A major "loophole" in Einstein-Podolsky-Rosen (EPR) experiments has been almost totally neglected, that of the "subtraction of accidentals". This is quite different from the detection loophole, and depends on assumptions about the emission process, not the detection one. If the emissions are not independent, being spaced out in time rather than purely random, then the customary estimate of accidentals may be much too large. Assuming that the true number is negligible, we find that many experiments, both old and new, would be easily explained by local realist models. The data is compatible with wave models of light - a fact that has been obscured by the presentation of derived and modified data only, and failure to publicise doubts and anomalies. The quantum theory interpretation, with its implications of nonlocality, amounting to magic when operating on the macroscopic scale of real experiments, must be questioned. Study of EPR experiments involving atomic cascade and parametric down conversion sources suggests new models for the behaviour of light, and new ways of testing them.
Thus this paper is about some details of experimental imperfections that can make all the difference between the appearance of agreement with quantum theory (QT) and of agreement with local realism. It will be illustrated primarily from the work of Alain Aspect, as it is here that I have access to a certain amount of raw data. The imperfections I shall discuss are of similar numerical importance in recent experiments, such as those of Tittel et al [4], involving apparent non-locality over a distance of 10 km, but the experiment I shall concentrate on is Aspect's first, which uses a Bell test that cannot be infringed by the detection loophole alone. From the point of view of my realist model of light (similar in most ways to Stochastic Optics [5, 6] and to CWN [7]), his last experiment, involving time-switching, is identical and so susceptible to the same explanation.
All EPR experiments share the same basic problems, in that the experimenter has considerable freedom in the choice of equipment, its settings, and how the quantum theory prediction is to be adapted to allow for recognised imperfections. Aspect supported all his choices by eloquent reasoning, but none-the-less there was an open bias towards choices that favoured quantum theory. In his thesis, he states that he is convinced by earlier experiments such as that of Freedman [8] that QT is correct: his role was not, as many have assumed, to prove that Nature supported QT. His stated primary aim was to show that the high correlations that he was confident of obtaining could not be the result of exchange of signals unless these were at superluminal velocities. He therefore felt, as have many others both before and since, that he could use agreement with the QT predictions as a "privileged method" for judging when his apparatus was "correctly" set. He was careful not to invoke this licence explicitly for the actual coincidence rates used in the Bell tests, but there are intermediate decisions where it must have played a part. Another common way in which experimenters restrict their search for realist explanations is that they know that, under QT, certain imperfections, such as low "quantum efficiency" of detectors, will decrease the significance of the Bell tests. They therefore, sometimes quite explicitly, assume that all imperfections must have this effect, so that it is impossible, they think, for an imperfection to bias the results towards QT. Unfortunately, this is entirely wrong. Under realist models [9,10], the very example they understand the best, the low efficiency of the detectors, can increase correlations, pulling the realist prediction closer to QT.
The QT story is that pairs of "photons" are produced by the source in an atomic cascade, pass through the polarisers if their polarisation is suitable, and a fixed proportion of those that pass through are detected. The coincidence monitor analyses the differences in detection times and, using parameters at the discretion of the experimenter, assesses the coincidences, supposed to represent an estimate of the number of occasions when both photons are detected. The problem arises, as you will see, from the fact that the two photons are not (under the QT story) emitted simultaneously. (Under my own model, they are emitted simultaneously but each is extended in time, with similar, but not, I think, identical, consequences.)
My information comes largely from Aspect's PhD thesis [2] - a document that, had it been in English, might by now have become a best-seller. I have made translations of two sections available through my Web pages.
| Angle between polarisers | 0 | 22.5 | 45 | 67.5 | 90 | One polariser absent | Both absent | |
| Raw | 96 | 87 | 63 | 38 | 28 | 126 | 248 | |
| Accidental | 23 | 23 | 23 | 23 | 23 | 46 | 90 | |
| Adjusted | 73 | 64 | 40 | 16 | 5 | 81 | 158 |
As you will see, the adjustment subtracting "accidentals" is large. The raw data (for both polarisers present) follows a nice sine curve, displaced upwards; the adjustment shifts it down, which increases the visibility ((max - min)/(max + min)) from 0.55 to 0.88. If you calculate the Bell test you will find that the raw data does not infringe it. The visibility is in fact only slightly greater than the prediction (0.5) of the simplest realist model - the model that assumes the counts obey Malus' Law exactly and that you get the expected coincidence probabilities by multiplying the two singles probabilities, for fixed polarisation angle. The basic QT prediction for visibility is 1.0, but making allowance for real transmission factors etc. one can quite easily justify a reduction to near the figure observed.
Now one might be forgiven for thinking that such an evidently important adjustment would have been the subject of vigorous debate and fully investigated. As I mentioned earlier, it has been discussed - by Marshall, Santos and Selleri on the one hand, and Aspect and Grangier on the other. But Aspect used theoretical arguments that few, if any, could be expected to follow. Moreover, he quoted figures not for the experiment I am concerned with but for his second experiment, which had two outputs for each polariser and used a different Bell test, one in which each term is estimated using a ratio in which the denominator is the sum of four coincidence rates. The use of this denominator invalidates the test - it can produce a bias that allows it to be infringed relatively easily, whenever there are "variable detection probabilities" (of which more later). I have published a paper [10] explaining by means of an intuitive analogy (a "Chaotic Ball") this well-known but frequently forgotten fact.
To return to our story: Aspect was able to find a case in which a Bell test was violated even when he did not do the subtraction, but this was irrelevant to the real problem, which concerns violation of the more stringent tests used in certain single-channel experiments. (These tests have the weakness that they involve comparisons between results with and without polarisers in place, but there is no a priori reason to expect bias in any particular direction.) To my knowledge, no experimental investigation has been done on the subtraction. If QT is correct in its description of the atomic cascade, then there is no theoretical difficulty barring investigation: we just have to decrease our source emission rate so as to have negligible accidentals and see if we still get violations of Bell inequalities. Freedman [8] was able to do this, though his source was slightly different (his experiment suffered from different problems, including some relating to timing [11]). If there prove to be difficulties using Aspect's source with a low emission rate then this may give us new insight into the atomic cascade process, as well as confirming that it is not suitable for use in EPR experiments.
The subject, it would appear, has attracted little attention because the data I used above has not been made public. I summarised it from a table (table VII-A-1) in Aspect's thesis that was presented in confused order, that in which it was collected, which was impossible to assess by eye. On the rare occasions when such data is published (for example, in graphical form in Tittel et al's paper), the subtraction is done with such confidence, no hint of the assumptions behind it, that the reader has no reason to query it.
The experiment outputs a time-spectrum, of which one is shown below. Those displayed on a VDU during the running of each subexperiment would have had rather greater scatter as the accumulation time was shorter.
Thus it is natural for Aspect to assume that the falling part of the spectrum simply mirrors the time of emission of B, with the peak at 0, corresponding to zero delay. Time is measured from the time of emission of A. He followed established practice in assuming the rising part represented just error.
The basic idea in defining "coincidences" is to chose a start and end time relative to the peak and count the number of events in this "window". Note, incidentally, that the QT model is inadequate in practice, as it only ever mentions one parameter, the window size.
But what of the "shoulders"? For a valid EPR experiment (see footnotes in Clauser and Horne's 1974 paper [11]), we should have organised the source so that coincidences are easy to identify, with the shoulders negligible. Aspect's clearly were not, so he had to fall back on a model (Fig. 4 (a)), valid only if emissions are stochastically independent.
In this model, the shoulders, together with the whole base of the spectrum, correspond to cases where the A and B photons come from different atoms. To estimate these "accidentals", we need to know the probability of events happening this close together just by chance, when there is no synchronisation, so we artificially delay one channel (the B one) by 100 ns, sufficient to destroy this synchronisation. The "coincidences" as measured using the delayed channel give the required estimate.
In this semi-classical model, the whole spectrum is interpreted quite differently from QT. The pair of signals approaching the two detectors might be visualised as shown below:
Remember that we are talking about the very weakest (visible range) light we can detect. This light is so weak that it requires the addition of EM noise to push its intensity over a threshold and cause the emission of a photoelectron. It is quite different from the case of X-rays, say, which come in pulses that are each capable of causing a whole shower of electrons to be emitted.
The noise has the effect of making the detector take samples of the intensity, with the likelihood of a sample being closely related to the magnitude of the intensity, but there is no a priori reason to suppose the probability of detection to be exactly proportional to intensity. The detector could, of course, take several sample from the same signal. It does not generally do so because in practice there is a "dead time" after each detection, controlled by the experimenter.
There is, in a sense, plenty of experimental evidence for the importance of this noise: we know that temperature is critical; screening is important (and it might not have been possible [7] to screen out EM noise from local electronic equipment); physical proximity of detectors can increase correlations. The assumption that it is only intensity that matters seems adequate for our purpose. It lends itself to computer simulation, which can easily confirm that the output will be at least qualitatively as observed. There is more than one way in which the shoulders might arise, and further experimentation is needed. They are very likely for Aspect's experiments to include a large contribution from signals that were emitted at only the one frequency, unpaired, as the system produced three times as many B detections as A ones. (Aspect may well have been wrong, incidentally, in assuming this was just because detection efficiencies were lower for A.)
This model is entirely compatible with real experimental results that purport to demonstrate the particle behaviour of light [13]. Basic classical theory may seem to imply that light is more smoothly distributed than it actually is, but the leap from smooth distribution to point particles was an over-reaction.
There will be a large element of random error in the time of detection, but there will also be systematic effects, with higher intensities (per signal) being detected sooner (only the first detection counts, as a result of dead times). This feature gives a testable difference from QT: if we insert an extra polariser, say, to decrease the intensity, in the QT model we decrease the number of photons. The shape of the time spectrum stays the same. In my model, the shape will change. I must emphasise that this effect is only expected with detection systems similar to that used in this particular experiment, in which dead times would have been large enough to suppress later detections and only the first is registered.
To return to the matter of "accidentals", it seems to me that the assumption of independence that is used in their estimation needs experimental testing. In experiments using parametric down conversion sources (all recent ones), classical theory is quite firm that the crystal produces only one pair at a time, so that independence is out of the question. Under realist reasoning, the infringement of Bell inequalities is always evidence of false assumptions, and independence seems a prime suspect.
The subtraction of accidentals is numerically important whenever it is applied, but it is not sufficient on its own to account for violation of the Bell inequality in Aspect's second, two-channel, experiment. As mentioned earlier, the Bell test appropriate here is easily violated if we have "variable detection probabilities", which we can have if Malus' Law does not apply exactly to counts .
For weak signals, we might have something like the dashed curve of Fig. 6, which has a wider trough than a sine curve does.
There is, I believe, some experimental evidence that Malus' Law does not hold perfectly for counts. Aspect reports in his thesis several "anomalies", such as reversal of the roles of A and B producing changes in coincidence rates, and the total of the four coincidence rates in his two-channel experiment not being quite constant. (In relation to the former, he makes the highly questionable decision that it does not matter that the separate values do not conform to QT since it is only the total that is needed for his Bell test.) Considering that he has made settings so that all the singles rates appear to be behaving correctly and he gets, as expected, a doubling of coincidence rate when he removes a polariser, the most likely explanation of the anomalies is slight deviations from the rule. They are all small, but they must have been reproducible or he would not have felt the need to report them at all. I can see no good reason why they should not be investigated, as only a modest increase in the scale of the experiment would be needed.
In EPR experiments, the experimenters do what they are asked to do: find conditions in which Bell inequalities are infringed! There is no independent check on the methods they use, or requirement to publish full data, including the runs that do not quite work. The magicians know how to produce their illusions (albeit not quite perfectly - witness those "anomalies"), but why do they still not understand them? The true explanation demands a pure wave model of light.
[2] Aspect, A: "Trois Tests Expérimentaux des Inégalités de Bell par mesure de corrélation de polarisation de photons", PhD thesis No. 2674, Université de Paris-Sud, Centre D'Orsay (1983).
[3] Aspect, A, Grangier, P and Roger, G: Physical Review Letters 47, 460 (1981) and Physical Review Letters 49, 91-94 (1982); Aspect, A, Dalibard, J and Roger, G: Physical Review Letters 49, 1804-1807 (1982).
[4] Tittel, W et al: "Experimental demonstration of quantum-correlations over more than 10 kilometers", submitted to Physical Review Letters 1997. Available at http://xxx.lanl.gov, ref quant-ph/9707042.
[5] Marshall, T W, Santos, E and Selleri, F: "Local Realism has not been Refuted by Atomic-Cascade Experiments", Physics Letters A 98, 5-9 (1983).
[6] De la Peña, L and Cetto, A M: "The Quantum Dice: an Introduction to Stochastic electrodynamics", Kluwer (1996).
[7] Gilbert, B and Sulcs, S: "The measurement problem resolved and local realism preserved via a collapse-free photon detection", Foundations of Physics 26, 1401 (1996).
[8] Freedman, S J: "Experimental Test of Local Hidden-Variable Theories", PhD thesis (available on microfiche), University of California, Berkeley (1974).
[9] Pearle, P: "Hidden-Variable Example Based upon Data Rejection", Physical Review D 2, 1418-25 (1970).
[10] Thompson, C H: "The Chaotic Ball: An Intuitive Analogy for EPR Experiments", Foundations of Physics Letters 9, 357 (1996). Available from my Web page or from http://xxx.lanl.gov, where it is ref 9611037 of the quant-ph archive.
[11] Thompson, C H: "Timing, "Accidentals", and Other Artifacts in EPR Experiments", submitted 4 November 1997 to Physical Review Letters. Available from my Web page or as ref 9711044 of http://xxx.lanl.gov quant-ph archive.
[12] Clauser, J F, and Horne, M A: Physical Review D 10, 526 (1974).
[13] Grangier, P, Roger, G and Aspect, A: "Experimental Evidence for a Photon Anticorrelation Effect on a Beam Splitter: a New Light on Single-Photon Interference", Europhysics Letters 1, 173 (1986).