Non-orthogonal domains in phase space of quantum optics and their relation to fractional Fourier transforms

Abstract

It is customary to define a phase space such that position and momentum are mutually orthogonal coordinates. Associated with these coordinates, or domains, are the position and momentum operators. Representations of the state vector in these coordinates are related by the Fourier transformation. We consider a continuum of “fractional” domains making arbitrary angles with the position and momentum domains. Representations in these domains are related by the fractional Fourier transformation. We derive transformation, commutation, and uncertainty relations between coordinate multiplication, differentiation, translation, and phase shift operators making arbitrary angles with each other. These results have a simple geometric interpretation in phase space and applications in quantum optics.