It is time to conclude. You have learned a lot today. You have learned how to describe in the Quantum Optics formalism, an important device: a Beam Splitter. More generally you now know how to treat the case of an optical instrument by expressing the output field operators as a function of the input operators. In fact, you have learned that the relations are the same as for classical amplitudes. One can hardly overestimate the importance of that method, which plays a major role in many quantum optics calculations. This method of transforming the operators is yet more powerful, since it also provides a method to transform the state of radiation between the input and the output. As I have insisted, this is not a convenient method in general but there are situations where it may be interesting to do it. You will learn in your homework how to use a transformation of the creation operators to obtain the output radiation state. We have applied the formalism of quantum optics to the simplified case of one photon in a single mode. And we have encountered wave-particle duality for a single photon. We will come back to this fascinating quantum property in a future lesson where I describe real experiments performed with real one photon pulses, whose description demands a more elaborated model involving several modes of the field. With the simplified model of today, we have already obtained intriguing predictions of the quantum optics formalism, consistent with the experimental results. Firstly, if we consider double detection measurements, the result of the calculation is consistent with the image that there is only one particle, that goes either in the reflected or in the transmitted channel. In contrast, the result of calculations in the interference scheme is consistent with the image of the wave split in two on the first beam splitter, traveling simultaneously along the two channels 3 and 4, and recombine at the second beam splitter. You find this intriguing or even mysterious? You are in good company. This is what the great physicist Feynman wrote in the first chapter of the volume on quantum mechanics of his famous lectures on physics. It was about electron interferences, at that time there was no single photon source. I cannot resist reading it with you. "In this chapter we shall tackle immediately the basic element of the mysterious behavior in its most strange form. We choose to examine a phenomenon which is impossible, absolutely impossible to explain in any classical way and which has in it the heart of quantum mechanics. In reality it contains the only mystery. We cannot make the mystery go away by explaining how it works. We will just tell you how it works." Wonderful citation. My only reservation about that citation of Feynman is when he quotes wave particle duality as the only mystery. In fact, two decades later, Feynman changed his mind and acknowledge another quantum mystery, entanglement. We will discover this second quantum mystery in a future lesson. The wave-particle duality mystery lies in the contradictory character between the two images. In an attempt to surmount that contradiction, Niels Bohr, one of the founding fathers of quantum mechanics, introduced the notion of complementarity: the fact that we cannot perform the two experiments simultaneously, we need to choose between the two setups. Not everybody is fully satisfied with this comment, and I will come back to it when I describe the real experiments. But today I want to emphasize one remarkable feature of the quantum optics formalism which you can check in the calculation we have done today. We use the same quantum optics formalism to describe the two experiments. The formalism has in it both the wave-like and the particle-like aspect. This unity of the formalism must be contrasted with the too often read statement that one needs to change the description of light from a wave-like model to a particle-like model, when going from one experiment to the other one. Yes, we must change our images, but no we must not change our theoretical formalism. With this comment, I do not mean to imply that all mystery has disappeared. Feynman new perfectly well the formalism and he any way considered the phenomenon a great mystery. I have devoted most of my research to investigate experimentally quantum mysteries with photons and atoms. And I am still fascinated. And I know that many of the scientists who have invented quantum information schemes, Feynman himself, but also Brassard, Bennett, Ekert, Shor, Zoller, Cirac and many others acknowledge that quantum physics is weird. So If you are intrigued, if you think you do not understand how there can be such strange behaviors, you are in good company. But it is reinsuring to know that you can count upon a consistent formalism to predict or render an account of experiments in which the intriguing quantum behavior manifests itself. Bye-bye and come again next week for quantum mysteries revealed in real experiments.