Information Technology Reference
In-Depth Information
where H
0
is the interaction Hamiltonian of the quantum system, H
1
is the control
Hamiltonian of the quantum system, and f.t/is the real-valued control field for the
quantum system. The control problem consists of calculating the control function
f.t/ such that the system's state (probability transition matrix .t/) with initial
conditions .0/ D
0
converges to the desirable final state
d
for t!1.Itis
considered that the initial state
0
and the final state
d
have the same spectrum
and this is a condition needed for reachability of the final state through unitary
evolutions.
Because of the existence of the interaction Hamiltonian H
0
it is also considered
that the desirable target state also evolves in time according to the Lindblad
equation, i.e.
P
d
.t/ DiŒH
0
;
d
.t/
(13.24)
The target state is considered to be stationary if it holds ŒH
0
;
d
.t/ D 0, therefore
in such a case it also holds P
d
.t/ D 0. When ŒH
0
;
d
.t/¤0 then one has P
d
¤0 and
the control problem of the quantum system is a tracking problem. The requirement
.t/!
d
.t/ for t!1 implies a trajectory tracking problem, while the requirement
.t/!O.
d
/.t/ for t!1 is an orbit tracking problem.
It will be shown that the calculation of the control function f.t/which assures
that .t/ converges to
d
.t/ can be performed with the use of the Lyapunov method.
To this end, a suitable Lyapunov function V.;
d
/ will be chosen and it will be
shown that there exists a gradient-based control law f.t/such that V.;
d
/0.
The dynamics of the state of the quantum system, as well as the dynamics of the
target state are jointly described by
P .t/ DiŒH
0
C f.t/H
1
;.t/
P
d
.t/ DiŒH
0
;
d
.t/
(13.25)
A potential Lyapunov function for the considered quantum system is taken to be
V D 1
Tr
.
d
/
(13.26)
It holds that V>0if ¤
d
. The Lyapunov function given in Eq. (
13.26
) can
be also considered as equivalent to the Lyapunov function V. ;
d
/ D 1 j <
d
.t/j .t/ > j
2
, which results from the description of the quantum system with
the use of Schrödinger's equation given in Eq. (
13.1
). The term <
d
.t/j .t/ >
expresses an internal product which takes value 1 if
d
.t/ and .t/ are aligned.
The first derivative of the Lyapunov function defined in Eq. (
13.26
)is
V D
Tr
. P
d
/
Tr
.
d
P /
(13.27)