A multimode quasi-classical state is a tensor product of quasi-classical states in different modes. In lesson one, you learned that the quasi-classical state of a mode ℓ at time t equals 0 remains a quasi-classical state of that mode with alpha_ℓ evolving with time at frequency omega_ℓ. This can be generalized immediately. Do not hesitate to go back to lesson one to do this generalization yourself. Another interesting result of lesson one, is the fact that the quasi-classical state alpha_ℓ is obtained by applying the operator exponential of alpha_ℓ a^dagger_ℓ to the vacuum, with a necessary normalizing factor. Since all a^dagger operators commute, this generalizes to a remarkably compact formula. In fact, this is a particular case of a general result based on the so-called displacement operator. A quasi-classical state is nothing else than a displaced vacuum. This formula is yet simpler if you replace the sum of alpha_ℓ squared by the average of the total number of photons, whose value you will see in a few minutes. The effect of the electric field observable onto a multimode quasi-classical state generalizes immediately a remarkable result that you have learned in lesson one. The multimode quasi-classical state is an eigenstate of the positive frequency part E plus hat of the electric field observable. The eigenvalue has the form of the complex amplitude of a multimode classical field and we call it E plus class for classical of r and t. As in the singlemode case, a multimode quasi-classical state is not an eigenstate of the whole electric field observable, and we must consider separately the actions of E plus hat and E minus hat. The eigenvalue of E minus is the complex conjugate of E plus. Finally, the quantum average of the electric field observable assumes the form of a classical field whose complex amplitude is E plus class. You know that probabilities of photo-detections, single and double, are very important quantities in quantum optics. In the case of quasi-classical states, they assume forms identical to what would be obtained with a classical field. I suggest you try to show it yourself as an exercise. Did you do it? Then check that you were right. Let us start with the probability per unit time and unit of phase of a single detection w^(1). Since the quasi-classical state is an eigenstate of the operator E plus, you immediately obtain the square modulus of E plus class, a result identical to the classical expression. Note that you would have obtained the same result yet more easily using the Heisenberg formalism. When it comes to the rate of double photo-detection at two different times, you must use the Heisenberg formalism, where the electric field operators depend on time and the state is taken at time t equals 0. To keep the writing simple, I have assumed here that the spectrum of the radiation state is narrow enough that the sensitivity is the same for all components involved. Here again, the result is the same as the result of the classical calculation for a classical field with complex amplitude E plus class. Note that this result remains valid even for feeble light, with an average number of photons smaller than 1. We will come back to this point in next section. This is in stark contrast with the null value of a double detection for one photon wave-packets as shown in Quantum Optics 1. Let us now address the question of the photon number in a multimode quasi-classical state. As a singlemode quasi-classical state, a multimode quasi-classical state is not an eigenstate of the number of photons observable. So we will calculate the average of the photon number and its rms dispersion. The average is nothing else than the sum of the average photon numbers in each mode. You know how to calculate the variance. Calculate first the average of the square, then you have to be careful, it is non trivial. The result is equal to the square of the average plus a term that is the variance. It is equal to the average number of photons. The rms deviation of the total number of photons is thus the square root of the average total number of photons. It means that the variance of the total number is the sum of the variances, in other words, the rms deviations add quadratically. This non trivial result suggests that the distribution of the total number of photons is a Poisson distribution. This statement is true, but the demonstration is not easy. I propose to do it however, because it will allow me to introduce an important tool already encountered in Quantum Optics 1, I mean a multimode creation operator. Consider this expression of a multimode quasi-classical state that we have encountered a few slides ago. It is tempting then to introduce a linear combination of singlemode creation operators that we call a^dagger multi. of the moduli of alpha_ℓ. The constant alpha multi is obtained by a quadratic sum It is equal to the average number of photons. The multimode quasi-classical state can thus be obtained by an expression similar to an expression established for a singlemode quasi-classical state. This suggests that a^dagger multi can be considered a creation operator. It turns out indeed that a^dagger multi, and its hermitian conjugate a multi, have the properties of a creation and a destruction operator. I suggest you establish yourself these properties using the expansion of a^dagger multi and I only give the relevant properties. Firstly, their commutator is equal to 1, then one can define an observable number of multimode photons N hat multi equal to a dagger multi a multi. You can also obtain the commutators of N multi with a multi and a dagger multi. All these relations are the same as the ones for the operators a_ℓ and a^dagger_ℓ in a mode ℓ. One can then use the Dirac method to obtain the eigenstates and eigenvalues of N hat multi. The reasoning is strictly identical to the derivation for an elementary harmonic oscillator since all the commutation relations are the same. The goal is to determine the eigenstates and eigenvalues of the observable N hat multi. Using the commutation relations of N hat multi with a multi and a dagger multi, one finds the usual relations giving the result of the action of a multi and a dagger multi on the eigenstates of N hat multi. One can then show that the eigenvalues of n multi are positive or null integers. We can call these eigenstates multimode number states. They can be generated by applying n times a dagger multi to the vacuum. These multimode number states have a remarkable property: they are eigenstates of the total number of photons observable with the eigenvalue n multi. If you want to demonstrate it, I give you a hint. Use the commutation relation between N and a dagger multi to show the property for n multi equals 1, and proceed by recurrence. By the way, using the expression of a dagger multi, you will find that the state n multi equals 1 is nothing else than the one photon state we have already mentioned in section one and studied in detail in Quantum Optics 1. We can now use the formalism of multimode photon states as a basis to express multimode semi-classical states. We have used the formula with the average number of photons. Expanding the exponential as usual, we recognize terms a dagger multi applied n times to the vacuum, that is to say number states n multi within a factor. Wrapping up, we obtain the expansion of the multimode quasi-classical state on the basis of multimode number states. The square moduli of the coefficients in that expansion yield the probabilities of having a number n multi of multimode photons. It is a Poisson distribution as announced. This result is an important result since it applies to real situations, for instance to wave packets such as we will study now.