A cornerstone of modern physics is mathematics. Like it or not, without the tools provided to physicists by mathematics, physics would be dead in the water. But (and this is something that all of us forget on occasion) solving equations is not the same as understanding the physics. A critical step in the development of physical insight is to recognize which solutions to an equation might correspond to reality, and which do not.

To give a concrete example, the equations of physics are blind to the direction of time, yet we know that solutions that involve time going backward are usually (but not always) invalid.

Unfortunately, as a recent publication in Physical Review Letters shows, even the brightest and best can get this wrong, and do so repeatedly over the course of years. A team of physicists has shown that light with a negative frequency (thought to be a quirk of the equations) actually, in some sense, exists.

The equations in question are Maxwell's, and they describe the propagation of light. When describing the propagation of light, the equations require that we describe the light field as having both positive and negative frequencies. A negative frequency would indicate a wave made up of photons that have a negative energy, something that doesn't necessarily make a lot of sense.

Using light to make light

A team of researchers has shown that, in some sense, negative frequencies can be observed through the generation of radiation with a positive frequency. To explore this idea, they looked at very intense light fields moving through certain types of glass and glass fibers. When the light field is very intense—as is the case when a very short, intense burst of light is created—this can lead to some very cool effects. In particular, when the light is passing through a material, the light field pushes the electrons around so hard that the electrons start to push back.

In a material like glass, the electrons can only move so far before they will be ripped away from the atom they are bound to. The harder you push them, the more they resist. So, for a weak light field, the electrons move smoothly back and forth in exact imitation of the light field that is pushing them around. As they move in response to the field, they radiate light at exactly the same color.

But when the field is very strong, the electrons don't follow the field exactly. If they did, they would be ripped away from their parent atom, and the field isn't strong enough to give the electrons enough energy for that. Instead, they just stop moving at some point. The result is that the electrons radiate light at all colors. Or, more simply, our pulse of light with one color generates another pulse of light with a different color. As the two pulses travel together, energy is drawn from the input pulse and placed into the new pulse, so as long as they overlap, the second pulse will grow brighter and brighter.

Warning: Things are about to get complicated

This process is all described by Maxwell's equations for the propagation of light through a material. But the solutions to Maxwell's equations are rather weird. Remember, every light field is described mathematically by a positive and a negative frequency. If we just consider the positive frequency component, then there are four solutions to the equation. These correspond to light waves that have positive and negative frequencies, and waves that are travelling in the same and opposite direction to the initial pulse of light.

The researchers ignore the two solutions corresponding to waves travelling in the opposite direction to the input pulse because they are not amplified. Any pulse that travels in the opposite direction does not overlap with the input pulse for very long, and there is no time to transfer much energy to the generated pulse.

Of the remaining two solutions, one has a positive frequency and travels with the generating pulse, allowing it to be amplified. This is commonly observed. The last solution corresponds to a negative frequency, also travelling with the input pulse. This solution, which should produce photons with negative energy, was thought to be an artifact of the equations and did not correspond to anything physical.

But that field itself consists of positive and negative frequencies. And (believe it or not) the negative frequency component of the negative frequency solution is a positive frequency. Or, you might think of it like this: the negative frequency element of the input pulse also generates four solutions that have both positive and negative frequencies.

In either case, what this tells us is that there should be a third pulse of light—remember, we have our input pulse of light, which generates a second pulse of light at a different (bluer) color, and now, thanks to the negative frequency solution, we get a third pulse of light at an even bluer color than the rest.

The researchers performed experiments showing that this extra pulse of light is indeed generated. The experimental results show that negative frequency radiation, in some sense, exists. But it also shows that these frequencies are only observed by their generation (either directly or indirectly) of positive frequency radiation. However, the researchers do not go further in interpreting the physical realization of negative frequency modes. I suspect that in the context of classical electrodynamics, this is actually impossible, and one has to resort to using quantum electrodynamics to interpret the negative frequency modes.

Physical Review Letters, 2012, DOI: 10.1103/PhysRevLett.108.253901