Chemistry Reference
In-Depth Information
In this case, we can assume that the state belongs only to one sphere since the
dynamics on the other sphere is trivial. Using this description, one deduces that if
the system is initially on one of the two spheres, then it will remain on this sphere
by unitary evolution and it will not reach a state belonging, for example, to the other
sphere. The description of the control is more difficult if the two radii are different
from zero. In particular, due to the symmetry of Eq. (15), the optimal trajectories
on the two spheres are the same and cannot be controlled independently.
B. Optimal Control of a Three-Level Quantum System:
The Grushin Model
In this section, we consider the optimal control on one of the two spheres. This
corresponds to the optimal control problem associated with the Grusin model on the
sphere [14, 34]. The geometric properties of this model were presented in Section
II.C, but we show here that this problem can be completely solved analytically. We
analyze the optimal control of this three-level system either with the constraint of
minimizing the duration of the control or the energy of the laser field. We denote
by
U
2
the manifold of admissible values of the control fields. For the time
minimum cost, we have the condition
u
1
+
⊂ R
u
2
≤
1 on the control field, whereas
there is no restriction on laser fields if the cost minimizes the energy. The total
duration
T
of the control is fixed for the energy cost problem.
We begin with the standard formulation of the PMP. We introduce the pseudo-
Hamiltonian
H
P
, which can be written in our case as follows:
H
p
=
p.
(
u
1
F
1
+
u
2
F
2
)
+
p
0
f
0
(
u
1
,u
2
)
6
is the adjoint state and
p
0
is a negative constant such that
p
and
p
0
are not simultaneously zero. Here,
f
0
is a function of
u
1
and
u
2
whose integral
over time gives the associated cost
C
.Wehave
where
p
∈ R
T
T
[
u
1
(
t
)
u
2
(
t
)]
dt
C
E
=
f
0
(
u
1
(
t
)
,u
2
(
t
))
dt
=
+
0
0
for the energy minimization problem and
T
C
T
=
dt
=
T
0
for the time-minimum optimal control. The PMP states that the coordinates of
the extremal vector state
x
and of the corresponding adjoint state
p
fulfill the
Hamiltonian's equations associated with the Hamiltonian
H
P
x
∂H
P
∂p
=
∂H
P
∂x
p
=−
