EXPERIMENTAL TESTS OF BELL'S INEQUALITIES

Ala in Aspec t

Institut d'Optique Theorique et Appliquee Batiment 503 - Centre Universitaire d'Orsay - B.P. 43 F 91406 ORSAY Cedex France

My aim is to speak of the experiments related to Bell's inequa­lities. But since I am the first speaker who invokes Bell's inequalities, I must first introduce Bell's theorem (at least the way in which I understand it). Then, I will present a brief review of the experiments carried out between 1970 and 1976. In the third part, I will give some details on experiments that are in progress in the Institut d'Optique d'Orsay.

1. BELL'S THEOREM

1.1. Einstein-Podolsky-Rosen Correlations

I II

~ x

+1 +1 L y -1 -1 -a

-+

b

F~une 1. Optica{ ve~~on 06 the E~~t~-Podo~ky-Ro~en-Bohm thought expeJU.mmt. The two photoM v 1 and v'l' emJ:;t~d m -6tate (1)

aJte artuyzed by ~eaJt po{atUze~ -<-rt otUentation~ a and 6. Orte. cart mea~uJte the PJtobabdiliu 06 ~~g{e. Oft jomt de.te.ctiOM aOte.Jt

the pofutize.~.

377

s. Diner et al. reds.J, The Wave-Particle Dualism, 377-390. © 1984 by D. Reidel Publishing Company.

378 A.ASPECT

Let us consider the optical version of the Einstein-Podolsky­Rosen 1 Gedankenexperiment as modified by Bohm2 • Two photons with different energies, v1 and v 2 , are emitted by the source S in a state

1 --- {Ix,x> + Iy,y>} 12

where Ix> and \y> are linear polarization states. Photons '4l is propagating along + O~, whilev2 is propagating towards - Oz.

The polarization analyzer I, in orientation ~, yields two possib­le results, + 1 or - 1, corre!ponding to a polarization found parallel or~perpendicular to a. Similarly, the analyzer II measu­res (along b) the linear polarization of photon v 2 .*

Standard Quantum Mechanical calculations yield, as probabilities of single or joint measurements

~ (~) 1 P+(a) P 2 (2)

~ Pj6) 1 P + (b) 2

and

~ ~ ~ ~ 1 2 ~ ~

I P++(a,b) P __ (a,b) 2 cos (a,b)

~ ~ ~~ 1 . 2 (~ ~) P (a, b) = P_+(a,b) =2 Sln a,b +-

(3)

(p (~,b) being the probability of finding + 1 for v 1 a~d -fot-v 2 with polarizers I and II in orientations ~ and b).

~ ~

In the particular orientations (a,b) = 0, these formulas become

~ ~ ~ -I- 1 I P++(a,b) = P __ (a,b) = "2

~ ~ ~-+

P+_(a,b) = P_+(a,b) = 0

while the single detection probabilities remain ~ (Eq (2» .

* These measurements are obviously similar to measurements of spin components by Stern-Gerlach filters acting on spin 1/2 particles.

EXPERIMENTAL TESTS OF BELL'S INEQUALITIES 379

So, there is 50 % chances of getting \1 1 in channel + 1, but if it is so, then we are sure to find \1 2 in channel + 1. There is a complete correlation between the results of measurements at I and II, although each result looks random.

A convenient way of exhibiting these correlations is the coeffi­cient of correlation of polarization

Its predicted value (according to Quantum Mechanics) is

++ ++ EMQ(a,b) = cos2(a,b)

It ~eaches ± (a,b) = rr/2.

. ++ (complete correlat10n) for (a,b) o and

(5)

There is a classical picture for understanding correlations between remote measurements on two particles that have interacted. The two particles of a given pair bring some common property (related to their common past) that determines the results of the measurements.

For explaining the randomness of individual results~ we have just to admit that this property is different for the various pairs. By averaging over all the pairs, we can hope to recover the Quantum Mechanical predictions. On the other hand, in Quantum Mechanics, all the pairs are described by the same state vector (1). So, our classical looking explanation for the EPR correla­tions leads to complete Quantum Mechanics with some supplementary properties (supplementary parameters, or hidden variables).

1.2 Bell's inequalities

Bell tried to express in mathematical terms this classical­looking picture. Each emitted pair is characterized by a supple­mentary parameter A, which determines the results of the measure­ments by the analyzers. These results are:

+ A (A, a) ± at analyzer I

+ B(A,b)

(6) ± at analyzer II

The pairs are emitted with differents A. The distribution of the supplementary parameters is specified by a probability distribu­tion p(A), such that

p(>,) ;;: 0 and fdA p(A) (6' )

380 A. ASPECT

Any particular supplementary par~meters the~ry will explicity specify the functions p(A), A(A,a) and B(A,b). It will then be easy to compute, in the frame of this theory, the probabilities of the various possible results of measurements. The function of correlation of polarization can for instance be written

+-+ f + + E(a,b) = dA p(A) A(A,a) B(A,b) (7)

The formulas (6), (6') and (7) are sufficient for derivating Bell's inequalities 34 • For any theory embeded in the formalism (6) and (6'), there are restrictions on the function of correla­tion. A useful form for these rectrictions is the so-called BCHSH inequalities (Bell-Clauser-Horne-Shimony-Holt inequali­ties")

- 2 :;; S :;; 2

with (8)

+ + + + + + + + S = E(a,b) - E(a~b') + E(a',b) + E(a'.b')

The quantity S involves four measurements of the function of correlation of polarization, in four different orientations of the polarizers.

The demonstration is very straight forwards. Let us consider four numbers x, x'. y and y' with an absolute value equal to 1. It is then obvious to show that

s = xy - xy' + x'y + x'y' = ± 2 (9)

We can apply this result to the quantities

+ +

1 : A(A,a)

{ x' = A(A,a')

B (A, b) + y' = B(A,b')

The quantity s can be averaged over A by multiplication by p(A) and integration, and this leads to the inequality (8).

1.3 Conflict with quantum mechanics

As we have already said, there is an a priori hope that a supplementary parameter theory yields, on the average, the same predictions as Quantum Mechanics. Thanks to the Bell's inequali­ties, we can show that it is impossible. Let us consider the particular set of orientations displayed of Fig. 2.a.

EXPERIMENTAL TESTS OF BELL'S INEQUALITIES

-a

-b

--b" ( b )

Figune 2. O~entation~ g~v~ng the maximum vio~on 06 Belt'~ inequa~~.

381

The quantity S computed for these orientations, with the Quantum Mechanical results (5), takes the value

SQM = 212

This value clearly violates the BeHSH inequalities (8).

1.4. Discussion

We are then allowed to conclude that no supplementary parameter theory following the formalism (6) and (6') can yield results in agreement with all the Quantum Mechanical predictions. Since the formalism (6) and (6') considered up to now might appear restrictive, it is important to know that generalizations of this result exist S/ 9 , that deal with larger classes of theories, for instance stochastic (non-deterministic) supplementary parameters theories. Thanks to these discussions, we can point out a crucial hypothesis for derivating Bell's inequalities (and thus obtaining a conflict with Quantum Mechanics) : it is the locality assumption already stressed by Bell in his first paper. The locality condi­tion clan,ls. that the resuls.s of measurements at I, A(A,:i), do not depend on the orientation b at II, and vice-versa, nor does p(A) (i.e. the way in which pairs are emitted) depend on ~ and h. The formalism (6) (6') obviously obeys this condition.

The locality condition can be taken as an hypothesis, as we have done here. But, it would be better to derive it as a consequence of a basic physical law. In his first paper, Bell remarked that this is possible, if we consider experiments in which the orienta­tions of the polarizers are changed quickly and at random. If, according to Einstein's causality, we admit that no influence can

382 A.ASPECT

travel faster than light, then we see that the result of a measurement on one side cannot depend on the orientation of the other (remote) polarizer. So, in such an ideal experiment with time-varying analyzers, the Bell's locality condition would be the consequence of Einstein's causality.

1.5 Conclusion

As a summary, Bell's theorem states that no local supplementary parameter theory can mimick all the predictions of Quantum Mechanics. The locality condition, essential for the conflict, can be taken

either as a reasonable assumption ; - or as a consequence of Einstein's causality, in an

experiment with random time-varying analyzers.

2. FROM GEDANKENEXPERIMENT TO ACTUAL EXPERIMENTS

Quantum Mechanics is such a successful theory that one might believe that Bell's theorem states the impossibility of local supplementary parameters theories. But situations in which there is a conflict (we will call them "sensitive situations") are very rare; in 1965, none had been realized. One could then think that Bell's theorem points out a limit of the validity of Quantum Mechanics. By yielding a quantitative criterium' (Bell's inequa­lities), Bell's theorem allowed to give an experimental answer to the problem.

As a matter of fact, sensitive situations are rare, even when we consider ideal experiments. Moreover, when considering an actually workable experiment, the possibility of a sensitive experiment looks even more seldom since all the know defects, lead to a decrease of the correlations predicted by Quantum Mechanics. With experimental imperfections, the conflict thus decreases, or even disappears.

A first series of experimentS~made use of pairs of y photons emitted by annihilation of positronium in its fundamental (singlet) state. As shown by Bohm and Aharonov10 , such pairs of photons are in an EPR type state, similar to state (1). Unfortunately, there are no true analyzers of polarization for y photons at such an energy (0.5 MeV). So the polarization measurements were replaced by the study of Compton scattering of the photons (which depends on the polarization). Then, by an indirect reasoning (using a Quantum Mechanical calculation), it is possible to test the experimental results versus Bell's inequalities.

Although the first results were conflicting, most of these experiments finally gave results in agreement with Quantum

EXPERIMENTAL TESTS OF BELL'S INEQUALITIES 383

Mechanics 1l , and it was claimed that Bell's inequalities were violated. Concerning this last claim, we must remember that the reasoning is very indirect since we have not true measure­ments of polarization, and this conclusion has been challenged9'~2

An experiment subjected to the same kind of criticisms has been performed onto pairs of protons scattered in a singlet state13 •

The results are in good agreement with Quantum Mechanics.

Stimulated by the famous CHSH paper4, another series of experi­ments was performed, using visible photons emitted in atomic radiative cascades. For visible photons, there are true polari­zers, and the situation looks better. However, for practical reasons, the four experiments carried out between 1972 and 1977 made use of single channel polarizer. Such polarizers transmit light polarized along; (cf. Figure 1) but they block light pola­rized perpendicular to ~. The measurements are thus incomplete and one must use an indirect reasoning (and admit a supplementary assumption)4 to be able to test the experimental results versus Bell's inequalities.

Although the two first results were conflicting, these experi­ments9'l~lSclearlyfavour Quantum Mechanics, and show a violation of Bell's inequalities (modified for the one-channel-polarizers case). Even if they do not exactly reproduce the ideal experimen­tal scheme of Fig. 1, these experiments are quite convincing, and closer to the thought experiment than the y photons experiments. The discrepancies between the first results can be easily understood as due to the very low signal obtained. But in the last of these experiments 1S the signal was larger, and the experiment seems very conclusive i the results are in excellent agreement with Quantum Mechanics, and they violate Bell's inequalities by 4 standard deviations.

3. ORSAY EXPERIMENTS

In designing a new experiment~ our main purpose was to realize more sophisticated experimental schemes, closer to the ideal thought experiment. But we had first to build a high efficiency source of pairs of photons correlated in polarization.

3.1. Source of pairs of photons

As shown by CHSH4, a J = 0 + J = 1 + J = 0 atomic cascade yields pairs of photons in a state (1), i.e. good candidates for a sensItIve experimen~When one takes into account the finite solid angles of detection of the fluores~ence light, one finds as corre-

384 A.ASPECT

lation coefficient

-+-+ -+-+ EMQ(a,b) = F(u) cos2(a,b) (10)

instead of formula (5). With an half-angle of detection u = 32° (our experiments), the function F(u) remains close to 1 (0.984) and the correlation remains large enough for a sensitive expe-riment.

2 1 1 2 1 We have chosen the 4p S - 4s4p P1 - 4s S cascade of Calcium, already used by °Clauser and Freedma814 • The atoms in the ground state (in an atomic beam) are excited to the upper state by a two-photon absorption, involving a Krypton ion laser at 406 nm and a tunable dye laser at 580 nm (Figure 3).

F -igWte 3. Two photon.;., exc.-<-:ta.tion 06 :the e -+ Jt -+ 6 M..cU.a:t.J..ve c.Mc.a.de. The ex.c.-<-:ta.tion pJtOc.u,.6 -i...6 e6Mc.-ien:t when v' + v" = vI + v 2

The two lasers are focussed onto the calcium atomic beam, and the resulting source has a small size (0,5 mm x 0,05 mm x 0,05 mm). This feature is convenient for the optics of the detection channels.

A feedback loop controls the wavelength A' of the tunable laser for ensuring the resonance conditon (v' + v" = v 1 + v 2). A second loop controls the power of one laser, for a constant rate of emission of the pairs of photons. The remaining fluctuations, and drifts are less than 1 % for several hours.

With a few tens of milliwatts from each laser, we easily obtain a cascade rate N of several 107 s-1. An increase of this rate would not significantly improve the signal-to-noise ratio for coincidence counting, since the accidental coincidence rate varies as;lN2, while the true coincidence rate varies as N.

EXPERIMENTAL TESTS OF BELL'S INEQUALITIES 385

-1 With such a signal, we obtain a coincidence rate over 100 s In only 100 s of data accumulation, we reach a statistical accuracy of 1 % (in the first experiments, a similar accuracy needed several tens of hours of experiment). This allows us to perform various test and auxiliary checks.

3.2 Experiment with one channel polarizers 16

Similarly to previous experiments, our first experiment used one-channel polarizers, that were pile-of-plates polarizers (ten glass plates at Brewster angle). The optical and mechanical quality of these polarizers was good enough to avoid any displa­cement of the light beams when rotating or withdrawing the polarizers, avoiding any spurious effect related to the hete­rogeneity of the photocathodes.

A test of the Bell's inequalities suited for one channel polarisers 4 exhibited a violation by 9 standard deviations. Figure 4 displays a comparison of our measured values with the Quantum Mechanical predictions taking into account the solid angles and the measured actual efficiencies of our polarizers.

The agreement is obviously excellent.

R(S)/Ro Q1

-:;; .5 ... Q1 t.l C Q1 .., t.l C

C t.l

.., III N

'" e ... c

Z

o 90 180 270 Relative angle oT polarlzers

8 360

Fig~e 4. Expeniment w~h one ~hannel potanize~. Indi~ed vuwM M.e ± 1 ).,.tandMd devia..:tion. The Mud ~UJ!.Ve ~ no~ a 6~

~o ~he da~a but ~he pkedi~on)., by Quantum Me~hani~).,. Additionally, we could perform several auxiliary tests. The most inceresting consisted in removing both polarizers at 6.5 meters from the source. This distance is larger than the coherence length associated to the photon v 2 (CT r = 1,5 m), and it had been

386 A.ASPECT

suggested that the correlations could decrease at such distances (this point had been subject to controversy in experiments with y photons)17. Actually, we observed no modification of the experi­mental results.

3.3 Experiment with two channel polarizers*

We have already seen that measurements of polarization with single channel polarizers are incomplete, since one does not detect the -1 result.

By using two-channel polarizers, we have performed an experiment following much more closely the ideal scheme. of Figure 1. 18

I I - (".J

:E: :E: 0- 0-

r-ol -1 -1 ol

/I "'- VI V2 V /I if PMl+ S PM2+

+1 "'- - V +1

IC-a) II (\))

COINCIDENCES .......

N ±± (a, b)

UgWte. 5. ExpeJU.me.nt wdh po.tOv'Uzhl.g c.ube:i. One. me.a.6Wtu hi. :the Mme. ,!tun. :the. nOM c.ohtUde.n.c.e. !ta:te.~ N t:t (a, 6). B~ !to:ta:thtg any

po.e.a.M.ze.!t, one. c.an me.a.6 Me. :the. c.oMe..to.::v.on nunC-tion a:t ano:the.Jt o JUe.nta.:ti.o n .

Each polarizer is a polarizing cube, using the properties of dielectric thin films, th~t transmits one polarization (respecti­vely parallel to a or to b) and that reflects the orthogonal one. This polarization splitter is mounted in a rotatable mechanism, with two photomultipliers. One thus actually gets the + 1 and - 1 results for a measurement of linear polarization. With a fourfold coincidence system, we are able to measure the four coincidence

~similar experiment, using calcite two-channel polarizers, is in progress at the university of Catania (cf. the paper by Falciglia et al. in the same volume).

EXPERIMENTAL TESTS OF BELL'S INEQUALITIES 387

rates N (;,b) (the polarizers I and II being orientated along ; and 1;f:±This yields directly the polarization correlation coef­ficient.

+ + E(a,b)

+ + N++(a,b) + N

+ + N++(a,b) + N

+ + + + (a,b) - N (a,b)

+-++ ++ ++

(a,b) + N (a,b) + N (a,b) +- -+

(11)

Repeating this measurement for the four sets of orientations of Figure 2, we can directly test the B.C.H.S.H. inequalities (8).

The results of this measurement are

S 2.697 ± 0.015 (12) exp

The inequalities (8) are violated by about 40 standard deviations, and the result is in excellent agreement with the predictions of Quantum Mechanics taking into account the polarizers efficiencies and the solid angles of detection :

SQM = 2.70 ± 0.05

Here, we must remark that we can use the formula (11) for testing inequalities (8) only if we admit that the ensemble of actually detected pairs is a faithfull sample of all emitted pairs (only a weak fraction of which is detected, since the detection effi­ciencies are low). This is a reasonable assumption, but it cannot be proved to be true. However, we have performed some tests that are in agreement with this assumption (and that could have dispro­ved it). The most convincing consist~d in checking that the sum of the four coincidence rates I N++(a,b) is constant, i.e. that

~ + --the size of the selected sampLe-is constant when the orientations are changed. (it is important for this test that the source is very stable).

In addition to this test of Bell's inequalities, we have also measured the polarization correlation coefficient in various orientations, for comparison with the Quantum Mechanical calcu­lations. Figure 6 displays the results: the agreement is obviously excellent.

388

c: c ~ jd ,... ~ jd

o a.. -o c: o .... jd

III ~ ~ o

u

[(8) 1 ....

.5

-.5

-1

A.ASPECT

8

30 60 90

.. b

•.•• '1:1

Relative orientation of polarizers

UguJte. 6. ExpeJUment wah :two -c.hanneA'. po-taJUZe.M. The ,tncitc.a.-te.c... eJUto/t.6 aJl.e. ±Z -b.ta.ndaJtd de.V,ta:U..OM. The. dMhe.d c.Wtve. ~ .the. p~e.citc..t,ton 06 Quan.tum Me.c.han-<-c.-b 60~ .the. ac..tuaR. e.xp~e.n.t.

3.4 Experiment with time-varying polarizers

As emphasized in § 1, it would be interesting to carry out an experiment in which the orientations of the polarizers are changed quickly at random. A step towards such an experiment can be done by using a modified experimental scheme 19 (Figure 7).

Each polarizer is replaced by an optical switch which is able to rapidly redirect the incident light to one of two polarizers. Each set up is obviously equivalent to a variable polarizer switched between two orientations.

In the experiment now in progress*, the switches use the acousto­optical interaction of the light with an ultrasonic standing wave at 25 MHz. A switching occurs about each 10 ns, which is smaller than L/c (40 ns). Unfortunately, due to this principle, the switching is not effected at random but rather in a periodic way (however, the two switches function in an uncorrelated way).

* Note added in proof. The experiment has been completed an yields a result in good agreement with Quantum Mechanics, violating Bell's inequalities by 5 standard deviations (Alain Aspect, Jean Dalibard, and Gerard Roger, Phys. Rev. Lett 49, 1804 (20/12/82)).

EXPERIMENTAL TESTS OF BELL'S INEQUALITIES 389

COINCIDENCES

HgU/Le r. ExpeJUmenJ: wilh optic.a)!. -6wdc.he-6.

In spite of its defects this experiment is, as f~r as I know, the first one that includes time-varying polarizers

4. CONCLUSION

We have reviewed a series of experiments of increasing accuracy and/or leading closer to the ideal thought-experiment. Actually, all these experiments have some imperfections that leave open some loopholes for the advocates of local supplementary para­meters theories. Indeed, one can imagine theories that would exploit these defects for restoring results in agreement with the Quantum Mechanical predictions. But, such theories appear somewhat unnatural since the results should change with improved apparatus.

So, it is more natural to conclude that local supplementary para­meters theories are not a possible picture for understanding the E.P.R. correlations that appear as a typically quantum phenomenon. A little more generally, we can say that these experiments show that we must renounce interpreting E.P.R. correlations with a picture in the spirit of Einstein's ideas of separability and causality. Quantum Mechanics demand pictures more alien from our common sense.

REFERENCES

1 A. Einstein, B. Podolsky et N. Rosen, Phys. Rev. ~, 777 (1935)

2 D. Bohm, Quantum Theory (prentice Hall, Englewood Cliffs, N.J., 1951).

J.S. Bell, Physics~, 195 (1964).

390 A. ASPECT

4 J.F. Clauser, M.A. Horne, A. Shimony et R.A. Holt, Phys. Rev. Lett. ~, 880 (1969).

5 This demonstration is inspired from the paper by P. Eberhard, Nuovo Cim. B 46, 392 (1978).

6 J.S. Bf~l in Foundations of Quantum Mechanics, proceedings of the 49 course Enrico Fermi, B. d'Espagnat edit., Academic (N.Y. 1971).

7 J. Clauser and M.A. Horne, Phys Rev. p~, 526 (1974).

8 B. d'Espagnat, Phys. Rev. ~, 1454 (1975) ; D 1~, 349 (1978).

9 J.F. Clauser et A. Shimony, Rep. Progr. Phys. 41, 1881 (1978). F .M. Pipkin, Advances in Atomic and Molecular Physics, D.R. Bates and B. Bederson ed., Academic (1978) • F. Selleri and G. Tarozzi, Riv. Nuovo Cim !!..., 1 (1981).

10 D. Bohm et Y. Aharonov, Phys. Rev. 108, 1070 (1957) .

11 In addition to the results reported in the reviews 9, see also K. Meisenheimer, Diplomarbeit (Freiburg 1979).

12 M. Froissart, Nuovo Cim. B 64, 241 (1981)

13 M. Lamehi-Rachti and W. Mittig, Phys. Rev. ~, 2543 (1976).

14 S.J. Freedman and J.F. Clauser, Phys. Rev. Lett 28, 938 (1972). J.F. Clauser, Phys. Rev .. Lett. 36, 1223 (1976). For the Holt Pipkin experiment,-See F. Pipkin, ref. 9.

15 E. S. Fry et R.C. Thompson, Phys. Rev. Lett .. ri, 465 (1976).

16 A. Aspect, P. Grangier et G. Roger, Phys. Rev. Lett. ~, 460 (1981).

17 D. Bohm et B.J. Hiley, Nuovo Cim. B 35, 137 (1976).

18 A. Aspect, P. Grangier et G. Roger, Phys. Rev. Lett. 49, 91 (1982) •

J. 9 A. Aspect, Phys. Lett. 54 A, 117 (1975) A. Aspect, Phys. Rev. D 14, 1944 (1976).