[MUSIC] Hello everybody. I'm now going to introduce the homework related to the first course. As you may have seen, each course is complemented by a quiz. So that you can evaluate your degree of understanding of the video lecture. In addition, we provide homework with correction will be published online one week later. In the first course, we presented general formalism for quantizing the single mode plane wave. So first, homework consists in applying this formalism to the case of the standing width defined by cavity mirrors. As seen in the first lecture quantization of plane waves, results in the introduction of operators for describing filled of valuables, like the electric field amplitude of a wave at the given position of space. This operator is shown here. It involves two Non-Hermitian operators, a and a dagger. They are respectively interpreted as photon annihilation and creation operator. They obey the fundamental relation a, a dagger = 1, which is characteristic of an harmonic oscillator. Why are you considering plane waves? Well, first, by simplicity. Plane waves are the simplest solution of Maxwell equations for the free classical field. They additionally form a complete set of modes allowing for the decomposition of an arbitrary field by Fourier transform. In other words it is a good simple basis for expanding any classical field. Our goal in the homework, consists in applying the presented field quantization method. To other field modes. You already encountered plenty of useful modes, in classic electromagnetism. Let us first mention spherical waves, whose wavelengths are spheres instead of planes. The fields are radiated by dipole, is another example its radial dependence looks like a spherical wave. In addition to the amplitude of the field, there is as the sine of the angle between the dipole and the observation direction. Of course, this mode could be quantum mechanically described a super position of quantized plain waves, however this approach may be mathematically very heavy. Expanding the spherical wave on plane waves, is something that you only do if you have no other choice. However, you can also choose an alternative solution. You can directly quantize the field modes which are of interest for you. Another very interesting situation, is the case of standing wave defined by the reflection of a wave on mirrors forming a cavity. Let us consider the simplest case of a Fabry-Perot cavity, made of two parallel mirrors, separated by distances. In this situation, the state size of the wave vector, is set by the boundary condition in the mirrors. More precisely, the field with the polarization parallel to the mirror, should vanish at the surface of the mirror. The sign function here, effectively vanishes at z = 0. It also vanishes at z = l, provided k times x is an integral multiple of pi. This condition, is nothing else than the constructive interference condition for the wave of the 100 round cavity. The condition on k times L indeed, defines the wave vectors and choose the frequencies of the modes defined by the cavity. Before starting the homework, let me now tell a few words about the interest of quantizing a cavity mode. Indeed, many recent experiments are based on the manipulation of the quantum state, fulfilled stored in a cavity. So a single mode quantization model developed in the homework, implicitly assumes perfectly reflecting mirrors, so that dumping is negligible. Of course, real cavities are never perfect, but say can indeed be rather extremely good. The best optical mirrors the construction of the Fabry-Perot cavities, with a finesse of about one million. One million of reflection of light before photon disappear. One can do even better if you switch to the microwave domain, involving photons with millimetric wavelengths. In this domain, mirrors can even be fabricated out of superconducting material which behave as loss-less, perfect metallic mirrors. The best mirror ever fabricated are shown on the picture on the left. They are made out of high quality copper minerals, covered by a thin layer niobium. When cooling them down to 9.2 Kelvin, niobium becomes superconducting. While cooling it further down to 0.8 Kelvin. It becomes such good material, that a single photon can bounce back and forth billion of times before disappearing. The corresponding lifetime of one photon in to cavity is 1.10 of a second, a time scale which is even accessible to your perfect perception. So experiments in the group of Serge Haroche, at the Ecole Normale Superieure in Paris, rely on the manipulation of the quantum state of a few photons trapped in such a cavity, and probed by very sensitive made of single rebel atoms. More recently, another branch of cavity QED, named circuit QED, showed up. It is based on full solid state superconducting devices. These devices involve a cavity, consisting in a so called straight line. Which is nothing else than the two dimensional invention of a coaxial cable, on which acts as a microwave guide. The wave guide is get at both end, and the standing wave melts up between the two ends of the line, defining your high finesse mode. In this experiment, microwave photons are manipulated and probed by artificial atoms. They consist in the superconducting electrical circuit, involving capacitors in UNKNOWN] and. These elements are positively coupled to the guided standing wave movement. We saw the two devices are very different. They both involve strong interaction, between individual atoms and individual photons. Confined in this small volume of a super conductive resonator. Since physics attracts high interest, either for the testing of the most fundamental aspects of quantum theory, such as quantum theory of measurement. As well as for applications in the field of manipulation of quantum information. This a story you will hear more about in next lectures. In order to conclude this introduction, let us come back to the homework, which deals with field quantization of a cavity mode. There is indeed, a very general method of single mode field quantization in a cavity. It consists in two steps. First, you have to solve the Maxwell equation in the presence of boundary conditions, set by the position of mirrors. The first step is a classical calculation. There is nothing quantum at this level. The outcome of this step, is the determination of the spatial shape and of the frequency of the cavity modes. The 2nd step, consists of quantizing each mode by introducing harmonic oscillator, emulation and creation operators a and a-dagger. This operator replace alpha and its complex conjugate, in the expression of the field. So, classical field amplitude, then becomes an emission operator. E(z) represents the amplitude of the field at position z. Note that in this expression, z is a parameter and not an operator. Indeed, you should realize that in the process of field quantization, it never shows up anything looking like the position operator of a photon. The position operator of photon, is indeed not an appropriate concept. Another important feature of the field operator, is the absence of time dependence. So time parameter disappear in the field quantization process. Of course, it does not means, that quantum field do not evolve. Indeed, time evolution will show up again in the quantum framework when applying the Schwinger equation, to the state vector of the system. The main outcome of the homework states the very simple result. The quantum description of cavity mode, is no more complex than the quantization of a free plane wave. Note that this is not a trivial result. The cavity mode dynamic, indeed, involves interactions coupling the cavity field to the motion of all charges. Located close to the mirrors surfaces. We solved this problem without need of describing quantum mechanically neither the motion of charges, nor that interaction with the field. The strength of the cavity mode quantization formalism relies in describing the interaction between the field and mirror charges. At a microscopic, classical level, by simply solving Maxwell equation. When defining a cavity mode in this way, a mode is implicitly a combination of field oscillations in the free space between the mirrors, and of current oscillations and the mirror surface. This coping motions are described by single normal variable, with quantization follows the same mode as for if we moved. And now, you can learn more about this by taking time to go into the homework. Goodbye and see you next week. [MUSIC].