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In-Depth Information
dzdz
C
D„H
c
dt
;
dzdz
T
D„Y
dt
. In the above equations matrix Y is a symmetric
complex-valued matrix. Variable H
c
is defined as m
1
DfH
c
D
diag
.n
1
; ;n
L
/ W
8l;n
l
2Œ0;1g, where n
l
can be interpreted as the possibility of monitoring the
l-th output channel. There is also a requirement for matrix U to be positive semi-
definite. As far as the
measured output
of the Belavkin equation is concerned one
has an equation of complex currents J
T
dt
D< cH
c
Cc
C
Y>
c
C
dz
T
, where
<> stands for the mean value of the variable contained in it [
210
]. Thus, in the
description of the quantum system according to Belavkin's formulation, the state
equation and the output equation are given by Eq. (
13.14
).
„d
c
D
dt
DŒc
c
C HŒi H
dt
C
dz
C
.t/c
c
J
T
dt
D< c
T
H
c
Cc
C
Y
c
>
dt
C
dz
T
(13.14)
where
c
is the probability density matrix (state variable) for remaining at one of
the quantum system eigenstates, and J is the measured output (current).
13.3.3
Formulation of the Control Problem
The control loop consists of a cavity where the multi-particle quantum system is
confined and of a laser probe which excites the quantum system. Measurements
about the condition of the quantum system are collected through photodetectors and
thus the projections of the probability density matrix of the quantum system are
turned into weak current. By processing this current measurement and the estimate
of the quantum system's state which is provided by Lindblad's or Belavkin's
equation, a control law is generated which modifies a magnetic field applied to the
cavity. In that manner, the state of the quantum system is driven from the initial
value .0/ to the final desirable value
d
.t/ (see Fig.
13.1
).
When Schrödinger's equation is used to describe the dynamics of the quantum
system the objective is to move the quantum system from a state , that is associated
with a certain energy level, to a different eigenstate associated with the desirable
energy level. When Lindblad's or Belavkin's equation is used to describe the
dynamics of the quantum system, the control objective is to stabilize the probability
density matrix .t/ on some desirable quantum state
d
.t/2C
n
, by controlling the
intensity of the magnetic field. The value of the control signal is determined by
processing the measured output which in turn depends on the projection of .t/
defined by
Tr
fP.t/g.
