It’s been a long and brutally busy week here, so I really ought to just take a day off from blogging. But there’s a new paper in Science on quantum physics that’s just too good to pass up, so here’s a ReasearchBlogging post to close out the week.

**Aw, c’mon, dude, I’m tired. What’s so cool about this paper that it can’t wait until next week?** Well, the title kind of says it all: they measured the average trajectories of single photons passing through a double-slit apparatus. By making lots of repeated weak measurements at different positions behind the slits, they could reconstruct the average trajectories followed by photons on their way to form the interference pattern. They look like this:

**Whoa! That got my attention. So, like, these guys are going to get a Nobel, right?** What do you mean?

**Well, they’ve measured definite positions and momenta for photons in a double-slit experiment. Which means they must’ve proved orthodox quantum mechanics wrong, because you’ve always said that quantum particles don’t have a well-defined position and momentum. And if orthodox quantum physics is wrong, that’s got to be good for some dynamite money, right?** Not so fast. What I said, and what orthodox quantum physics tells us, is that a quantum particle like a photon of light doesn’t have a well-defined *instantaneous* position and momentum. You can perfectly well construct an *average* position and momentum for a quantum particle, though, by making lots of measurements of identically prepared systems. If you string a whole bunch of those average measurements together, you can construct an average trajectory, which is what they did here.

They haven’t done anything to prove orthodox quantum mechanics wrong, though I can predict with confidence that there will be at least one media report about this that is so badly written that it implies that they did. In reality, though, their measurements are completely in accord with ordinary quantum theory.

**Yeah? Then how did they manage to measure both the position and momentum without destroying the interference pattern?** They made use of an idea called “weak measurement,” which is more or less what it sounds like. If you set up a situation where you measure the exact position or momentum of a quantum particle, you lose all ability to measure the other of those two quantities, and destroy your interference pattern. That sort of measurement won’t allow you to reconstruct these kinds of trajectories.

If you set your experiment up in such a way that you only get a little bit of information about either the position or momentum of a single quantum particle, though, you only disturb the other quantity by a tiny bit, which means you can go ahead and measure it. This doesn’t let you say what the exact position and momentum of a single photon is, but you can measure a fairly exact position with a tiny bit of information about the momentum. If you repeat that many times, you can determine the average value of the momentum at a particular position without upsetting the trajectory.

**How do you only measure a tiny bit of the momentum? Isn’t that a “little bit pregnant” sort of contradiction?** The system they used for this is really ingenious: they use the photon polarization as a partial indicator of the momentum. They send their original photons in in a well-defined polarization state, then pass them through a calcite crystal. Calcite is a “birefringent” material, which changes the polarization by a small amount depending on the amount of material the photon passes through.

Photons that enter the calcite perpendicular to the surface pass straight through, and travel a distance equal to the thickness of the calcite. Photons that enter at a shallower angle follow a longer path through the calcite (think of it like cutting a loaf of French bread on the bias– the angle-cut pieces are longer than the thickness of the loaf), and thus experience a greater change in polarization. The polarization of an individual photon then depends on the angle it took through the calcite, which tells you the direction of its momentum. The magnitude of the momentum is determined by the wavelength, which is the same for all the photons, so this gives you the information you need for the trajectory.

**So, you measure the polarization, and you know the momentum?** Sort of. You see, you can’t really measure the polarization of a single photon– when you send it into a polarizer, it either makes it through or doesn’t, and that’s all you can say. If you want to know the polarization, you need to measure a whole bunch of photons, which lets you determine the probability of transmission, which then tells you the exact polarization.

So, the polarization of each individual photon is correlated with its momentum, but in a way that only lets you get a tiny bit of information out of a single photon. This is a weak measurement of the momentum, and that doesn’t perturb the quantum state of the particle by enough to destroy the interference pattern. If you measure a whole bunch of photons, you can reconstruct the average momentum of all the light at a particular position, but that doesn’t necessarily tell you anything about the momentum of an individual photon detected at that position.

**That sounds really complicated. Really clever, but really complicated.** It is. It’s especially clever because they came up with a nice way to get a cross-section of all the momenta of photons along a line across the interference pattern: they use a set of three lenses to create an image of the light pattern at a particular distance behind the slits on a CCD camera. Just before the camera, though, they insert a special polarization-dependent optic that shifts two different polarization components in the vertical direction (that is, perpendicular to a line drawn between the two slits)– one polarization shifts up, and the other shifts down. This gives two versions of the intensity pattern, one right above the other, and by comparing the relative brightness of two pixels in a vertical line, they can determine the average polarization of the light at that horizontal position. When they repeat that for all the horizontal pixels, they can make an intensity and momentum plot like the long figure at right.

**That’s a pretty complicated figure, dude. This is gonna need some unpacking.** Yeah, there’s a lot going on in this one. The four parts of this figure plot two different things for each of four different distances behind the slits, with the distance increasing as you go down.

The top sub-graph in each lettered part of the figure just shows the total intensity of the light (red is one polarization, blue the other). You can see the development of the interference pattern in these graphs as you gown down the line: at the top, there are just two lumps, corresponding to the two photons that went through each slit. As you move down, those lumps widen out and overlap, and eventually form a complicated interference pattern in the last graph.

The much scarier looking sub-graphs with all the black points are the measured momentum for each position, with positive values being directed upwards (in the trajectory plot at the top of this post) and negative values directed downwards. These are a little noisy, but you can see a clear pattern. The magenta line is a sort of smoothed-out version of the momentum distribution, done by basically requiring that the total number of photons at one slice through the pattern be the same as the total number in the previous slice.

**And each of these graphs represents a whole bunch of detected photons?** Exactly. For each position, they recorded an image for 15s, which corresponded to about 31,000 photons. Those photons are sent in one at a time– they did an antibunching experiment to confirm that they’re really only sending single photons into the apparatus at a time.

**That seems like it would take kind of a long time, doesn’t it?** Yes, but not nearly as long as it would take if they didn’t have the clever polarization-shifter measurement trick, and had to scan a single-photon detector across the pattern.

**Good point. So, how do they get from the messy black curves to the nice set of trajectories at the top of the post?** Well, those messy black curves give you the direction of the average momentum at each pixel position across the interference pattern. Which basically gives you a lot of little arrows pointing along the direction of the average photon trajectory passing through that horizontal pixel at that distance from the slits. You just set all these sets of arrows next to one another, and draw lines connecting them.

**I bet it took quite a while to draw all those lines. They must have some grad students with really steady hands.** I think they probably used a computer for the data analysis. It is the ~~century of the anchovy~~ 21st century, you know.

**Good point. So, are you sure this doesn’t disprove orthodox quantum mechanics? Because those trajectories up top look a whole lot like a picture I saw in an article about Bohm’s hidden variable version of quantum mechanics.** In the Bohmian picture, these trajectories would be the absolute and true trajectory of the photons passing through the apparatus. In the Bohmian approach, all quantum particles have a well-defined position and momentum at all times, but we’re not able to track it directly because the uncertainty in the initial position and momentum prevents us from selefting a single trajectory to follow on repeated measurements.

The more orthodox approach to quantum mechanics holds that the photons do not have well-defined position and momentum before they are detected at a particular position or with a particular momentum. In this version, the particular results of a given measurement of position or momentum is picked at the instant of measurement from a probability distribution determined from the wavefunction at that position at that time. If you repeat this process lots of times, you’ll get a different position and momentum every time, but those values will follow a particular pattern. If you average together a whole slew of these measurements, you can come up with an average momentum and average position, which is what they’ve done here.

When you do that, the result is a set of *average* trajectories that look exactly like the actual trajectories in the Bohmian picture. So, this experiment would produce exactly the same results, no matter what interpretation of quantum mechanics you favor. In some sense, the only difference between the models is in when you do the averaging: In the Bohmian picture, you take an average because you start with a distribution over all the possible starting positions and momenta, even though each particle follows a well-defined path at all times. In the more orthodox interpretations, you take an average because the final position and momentum that you measure is chosen from a rage of possible values determined from a probability distribution, and the only way to find probabilities is by taking averages.

Any way you look at it, though, you expect trajectories that look exactly like this.

**So, no Nobel for the authors, then?** As cool as that would be (the PI, Aephraim Steinberg, is a friend from my days at NIST), they shouldn’t go booking flights to Stockholm just yet. Unless, you know, they really want to vacation in Sweden.

This is an extremely cool example of the art of experimental physics, and a spectacular demonstration of the power of weak measurements, but it’s not *that* revolutionary. Though, as I said above, I confidently predict that there will be no shortage of crazy people trying to claim this as conclusive proof for their particular favorite interpretation of quantum theory.

(Which is probably appropriate, as Aephraim’s Ph.D. thesis research involved “superluminal” propagation of light, which has spawned no end of cranks claiming that Special Relativity has been overthrown. If he can find a really clever test of General Relativity to do, he can complete the trifecta of modern physics kookery.)

So, anyway, was that worth staying awake through Friday?

**Yeah, that was pretty cool. Thanks. Now can we call it a week?** Yeah, that’s about enough for me, too. Stick a fork in this week, and I’m outta here.

Sacha Kocsis, Boris Braverman, Sylvain Ravets, Martin J. Stevens, Richard P. Mirin, L. Krister Shalm, & Aephraim M. Steinberg (2011). Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer Science, 332 (6034), 1179-1173 : 10.1126/science.1202218