Do you remember the scheme I described in a previous lesson about a one photon wave packet on a beam splitter? I insisted that a single photon can be detected only once in one of the two output channels, but not in the two channels simultaneously. These may seem obvious when said with these words, but it is a behavior emblematic of quantum optics absolutely impossible to understand in the framework of classical electromagnetism. When I described that scheme, I warned you that I was using a toy model, an oversimplified situation that does not exist in the real world. Now you have learned enough of quantum optics to be able to do the full calculations associated with real one photon sources. So I invite you to embark with me into such a calculation. The scheme shown here, is based on a heralded one photon source, with the gating scheme allowing us to isolate one photon wave-packets. The one-photon wave packet is sent into the input 1 of a beam splitter and the input 2 is void. The radiation in input 2 is vacuum. Two photo-detectors, D3 and D4 are in output channels 3 and 4. These detectors are activated only during gates opened by the detection of a heralding signal. Electronic devices allow us to monitor the number of single detections, N3 and N4, for an experiment repeated many times. We also detect and count coincidences, defined as the fact that the two detectors are activated during the same gate. The total number of gates NH is also counted. It is equal to the number of counted heralding photons. We want to relate the number of counts obtained at each counter to the calculated values of the single and double photo detection signals. Let us then recall the values of the rates of single and coincidence detections. The calculations are almost the same as the ones we did in the previous lesson. Do not hesitate to go back to these calculations. Radiation in the input channel 1, is a superposition of a one photon multimode state plus some vacuum component. The global state in the input channels 1 and 2 of the beam splitter is tensor product of the state of input 1 by vacuum of input 2. Here, be careful about the notations. The indices 1 and 2, do not refer to a particular mode, but to the inputs 1 and 2 of the beam splitter. I could have use a different notation, but experience shows that it is a useless complication, and that the risk of confusion is weak when one uses the notation here, which by the way is the one you will find in many papers and books of quantum optics. You remember that the effect of the beam splitter is conveniently described by a unitary transformation of the field operators, more precisely, relations identical to the ones for classical field amplitudes. This relations written on the beam splitter at point O, are the same in the Schrodinger formalism and in the Heisenberg formalism. Here, we choose the Heisenberg formalism. The field operators are thus depending on time while the radiation state keeps its initial value. Note that I have written the amplitude reflection and transmission coefficients as the square roots of the intensity reflection and transmission coefficients, capital R and capital T. The unitarity of the transformation is guaranteed by the minus sign in the second line. The calculation of the rate of single detections in any output channel is analogous to calculations done in the previous lesson, except that the electric field must be expressed as a function of the input electric fields before the beam splitter. Consider for instance the rate of single detection at D3. You will obtain it by expressing E3+ at r_3 as a function of E1+ and E2+ on the beam splitter. Note that in the Heisenberg formalism the propagation from O to r_3 can be described in a very simple way, just introducing the propagation time, r_3/c. This retardation is the same for all frequencies provided that the medium is not dispersive. When the operator, E3+ is applied to the input state |psi_12>, the term E2+ plays no role, since it involves annihilation operators applied to the vacuum. The result is thus the same as we obtained without a beam splitter with a multiplying factor capital R, the intensity reflection coefficient. In this formula, tau is the delay from the time of emission of the one photon wave packet corrected for the propagation times. The factor modulus of gamma squared corresponds to the limited collection efficiency as seen in previous lesson. The total probability P_3 to detect a count during the gate is obtained by integration over the gate. If the gate is wide enough to cover the whole decaying exponential, the integral can be approximated by taking tau from 0 to infinity. The result has then a simple form which we can write as epsilon 3, the total quantum efficiency, including the reflection coefficient R. A similar reasoning yields the probability of a detection per gate at detector 4 under the assumption that the gate covers completely the pulse. This result can be written as epsilon 4, the overall quantum efficiency, taking into account the intensity transmission coefficient capital T. Let us now evaluate the probability of a coincidence that is to say a joint detection at D3 and D4 during the same gate. We must first calculate the rate w_2 of double detections at D3 and D4. I suggest you stop the video, and try to start the calculation yourself. Even if you don't finish that calculation, make the effort to write the initial equations, and you will understand much better when you resume the video. Do not hesitate to stop again the video at each step of the calculation. To start the calculation you must first express a field operator E3+ and E4+ as a function of the input fields E1+ and E2+, then apply their product to the state expressed in the input space and take the squared modulus. I've not indicated explicitly the time dependence and the propagation factors, because they will not play any role in the final result, as you can check yourself. Indeed, when you develop the product of the two parenthesis, you get three terms with E2+ which yield 0 since E2+ involves only destruction operators acting in the input 2, where there is vacuum only. The fourth term involves E1+ applied twice to the one photon state. As you know this yields 0, because you apply twice a destruction operator to a one photon state. As expected the probability of a coincidence is null for a one photon wave packet. I hope most of you have been able to do that calculation and realize how simple it is although it describes a phenomenon emblematic of modern quantum optics. In a real experiment, the number of coincidences N_C, is not strictly null. The reason is that the detectors may fire for reasons other than a detection from the single photon wave packet associated with the gate. We call such a signal a supplementary count. A first kind of supplementary counts is dark counts, which are electric pulses produced even in the absence of light. They are mostly due to thermal activation of the electrons of the photo cathodes. These thermal dark counts can be reduced to a negligible level by cooling the detectors, and we will ignore them in this calculation. Another kind of supplementary counts specific of the heralding scheme is due to the radiative cascades other than the one responsible for the gate, happening by chance during the gate. We will take this supplementary photons into account in our analysis of the real experiment, and show that if their rate remains small enough they do not hide the anticorrelation phenomenon. But how can we characterize quantitatively anticorrelation when the probability of a joint detection is not strictly null? We can do it with the correlation parameter alpha which is a measurable quantity that we are going to define now. Let us first note that the probabilities of single detections per gate P3 and P4 are obtained experimentally as the ratios of the numbers of single counts, N3 and N4 divided by the number of gates NH, provided that the numbers of counts are large enough so that the law of large numbers does apply. Similarly, the measured probability of a coincidence per gate is NC divided by NH. If there were no correlation between detection at D3 and D4, the probability of coincidence would be equal to the product of the probabilities of single detection, as it is the rule for uncorrelated random variables, We can thus define anticorrelation as the situation when the probability of coincidence per gate is less than the product of single detections probabilities. It is thus convenient to define the parameter alpha as the ratio between PC and the product P3 times P4. The experimental value of alpha can be directly expressed as a function of the measured number of counts as NC times NH divided by N3 times N4. Anticorrelation will be characterized by a measured value of alpha less than 1. A value of alpha less than 1 is a fully quantum phenomenon which cannot be understood in the framework of the semi-classical model of light, as I will show you now.
