Let us take the formula for double detection applied to the case of a one-photon state in a single mode ℓ. Once again, since a number state is a stationary state, it evolves as a complex exponential of time, which disappears when taking the square modulus. The same happens for the complex exponentials associated with the positions r and r' where the electric field operators are taken. So the rate of double detection depends neither on time nor on position. It is constant in the volume of the wave packet. We should not be surprised, since our model is a uniform wave packet in the volume S c T. Expressing the E plus operator as a function of a, we find that we must calculate the result of applying twice the annihilation operator to the one-photon state vector. Now comes the central result, mathematically very simple, but with far reaching consequences. Can you evaluate a applied twice to the one-photon state? Remember, the fundamental properties of the annihilation operator applied to number states. If you don't remember, it's time to make a note that you need to go back to that point and memorize it. But let us use these properties, a applied to one returns a vacuum state. A number state associated with zero photon. A perfectly legitimate state whose modulus is one. But when you apply a another time to the zero photon state, you get 0, the number 0. So that the probability of a double detection is null. It is possible to measure the single and double photo-detection signals as shown here. Two detectors positioned in the light beam are connected to an electronic circuit that monitors and counts the pulses. It also registers coincidences that is to say the fact that the two detectors, D1 and D2 have fired in coincidence, which means during the same time window which can be chosen at will. Here, we choose a coincidence window equal to the duration of the wave packet, so that the coincidence counter registers an event if the two detectors fire during the same wave packet. The experiment is repeated for a large number of times, noted N_wp, number of wave packets. For a constant amplitude wave packet, the average number of detections w1 times the area of the detector times the duration of the wave packet T, and we multiply by the number of wave packets to obtain the total number of counts. A similar result is obtained for the second detector. The number of coincidences during a wave packet is w2 times the area of each detector times T squared. Multiplying by N_wp, we obtain the number N_c of coincidences. For a one photon wave packet, we have calculated w1 and w2. Using these results we obtain N1 equals eta times S1 over S_ℓ times N_wp, where eta is the quantum efficiency, and similarly for N2. But we have found that w2 is null, since one cannot detect twice a single photon and the number of coincidences is null. In order to better appreciate how this result of a null coincidence rate is surprising from the point of view of traditional optics, let us compare it to the prediction of the semi-classical model for a classical wave packet. The difference is that w2 equals w1 squared in the semi-classical model. We take a constant amplitude over the wave-packet volume and the formulae of the semi-classical model give constant rates of single and double photo-detections. After integration over the size S1 and S2 of the detectors and over the whole duration of the wave-packet, we obtain the average number of counts during each wave-packet, which we then multiply by the number of wave-packets to obtain the total number of single counts and of coincidences. The single counts expression closely resembles the ones obtained with the semi-classical model. But the result is dramatically different when it comes to the coincidences, since it is not null. Note that this result is independent of the amplitude of the classical field and remains unchanged, even with extremely weak classical wave-packets. We will come back to this point when we describe real experiments.