Real one-photon sources are all based on a single quantum emitter put at time t0 in an excited state |e> of lifetime 1 over gamma. The phenomenon of spontaneous emission will be described in a future lesson. Today, all we need to know is that after a delay longer than gamma^ -1 we are sure that one photon has been emitted, more precisely, one has free radiation of the form |psi(t0)> = sum of one photon components. Note the I use here the Heisenberg formalism. The scheme that you see here is reminiscent of Niels Bohr formula, according to which the atom jumps from |e> to |g>, while a photon is emitted with energy hbar omega_0 = E_e- E_g. This corresponds to a frequency omega_0, but in fact, because of the finite lifetime of the upper level, there is a spread of frequencies of width gamma around omega_0. More precisely, the distribution of the squared modulus of the coefficients c_ell is a Lorentzian of width gamma at half maximum. Now, I am going to show you in detail how to calculate the photodetection signal w1 for such a spontaneous photon. It will be somewhat lengthy, because I will show you the details of methods that are used quite often in quantum optics. Remember that one of the goals of this course is to allow you to read advanced books or papers of quantum optics, so you need to know these methods. To learn them you must be equipped with a notepad and a pencil and do yourself the detailed calculation that I show you. Of course, if you do not want to do that effort, you can jump directly to the end of the sequence. But let me emphasize that to fully appreciate the wonders of quantum optics, you must be able to do such calculations. It is not a theorist who tells you that, remember, that I am an experimentalist. In free space, emission happens in all directions, and to describe the emitted radiation, one should use modes of the electromagnetic field well adapted to such geometry. It turns out however, that such modes are not simple to handle. In order to simplify the mathematics, I thus just consider a situation where the emitter is at the focus of a deep parabolic mirror of transverse dimensions large compared with the wavelength, so that the resulting one photon state can be safely expanded on a basis of plane waves with k vectors along z. The discrete values of k are determined by a periodic boundary condition along z with length L that can be arbitrarily large. We associate a duration T with the length L. The quantization volume is the product of the transverse surface S by L. As usual, the final results relevant to experimental quantities should not depend on L. For simplicity, we consider only one polarization perpendicular to the z-axis, for instance, along the x-axis. So the index ell is equivalent to the integer n_ell. A one photon wave packet resulting from spontaneous emission is described by a superposition of one photon states of several modes, as you already know. One can show that the coefficients c_ell have a value given by the expression here with capital K, a normalization constant. Starting from that form, which you will be able to demonstrate later, you can immediately derive interesting results, but you have to learn some methods of calculation. The square modulus of the coefficients c_ell yield the frequency distribution that is the spectrum of the radiation. As announced this distribution assumes a Lorentzian form with half width half maximum gamma over 2. To do complete calculations, we need to determine the value of coefficient K taken as a positive number, fulfilling the normalization condition. Let me show you this calculation as an example of a calculation that frightens many students in quantum optics. I mean, a calculation involving the density of states. I want to convince you that this kind of calculation is not really difficult if you make the effort to understand at each stage the meaning of the mathematics you use. Let us start. The first trick frequently used in such calculations consists of replacing the discrete sum over ell with an integral over omega_ell. To write the formula written here, you have just to ask yourself the question, how many values of n_ell are in the interval d omega_ell? Asking the question is answering it. It is d omega_ell times d n_ell over d omega_ell. Easy, isn't it? The quantity d n_ell over d omega_ell is nothing else than the famous, should I say infamous, density of modes. Its value is obtained straight from the periodic boundary condition. You can now finish the calculation. Remembering from your calculus course that remarkable integral, this is the final result, which allows you to obtain the value of K. If you are not comfortable with calculus, you may find this calculation somewhat difficult, and you can jump over it. Let me tell you, however, that I have seen many students unable to do such a calculation, not because they did not master the calculus technique, but because they did not understand the meaning of the math. If it is your case, if you know the math but are not sure to understand its meaning in a physics context, make the effort to follow me. You will be rewarded by becoming able to understand and do yourself many advanced calculations in quantum optics. Let us now proceed with a calculation of the rate of single photodetection at time t, if we have excited our emitter at time t0. This corresponds to measurements we can make using a photodetector at position z and a device to measure the delay t- t0. Each measurement will yield one value only. But if we repeat the experiment many times, we will be able to build the histogram showing the probability to have a count at each value of the delay. This is nothing else than the rate of photodetection as a function of t- t0. So let us take the expression of w1 in the Heisenberg form. To calculate E+ applied to |psi(t0)>, we take the expansion of E+ over the modes and similarly for the state |psi(t0)>. As in a previous calculation, the double sum condenses into a simple sum over ell, and the vacuum eventually comes in factor. We will now calculate this expression substituting c_ell with its expression. Before doing it, we first replace E1_ell, which depends on omega_ell by its value at omega_0 in the expression of E+. It is a fact that this quantity is almost constant over the bandwidth of a few gamma around omega_0 corresponding with non null values of c_ell. For typical spontaneous emission lines, gamma is not more than 10 to the 9th sec^-1, while omega_0 is of the order of 10 to the 15th sec^-1. Then we replace the discrete sum with an integral as you have just learned, that is, using the density of modes L over 2 pi c. A simple transformation allows us to obtain an expression proportional to a Fourier transform. To see it, just replace omega_ell- omega 0 with capital omega and t- t0- the retardation z over c with tau. This Fourier transform is known and allows us to evaluate the integral as shown here. The Heaviside function H of tau, also called the step function, is 0 for negative tau and 1 for positive tau. Using this expression, we obtain w1. It is proportional to the step function times a decaying exponential, with an inverse time constant gamma. Now, remember the expression of E1 squared, it is proportional to 1 over L. So that L disappears in the expression w1 as it should. Moreover, using the expression of the sensitivity s as function of the quantum efficiency eta given in a previous lesson, we obtain a simpler result. You can check that the total probability to detect a count after integration over time and transverse surface is equal to eta, the global quantum efficiency of detection. Let us interpret this result. At time t0 the emitter is excited, and at time t we register a photodetection event on a detector situated at a distance z from the emitter. A monitoring device using a time to digital converter allows one to obtain the delay between the excitation time t0 and the detection time t. Repeating this many times, the apparatus builds a histogram of the delays. At the limit of a large number n of such measurements, the histogram tends towards w1 of t at position z, of which we have calculated the expression. Actually what we measure is the integral of w1 over the detections surface, that is the probability of detection between t and t plus dt. If the detector is larger than the beam, the detection surface is a surface of the beam S, so that the probability of detection between t and t plus dt is simply w1 times S. And we obtain a simple result, dP over dt is a normalized decaying exponential times a quantum efficiency, eta. The rising step of the exponential happens at t equals t0 + z over c, that is, the excitation time plus a delay due to the propagation from the emitter to the detector. In fact, one must also take into account the time of propagation in the electric cables. This is a real signal, which you have already seen. In the late 1970s, I observed for the first time such a signal building one point after another one until one progressively distinguishes the rising edge and the decaying exponential. It is an experience I still remember. One has a feeling to directly observe the elementary quantum world. Do not miss such an experience, if you have a chance.
