In Quantum Optics 1, we introduced one photon wave packets with a finite extension in space. The meaning of one photon is therefore intrinsic and independent of the arbitrary quantization volume, provided that this fictitious volume is much larger than the size of the wave packet. In such a wave packet, the number of photons is thus an intrinsic notion. We will find here a similar situation where we build a quasi-classical wave packet with a finite extension and the average number of photons will be an intrinsic quantity. Let us consider a beam propagating along Z. To simplify, we assume that the transverse dimensions are large enough that the diffraction is negligible over the length L that we take to define the quantization volume. The elementary modes that we use are plane waves of section S propagating along Z and linearly polarized along X. The length L allows us to define longitudinal modes by taking periodic boundary conditions. The state defining the quasi-classical wave packet is a tensor product of modes with frequencies spread around omega zero. For definiteness, we take the example of coefficients encountered in Lesson 5 of Quantum Optics 1. You can find there many calculations regarding that form, and I encourage you to go there if you want to know all details. The distribution of the average photon number in the various modes assume a Lorentzian form. The sum is equal to the average number of photons in the wave packet which is an intrinsic quantity. This is a normalization condition that will determine the value of K which we take real and positive. As explained in Lesson 5 of Quantum Optics 1, the discrete sum can be replaced by an integral introducing the density of states. Using a standard integral we find the relation that determines K as a function of the average number of photons and of the square root of the arbitrary length L. The arbitrary length L should disappear when we evaluate intrinsic quantities such as, the average electric field or the photon detection signals. Let us show it. As we have seen earlier in this lesson, the average electric field assumes a classical form with a complex amplitude given by a sum over the various components. The expression of the field at position z is delayed by z over c, as it should. A calculation not too simple, analogous to one done in details in Quantum Optics 1 Lesson 5, shows that it corresponds to an oscillating function multiplied by an envelope starting suddenly at t equals t zero plus z over c, and decaying exponentially with an inverse time constant Gamma over two. Tau is a reduced time taking into account the propagation. It also contains t zero which was introduced in the coefficients alpha_ℓ. This formula shows that t zero is the instant of the beginning of the wave packet at z equals zero. As expected, the result does not depend on the arbitrary length L, this is because the density of states scales as L, while the constant K scales as L to the minus one-half, and the one photon amplitude E^(1) also scales as L to the minus one-half. Note that since E^(1) varies slowly with omega_ℓ and the bandwidth Gamma is small compared with the frequency omega_ℓ, we have taken its value at omega zero and put it out of the sum. A photo-detector perpendicular to the direction of propagation and covering the whole beam allows one to determine the probability of single photo detection at position z and time t, provided that the experiment is repeated many times. More precisely, what is recorded is the probability of photo detection as a function of the delay between the time of emission t zero of the wave packet and the time of detection t taking into account the propagation delay z over c. We know that for a quasi-classical state the photo-detection rate is expressed as a function of the classical field associated with the state. Taking the example just presented, it is a function rising at the beginning of the wave packet and decaying exponentially with inverse time constant Gamma. It is convenient to express the sensitivity of the detector as a function of the fraction eta of the ideal sensitivity of a perfect photo detector, that is to say, a detector yielding one-click for each photon. The probability of single photo-detection per unit time and surface then assumes a simple form whose integral is nothing else than the average photon number in the wave packet times the quantum efficiency eta. As already emphasized, for a quasi-classical state the photo-detection probabilities assume the same form as what would be obtained in the semi-classical model, with a classical field identical to the average field associated with the semi-classical state. The rate of double photo-detection is thus equal to the product of the single photo detections. It means that the probability of double detections is not null even if the average number of photons is small compared to one. This is intriguing since one could argue that such a wave packet should behave almost as a single photon wave packet for which there is no double photo detection. Actually, such a reasoning was frequently made until 1986 when an experiment was performed with a pulse emitted by a light emitting diode and attenuated so much that the average number of photons was about 10 to the minus two. The result was found in agreement with what we have calculated here, the probability of double photo-detection was found definitely different from zero. This experiment, which involves a beam splitter is presented in detail in Lesson 6 of Quantum Optics 1. But in this lesson, I do not use the notion of quasi-classical states and I ask you to admit this surprising result. Today, I want to show you the full quantum calculation.