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Physical Review A

Generalized squeezing

Samuel L. Braunstein, Robert I. McLachlan

Published Feb 1, 1987• 133 citations
Publisher

Abstract

We consider a generalized form of parametric amplification which produces k-photon correlations. We show numerically that this process is well-defined quantum mechanically, and we explain the quantum phase-space structures produced by such parametric amplification.
DOI
10.1103/physreva.35.1659

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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.
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The process of quantum-injection into an amplifier operating in a novel entangled configuration is adopted to “amplify” into a large dimensionality spin-12 Hilbert space the superposition properties of the entangled photon-couples generated by parametric down-conversion. The process is applied to the cases of mode-degenerate and mode non-degenerate Parametric Amplification. The quantum structure of the Wigner function and of the field correlation functions shows in both cases that the Schroedinger-cat behavior of the emitted field is largely detectable against the squeezed-vacuum noise. The absence of decoherence is explained in terms of a basic property of nonlinear complex systems.