Single Electron Interference and Decoherence

Jennifer Ouellette meantions it as the jumping-off point for her particle-wave duality post, but I want to spend a little time talking about this paper on single-electron interference (Science 318, 949 (2007)), because it's a very nice piece of work. There's also a Physics World news article about the experiment, which is pretty good, but slightly understates the coolness of the experiment.

The experiment here is extremely simple: They take a jet of molecular hydrogen, and hit it with extremely high-energy photons from the Advanced Light Source at Lawrence Berkeley National Lab. These photons have enough energy that when one hit a molecule, it blows it apart completely, into two protons and two electrons. They then use a technique called "Cold Target Recoil Ion Momentum Spectrcoscopy (COLTRIMS)" that's all the rage on the "M" side of AMO physics to detect all four particles, and figure out their initial velocites.

Using this, they see what they modestly call the "simplest double slit" experiment possible. They also provide a beautiful illustration of the process known as decoherence, something I was laboriously explaining to the dog just a week or two ago.

Let's start with the "simplest double slit" part of things. When they blast the molecules apart, and look at the pieces that come out, this is what they see. You're looking at a polar plot, here, showing the probability of finding the electron with the higher energy coming out at a given angle from the axis of the molecule (the little dumbbell in the upper right). This is reconstructed from the COLTRIMS data-- the molecules are all pointing in different directions, but by measuring where the protons come out for each individual molecule, they can figure out what direction it was pointing, and combine data from lots of different shots.

The probability increases as you move out from the center, so the odd four-lobed shape in this figure (which is part D of Figure 1 in the paper) shows a clear interference pattern. There are certain directions in which you're very likely to see an electron (roughly, along the horizontal or vertical axes), and other directions in which the odds of finding an electron aren't very good. This is the result of interference as the electron leaves the molecule-- the two protons act sort of like the slits in a double-slit experiment, and you see interference between the part of the electron wavefunction that started close to one of the protons with the part that started close to the other. The pattern is in pretty good agreement with the predictions of a simple theoretical model of this process (which you'll have to read the actual article to learn about).

"But wait," you say. "There are twoelectrons here, and you're only talking about one. What happened to the other?"

The data shown in the first figure are for those cases in which one electron takes away most of the energy from the photons. The high-energy electron goes screaming out of the wreckage of the molecule, leaving the other one in the dust. For the purposes of calculating the interference pattern, you can pretty much ignore the slow electron.

They also have data for cases where the two electron energies are more similar, though, and those are shown in the figure on the right (Parts A and B of Figure 2 from the paper). When the energies are a little more comparable, you can no longer ignore the influence of the second electron. The two electrons interact rather strongly with one another, and that interaction changes the interference pattern. As you decrease the energy of the fast electron, and increase the energy of the slow electron, you see the interference pattern become less obvious (Fig. 2A), and eventually disappear altogether (Fig. 2B).

This is an example of the process called "decoherence," in which interactions with the environment destroy quantum interference effects, and create a situation that appears very classical. In Fig. 2B, the electrons spray out in all directions, with not much difference in probability, which is what you would expect if the electrons were classical billiard-ball type particles. This is due to the interactions between the fast and slow electrons-- what they're plotting is just the probability distribution for the fast electron, and the interaction with the slow electron wipes out the interference pattern.

It's important to note how this is done, here. These are experiments that detect the electrons as particles in specific places, so these probability distributions are built up from repeated measurements. They do the experiment once, record where the electron is, and then start over with another molecule. Those graphs reflect hundreds or thousands of different measurements, each detecting an electron in a single position. They don't see the probability distribution all in one shot, but build it up over many times.

In the first graph above, Fig. 1D, this works nicely because all of the electrons come out with exactly the same probability distribution, and you clearly see the interference. In Fig. 2B, on the other hand, exactly what the probability distribution looks like for a given fast electron will depend on the interaction with the slow electron, which isn't measured. Those slow electrons go out in all different directions, and each different direction causes a different shift in the probability distribution. When you add together the results of hundreds or thousands of different measurements with hundreds or thousands of different probability distributions, the distribution you end up with is a big smear, with no clear interference at all.

This doesn't mean that the electron doesn't undergo any interference, though. Each individual electron still has a wavefunction that starts with two pieces corresponding to the two different protons, and those two pieces still interfere with one another. It's just that the patterns depend on the slow electron, and without information about what happened to the slow electron, the fast electron interference patterns are randomly oriented, and wash out.

If you keep track of the slow electron, though, the interference is still visible, as seen in the figure at left, taken from Fig. 3A of the paper. Here, the horizontal axis gives the direction of the fast electron, the vertical axis the direction of the slow electron, and the color shows the probability of finding the fast electron at that particular angle, with reds and yellows indicating high probability and blues and greens low probability. The overall conditions are the same as for Fig. 2B, but in this two-dimensional plot, you can see a collection of red and yellow spots, at particular directions for both fast and slow electrons. These are the interference pattern for the fast electron.

If you keep track of the momentum of both electrons, you still see interference. If you take a slice through Fig. 3A, and only look at the probabilities for those cases where the slow electron came out in a particular direction, you see interference just as in Fig. 1D. If you take that slice, and make a polar plot of it, the figures are almost identical (they do this in the article, but I'm trying to limit my use of their figures, so you'll have to go there to see it). The decoherence that wipes out the pattern in Fig. 2B requires two things: an interaction with the environment (the other electron), and that that interaction is unmeasured. If you have information about the interaction, though, you can un-do the decoherence, and recover the interference pattern.

This isn't anything all that shockingly new-- people have known about decoherence for quite a while. This is a particularly nice demonstration of the important features of the effect, though. It also finally lets me understand a little bit about why all the molecular physics types are all gaga over this COLTRIMS business.

The paper itself is fairly clear and readable, and I recommend looking at it if you have access. It's cool stuff. The dog loves it.