Quantum Eraser

Quantum eraser experiments are variations of the double slit experiment intended to highlight the strange properties of quantum mechanics. If the double slit experiment was strange and the measurements on Bell states were mind-bending, then the quantum eraser experiments are intended to be truly mind blowing. However, what we will see here is that one’s mind is only blown if one insists on telling a story based on ideas of measurement-induced collapse. If we use the picture developed here, the whole thing becomes pretty mundane. In fact, the picture is very similar to what we have already seen in previous sections.

The general idea of a quantum eraser is as follows. Start with a double slit setup, or perhaps the equivalent Mach-Zehnder interferometer that we have been using. Under normal operation, interference can be observed. As we know, if one obtains information (i.e. makes a measurement) of which path the particle takes, the interference goes away. The idea of the quantum eraser is that if we can somehow erase the “which path” information, we can get the interference back. Furthermore, even if “erasure” takes place after the particle has been “observed” to interfere or not, the interference can “retroactively” be “restored.” (The number of scare quotes in that sentence should indicate that this story is a little questionable.) This last case is called the “delayed-choice quantum eraser.”

Let’s first look at the regular quantum eraser (not delayed choice) using our picture of micromeasurements. To review: we start with the usual Mach-Zehnder situation of a particle split into two packets. If we overlap the two packets without entangling with any other particles, we get the interference as usual. But if we reflect one packet off of a mirror that recoils unambiguously, we obtain an entangled state where interference does not occur when the two packets of the particle overlap. This occurs because the mirror has now split into two packets — each correlated with one one packet of the particle. This recaps the situation first shown back in the Double slit section, and illustrated again in Fig. 7.1.

Figure 7.1

Video example of a double slit “which path” measurement. Particle 1 is initially in a superposition of two wave packets which separately reflect off of a mirror particle (particle 2). Due to the resulting entanglement, the two packets of particle 1 do not interfere when they cross.

In our double slit picture, the interference disappears after “which path” information has been obtained because the two blobs of the entangled state do not overlap, even when one of the particle’s wave packets cross each other. In Fig. 7.1, we see this crossing occur with no interference when the blobs cross in the horizontal direction. We could see it again if we reflected the leading packet back in the horizontal direction – the blobs would still not overlap. But can we do something to the mirror particle to get the interference back? Sure! We just need to arrange things so that the two blobs do meet up when the particle’s packets cross. Perhaps most simply, we could imagine somehow “repairing the damage” to the mirror: that is, bringing the two packets of the mirror particle back together. If we could, then the the two blobs would again be aligned in the horizontal direction, and we would again get interference when the particle’s packets cross. And since the “which path” information was stored in the two distinct mirror packets, by bringing them together we have erased the “which path” information.

While it might be experimentally feasible to bring the two packets of the mirror back together with some additional mirrors or something, a more common way to go about it is instead to perform a complementary measurement on the mirror particle. The “which path” information is stored in the position of the mirror particle. We know that a complementary measurement, say of the phase difference between the packets, should in some sense scramble the position information. We have looked at this type of situation several times already. Here is another look. Figure 7.2 shows the situation starting from a state like the end of Figure 7.1, but after the leading packet of the mirror particle has been reflected back towards the trailing packet. This causes the blobs to be approaching each other in the vertical direction. An atom phase sensor is placed at the position where the two mirror packets will cross. This is essentially identical to what we saw in Figure 6.3 in the context of measuring Bell states.

Figure 7.2

Video example of a phase micromeasurement performed on the mirror particle which had obtained “which path” information.

Just as we saw previously, when the entangled state interacts with the atom phase sensor, each packet splits up into pairs correlated with both the \(|+\rangle\) and \(|-\rangle\) states of the atom. This action on the mirror particle has now brought the blobs into an alignment where the particle 1 packets can now interfere if they cross.

We see just that interference in Fig. 7.3. This video starts right from the end of the previous video, with an external barrier placed to reflect the leading packet of Particle 1 back towards the trailing packet. We see clear interference fringes when the Particle 1 packets cross.

Figure 7.3

Video example of interference of particle 1 following “erasure” of the which path information by a phase measurement on the mirror particle.

A caveat about the interference that we have now restored: there are really multiple interference patterns associated with the various pairs of blobs that overlap. As we have seen before, the interference patterns correlated with the \(|+\rangle\) state of the atom (left panel) are opposite from those correlated with \(|-\rangle\) (right panel.) Where one has a peak the other has a zero, because of the pairs are out-of-phase on the left and in-phase on the right. We can clarify this, just as we did in the previous section, by performing a second phase measurement to detect the now-restored interference of the particle. As in the Bell state measurement section, we show the phase measurement of the particle in Fig. 7.4. This is exactly the same as Fig. 7.3 except now we have placed a second atom sensor where the particle’s packets cross. Once again, the two outcomes of the phase measurements are correlated with each other. A \(|+\rangle\) outcome of the phase measurement on the mirror is correlated with interference of the particle that leaves the second atom also in the \(|+\rangle\) state. Or a \(|-\rangle\) outcome of the phase measurement on the mirror is correlated with an interference pattern of the particle that rotates the second atom’s state also to \(|-\rangle\).

Figure 7.4

Video example of detection of the now-restored interference.

In order to see interference in quantum eraser experiments, it is necessary to separate the interference patterns correlated with the two outcomes of the mirror phase measurement. Otherwise you get two opposite interference patterns added together that cancel each other out to look like no interference at all. Or in our example, just looking at atom 2 we would see 50% \(|+\rangle\) and 50% \(|-\rangle\), just like we had a single packet and no interference. This means that in order to see that interference has happened, we require an interaction between the two atoms sensors to find out if they are both \(|+\rangle\) or both \(|-\rangle\). Hypothetically one could imagine something where if the first atom sensor is \(|-\rangle\), it launches another particle towards the original particle, deflecting it away from the second atom sensor. In that way, the second atom would only register one interference pattern — that correlated with the atom 1 \(|+\rangle\) state.

The above examples show that the quantum eraser experiment is nothing more than the behavior we saw for measurements of different observables on a pair of entangled particles. In this micromeasurement picture, the point is that a measurement doesn’t really erase the information about the complementary observable, it just makes it less readily accessible. We can get it back if we can manipulate the system skillfully enough.

We can contrast the quantum eraser example just shown with the standard measurement axiom picture involving collapse. There, when the “which path” information is measured, say by the recoil of a mirror [1], the state is said to randomly collapse to one state or the other. That is, there is a 50% chance that the leading packet disappears, or a 50% chance that the trailing packet disappears. Since only a single packet exists, there is nothing for it to interfere with. Then we perform a complementary measurement on the mirror, such as a phase measurement. Prior to this point the mirror is in a well-defined state of position, so the complementary measurement results in a 50/50 randomly chosen outcome of well-defined phase, erasing the position information. Now the bizarre thing is that this erasure of the “which path” information somehow grants us a do-over and the lost wave packet pops back into existence, and goes on to interfere with its partner. Furthermore, the lost wave packet pops back into existence with the correct phase to be consistent with the phase just measured on the mirror.

As nonsensical as the standard collapse picture sounds here, it gets even crazier in the delayed-choice version. In this case, the order of the two phase measurements is reversed. Assume the “which path” measurement has occurred. Then, we first use atom 2 to see if there is interference of the two particle packets. Then after that, we can choose to use atom 1 to measure the phase of the mirror or not. In the micromeasurement picture, it is not even worth dwelling on this. It plainly doesn’t matter which order we use the atom sensors. The first will set up the aligned rectangle of blobs, and the second will cause them to interfere, leading to same final set of correlations. But this is very hard to justify in the collapse picture. That is because one would say that the measurement outcome on the first particle has already been recorded. How can the outcome that has already been recorded depend on whether one later chooses to do something to a remotely located particle? This seems like something is going backwards in time! Holy crap!

While it would be cool to have a time machine [2], perhaps we should consider our picture of micromeasurements to be more plausible than the story told about wave packets popping in and out of existence depending on things that have not happened yet. Keep in mind that the standard formulation of quantum mechanics also predicts the behavior of micromeasurements we are showing here. We are not looking at some wacky alternative to quantum mechanics. All the simulations done to make the videos here are basic, standard quantum mechanics following from the Schrödinger equation. The only thing we are avoiding is the addition of the extra axioms about measurements that have to be tacked on to the standard quantum theory.

At this point you might be wondering why anyone takes the measurement axiom seriously in light of the quantum eraser experiments, if the behavior can be more simply and clearly explained just using the Schrödinger equation. I think there are two reasons. One is that here we are showing a very simple example of a quantum eraser experiment. In reality, the experiments are more complicated and cannot be directly simulated using the Schrödinger equation. Of course, I would argue that if it works in one case, it should probably also work in more complicated cases that are effectively the same. But a more substantial issue is the potential difference between a “measurement” and a “micromeasurement.” Few people would dispute that the Schrödinger equation describes the evolution and interaction of particles’ wave functions as we have shown here. But an actual real-world measurement device involves more than a few microscopic particles. Maybe something different happens when lots of particles are involved, as compared to just a few.

It is entirely possible for different phenomena to occur in complex systems of large numbers of particles, than in simple systems of a few particles. This is the concept of “emergence” encapsulated by Philip Anderson’s phrase “more is different.” But in that case, the phenomena emerge from the same underlying rules. We don’t add new fundamental rules that only apply once the number of particles exceeds some threshold. We can describe emergent phenomena using new effective rules, but just as a shortcut to make our lives simpler. So maybe what we call the measurement axiom should not be a fundamental axiom, but instead is a useful shortcut that allows us to think about the relationship between the quantum microscopic world and our effectively classical macroscopic world.

A final take-away from this section. We have occasionally alluded to the “many worlds” idea that once particles have become entangled, we might want to think about the different components of the entangled state (the misaligned blobs in our plots) as existing in separate worlds. But the quantum eraser example shows a case where we need to be careful about that. If we assumed that once the mirror had initially performed a “which path” measurement on the particle that the two resulting blobs were now in separate worlds, then it would have been surprising when those parallel worlds somehow merged back together to produce the restored interference. There’s no rule that says we can’t picture worlds separating and coming back together, but the picture we are driving towards is one in which separate worlds do not affect each other.

In the next section, we will think about how the behavior of microscopic particles can be transferred to a macroscopic effect that we can perceive. This will set the stage for a picture in which the evolution of entangled states can be viewed as a branching into many separate worlds.

Footnotes

[1]

In reality, these experiments are usually done with photons, where one manipulate the polarization to mark or erase which path the particles are taking, and photon detectors to measure the interference.

[2]

In Back to the Future, Doc Brown chides Marty McFly for “not thinking fourth-dimensionally!” In that case, he is referring to time as the fourth dimension. Here, where we do not have a time machine, we see that we also need to think in higher dimensions to understand the state of multiple entangled particles.