Abstract

We study a single mode of the radiation field subjected to ideal k-photon parametric amplification (photons created or destroyed k at a time). In an attempt to generalize ordinary squeezing (k = 2), Fisher et al.¹ considered the case k > 2. They concluded that there is something seriously wrong with the evolution operator after discovering that for k > 2 its matrix elements in the number-state basis have divergent Taylor series expansions in time. We show that this divergence is due to a branch cut along the negative time axis, and we obtain useful information by treating these Taylor series as asymptotic expansions. We investigate the states generated by a k-photon paramp by calculating the evolution of the quantum O-function. We watch the vacuum evolve into a state whose O-function has k-fold symmetry. The classical behavior which corresponds to a k-photon parametric interaction is described by phase-space trajectories near an unstable fixed point. We compare the classical and quantum behavior and find that the quantum corrections smear out classical phase-space features which are smaller than allowed by the uncertainty principle. Finally, we consider the evolution of an initial coherent state which has large amplitude, and we find that a three-photon paramp generates squeezing at a rate proportional to the amplitude of the initial coherent state.

© 1986 Optical Society of America