In the sixth lesson , we studied the estimation and reconstruction of states of a harmonic oscillator evolving in a space subtended by an infinite number of states. We began with a reminder of the Wigner function W, which gives a description of the oscillator in its phase space, as close as possible to the classical description. This description is perfectly equivalent to that of the density operator 𝛒, since the Wigner function W and 𝛒 can be deduced from each other by simple mathematical transformations. We recalled the essential properties of W and described the Wigner functions of coherent states and a few non-classical states of the field (Fock states and Schrödinger cat states). In particular, we emphasized the signature of "non-classicality" constituted by the existence of negative-valued non-Gaussian structures of the Wigner function.
We then described two standard methods for directly reconstructing the Wigner function of a field mode state when a large number of copies are available. The so-called quantum tomography method relies on the measurement of field quadrature distributions along multiple directions in phase space. Having made these quadrature measurements, the Wigner function is calculated using an inversion procedure analogous to that used to reconstruct medical tomography images from absorption measurements of X-rays passing through the body at different angles. Quadrature measurement is performed using a homodyning method that mixes the field to be measured with a reference field (local oscillator) of variable phase. This method is well suited to the measurement of free-propagating optical fields or optical fibers. Another method involves translating the field into phase space (which, like homodyning, involves mixing the field to be measured with a coherent reference) and then measuring the parity of the photon number in the resulting field. This method directly gives the value of the Wigner function at various points corresponding to the amplitudes of the translational fields used. It requires no inversion procedure and is well suited to the study of microwave fields trapped in a cavity.
