Last time, I showed you how you could construct a photon, a light particle, in a configuration of mirrors called a ring cavity. This time I’ll show you that sometimes, you can’t make just one particle—they only come in pairs. And sometimes, the notion of a particle doesn’t make any sense at all. (This post relies heavily on last week’s post, so if you haven’t read that, I recommend you do so.)

Disclaimer: What I’m about to describe is only the simplest case, and I make simplifications for the sake of exposition. It is possible to capture and manipulate single photons between two mirrors for short times if you play tricks. In fact, that work recently won the Nobel prize.

Last time, I showed you what happens when you arrange three mirrors to make a ring. Now let’s see what happens when we bounce light between two parallel mirrors, as shown in Figure 2. This is called a Fabry-Perot cavity.

We’re going to put waves into our Fabry-Perot cavity and see if we can make just one particle. (Spoiler alert: it won’t quite be possible!)

As I’ve discussed before in some detail, light is an electromagnetic wave made up of electric and magnetic fields. To draw a parallel to our example last week, the strength of the electric field can very roughly be thought to correspond to the probability of measuring a photon. However, there are complications; for example, the quantum-mechanical wavefunction can be imaginary. It’s an experimental and theoretical fact that electric fields are zero inside conducting materials like metals. (This isn’t quite true…the field actually falls off slowly, based on something called the plasma frequency. But we’re making approximations.) Therefore, the electric field that makes up a photon must be zero at the metal mirrors.

This means that if we put a wave of light between the two mirrors and look at the strength of the electric field (which wiggles), it has to go to its center position at the mirrors, as shown in Figure 3. This restricts the type of wave that can fit in between the mirrors. (On our plot, the field is zero when it’s smack dab in the middle of the plot. Above the center line, it’s positive. Below the center line, it’s negative.)

Last time, when we added a wave to our ring cavity, the wave travelled uniformly to the right with some speed. That’s not what happens now. Now the wave can’t travel, so the height just grows and shrinks. Let’s look at the longest possible wave that can fit in the cavity, shown in Figure 4. This is called a standing wave.

(There are complications, of course; I’m completely ignoring what the magnetic field is doing. But for explanatory purposes, this is enough.)

Now we can add additional waves to the cavity. If we add the first five that fit (in special amounts based on a mathematical calculation using Fourier analysis), we get a plot that looks something like Figure 5.

We seem to have some complicated wave motion here! Let’s add even more waves! If we add nine waves to the cavity, we get something like Figure 6.

If we add nineteen, we get something like Figure 7.

Now what’s happening is beginning to become clear. If we extrapolate to Figure 1, we see this:

We attempt to put a wave with a particle-like shape into our cavity, but it splits into two waves which fly apart, reflect off of the mirrors, pass through each other, and continue reflecting for all eternity.

In this case, it’s not possible to put just one particle in between the mirrors.

Sometimes Particles Just Don’t Make Sense

The example I’ve just described highlights a problem with the standard popular narrative of particle-wave duality. We’re told that particles sometimes act like particles and sometimes act like waves. But if this were true, a single particle would never split into two just because we dropped it between two mirrors. The truth of the matter is that everything is a wave. It’s just that sometimes, like in last week’s experiment, waves can be made to act like particles.

But this week’s experiment shows us that sometimes, waves can’t be made to act like particles–at least, not a single particle. And sometimes they refuse to behave like particles at all! What all of this means is that there are conditions where particles cannot exist. For example: We think that, about 13.8 billion years ago, the universe underwent a period of rapid inflation. During this expansion, for reasons that I promise to try to address in the future (see Mukhanov and Winitzky), the very notion of a particle broke down. In the inflationary period, the packets of waves that make up particles simply could not form.

Related Reading

• This article was a sequel to the article I wrote last week on how waves can be made to behave like particles. If you didn’t understand something this week, last week’s article might clear some things up.
• One consequence of light being a wave is that it bends when it passes from air to glass or water and vice versa. This is called refraction and I wrote an article about it.
• Sometimes when you bounce light off of a material, the reflected light isn’t the same color as the incident light. This is called Raman scattering and I wrote an article explaining how it works using the wave nature of light.
• Fabry-Perot cavities are often used in lasers. If you’d like to know how they’re related, check out this article I wrote about how lasers work.
• If you want to know more about the inflationary period in the early universe, you may be interested in the three part series I wrote about it. You can find the parts here, here, and here.

Sources

I know I’ve been lazy about citing my sources on this blog and I should be better about it…even when the sources are not layperson-legible. So, for that reason, I offer the intrepid student a list of introductory texts he or she can use to learn more.

• Introduction to Electrodynamics by Griffiths offers a comprehensive introduction to electromagnetic theory (e.g., how light behaves).
• You can also find a simpler, more accessible introduction in the Feynman Lectures on Physics, which are available for free online.
• A Fabry-Perot cavity can be approximated as a particle in an infinite square well. This problem, as well as particle wave duality and basic Fourier analysis, are all covered at an introductory level in the excellent text Modern Physics For Scientists and Engineers by Taylor, Zaphiratos, and Dubson.
• A more advanced student may want to check out Introduction to Quantum Mechanics by Griffiths.