This is the scheme to measure the anti-correlation parameter with a heralded source of one-photon wave packets. Detectors D3 and D4 are in the output channels of a beam splitter receiving the one photon pulses in input 1. These detectors are activated only during gates triggered by heralding photons. After a large number of gates, N_H, one has counted N3 and N4 single detections at D3 and D4, and N_c coincidences, that is to say, double detections during the same gate. These numbers allow us to determine the probabilities per gate of single and double detections, and thus, of alpha. Let us now proceed with the calculation of alpha taking into account the possibilities that supplementary photons are emitted during a gate. Remember that during a gate, the probability to detect the photon associated with that gate is equal to the global quantum efficiency epsilon 3. If now, there is a supplementary emission during the gate with probability nu, the probability that this supplementary emission be detected in channel three is epsilon 3 times nu, so that the total probability of a single detection in channel 3 is epsilon 3 times 1 plus nu. The probability of a single detection in channel 4 assumes a similar form. Let us now look into the probability of a coincidence. It is tempting to write it as the product of the single probabilities. Do you think it is the case? Is the probability of a coincidence the product of single probabilities, as written here? The answer is clearly no. The term 1 would correspond to detecting twice the photon associated with the gate. We know that it is not possible. In contrast, it is possible to detect the photon associated with the gate in one channel and another photon in the other channel and vice versa. This is the term nu times 2. Finally, there may be two supplementary photons emitted by chance by two different atoms during the gate: this is the term, nu squared. Hence, this expression of the probability of joint detection, which is not null. We can now write the expression for the predicted value of alpha, defined as P_c divided by the product of the single probabilities P3 and P4. This is a quantity, smaller than 1. In conclusion, the quantum optics description of this experiment, predicts a clear anti-correlation for a heralded single photon source, even if there are supplementary counts. At values less than 1 of the average number of supplementary photons per gate, alpha is proportional to 2 nu and tends to 0 with nu. Look again at the experimental results. The solid line is a plot of the formula we have just established. It agrees perfectly well with the results taking into account the statistical uncertainty. The anti-correlation that is to say the single photon character is undisputable especially if this source is used at values of nu significantly less than 1. I already told you that nowadays, there exists single photons sources based on single emitters, that are triggered by short, electric, or light pulses. Although the emission can be triggered at any moment, these sources are usually excited periodically. Here is an example, with a single emitter that re-emits a fluorescence photon at 637 nanometer, when excited at 532 nanometer by perioidic pulses. The fluorescence light is split and sent to two photo-detectors. The double detection signals are recorded as a function of the delay between the two detections with a time interval analyzer. After accumulating enough data, one obtains the time spectrum shown here. This signal is proportional to the probability of observing a double detection at D3 and D4 as a function of the delay. We first observe a periodicity with a period exactly equal to the time interval between the exciting pulses equal to 436 nanoseconds. The peaks associated with delays equal to a non-null integer number of such periods correspond to detection of photons emitted independently, and the rate of joint detections of such independent events is just the product of the rates of single detections. Each of this single rate is a decaying exponential, rising suddenly at the time of the quasi-instantaneous excitation and decaying exponentially with a time constant equal to the lifetime of the e level, approximately 45 nanoseconds. The signal is in fact a symmetric decaying exponential. This is due to the fact, that the time interval analyzer registers double detections with D3 firing first as well as double detections with D4 firing first. You can do yourself the not too difficult calculation showing that what we observe is what is expected. The situation is obviously different around the null delay. Here we are dealing with a double detection of photons emitted after the same exciting pulse and we expect a null double detection signal for a one photon wave packet. In fact, there are some supplementary photons due to scattered light, and it gives rise to the small peak around zero. We can proceed with the same analysis as in the heralded source, and determine the alpha parameter using these data. The correlation in the pulse is obtained by integrating the small peak around zero. The product of single detection rates is obtained by integrating any peak different from the one around zero. The result is a parameter alpha = 0.13, that is to say the signature of a strong anti-correlation. This source is a good source of one photon wave packets. It has been used for one photon interference experiments. To conclude this section, I would like to ask a question, what is the difference between the formalisms of classical and quantum optics that leads to so different values of the parameter alpha? In classical optics, the rate of single photo-detections is proportional to the light intensity, while the rate of double detection is proportional to the square of the intensity. Then the Cauchy–Schwarz inequality entails the fact that alpha cannot be smaller than 1. If now we consider quantum optics, in the case of a number state, the rate of single detections is proportional to the number of photons n. This should not come as a surprise. But if we consider now the expression for double photo detections, we must take into account the commutation relation of a and a_dagger to have an expression as a function of the number of photons. We then find an expression always less than 1 and equal to 0 for n equals 1, that is to say for a one photon state. The major difference between classical and quantum optics is the fact that classical numbers are replaced by operators that do not commute. In the particular case considered here, this entails the fact that the joint photo detection rate is proportional to n times n minus 1. In ordinary words, we can say that the detection of a photon destroys it, and leaves one photon less.