Tomographic reconstruction of the density matrix via pattern functions

We propose a general method for reconstructing directly the density matrix of a single light mode in optical homodyne tomography. In our scheme the density matrix 〈a‖ρ^‖${\mathit{a}}^{\prime }$〉 is obtained by averaging a set of pattern functions ${\mathit{F}}_{\mathit{aa}}^{\prime }$(${\mathit{x}}_{\mathrm{\theta }}$,θ) with respect to the homodyne data ${\mathit{x}}_{\mathrm{\theta }}$. The functions show the typical features of the quadrature distributions for the corresponding density-matrix elements. It is also possible to compensate the effect of detection losses which requires, however, extra effort in both experimental and numerical precision. We calculate the pattern functions for the coherent-state and Fock representations and study their properties. We believe that our method is the most efficient way for reconstructing the density matrix from homodyne measurements.