We are now going to use the formalism of multimode quantized radiation which you have studied in a previous lesson, to describe realistic one-photon wave-packets. Sources of one-photon wave-packets are now routinely used in laboratories working in quantum optics and quantum information. Consider that state. It is a sum of one photon states in many different modes. Recall that the short notation state |1_ell> means in fact that we have one photon in mode ell and the vacuum in all other modes. Applying the number operator N_hat to the state |1>, we find, after some calculation, 1 times the state |1>. It means the state |1> is an eigenstate of the number of photons observable N with eigenvalue one. It is a one-photon-state. If you do not understand that calculation, you can go back to the lesson about multimode quantum radiation. I have detailed there a similar calculation, where the double sum reduces to a simple sum because a_dagger_m a_m applied to |1_ell> gives a number 0, unless m is equal to ell. You remember that such a superposition of one-photon-states is not an eigenstate of the Hamiltonian, unless all the modes have the same frequency omega_ell, a hypothesis we do not make here. That state has thus a time evolution that we can write in the Schrödinger formalism in the standard form. You can check that the state |1(t)> remains a one-photon-state at all times. Let us write the expression of the single photon detection rate for a one-photon wave-packet in the Schrodinger formalism. Once again, expanding E+ and the state vector leads to a double sum with two indices, which condenses into a single sum since there is only one term of the electric field, different from zero for each component of |1(t)>. Check that you can do yourself this first step of the calculation. Now comes the part of the calculation that is specific to a one-photon state. When you apply the a_ell operators to each |1_ell>, you obtain the vacuum. Since vacuum is the same for all terms, you can put it in factor before taking the squared modulus of the expression. The result then appears as a squared modulus of the sum of classical functions of position and time. This sum, noted E+(r,t), is a complex amplitude of a classical field which can be considered the spatio-temporal mode associated with our one-photon state. You can think of that classical electromagnetic field as a wave function of the single photon in the state 1. As it should be, the probability to find the photon in a small volume around point r, at time t, is proportional to the squared modulus of the wave function. The notion of a wave function associated with a single photon must be taken with a grain of salt, but I find it useful. Note that if we had used the Heisenberg formalism to express the photodetection signal, we would have directly obtained this expression. The Heisenberg formalism provides a slightly simpler calculation. I encourage you to verify it yourself. You are now going to calculate yourself the double photodetection signal at two different times. For that you have no choice, you must use the Heisenberg formalism. Now take the expression of the one-photon state at time t0 and develop each E+(r, t) operator. You then obtain a sum of terms with two annihilation operators acting upon a one-photon state. The first annihilation yields the vacuum, and the second gives a number, 0. We find again, now in the multimode case, that the double detection signal is null for a one-photon state. A single photon cannot be detected twice. Remember that this is a dramatic difference with the semi-classical prediction We will come back to this point when we describe a real experiment.