In my post about how we know photons exist, I make reference to the famous Kimble, Dagenais, and Mandel experiment showing “anti-bunching” of photons emitted from an excited atom. They observed that the probability of recording a second detector “click” a very short time after the first was small. This is conclusive evidence that photons are real, and that light has discrete particle-like character. Or, as I said in that post:

This anti-bunching effect is something that cannot be explained using a classical picture of light as a wave. Using a wave model, in which light is emitted as a continuous sinusoidal wave, you would expect some probability of a detector “click” even at very short times. In fact, you can easily show that any wave-like source of light must have a probability of recording a second click immediately after the first one that is at least as big as the probability of recording a second click after a long delay. Most of the time, the probability is actually higher at short times, not lower. A decrease in the probability of a second detection at short times is something that can only be explained by the photon model.

Since then, I have had a steady trickle of comments asking for the detailed explanation of why you can’t, or protesting that you can too explain anti-bunching with a wave model. I generally try to avoid posting lots of equations on the blog, but this is unavoidably mathematical, so I will put the explanation below the fold.

If you want to characterize the detector signal from a wave model of light, you need to look at the intensity of the light, and in particular at the fluctuations of that intensity. For a classical source of light waves, we can describe this mathematically as a constant average intensity *I _{0}* plus a fluctuating term that is a function of time

*δ I*(

*t*). So, the intensity at any given time

*t*is:

I(t) =I+_{0}δ I(t)

If you just had the constant intensity, you would always have exactly the same probability of measuring a “click” at your detector in any given instant of time. The fluctuating term *δ I*(*t*) gives you a way to describe changes in this probability, and can be either positive (indicating an increased chance of a “click”) or negative (indicating a decreased chance of a “click”). It’s equally likely to be positive or negative at any given instant in time.

Of course, your detector can’t give you a perfect and instantaneous snapshot of the intensity, so what we really measure is the *average* intensity over some measurement interval, which we write as:

<

I(t)> = <I+_{0}δ I(t)> = <I> + <_{0}δ I(t)>

(Basically, angle brackets indicate an average over some time interval, and the average of the sum of two quantities is equal to the sum of the averages.) Looking at our definitions above, we can immediately see that <*δ I*(*t*)> = 0– if it didn’t, that would just represent a change in the average intensity *I _{0}*, so we could trivially redefine everything so that the average of the fluctuating term

*is*zero.

With me so far? This is just the definition of the way you deal with characterizing the energy flow in a classical wave model of light. You can do the whole thing in terms of the electric field amplitude if you really want to– the math is slightly more tedious (because intensity is the average square of the electric field amplitude), but works out exactly the same in the end. I haven’t said anything about the mathematical details of <*δ I*(*t*)>, so this is all perfectly general.

If you want to know the probability of getting a *second* detector “click” a short time *τ* after the first, the quantity you need to describe this is the intensity correlation:

<

I(t)I(t+τ)> = <(I+_{0}δ I(t))(I+_{0}δ I(t+τ))>

This is just saying that you take the intensity at time *t* and multiply it by the intensity a short time *τ* later, and take the time average of that product. The right-hand side just plugs in the definition of those two intensities in the mathematical notation we used earlier. If we multiply out the right-hand side, and use the fact that the average of a sum is the sum of the averages, we get:

<

I(t)I(t+τ)> = <I_{0}I> + <_{0}I_{0}δ I(t)> + <δ I(t+τ))I> + <_{0}δ I(t)δ I(t+τ)>

This may look a little scary, but if you go through it term by term, it’s really pretty simple. The first term is just the average intensity squared, which is some positive number. The next two terms are the average of the intensity (just a number) multiplied by the fluctuating term, which by definition averages to zero. So, the next two terms are zero. That leaves us with just the first and last terms to deal with:

<

I(t)I(t+τ)> = <I_{0}I> + <_{0}δ I(t)δ I(t+τ)>

So, what is <*δ I*(*t*)*δ I*(*t*+*τ*)>? This is the average of the product of the fluctuating term at time *t*, and the fluctuating term a time *&tau* later. We can’t give a precise value of this without specifying the mathematical form of *δ I*(*t*), but we can say a couple of things about limiting cases.

For large values of *τ*, we expect the average <*δ I*(*t*)*δ I*(*t*+*τ*)> to be equal to zero, because the fluctuations are supposed to be random. That means that there shouldn’t be any particular correlation between the fluctuation at one instant and the fluctuation at any later time– if you could use the value of *δ I*(*t*) to predict *δ I*(*t*+*τ*), then the fluctuations wouldn’t be random, would they?

The other time whose value we can specify <*δ I*(*t*)*δ I*(*t*+*τ*)> for is *τ*=0, for which we have:

<

δ I(t)δ I(t+ 0 )> = <δ I(t)^{2}>

That is, for zero delay between “clicks,” the final term in our correlation is the average of the square of the fluctuating term at time *t*. Since the intensities we’re dealing with are by definition real numbers, this has to be a number greater than or equal to zero. There’s no way for the square of a real number to average out to be a negative number.

This means that, for a classical wave picture of light, the probability of getting a second detector “click” immediately after the first “click” has to be *at least as big* as the probability of getting the second click a long time later (large values of *τ*, where the correlation term averages to zero). The very best you can do with a classical source is to find a flat correlation function– that is, exactly the same value everywhere– which you can get using a laser. Thermal sources tend to show “bunching” effects– that is, a correlation at short times that is significantly greater than the correlation at long times (as in the famous Hanbury Brown and Twiss experiments).

The anti-bunching effect observed by Kimble, Dagenais, and Mandel, and every other correlation experiment with single-photon sources that’s been done shows a *lower* value of this correlation at short times, which is impossible unless you know a way to make the square of a real number turn out negative. You can find more detailed versions of this calculation in any textbook on quantum optics. As it makes no assumptions about the mathematical form of the fluctuations beyond insisting that the intensity be a real number, there’s no way to get around the problem while still using a wave model of light. Thus, anti-bunching measurements conclusively show that light has particle nature, and thus photons are real.