I already mentioned the name of Roy Glauber who established in the 1960s the formalism of quantum optics that we still use. Among the most important features of that formalism, you must know the formulae that will allow you to express photo-detection signals for any state of the quantized radiation. I will now introduce these expressions, and you will discover what is predicted when we apply this formulae to the case of one photon wave packet. Glauber established the expressions of the photo-electric detection signals, with quantized radiation by using the expression of the interaction Hamiltonian between quantized radiation and a quantized detector. That is to say an atom with a series of discrete levels and a continuum of ionized states. This interaction Hamiltonian resembles the semi-classical one. But in fact, it differs dramatically. Both quantities are quantized, not only the atom, but also the electric field represented here by an operator as indicated by the hat over E. The physical effect is, of course, the same. It does not change when we use different mathematical tools to describe it. So again under the effect of the interaction, a bound electron is excited to the continuum and escapes from the atom. I explained you earlier that with an electron multiplier we can then obtain a detectable electric pulse for each released electron. I will teach you that quantum interaction formalism in a future lesson. Today, I only ask you to accept Glauber's formulae and to remember them since you will have to use them. The rate of photo-detection per unit surface and unit time has been defined earlier in this lesson, in the context of the semi-classical model. In the fully quantum description it is given by this expression, in which E+ is the part of the electric field operator involving the destruction operators a_ℓ. The operator E minus is its hermitian conjugate involving the creation operators a^†_ℓ. Note carefully, I told you that already, that E plus involves a while E minus involves a^†. I must admit that this may be confusing. But everybody uses this notation, which was introduced by Glauber, so we have no choice. The coefficient s is the sensitivity of the detector. Its value can be obtained by a full calculation à la Glauber, provided one knows the exact expressions of the electronic states in the detector. We will see in a minute what is the value of s for an ideal detector. Notice that you can write w1 as the squared modulus of E+ applied to psi. An expression that may evoke the semiclassical expression. Let us now discover the very important formula giving the rate of double detections around (r,t) and around (r',t). Please note carefully the order in which the electric field operators appear with E minus on the left and E plus on the right. One way to remember it is to write it as the square modulus of E+(r') times E+(r) applied to the state |psi>. If you remember that when taking the hermitian conjugate of a product you must reverse the order, you'll get the right ordering. This ordering with a operators on the right and a^† on the left plays a crucial role. I've already told you in the first lesson that it is called the normal ordering. You must remember that in the photo-detection signals formulae, one has normal ordering. We will immediately see the consequences of that ordering when calculating the double photo-detection signal for one photon state. You can already notice that in contrast to semi-classical formula, w2(r,r') is not equal to the product of w1(r) and w1(r'). The reason is that the first photo-detection destroys a photon, and thus changes the state of radiation before the second photon is detected. This is in contrast to classical physics, where we implicitly assume that the measurement can be gentle enough not to perturb the system that we observe. I am sure that some of you would like to ask what is the formula giving the expression of the double detection at two different times. I must admit that at this stage, I cannot answer the question. The Schrodinger formalism that we are using here is not convenient to answer the question. Because here, we have only one state vector evolving with time. In fact it is one of the reasons why I will teach you the Heisenberg formalism in a future lesson. With this formalism you will be able to calculate the rate of double detections at two different times t and t'. But we can learn a lot by applying this formulae to the case of the one photon state.
