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Fig. 13.1
Feedback control loop for quantum systems
13.4
A Feedback Control Approach for Quantum
System Stabilization
13.4.1
Control Law Calculation Using Schrödinger's Equation
It is assumed that the dynamics of the controlled quantum system is described by a
Schrödinger equation of the form
i„
P
.t/ D ŒH
0
C f.t/H
1
.t/ .t/ 2 C
n
(13.15)
where H
0
is the system's Hamiltonian, H
1
is the control Hamiltonian, and f.t/is
the external control input. The following Lyapunov function is then introduced [
9
]
V. /D .
C
Z Z
d
/
2
(13.16)
where C stands for the transposition and complex conjugation, Z is the quantum
system's observable and is associated with the energy of the system. The term
C
Z denotes the observed mean energy of the system at time instant t and
Z
d
is the desirable energy value. The first derivative of the Lyapunov function of
Eq. (
13.16
)is
V. /D 2Œ
C
Z Z
d
Œ
P
C
Z C
C
Z
P
(13.17)
