With the current detectors, it is possible to measure the fields in the radio frequency range up to hundreds of gigahertz. But there's no detector fast enough to follow the fields at frequencies corresponding to visible light, typically several 10^14 hertz. At such frequencies, one can detect radiation with bolometers, which absorb radiation and measure the resulting increase of temperature. There is, however, a much more useful method to detect radiation in the visible range and around. It is based on the photoelectric effect. I presented it in the lesson on single mode radiation and I will show you the formulae for the multimode case. The formulae are pretty simple, but you must know that their demonstration which was published first by Roy Glauber is quite involved, and I will ask you to simply accept the result. You remember that the photoelectric effect is the fact that the electric field of radiation can extract an electron from an atom. That is to say, promoting it from a bound state to a state of the continuum where the electron can escape. In 1905 Einstein interpreted this effect as the absorption of a photon which communicates its energy to the bound electron. The escape of the electron is thus possible only if the photon energy, hbar omega, is larger than the threshold energy W_T which is the difference between the energy of the bottom of the continuum and the energy of the bound electron. This threshold energy is typically of the order of a few electron volts, so the photo-electric effect exists only for light in the visible or ultraviolet range. Each individual electron released by absorption of a photon can be multiplied and detected. I already described a photomultiplier tube which is an evacuated tube with a photocathode and several electrodes called dynodes. An electron extracted from the photocathode by the incident radiation is accelerated towards a first dynode, gaining an energy typically of the order of 100 electronvolts. The collision on the first dynode then extracts several electrons, typically three, which are in turn accelerated toward the second dynode. After ten stages of amplification, one gets a pulse of charges large enough to be detected by standard electronics, so that each initial individual electron is detected. Modern electronics permits to determine the time of detection with accuracy of the order of one nanosecond or better, in some cases. One can then measure the statistical properties of the photo-detection events, and compare them to theoretical predictions for various kinds of radiation. We give now the expressions of the photo-detection signals for a multimode quantum radiation. Let us first express the average rate, w1 of single photo-detections around the point r. It is such that the probability to have a single photo-detection at a detector of surface dS placed at r during the interval of time dt around t, is: dP equals w1 times dS times dt. In order to derive w1, one must use a formalism describing the interaction between the quantized radiation and matter. I will teach you this formalism in a future lesson but at this stage I ask you to accept the result which is a generalization of what you have learned in the lesson on single mode quantized radiation. w1 is proportional to the quantum average of E minus times E plus, taken in the state psi, describing the radiation. E plus and E minus involve the infinite number of modes over which we have decomposed radiation. Note that although E minus and E plus are not hermitian, the product E minus times E plus is hermitian, so its average is real. In fact, it is a positive number since we can also write it as the norm of the vector E plus applied to psi. The coefficient s is the sensitivity of the detector. As it is defined here, it is not exactly what you find in data sheets, which mention sensitivities in units of amps per watt. In fact, both quantities are proportional within universal constants, since w1 is a number of electrons per unit time and surface and E squared is proportional to the Poynting vector that is watts per unit surface or Joule per unit surface and per unit time. In the expression of w1, I have assumed that the sensitivity is constant with frequency which is a good assumption if the frequency spectrum of the radiation is not too broad. If it is not the case one must use a sensitivity s of omega_ell, depending on the frequency omega_ell, and write the factor square root of s of omega_ell in each ell term of the expression of w1. In this formula, there is another quantity varying with the frequency. It is the one photon amplitude, E1_ell. I will introduce now, a quantity that takes into account all these variations and which has a clear physical meaning. It is the quantum efficiency of the detector at frequency omega_ell. Let us write again, the expression of w1 with a frequency dependent sensitivity. A perfect photo detector should detect a single photon with a probability of 1, and we have seen in the single mode case that a perfect detector at frequency omega_ell must have a sensitivity s_perfect equals 2 epsilon_0 c divided by hbar omega_ell. Replacing E1_ell by its value, we find that for an ideal detector, the frequency dependencies cancel each other, and w1_perfect, assumes a simple form. You can check that if psi describes a one photon state in a single mode with a volume of quantization L squared times cT, the integral of w1 over the one photon wave packet leads to a probability of 1. If now, we have a non-ideal detector, the ratio between its sensitivity, and the sensitivity of a perfect detector is the quantum efficiency eta. This quantity has a clear meaning: It is the probability to detect one photon at frequency omega_ell when the detection is integrated over the whole one-photon wave packet. We can use it to express w1 for an imperfect detector. In most detectors the quantum efficiency varies slowly with the frequency. You can see here, the quantum efficiency of a camera installed on the Hubble Space Telescope over a wavelength range, extending from 0.1 to 1 micrometer from far ultraviolet to infrared. Variations are slow in most of the range. Provided that we have radiation with a frequency band not too wide, we can just take eta_ell as a constant and w1 assumes a form without any frequency-dependent coefficient inside the sum. Such a form, is often used in quantum optics calculations. I already told you, that major progress in quantum optics came from the possibility to obtain signals of joint, or coincidence detections, on two photo detectors. We are thus interested in the probability to have a joint detection on D1 during the interval dt_1 around t, and on D2 during the the interval dt_2 around t. This probability can be measured with a coincidence circuit. It is proportional to the rate of joint photo-detections w2. The expression of w2 as found by Glauber, is the quantum average over the radiation state psi(t) of E minus(r_1), times E minus(r_2), times E plus (r_2) times E plus (r_1). Note immediately that w2 is not equal to the product of the corresponding w1, which means that the probability of a joint detection is not equal to the product of the probabilities of single detections. This is in contrast to the semi-classical description where the equality holds. This is a crucial feature of quantum optics. It can be related to the fact that the first detection entails a change in the state of radiation, the so-called collapse of the wave function, when a measurement happens. Mathematically, it is related to the normal ordering of the operators, with the a operators on the right-hand side and the a_dagger on the left-hand side. Since we know that vacuum may have surprising properties, it is legitimate to ask about photo-detection signals in the vacuum. You can check that photo-detection signals are null in the vacuum, because the operators are ranked in the normal order, with the a operators on the right-hand side and the a_dagger on the left-hand side. Remember that E plus involves a, and E minus involves a_dagger. We can conclude that vacuum fluctuations cannot be detected directly. We know that they exist only by their indirect manifestations.