Let us start with a discussion of what is fully quantum optics by contrast to classical optics. I already told you that until the 1970s there was no optics phenomenon that could not be described by the semi-classical model, in which light is described by a classical electromagnetic wave and only matter is quantized. More precisely in the semi-classical model, light can be considered the sum of modes of the electromagnetic field, which are elementary solutions of classical Maxwell's equations. It can always be written as a sum of a complex field and its complex conjugate, E plus of r and t, and E minus of r and t. For a single mode the complex amplitude E plus involves only one frequency and we have here its expression for a polarized plane travelling wave. Let me drew your attention to a not so clear vocabulary E plus is called the positive frequency part of the electric field amplitude, although omega comes with a minus sign. I could invoke a reason for this surprising vocabulary. It is the same evolution as for a quantum state with a positive energy. But it is not worth focusing on this vocabulary question and you only need to know the vocabulary because you will find it in books or articles of quantum optics. Let us rather continue our description of the semi-classical model. In contrast to radiation, matter is quantized so that atoms, molecules, ions, or even solids are characterized by the fact that the electronic motions around the nucleus are quantized. I have given here a pictorial representation not totally correct but very suggestive anyway. In the spirit of the Bohr's atom. There is also a representation of the energy levels showing a set of discrete levels, where the electrons are bound, and a continuum of states where one electron is free above the ionization threshold. In the semi-classical model the interaction between light and matter results from the interaction of the classical electromagnetic field, with the quantized atom and can be described by an interaction Hamiltonian involving a quantum observable associated with the electrons. Here, the quantum electric dipole D_hat and a classical field here, the electric field, this is semi-classical electric dipole Hamiltonian which is often used. The semi-classical model allows one to describe the photoelectric effect that is the fact that an electron of an atom or of a metal, can be ejected under the effect of radiation of high enough frequency, typically visible or even ultraviolet light. More precisely, the electron is excited from a bound state to a state of the continuum of free states where it can escape, far from the nucleus. Contrary to what you can read sometimes, the photoelectric effect can be quite well described by the semi-classical model, even though the fully quantum descriptions gives a more intuitive explanation of its characteristics. When it is associated with an electron multiplier, the photoelectric effect allows one to work in the so called photon counting regime. Here, is the example of a photomultiplier tube, an instrument that has a photocathode where a primary electron is ejected under the effect of the radiation, and a series of dynodes, which act as electron multipliers will produce a signal. More precisely, the initial electron is accelerated by a voltage of 100 volts or more. And when it impinges on the first dynode, several electrons are extracted. Typically, three, four, five. Each of them is accelerated towards a second dynode where the same phenomenon happens. After 12 dynodes, one has a bunch of more than 1 million electrons delivered in a few nanoseconds enough to produce a pulse of 10s of millivolts on a 50 ohm load. Such pulses can easily be monitored and one can for instance determine the rate of these pulses. That is the rate of extraction of primary electron, since each pulse is associated with one primary electron. The semi-classical model allows one to calculate the probability of extraction,of an electron from the photocathode, due to the photoelectric effect. The probability for an electron extraction in a small surface dS around r, for a time interval dt around t, is found to have the form dP equals w1 of r and t times dS times dt with w1 proportional to the square of the electric field avaraged over many periods, which can be expressed as the square modulus of the complex electric field amplitude, at (r,t). The proportionality coefficient, s, called the sensitivity, depends on the light frequency, as predicted by Einstein and on the wavefunction of the electrons in the atoms of the photocathode. The superscript 1 in w1 refers to the fact that we are interested in single detection events where 1 electron only is extracted from the photocathode. A yet more important quantity in quantum optics is the probability of the double detection at (r,t) within dS and dt and at (r',t') within dS' and dt'. The ability to measure it was a starting point of modern quantum optics just after World War II. In the semi-classical model the rate w2 of joint detections is nothing else than the product of single detection rates w1. Since random electron emissions are independent events once the value of the electric field is given. This is all we need to know about the semi-classical model of optics. It is only in the late 1970s and in the beginning of 1980s that experimentalists were able to produce new states of free radiation whose behavior could not be understood within the semi-classical description. A one photon state of light is the simplest example of these fully quantum states of light. We will apply the formalism developed in the first lesson, to such a state to discover how its behavior is definitely non-classical.