So, I’ve put myself into a position where I need to spend a substantial amount of time thinking about weird foundational issues in quantum mechanics. This has revealed to me just why it is that not that many people spend a substantial amount of time thinking about weird foundational issues in quantum mechanics.
Let’s consider a Mach-Zehnder interferometer, of the type shown in the figure at left (click the figure to see the original source). A photon (or an electron, or an atom, or any quantum particle) enters the interferometer at the lower left, and is split onto two paths by a 50-50 beamsplitter– half the light is reflected up, half is passed straight through. Those two paths separate, then are brought back together on a second beamsplitter, with half of each beam being transmitted, and half reflected, so that light from each path falls on each of the two detectors at the upper right. Those two beams interfere with each other, and determine the outcome of the experiment.
Now, if you do this with a single photon, you will get a single click, either from detector 1 or detector 2. If you repeat the experiment many times, you’ll build up some number of counts in each of the detectors, and if you shift one of the mirrors slightly (thus making one path longer than the other), you’ll find an interference pattern in the fraction of the total counts on each detector. At some positions of the mirror, you’ll find absolutely no counts on detector 1, and in other positions you will never see a count on detector 2.
So far, so good. Now, we can start to make this weird by replacing the second beamsplitter…
Let’s say we replace the second beamsplitter with a variable beamsplitter that can be set to either the 50-50 configuration of the original set-up, or a 100-0 configuration, where it transmits absolutely everything (this is relatively easy to do, using polarization optics, but the details aren’t important). In the 100-0 configuration, the only light that reaches detector 1 is light that was reflected up at the first beamsplitter, and the only light that reaches detector 2 is light that was transmitted at the first beamsplitter. In this configuration, it doesn’t matter where you position the mirrors, because there’s nothing for the light to interfere with– no matter what you do, you get half of your counts on detector 1, and half on detector 2.
The usual description of this is in terms of particle and wave behavior. In the first case, with the 50-50 beamsplitter, we have light behaving like a wave, with some of the light going on each path, and recombining to interfere with itself. Which detector records the count depends on both possible paths to the detector. In the second case, with the 100-0 beamsplitter, we have light behaving like a particle, with the photons choosing a definite path at the first beamsplitter, and staying on that path. Which detector records the count depends only on what happens at the first beamsplitter.
Now, we can make this even stranger, by allowing the switch between configurations to be very quick (again, this is relatively easy to do experimentally). And then we set it up so that we determine whether we have a 50-50 or 100-0 beamsplitter only after the photon has passed the first beamsplitter. This isn’t trivial, but it’s been done by a few different groups, most recently the Aspect group in France. What you find in this case is remarkable– if you repeat the experiment many times, and keep track of the configuration, you find that when the beamsplitter was 50-50, the counts add up to give you an interference pattern, and when it was 100-0, you don’t see any interference. Even though the configuration isn’t set until after the photon has passed the first beamsplitter, somehow, it still turns out to have gone both ways when the second beamsplitter is 50-50, and only one way when the second beamsplitter is 100-0.
Spoooooky.
Now, here’s the part that’s bugging me: How do you think about this in the Many-Worlds Interpretation? The Copenhagen-ish version of the experiment is fairly simple– you have a wavefunction that describes the whole apparatus and both light paths, and when you detect a photon, it collapses into one of the possible states. It’s weirdly retroactive, but the collapse happens at a definite time, the moment when the detector records a count.
In the Many-Worlds picture, you ought to end up with four different universes, corresponding to the four possible paths the particle could take: reflected up, to detector 1; transmitted down to detector 2; both ways, ending up in detector 1; and both ways, ending up in detector 2. But when does the split happen? Does it split into four parts when passing the first beamsplitter? Two at the first beamsplitter, and two again at the second? Three at the first (one for each definite path, and one “both ways”) then one of those splitting at the detectors? All four only when the light reaches the detectors?
I sort of lean toward a decoherence-based interpretation of the interpretation (if you follow), so I think that the right way to look at it is that you get all four after the light passes the first beamsplitter, and then those different paths experience decoherence as the photons move through the interferometer. They end up being completely separate by the time the photon hits one of the detectors, and you find yourself in one of the four through whatever mystical mechanism it is that usually causes you to see only one of the possible paths.
But then, that doesn’t really seem to account for the delayed choice of what configuration to put the second beamsplitter in. So maybe you really want the 3-way split at the first beamsplitter, and then one of the three splitting at the second. Or maybe the two 2-way splits.
But then, both of those just seem weird. So maybe the way to do it is to go for the full-on Heisenberg treatment, in which nothing exists until it’s detected, and say that the universe splits into four at the instant of detection. But then, that doesn’t really seem to fit with the “decohering branches of a single unitarily evolving wavefunction” thing, which is sort of the whole point of Many-Worlds in the first place, and, well…
Well, now I know why more people don’t write books about this stuff.