Abstract
In the fourth lecture , we established the expression of field operators in frequency space, based on the decomposition of the signal into waves of well-defined wave vectors. The commutator of these field operators is singular : it is given by a Dirac function involving the sum of frequencies. Similarly, in the thermal state, the average value of the anti-switch is given by the same Dirac function, but multiplied by a function analogous to the average photon number of an oscillator. This leads to expressions that are convenient for calculations ; but to recover the physical meaning of the operators, we need to introduce mode creation and annihilation operators, based on the wavelets defined in the previous lesson. These mode operators make it possible to rigorously specify the state of the field in a transmission line, e.g. a semi-classical state of the field. A semi-classical state can be represented by a generalization of the Fresnel vector, sometimes nicknamed " Fresnel lollipop " : the segment of the vector, which represents the amplitude and mean phase of the state in the plane of the quadratures, is fitted not with an arrowhead, but with a disk whose radius gives the standard deviation of the fluctuations, in this case the zero-point fluctuations. On the other hand, a state with a well-defined number of photons (the so-called Fock state) corresponds to a rotationally symmetrical figure comprising a series of rings, the number of rings being equal to the number of photons.
