We define a quasiprobability distribution ${\mathit{S}}_{\mathrm{NP}}$(n,θ) which describes the quantum statistics of the photon number and phase observables of a single-mode field (or, equivalently, a harmonic oscillator). The properties of ${\mathit{S}}_{\mathrm{NP}}$(n,θ) are the photon number and phase analogies of the properties of Wigner’s original function, which describes the position and momentum observables. For example, the marginals of ${\mathit{S}}_{\mathrm{NP}}$(n,θ) are the continuous phase and the discrete photon-number probability distributions. We give examples of the ${\mathit{S}}_{\mathrm{NP}}$(n,θ) representation of various states and show, in particular, that ${\mathit{S}}_{\mathrm{NP}}$(n,θ) displays the quantum interference associated with Schrödinger cat states. We also describe how ${\mathit{S}}_{\mathrm{NP}}$(n,θ) can be determined from quantities that are, in principle, measurable.

• Received 21 March 1995

DOI:https://doi.org/10.1103/PhysRevA.52.3474