In this lesson, you have encountered for the first time the formalism of multimode radiation. This is a formalism you must know to describe real experiments, real situations happening in nature. The lesson was somewhat formal, but there is no choice, you must know all these tools to be able to understand quantum optics phenomena, and to do yourself calculations that will show you how subtle and intriguing they may be. To make a long story short, you have learned that quantized radiation, far from charges is a collection of quantum harmonic oscillators. These oscillators are independent as shown by the fact that the total Hamiltonian is a sum of each individual Hamiltonian and that a and a_dagger of different oscillators do commute. This independence allows one to build a basis of the space of states by simply taking tensor products of number states of each mode. Building such a basis of the Fock states is very simple, but if you try to express in this basis the most general state of radiation, you find that the number of components is a priori unbelievably large. For each mode ell you have an infinity of basis states, 1_ell, 2_ell, 3_ell, n_ell, and there is an infinite number of modes. To better appreciate the immensity of that space, we can restrict the situation to M modes with not more than N photons in each of these modes. The dimension of the space, is then, N to the power M, and a priori, N and M are as large as we want. Even if we take numbers as small as N = 2 and M = 10, the result is 2 to the 10th, about 1,000. For M = 20, it is 1 million. If you have 500 modes, it is a number of the order of the number of atoms in the universe. Having that many modes is not unusual, a mode-locked Ti-Sapphire laser has more than 10,000 mode. So there is no doubt, the space of states of quantum radiation is big. This fantastic size of the space of states is at the root of quantum computing, which operates in such a space. One way to understand why quantum computing is potentially much more powerful than classical computing is to compare the number of variables necessary to describe the most general state in that quantum space with the number of variables necessary to describe the most general state in classical electromagnetism with the same number of modes. In the former case, we need N to the M complex coefficients. In classic electromagnetism, we only need M complex numbers. Only one complex amplitude per mode. So, the amount of information that can be encoded in a quantum state is incommensurably larger than the corresponding classical information. We have seen today another result. The fact that we can build one photon states as superpositions of elementary one-photon states. We will elaborate more in a future lesson and discuss one-photon wave packets which can be produced and studied in quantum optics laboratories. When discovering the expressions of quantum observables of multi-mode radiation, there was a surprise. The appearance of infinite quantities even in the vacuum for the energy or for the fluctuations of the fields. These are vacuum fluctuations. This fluctuation akin to the fluctuation of a quantum harmonic oscillator in its ground state are finite in each mode, but we have an infinite number of modes and a priori, you must take all of them into account even in the modes which are empty. Getting rid of these infinities demands a procedure known as renormalization, which is out of the scope of this course except for the case of the energy as you've seen. But you must remember that these fluctuations have physical consequences. I cited the Lamb shift, a slight correction to the position of the energy levels of atoms which can be interpreted as due to the quivering of the electrons under the effect of the fluctuations of the electric field. This quantity can be measured with an amazing precision and compared to the results of quantum electrodynamics calculations with renormalization. The agreement is impressive. Other effects of the same kind do exist, such as the correction to the gyro-magnetic ratio of the electron or the Casimir effect, that is the attraction between two ideal conductors in the vacuum. I will not comment more about these effects but there is an effect that we will study in a future lesson, it is spontaneous emission. The fact that an atom in an excited state eventually decays to a lower lying state. There is no way to understand spontaneous emission if we do not consider the coupling of the atom with quantized radiation in the vacuum. One can say that spontaneous emission is stimulated by vacuum fluctuations. These words will take a precise meaning in the lesson about spontaneous emission. If you want to understand how quantum fluctuations stimulate spontaneous emission, as well as the properties of one photon wave packets, entanglement, and other quantum wonders, you must study the lesson of today. It is worth the effort. See you at the next lesson.