---
title: "Freedman and Clauser test Bell’s inequality with calcium light"
description: "How did the calcium photon experiment test a numerical limit on local explanations?"
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last_modified: "2026-10-24"
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[Quantum physics: reading the evidence](https://sandboxphysics.com/stories/quantum-evidence/)

Nobel A · Evidence and control

# Freedman and Clauser test Bell’s inequality with calcium light

Sandbox Physics · 2026-10-16

How did the calcium photon experiment test a numerical limit on local explanations?

## A cascade supplies photon pairs linked in time

In 1972 Freedman and Clauser used calcium atoms that emitted two photons in succession. The paired emissions gave them a way to compare polarization measurements on opposite sides of the apparatus. Each side held a removable polarizer, which selected a direction of polarization, followed by a photon detector. The electronics counted a coincidence when the two detectors clicked within the accepted time interval.

They compared coincidence rates at selected relative angles with reference runs made after both polarizers had been removed. The reference measured how many pairs the apparatus could collect without that polarization selection. Dividing by it put the angled measurements on a common scale, ready for comparison with the Bell bound.

Read the evidence: S. J. Freedman and J. F. Clauser (1972), Experimental Test of Local Hidden-Variable Theories — LBL-392 preprint, January, University of California archive

## Define the rates before reading the number

In the expression, the rate with subscript zero is measured with both polarizers removed; the other rates are taken at the stated angle between the transmission axes. The bound is the experiment’s simplified inequality, which relies on an extra detection assumption. It differs from the binary CHSH expression normalized by coincidences.

The authors reported a value 0.050 above zero, with a one-standard-deviation uncertainty of 0.008. That figure is quoted from the paper, not derived here. The central value can be checked against the rounded normalized rates printed in the preprint, but those rounded numbers are not enough to rebuild the full uncertainty analysis.

Take a separate ideal model with perfect polarizers, in which the normalized transmitted coincidence rate is one quarter of one plus the cosine of twice the relative angle. At the two angles in the formula, the rates are about 0.427 and 0.073, and their difference exceeds one quarter by about 0.104.

The ideal prediction is about 0.104; the historical result is 0.050 with its quoted uncertainty. Real polarizer transmission, finite collection angles and the reference procedure change the predicted comparison. Matching a prediction to those actual components is part of understanding why the experimental number differs from the perfect-optics example.

$$
\begin{aligned}\delta&=\frac{|R(22.5^\circ)-R(67.5^\circ)|}{R_0}-\frac14\\ &\leq 0\end{aligned}
$$

Read the evidence: S. J. Freedman and J. F. Clauser (1972), Experimental Test of Local Hidden-Variable Theories — LBL-392 preprint, January, University of California archive

## The reference rate has its own uncertainty

Suppose two measured rates are both divided by the same uncertain reference rate. If that denominator fluctuates, both ratios move together. Treating the two ratios as independent observations can then misstate the uncertainty of their difference.

A careful analysis keeps the raw counts, counting times and shared reference periods, and propagates their uncertainties through one joint statistical model. Ratios printed with separate error bars summarize the analysis. The raw counts and reference schedule supply the information about shared denominators and drift corrections.

Read the evidence: Larsson (2014), Loopholes in Bell inequality tests of local realism

## Detection and locality need separate evidence

The preprint stated outright that it assumed a photon’s chance of detection did not depend on whether it had passed through a polarizer. That assumption is what lets the detected subset test the bound. Later loophole analysis asks how inefficient detection could pick out samples that do not represent all trials.

Fast setting choices address a different possibility: that one part of the apparatus communicates using information available elsewhere. Better timing does not by itself show that the detected photons are representative, and efficient local readout does not show spacelike separation. Each requirement needs its own evidence.

Read the evidence: Larsson (2014), Loopholes in Bell inequality tests of local realism

## What the result showed, and what came later

The measurement brought a conflict between quantum predictions and a class of local models into the laboratory, under its stated detection assumption. Later experiments improved the source, changed settings during flight and counted losses more completely. Each improvement addressed a specific way the earlier apparatus could leave room for a local explanation.

The 1972 result is quoted from the authors’ report; the original raw counts and coincidence time series are not supplied here. Its value lies in the comparison it made possible: photon-counting experiments could now challenge a numerical limit on local explanations.

Read the evidence: Aspect (2022), Experimental tests of Bell’s inequalities: a first-hand account

## Continue with the models

These teaching Labs produce simulated data, which are separate from the historical observations discussed above.

- [Compare stated sampling rules](https://sandboxphysics.com/labs/bell-chsh/)
- [Separate a state estimate from a Bell test](https://sandboxphysics.com/labs/quantum-state-process-tomography/)

## Sources and reading notes

We read the University of California copy of the authors’ January 1972 preprint, LBL-392, including its detection assumption, coincidence normalization and reported result, and the APS author abstract of the published paper. The later loophole terminology comes from Larsson’s review. The ideal-polarizer calculation is only an illustration; it is not a fit and does not reconstruct the uncertainties. The linked Bell model uses binary outcomes, not the historical pile-of-plates polarizers or rate normalization.

1. [S. J. Freedman and J. F. Clauser (1972), Experimental Test of Local Hidden-Variable Theories — LBL-392 preprint, January, University of California archive](https://escholarship.org/content/qt5qs9830k/qt5qs9830k.pdf)
2. [Aspect (2022), Experimental tests of Bell’s inequalities: a first-hand account](https://arxiv.org/abs/2212.04737)
3. [Larsson (2014), Loopholes in Bell inequality tests of local realism](https://arxiv.org/abs/1407.0363)

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