1.2One photon at a time

The experiment in this lesson needs a lamp, a sheet of the plastic used in polarising sunglasses, and a clear crystal of calcite. It can be done on a classroom projector. It is also enough, once you accept that light comes in photons, to end the idea that the same cause always has the same effect.

Two spots of light

Light is a wave in the electric field, and the field swings from side to side, across the direction the light is travelling. The line along which it swings is called the polarisation. Light from a lamp is a jumble of every direction at once. A polariser tidies it up: it lets through only the part of the swing that lies along one line, its axis, so what comes out is polarised along that axis.

Calcite does something odder. It sends light two different ways depending on its polarisation: the horizontal part of the swing goes one way, the vertical part another. Look at a printed page through a lump of calcite and you see every letter twice. Put a screen behind it and you get two spots of light.

Now turn the polariser, so that the light reaching the crystal swings along a line tilted at an angle $\alpha$ from the horizontal. A tilted swing is part horizontal and part vertical. If the swing has size $E$, the horizontal part has size $E\cos\alpha$ and the vertical part $E\sin\alpha$. The brightness of a beam goes as the square of the field, so the two spots have brightness

Since $\cos^2\alpha + \sin^2\alpha = 1$ whatever the angle, the two spots always share out all of the light between them. Nothing is lost in the crystal.

So far this is nineteenth-century physics, and it is entirely satisfactory. The rule for the brightness of a beam that has passed a polariser, $\cos^2\alpha$, has been in textbooks since 1809.

Now turn the lamp down

The previous lesson ended with the evidence that light arrives in lumps. Light of frequency $\nu$ comes in photons, each carrying energy $h\nu$, and a detector either registers a whole one or registers nothing. So make the lamp so dim, and the detectors so quick, that photons can be counted one at a time, and ask the obvious question.

Two things are on show here, and they pull in opposite directions. Over thousands of photons the fraction going up is $\cos^2\alpha$, exactly the old brightness rule, as it must be: a bright beam is nothing but a great many photons. But the printout itself, the order of the pluses and minuses, has no pattern at all.

Can the next click be predicted?

Perhaps there is a pattern and you have not found it. The way to find out is to bet on it.

Every photon in that run was prepared in the same way: same lamp, same polariser at the same angle, same pinhole. The crystal and the detectors did not change either. And still some went up and some went down. There is nothing we know how to control, or even to look at, that says which way the next one will go.

This is the break with classical physics, and it is a deeper one than the fuzziness of the last lesson. Classical physics rests on determinism: set up the same conditions exactly and you get the same result. Here the same conditions give different results. What is reproducible is not the outcome but the fraction of each outcome over many trials.

That fraction is what we shall mean by probability throughout the course: prepare the same thing again and again, and the probability of an outcome is the fraction of trials in which it happens, in the limit of very many trials. The theory we are building predicts those fractions and nothing else.

A fair objection: maybe each photon carries some hidden detail that decides its fate, and the lamp simply hands out that detail at random. For polarised photons on their own, such a story can be made to work. Later in the book (chapter 6, Bell’s theorem) it turns out that any story of this kind has to pay a price most physicists find worse than randomness. For now, take the randomness at face value.

A filter is also a preparation

There is one more thing to learn from this apparatus, and it will matter in every later lesson. Put two polarisers in a row with their axes at right angles. The first lets through light polarised along 0°. The second passes the fraction $\cos^2 90^\circ = 0$ of that. The pair is black.

Now slide a third polariser between them, at 45°. It seems this can only make things darker. It does the opposite: light comes through.

Think of it one photon at a time. A photon polarised at 0° meets the middle filter at 45° and passes with probability $\cos^2 45^\circ = \tfrac12$. If it passes, it is now a photon polarised at 45°: it behaves in every later experiment exactly like a photon that a lamp and a 45° polariser had prepared. The filter has not merely let it through; it has prepared it afresh. At the last filter, 45° away again, it passes with probability a half once more. Half of a half is a quarter.

With $N$ equal steps of $90^\circ/N$ the fraction that survives every one is

which is 0.25 for two steps, 0.42 for three, 0.78 for ten and 0.97 for ninety. A photon asked a long series of questions, each only slightly different from the last, almost always says yes to all of them, and is led gently round to a polarisation that it would never have reached in one jump. Peres sets this as the first exercise in the book, with crystals in place of filters.