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Physical Review A

Stochastic optics: A local realistic analysis of optical tests of Bell inequalities

Trevor W. Marshall, Emilio Santos

Published Jun 1, 1989• 33 citations
Publisher

Abstract

Stochastic optics may be considered as simply a local realistic interpretation of quantum optics and, in this sense, it is a first step in the reinterpretation of the whole of quantum theory. However, as it is not possible to interpret all the details of quantum theory in a local realistic manner, as shown by Bell's theorem, minor changes are introduced in the formalism with the consequence that the new theory makes different predictions in some special cases. In stochastic optics, the quantum-operator formalism is simply considered a formal way of dealing with stochastic fields. In particular, the quantum zero point is taken as a real random electromagnetic radiation filling the whole of space. This radiation noise has the same nature as light signals, the only difference being the greater intensity of the latter. We assume that photon detectors have an intensity threshold just above the level of the noise, thus detecting only signals. Transmission of radiation through polarizers follows Malus's law, but the interplay of signal and noise leads quite naturally to the prediction that the detection probability of some signals is enhanced, which is known to be a necessary condition for the violation of the empirically tested Bell inequalities. In our view, correlated photon pairs are pairs of light signals supercorrelated in polarization, in the sense that, as well as the signal, the accompanying noise is also correlated. Thus stochastic optics allows predictions for the empirical correlations very close, but not identical, to the quantum ones. The theory is applied to the analysis of all experiments designed to test the Bell inequalities by measuring polarization correlations of photon pairs. The predictions agree with quantum optics and experiments within statistical errors, except for the Holt-Pipkin experiment. In this case, the experimental results agree with stochastic optical predictions within two standard deviations while violating quantum optics by four.

DOI
10.1103/physreva.39.6271

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Stochastic optics may be considered as simply a local realistic interpretation of quantum optics and, in this sense, it is a first step in the reinterpretation of the whole of quantum theory. However, as it is not possible to interpret all the details of quantum theory in a local realistic manner, as shown by Bell's theorem, minor changes are introduced in the formalism with the consequence that the new theory makes different predictions in some special cases. In stochastic optics, the quantum-operator formalism is simply considered a formal way of dealing with stochastic fields. In particular, the quantum zero point is taken as a real random electromagnetic radiation filling the whole of space. This radiation noise has the same nature as light signals, the only difference being the greater intensity of the latter. We assume that photon detectors have an intensity threshold just above the level of the noise, thus detecting only signals. Transmission of radiation through polarizers follows Malus's law, but the interplay of signal and noise leads quite naturally to the prediction that the detection probability of some signals is enhanced, which is known to be a necessary condition for the violation of the empirically tested Bell inequalities. In our view, correlated photon pairs are pairs of light signals supercorrelated in polarization, in the sense that, as well as the signal, the accompanying noise is also correlated. Thus stochastic optics allows predictions for the empirical correlations very close, but not identical, to the quantum ones. The theory is applied to the analysis of all experiments designed to test the Bell inequalities by measuring polarization correlations of photon pairs. The predictions agree with quantum optics and experiments within statistical errors, except for the Holt-Pipkin experiment. In this case, the experimental results agree with stochastic optical predictions within two standard deviations while violating quantum optics by four.
I assume that everywhere in space there is a real random electromagnetic radiation, or zeropoint field (ZPF), which looks similar for all inertial observers, so that the stochastic properties of the field should be Lorentz invariant. This fixes the spectrum except for a single adjustable parameter measuring the scale, which is identified with Planck's constant, so making the ZPF identical to the quantum electromagnetic vacuum. Photons are just fluctuations of the random field or, equivalently, wavepackets in the form of needles of radiation superimposed to the ZPF. Two photons are "classically correlated" if the correlation involves just the intensity above the average energy of the ZPF, but they are "entangled" if the ZP fields in the neighbourhood of the photons are also correlated. These assumptions may explain all quantum optical phenomena involving radiation and macroscopic bodies, provided the latter may be treated as classical. That is, we have an interpretation of quantization for light but not for matter. Detection of photons involves subtracting the ZPF, which cannot be made without a fundamental uncertainty. This explains why photon counters cannot be manufactured with 100% efficiency and no noise (dark rate), which prevents the violation of a genuine Bell inequality (this is the so-called detection loophole). The theory thus obtained agrees very closely with standard quantum optics if this is formulated in the Wigner representation.
Bell's theorem of 1965 is a proof that all realistic interpretations of quantum mechanics must be non-local. Bell's theorem consists of two parts: first a correlation inequality is derived that must be satisfied by all local realistic theories; second it is demonstrated that quantum mechanical probabilities violate this inequality in certain cases. In the case of ideal experiments, Bell's theorem has been proven. However, in the case of real experiments where polarizers and detectors are non-ideal, the theorem has not yet been proven since the proof always requires some arbitrary and {\em ad hoc} supplementary assumptions. In this paper, we state a new and rather weak supplementary assumption for the ensemble of photons that emerge from the polarizers, and we show that the conjunction of Einstein's locality with this assumption leads to validity of an inequality that is violated by a factor as large as 1.5 in the case of real experiments. Moreover, the present supplementary assumption is considerably weaker and more general than Clauser, Horne, Shimony, Holt supplementary assumption.
John Bell’s theorem of 1964 states that local elements of physical reality, existing independent of measurement, are inconsistent with the predictions of quantum mechanics (Bell, J. S. (1964), Physics (College. Park. Md). Specifically, correlations between measurement results from distant entangled systems would be smaller than predicted by quantum physics. This is expressed in Bell’s inequalities. Employing modifications of Bell’s inequalities, many experiments have been performed that convincingly support the quantum predictions. Yet, all experiments rely on assumptions, which provide loopholes for a local realist explanation of the measurement. Here we report an experiment with polarization-entangled photons that simultaneously closes the most significant of these loopholes. We use a highly efficient source of entangled photons, distributed these over a distance of 58.5 meters, and implemented rapid random setting generation and high-efficiency detection to observe a violation of a Bell inequality with high statistical significance. The merely statistical probability of our results to occur under local realism is less than 3.74×10-31, corresponding to an 11.5 standard deviation effect.
Conventional tests of Bell’s inequality rely on entangled photon pairs. Here, we replace entangled pairs with two independent photons of orthogonal polarization, and demonstrate that Bell’s inequality is still violated. Given the inherent local realism of independent photons, this experiment proves that Bell’s inequality cannot falsify the local realism of photons. We thus conjecture that the violation of Bell’s inequality by entangled photon pairs originates from their orthogonal polarizations, rather than the breakdown of local realism. To interpret this unexpected violation with independent photons, we further substitute the two photons with two monochromatic light beams, and calculate the transmittance correlation through polarizers via Malus’s law and Karl Pearson’s correlation formula. We show that this correlation also defies Bell’s inequality. Retracing the derivation of Bell’s inequality reveals its validity is restricted to binary events, which accounts for the observed violation with light beams. Finally, we propose a thought experiment involving gradual attenuation of light intensity down to the single-photon regime, and hypothesize that single-photon transmission through a polarizer does not constitute a binary event. This hypothesis provides a unified interpretation for both our experimental findings and all canonical Bell inequality tests reported to date.
Over the past three decades or so, experiments of the type called Gedanken have become real. Recall the Schrödinger quote from Chapter 8: “… we never experiment with just one electron or atom or (small) molecule.” This is no longer true. We can do experiments involving single atoms or molecules and even on single photons, and thus it becomes possible to demonstrate that the “ridiculous consequences” alluded to by Schrödinger are, in fact, quite real. We have already discussed some examples of single-photon experiments in Chapter 6, and in Chapter 10 we shall discuss experiments performed with single atoms and single trapped ions. In the present chapter, we shall elaborate further on experimental tests of fundamentals of quantum mechanics involving a small number of photons. By fundamental tests we mean tests of quantum mechanics against the predictions of local realistic theories (i.e. hidden variable theories). Specifically, we discuss optical experiments demonstrating violations of Bell's inequalities, violations originally discussed by Bell in the context of two spin-one-half particles. Such violations, if observed experimentally, falsify local realistic hidden-variable theories. Locality refers to the notion, familiar in classical physics, that there cannot be a causal relationship between events with space-like separations. That is, the events cannot be connected by any signal moving at, or less than, the speed of light; i.e. the events are outside the light-cone. But in quantum mechanics, it appears that nonlocal effects, effects seemly violating the classical notion of locality in a certain restricted sense, are possible.