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P
.t/ D
i
„
while from Eq. (
13.15
) it also holds
ŒH
0
C f.t/H
1
.t/, which results
into
V. /D
2i
„
.
C
Z Z
d
/
C
fH
0
Z ZH
0
C f.H
1
Z ZH
1
/g (13.18)
Choosing the control signal f.t/to be proportional to the gradient with respect to
f of the first derivative of the Lyapunov function with respect to time (velocity
gradient), i.e.
f.t/D kr
f
f V. /g
(13.19)
and for Z such that
C
H
0
Z D
C
ZH
0
(e.g. Z D H
0
) one obtains
2i
„
.
C
Z Z
d
/
C
.H
1
Z ZH
1
/
f.t/D
(13.20)
Substituting Eqs. (
13.20
)into(
13.18
) provides
V. /D k
2i
„
.
C
Z Z
d
/
C
Œ
2i
„
.
C
Z Z
d
/
C
.H
1
Z ZH
1
/ .H
1
Z ZH
1
/
(13.21)
and finally results in the following form of the first derivative of the Lyapunov
function
V. /Dk
4
„
2
.
Z Z
d
/
2
C
2
.H
1
Z ZH
1
/
2
2
0
(13.22)
which is non-positive along the system trajectories. This implies stability for the
quantum system and in such a case La Salle's principle shows convergence not to
an equilibrium but to an area round this equilibrium, which is known as
invariant
set
[
92
].
Consequently, from Eq. (
13.22
) and LaSalle's theorem, any solution of the
system .t/ remains in the invariant set M Df W V. /D 0g.
13.4.2
Control Law Calculation Using Lindblad's Equation
Next, it will be shown how a gradient-based control law can be formulated using the
description of the quantum system according to Lindblad's equation. The following
bilinear Hamiltonian system is considered (Lindblad equation)
P .t/ DiŒH
0
C f.t/H
1
;.t/
(13.23)
