Unified Treatment of Heterodyne Detection: the Shapiro-Wagner and Caves Frameworks

A comparative study is performed on two heterodyne systems of photon detectors expressed in terms of a signal annihilation operator and an image band creation operator called Shapiro-Wagner and Caves frame, respectively. This approach is based on the introduction of a convenient operator $hat psi$ which allows a unified formulation of both cases. For the Shapiro-Wagner scheme, where $[hat psi, hat psi^{dag}] =0$, quantum phase and amplitude are exactly defined in the context of relative number state (RNS) representation, while a procedure is devised to handle suitably and in a consistent way Caves framework, characterized by $[hat psi, hat psi^{dag}] eq 0$, within the approximate simultaneous measurements of noncommuting variables. In such a case RNS phase and amplitude make sense only approximately.