Physica A 280 (2000) 374–381www.elsevier.com/locate/physa

The Zeno e�ect for coherent statesD. Bar

Department of Physics, Bar Ilan University, Ramat Gan, Israel

Received 3 December 1999

Abstract

It is known that by performing dense measurement, that is, by repeating the same measure-ment a very large number of times in a �nite time we can preserve and maintain an initialquantum state. We show here the same e�ect applying even for the case of the quantum coher-ent states. Moreover, by taking the complete system of (measured system + measuring system)we prove the Zeno e�ect not only for the measured system, but also for the complete system.That is, we include here, in our optical treatment of the Zeno e�ect, the relevant equations ofthe detecting device, so as to show that the general system (of observed system + observingsystem) demonstrates the same behaviour under dense measurement as it is regularly demon-strated by the observed system. But unlike the case for noncoherent states where the Zeno e�ectis proved by applying approximate calculations, here we prove this e�ect rigorously without anyapproximation whatsoever. We prove it �rst for the crosscorrelation case where two points areinvolved: the point source, and the point of observation where a point detector is located. Thenwe generalize it to the more realistic case of an extended light source emanating light frommore than one spot which is detected by more than one detector. c© 2000 Elsevier ScienceB.V. All rights reserved.

1. Introduction

It is known that harmonic oscillator coherent states are quantum states which amongother things are solutions to the minimum-uncertainty conditions, and they evolvequasi-classically [1–5]. The question that is frequently raised is whether we can main-tain the quasiclassical coherence evolution even under more general conditions [2–5].A method often employed to construct coherent states is to exploit the use of somede�nite groups which, by themselves are not symmetry groups of the Hamiltonian,but whose algebras are spectrum generating algebras [2–5]. We show here that theinitial quantum coherent states can be preserved in time by employing the dense mea-surement limit, commonly known as the Zeno e�ect [6–8], that is, by repeating thesame measurement a very large number of times in a �nite time the initial coherentstate is maintained. In Section 2 we show that even when the quantum coherent state

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D. Bar / Physica A 280 (2000) 374–381 375

function is acted upon by the Hamiltonian (the undisplaced Hamiltonian) [1] that ac-tually transforms it to one with classical traits (that is, its phase space variables q(t)and p(t) evolve in time classically), nevertheless, by repeating the same experimenta very large number of times in a �nite time we can preserve its initial quantumstate, not only for the measured state, but also for the measuring apparatus whichis represented here by a point detector. We follow here the point of view taken byHartle and others [9] which represents the point detector itself as a harmonic oscil-lator, in which the parameters P and Q depend upon the relevant conditions of theexperiment such as temperature, etc. In Section 3 we generalize from this �rst-ordercoherence to nth-order coherence, that is, to the coherence of the light emanating frommore than one spot in an extended light source which is detected by more than onedetector.The Zeno e�ect has been shown theoretically especially for two states wave func-

tions like 12 spin particles [10], right and left isomers [11], the two-state model of

the localized Born–Oppenheimer states [12], etc. Here we show the Zeno e�ect forthe quantum coherent state which is a multi-level system (actually in�nite-level sys-tem). Since we deal especially with one such system we do not use the density matrixformalism which is destined to be used for many systems complexes [13,14]. More-over, since we are dealing with the complete system of (observed system + observingsystem) in which we are going to keep both of its components, then the densitymatrix formalism is not appropriate for us [8,14]. Thus our analysis will be basedon the wave function and Schrodinger equation. As will be shown in the sequel ourderivation of the Zeno e�ect for the coherent states will be exact 1 without taking re-sorting to any approximation, in contrast to the usual derivation met in the literature[6,7,10–12].

2. Dense measurement of the harmonic oscillator system

The coherent states are usually written as [1]

|z〉= e−(1=2)|z|2∞∑n=0

zn

(n!)1=2|n〉 (1)

1 In the usual derivation [6–8,10–12] one �rst gets the expression cos2 �t (not under the exponentialsign as in the treatment here (see Eq. (13))), then using the fact that �t.1, cos �t is expanded in aTaylor series and only the �rst two terms are kept, so that P()=cos2 �t=1− (�t2)=2. This is the probabilityfor getting the initial state in a single measurement. Repeating the same measurement n times at intervalsof �t in a total �nite time T one get Pn() = (1− (�t)2=2)n. Using the relation n= T=�t, and passing to thelimit of dense measurement, that is, n→ ∞ (so that �t → 0) we get limn→∞ Pn()= limn→∞ (1−1=n2)n=limn→∞ exp(−1=n)→ 1. This is the usual approximation (see especially Ref. [10]) where one expands thefunction cos �t in a Taylor series and keep the �rst two terms only so as to have the term 1 which is thevalue taken by the probability in the limit of dense measurement, and so as to get the exponent term. Herein our derivation we get the unity term and the exponent term without any approximation (see the last resultof Eq. (13)).

376 D. Bar / Physica A 280 (2000) 374–381

for all complex numbers z. If we de�ne

z =q+ ip(2˝)1=2 ; (2)

where q and p are two arbitrary real c-numbers then we can write (1) as

|z〉 ≡ |p; q〉 ≡ exp(− 14˝ (p

2 + q2)) ∞∑n=0

(q+ ip)n

(2˝)n=2(n!)1=2 |n〉 : (3)

We treat here an oscillator of unit mass and angular frequency for convenience ofnotation. The undisplaced Hamiltonian operator can be written, in our simpli�ednotation as

H = 12(P

2 + Q2 − ˝) : (4)

This is the Hamiltonian for a system which in its ground state has vanishing meanposition and momentum

〈0|Q|0〉= 〈0|P|0〉= 0 : (5)

The operator which has the e�ect of translating both the operators P and Q by thec-numbers p and q is denoted by U [p; q] and is equal to

expi(pQ − qP)

˝ : (6)

Our coherent state |p; q〉 can be generated from the ground state |0〉 by the unitaryoperator U [p; q],

|p; q〉 ≡ U [p; q]|0〉 : (7)

That is |p; q〉 is the ground state of a similar oscillator in which the coordinate isdisplaced by q and the momentum by p. We �rst demonstrate the Zeno e�ect for thesimple example of an oscillator which is described by the undisplaced Hamiltonianfrom Eq. (4). We show that if its initial state is a coherent one, then if we repeat avery large number of times in a �nite time the experiment of checking its coherence,then we can preserve its initial coherence as the time passes. We make use of thefact [1] that the time evolution of the displaced oscillator ground state |p; q〉 under theaction of the undisplaced Hamiltonian operator H from (4) is

e−i(Ht=˝)|p; q〉= |pcl(t); qcl(t)〉 ; (8)

where pcl(t) and qcl(t) are the classical oscillator solutions given by

qcl(t) = q cos t + p sin t ; (9)

pcl(t) =−q sin t + p cos t (10)

D. Bar / Physica A 280 (2000) 374–381 377

with the boundary conditions pcl(0) = p and qcl(0) = q. As is remarked in Ref. [1]the evolution of |p; q〉 according to the classical equations of motion does not entail aclassical interpretation of such solutions, that is, a nonprobabilistic one. The reason isthat as long as ˝¿ 0 there is still a nonvanishing variance to measured momentum andcoordinate values. Now the probability that on a measurement (of duration �t where�t.1) of the coherence of this initial coherent state, interacted upon by the undisplacedHamiltonian, we �nd it still in its initial coherent state

P(|p; q〉) = |〈p; q|e−i(H�t=˝)|p; q〉|2 = |〈p; q|pcl(�t); qcl(�t)〉|2 : (11)

The last result was obtained by using (8). We substitute now from (3) into (11) to get

P(|p; q〉) =∣∣∣∣exp

[− 14˝ (p

2 + q2)− 14˝ (p

2cl(�t) + q2cl(�t))

]

×∞∑

m;n=0

1(2˝)(m+n)=2(n!m!)1=2 (q− ip)

m(qcl(�t) + ipcl(�t))n〈m|n〉∣∣∣∣2

=∣∣∣∣exp

[− 14˝ (p

2 + q2)− 14˝ (p

2cl(�t) + q2cl(�t))

]

×∞∑n=0

1(2˝)nn! (q− ip)

n(qcl(�t) + ipcl(�t))n∣∣∣∣2

=∣∣∣∣exp

[− 14˝ (p

2 + q2)− 14˝ (p

2cl(�t) + q2cl(�t))

+12˝ (q− ip)(qcl(�t) + ipcl(�t))

]∣∣∣∣2

: (12)

Substituting for pcl(t) and qcl(t) from (9) and (10) we get

P(|p; q〉) =∣∣∣∣exp

[− 12˝ (p

2 + q2) +12˝ (q− ip)((q cos �t + p sin �t)

+ i(p cos �t − q sin �t))]∣∣∣∣2

=∣∣∣∣exp

[− 12˝ (p

2 + q2) +12˝ ((p

2 + q2) cos �t − i(p2 + q2) sin �t)]∣∣∣∣2

=∣∣∣∣exp

[− 12˝ (p

2 + q2)(1− cos �t)− i2˝ (p

2 + q2)sin �t]∣∣∣∣2

= exp[−1˝ (p

2 + q2)(1− cos �t)]: (13)

This is the probability for getting the coherent state |p; q〉 in a single measurement.Repeating this measurement n times and using the relation n= T=�t we obtain

Pn(|p; q〉) = exp[−n˝ (p

2 + q2)(1− cos �t)]

= exp[−T˝ (p

2 + q2)(1− cos �t)

�t

]: (14)

378 D. Bar / Physica A 280 (2000) 374–381

In passing to the dense measurement limit where �t → 0 we take into account thecontinuity of the exponent function, and we have also to apply the known L’hopitaltheorem [15] so as to get

limn→∞ Pn(|p; q〉) = lim

�t→0PT=�t(|p; q〉)

= lim�t→0

exp[−T˝ (p

2 + q2)(1− cos �t)

�t

]

= exp[−T˝ (p

2 + q2) lim�t→0

(1− cos �t)�t

]

= exp[−T˝ (p

2 + q2) lim�t→0

sin �t]→ 1 : (15)

All the above discussion establishes the Zeno e�ect for the autocorrelation case inwhich we take into account only one point where the point source is located. Wenow generalize to the crosscorrelation case in which we consider two points: the pointwhere the point source is located and another point where a point detector is locatedcalled “the point of observation”. The Hamiltonian for this case must include, besidesthe regular terms of the harmonic oscillator representing the system, other terms repre-senting the point detector and also the interaction between the system and the detector.An appropriate model is that in which the point detector itself is treated as a harmonicoscillator [9] where its variables depend of course upon the relevant conditions of themeasuring system like temperature, etc.; the Hamiltonian is

�H = 12(P

2 + Q2 − ˝) + 12 ( �P

2+ �Q

2 − ˝) + �PP + �QQ : (16)

The variables with the “ �” denote the point detector, and the last two terms representthe interaction of the system and detector. The separate Hamiltonians for the systemand detector are each of the undisplaced Hamiltonian kind. The initial state is that inwhich the system and detector did not interact yet and they are both in coherent statesso this state is simply given by the product |p; q〉| �p; �q〉. The probability that upon ameasurement of duration �t we remain with the same initial state that we begin with is

P(|p; q〉| �p; �q〉) =∣∣∣∣〈p; q|〈 �p; �q| exp

[− i

�H�t˝

]|p; q〉| �p; �q〉

∣∣∣∣2

=∣∣∣∣〈p; q|〈 �p; �q| exp

[−(i˝

(12(P2 + Q2 − ˝)

+12( �P2+ �Q

2 − ˝) + �PP + �QQ))

�t]|p; q〉| �p; �q〉

∣∣∣∣2

=∣∣∣∣〈p; q|〈 �p; �q| exp

[−(i2˝ ((P

2 + Q2 − ˝)

+ ( �P2+ �Q

2−˝)))�t]exp

[− i˝ (

�PP+ �QQ)�t]|p; q〉|| �p; �q〉

∣∣∣∣2

:

(17)

D. Bar / Physica A 280 (2000) 374–381 379

The last result was obtained because the separate parts of the Hamiltonian �H from (16)

commute with each other, that is, we have [ 12 (P2+Q2−˝)+1

2 ( �P2+ �Q

2−˝); �PP+ �QQ]=0.Making use of the following properties of the coherent states [1]:

〈p; q|P|p; q〉=p ;〈p; q|Q|p; q〉= q ; (18)

we get for (17)

P(|p; q〉| �p; �q〉) =∣∣∣∣〈p; q|〈 �p; �q| exp

[− i˝

(12(P2 + Q2 − ˝)

+12( �P2+ �Q

2 − ˝))�t]exp

[− i˝ ( �pp+ �qq)�t

]|p; q〉| �p; �q〉

∣∣∣∣2

=∣∣∣∣〈p; q|exp

[− i2˝ (P

2 + Q2 − ˝)�t]|p; q〉

× 〈 �p; �q|exp[− i2˝ (

�P2+ �Q

2 − ˝)�t]| �p; �q〉

∣∣∣∣2

= |〈p; q|pcl(t); qcl(t)〉〈 �p; �q| �pcl(t); �qcl(t)〉|2

=∣∣∣∣exp

[− 12˝ (p

2 + q2)(1− cos �t)− i2˝ (p

2 + q2)sin �t

− 12˝ ( �p

2 + �q2)(1− cos �t)− i2˝ ( �p

2 + �q2)sin �t]∣∣∣∣2

= exp[−1˝ (p

2 + q2 + �p2 + �q2)(1− cos �t)]: (19)

We make use of (8)–(11) and (13). This is the probability for getting the coherentstate |p; q〉| �p; �q〉 in a single measurement where also the point detector remains in itsinitial coherent state. If we repeat the same measurement n times at intervals of �t ina total time T we get, by using the relation n= T=�t,

Pn(|p; q〉| �p; �q〉) = exp[− T2˝ (p

2 + q2 + �p2 + �q2)1− cos �t

�t

]: (20)

As we have done in the autocorrelation case we use here the continuity of the exponentfunction, and apply the L’Hopital theorem in the limit of dense measurement. Thuspassing to this limit, that is, n→ ∞, so that �t → 0 we get the desired result:

limn→∞ Pn(|p; q〉| �p; �q〉)= exp

[− T2˝ (p

2+q2+ �p2+ �q2) lim�t→0

1−cos �t�t

]→ 1 : (21)

We have thus shown that we can preserve the initial quantum coherent state of thegeneral system of (observed system + observing system).

3. The nth-order coherence case

We now generalize from the �rst-order coherence in which the light emitted fromone point (the point source) was detected at another point (the point detector) to the

380 D. Bar / Physica A 280 (2000) 374–381

more realistic and general nth-order coherence case, 2 in which the light emanates frommore than one point in the light source and detected at several points by more thanone detector. We make use of the following relation for the coherence function [1]:

�(r; t; �r; �t) = 〈V ∗(r; t)V ( �r; �t)〉 : (22)

V (r; t) is the analytic signal representation of the electromagnetic �eld. The angularbrackets denote the ensemble average. When �r = r, � measures the autocorrelation ofthe signal V (r; t), and when �r 6= r, � measures the crosscorrelation of the two signalsat r and �r, where �r is the point light source and r is the point of observation. SinceV (r; t) is analytic, � also is analytic, and we use this property to eliminate t and �t infavor of � [1]:

�(r; t; �r; �t) =∫ ∞

0�̃(r; �r; �) e−2�i�(t− �t) d� : (23)

We now write the following two known relations [1]:

|�̃(r; �r; �)|2 = �̃( �r; �r; �)�̃(r; r; �) ; (24)

|�̃(r; r(1); r(2); : : : ; r(2n); �)|2 =2n∏k=1

�̃(r(k); r(k); �) : (25)

Eq. (24) tells us that the modulus of the crosscorrelation of the signals at r and �requals the product of the autocorrelation of the signal at r by the autocorrelation ofthe signal at �r. Eq. (24) is the known formal condition of �rst-order coherence, and(25) is the analogous condition for nth-order coherence. But as remarked in Ref. [1]we must note that these conditions are only necessary conditions and not su�cientones. Substituting now from (24) into (25) we get

|�̃(r; r(1); r(2); : : : ; r(2n); �)|2 =n−1∏k=0

�̃(r(2k+1); r(2k+1); �)�̃(r(2k+2); r(2k+2); �)

=n−1∏k=0

|�̃(r(2k+1); r(2k+2); �)|2 : (26)

On the extreme right-hand side we have n products of the moduli of the �rst-ordercoherence functions. Since, as we have proved, each �̃(r(k); r(k+1); �) approach 1 inthe limit of dense measurement so does its norm, and thus we conclude that whenwe perform dense measurement for all the di�erent pairs of points (where for eachsuch pair one point is in the extended light source, and the other point is occupiedby a detector) thereby causing the extreme right-hand side of (26) to approach 1 andwith it the extreme left-hand side likewise approach 1. Thus, we see that performingsimultaneously n �rst-order dense measurements (for n di�erent pairs of points) wepreserve the initial quantum coherence of the nth-order system.

2 See Eqs. (8:36)–(8:37) in Ref. [1] for �rst-order coherence and (8:35) in Ref. [1] for nth-order coherence.Eq. (8:37) is |�(r; t; �r; �t)|2 =�(r; t; r; t)�( �r; �t; �r; �t) where � is the Fourier transform of �̃. Taking the Fouriertransform of both sides of the last equation yields Eq. (24) here.

D. Bar / Physica A 280 (2000) 374–381 381

4. Concluding remarks

We show that by performing dense measurement upon the observed system, whenwe take into account the observing system and include it, as an inherent part, in ourequations of motion, if the general system (of observed system + observing system)was initially in some quantum coherent state, we can preserve in time the initial quan-tum coherent state of the total system and bypass any transition of this quantum stateto another one with classical characteristics which arise when the undisplaced Hamil-tonian acts upon the general initial coherent state. We show it for the crosscorrelationcase where two points are taken into account (the point source and the point detectortreated like a harmonic oscillator). Finally, using the known coherence conditions forthe �rst-order and nth-order coherences we have generalized it to include the morerealistic case of an extended light source emanating light from more than one pointwhich is detected at more than one point.

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