The clicking detectors

The detection of clicks or spots on a photographic plate has long been held to be clear evidence for the particle nature of light. But attempts to make a mathematical case have always fallen just short. The reason is that the statistics are ambiguous. It turns out the classical theory supports the facts just as well as the quantum theory.

By "classical" I'm talking about the theory where the waves are classical but the atoms are quantum. You have to make one assumption: that the detection probability, say of a silver halide crystal changing state, is proportional to the intensity of the incident wave. Why should a crystal behave this way? That's another question, but at least it's a reasonable working theory. It is of course the only theory which makes the classical wave agree with the photon picture, and it happens to be a reasonably plausible theory.

It works fine when you have lots of photons, but gets tricky when you go down to "single photons". You do this by attenuating your source so the photons only strike once every ten minutes. Wouldn't the waves be too weak to concentrate their power into a detection event? Not according to our working assumption. Remember, by the way, the appearance of a dot on a photographic plate is what we call thermodynamically spontaneous - you don't need a net energy input to drive the transition. The wave theory still works even down to the very lowest levels.

But now let's add a twist. We're going to split the beam with a half-silvered mirror. Now the wave and particle theories make different predictions. The wave procedes at half strength to each detector, so there's some chance that both will click. But the particle goes one way or the other, so both detectors can never click.

Or can they? The problem is you're never quite sure that you have only one photon, no matter how weak the beam. Let's do the statistics. Take a beam with one photon every ten seconds. Use a one-second "coincidence" window. The beam splits in two so each detector fires every twenty seconds (100% efficiency). The wave theory says that there is a certain, constant wave intensity at each detector, that they fire on average every twenty seconds, and it doesn't matter if one or the other fires. In any given second, there is a one-in-twenty chance of either detector firing, and a one-in-four-hundred chance of both.

What do photons do? Photons are different, because if Detector A fires, it "uses up" the energy that "might have" also fired Detector B. That's the theory anyways. What do the statistics say?

Let the photons obey Poisson statistics (typical for laser). You can see where this is going. In one second there is a one-in-ten chance of a photon. What are the chances of two photons within the same "window"? One in...get this...two hundred. (Write the formula for Poisson if you don't believe me). But wait...just because there are two photons doesn't mean that you get two clicks. They might both go to the same detector. Four ways...AA, AB, BA, and BB. Only half the options give you coincidence counts. The odds? One in four hundred.

You can see that we can make the beam weaker and the window tighter, and it still doesn't hlep. The photon theory and the wave theory will always predict exactly the same number of clicks.

Now...IF you could fire one photon at a time, there is a very simple experiment which would totally go against the wave theory. You just put up a beam splitter and wait. With single photons, you can NEVER get two clicks. And the wave theory predicts that whatever your detection efficiency, call it n, you should get two clicks with a probability of n-squared/4. (25% of the time for 100% efficiency). It's a very simple experiment.

Interestingly enough, the much-referenced paper by Thorne et al http://physics.princeton.edu/~mcdonald/examples/QM/thorn_ajp_72_1210_04.pdf

proposes an experiment which is generally similar to what I've described except it is decidedly more complicated. There is a third beam used for gating, and there are electronic coincidence detectors. Then you have to calculate something called "second-order coherence". It's just overly complicated enough that I might plausibly claim to not be convinced by the results. But of course that would be unreasonable of me.

Yet I have to wonder. The authors of this experiment make it a major selling point that it clearly demonstrates the particle nature of light at a level suitable for an undergraduate lab. Yet their version is far more complicated than my bare-bones version. If the case is so clear-cut, why the complications?