These two
extremals intersect on this axis at the same time. Then, we determine the parameters
of the trajectories initiated from (
θ
=
3
π/
4
,ϕ
=
0) that are symmetric with respect to the axis
θ
=
=
π/
2
,ϕ
=
π/
2) and (
θ
=
π/
4
,ϕ
=
0) such
that they intersect on the axis
θ
π/
2 at the same time. We have solved this
problem numerically. We have obtained
p
θ
(0)
=
0
.
6272 and
j
=−
1 for the first
extremal and
p
θ
(0)
−
0
.
9245 and
j
=
1 for the second one. The two extremals
intersect in
ϕ
1
.
668. Let
T
0
be the cost corresponding to these
three trajectories. The total cost
C
T
is given by
1
.
091 at time
t
C
T
=
C
0
+
T
0
Since by definition
C
0
≥
0, the optimal solution is
C
0
=
0 (i.e., the measurement
has to be performed at time
t
=
0). The result is thus independent of the initial
state of
S
i
.
Note that the same work can be done for the minimization of the energy.
Case (d)
Here, again, we only consider the minimization of the time. Using Eq. (28), the
time to go from (
π/
2
,
0) to the state (
θ
i
,ϕ
i
) where the measurement performed is
arcsin(
1
−
1
m
i
cos
θ
i
)
1
C
0
=
+
(33)
m
i
+
where
m
i
=
j/ h
T
. To establish Eq. (33), we have assumed that the function
θ
(
t
)is
an increasing function of time. In addition, we notice that the value
ϕ
i
depends on
the constant
m
i
chosen. In other words, minimizing the cost with respect to (
θ
i
,ϕ
i
)
is equivalent to minimizing the cost with respect to
θ
i
and
m
i
. Next, we determine
the cost to reach the target state from each state of the measured operator. From
