Let us start with a little bit of formalism about multimode radiation. It is a simple extension of what we have done in the lesson on quantization of a single mode. Do not hesitate to go back to that lesson if you have difficulties to understand what I am going to teach you now. In standard books on electromagnetism, you find that the most general solution of classical Maxwell's equations in vacuum can be expressed as a sum of plane monochromatic waves, that is to say, the modes I have used to show you how to quantize radiation. You remember that each mode is characterised by a wave vector k_ell and a polarization epsilon_ell, perpendicular to k_ell. The classical state of radiation in that mode is fully determined by the complex amplitude E_ell. The real and imaginary parts of E_ell are two dynamical variables describing the state of classical radiation in mode ell. The most general state of radiation can thus be fully determined by the ensemble of pairs of dynamical variables associated with each mode. We know that the evolution of each mode is sinusoidal with the frequency omega_ell equals c times the modulus of k_ell. In fact, this dynamics can be considered as the consequence of the dynamical equation dE_ell over dt equals -i omega_ell E_ell. As in the case of single mode radiation, we introduce a fictitious volume of quantization: L_cube, and we impose that the field reproduces itself after a length,L, along each direction of space. These are the so-called periodic boundary conditions, well adapted to the case of travelling wave modes. The k-vectors of the modes can only take discrete values, which lie on a 3-dimensional cubic lattice of the reciprocal space. The frequencies omega_ell, also belong to a discretized set of values. Finally, a mode is fully determined by a k-vector k_ell, plus the polarization epsilon_ell. Since the polarization is a unit vector, necessarily perpendicular to k_ell, it belongs to a two dimensional space and can be defined by a two-valued index s with conventional values one or two. Finally, a mode can also be defined by three positive or negative integers n_X, and n_Y, and n_Z, and the two valued index, s. Let us elaborate a little more on the polarizations associated with a given k vector. It is obviously a good idea to take the two polarization vectors epsilon_ell_1 and epsilon_ell_2 orthogonal to each other, so that they form a basis of the two dimensional space of the polarizations associated with the k-vector, k_ell. In addition, we choose them so that epsilon_ell_1, epsilon_ell_2, and k_ell form a direct triad. Up to now, we have implicitly considered real unit vectors, epsilon_ell, associated with linear polarizations. But it is also useful to introduce complex polarization vectors, which allow one to describe circular polarizations. For each k_ell, we can thus introduce two complex unit vectors epsilon_ell+ and epsilon_ell- . If you write the corresponding real-electric field with explicit time evolution of E_ell of t, and a complex polarization vector, you can check that this real electric field at any point r is a vector rotating around k_ell, in the direct or the reciprocal sense respectively. You may be surprised by the minus sign in front of epsilon_ell plus. Here you could remove it without changing anything. But this form is consistent with the general formalism of so-called vector operators, and facilitates the use of general theorems you may encounter, such as the Wigner-Eckart theorem. This is why I prefer to use that form. Introducing a quantization volume allows one to simplify the mathematics, but it also permits one to work with a finite energy radiation, and hence to define the number of photons in the quantization volume. Let us first calculate the energy of a classical electromagnetic field in the quantization volume V. Its expression involves both the electric field and the magnetic field. When we expand the fields on the modes ell, and take the integral over the quantization volume, we have terms, which are null since the integral is taken over an integer number of periods. They are null except if the two integers n_ell_x and n_ell'_x are equal. Similarly, terms with a plus rather than a minus sign are null unless the integers are exactly opposite. Finally, when we take the three dimensional integral of E squared, the only surviving terms are those involving the complex amplitudes of the same mode ell or of exactly opposed mode ell and minus ell. This form is pretty simple, but its demonstration demands some attention, as you have seen. If you do the same calculation with B, you find almost the same result, but with two differences. First, there is obviously a one over c squared factor since the magnitude of k_ell is omega_ell over c. More subtle, is the change in sign in the second term. This minus sign is due to the fact that the magnetic fields associated with the modes ell and minus ell have opposite signs, because of the properties of the vectorial product. As a result, when you add the electric and the magnetic contributions, the crossed terms involving modes ell and minus ell disappear, and you are left only with the sum of the energies H_ell of each individual mode. I must admit that the calculation that we have just done is not simple, but it is the simplest correct demonstration I know to obtain this remarkable result, far from trivial. The total energy of radiation is the sum of the energies of each mode without any cross term. It corresponds to the fact that the various modes are decoupled, that their dynamics are independent. If they were coupled, one would have in the expression of the energy, products of amplitudes of different modes. It is this decoupling, which allows us to easily effect the canonical quantization. Last, and not least, we have shown that the energy in each mode is strictly constant, because the electric and the magnetic contributions have terms that oscillate out of phase and exactly compensate each other. This is another way of seeing the decoupling of different modes. In order to identify pairs of canonical conjugate variables, we will proceed, as in the single mode case. We will express the energy as a function of variables, which we think are canonically conjugate of each other. Then we will write the corresponding Hamilton equations. If we find the evolution equations that we already know, We will conclude that we have indeed identified pairs of canonically conjugate variables. Let us then express the energy with the same variables alpha_ell as for the single mode case. Remember that the dimensionless variables alpha_ell are proportional to the complex amplitude E_ell. And that the one photon amplitude E1_ell is the classical amplitude associated with an energy of hbar omega_ell in the quantization volume. Each energy term then assumes the form we already encountered. We now introduce the real and imaginary parts of each alpha_ell, and obtain the same form as previously for the energy of each mode. We have now the total energy of the radiation in the volume V, expressed as a function of the pairs of variables Q_ell and P_ell. In the single mode case, we have shown that the two dynamic variable Q_ell and P_ell are canonically conjugate of each other. Since we have decoupling of the terms associated with each mode, you probably anticipate that the various pairs Q_ell, P_ell, are independent pairs of canonically conjugate variables. Let us verify it. We thus make the assumption that these variables are canonically conjugate of each other and write the corresponding Hamilton equations. The first order differential equation associated with each pair can be written as a single differential equation on the complex variable, (Q_ell + iP_ell), or equivalently, alpha_ell. As expected, we find the known dynamical evolution of each mode. We thus conclude that the pairs Q_ell and P_ell are independent pairs of canonically conjugate variables, and we can proceed with canonical quantization. You remember that canonical quantization consists of replacing each pair of canonical conjugate variables with a pair of hermitian operators whose commutator is i hbar. The operators associated with different pairs do commute, as indicated by the Kronecker symbol whose value is zero, except if l equals m. Starting from the classical expression of the Hamiltonian, we then immediately obtain an expression of the quantum Hamiltonian. But you remember that it is very useful to introduce the operators, a_ell and a_ell_dagger, Which can be considered the quantum counterparts of alpha_ell, and its complex conjugate. You remember also that their commutator has a value of one. Here again, the Kronecker symbol indicates that the commutators associated with different modes are null, reflecting the decoupling of the various modes. Although, a_ell and a_ell_dagger are not physical observables, they allow one to express observables in a very useful form. In the case of the Hamiltonian, we obtain an expression that you must remember: H_R equals sum over all the modes ell of hbar omega_ell time a_dagger_ell a_ell plus one half. Do you remember that the one-half terms stems from the commutators? If you are not sure, I recommend that you write again the calculation leading to that central expression. Maybe you want to know why I use the underscript capital R in H_R, well, it stands for radiation. H_R is the Hamiltonian of radiation. We have now established the main result which is the starting point of all quantum optics: quantized radiation in the absence of charges, can be considered a set of quantum harmonic oscillators independent of each other.
