[MUSIC] Arrived at the end of the first lesson, we have totally entered into the territory of quantum optics. We have in hand the basic formalism that will allow us to describe emblematic quantum phenomena, such as single photon behavior, spontaneous emission, quantum correlations between pairs of entangled photons, squeezed light. In a nutshell, this formalism can be summarized by the Hamiltonian of a mode of radiation in Dirac formalism, and its eigenvalues. The Hamiltonian eigenstates, the number states |n_l>, are related together by the annihilation and creation operators. With the latter relation we can generate all number states from the vacuum. The ensemble of number states constitute a complete basis of the space of the states of radiation in the mode l. We have seen that the number state |0_l>, the vacuum, has properties. In particular, the field is not null in it, although it is null in the average. Last and not least, we have encountered the photon, a particle of energy hbar omega and momentum hbar k. Actually, I've warned you the latter property, the value of the momentum, holds only if we have quantized on a basis of traveling waves. If we use a basis of standing waves, each wave of the basis is composed of two counter-propagating traveling waves, and the photon can be considered as the linear superposition of two photons traveling in opposite direction. Intriguing, isn't it? Yes, it is intriguing. We are in the heart of quantum mechanics. So what is really a photon? [MUSIC] A photon is an elementary excitation of the quantized electromagnetic field. Its properties are linked to the properties of the creation operator, a dagger, that yields a photon from vacuum. If the creation operator involves only one momentum component, the photon has a well-defined momentum. But if it involves several momentum components, then the photon is a superposition of components with different momenta. Similarly, if the creation operator has only one frequency component, omega, the photon has a well-defined energy, hbar omega, but we will see examples of creation operators with several different frequency components. In these cases, the photon will not have a well-defined energy, hbar omega. In fact, the formalism we have introduced here is just an example of a formalism that can be used for the quantization of other kinds of waves. A well-known example is the quantization of acoustic waves in solids which leads to the notion of phonons. More generally, this formalism is a particular case of what is called second quantization, which allows us to describe ensembles of indistinguishable particles, either fermions or bosons. Photons are bosons. Then we can put more than one in an elementary mode of the electromagnetic field. So learning quantum optics, you learn also the second quantization formalism for bosons. It is an added value to this course. Putting back our feet on Earth, let me tell you a few words on what we are going to discover in the next lesson. [MUSIC] In the next lesson, we will use a formalism to discover an emblematic quantum optics phenomenon, wave particle duality for a single photon. [MUSIC] See you for the next lesson! [MUSIC]