Hi. Welcome to this new lesson of our course on quantum optics. In the previous lesson you have discovered how to apply the formalism of quantum optics to the case of a number state with a number of photons equal to one. More precisely, you have used the quantum expressions of the photo detection signals to show that the probability of a double photo detection is null, although each photo-detector has a finite probability to detect the photon. We have compared this prediction with the one of the semi classical model in which light is not quantized, and we have found that the predicted probabilities are identical in the two models regarding single photo detection signals, but they are different when it comes to double photo detection signals: the latter is different from zero in the semi-classical model while it is null in the quantum description of the one photon wave packet. We will see in a future lesson that the corresponding experiment has been performed. In the conclusion of the previous lesson, I commented that if words have a meaning we should not be surprised that the double photo-detection signal is null for one photon wave packet, since the first photon detection destroys the photon, and nothing is left to be detected. The quantum description is thus perfectly consistent with a particle-like model of light in which one photon wave packet contains one, and only one, particle. But on the other hand we cannot ignore that light should also be considered as a wave in order to describe the phenomena of interference and diffraction. We encounter here the question of wave particle duality of light, which was raised by Einstein in the Salzburg conference of 1909. You may tell me that you have already encountered wave particle duality in your general course of quantum mechanics, that the concept was clarified by Louis de Broglie in 1923, and that Schroedinger equation supplemented by the Born Interpretation gives a consistent description of it. But hold on: here, we do not have a material particle, for instance an electron, described with the Schroedinger equation. We have quantum radiation described by the formalism of quantum optics. So it is legitimate to ask the question: does this formalism render an account of wave-particle duality? More specifically, is there any wave-like behavior predicted for one-photon wave packet? This is the question we will address in this lesson. We will do it using the simple model for one photon wave packet in a single mode, which you have seen in the previous lesson. This one photon wave packet will be sent at the input of a Mach-Zehnder interferometer. This type of interferometer is much used in experiments and in applications and also in theoretical discussions. As many other optical devices, a Mach-Zehnder interferometer, contains beam splitters. A beam-splitter is a very important component in optics in general and in quantum optics in particular. In this lesson, you will learn how to describe a beam splitter in the formalism of quantum optics. As usual, I'm not only presenting fascinating phenomena, I also do my best to teach you fundamental techniques in quantum optics calculations in order to provide you with a tool box that you will be able to use when reading books and articles of quantum optics, and doing calculations yourself. The description of beam splitters is one of these fundamental techniques. Let us then, have an overview of this lesson. I will first show you how to describe the action of a beam-splitter in the quantum optics formalism. You will then be able to use the beam-splitter formalism to revisit in a different way the fully quantum behavior of a one photon wave packet. You are now armed to address a question that has fascinated physicists since the introduction of the quantum of light by Einstein in 1905: if we send a single photon in an interferometer will we observe interferences? In other words, can the photon ''follow two path'' simultaneously? Before answering that question, you need to be sure that you know how to describe a Mach-Zehnder interferometer in classical optics. The reason is that the quantum optics calculation borrows many elements from the classical calculation, so it is necessary to master the details of the classical calculation. This is a quite general feature of quantum optics calculations: a significant part of them is identical to classical calculations. You must learn to use this property. In this section you will get your reward. You are now able to calculate yourself, what happens to a one-photon wave packet in an interferometer, and answer unambiguously a question that has lead to a lot of confusion. At this point it will be enough for today, but I cannot quit without sharing with you some reflections about wave-particle duality, one of the great mysteries of quantum mechanics according to Richard Feynman. It will be an opportunity to comment about the power of the formalism of quantum optics, which can describe consistently such counterintuitive phenomena.