About integration within ordered products in quantum optics

The fundamentals of Fan's method of integration within ordered products are investigated and it is shown that it is a reliable calculation method in quantum optics. Differentiation within ordered products is included into the considerations. The integrations are, in applications to single boson modes, mostly one-dimensional or two-dimensional integrations over phase-space variables. We examine some examples of integration within ordered products and compare them with calculations of results by other methods such as Lie group methods. A new representation of the main class of quasiprobabilities including the Wigner and the Husimi-Kano quasiprobability is derived by the method. Different representations of squeezing operators are related by the method and discussed. As a further new result, the position and momentum representation of the general unitary squeezing operators are derived and specialized to cases with squeezing in the direction of the coordinate axes which were earlier dealt with by the method under consideration.