We consider a situation with one photon state in mode one, and zero photon in mode two. We put the detector in the output 4 and want to calculate the rate of single detections w1 proportional to the squared modulus of E4 plus applied to the radiation state in the output space. Note that here, again, the complex exponentials of position and time have disappeared when taking the square modulus, as well as the polarization unit vector, epsilon. So that we can use a scalar operator, E4 plus at r equals 0 and t equals 0. Now, we apply the beam splitter recipe. Try to remember it. You will use it quite often. Here, you must express E4 plus as a function of the field operators in the input space E1 plus and E2 plus, and replace the output radiation state |psi_out> by its expression in the input state |psi_12>. Then write the E plus fields as functions of the annihilation operators a. You have then to evaluate the term a1 applied to one photon in mode 1 and 0 photon in mode 2. In fact, it implicitly means that you must use the operator a1 times the identity operator in mode 2 so that you can apply a1 to the mode 1 and leave the term 0 photon in mode 2 unchanged. You then obtain the 0 photon state vacuum, both in mode 1 and mode 2. Applying the same reasoning for a2 applied to 1 photon in mode 1 and 0 photon in mode 2, you leave the term 1 photon in mode 1 unchanged and apply a2 to the state 0 photon in mode 2. This latter operation yields 0, the number 0. So the term involving E2 plus will disappear from the result. Finally, the rate of detection in the output 4 is proportional to capital T. So square of the amplitude transmission coefficient. Be careful not to confuse the lowercase t with time. Note also that the intensity transmission coefficient capital T must not be confused with the duration of the wave packet used earlier in this lesson. There are not enough letters in our alphabet. Take a detector larger than the transverse area of the beam, and integrate the signal over the whole duration of the wave packet. You get the probability P4 of the detection per one photon wave packet. P4 equals eta the quantum efficiency times capital T, the intensity transmission coefficient of the beam splitter. Multiplying by the number N_wp of wave packets sent during an experiment, you get the number of counts at detector 4, N4 equals eta T N_wp. A similar calculation yields the rate of single detections w1 for detector D3 in the output channel 3. I suggest that you do this calculation yourself. You only have to repeat the calculation just done for the output 4. Rewind and watch it again if you need it. I will not give as many details in future calculations. This is the result you should have obtained, as rate of single photo detection in output channel three, and as probability P3 of a detection per one photon wave packet. From this you immediately get a number N3 of counts if N_wp wave packets have been sent into input one of the beam splitter. We now calculate the coincidence photo-detection signal in the two output channels of the beam splitter. You may expect to find the specifically quantum behavior. Remember that specific quantum results are found on double detection signals. We still have a 1 photon wave packet entering the input 1 of the beam-splitter, 0 photon in mode 2, and 2 detectors in the outputs 3 and 4. But, we now add a coincidence circuit in order to be able to detect the probability of a double detection, and thus the total number of double detections for N_wp wave packets sent to the beam splitter. Can you guess what is expected? Write your guess on a notepad, and then embark with me in the calculation, using the tools we have learned. We first write the expression of the rate of double detection in 3 and 4. With two E plus operators applied to the radiation state. Replacing the field operators by their expression in the input space and the radiation state by its expression in the input space, we obtain the rate of double detection in 3 and 4 expressed as a function of quantities of the input space. As in the previous calculation, all terms containing E2+ cancel when applied to 0 photon in mode 2, and we obtain an expression where E1+ is applied twice to the input state of radiation. The term 0 photon in mode 2 is not affected and gives 1 when we take its square modulus. We are just left with E1+ applied twice to 1 photon in mode 1. Once again we obtain 0. A single photon cannot be detected twice. No need then to add that when integrating over the whole wave packet. The probability Pc of double detection is null. Had you guessed that result? In order to fully appreciate it, we now compare it to the result of the classical calculation for a classical wave packet. We model the classical wave packet in the input one by a constant amplitude in the volume V1 defined as in the beginning of this lesson. Out of the wave packet the amplitude is null. In section one of this lesson you have seen the formula giving the rates of photo detection in the semi-classical model. So signal detection rate in channel 3 is proportional to the squared modulus of the complex amplitude E3 plus. Taking into account the reflection coefficient, we obtain the rate of photo-detection in output three as a function of the entering field amplitude. After integration over the volume of the wave packet and multiplication by the number of N_wp of wave packets. We obtain the number N3 of single detections in the reflected output, 3. A similar calculation yields the number N4 of single detection in the transmitted output, 4. The reflection coefficient, capital R, is replaced by the transmission coefficient, capital T. Let us now calculate the number of double detections in channels three and four. In the semi-classical model, the double detection rate is proportional to the products of the singles detection rates. So the number of coincidences per wave packet is obtained after integrating twice on the wave packet volume V1 and multiplying by the number N_wp of wave packets yields a total number of coincidences Nc. Using the expression of N3 and N4, we obtain a remarkable formula characteristic of classical wave packets of constant amplitude. For a coincidence window, covering the whole wave packet, the number of double detections is equal to the product of the number of single detections divided by the number of wave packets. The comparison with the single-photon case is striking. While the number of single detections assume similar forms, the number of coincidence is null for a single photon. At this point, you may wonder whether this discussion is purely academic or if it corresponds to real experiments. In fact, such an experiment was performed for the first time in 1985 to characterize the first source of single photon wave packets. I will describe that experiment later. A quite important point about that scheme is that it is nowadays a standard scheme to characterize single photon sources, an important resource in modern quantum optics and quantum information. I will come back to this point later. For the time being, let us continue to explore the intriguing properties of a one photon wave packet as predicted by the quantum optics formalism.
