Phase-space-interference approaches to quantum superposition states

Abstract

Phase-space-interference approaches employ interference on the phase plane to explain a quantum state's photon-number distribution. This paper generalizes the approach so that knowledge of the photon-number distribution explains the state's quantum phase-plane distribution. The required generalization is as follows. The Bohr-Sommerfeld ``band'' or trajectory of the mixture of number states \ensuremath{\sim}(\ensuremath{\Vert}N〉〈N\ensuremath{\Vert}+\ensuremath{\Vert}M〉〈M\ensuremath{\Vert}) consists of two annuli centered on the origin. In contrast, the trajectory of the pure number state superposition \ensuremath{\sim}(\ensuremath{\Vert}N〉+\ensuremath{\Vert}M〉) (with M>N) contains additional interference structures, namely, (M-N) symmetrically positioned radial spokes connecting the annuli. These spokes intersect the circular band attributed to a photon-number state at (M-N) locations, with the resulting destructive interference eliminating states \ensuremath{\Vert}N+1〉 to \ensuremath{\Vert}M-1〉. These structures build state phase-plane quantum probability distributions, as we illustrate for the squeezed vacuum. This state is missing every odd-numbered photon number state, and thus its trajectory consists of a repeated pattern of two heavily emphasized bars eliminating every odd-numbered state. These bars sum to generate the traditional ``cigar'' shape of the squeezed state.