VOLUME 49, NUMBER 25 PHYSICAL REVIEW LETTERS 20 DECEMBER 1982

for trapping as 6 decreases is reflective of theincorporation of periodic components into thesequence of numbers generated.To summarize the motivation and principal con-clusion of this Letter, we restate' that for valuesof b where numerically generated sequences ~PPear to be chaotic, it has not been settled wheth-er those sequences "are truly chaotic, or wheth-er, in fact, they are really periodic, but withexceedingly large periods and very long tran-sients required to settle down. " On the one hand,Grossman and Thomae" have suggested that(only) the parameter value b =1 generates purechaos [see the discussion following Eq. (31) ofRef. 5 and the correlations plotted in their Fig.9]. On the other hand, for certa, in other valuesof b, numberical results of Lorenz (reported inRef. 1) "strongly suggest that the sequences aretruly chaotic. " The purpose of this communica-tion was to use an independent and exact resultfrom the statistical-mechanical theory of d =1random walks to test the randomness of the para-bolic map for parameter values where the exis-tence of "true chaos" is still an open question.

Our results strongly support the conclusions ofGrossmann and Thomae.This research was supported by the Office ofBasic Energy Sciences of the U. S. Departmentof Energy.

Permanent address: Miles Laboratories, Elkhart,Ind. 46652.( )Present address: Department of Chemistry,

Stanford University, Stanford, Cal. 94305.'E. Ott, Bev. Mod. Phys. 53, 655 (1981).'C. A. Walsh and J. J. Kozak, Phys. Rev. Lett. 47,

1500 (1981).3E.W. Montroll, Proc. Symp. Appl. Math. Am. Math.

Soc. 16, 193 (1964); E. W. Montroll and Q. W. Weiss,J. Math. Phys. 6, 167 (1965); E.W. Montroll, J. Math.Phys. 10, 753 (1969).4K. Tomita, in Pattern Formation by Dynamic Sys-

tems and Pattern Recognition, edited by H. Haken(Springer-Verlag, Heidelberg, 1979), pp. 90-97.5S. Thomae and S. Grossmann, J. Stat. Phys. 26,

485 (1981).S. Qrossmann and S. Thomae, Z. Naturforsch. 32a,

1353 (1977).

E*perimen&al Tes& of Bell's Inequalities Using Time-Varying AnalyzersAlain Aspect, Jean Dalibard, ' and Gerard Roger

Institut d'Optique Theomque et APPliquee, F-9j406 Qxsay Cedex, France(Received 27 September 1982)

Correlations of linear polarizations of pairs of photons have been measured withtime-varying analyzers. The analyzer in each leg of the apparatus is an acousto-opti-cal switch followed by two linear polarizers. The switches operate at incommensuratefrequencies near 50 MHz. Each analyzer amounts to a polarizer which jumps betweentwo orientations in a time short compared with the photon transit time. The resultsare in good agreement with quantum mechanical predictions but violate Bell's inequal-ities by 5 standard deviations.

PACS numbers: 03.65.8z, 35.80.+s

Bell's inequalities apply to any correlated meas-urement on two correlated systems. For in-stance, in the optical version of the Einstein-Podolsky-Rosen-Bohm Gedankenexperiment, ' asource emits pairs of photons (Fig. 1). Measure-ments of the correlations of linear polarizationsare performed on two photons belonging to thesame pair. For pairs emitted in suitable states,the correlations are strong. To account for thesecorrelations, Bell considered theories which in-voke common properties of both members of the

PM1 ~I(a)

~ PM2

I I (b)

COINCIDENCEMONITORING

FIG. l. Optical version of the Einstein-Podolsky-Bosen-Bohm GedankenexPeximent. The pair of photonsv, and v, is analyzed by linear polarizers I and II (inorientations a and b) and photomultipliers. The coin-cidence rate is monitored.

1804 1982 The American Physical Society

VOLUME 49, NUMBER 25 PHYSICAL REVIEW LETTERS 20 DECEMBER 1982

pair. Such properties are referred to as supple-mentary parameters. This is very different fromthe quantum mechanical formalism, which doesnot involve such properties. With the addition ofa reasonable locality assumption, Bell showedthat such classical-looking theories are con-strained by certain inequalities that are not al-ways obeyed by quantum mechanical predictions.Several experiments of increasing accuracy

have been performed and clearly favor quantummechanics. Experiments using pairs of visiblephotons emitted in atomic radiative cascadesseem to achieve a good realization of the idealGedankenexperiment. ' However, all these ex-periments have been performed with static setups,in which polarizers are held fixed for the wholeduration of a run. Then, one might questionBell's locality assumption, that states that the re-sults of the measurement by polarizer II does notdepend on the orientation a of polarizer l (andvice versa), nor does the way in which pairs areemitted depend on a or b. Although highly rea-sonable, such a locality condition is not prescribedby any fundamental physical law. As pointed outby Bell, it is possible, in such experiments, toreconcile supplementary-parameter theories andthe experimentally verified predictions of quan-tum mechanics: The settings of the instrumentsare made sufficiently in advance to allow them toreach some mutual rapport by exchange of signalswith velocity less than or equal to that of light.If such interactions existed, Bell's locality condi-tion would no longer hold for static experiments,nor would Bell's inequalities.Bell thus insisted upon the importance of ex-

periments of the type proposed by Bohm andAharonov, ' in which the settings are changed dur-ing the flight of the particles. " In such a timingexperiment, " the locality condition would then be-come a consequence of Einstein's causality, pre-venting any faster-than-light inf luenee.In this Letter, we report the results of the first

experiment using variable polarizer s. Followingour proposal, ' we have used a modified scheme(Fig. 2). Each polarizer is replaced by a setupinvolving a switching device followed by two po-larizers in two different orientations: a and a'on side I, and b and b' on side II. Such an opticalswitch is able to rapidly redirect the incidentlight from one polarizer to the other one. If thetwo switches work at random and are uncorrelat-ed, it is possible to write generalized Bell's in-equalities in a form similar to Clauser-Horne-

I (~a) C( CII I I (b)

J«~l- ~~ = Q~ - "" +«&1L

I'(a')

FOLIRFOLD COINCIDENCE

NONI TOR ING

FIG. 2. Timing experiment with optical switches.Each switching device (C&, C&&) is followed by two po-larizers in two different orientations. Each combina-tion is equivalent to a polarizer switched fast betweentwo or ientations.

Shimony-Holt inequalities':—1~S~O

with

~(a,b) ~(a, 6 ) V(a, b)Q(&x& ao) Q(oo m') Q(oo' ao)

V(a', b ) N(a, , ) V(-, b)P7(~' ~~) ~(~~ ~) N(~ ~)'

The quantity & involves (i) the four coincidencecounting rates [N(a, b), W(a', b), etc.] measuredin a single ~n; (ii) the four corresponding coin-cidence rates [Ã(~, ~), V(~', ~), etc.] with allpolarizers removed; and (iii) two coincidencerates [N(a', ~), N(~, b)] with a polarizer removedon each side. The measurements (ii) and (iii) areperformed in auxiliary runs.In this experiment, switching between the two

channels occurs about each 10 ns. Since this de-lay, as well as the lifetime of the intermediatelevel of the cascade (5 ns), is small comparedto L/c (40 ns), a detection event on one side andthe corresponding change of orientation on theother side are separated by a spacelike interval.The switching of the light is effected by acousto-

optical interaction with an ultrasonic standingwave n water. ' As sketched in Fig. 3 the inci-dence angle is equal to the Bragg angle, 6 g =5&10 ' rad. It follows that light is either trans-mitted straight ahead or deflected at an angle26 &. The light is completely transmitted whenthe amplitude of the standing wave is null, whichoccurs twice during an acoustical period. Aquarter of a period later, the amplitude of thestanding wave is maximum and, for a suitablevalue of the acoustical power, light is then fully

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VOLUME 49, NUMBER 25 PHYSICAL REVIEW LETTERS 20 DECEMBER 1982

eI IdeAt

fYl

nsducer 20nsz„ li

h'le

~ ~

l transmitted:, ~ ~

. 5

generator25 Mhz

~ i oi~ ~t ~ ~~i

is ~ I+~ A

~ ~~ ~ i ~

~ ~st!

FIG. 3. Optical switch. The incident light is switchedat a frequency around 50 MHz by diffraction at theBragg angle on an ultrasonic standing wave. The inten-sities of the transmitted and deflected beams as a func-tion of time have been measured with the actual source.The fraction of light directed towards other diffractionorders is negligible.

68

FIG. 4. Average normalized coincidence rate as afunction of the relative orientation of the polarizers.Indicated errors are + 1 standard deviation. The dashedcurve is not a fit to the data but the predictions by quan-tum mechanics for the actual experiment.

deflected. This optical switch thus works attwice the acoustical frequency.The ultrasonic standing-wave results from

interf erenee between counterpropagating acousticwaves produced by two electroacoustical trans-ducers driven in phase at about 25 MHz. Inauxiliary tests with a laser beam, the switchinghas been found complete for an acoustical powerabout 1 W. In the actual experiment, the lightbeam has a finite divergence, and the switchingis not complete (Fig. 3).The other parts of the experiment have already

been described in previous publications. ' Thehigh-efficiency well-stabilized source of pairs ofcorrelated photons, at wavelengths &, =422.7 nmand &, =551.3 nm, is obtained by two-photon exci-tation of a (&=0)- (&=1)-(&=0) cascade in calci-umeSince each switch is 6 m from the source,

rather complicated optics are required to matchthe beams with the switches and the polarizers.We have carefully checked each channel for nodepolarization, by looking for a cosine Malus lawwhen a supplementary polarizer is inserted infront of the source. These auxiliary tests areparticularly important for the channels which in-volve two mirrors inclined at 11'. They alsoyield the efficiencies of the polarizers, requiredfor the quantum mechanical calculations.The eoineidence counting electronics involve

four double- coincidence- counting circuits withcoincidence windows of 18 ns. I'or each relevantpair of photomultipliers, we monitor nondelayedand delayed coincidences. The true coincidence

rate (i.e., coincidences due to photons emittedby the same atom) are obtained by subtraction.Simultaneously, a time-to-amplitude converter,followed by a fourfold multichannel analyzer,yields four time-delay spectrums. Here, thetrue coincidence rate is taken as the signal inthe peak of the time-delay spectrum. 'We have not been able to achieve collection ef-

ficiencies as large as in previous experiments, "since we had to reduce the divergence of thebeams in order to get good switching. Coinci-dence rates with the polarizers removed wereonly a few per second, with accidental coincidencerates about one per second.A typical run lasts 12 000 s, involving totals of

4000 s with polarizers in place at a given set oforientations, 4000 s with all polarizers removed,and 4000 s with one polarizer removed on eachside. In order to compensate the effects of sys-tematic drifts, data accumulation was alternatedbetween these three configurations about every400 s. At the end of each 400-s period, the rawdata were stored for subsequent processing withthe help of a computer.At the end of the run, we average the true coin-

cidence rates corresponding to the same config-urations for the polarizers. We then compute therelevant ratios for the quantity S. The statisticalaccuracy is evaluated according to standard sta-tistical methods for photon counting. The proces-sing is perf ormed on both sets of data: that ob-tained with coincidence circuits, and that obtainedwith the time-to-amplitude converter. The twomethods have always been found to be consistent.

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VOr. UME 49, NUMSER 25 PHYSI GAI. R K VI Z W I.K TTKR S 20 DECEMBER 1982

Two runs have been performed in order to testBell's inequalities. In each run, we have chosena set of orientations leading to the greatest pre-dicted conflict between quantum mechanics andBell's inequalities [(a,5) = (6, a') = (a', b') =22.5;(i, 6') =67.5']. The average of the two runs yields

Sgyp t 0 101+ 0 020 p

violating the inequality S ~ 0 by 5 standard devia-tions. On the other hand, for our solid anglesand polarizer efficiencies, quantum mechanicspredicts SQM 0 112.We have carried out another run with different

orientations, for a direct comparison with quan-tum mechanics. Figure 4 shows that the agree-ment is excellent.The new feature of this experiment is that we

change the settings of the polarizers, at a rategreater than c/&. The ideal scheme has not beencompleted since the change is not truly random,but rather quasiperiodic. Nevertheless, the twoswitches on the two sides are driven by differentgenerators at different frequencies. It is thenvery natural to assume that they function in an un-correlated way.A more ideal experiment with random and com-

plete switching would be necessary for a fullyconclusive argument against the whole class ofsupplementary-parameter theories obeying Ein-stein's causality. However, our observed viola-tion of Bell's inequalities indicates that the exper-imental accuracy was good enough for pointingout a hypothetical discrepancy with the predic-tions of quantum mechanics. No such effect wasobserved. '0This work has been carried out in the group of

Professor Imbert, who we thank for his support.We thank all the technical staff of the Institutd'Optique, especially Andre Villing for the elec-

tronies. We are indebted to Jean-Pierre Pas-serieux, from the Departement de Physique Nu-cleaire et Basse Energie at Centre O'Etudes Na-tional de Saclay, for his assistance in fast-coin-cidence techniques, and to Dr. Torguet and Dr.Gazalb, from Laboratoire d'Acoustooptique deValenciennes, for help with the optical switches.We also acknowledge the valuable help of PhilippeGrangier during the final runs.

)Permanent address: Laboratoire de Physique del'Ecole Normale Superieure, F-75231 Paris Cedex 05,France.'A. Einstein, B. Podolsky, and N. Bosen, Phys Hev.

47, 777 (1935); D. Bohm, Quantum Theory (Prentice-Hall, Englewood Cliffs, N.J., 1951).2J. S. Bell, Physics (N.Y.) 1, 195 (1965).3J. F. Clauser and A. Shimony, Rep. Prog. Phys. 41,

1981 (1978). (This paper is an exhaustive review ofthe subject); F. M. Pipkin, inAdvances in Atomic andMoLeculax Physics, edited by D. B. Bates and B. Beder-son (Academic, New York, 1978) {acomprehensivereview).A. Aspect, P. Grangier, and G. Roger, Phys. Bev.

Lett. 47, 460 (1981).5A. Aspect, P. Grangier, and G. Roger, Phys. Bev.

Lett. 49, 91 (1982).'D. Bohm and Y. Aharonov, Phys. Bev. 108, 1070

(1957). The suggestion of this thought-experiment wasalready given by D. Bohm, Ref. l.7A. Aspect, Phys. Lett. 54A, 117 (1975), and Phys.

Bev. D 14, 1944 (1976}.J. F. Clauser, M. A. Horne, A. Shimony, and B.A.

Holt, Phys. Rev. Lett. 23, 880 (1969).9A. Yariv, Quantum E/ect~onics (%'iley, New York,

1975).' Let us emphasize that such results cannot be takenas providing the possiblity of faster-than-light com-munication. See, for instance, A. Aspect, J. Phys.(Paris), Colloq. 42, C263 {1981).

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