Historical research guide · 0869 · 1989
Stochastic optics: A local realistic analysis of optical tests of Bell inequalities
Trevor W. Marshall · Emilio Santos
In one minute
Trevor Marshall of Manchester and Emilio Santos of Cantabria ask what optics looks like if the zero-point field is a real radiation rather than a bookkeeping device. Their answer, stochastic optics, leaves Maxwell’s equations untouched and adds one physical ingredient: space is already filled with random electromagnetic radiation of the same nature as light, differing only in being weaker. A single photon becomes a needle-shaped wave packet riding on that sea, and a photodetector becomes a device with an intensity threshold set just above the noise — which is why every real detector shows a dark count rate, and why pushing its efficiency up pushes that rate up with it. With two adjustable numbers fixed once and for all, the authors compute what this fully classical picture predicts for every optical test of Bell’s inequalities performed up to 1989, eight in total. Seven agree with quantum optics and with the measurements. The eighth, the Holt-Pipkin result long set aside as a systematic error, is the one their model reproduces — and they give a physical reason why.
Connection
Chapter 2 says the vacuum is a real, structured medium rather than a formal device, and this paper is one of the sharpest tests anyone put that idea to: take the zero-point radiation literally, follow it through a polarizer and into a photomultiplier, and then ask what the Bell experiments actually establish. It is also a chapter 1 lesson in how much of an experiment’s meaning rides on what the detectors are assumed to do.
From the source
Stochastic optics takes the quantum zero point to be a real random electromagnetic radiation filling the whole of space, of the same nature as light signals and differing only in intensity. Formally this means keeping the symmetric Hamiltonian, in which every vacuum mode carries an average energy of half a Planck quantum, rather than normal-ordering it away — and the authors argue that the reordering which removes that half quantum is the ad hoc step, not its retention.Sect. II, Principles of stochastic optics, Eq. (2.5)
A single photon signal is not a particle but a wave packet: a needle of radiation of the kind Einstein called Nadelstrahlung, of the order of metres long in the direction of propagation and a few wavelengths across, superposed on the zero-point field. Everything the theory covers is the propagation of that packet, plus noise, through macroscopic optical devices.Sect. II, following Eq. (2.7)
Because the zero-point radiation is random, no detector can have a sharply defined sea level, so an ideal detector is impossible in principle. The authors replace the quantum detection rule with a threshold rule: the detection probability rises with the signal intensity above a threshold energy. Two consequences follow that they present as merits rather than defects, because both are what laboratories actually see — every detector has some dark rate, and detector efficiency must stay low if the dark rate is to stay low. Their threshold parameter corresponds to the adjustable voltage bias on a real photomultiplier.Sect. II, Eq. (2.10) and the discussion following it
The theory has only two adjustable numbers, and they are fixed once by matching the first two Legendre coefficients of the polarizer transmission function to the quantum result, giving an amplitude parameter of 2.204 and a threshold of 1.431. The interplay of signal and noise then makes the detection probability of some signals enhanced rather than merely attenuated — which is exactly the property known to be necessary for a local realistic model to reproduce a violated Bell inequality.Sect. III, Action of a polarizer, Eqs. (3.9) to (3.11); Abstract
Applied to all eight optical Bell tests performed up to 1989 — Freedman-Clauser, Holt-Pipkin, Clauser, Fry-Thompson, Aspect and colleagues I, II and III, and Perrie and colleagues — the model’s predicted values of the tested parameters sit within statistical error of the measurements in every case. Marshall and Santos note that Aspect’s no-enhancement auxiliary hypothesis amounts to requiring the two channel probabilities to sum to a constant, which their model does not do, so a predicted value above the bound of two carries no contradiction.Sect. IV, Table II; Eqs. (4.22) to (4.28)
The model predicts that calcite polarizers and pile-of-plates polarizers act differently on signal plus noise, and that difference is where the one outlying experiment lives. For the Holt-Pipkin measurement, which used calcite and has usually been set aside as containing an unidentified systematic error, the measured Freedman parameter of 0.216 plus or minus 0.013 sits within two standard deviations of the stochastic-optics value of 0.242 while lying four standard deviations from the quantum-optics value of 0.266. What to watch: a modern repetition of the calcite-polarizer configuration, run against the pile-of-plates configuration in the same laboratory, is the measurement that would decide between the two readings.Sect. IV, Table II and the discussion at Eq. (4.17)
The way in
https://doi.org/10.1103/physreva.39.6271How to cite it
Trevor W. Marshall, Emilio Santos (1989) Stochastic optics: A local realistic analysis of optical tests of Bell inequalities doi:10.1103/physreva.39.6271