Chemistry Reference
In-Depth Information
a basis of
H
. A possible choice is given by the vectors
⎧
⎨
1
|
ϕ
1
=
√
2
(
|
1
+|
3
)
|
ϕ
2
=
i
|
2
⎩
i
|
ϕ
3
=
√
2
(
|
1
−|
3
)
=
i
=
1
q
i
|
The corresponding observable
Q
is equal to
Q
ϕ
i
ϕ
i
|
, where the
q
i
's
=
i
=
1
a
i
|
are real numbers. Let
be the state of the system at time
t
. Then
after the measurement, this state becomes
|
ψ
ϕ
i
|
ϕ
i
(
i
∈{
1
,
2
,
3
}
) with the probability
2
. Since all the eigenvectors of
Q
belong to
S
f
, the sphere of the target state,
one sees that, whatever the result of the measurement, the target state can now be
reached by unitary evolution. Note also that the control by laser field on the sphere
S
f
will depend on the result of the measurement. We assume that the operator
knows this result and can modify the control field according to the result of the
measurement.
To solve this problem, we begin by the standard formulation of the PMP, (i.e.,
without measurement), as shown in Section III. B. Then, we assume that a mea-
surement is performed at a time
t
|
a
i
|
[0
,T
]. The definition of the cost given below
can be extended straightforwardly to the case of several measurements. We denote
by
∈
|
ψ
(
t
)
the state of the system at time
t
at which the measurement is performed.
Since
{
ϕ
i
}
(
i
=
1
,
···
,
3)
is a basis of
H
,
|
ψ
(
t
)
can be written as:
|
ψ
(
t
)
=
a
i
(
t
)
|
ϕ
i
i
Let
C
0
(
t
) be the cost corresponding to the optimal path from
|
ψ
i
to
|
ψ
(
t
)
. Note
that
C
0
is equal to zero if
t
=
0. We also introduce the costs
C
i
,(
i
=
1
,...,
3)
which are, respectively, associated to the optimal passage from
|
ϕ
i
to
|
ψ
f
. The
total cost of the control
C
(
t
) is then defined by
3
2
C
i
C
(
t
)
=
C
0
(
t
)
+
1
|
a
i
(
t
)
|
i
=
The choice of
C
(
t
) is related to the fact that the operator knows the result of the
measurement and can modify the electric field accordingly. Here,
i
=
1
|
2
C
i
can be viewed as an average of the three costs
C
i
. From the cost
C
(
t
), the goal
is then to determine the control fields, the observable
Q
and the time
t
at which
the measurement is performed to minimize
C
(
t
). We will solve this problem by
using twice the PMP: Once on
S
i
and once on
S
f
. Indeed, for a fixed observable
Q
, it is clear that the trajectories that minimize
C
correspond to the concatenation
of extremal trajectories on
S
i
and
S
f
. We solve this optimal control problem in
a
i
(
t
)
|
