Chemistry Reference
In-Depth Information
3
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
2
2.5
3
φ
Figure 11.
π/
4
and
ϕ
(0)
=
0.
p
θ
(0)
=±
5 for the two trajectories in solid lines and
p
θ
(0)
=
0 for the extremal in dashed
lines.
Flow of the Hamiltonian
H
T
. Numerical values are taken to be
p
ϕ
(0)
=
2,
θ
(0)
=
can be determined by using the previous analytical computations. From geometric
arguments of Section II.C, we recover also the notion of cut locus where two
trajectories starting from the same initial point intersect with the same cost. The cut
locus is here a subset of the antipodal parallel. Reference [35] gives a mathematical
description of the cut locus for a class of metrics similar to the Grushin one on a
two sphere.
C. Optimal Control of a Three-Level Quantum System by Laser Fields
and von Neumann Measurements
Now, we describe the control assisted by von Neumann measurements (VNMs).
The VNMs can be classified into two types: the instantaneous measurements and
the continuous ones. Among instantaneous measurements, we also distinguish
the selective ones where the state after the measurement is known (with a given
probability) and the nonselective ones where this state is unknown [36]. Here,
we will only consider selective instantaneous measurements. The question of the
measurement driven quantum evolution has already been discussed in a series of
work mainly from a numerical point of view (see [36-40] to cite a few), either
with or without a laser field assisting the control. In this section, we revisit this
problem by using the PMP and a more geometrical point of view.
