Chemistry Reference
In-Depth Information
C
T
is then minimized with respect to
θ
i
and
ϕ
i
. We have checked numerically
that the optimal solution corresponds to
C
0
=
0. The total cost is given by
C
T
C
. We have also obtained the same result when the initial angle is different
from
π/
2.
=
IV. APPLICATION TO THE TIME-OPTIMAL CONTROL OF
TWO-LEVEL DISSIPATIVE QUANTUM SYSTEMS
The objective of this section is to describe the geometric aspects of the optimal
control of two-level dissipative quantum systems. When a quantum system inter-
acts with a Markovian or a non-markovian bath, the control field cannot generally
fully compensate the dissipation effect that largely enhances the difficulty of the
control. This point has been rigourously shown in [21] for a dynamics governed
by the Lindblad equation [41-43]. In this context, several studies using numerical
optimization techniques have proved that efficient control can still be achieved (see
[44-50] to cite a few). We propose here to give a geometric analysis of this ques-
tion in the time-minimal case, the maximum of the control field being fixed to an
arbitrary value. Note that this cost functional is particularly relevant in the context
of a dissipative environment especially when the dissipation effect is undesirable
to reach the target state.
A. The Kossakowski-Lindblad Equation for
N
-Level Dissipative
Quantum Systems
First, we recall the form of the Kossakowski-Lindblad equation for an
N
-level
quantum system described by a density operator
ρ
[41, 42, 51, 52]. The dynamics
of
ρ
is governed by the following first-order differential equation:
i ρ
(
t
)
=
[
H, ρ
(
t
)]
+
iL
D
[
ρ
(
t
)]
(34)
This equation differs from the standard von Neumann equation
i ρ
(
t
)
[
H, ρ
(
t
)]
in that a dissipation operator
L
D
acting on the set of density operators has been
added. This linear operator, describes the interaction with the environment, can-
not be chosen arbitrarily. Under particular assumptions, such as the semigroup
dynamics, the norm continuity, and the conservation of probability [41, 42], the
form of
L
D
can be deduced from a rigorous mathematical analysis. The dissipation
operator
L
D
can be written as follows:
=
2
k
1
L
D
[
ρ
(
t
)]
=
([
L
k
ρ
(
t
)
,L
k
†
]
+
[
L
k
,ρ
(
t
)
L
k
†
])
(35)
