# Subtracting Photons from Arbitrary Light Fields

There’s been a fair bit of press for the article Subtracting photons from arbitrary light fields: experimental test of coherent state invariance by single-photon annihilation, published last month in the New Journal of Physics, much of it in roughly the same form as the news story in Physics World (which is published by the same organization that runs the journal), which leads with:

A property of laser light first predicted in 1963 by the future Nobel laureate Roy Glauber has been verified by physicists in Italy.

These stories can be a little puzzling, though. After all, Glauber got his Nobel a few years ago– why is one of his predictions just being verified now? Shouldn’t it be experimental verification first, dynamite money second? And what, exactly, did he predict, anyway?

The subject of this experiment is necessarily a little esoteric, but let’s take a whack at explaining it all the same. The news stories talk about proving that “the addition and subtraction of single photons from coherent light does not affect its coherence,” which might seem obvious, but is actually kind of tricky, for reasons having to do with the “quantum” in Quantum Physics.

Quantum mechanics takes its name from the idea that things like energy come in discrete lumps. If you have an atom, for example, there are certain special energies that an electron in that atom can have, and every other possible energy is forbidden. You can assign numbers to all the possible states, and be sure that you will only ever find an electron in one of the integer number states.

What determines which states are “allowed states?” In technical language, they’re described as “eigenstates,” which is a German-derived mathematical term meaning that when you take the mathematical representation of one of these states, and do the mathematical operation that corresponds to asking “what is the energy of this state?” the thing that you get back after the operation is the same as what you started with, but multiplied by the energy (which is just a number). For any quantum system, there are a limited number of these states, and much of the theoretical activity in modern physics is concerned with determining what those states are.

Quantum mechanics, applied to light, says that light comes in individible chunks as well. These are called photons, and each photon carries a discrete amount of energy determined by its frequency. If you have a beam of light, it can contain one photon, ten photons, or three billion photons, but never “ten and a half” or “three billion, plus a quarter.” You’ll always have an integer number of photons, and the beam will have a total energy equal to some integer multiple of the energy of a single photon.

So far, so good. If you start thinking about this carefully, though, you’ll notice a weird sort of problem, namely that these states of well-defined numbers of photons are very different from one another. They’re not very much like the classical light waves that we all learn about in grade school.

In particular, if you take one photon out of the state, you get a state that is, mathematically speaking, very different from the state you started with. This doesn’t quite fit with our usual experience of light. In particular, it doesn’t really fit with the behavior of lasers, where splitting off a small part of the beam doesn’t really change anything.

What you want, if you want to describe a laser, is a mathematical object that doesn’t change when you take one photon away. You want something that’s an eigenstate, but not an eigenstate of the energy operation. Rather, you want an eigenstate of the “take one photon away” operation– when you take one photon away, you get back the thing you started with, multiplied by a number.

That’s what Glauber developed, that got him his share of the Nobel. He showed that it’s possible to construct states of light that are eigenstates of the photon annihilation operator, and moreover, that these “coherent states” have the properties that coherent light from a laser has. “Coherence” is kind of a slippery word, in physics, but the basic idea in this context is that the entire beam acts like a good approximation of those classical light waves we know and love– it has a reasonably well-defined amplitude, and a reasonably well-defined phase.

Glauber’s theoretical framework for talking about this stuff makes a lot of other predictions as well, many of which have been confirmed, so his Nobel Prize was well earned even without a direct demonstration that coherent states don’t change when you take away one photon. Which is good, because it’s a rather tricky experiment to do well. It’s always nice to have a good demonstration of these basic effects, though, and that’s what Marco Bellini and his colleagues have done.

Their experiment is much simpler than the rather crowded diagram in their paper (Figure 2a) makes it seem. They did three different measurements in the course of their experiment, and all three are sort of mashed together in that figure. The basic idea of all of three is the same, though: they prepared a particular state of light, and sent it through a really weak beamsplitter that only occasionally diverted a photon out of the beam into a detector. Whenever they recorded a photon hitting that detector, they used that signal to trigger another measurement, on the part of the beam that made it through the beamsplitter.

The simplest of the three experiments is the second one reported, where they sent a beam containing a definite number of photons (usually 1) at their beamsplitter. If they detected a photon at their trigger detector, they knew that they had gone from 1 to 0 photons in the beam, and indeed, their measurement reflected this. The results are plotted a couple of different ways in Figure 4 of the paper.

The second thing they did (which, weirdly, is reported first, but then they didn’t ask me how to organize their paper) was to measure a “thermal” beam of light, which is to say, light that acts like light from a lamp, rather than light from a laser. They made the thermal beam by sending laser light through a rotating disk of glass, which acts to mess up the phases of the different parts of the beam in a way that makes it look just like thermal light.

The beam they sent in had an average photon number of 0.36 photons, which doesn’t represent fractional photons, but rather the probability of detecting one photon. Most of the time, when they looked with the detector at the end of the experiment, they found no photons. Occasionally, they’d get one, very rarely two, once in a very great while three, and so on.

When they detected one photon at the trigger detector, the average photon number after the detection was 0.69 (the graphs are in Figure 3). Yes, it increased, which may seem weird, but reflects the fact that photons tend to “bunch”– they like to be together, so if you’ve got one, the chances are pretty good that you’ve got two. Their result matches the prediction pretty well. They also did an experiment where they looked for two photons one right after the other, and found the average increased to 1.03, again, in good agreement with the theoretical prediction.

The final experiment, the one that all the news stories mentioned, was to send in a beam of laser light, and measure what happened when they took one photon out of that beam. The prediction here is that nothing should change, and when you look at the graphs in Figure 5 of their paper, it’s fairly clear that nothing has changed. They measured the degree to which these are the same, and putting it a little roughly, the state after the subtraction of one photon is 98% identical to the state they started with.

(Granted, that’s the difference between a human and a chimpanzee, more or less, but given the technical challenges of doing the experiment at all, it’s pretty darn impressive.)

So, that’s what they did, and why. This isn’t an earth-shaking result– everybody in the field already knew that Glauber’s theoretical framework for coherent states works just fine. It’s a nice, clean demonstration of the crucial mathematical property that these states need to have, though, and well worth having just for that.

A Zavatta, V Parigi, M S Kim, M Bellini (2008). Subtracting photons from arbitrary light fields: experimental test of coherent state invariance by single-photon annihilation New Journal of Physics, 10 (12) DOI: 10.1088/1367-2630/10/12/123006

January 13, 2009

This is cool, and thanks for the description (I had never thought to think of a laser beam as an eigenstate of the photon annihilation operator)

I am going to reveal my ignorance, with a question.

Why do photons bunch?

2. #2 Jamie Bowden
January 13, 2009

Protection…it’s all about safety in numbers.

Only the weak get captured by those wily physicists to be experimented on, and the rest of the herd is safe.

3. #3 Demian Cho
January 13, 2009

Thanks for a very nice post.

I especially love “take one photon away” operation for annihilation operator. I will probably begin using in my QM class.