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The Journal of Chemical Physics

Approximate and exact treatments of second-harmonic generation: Regular vs ‘‘stochastic’’ behavior

Jacob Katriel, Nimrod Moiseyev

Published Jan 15, 1983• 2 citations
Publisher

Abstract

The dynamics of the second-harmonic generation Hamiltonian is studied both exactly and using an approximation involving the application of generalized Bose operators. The limits of validity of the approximate treatment, which is of interest in the context of quantum optics, are established. The transition from a regular to a quasistochastic behavior in this nonlinear system is investigated.

DOI
10.1063/1.444788

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The dynamics of the second-harmonic generation Hamiltonian is studied both exactly and using an approximation involving the application of generalized Bose operators. The limits of validity of the approximate treatment, which is of interest in the context of quantum optics, are established. The transition from a regular to a quasistochastic behavior in this nonlinear system is investigated.
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We address the second-harmonic generation process in a quantum frame. Starting with a perturbative approach, we show that it is possible to achieve a number of analytic results, ranging from the up-conversion probability to the statistical properties of the generated light. In particular, the moments and the correlations of the photon-number distribution of the second-harmonic light generated by any initial state are retrieved. When possible, a comparison with the results achieved with the classical regime is successfully provided. The nonclassicality of some benchmark states is investigated by inspecting the corresponding autocorrelation function.
With advanced micro- and nanophotonic structures, the vacuum photon-photon coupling rate is anticipated to approach the intrinsic loss rate and lead to unconventional quantum effects. Here, we investigate the classical-to-quantum transition of such photonic nonlinear systems using the quantum cluster-expansion method, which addresses the computational challenge in tracking large photon number states of the fundamental and harmonic optical fields involved in the second-harmonic generation process. Compared to the mean-field approximation used in the weak-coupling limit, the quantum cluster-expansion method solves multimode dynamics efficiently and reveals the quantum behaviors of optical parametric oscillations around the threshold. This paper presents a universal tool to study quantum dynamics of multimode systems and explore the nonlinear photonic devices for continuous-variable quantum information processing.