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Physical Review A

Quantum-noise-induced macroscopic revivals in second-harmonic generation

M. K. Olsen, R. J. Horowicz, Lev I. Plimak, Nicolas Treps, Claude Fabre

Published Jan 5, 2000• 39 citations
Publisher

Abstract

We investigate the behavior of the fundamental and second-harmonic fields in phase-matched traveling plane-wave second-harmonic generation, using the full-operator equations of motion. We find that, after a certain interaction length, both the macroscopic and quantum-statistical properties of the harmonic and fundamental fields are qualitatively different from those found in previous analyses. The mean fields do not vary in a monotonic way, but oscillate with the propagation length, leading to an unexpected periodic revival of the fundamental field, triggered by the quantum fluctuations always present in the mode. Accordingly, the amplitude noise of the fundamental, previously predicted to be perfectly squeezed for long interaction lengths, actually reaches a very small minimum for a definite length, then increases again.

DOI
10.1103/physreva.61.021803

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We analyze the evolution of quantum noise in both the fundamental and the harmonic fields that are undergoing traveling-wave interaction in a second-order [χ(2)] nonlinear medium. Assuming perfect phase matching between the fundamental and the harmonic fields and arbitrary input boundary conditions, the behavior of quantum noise in the propagating fields is studied by linearization of the nonlinear-operator equations around the mean-field values. We first consider the degenerate case that is applicable to type-I phase-matching geometries, obtaining expressions for squeezing in both the fundamental and the harmonic fields. We then analyze the polarization-nondegenerate case that applies to type-II phase-matching geometries. In the special case, when the two orthogonally polarized fundamental inputs are of equal amplitude, we obtain analytical results and show that the type-II phase-matched second-harmonic-generation process can be configured to generate sub-Poissonian light in both polarization components of the fundamental field. Finally, we numerically solve the linearized quadrature-operator equations along with the nonlinear mean-field equations for the general case of type-II phase matching. For both type-I and type-II processes, we find that whenever the fundamental field experiences deamplification it is associated with amplitude squeezing and phase desqueezing. If, in contrast, the fundamental field experiences amplification, then it is accompanied by amplitude desqueezing, but with squeezing in the phase quadrature. The harmonic field is amplitude squeezed if the input boundary condition leads to harmonic conversion and is phase squeezed if the input boundary condition leads to parametric amplification.
We analyze squeezing of both the fundamental and harmonic fields undergoing traveling-wave, second-harmonic generation (SHG) in second-order (${\mathrm{\ensuremath{\chi}}}^{(2)}$) nonlinear media. We take into account depletion of the fundamental field as well as the phase mismatch between the fundamental and harmonic fields. The behavior of the quantum noises on the propagating fields is studied by linearizing the nonlinear operator equations around the mean-field values. We first consider the degenerate case that is applicable to type-I phase-matching geometries, obtaining expressions for squeezing in both the fundamental and harmonic fields under the conditions of perfect phase matching and large phase mismatch. We show that in the case of a large phase mismatch, the intensity-dependent self-phase shift of the fundamental field, arising due to cascading of the ${\mathrm{\ensuremath{\chi}}}^{(2)}$ nonlinearity, is responsible for the squeezing generation. We also numerically solve the linearized quadrature-operator equations together with the nonlinear mean-field equations. We find that in the case of a finite phase mismatch the harmonic field can be highly squeezed. This is in contrast to the perfectly phase-matched case where the maximum squeezing is limited to 50%. Finally, we analyze the nondegenerate case that applies to type-II phase-matching geometries. Here we show that the commonly used type-II phase-matched SHG process with the input harmonic field in the vacuum state is equivalent to a type-I SHG process in parallel with a degenerate optical-parametric process. The latter causes squeezing in the mode that is polarized orthogonal to the fundamental beam. In the perfect phase-matching case, the squeezing in the orthogonally polarized mode follows the simple expression S=1-\ensuremath{\gamma}, where \ensuremath{\gamma} is the harmonic conversion frequency.
Using the full nonlinear equations of motion, we calculate the quantum-nondemolition (QND) correlations for the traveling-wave second-harmonic generation. We find that, after a short interaction length, these are qualitatively different from results calculated previously using a linearized fluctuation analysis. We demonstrate that, although individual QND criteria can be very good in certain regions, there is no region where all three of the standard criteria are perfect, as has previously been claimed. We also show that only the amplitude quadrature of the output field can be considered as a QND quantity, with the phase quadrature not satisfying all the criteria.
By solving linearized wave equations of nonlinear polarization for a medium, we derive the input-output relationship for the propagation of the quantum fluctuations in single-pass second-harmonic generation with arbitrary interaction length. Spectra of quadrature-phase squeezing are calculated from the input-output relations. It is found that the intensity fluctuations of both harmonic and fundamental fields are squeezed below the vacuum fluctuation level. For large interaction length, an arbitrary amount of squeezing can be achieved for the fundamental field whereas only 50% squeezing is possible for the harmonic field. The bandwidth of squeezing is determined by the phase-matching condition and can be very broad in this case. The possibility of a quantum-nondemolition measurement in the process is discussed. Some interesting features in quantum fluctuations of the fields are presented.
We investigate theoretically the spatial quantum correlations of the light produced by spontaneous emission in type-I second-harmonic generation in the traveling-wave configuration. It is first shown that spontaneous emission occurs at both fundamental and second-harmonic frequency. Considering the quantum fluctuations of the intensities collected in symmetrical parts of the far-field plane, nonclassical correlations below the shot noise are predicted not only at fundamental frequency, but also at second-harmonic frequency. The latter cannot be traced back to any twin-photon emission mechanism, but are generated by a secondary process acting on twin photons. This mechanism also creates correlations between fundamental and second-harmonic field, at a given transverse wave number, and at opposite wave numbers. The analysis of a simplified few-mode model, on a quantum level, provides a good qualitative understanding of these correlations.
We investigate the spatial behavior of nonclassical light produced by type I second-harmonic generation in the traveling-wave configuration. An input-output transformation for the system of fundamental and second-harmonic field is derived in the framework of a linearization approach and used to investigate the properties of an optical system, which consists of a crystal with a quadratic nonlinearity pumped at frequency $\ensuremath{\omega}$ and enclosed in a two-lens telescopic system. If a faint input image at frequency $2\ensuremath{\omega}$ is injected into this device, for a sufficiently large interaction length, the output displays a pair of symmetric amplified versions of the input image at both fundamental and second-harmonic frequency. The analysis of the quantum fluctuations in the output images shows that under certain conditions, this optical device operates noiselessly with respect to the output at frequency $\ensuremath{\omega},$ whereas the output images at frequency $2\ensuremath{\omega}$ are affected by a slight degradation of the signal-to-noise ratio.