We have just seen that a one-photon wave-packet is characterized by anti-correlation on a beam-splitter. In the ideal case, it means that no joint detection is expected. In the real world anti-correlation means that the probability of a joint detection is less than the product of the probabilities of single detections. This is expressed by the alpha parameter taking a value less than 1. We will first show that a value of alpha less than 1 is incompatible with a classical description of the experiments, using the semiclassical model of matter-light interaction. Let us consider a classical electromagnetic wave-packet, synchronized with a heralding signal that opens the gate covering the whole wave-packet. You remember that in the semi-classical model of matter-light interaction, the rate of photo detection is proportional to the squared modulus of the complex electric field. Low case "s" is the sensitivity of the detector. Integrating in time over the gate, and in space, over the transverse area of the beam, capital S, we obtain the average number of photodetections during a gate. We assume that the electromagnetic energy per pulse is less than the energy of a photon, so that the average number of detections during a gate is less than one. We can thus consider it as the probability of a detection during a gate, for a detector that would be inserted in the input channel 1, before the beam splitter. The number of counts per gate expected at each detector is thus also less than one, and can be considered the probability of a detection during a gate. We note these probabilities p_3 and p_4, using low case letters for probabilities associated with a specific gate. In the semi classical model of light, the rate of a double detection at times t' and t'' is the product of the single detections at t' and t''. In order to obtain the probability of a coincidence during a gate, we thus integrate twice over the gate the rate of double detections at t' and t''. The two integrals separate, and recalling the values of the single probabilities, we find that the probability of a coincidence during a gate is equal to the product of the probabilities of single detection in each channel. To obtain the probabilities per gate for a long experiment, we must consider the possibility of fluctuations in the pulse amplitudes. We thus need to take the average of the probabilities associated with each individual gate which may fluctuate from gate to gate. This averaging is indicated with an overbar. The correlation parameter alpha can thus be expressed as the ratio of the average probability p_C bar by the product of the average probabilities p_3 bar and p_4 bar. Let us consider first the case where all the pulses are identical without any fluctuation from one pulse to the next one. The averages associated with the overbar can then be ignored and alpha is equal to one. There is no anti-correlation. Let us now suppose that there are fluctuations from pulse to pulse. In other words, the integral w over the pulse of the squared modulus of the classical electric field is not the same for each pulse. We must now take into account the overbar averages to obtain the average probabilities per gate, curl P_3 and curl P_4. The single probabilities are proportional to the average of w, while the coincidence probability is proportional to the average of w^2. Alpha is thus equal to the ratio of w^2 and then averaged, to w averaged and then squared. It is well-known that the average of the square is not equal to the square of the average. Could it be, then, that for some specific models of fluctuations, one recovers a value of alpha small enough to be compatible with the quantum prediction? In fact there is a mathematical relation between the average of the square and the square of the average of any fluctuating positive classical quantity. Do you know it? This is the relation between the average of the square and the square of the average. For any positive random variable, the average of the square is larger than or equal to the square of the average. The equality holds if the variable has no fluctuation. We can thus conclude that alpha cannot be less than one. Classical fluctuations can only produce correlations, not anti correlations. This inequality, which is a particular case of a general inequality known as the Cauchy-Schwarz inequality, can be obtained directly in two lines of calculation. You should try to demonstrate it yourself, since this inequality, or equivalent ones, are often invoked in papers of quantum optics, when one wants to distinguish a behavior that can be described classically, from a fully quantum behavior. We measured the alpha parameter to characterize the source of heralded one photon wave-packets that I described in the previous lesson. You see that we found alpha less than one for all the working conditions, But anyway it was not null and, it depended on a parameter nu that I will define now. Remember the scheme of our source: an atom excited to the level u, re-emits a cascade of two photons, first a photon hbar omega_H, the heralding photon, and second a photon hbar omega_0 described as a one photon wave-packet. Photons omega_0 are observed only during time windows of 10 ns, gates triggered by the detection of the heralding photon. The gating scheme allows one to isolate single atom emissions although many atoms are submitted to the excitation scheme. In fact it happens sometimes, by chance, that another atom is excited during a gate already open, and emits another photon hbar omega_0. The parameter nu is the average number of supplementary photons emitted by chance during a gate. The plot shows the results of our measurements of alpha for various values of nu. The boxes represent the uncertainty due to the statistical accuracy of the measurements. You see that for all values of nu, we found alpha less than 1: the wave-packets have a definitive non-classical behavior, more precisely the kind of behavior expected for one-photon wave-packets. Observe, however, that when nu is not small compared to one, alpha approaches the value of 1, which is the frontier of the classical domain. The solid line is a calculation of what is expected as a function of the cascade excitation rate. The fact that alpha is not strictly equal to 0 is due to the emission of supplementary photons during a gate. You know enough of quantum optics to appreciate that calculation which I am going to sketch in the next section.