As soon as we have a quantum formalism, we have to manipulate two different types of objects. The states, represented by vectors of a Hilbert space, and observables. Which are represented by hermitian operators. We review now some observables of quantum optics. Let us recall the main results we have obtain and write quantized observables that are important in quantum optics, I will emphasize the most important results, the ones that you should remember. We have introduced the observables Q_ℓ hat and P_ℓ hat, whose commutator equals iħ, and the operators a and a^† which are hermitian conjugates of each other. Being non-hermitian, a and a^† do not correspond to physical observables, but they play a major role. Their commutator is equal to 1. I urge you to remember that commutator. a and a^† play a central role because we will express all observables as a function of a and a^†. For instance, by inverting the expressions of a and a^†, we could write Q_ℓ hat and P_ℓ hat as a function of a and a^†. We will do it soon. We have already seen the case of the Hamiltonian which has a remarkable simple expression as a function of a^† and a. This expression also is very important, you better remember it. Remember also that expression with a on the right and a^† on the left, the so-called normal ordering, are privileged forms for calculations. The field observables are also expressed as a function of a and a^†. You should not be surprised that the field operators do not depend on time. We do Schrödinger quantum mechanics in which the operators do not evolve, it is the states that evolve with time. When you know the state of the field, which is described by a state of vector psy(t), you can calculate the average value of the field. This average may evolve with time, we will see important examples in the next lesson. You can also calculate the fluctuations of the field by calculating the average of its square, and comparing it to the square of the average. The fluctuations may also depend on time. Actually, it's not fully true that you can measure the field observables. In the domain of radio waves and up to frequencies of hundreds of Gigahertz, you can indeed detect directly the electric or magnetic field using antennas and process signals oscillating at these frequencies. But in the domain of visible light, around 5x10^14 hertz, the frequency is so high that there exists no detector able to measure directly the electric or the magnetic field. Even the fastest detectors average a signal over a response time much longer than the period of the field and return a null result. This is not to say that you cannot measure anything, for instance you can measure the energy with a bolometer which is an operator that absorbs the radiation and you measure the resulting increase of temperature. The corresponding observable is the Hamiltonian. You can also use a photo-detector based on the photoelectric effect. Such a detector yields a signal proportional to the probability of detection of a photon, we will write the expression of that probability in the next lesson, and you will learn that these detectors, used in the so called photon counting mode, play a major role in the study of typical phenomena of quantum optics. In this lesson, I can already tell you that with modern photodetectors, which are almost perfect, we can measure the number of photons in the volume of quantization by letting all the radiation impinge on the detector. We know the corresponding observable N hat ℓ equals a_ℓ^† a_ℓ. It should not be confused with a Hamiltonian when there is more than one mode. We will come back to that subtle point. We will also see in this course that photo-detectors allow one to measure the observable Q_ℓ and P_ℓ using the so-called homodyne detection technique even for visible light. In fact, this is a good substitute to the direct observation of the field, since one can express a field operator as a function of P_ℓ and Q_ℓ. You can calculate yourself this expression by replacing a and a^† by their expressions as a function of Q_ℓ and P_ℓ, and expanding the complex exponentials in cosine and sine. But enough of observables. We need also to speak of states of the quantized radiation. We will describe important states of the quantized radiation more in detail in the next lesson. But today, I will tell you a few words about some of these important states. They are called number states of the quantized field. I will use them to explain how the notion of photon emerges from our formalism and we will also see the importance of a particular number state: vacuum.
