VOLUME 28, NUMBER 14 PHYSICAL REVIEW LETTERS 3 APRIL 1972

400-

200-0UJ

UJK

z —200-

—40Q-

'"'ll I ~ ''s

II III litil,

I I I I

(o)6 PRONGS

- Ip

OJ

O0

- -IO

Fraser and Rudolph Hwa. He is indebted to thefollowing members of Group A at the LawrenceBerkeley Laboratory for generously allowing himto participate in the analysis of the K' exposure:M. Alston-Ganjost, A. Barbaro-Galtieri, P. J.Davis, S. M. Flatte, J. H. Friedman, G. R. Lynch,M. J. Matison, J. J. Murray, M. S. Rabin, F. T.Solmitz, N. J. Uyeda, V. Waluch, and R. %ind-molders.

40Q-- Ip

200-0

p. ~+UJCO

—200-

Op

—400-(b)

8 PRONGS-- ~ . -. - ~ - ~ . -- ~ -I-4 -2 0 2 4

Y —Y2 I

- -10

F1G. 4. (a) G& and (b) G3 as defined in the text.

The statistics on the eight-prong data are notgood but show characteristics similar to thosefor six-prong.%e present this dramatic behavior of the two

n 's as functions of their rapidity separation as achallenge to any theory of inclusive reactions.The author wishes to thank Richard Lander for

his support and encouragement. The author alsoappreciates the many useful discussions with him,David Pellett, and Philip Yager. He also benefit-ted from stimulating conversations with William

*Work supported by the U. S. Atomic Energy Commis-sion under Contract No. AT(04-8)-84 PA 191.K. G. Wilson„Cornell University Report No. CLNS-

131, 1970 (to be published).W. B.Fraser et al. , to be published.H. D. I. Abarbanel, Phys. Rev. D 8, 2227 (1971).R. C. Hwa, to be published.D. Z. Freedman, C. E. Jones, F. E. Low, and J. E.

Young, Phys. Rev. Lett. 26, 1197 (1971).C. K. DeTar, Phys. Rev. D 8, 128 (1971).VA. Bassetto, M. Toner, and L. Sertorie, Nucl. Phys.

B34, 1 (1971).8A. Mueller, Phys. Rev. D 4, 150 (1971).W. Ko and B. L. Lander, Phys. Bev. Lett. 26, 1064

(1971}.J. Erwin, W. Ko, R. L. Lander, D. K. Pellett, and

P. M. Yager, Phys. Rev. Lett. 27, 1534 (1971).The correlation length of about 2 is even shorter

than the short-range Mueller-Begge-theoretical valueI.R. C. Arnold, ANL Report No. ANL-HKP 7189, 1971(unpublished), and Ref. 5J. In that theory a correlationlength of 2 is only achieved for the center-center corre-lation if the intercept of exotic trajectories (as the &—channel has exotic quantum numbers) is —1 [W. Ko,

R. L. Lander, and C. Risk, Phys. Bev. Lett. 27, 1476{1971)].The correlation length of 1 or 2 is usually pre-dicted for fragment-center or fragment-fragment cor-relations.H. T. Nieh and J. M. Wang, to be published.

Experimental Test of Local Hidden-Variable Theories*

Stuart J. Freedman and John F. ClauserDepartment of Physics and Lagerence Berkeley Laboratory, Unioersity of California, Berkeley, California 94720

(Received 4 February 1972)We have measured the linear polarization correlation of the photons emitted in an atom-ic cascade of calcium. It has been shown by a generalization of Bell's inequality that theexistence of local hidden variables imposes restrictions on this correlation in conflictwith the predictions of quantum mechanics. Our data, in agreement with quantum me-chanics, violate these restrictions to high statistical accuracy, thus providing strong evi-dence against local hidden-variable theories.

'Since quantum mechanics was first developed,there have been repeated suggestions that its sta-tistical features possibly might be described byan underlying deterministic substructure. Such

features, then, arise because a quantum staterepresents a statistical ensemble of "hidden-variable states. " Proofs by von Neumann andothers, demonstrating the impossibility of a hid-

938

Vuz. UMj.:28,NUMszR 14 PHYSICA L RKVIKW LKTTKRS 3 APRrj. 1972

den-variable substructure consistent with quantummechanics, rely on various assumptions concern-ing the character of the hidden variables. ' Bellhas argued that these assumptions are unduly re-strictive. However, by considering an idealizedcase of two spatially separated but quantum-me-chanically correlated systems, he was able toshow that any hidden-variable theory satisfyingonly the natural assumption of "locality" alsoleads to predictions ("Bell's inequality" ) in con-flict with quantum mechanics. 'Bell's proof was extended to realizable systems

by Clauser, Horne, Shimony, and Holt, ' who alsopointed out that their generalization of Bell's in-equality can be tested experimentally, thus test-ing all local hidden-variable theories, but thatexisting experimental results were insufficientfor this purpose. This Letter reports the resultsof an experiment which are sufficiently preciseto rule out local hidden-variable theories withhigh statistical accuracyIn the present work we measured the correla-

tion in linear polarization of two photons y, andy, emitted in a J=0-J=1-J=0 atomic cascade.The decaying atoms were viewed by two symmet-rically placed optical systems, each consistingof two lenses, a wavelength filter, a rotatableand removable polarizer, and a single-photon de-tector (see Fig. 1). The following quantities weremeasured: R(q), the coincidence rate for two-photon detection, as a function of the angle p be-tween the planes of linear polarization defined bythe orientation of the inserted polarizers; R„ thecoincidence rate with polarizer 2 removed; R„the coincidence rate with polarizer 1 removed4;Ro, the coincidence rate with both polarizers re-

Ca-OVEN

moved. Quantum mechanics predicts that R(p)and R, are related as follows":

R«)/RD =.(Eu'+ &.')(&u'+ E.')+ 4 (Eu'- '.')while

x( — )E (0) o 2p, (la)

R,/RD=~(e„'+e '), (lb)

R,/RD =2 (eu + e„'). (1c)

Here e„' (e ') is the transmittance of the ith po-larizer for light polarized parallel (perpendicu-lar) to the polarizer axis, and E,(8) is a functionof the half-angle 8 subtended by the primary 1ens-es. It represents a depolarization due to noncol-linearity of the two photons, and approaches unityfor infinitesimal detector solid angles. [For thisexperiment, 8=30, and E,(30 ) =0.99.]We make the following assumptions for any lo-

cal hidden-variable theory: (1) The two photonspropagate as separated localized particles. (2) Abinary selection process occurs for each photonat each polarizer (transmission or no-transmis-sion). This selection does not depend upon theorientation of the distant polarizer.In addition, we make the following assumption

to allow a comparison of the generalization ofBell's inequality with out experiment: (3) Allphotons incident on a detector have a probabilityof detection that is independent of whether or notthe photon has passed through a polarizer. 'The above assumptions constrain the coinci-

dence rates by the following inequalities':-1-Z(q)-O,

where

WP.M.2I

LENS

POLARIZER 2

ER I

POLARIZER I

LENS

IP. M. I I—3R(q) R(3q ) R, +R,

=IIDISC.I

CIDISC. I-- IAMPI-I

COINC. i=

P. H. A,

t

+DELAY' -' COINC.

TA.C. I-I

' ' 'I

FIG. 1. Schematic diagram of apparatus and associat-ed electronics. Scalers (not shown) monitored the out-puts of the discriminators and coincidence circuits dur-ing each 100-sec count period. The contents of thescalers and the experimental configuration were record-ed on paper tape and analyzed on an IBM 1620-II com-puter.

~ = IR(22-,")/R, -R(67-,")/R, I- —,' - 0, (3)

which does not involve B, or 8,.The experimental arrangement was similar to

For sufficiently small detector solid angles andhighly efficient polarizers, these inequalities (2)are not satisfied by the quantum-mechanical pre-diction (1) for a range of values of y. Maximumviolations occur at y =22-,' ' [A(cp) & 0] and y =67&2'[A(y) & -1]. At these angles of maximum viola-tion, inequalities (2) can be combined into thesimpler and more convenient expression

VOLUME 28, NUMBER 14 PHYSI C A I. R E V I KW I.E TTKR S $ APRIL 1972

4p2 ls3d4p Pj

4p4s Pi

that of Kocher and Commins. ' A calcium atomicbeam effused from a tantalum oven, as shown inFig. 1. The continuum output of a deuterium arclamp (ORIEL C-42-72-12) was passed through aninterference filter [250 A full width at half-maxi-mum (FWHM), 20/o transmission at 2275 A] andfocused on the beam. Resonance absorption of a2275-A photon excited calcium atoms to the Sd4p'P, state. Of the atoms that did not decay direct-ly to the ground state, about 7~/0 decayed to the4p''8, state, from which they cascaded throughthe 4s4P 'P, intermediate state to the groundstate with the emission of two photons at 5513 A(y, ) and 4227 A (y, ) (see Fig. 2). At the interac-tion region (roughly, a cylinder 5 mm high and 3mm in diameter) the density of the calcium wasabout 1X10"atoms/cm'. To avoid sphericalaberrations which would have reduced counter ef-ficiencies, aspheric primary lenses (8.0 cmdiam, f=0.8) were used. Photons y, were select-ed by a filter with 10 A FWHM and 50% transmis-sion, and y, by a filter with 8 A FWHM and 20/otransmission. The requirement for large effi-cient linear polarizers led us to employ "pile-of-plates" polarizers. Each polarizer consisted often 0.3-mm-thick glass sheets inclined nearly atBrewster's angle. The sheets were attached tohinged frames, and could be folded completelyout of the optical path. A Geneva mechanism ro-tated each polarizer through increments of 222 '.The measured transmittances of the polarizerswere ~„'=0.97+0.01, ~ '=0.038+0.004, ~„'=0.96+0.01, and ~ ' =0.037+0.004. The photo-multiplier detectors (RCA C31000E, quantum ef-ficiency = 0.13 at 5513 A; and RCA 8850, quantumefficiency=0. 28 at 4227 A) were cooled, reducingdark rates to 75 and 200 counts/sec, respective-ly. The measured counter efficiencies with po-

4s~ iSO

FIG. 2. Level scheme of calcium. Dashed lines showthe route for excitation to the initial state 4p Sp.

larizers removed were q~ =1.7X10 and g2 =1.5xj0 3.'A diagram of the electronics is included in Fig.1. The overall system time resolution was about1.5 nsec. The short intermediate state lifetime(-5 nsec) permitted a, narrow coincidence window(8.1 nsec). A second coincidence channel dis-placed in time by 50 nsec monitored the numberof accidental coincidences, the true coincidencerate being determined by subtraction. " A time-to-amplitude converter and pulse-height analyzermeasured the time-delay spectrum of the twophotons. The resulting exponential gave the in-termediate state lifetime. "The coincidence rates depended upon the beam

and lamp intensities, the latter gradually decreas-ing during a run. The typical coincidence ratewith polarizers removed ranged from 0.3 to 0.1countx/sec, and the accidental rate ranged from0.01 to 0.002 counts/sec. Long runs required bythe low coincidence rate necessitated automaticdata collections.The system was cycled with 100-sec counting

periods. Periods with one or both polarizers in-serted alternated with periods in which both po-larizers were removed. Both polarizers rotatedaccording to a prescribed sequence. For a givenrun, R(y)/8, was calculated by summing countsfor all configurations corresponding to angle yand dividing by half the sum of the counts in theadjacent periods of the sequence in which bothpolarizers were moved. Data for R,/R, and A, /Bp w er e analyz ed in a similar fashion. The val-ues given here are averages over the orientationof the inserted polarizer. This cycling and aver-aging procedure minimized the effects of driftand apparatus asymmetry.The results of the measurements of the corre-

lation A(cp)/R„corresponding to a total integra-tion time of -200 h, are shown in Fig. 3. All er-ror limits are conservative estimates of 1 stan-dard deviation. Using the values at 22~2' and67-,' ', we obtain & =0.050+0.008 in clear violationof inequality (3).'2 Furthermore, we observe noevidence for a deviation from the predictions ofquantum mechanics, calculated from the mea-sured polarizer efficiences and solid angles, andshown as the solid curve in Fig. 3. We considerthese results to be strong evidence against localhidden-variable theories.The authors wish to express their sincerest ap-

preciation for guidance and help from ProfessorEugene Commins, to Professor Charles Townesfor his encouragement of this work, and to M. Sim-

940

VOLUME 28, NUMBER 14 PHYSICAL REVIEW LETTERS 3 APRIL 19/2

.5

4

O

~2

O I I ~ } i I i I

0 22' 45 67y 90ANGLE Q IN DEGREES

FIG. 3. Coincidence rate with angle p between thepolarizers, divided by the rate with both polarizers re-moved, plotted versus the angle p. The solid line isthe prediction by quantum mechanics, calculated usingthe measured efficiencies of the polarizers and solidangles of the experiment.

mons for helpful suggestions.

*Work supported by U. S. Atomic Energy Commission.The best-known proof is by J. von Neumann, Mathe-

matische Gyundlagen dew Quantemechanih (Springer,Berlin, 1932} [Mathematical Eoundations of QuantumMechanics (Princeton Univ. Press. Princeton, N. J.,1955)]. For a critical review of this and other proofssee J. S. Bell, Rev. Mod. Phys. 38, 447 (1966).J. S. Bell, Physics (Long Is. City, N.Y.) 1, 195

(1964).J. Clauser, M. Horne, A. Shimony, and R. Holt,

Phys. Rev. Lett. 23, 880 (1969).A hidden-variable theory need not require that A~ and

A2 be independent of the orientation of the inserted po-larizer, and we do not assume this independence in ourdata analysis. However, the results are consistent withA ~ and 82 being independent of angle, and for simplicitythey are so denoted.M. Horne, Ph. D. thesis, Boston University, 1970

(unpublished). See also A. Shimony, in "Foundations ofQuantum Mechanics, Proceedings of the InternationalSchool of Physics 'Enrico Fermi, ' Course IL" (Academ-ic, New York, to be published).

This assumption is much weaker than the assumptionmade by L. B.Kasday, J. Ullman, and C. S. Wu, Bull.Amer. Phys. Soc. 15, 586 (1970), in their discussion ofthe two-p decay of positronium; see L. R. Kasday, in"Foundations of Quantum Mechanics, Proceedings ofthe International School of Physics '.Enrico Fermi, 'Course IL" (Academic, New York, to be published).The inequality A(P) - 0 is derived in Refs. 3 and 5.

The other forms of the hidden-variable restriction areobtained by similar arguments; see S. Freedman, Ph. D.thesis, University of California, Berkeley, LawrenceBerkeley Laboratory Report No. LBL-391, 1972 (un-published) .C. A. Kocher and E. D. Commins, Phys. Rev. Lett.

18, 575 (1967); C. A. Kocher, Ph. D. thesis, Universityof California, Berkeley, Lawrence Berkeley Labora-tory Report No. UCRL-17587, 1967 (unpublished).The counter efficiencies are given by I7; = (II,/4w}T;

&&)L], where 0; is the solid angle, T; is the transmis-sion of the filter, &; is the quantum efficiency, and L&accounts for other losses. The measurement of q2 wasmade, employing the properties of the calcium cascade,by comparing the coincidence rate and the p~ singlesrate after suitable background correction; g& was theninferred from the known quantum efficiencies and filtertransmissions assuming that 0& and L

&were the same

for both detector systems.An estimate of the accidental rate was also obtained

from the singles rates. The two estimates gave consis-tent results. In fact, our conclusions are not changedif accidentals are neglected entirely; the signal-to-ac-cidental ratio with polarizer removed is about 40 to 1for the data presented.Resonance trapping, encountered at high beam densi-

ties, resulted in a lengthening of the observed lifetimeand a slight decrease in the polarization correlationam-plitude, see J. P. Barrat, J. Phys. Radium 20, 541, 633(1959). At low beam densities the measured lifetimeis consistent with previously measured values. SeeW. L. Weise, M. W. Smith, and B.M. Miles, AtomicTyansition I'xobabilities, U. S. National Bureau of Stan-dards Reference Data Series—22 (U.S. GPO, Washing-ton, D.C., 1969), Vol. 2.The results that are of interest in comparison with

the hidden-variable inequalities are 8 f/Ra 0.497+0.009,82/R0=0. 499 +0.009, R (22$')/Ra=0. 400 +0.007, andR (67y')/RD =0.100 +0.003. We thus obtain D(22 2

')=0.104+0.026 and 6(67'') =-1.097+0.018, in violationof inequalities (2}.

941