I have written here the expression of the electric field observable that we have obtained in the first lesson of Quantum Optics 1. It does not evolve with time since we use the Schrodinger point of view. The constant E^(1)_ℓ, is the so-called one-photon electric field amplitude. That is to say, the value of the classical electric field amplitude of a classical wave whose energy is ħω_ℓ in the volume of quantization. Let us now calculate the average value of the quantum electric field observable for a quasi-classical state. We first calculate the averages of a and a^†. The calculation is straightforward. Since the state alpha_ℓ is an eigenstate of a_ℓ, and the state is normalized, we obtain the complex number alpha_ℓ as the average of a_ℓ. Taking the hermitian conjugate, we obtain the complex conjugate of the number alpha_ℓ as the average value of a^†. So the average value of the electric field is real as it should. In fact, it has a form that looks familiar. Don't you think? It will be yet more striking if we take into account the time evolution of the quasi-classical state. So let us calculate the average of the electric field at position r and time t, using the time-dependent expression of the quasi-classical state. The calculation is straightforward and we obtain a form identical to the form of a classical field in the mode ℓ with a complex amplitude E^(+) classical of r and t. More precisely, it has the form of a classical monochromatic plane wave propagating with the k-vector k_ℓ, and polarized along epsilon_ℓ. Do not hesitate to go back to lesson 1 of Quantum Optics 1 if you don't feel comfortable with such a classical form. You understand now the word classical in the name quasi-classical state. But why quasi? Wait for what is coming now. The result we have obtained is remarkable. For a quasi-classical state of a mode ℓ, the spatial temporal dependence of the average value of the field is exactly the same as for a classical field. This is to be contrasted with the case of a number state of the same mode ℓ, which does not evolve, neither in time nor in space. Do you remember that property of number states, that is states in a well-defined number of photons? A little bit of thinking should allow you to find that statement obvious. Let us come back to the quasi-classical state, of which we know the average value. Now, you know that in quantum mechanics, the average value is not the end of the story. To know more, you need to calculate the dispersion of the result of many measurements of the electric field, assuming that we can repeat the experiment many times. Remember indeed what is the meaning of a quantum average? Each measurement yields a result, only one, and the average value is obtained after repeating many times the whole operation, that is preparing again the system in the same state and doing the measurement again. So for each value of position and time, you have a series of results. From this series, you can obtain the mean value, which is the average, but also the dispersion around the mean, which is characterized by the standard deviation. The square of the standard deviation of any random variable, classical or quantum, is the variance. The variance is the average of the square of the difference to the mean. But you should also know this very useful relation which yields a convenient way to calculate the variance. Can you demonstrate that formula? You should know. Let us use this formula to calculate the variance of the electric field in the state alpha_ℓ at point r, and time t. We need first to calculate the average of E squared in the quasi-classical state at time t. I've recalled here the expression of the observable E. Let us first do the calculation at t equals zero and r equals zero. We just have to evaluate the average of a_ℓ minus a^†_ℓ squared, in the state alpha_ℓ. It looks quite simple, but one must anyway be a little careful. Why? Try to do the calculation yourself. You should have realized that you cannot use the usual expansion of the square of a difference, because a and a^† do not commute. So this is what you must write. Now, I'm going to share with you a trick that we learned from Roy Glauber. To take any quantum average of an expression of a and a^†, it is a very good idea to reorganize the operators in the normal order with a_ℓ on the right and a^†_ℓ on the left. Here, we have one term, a a^†, which is not in the normal order. We have to fix that problem. Do you have a suggestion to do it? Think about it. To obtain an expression in the normal order with a on the right-hand side, the solution is simple, just reverse the order using the commutator of a and a^†. You obtain then the whole expression in normal order. Now you get the benefit of the normal ordering. The operator a_ℓ acting on the right yields the complex number alpha_ℓ, while a^†_ℓ acting on the left yields the complex conjugate alpha_ℓ^*. This is a very simple calculation, isn't it? Since you have ordinary numbers, you can use the usual formula to put the result in a form that will immediately lead to that expression of the average of the squared electric field operator. Subtracting the square of the average, we obtain that remarkable result, the variance is equal to E^(1)_ℓ to the square. Now, look at the whole calculation. How has it changed if you want to repeat it at anytime t and position r? In fact, the only change is complex exponential factors and their complex conjugates whose product is one, so the final result is exactly the same. The standard deviation at any time and any position is thus nothing else than E^(1), the one-photon amplitude. Do you remember what is E^(1), the so-called one-photon amplitude? It is the amplitude of a classical field whose total energy in the volume of quantization is ħω_ℓ, the energy of one photon. I have plotted here the results of the calculation we have made, for alpha with a modulus equal to five. The red line is the mean value of the field as a function of time at a given point. The two green lines are obtained by adding plus or minus the standard deviation of the field, which is constant over time. You may have the impression that it is not constant, but if you look carefully at the figure, you will see that indeed the height of the band between the green lines is constant. If we could measure the value of the electric field at a precise time and position, and repeat many times the measurements, most of the results would fall in the band between the green lines. As soon as the width of the band is small compared with the amplitude of the sinusoid, any measurement fall so close to the sinusoid that the field behaves as a classical field. It is almost a classical field, this is why it is called a quasi-classical field. The ratio of the dispersion to the amplitude of the sinusoid is about one over the modulus of alpha_ℓ. So the larger the modulus of alpha_ℓ, the more classical the quasi-classical state. It corresponds to a situation where the number of photons in the volume of quantization is large, as we will see soon. You have certainly already met the so-called phasor representation, which applies to any oscillating classical quantity. It is a graphical representation in the complex plane, where one has a vector rotating at the angular frequency ω, represented by a complex number with argument plus or minus ωt + phi. The projection on the real or imaginary axis yields a quantity varying as the cosine or the sine of plus or minus ωt + phi. On the figure here, we have used this representation of the average value of the electric field for a quasi-classical state. It is twice the projection of the red phasor on the imaginary axis. This representation also allows us to represent the dispersion of the measurement results, using a disk of radius E1 centered on the phasor of the average electric field. The projection of the disk onto the electric field axis yields the dispersion of the field at each moment. You may find all that trivial, but this kind of representation will turn out to be very useful when we study squeezed light, so it cannot hurt to get familiar with it in a simple situation. Let us see here another way to look at the dispersion of the electric field measurements. When we quantize the field, we introduced quadrature observables which are canonically conjugate observables, so their commutator equals iħ. They are analogous to the position and the momentum of the mechanical harmonic oscillator. This is why one uses the notations Q and P. Remember that the electric field can be expressed as a function of these observables. We will see in a future lesson that quadrature components of the field can be measured using the so-called homodyne method. It is possible even for visible light for which the field itself oscillates too fast to be directly observable with existing detectors. As for any pair of canonically conjugate variables, the dispersions of Q_ℓ and P_ℓ must obey the Heisenberg inequalities, Delta Q times Delta P larger than or equal to ħ over 2, whatever the state of radiation. Remember that the exact value depends on the particular state considered. So what is the value of Delta Q, Delta P for a quasi-classical state? The calculation follows exactly the same line as the one we have done for the dispersion of the electric field, you should do it yourself. The result is here. For a quasi-classical state, the dispersions are equal and their product is exactly equal to the lowest value permitted by the Heisenberg relations. One says that this is a minimum dispersion state. The minimum values of Delta Q and Delta P found here are called the standard quantum limit, meaning minimum quantum noise. We will come back to this point in a future lesson when we learn more about quadrature components. In particular, we will see how it is possible to get around the standard quantum limit.