Bell’s Theorem on the Cheap

This past spring, I helped one of my students carry out a quantum entanglement demonstration using Sodium-22, Geiger counters, and blocks of Aluminium to act as polarizers. The inspiration was this blog post on Scientific American. We used a 10 uCi source — an order of magnitude stronger than in the blog — set up the experiment to reduce the number of non-polarized coincidences (more on this later). The student saw the detection of about 5 entangled photon-pairs per minute.

This got me thinking about the further utility of this low-cost approach to quantum physics. Specifically, I wondered whether I could use this approach to demonstrate Bell’s inequality. The first experimental work was done in the 1960’s, with Alain Aspect’s 1982 paper considered the definitive result, and further “clean-up” continuing into the current century. Here’s a good overview by Aspect.

A note on the equipment. Na-22 can be purchased from Spectrum Techniques or United Nuclear. I had about a month’s delay before the product shipped with the former, but the latter doesn’t ship outside the USA. Check you country’s laws on importing radioactive isotopes. Aware Electronics make good Geiger counters and have good customer service, although their web site is a bit of a mess. We also bought their coincidence box, but you could save yourself a few dollars and wire something together cheaply. For the Aluminium blocks, we ended up chopping up some door frame material, but you can do better at any number of online metal retailers. The lead plates are half-inch pieces that came as part of the packaging with the Na-22.


Image 1: Apparatus for entanglement demonstration

In the entanglement experiment, positrons emitted by the Na-22 annihilate with passerby electrons to produced pairs of entangled photons. If measured, these photons should have orthogonal polarization (note to self: why?). Each photon, when it hits the Aluminium block, will undergo Compton scattering against a free electron in the block. The direction of this scattering is dependent on the incoming polarization of the photon: scattering is maximized in the direction perpendicular to the photon’s plane of polarization, and approaches zero in the direction that is parallel to the plane of polarization and perpendicular to the direction of propagation.

Thus, in the image above (and in an ideal world!) no coincident photons will be measured by the two Geiger counters. If one of the Geiger counters is rotated to point into the screen, coincidences should be measured. In the student’s experiment, he found about 40 “noise” and 3 “genuine” coincidences in 30 seconds.

The most straightforward way to test Bell’s theorem would be to add two more Geiger counters to the apparatus from the entanglement demonstration. As Aspect shows in his paper (linked above), Bell’s inequality can be applied to a situation in which the polarizations of entangled photons are measured with respect to polarizers that can be set at a variety of angles.

Image 2: Geometry of a a test of Bell's theorem

Image 2: Geometry of a a test of Bell’s theorem

As you see in Image 2, the entangled photons are incident on polarizers a and b. Geiger counters detect photons that emerge parallel to, and perpendicular to, a chosen axis.

Image 3: Rotation of the polarizers

Image 3: Rotation of the polarizers

The two polarizers are set to a particular relative azimuthal angle, and coincidences are measured in the two perpendicular directions for each. Aspect (link above) shows that the optimal relative angle is 22.5 degrees. Equations 28 and 21 allow calculation of the value of a particular parameter S.

My first attempt, without trying to correct for “noise” or any systematic errors, produced the rather unconvincing value of S = 0.1. For comparison, Bell’s inequality is that |S| < 2 for naive theories, and |S| < 2*sqrt(2) ~ 2.283 according to quantum mechanics. I’ll have to do a lot better if I am to have any chance to show that this theory is correct.

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