Ordered moments and relation to Radon transform of Wigner quasiprobability

Abstract

It is shown that the symmetrically ordered moments of boson operators for a single boson mode can be reconstructed from the corresponding moments of the Radon transform of the Wigner quasiprobability for discrete sets of equidistant inequivalent angles which solve the circle division problem. This reconstruction is sometimes simpler than the corresponding reconstruction of the normally ordered moments where one first has to multiply the Radon transform with Hermite polynomials in comparison to power functions for symmetrically ordered moments and then to integrate. The connection to the reconstruction for the general class of s-ordered moments is established. The transition from discrete sets of angles to integration over angles via averaging over the discrete angles is made. The results are applied to displaced squeezed thermal states. It is shown how the ordered moments for these states can be explicitly found from the calculated Radon transform of the Wigner quasiprobability. The obtained formulae for these moments possess independent interest since they contribute to the discussion of the properties of the most general class of states with quasiprobabilities of Gaussian form with many possible special cases as, for example, squeezed coherent states and squeezed thermal states.