As explained in Quantum Optics 1, any classical free electromagnetic field, can be considered a sum of independent modes that evolve as harmonic oscillators. The global classical Hamiltonian, is thus a sum of Hamiltonians of classical harmonic oscillators, and the quantized Hamiltonian assumes the form of a sum of quantum harmonic oscillators. Remember the commutation relations where delta_l,m is the Kronecker symbol, whose value is zero except if l equals m. It means that all commutators are null except for a, a dagger of the same mode, whose commutator is one. In free space, it is convenient to use plane traveling waves as elementary modes. A discrete series of modes is defined by introducing an arbitrary volume of quantization, and imposing periodic boundary conditions. One obtains a discrete series of modes that are plane waves. Each mode is characterized by three integers n_x, n_y, n_z and a fourth index characterizing the polarization, taking one value among two. The set of four indices characterizing a mode is collectively denoted ℓ. The Global Hamiltonian is a simple sum of Hamiltonians of each mode. The eigenstates of the Hamiltonian H_ℓ of mode ℓ, are number states n_ℓ. The corresponding eigenvalues are E of n_ℓ equals hbar omega_ℓ times n_ℓ plus one half. Remember that the n_ℓ are positive or null integer numbers. The eigenstates of the global quantum Hamiltonian are the tensor products of all eigenstates of each single mode Hamiltonian. They constitute a complete basis of the space of states of the system, here quantized radiation. Each eigenstate of each mode can be generated by applying n_ℓ times the creation operators to the vacuum. There is only one vacuum. So the most general eigenstate of the Hamiltonian can be generated from the vacuum. The space of states has an enormous dimension. To get a feeling of that dimension, let us consider the case of only M modes, containing each a maximum number N of photons. The dimension is then N plus 1 to the power M. The size grows exponentially with the number M of modes. For 20 modes with a maximum of one photon per mode, the dimension is two to the 20th power. That is to say about one million. Hard to swallow! But we have no indication that it is not true, and several ideas of quantum technologies are based on that huge dimension. Another surprising feature of multimode quantized radiation is the energy of the vacuum. Can you figure out what is the problem? In each mode, there is a lowest state energy of one-half hbar omega_ℓ. Since there is an infinite number of modes, this energy is infinite. A standard way to get out of this difficulty is to take the energy of the vacuum as a reference, and to count the energy of radiation states above that reference. The resulting energy is then finite, provided that they are a finite number of photons. You should not think that the energy of vacuum fluctuations is a fictitious notion without any real consequence. In fact, the vacuum energy of empty modes is associated with the so-called vacuum fluctuations, that is to say, fluctuations of the electric and magnetic fields that exist even in zero photon states. These fluctuations associated with the Heisenberg dispersion relations have observable effects, for instance the famous Lamb shift, which is an additional term in the energy of atomic levels. In order to keep this term finite, one must limit the number of modes involved by introducing a cutoff, that is to say a maximum wave vector above which the modes do not count. Taking that cutoff at the Compton k vector, m c over hbar, Hans Bethe could calculate in 1947 a numerical value, which was in agreement with the result of the measurement by Lamb and Retherford, the same year. This was an example of the renormalization methods which have allowed Feynman, Schwinger, and Tomonaga to develop a sophisticated quantum electrodynamics theory, whose numerous predictions have been found in agreement with the most accurate measurements. Let us come back to the basic quantum optics, and address the question of photon number. When several modes are involved, the total number observable is the sum of the photon number observables in each mode. The eigenstate of the Hamiltonian, that is to say number states n_1, n_2, etc, are obviously eigenstates of the photon number observable. But the reciprocal is not true if the various components have different frequencies. For instance, in Quantum Optics 1, we studied the case of a one photon multimode field, that is to say a superposition of one photon states in modes associated with different frequencies. This is an eigenstate of the photon number observable with the eigenvalue one. But it is not an eigenstate of the Hamiltonian since the various terms hbar omega_ℓ do not come as a common factor. Remember this expression of the photoelectric effect in the case of single mode radiation. I use the Heisenberg formalism with the electric field observable depending on time, to be able to consider different times t_1 and t_2 in W2. These expressions can be generalized to the case of multimode radiation, but one must take care that the sensitivity s depends on the radiation frequency, so we note it s_ℓ. Now, something remarkable happens if one takes into account the fact that the one photon amplitude E one ℓ also depends on frequency. I have written here its expression in the case of a single mode at frequency omega_ℓ with a volume of quantization associated with the beam of transverse surface S and duration T. For an ideal detector at that frequency, we demand that the average number of photodetections in the quantization volume, that is to say the integral of w^(1) on S and T be equal to the average number of photons. Do that calculation and you will find that the product of the sensitivity by the square of the one photon amplitude, does not depend on the frequency. It can then be put out of the sum in the formulae expressing the rate of photo-detection, for instance w^(1). Note in passing that the quantum efficiency of an ideal detector does not depend on the quantization volume. Now, you know that a real detector has a sensitivity lower than the ideal sensitivity, by a factor eta less than or equal to one, which is the quantum efficiency. It turns out that usually the quantum efficiency varies slowly with the frequency of radiation. So quite often, you can write the rate of photo-detections for real detectors as the expression for an ideal detector, multiplied by the quantum efficiency of the real detector.
