This is the scheme of the experiment. A quasi-classical wave-packet with a duration of a few nanoseconds is sent on a beam-splitter. Detections are performed in channels three and four during time windows, called gates on the figure, synchronized with the expected arrival time of the wave-packet. The experiment is repeated many times, and data are accumulated. One then obtains the number of gates, the number of single detection in each channel, and the number of coincidences. A coincidence means a double detection in three and four during the same gate. Curled letters indicates the total number of events during the whole experiments. Dividing by the number of gates, one obtains the average number of single and coincidence detections per wave-packet. Since these numbers are small compared to 1, they are also the probabilities of single and joined detections. You can see here a typical series of measured numbers in an experiment where the light pulses emitted by a LED were much attenuated. In this experiment, the number of single detections per gate is of the order of 2 times 10 to the minus 3. With a quantum efficiency of about 10%, it corresponds to an average number of photons per pulse of about 2 times 10 to the minus 2. The observed probability of coincidences divided by the product of the singles probabilities is found equal to 1.05. The statistical uncertainty is dominated by the number of coincidences whose estimated relative standard deviation is 3%. The standard uncertainty on the measured ratio is also 0.03. It means that the difference between the observed value and the value of 1 expected for a classical behavior is less than two standard deviations. There is no significant deviation from the expected value of 1. I will show you now how to obtain this theoretical value of 1. We want to calculate the probabilities of single and joint detections in the output channels of the beam splitter for a quasi-classical wave-packet at the input 1. As explained in detail in Quantum Optics 1, when you know the expression of the radiation state in the input space, a convenient way to do the calculation is to express the observables of the output space as a function of the observables of the input space. We use here the Heisenberg formalism. That is why I have written the input state at time t equals 0. Let us use that formalism to calculate the rate of photo detection w^(1) at the detector three. The electric field operator depends on time, and takes into account the propagation delay. Since the input state in channel two is vacuum, the term E_2 can be ignored. The rest of the calculation is the same as in the previous section. The result corresponds to an exponentially decaying wave-packet, with a multiplying coefficient r squared, and the propagation delay. A similar result is obtained for the other channel. Integrating the rate of detection w^(1) in the reflected channel over the whole wave-packet yields the average number of detected photons per wave-packet. It is also the probability of a single detection since that number is small compared to 1. A similar result is obtained for the transmitted channel. Similar calculation yields the average number of coincidences in the reflected and transmitted channels for each pulse. Starting from the rate w two, which is equal to the product of w ones, a double integration splits into two separated integrals, so the probability of a coincidence per wave-packet is the product of the probabilities of a single detection in each channel. Replacing the probabilities by the counted numbers, we obtain the equality that has been checked experimentally. This result is intriguing in the case where the number of photon per pulse is small compared to 1. How can it be that there are double detections when the probability of a single detection is quite small? To answer the question, let us look into the probability distribution of the number of photons in the wave-packet. In section two, we have shown that for any multimode quasi-classical state, the photon distribution is a Poisson distribution. The formula applies in particular to a wave-packet. It is written here in the standard form for a Poisson distribution, where the only parameter is the average number of photons. Here, I used the letter Pi for the probabilities related to photon numbers since the curl p is already used for the probabilities of detection. If the average number of photons in the wave-packet is small compared to 1, the expansion at the lowest order of the Poisson formula gives a probability to have one photon equal to the average number of photons. This photon is detected in output three or four, with probabilities r squared and t squared. Taking into account the quantum efficiency eta, we obtain probabilities of single detection in each channel. They are identical to what we have calculated using the probability rates w^(1). Let us now address the case of double detections. They are possible provided that there are two photons in the same wave-packet at the input. According to the Poisson law, the probability to have a pair of photons is at the lowest order equal to the square of average of n divided by 2. If we assume that the two photons of a pair are redistributed randomly and independently in the outputs of the beam-splitter, the probabilities to find two photons in channel three is r to the fourth power, and the probability to have two photons in channel four is t to the fourth power. The probability to have one photon in each channel is r squared times t squared times 2. The factor of 2 represents the two ways to distribute two independent particles in two channels. Introducing the quantum efficiency eta, we obtain a probability of joint detection equal to the products of single detection. We have thus an interpretation in terms of photon distribution of the fact that the probability of a joint detection is equal to the product of the probabilities of single detections. It stems from the fact that there are pairs of photons with the probability given by the Poisson distribution. Thinking about it, you may be surprised that considering the photons as independent classical particles leads to the same result as the one calculated by standard methods of quantum optics. I must admit that the first time I did that calculation, I was surprised, since photons are bosons, which obey a specific quantum statistics, usually different from classical statistics. In order to double check that there is no mistake, I decided to calculate, with the solid methods of quantum optics, how two photons at the input are distributed in the two output channels. I show you that calculation on the next slide. I already told you that an efficient way to calculate the radiation state in the output channels of a beam splitter is to express the input state with the creation operator applied to the vacuum; transforming the creation operator then allows you to obtain the state in the output channel. Let us do it in the case of a two-photon state at the input. We then replace the creation operator in channel one by its expression as a function of creation operators in channels three and four, and we obtain an expression of the output state as a function of the vacuum. You should continue the calculation yourself. Developing and using the result of the action of the creation operators a^dagger_3 and a^dagger_4 on the vacuum, you will obtain an expansion on two photon states at the output. The square moduli of the coefficients in this expansion give the probabilities of the various outcomes. Probability r to the fourth to have two photons in output three. Probability t to the fourth to have two photons in output four. Probability r squared t squared multiplied by 2 to have one photon in three and one photon in four. The factor of 2 in the latter result means that there are two different ways of putting one particle in one channel and one particle in the other channel. The two input particles behave as if they were independent of each other. So two photon in the same mode do not tend to be found preferably in the same output channel, as one might have expected naively for bosons. This is what quantum optics calculations tell us, and what has been verified in the reported experiment.