Chemistry Reference
In-Depth Information
A second test is to compute the
n
Jacobi fields
J
i
(
t
)
=
(
δx
i
(
t
)
,δp
i
(
t
))
,i
=
1
,...,n
such that
δp
i
(0)
=
e
i
,i
=
1
,...,n
. In this case, the rank condition reads
rank[
δx
1
(
t
)
,...,δx
n
(
t
)]
≤
n
−
2
for a conjugate time. In the third test, one uses the dynamics
x
=
F
(
x, u
)ofthe
system. One considers
n
−
1 Jacobi field such that the relations (9) hold. At a
conjugate point, the following determinant
det[
δx
1
(
t
)
,δx
2
(
t
)
,
···
,δx
n
−
1
(
t
)
,F
(
x
(
t
)
,u
(
t
))]
is equal to 0.
Using these different tests, the numerical implementation of the computation
of conjugate points is relatively straightforward. One has to integrate differential
equations and to compute the rank or the determinant of a matrix. The singular
value decomposition of this matrix is an efficient way to determine the variation
of the rank of a matrix. The rank decreases by one when one of the singular values
vanishes. Some examples will be given in Section IV for the time-optimal control
of two-level dissipative quantum systems.
The different tests are implemented in the COTCOT software (Conditions of
Order Two and COnjugates Times [30]) that can be downloaded free from the
website of [31].
III. APPLICATION TO THE CONTROL OF A THREE-LEVEL
QUANTUM SYSTEM
The objective of this section is to apply the PMP to the control of a three-level
quantum system by laser fields in a first part and then by laser fields plus von
Neumann measurements. The results of this section are taken from [32].
A. Formulation of the Problem
We consider a three-level quantum system whose dynamics is governed by the
Schr odinger equation. The system is described by a pure state
|
ψ
(
t
)
belonging
to a three-dimensional (3D) Hilbert space
H
. The time evolution of
|
ψ
(
t
)
is
given by
=
H
0
+
E
2
(
t
)
H
2
|
i
∂
∂t
|
ψ
(
t
)
E
1
(
t
)
H
1
+
ψ
(
t
)
(10)
where
H
0
is the field-free Hamiltonian defined in matrix form as:
⎛
⎞
−
E
0
00
000
00
E
0
⎝
⎠
