Critically analyzing the so-called quantum Zeno effect in the measurement problem, we show that observation of this effect does not necessarily mean experimental evidence for the naive notion of wave-function collapse by measurement (the simple projection rule). We also examine what kind of limitation the uncertainty relation and others impose on the observation of the quantum Zeno effect

Namiki, Mikio

Pasaczio, Saverio

English

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Quantum Zeno effect in the measurement problem 1
Mikio Namiki
Department of Physics, Waseda University, Tokyo 169, Japan
Saverio Pascazio*
*Department of Physics, University of Bari, 1-70126 Bari, Italy
Abstract
Critically analyzing the so-called quantum Zeno effect in the measurement problem, we
show that observation of this effect does not necessarily mean experimental evidence for the
naive notion of wave-function collapse by measurement (the simple projection rule). We
also examine what kind of limitation the uncertainty relation and others impose on the
observation of the quantum Zeno effect.
1 Introduction
The quantum Zeno paradox, named after the famous Greek philosopher Zeno, states that an
unstable quantum system becomes stable (i.e. never decays) in the limit of infinitely frequent
measurements. However, we cannot observe this limit, practically and in principle, as will be seen
later on the basis of the uncertainty relations. Physically, its milder version, i.e. the quantum
Zeno effect (QZE) can be observed. On the other hand, one may consider as if observation of this
kind only of phenomenon were a clear-cut support for the naive notion of wave-function collapse
(WFC) (the simple projection) onto an observed state. The purpose of this paper is to contrast
this kind of misunderstanding by analyzing the mechanism of QZE and to discuss the important
role of the uncertainty relations in observation of QZE.
A quantum system that is initially prepared in an eigenstate of the unperturbed Hamiltonian
undergoes a temporal evolution that can be roughly divided into three steps [1]-[6]: A Gaussian-
like behavior at short times, a Breit-Wigner exponential decay at intermediate times, and a power
law at long times.
The first idea of the QZE was introduced under the assumption that the Gaussian short-time
behavior can be observed in a quantum decay and only the naive WFC takes place in quantum
measurements [7] [8]. In this context, the QZE is closely connected to the WFC by measurement,
so that we have to examine one of the central measurement problems: What is the wave-function
collapse? The authors have formulated a reasonable theory of measurement without resorting
to the naive WFC [9][10]. In this paper, however, we shall not discuss this problem (see refs).
Rather, we shall explain the QZE along the line of thought of the naive WFC.
Usually, the Gaussian decay is very difficult to observe. For this reason, the QZE was not
considered to be easily amenable to experimental test until Cook [11] proposed using atomic
1In collaboration with H. Nakazato, G. Badurek and H. Rauch.
133
https://ntrs.nasa.gov/search.jsp?R=19960025009 2020-06-16T04:07:17+00:00Z
transitions in two-level atoms. On the basis of this idea, Itano and his group [12] recently carried
out an interesting experiment, claiming that they experimentally justified the naive WFC by
observing the QZE. As opposed to this, Prigogine and his group asserted, by theoretically deriving
the same result only through a dynamical process without the help of the naive WFC, that this
experiment did not necessarily support the naive WFC. This provoked an interesting debate [13].
The present authors and others proposed to use neutron spin-flip, instead of atomic transitions,
in order to confirm and simplify (and generalize) Prigogine et al's theory [14]. Similar kind of
experiments were proposed and recently performed by making use of photon polarization [15][16].
It is also important to note that Itano et al's experiment did not observe the state every time
except the final one, and therefore was not exactly the same as Cook proposed. One of the main
interests in this paper is, therefore, to find the reason why their experiment could give the same
result as Cook's prediction, which was given by assuming the naive WFC (a simple projection) at
every step.
2 Naive formulation of QZE
We first formulate the QZE along the line of thought of the naive WFC, and discuss it as an
dynamical process in the next section.
For initial state u_ at t = 0 (an eigenstate of unperturbed Hamiltonian/2/0), the wave function
dynamically changes as
_2t : exp(-iE_t - _F_t2)u_ + O(t) • F_ - (u_,/2/fu_) (i)
at very short t(> 0), provided that (u,,/2/iu_ ) = 0.
According to the idea of the naive WFC, the system suffers such a sudden change as
1 2
Ct ==_ exp(-iE_t - _F_t )u_
for its wave function, or
_-- (1 - _F_t2)u_e -iE_t (2)
p, exp(-Fot:)poo (i- F t )p o : poo-luo >< (3)
for its density matrix, at very short t.
The probability of finding state u_ at very short t(> 0) is given by
P_(t) = (i - F.t2). (4)
Therefore, the probability of finding the same state N times by repeated measurements of this
kind in time intervals (O,T/N),...,((N- 1)TIN, T) (note that t = TIN for one step) during
(0, T = tN) is given by
= - ----+ I. (5)
We propose to distinguish the QZE from the quantum Zeno paradox in the following way:
134
Quantum Zeno paradox:
lim P_V(T)= 1 : only in the infinite N limit, (6)
N_oo
Quantum Zeno effect:
Pff(T) > P:V'(T), if N > N' : for finite N and N'. (7)
Remember that we have simply formulated the quantum Zeno paradox and the QZE by making
use of the Gaussian decay and the naive WFC.
3 Neutron spin-flip and discussion on QZE
In Cook's case, we observe only the temporal evolution of the type cos2(w/2), so that we obtain
(8)
for the probability of finding the initial state u_ at time T after N-step measurements, if we choose
T so as to give cos(coT/2) = 0.
In the neutron spin-flip case, we can also formulate the theoretical procedure in a similar way
as in Cook's case, if we use a polarized neutron beam along the z-axis and N magnetic fields with
strength B along the x-axis as shown in Fig.1 (Case A), where w = #B/h (# being the neutron
magnetic moment). Therefore, we can describe the one-step measurement as
p(t) _t wt
= p_ cos 2 _ + Pbbsin 2 -_-
wt . _t
--ip_b cos -_- sm _ + h.c.
wt
Paa cos 2- (naive WFC projection)
2
(9)
where a =T, b =J., t = T/N (or _t/2 = 7r/2N), and then the final density matrix after N-step
measurements becomes (N
\ Paa ,
PA(T) = cos2 7r (10)
and correspondingly, we can get the probability of finding the upward spin state at time T in the
same form as (8) with a =T.
In this case, we can explicitly write down the whole density matrix in the channel representation
before the final spin-detection in the following way:
p_(T) =
C 2N
$2c2N-2
S2c2N-4
0
0 s _
135
D2 DN
B B B
Figure 1: QZE in case A.
B B
Do
B
Figure 2: QZE in case B
where c = cos(_-/2N) and s = sin(_'/2N).
On the other hand, we know that a measurement process can be divided into two steps, the first
being the spectral decomposition and the second the detection. Usually, spectral decomposition
step is a sort of dynamical process that keeps coherence among the branch waves. In this case,
the experimental procedure is illustrated in Fig. 2 (Case B). We can easily show that, through
the N-step spectral decompositions, the density matrix of the system will dynamically change as
pN(T) = cos 2 _-_ P_a + other components . (11)
Therefore, we can explicitly write down the whole density matrix in the same channel repre-
sentaion before the final spin-detection as follows:
pN(T) =
c2N isc2N-1 isc2N-2 ... iSC N
_isc2N-I S2c2N-2 s2c2N-3 . . . s2cN-I
__SC 2N-2 S2c2N-3 s2c2N-4 . . . S2C N-2
: : : .. :
_isc N 82C N-1 s2cN-2 . .. S 2
Note that both the density matrices pN(T) and p_(T) have the same aa-component. Cor-
respondingly, we obtain the same formula as (8) (with a =Y) for the probability of finding the
upward spin-state by the final detector Do at time T. This is the answer to the question asked at
the end of the preceding section. That is, we cannot conclude that observation of PN(T) going to
unity is an experimental evidence in support of the naive WFC.
136
4Undoubtedly, one of the most important quantities is
¢=wt=--_-v t= 2-_) , in P_r(T)= 1-
The uncertainty relation and other situations
N
(12)
where l stands for the length of each magnet and v for the neutron speed.
Mathematically, ¢ is of order O(N-1), but we cannot take the infinite N limit for the following
reasons: (i) In practice, we cannot make the zero limit of the magnetic region, and (ii) in principle,
it is impossible to avoid uncertainties Av and Ax, because
#BI #BAx #B 1 AE_
¢ "" ¢0 ------ > -- > (13)
hvo hvo 2mvoAv 4 AEk
where v0, AEm = 2#B and AEk = A(mv_/2) are the mean neutron speed, the magnetic energy
gap and the neutron kinetic energy spread, respectively. Consequently, we should have
P(V(T) _ 1 - _ \_E-T] (14)
For this reason we can set the following limitation:
IYmax "" 104 •
Additionally, we have to take into account the probability of neutron leakage or absorption,
a < 1, at each step, which should modify the probability of finding the neutron as follows:
P (r) = (15)
We cannot take the limit N _ cc also for this reason, but we can estimate this kind of loss factor,
both experimentally or theoretically, in order to get the net effect.
5 Concluding remarks
We have shown that observation of the QZE does not signify any experimental evidence of the
naive WFC (the simple projection), and found the reason why Itano et al's experiment got the
same result as Cook's one, even though they did not exactly follow Cook's proposal. We have also
examined an important limitation arising from the uncertainty relations and other limitations to
be imposed on observation of QZE.
Acknowledgments
One of the authors (MN) is indebted for financial supports to the Ministry of Education, Culture
and Sciences and the other (SP) to the Japan Society for Promotion of Science and to Italian
Consiglio Nazionale delle Ricerche for financial support.
137
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Quantum Zeno Effect in the Measurement Problem

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