Once again there is a plate with two slits, only this time we use atoms instead of electrons. All are of the same type — Cesium-133, say — and all are prepared in the same excited state. In front of each slit there is a microwave resonance cavity. The two cavities are separated by a pair of electro-optically controlled shutters, which are initially closed. Between these shutters there is a photosensor.

The atoms enter the cavities from the left. Quantum mechanics allows us to calculate the probability with which any given atom, upon emerging from the cavities, is found in its ground state — provided that an energy measurement is made. The cavities can be — and are — designed so that this probability is 1. Each atom thus leaves with a lower energy than that with which it entered the cavities. Since energy is conserved, and since there is no way for the "lost" energy ΔE to escape the cavities, this energy is "trapped". It is customary to say that the atom has emitted a photon with this energy, and that this photon remains trapped inside the cavities.

The experiment of Englert, Scully, and Walther

First we keep the shutters closed. Atoms are launched in front of the cavities and detected one at a time. (Before an atom enters the cavities, the photon left behind by the previous atom is of course removed.) To keep the language simple, we will say that each atom leaves a mark where it is detected at the backdrop. How will the marks be distributed? Which Rule applies? Rule A or Rule B?

Rule A applies not only if the intermediate measurement is made but also if it is merely possible to infer from another measurement what its outcome would have been if it had been made. Rule B applies if the intermediate measurement is not made and if it is not possible to infer from another measurement what its outcome would have been.

The intermediate measurement has again the possible outcomes "through L" and "through R". Even though this measurement is not made, it is possible to learn what its outcome would have been: we simply insert a photosensor into each cavity and see which of them clicks. If the photon is detected in the left cavity, we can conclude that the atom would have been found emerging from the left slit if the appropriate measurement had been made.

Here is why we are entitled to draw this conclusion: whenever we determine both (i) the slit taken by the atom and (ii) the cavity containing the photon left behind by the atom, the outcomes are perfectly correlated. Whenever we find that the atom has taken the left slit, we find the photon in the left cavity, and whenever we find that the atom has taken the right slit, we find the photon in the right cavity.

Thus Rule A applies. Let's color the marks at the backdrop: those made by atoms which emitted a photon that is found in the left cavity green, and those made by atoms which emitted a photon that is found in the right cavity red. The left (right) dotted curve represents the distribution of green (red) marks. The black curve is the sum of the two.

The experiment with shutters closed

Observe that the green (red) marks are distributed exactly as we expect from atoms that went through L (R). The "green" atoms behave exactly like atoms that went through the left slit, and the "red" atoms behave exactly like atoms that went through the right slit. If something looks like a duck, walks like a duck, and quacks like a duck, then it is a duck, right?

Not everybody agrees. Some theorists insist that if the slit taken by the atom is not directly measured, then all that can legitimately be inferred from the detection of the photon in, say, the left cavity, is the truth of the following counterfactual: if a direct measurement of the slit taken by the atom had been made, then the outcome would have been L.

This uncompromising attitude is, however, self-defeating, for strictly speaking there is no such thing as a direct measurement. Measurement outcomes are inferred from pointer positions, digital displays, counter clicks, computer printouts, and so on and so forth. If we did not have the right to infer the value of an observable from an event such as the click of a counter, then it would be impossible to measure anything! But if we have the right to infer the position of a particle from the click of a counter, then we also have the right to infer the slit taken by an atom from the cavity in which the photon is found.