Let me describe first a one photon interference experiment. The one photon source was the heralded one photon source that I described earlier. Two lasers exactly tuned on a two photon resonance excited calcium atoms in an atomic beam and a gating scheme based on the detection of the first photon allowed us to isolate one photon wave packets. The interferometer with the size of about 50 x 40 centimeters had mirrors and beam splitters flat to 1/100th of a micrometer. This is the reason why they are so thick to avoid any deformation. Sophisticated mounts allowed us to change the path difference keeping the mirrors and beams splitter apart to each other within a fraction of a wavelength over the whole surface. The path difference was controlled within a small fraction of a wavelength by the electric tension applied to the piezo-transducer. We could keep it constant for a chosen duration and for each path difference we would register the number of counts at D5 and D6. The plots here show what we observed when we stayed only 0.1 second at each value of delta L. The mean total number of detected counts at each position is about 1. So there is a broad spread of the results with many occurrences of 0 count. The signal to noise ratio is not good enough to decide unambiguously whether there is an interference pattern or not. The second row of panels shows the results for a waiting time of one second at each value of delta L. You can now observe the clear modulation with a maximum value of about 20 counts. Assuming a Gaussian random variable the theoretical standard deviation is root of 20, that is to say about 4.5. You can check that the statistical fluctuation around the maxima is of the order of two times that standard deviation in agreement with the Gaussian model. The third row, where the time spent is ten times larger shows beautiful modulated signals with maxima of about 200 counts. The theoretical standard deviation is 14 and you can check that the fluctuations around the maxima are of the order of two times that standard deviation. The visibility of the fringes is close to one, actually slightly more than 98%. A remarkable value for a wide beam interferometer. In the previous lesson, I mention sources of one photon on demand. Here is an example where a single molecule is addressed by a confocal microscope allowing one to excite it and capture the re-emitted photon. After expansion, the beam is sent onto a double prism which deflects half the beam towards D1 and half the beam towards D2. For a real one photon source detection happens either at D1 or at D2. But there is no joint detection. It is obviously possible to define the alpha parameter as in the beam splitter case and the result of the measurement is α = 0.13. A clear evidence of one Photon wave packets. This is a source that I described at the end of section three and the alpha parameter is the same with the double prism or with a beam splitter. In fact this double prism is a device that was invented by Fresnel to observe interference fringes in the overlap of the two beams. At the time of Fresnel, the fringes were observed with the naked eye behind an eye piece. There was no electric bulb and the light was sunlight concentrated onto a slit and collimated. In this one photon interference experiment, one uses a CCD camera that can detect a single photon and show the place of detection. It is placed in the overlap of the two beams receiving the one photon wave packets from the one photon source on demand just described. The CCD camera allows one to observe single photons arriving at various positions. You can find the record of the signal as a document joined to this lesson. Panel (a) shows the accumulated data after 20 seconds. The 272 impacts of the photons seems to be randomly spread over the surface of the CCD chip. But after 200 seconds the 2240 photon-counts are clearly concentrated on vertical bands which correspond to bright fringes with the period of 0.12 millimeter. After 2000 seconds, we have registered almost 20000 counts and the interference pattern is yet more impressive. Binning all the points of the same vertical line we can plot the profile shown here in red dots. The solid line is the result of the calculation describing the experiment with only one fitting parameter the vertical scale. But how can we calculate what happens with one photon wave packets in that specific interferometer? Do we need to embark again in a lengthy quantum optics calculation as the one we have just seen for the Mach–Zehnder interferometer? The answer is fortunately no. I am going to teach you now a very important result of quantum optics for a one photon wave packet. The probability of a single detection in any optical device can be calculated with the classical model of light. The rate of photodetection at each point is proportional to the classical light intensity, that is to say to what would have been calculated in the semi-classical model. This is how we have obtained the solid line of panel (d) using the classical model of propagation of a beam with a transverse intensity profile corresponding to what is collected by the microscope objective. The only fitting parameter is a multiplying factor to adjust the vertical scale. Realizing that single photodetection signals for a one photon wave packet can be calculated classically was for me an important step in my understanding of quantum optics. And I do not resist to share this result with you and sketch the derivation. Let us consider the most general form of the one photon wave packet, a superposition of one photon states in many different modes. The rate of single photon detections around point r and time t is proportional to the square modulus of Ê⁽⁺⁾ applied to the state. I use here the Heisenberg formalism since the field operator depends on time in contrast to the state vector that keeps a constant value. Let us write the result of the application of the operator Ê⁽⁺⁾ upon the one photon state vector. Remember that the annihilation operator â_l applied to one photon state |1_l> returns a vacuum. So the only non-null terms that we get are all proportional to the vacuum state, which can be factored. The term in factor is a superposition of complex classical components which represent a complex classical field that we call Ecₗₐₛₛ⁽⁺⁾⁽¹⁾. To understand this notation, you can calculate the classical electromagnetic energy content in the wave packet by integrating over the volume of quantization L³ which enters into the expression of E⁽¹⁾l. The result is the average one photon energy, that is to say the energy hbar omega_l averaged over the square modulus of the coefficients c_l. We can thus call Ecₗₐₛₛ⁽⁺⁾⁽¹⁾ a classical one photon wave packet, although I must admit it sounds an oxymoron. Finally, the rate of single photon detections assumes exactly the form that would be obtained in the semi-classical model for the classical field that we have just defined. I already told you that it is possible with a grain of salt to consider such a classical field as the wave function associated with the one photon state. This is consistent with the fact that the square modulus of that function yields a density of probability to find the photon at time t and position r. From this point of view, the calculation of a one photon interference pattern is strictly equivalent to the calculation of the interference pattern of a particle such as an electron, a neutron or even an atom or a molecule. The property that we have just seen can be related to the fact that it is possible to associate the creation operator a^†_wp with any classical wave packet. The procedure is analogous with what we did when we quantized the elementary modes of the radiation field. We did it for plain traveling waves and in order to obtain the dimensionless creation and annihilation operators, it was necessary to normalize these modes. This led us to introduce the one photon amplitude Eℓ⁽¹⁾ defined over a fictitious box of volume L³. We want to normalize similarly the classical wave packet written here in order to obtain a classical one photon wave packet. In order to do it, we write each coefficient γ_ℓ as a constant μ times the one photon amplitude Eℓ⁽¹⁾ used to quantize the field times c_ℓ and we imposed to the c_ℓ coefficients a normalization condition. These defines each constant c_ℓ and the constant μ. You can check that the energy in the wave packet is the average one photon energy multiplied by the square modulus of μ. Dividing the classical field by μ, we then obtain a classical field Ecₗₐₛₛ⁽⁺⁾⁽¹⁾ of the form introduced on the previous slide. We can now use the coefficients c_ℓ to define an operator a^†_wp. When applied to the vacuum that creation operator returns a one photon wave packet formed with a c_ℓ coefficients. The classical field associated with that one photon wave packet is obviously the field Ecₗₐₛₛ⁽⁺⁾⁽¹⁾ just introduced. The operator a^†_wp can be considered a creation operator, if you take its hermitian conjugate a_wp you can easily show that a and a^† obey the canonical commutation relation. Applying that operator to the vacuum, we obtain a one photon wave packet. This is similar to the method which allowed us to obtain the n=1 number state of the mode ℓ using the creation operator a^†_ℓ. In the first lesson, I told you that it is possible to quantize the electromagnetic field using many different kinds of modes. In a sense, any classical wave packet can be considered a mode.