Commun. Theor. Phys. (Beijing, China) 44 (2005) pp. 455–458
c International Academic Publishers
Vol. 44, No. 3, September 15, 2005
Continuous Quantum Nondemolition Measurements of a Particle in Electromagnetic
and Gravitational Fields∗
ZHU Chun-Hua1,† and ZHA Chao-Zheng1,2
1
Department of Physics, Xinjiang University, Wulumuqi 830046, China
2
CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China
(Received January 27, 2005)
Abstract The detection of a particle in electromagnetic plus gravitational fields is investigated. We obtain a set
of quantum nondemolition variables. The continuous measurements of these nondemolition parameters are analyzed in
the framework of restricted path integral formalism. We manipulate the corresponding propagators, and deduce the
probabilities associated with the possible measurement outputs.
PACS numbers: 03.65.-w
Key words: continuous quantum nondemolition measurements, restricted path integral
1 Introduction
The problem of quantum measurement has provoked
a deep interest since the outset of quantum theory. From
the ordinary quantum mechanics we know that the measurement on a quantum system obeys the von Neumann’s
reduction postulate. Nevertheless, this approach has the
unpleasant feature of introducing an extra assumption in
the theory and deals only with instantaneous and perfect
measurement. During the past two decades many authors
attempted to solve these problems via various approaches
and a number of models have been proposed.
In this paper we investigate the quantum nondemolition (QND) measurement of a particle moving in
an electromagnetic plus gravitational field by means of
the restricted path integral (RPI) formalism, which was
proposed by Feynman[1] to describe continuous quantum measurement (CQM), and was developed latter by
Mensky.[2] The measurement system is considered in the
RPI theory of measurements as an open system, and the
back influence of the environment onto the measured system is taken into account by restricting the path integral.
The restriction is determined by the information which
the measurement imposes on the measured system and is
described by a weight functional. The RPI approach is
one of phenomenological descriptions of CQM, and does
not require an explicit model of the measuring device.
The RPI method has important applications in many
fields. For instance, it has been applied to the analysis
of QND measurements.[3−5] QND measurements were introduced by Braginsky, Vorontsov, and Khalili[6] and discussed later from different points of view.[7] Caves etc. defined the QND measurements of  as a sequence of precise
measurements of  such that the result of each measurement (after the first) is completely predictable from the
∗ The
result of the preceding measurement. In such a measurement one can monitor an observable with arbitrarily small
error, in principle, without quantum limit.
Camacho discussed QND measurement of a particle
in electric and gravitational fields.[8,9] In his model he ignored the magnetic fields. In this work we shall find a
set of QND variables for the case of a particle moving
in an electromagnetic and gravitational fields of a spherical body, for instance, in the Earth’s electromagnetic and
gravitational fields. In this sense, we would say our model
is a more realistic scenario. We will consider the continuous monitoring of these nondemolition parameters, and
calculate the corresponding propagators along the ideas
of the RPI formalism. Using these results the probabilities associated with the possible measurement outputs are
evaluated.
2 Our Model
Suppose we have a spherical body with mass M , radius R, and electric charge Q. We assume that the sphere
rotates around an axis of a sphere with an angular velocity ω0 . For the sake of simplicity, we suppose that the
sphere is provided with a constant electric charge density.
Now let us consider a particle whose mass and charge are
m and q, respectively. The Hamiltonian of the particle
moving in our electromagnetic and gravitational fields is
given by
1
~ )2 + k qQ − G mM ,
( p~ − q A
(1)
2m
r
r
where r denotes the distance from the center of the sphere,
~ is the vecp~ is canonical momentum of the particle, and A
tor potential of the electromagnetic field. Considering the
H=
project supported by National Natural Science Foundation of China under Grant No. 10265003
xdzch2002@yahoo.com.cn
† E-mail:
456
ZHU Chun-Hua and ZHA Chao-Zheng
~ is then written as
symmetry of the system, A
2
~ = µ0 ω0 QR sin θ ~ey ,
A
20πr2
where θ is the angle between rotational axis and vector ~r.
The corresponding Lagrangian is written as
p~ 2
qQ
n2 q 2 Q2 R4
mM
−k
−
+G
,
(2)
2m
2mr4
r
r
where n = µ0 ω0 sin(θ)/20π
Let r = R + z, where z is the height over the sphere.
When the height is much smaller than the radius of the
sphere z R, the Hamiltonian and Lagrangian become
L=
p~ 2
nqQ
2nqQ
1
−
py +
py z − F z + mω 2 z 2 ,
2m
m
mR
2
2
p~
1
L=
+ F z − mω 2 z 2 ,
2m
2
respectively, where
H=
(3)
(4)
qQ
mM
2n2 q 2 Q2
+k 2 −G 2 ,
(5)
mR
R
R
5n2 q 2 Q2
qQ
M
+
k
−
G
.
(6)
ω2 = 2
m2 R2
mR3
R3
From Eq. (4), it can be easily seen that the particle in our
model is a harmonic oscillator subjected to force F .
Compare our model with that of a particle moving in
electric and gravitational fields[8] in which the condition
for ω 2 > 0 is qQ > GM m/k, and the charge are only
of the same sign, or, if the charges are of opposite sign
(qQ < 0), the harmonic oscillator is then provided with a
complex frequency (ω 2 < 0). We find that in our model
the effect of magnetic field on the particle are represented
by 2n2 q 2 Q2 /mR and 5n2 q 2 Q2 /m2 R2 in Eqs. (5) and (6),
so the corresponding conditions is changed. Thus the condition for ω 2 > 0 is qQ > GM m/k − 5n2 q 2 Q2 R/mk and
the frequency
r
5n2 q 2 Q2
qQ
M
|ω| = 2
+
k
−
G
m2 R2
mR3
R3
is larger than that of the previous model. For ω 2 < 0, the
charge is of opposite sign. Moreover, not only the charges
should be opposite in sign for ω 2 < 0 in our model, the
condition 5n2 q 2 Q2 R/mk < GM m/k + |qQ| should also
be satisfied. It is worthy of mentioning that the previous
model is a one-dimensional system (having z component
only). While the present model is a two-dimensional system (having y component as well as z component) due to
~
the vector potential A.
F =
3 Continuous Quantum Nondemolition
Variables
Now we introduce the operator
A(t) = ρq + σp ,
(7)
where ρ, σ are real functions of time. Suppose now that
the continuous monitoring of the observable A is carried
Vol. 44
out during the time interval [0, T ], and the result of this
measurement being expressed in terms of function a[t]. An
effective Hamiltonian H̃ can be introduced,
ih̄
(A − a)2 ,
H̃ = H −
T ∆a2
where ∆a is the measurement error. In the case of multidimensional system, the condition for A to be a QND
variable is that the equation of motion for à = ρq̃ + σ p̃
(corresponding to the effective Hamiltonian H̃) coincides
with the equation for A (corresponding to the Hamiltonian H), all terms containing h̄/T ∆a2 being cancelled.[3]
We can apply this criterion in our case.
Our model is a two-dimensional systems because of
the magnetic field. So if ω 2 > 0, we obtain a set of QND
observable,
m
A1 = σ(t) − y + py ,
(8)
t
A2 = σ(t)(mω tan(ωt)z + pz ) .
(9)
And if ω 2 < 0, we can also have
m
A1 = σ(t) − y + py ,
t
A2 = σ(t)(−m|ω| tanh(|ω|t)z + pz ) .
(10)
(11)
Any choice for function σ(t) gives a set of QND variables.
In this work, we shall only consider the case ω 2 > 0, and
for ω 2 < 0 we can also deal with in the same way.
For the sake of simplicity we assume σ(t) = 1 . So we
obtain a set of QND variables,
m
(12)
A1 = − y + py ,
t
A2 = mω tan(ωt)z + pz .
(13)
Such a set of QND variables have the following properties. The variables can be measured simultaneously with
arbitrary precision, and in a sequence of precise, simultaneously measuring of all the variables, the results of each
set of measurements can be predicted from the results of
the preceding set.[7]
4 Propagator and Probability
We have already learned that a continuously measured
quantum system can be described by RPI method.[2,10]
The whole quantum theory of measurements can be derived from quantum mechanics by means of path integral
in the framework of the RPI approach. From the behavior of a set of QND variables we know that the variables
A1 and A2 can be measured simultaneously with arbitrary
precision.
First, let us consider the situation in which A2 (t) is
continuously monitored. The measuring progress is taken
into account through a weight functional in the path integral. We choose a weight functional in gaussian form,
Z T
n
o
1
2
[A
(t)
−
a
(t)]
dt
. (14)
ω[a2 (t)] [ p, q] = exp −
2
2
T ∆a22 0
No. 3
Continuous Quantum Nondemolition Measurements of a Particle in Electromagnetic and Gravitational Fields
Here a2 (t) is the measurement output, and ∆a2 is the
measurement error. According to RPI approach, if the
Z
U[a2 (t)] (W, T ; N, 0) =
n
d[ p]d[q] exp −
1
T ∆a22
Z
0
T
457
particle goes from point N to point W the corresponding
propagator is given by
n i Z T p~ 2
o
1
[A2 (t) − a2 (t)] dt exp
+ F z − mω 2 z 2 dt .
h̄ 0 2m
2
2
o
Considering expression (13), we obtain the propagator
Z T
o
n im
o
n
m 1
a22 (t)dt
exp
[(xW − xN )2 + (yW − yN )2 ] exp −
U[a2 (t)] (W, T ; N, 0) =
2
2πih̄T
2h̄T
T ∆a2 0
Z
Z
n i T h 1
i o
2ih̄
ih̄ 2
× d[z]d[ pz ] exp
pz −
(a2 − mω tan(ωt))z)pz dt
+
2
2
h̄ 0
2m T ∆a2
T ∆a2
n i Z T h ih̄
1
(mω tan(ωt))2 − mω 2 z 2
× exp
2
h̄ 0
T ∆a2
2
2a ih̄
i o
2
−
(mω tan(ωt)) − F z dt .
T ∆a22
(15)
(16)
This last path integral is gaussian in pz and z, then it can be calculated,
Z T
n im
o
n
o
T ∆a22 − 2imh̄
2
U[a2 (t)] (W, T ; N, 0) = N2 exp
[(xW − xN )2 + (yW − yN )2 ] exp −
a
(t)dt
2h̄T
(T ∆a22 )2 + (2mh̄)2 0 2
ia2 F
F2
a22 (T ∆a22 − 2imh̄)2 β 2
2
−
(T
∆a
−
2imh̄)
−
((T ∆a22 )2 + (2mh̄)2 ) )
2
2
(T ∆a2 )2 + (2mh̄)2
h̄
4h̄2
dt .
(17)
imω 2
0
(T ∆a22 − 2imh̄)β 2 +
((T ∆a22 )2 + (2mh̄)2 )
2h̄
For the sake of implicity, here we have introduced N2 , which is a normalized constant, and set β = mω tan(ωt).
The probability density, associated with the different measurement outputs, is given as (according to the expression
P[a2 (t)] = |U[a2 (t)] |2 )
Z T
n
o
2T ∆a22
2
P[a2 (t)] = exp −
a
(t)dt
(T ∆a22 )2 + (2mh̄)2 0 2
(Z
× exp
T
(Z
× exp
0
T
16a22 m2 h̄2 ω 2 T ∆a22 β 2
8h̄2 a22 T ∆a22 β 4
−
− 4a2 mω 2 F T ∆a22 − 2F 2 T ∆a22 β 2 )
(T ∆a22 )2 + (2mh̄)2
(T ∆a22 )2 + (2mh̄)2
dt .
4h̄2 β 4 − 8m2 h̄2 ω 2 β 2 + m2 ω 2 ((T ∆a22 )2 + (2mh̄)2 )
(18)
Second, if A1 (t) is measured simultaneously, we obtain the corresponding propagator with the same method,
n im
o
n
h
imω
2
U[a1 (t)] (W, T ; N, 0) = N1 exp
(xW − xN )2 exp
cos(ωT )(zW
+ zN2 ) − 2zW zN
2h̄T
2h̄ sin(ωT )
2
F 2 T sin(ωT ) io
2F
2
(1
−
cos(ωT
))(z
+
z
)
−
F
(1
−
cos(ωT
))
+
,
(19)
+
W
N
mω 2
m2 ω 4
m2 ω 3
where N1 is the normalized constant.
And probability is given as
P[a1 (t)] = |U[a1 (t)] |2 = 1 .
(20)
5 Conclusions
In this work we investigate the motion of a charged
particle moving in the electromagnetic and gravitational
fields of a spherical body, and find a set of QND variables.
Then it is assumed that the set of QND parameters are
continuously monitored simultaneously. Along the ideas
of RPI approach, the propagators and probabilities associated with the possible measurement outputs are manip-
ulated.
Looking at expression (18), the mass m of the particle
appears in the relation mh̄, which is consistent with the
result of QND measurement in electric and gravitational
fields.[8] It differs from the result of Colella, Werner, and
Overhauser experiment,[11] and differs from the case of a
quantum demolition measurement,[12] which has the combination m/h̄.
Obviously, we can see from expression (20) that there
is no standard quantum limit. Now let us consider two
different limits for expression (18). First, if ∆a2 → 0, we
obtain P[a2 (t)] → 1, namely, the probability does not de-
458
ZHU Chun-Hua and ZHA Chao-Zheng
Vol. 44
pend upon the measurement output a2 (t), and all the possible measurement outputs have the same probability. We
may notice that there is also no standard quantum limit,
in other words, we can measure variable A2 with an arbitrarily small error, and all the necessary information can
be extracted. Clearly, this quantum feature is associated
with QND measurement. Second, the opposite case, that
is, the limit of rough measurements, ∆a2 → ∞, renders
also the result P[a2 (t)] → 1. In this case weight functional
(which restricts the integration domain) ω[a2 (t)] → 1, that
is, all the paths have the same probability in path integral.
In this sense this case is as if there were no measuring
process, and all the possible measurement outputs have
the same probability. From our results (18) and (20), we
draw a conclusion that a set of QND variables A1 and A2
could be monitored simultaneously with arbitrary precision, which is consistent with the property of a set of QND
variables.
To finish, it is worthy to be mentioned that our case
was a two-dimensional system because we considered not
only the electric and gravitational fields but also magnetic
field. So we obtained a set of QND variables, expressed by
Eqs. (12) and (13). This obviously differs from the onedimensional case when only considering the electric and
gravitational fields,[8] where what we obtain is only one
QND variable.
References
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[11] R. Colella, A.W. Overhauser, and S.A. Werner, Phys.
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