Being Complementary About Uncertainty

This semester I took a course on quantum optics. I'm an AMO guy and quantum optics is one of our department's particular strengths, so it was both a very useful class and a pleasure to take. One of the graded requirements of the class is to write a paper from a list of quantum optics with the stipulation that it couldn't be from our own research. Essentially the paper is supposed to be sort of a review article / tutorial on that topic for our own edification, graded for clarity and grasp of the topic. I wrote on which-way detectors and quantum erasers. It's a bizarre and fascinating topic which explores the concept of complementarity in quantum mechanics from an experimental perspective.

Complementarity is a very general concept and not easy to define formally, though informally you might say it's the principle that the wave-like and particle-like aspects of an object can't be simultaneously observed. More formally you could say that each degree of freedom of a system corresponds to a conjugate pair of observables, which means these pairs (say, position and momentum) can't both be measured precisely at the same time.

We can look at this more closely with what's probably the simplest version of a which-way detector. In concept it works like this: direct a beam of electrons onto a double slit. Just like light waves, the electrons from each slit will interfere with each other and produce an interference pattern on the detector screen. (See the diagram below for a visual example)

Did I say "interfere with each other"? Well, while that sounds plausible it's wrong. If you turn down the beam so far it's just one electron at a time, you still get interference fringes, built up electron by electron probabilistically. After lots of electrons have hit the detector screen, their overall distribution is just the same interference pattern. How can this be? A single particle ought to go through one slit or the other, resulting in exactly two blobs on the screen, one for the electrons going through the top slit and one for the other. But instead we see interference, as though the wave nature of one electron has gone through both slits.

Weird, so let's try to get a better handle on this. Feynman proposed sticking a light between the slits. Light will scatter from electrons, so simply by looking at the apparatus with adequate equipment you can see which slit the electron went through. That ought to nail things down:

Let's look at the math of the experiment. We'd better lay out our variables. Call the horizontal direction x and the vertical direction y. The distance between the slits is D. We also know from quantum mechanics that the wavelength of the electron is inversely proportional to its momentum, with the value given by the de Broglie relationship λ = h/p, for Planck's constant h.

Now a little background on interference. Destructive interference - where the interference pattern is near zero - occur where the crests of one wave overlap with the troughs of the other. This means if you measure the distance from slit 1 to that point on the detector and from slit 2 to that same point, they have to differ by half a wavelength:

Where theta is the angle made between the line from the lower slip and the horizontal. (Though the definition works just as well using the vertical and in the far field both are the same anyway.) Using the old sin(x) = x approximation for small angles:

Where px is the momentum of the electron in the x direction, and as the angle is small it's essentially the entire momentum p.

But light has momentum, so when we "see" the electron we've bumped it out of its trajectory a bit. This bump had better be small or it'll wash the interference pattern out. How small? This small:

(Because py/px = tan(&theta), and we're using the small-angle approximation again.) Simplifying:

By conservation of momentum, this will be the same for the detecting photons. Now to find out which slit the electron came out of, we have to nail down the electron's position to within a distance Δy << D. So finally we have:

Which is impossible, a violation of the Heisenberg uncertainty principle. Anything we use to look at the electrons will certainly alter the electron's momentum too much to produce the interference pattern. Knowing which way the electron went and producing interference is not possible with this scheme.

Is it possible with any scheme? The paper from which I took the figure is from a 1991 review article* by Scully, Englert, and Walther on just this subject. The answer is no, but the details are quite subtle. It's possible to cleverly get which-way without perturbing the electron at all, and yet you still get a stark absence of the interference pattern if the experiment also provides which-way information by any method. That's pretty counterintuitive, and those experiments are definitely worth discussing.

*Nature vol. 351, 9 May 1991, 111-116.