In this lesson, we have seen that one-photon wave-packets, which can be detected in one place only can anyway lead to interferences, a phenomenon that implies that the single photon follows different trajectories simultaneously. The two statements are obviously contradictory, if we think of the photon as a classical particle. This leads us to discuss once again, the mystery of wave particle duality already encountered in a previous lesson. What is new in the lesson of today is the fact that wave particle duality is more than the result of a calculation, it is a reality that can be observed in real experiments. In our previous discussion of wave-particle duality I invoked Bohr's complementarity, which is the fact that one has to choose between two incompatible experimental schemes. Let us return to that discussion: we will find that it is more subtle than one sometimes thinks. I've recalled here the two schemes of the two experiments performed with the same one photon source. The one with the Mach-Zehnder interferometer shows high visibility interference and indisputable evidence of a wave-like behavior. The one with a beam splitter and a coincidence detection device shows the lack of double detection an undisputable evidence of particle like behavior. Complementarity is the fact that these two experimental schemes are incompatible: you must choose between the two setups. This was beautifully stressed by John Archibald Wheeler, a great physicist who contributed a lot to our understanding of quantum mechanics. He noticed that, in fact, you can switch from one scheme to the other one by removing the second beam splitter in the interferometer. Can you see it? This is the idea of Wheeler. If you remove the second beam splitter, there is no more modulation of the counting rates in D5 and D6 as a function of the mirror positions, but you can now check that the wave packet has traveled either on path 4, to D5, or on path 3, to D6, but not on both paths, as expected for a single particle. Wheeler invented that scheme to emphasize the fact that the choice can be done after the wave packet has passed the first beam splitter: this scheme is known as the Wheeler's delayed choice experiment, and it was carried out as a real experiment some years ago. This experiment is explained in a presentation that you can find as a supplementary document attached to this lesson. Here I want to focus on the following question, what is exactly the choice that we make when we select one scheme or the other one? There is no doubt that the two schemes here are incompatible, you cannot perform the two experiments simultaneously, but what are the incompatible quantities? The most frequent answer to that question is wave-like versus particle-like behavior. In fact, it is not that simple. Consider the first scheme with the interferometer. Observing the modulation of the rate of detection when the path difference is changed is no doubt a clear signature of a wave-like behavior. But while doing that experiment, it remains possible to look for joint detections at D5 and D6, and observe the lack of joint detections. So the answer cannot be that the incompatible observables are wave-like and particle-like behaviors. So what are the incompatible observables? A clear answer was given only in the 1980s when modern quantum optics was developing. Here is a summary of that answer: in the first experiment, the one with the interferometer, we test the coherence between the two paths. It can be measured by determining the visibility V of the fringes. For equilibrated beam splitters with equal transmission and reflection coefficients, the visibility is one. In the second experiment, we can tell which path was followed by the one photon wave packet. Only one detector fires, and we can tell which one. It is possible to define a quantity D the distinguishability characterizing the possibility to tell which path was followed. In this experiment, the distinguishability is 1. In fact, there are more sophisticated experimental schemes where it is possible to observe simultaneously an interference effect and to tell which path was followed. It is the case here if the second beam splitter is not equilibrated. When observing a photon in one of the two exit channels you can guess what was the path of the photon with a probability of success larger than 50%. This translates into a distinguishability D, with a value between zero and one. Since the beam splitter is not equilibrated, the visibility V is less than one, but it is not null. One can show that the two quantities D and V obey the inequality D squared + V squared less than or equal 1. The experiment could be more sophisticated. For instance, it is in principle possible to make in each arm of the interferometer a measurement called a quantum non-demolition measurement which allows one to detect the passage of a photon without destroying it. It is then possible to know with a certain probability which path was followed by the photon. What about the interference pattern? It turns out that the quantum non demolition measurement perturbs the state of the photon and this results in a decrease of the visibility of the fringes. Finally, we still have non-null but not perfect distinguishability and visibility obeying the relation shown here. This inequality is a quantitative expression of the notion of complementarity. There are quantities that cannot be measured simultaneously with an absolute precision, and these limitations are expressed by an inequality. Note that the two experiments considered previously were marginal cases of that inequality: with equilibrated beam splitters, the visibility V is 1 and the distinguishability is 0, while in the absence of the output beam splitter, V is 0 and D is 1. I suggest you make a pause and return to the previous slide to check this property. Notice that the inequality here is not a Heisenberg inequality. The operators associated with visibility and with which path, do not commute, so the two quantities cannot be determined simultaneously with certainty. But their commutator is not equal to i hbar. They are not canonically conjugate observables and the inequality has not the standard Heisenberg form. If you want to know more about this subtle point you will find several references attached with this lesson.