[MUSIC] Let us then consider a freely propagating beam of constant transverse area S, classically described as a plane wave with a constant electric field in the section of the beam. The amplitude of this classical electric field is a well-defined quantity with an intrinsic physical meaning. We can define several other intrinsic classical quantities, the power per unit surface Pi, also called the irradiance. It is equal to the modulus of the Poynting vector of the classical wave. The energy per unit volume, that is to say the density of energy D, equal to the modulus of the pointing Poynting divided by the velocity of light. The total power Phi flowing across any section of the beam equal to the Poynting vector modulus times S. It is an intrinsic quantity which can be measured, for instance, with a power meter covering the whole transverse section of the beam. It is sometimes called, in a loose way, the beam intensity, but there is no general agreement on what should be called intensity. In fact, there is no ambiguity provided that the unit is given. If you read that the beam intensity is 5 watts, you know that it refers to the laser power. We can express that power in unit of ħω_ℓ and the resulting quantity is called the photon flux Phi_phot. In spite of its name, it has no quantum character at this level. Any of these intrinsic quantities fully characterizes the radiation in the classical beam, provided that its section, capital S, is known. We look now for corresponding intrinsic quantities in the quantum formalism. We want to describe the radiation of such a beam in the quantum optics formalism as a quasi-classical state with parameter alpha_ℓ. How should we choose alpha_ℓ? To use a quantum formalism, we need a finite volume of quantization. We thus define an arbitrary volume, which is a portion of beam with an arbitrary length L = cT, with c the velocity of light and capital T an arbitrary time. The volume of quantization is then S c T. Let us first consider the quantum observable N hat associated with the number of photons in that volume. It is not an intrinsic quantity, but the quantum flux of photon, that is to say N hat divided by the time T associated with the length of the arbitrary quantization volume, is an intrinsic quantity. For the quasi-classical state alpha_ℓ, the average number of photons is the squared modulus of alpha_ℓ, and the average photon flux is thus that quantity divided by capital T. We want it to be equal to its classical equivalent, which is an intrinsic quantity. We thus conclude that the square modulus of alpha divided by T is an intrinsic quantity. You may be confused since T is arbitrary. Can you find how to get out of this paradox? The answer is that for a freely propagating beam, alpha_ℓ is also arbitrary, but T and alpha_ℓ are related in an intrinsic manner. With that choice of alpha, we can calculate the average values of the various quantum observables that should be intrinsic. Let us take the example of the electric field and write its average. Does it look independent of T? Take a while to think about it. Did you find the reason why this expression has an intrinsic value? You must remember the expression of the one photon amplitude E_1, which depends on the volume of quantization, it is inversely proportional to root of T. So the arbitrary quantity capital T will disappear in the value of the average electric field, which depends only on power per unit surface Phi over S, as it should. We conclude that in order to describe a freely propagating quasi-classical state in the quantum formalism, we must use a parameter alpha_ℓ proportional to the root of an arbitrary time T. The value of the squared modulus of alpha divided by T is chosen to agree with the average power of the beam expressed in number of photons per second, an intrinsic quantity that can be measured. The shot noise is the fluctuation that one observes in photo-detection measurements bearing on a perfectly stable beam, for instance, a beam emitted by a single-mode laser well above threshold. The theory of lasers shows that such a beam has an almost perfectly stable average intensity Phi. Observations show, however, that the photo-current has fluctuations. Similarly, if we rather decide to use a photon counting technique and measure the number of photo-detections in a given interval of time, we also find fluctuations. These fluctuations are responsible for the limited accuracy in measurements. For instance, the absorption of a sample obtained by looking at the ratio of the photo detection signals without and with the sample. This is the basis of absorption spectroscopic measurements. It is thus crucial to answer the question: what is the magnitude of the photo-detection fluctuations? A little thinking will show you that the question is too vague. If we think of photon counting, that is to say the measurement of the number of pulses detected with the photo-detector in the photon counting mode, we must indicate the duration capital T of the measurement. If we think of measuring the photo-current, we must take into account the finite response time of the photo-detector and of the electronic circuit used to read the photo-current. These quantities are averaged over the time T. So the good question is, what are the fluctuations of the photo-current observed with the circuit that has an integration time capital T? Note that for an ideal photo-detector, which produces one pulse for each photon, the photo-current observed in an integration time capital T is equal to the elementary charge q_e of the electron times the number of counts in T, all divided by T. We can now evaluate the fluctuation of the detection signals for a beam of constant average power Phi. We describe that beam as a quasi-classical quantum state with a parameter alpha_ℓ, whose squared modulus is proportional to Phi times the duration capital T associated with the quantization volume, as we have just seen. Here, capital T is taken equal to the time of integration of the signal in the detection circuit. In order to remember that alpha depends on capital T, we note that state alpha_ℓ,T. Let us first consider the case of a measurement in the photon counting regime with a perfect photo-detector. The calculation is straightforward if you use the results we have obtained in the previous sections, and I encourage you to do it yourself. This is how it works. The number of counts, N of T during the time T is nothing else than the result of the measurement of the quantum observable number of photons associated with the quantization volume of length: cT. Representing the radiation state by the quasi-classical state alpha_ℓ,T, one obtains the average value of the number of photons in that state, and thus the number of photo-counts in capital T. As expected, it scales as capital T. Similarly, the variance of the number of counts in T is equal to the variance of the number of photons in a volume of length cT. Using the results of section four, we obtain the variance delta N squared in that state. The standard deviation thus scales as root T. A measurement of the number of photons in the time capital T has an accuracy N over delta N. It is proportional to the square root of the measurement duration capital T. This is a well known result. The accuracy increases as the root of the duration of the measurement. It also increases with the power of the beam, which makes sense since it means more photons per unit time. [MUSIC]