Hi. Bonjour. Welcome to a new lesson of our Quantum Optics course. In the first course, Quantum Optics one, which you can watch at any moment, we have introduced the basic formalism of quantum optics and used it to describe a first example of radiation state that cannot be understood in the context of classical optics. That example was one photon states. That is to say, states for which the number of photons is perfectly determined and equal to one. Such fully quantum states of radiation are at the heart of modern quantum optics. They have played an important role in the progress about conceptual foundations of quantum physics, a still lively domain. Fully quantum states of radiation are also at the root of the development of many quantum technologies and we will encounter more examples in future lessons of this course. But today, I would like to address a question which I am sure bothers many of you. Why was classical optics so successful that modern quantum optics emerged only a few decades ago? It is a fact that most of the wonderful optics phenomena can be perfectly described by the classical model of light. That is to say, classical transverse waves as discovered by Young and Fresnel. Since the monumental achievement of Maxwell, we know that these waves are electromagnetic waves. Actually, all phenomena related to the behavior of light that were known in 1970, refraction, diffraction, interference, propagation in anisotropic crystals, you name them, could be fully explained in that classical framework. Of course, to describe light delivered by standard lamps or by the sun and stars, it was necessary to resort to the formalism of statistical optics, and use the notion of coherence. But this was still classical statistical optics. It was one of the achievements of Roy Glauber, who developed the formalism of modern quantum optics, to give an explanation of the success of classical optics in the framework of the quantum optics formalism. He showed that the quantum description of light emitted by all sources known in the 1960s including lasers could be done using specific quantum states named quasi-classical states of radiation or Glauber coherent states, which have properties very close to the properties of classical radiation, obeying classical Maxwell's equation. This was not a small achievement. Physics aims at describing all known phenomena of nature in a general framework, and physicists always want to understand how limited models efficient for a certain class of phenomena only, can be considered a particular case of a more general theory. A well-known example is the case of Newton mechanics, which is a particular form of relativistic mechanics at velocities small compared with the velocity of light. Similarly, it is interesting to understand how classical optics is a particular case of quantum optics. Moreover, knowing exactly why it is successful allows us to know when we can safely use the classical optics formalism. This is quite useful since classical optics calculations are often simpler than quantum optics calculations. So, understanding clearly what is classical will allow you to avoid the efforts of doing a full quantum calculation when it is not necessary. This is just like astronomers who use Newton mechanics rather than relativity to calculate the motion of planets. And conversely, understanding what is classical will allow you to identify situations which are not classical and where one must resort to the fully quantum formalism. More precisely, realizing where is the frontier between the classical and the quantum worlds allows one to understand and participate into the ongoing development of quantum technologies. Quantum technologies are methods allowing one to realize operations impossible to achieve in the context of classical technologies. Quantum optics is a major tool of quantum technologies. We have seen some examples of quantum technologies based on single photon wave packets in quantum optics one. It is hopeless to try to go beyond the limits of classical technologies if you have not a clear understanding of what is classical and what is fully quantum. The lesson of today will help you to develop your understanding and your intuition of where the frontier stands. Today, you will learn about quasi-classical states of radiation in the single mode case, leaving the multi-mode case for the next lesson. After a rapid overview of the essential of Quantum Optics formalism in the single mode case, I will first teach you the definition and some elementary properties of a quasi-classical state of a single mode of radiation. Some demonstrations are not too simple. So, I will let you discover them either in the homework or in supplementary documents of this week. We will then calculate the average value and the dispersion of the electric field in a quasi-classical state. This will allow us to better understand the meaning of the name "quasi-classical". We will also encounter Heisenberg dispersion relations for the electric field, more precisely for it's quadrature components. We will come back to this point in a future lesson, but you should already know that dispersions of the quadrature components are crucial quantities in modern quantum optics and in particular in quantum measurements. In section four, we will calculate the average value, the dispersion, and even the distribution of the number of photons. Here, you will see again in which sense you can understand the name quasi-classical. In section five, we will calculate the photo detection signals for a quasi-classical state of light, and we will find that these signals are exactly the same as what would be obtained in the semi-classical model of optics. This property is essential to understand the connection between classical and quantum optics. In section six, you will learn how to describe what happens to a quasi-classical state on a beam splitter. As you may expect, at the end, you get the behavior analogous to the one of a classical field. But the formalism to obtain this result is interesting and is useful in the description of many quantum optics phenomena. In section seven, we will apply the formalism of single mode quasi-classical states to the case of light in the cavity of a single mode laser. In section eight, we will raise the question of how to describe a freely propagating light beam, for instance, the beam emitted by a laser. You remember that in order to quantize a field, we have introduced a finite quantization volume, but most of light beams propagate without well-defined limits and we must be able to describe such situations. We will do it in this section and show how to introduce intrinsic quantities independent of the arbitrary volume of quantization. It will be also an opportunity to derive a celebrated formula about the shot noise of a perfectly stable beam, which constitutes the so-called standard quantum limit. To conclude, we will come back to the fact that semi-classical states of light with many photons per mode constitute a classical macroscopic limit of quantum radiation, and we will ask whether it is the only interesting application of these states. You may guess that the answer is no. The interest of quasi-classical states goes well beyond this property. So, there is no option. If you want to fully understand quantum optics, you must know a lot about quasi-classical states of radiation.
