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Physical Review A

Spatial quantum correlations in the fluorescence of traveling-wave second-harmonic generation

Pierre Scotto

Published Sep 29, 2003• 4 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.68.033814

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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.
We investigate the spatial quantum noise properties of the one-dimensional transverse pattern formation instability in intracavity second-harmonic generation. The Q representation of a quasi-probability distribution is implemented in terms of nonlinear stochastic Langevin equations. We study these equations through extensive numerical simulations and analytically in the linearized limit. Our study, made below and above the threshold of pattern formation, is guided by a microscopic scheme of photon interaction underlying pattern formation in second-harmonic generation. Close to the threshold for pattern formation, beams with opposite direction of the off-axis critical wave numbers are shown to be highly correlated. This is observed for the fundamental field, for the second-harmonic field, and also for the cross-correlation between the two fields. Nonlinear correlations involving the homogeneous transverse wave number, which are not identified in a linearized analysis, are also described. The intensity differences between opposite points of the far fields are shown to exhibit sub-Poissonian statistics, revealing the quantum nature of the correlations. We observe twin beam correlations in both the fundamental and second-harmonic fields, and also nonclassical correlations between them.
A quantum model for singly resonant second-harmonic generation in a cavity with transverse degrees of freedom is analyzed. An instability threshold for pattern formation exists in this system. Below threshold, a strong modulation of the noise is demonstrated in the transverse structure of the far field. The performed analysis encompasses both one- and two-point correlation functions. The noise in a single far-field point is generally above the classical vacuum noise level. In contrast, strong nonclassical two-point correlations are encountered between opposite emission directions in the far field. For the fundamental field the correlations are shown to be of the Einstein-Podolsky-Rosen type, thus establishing a remarkable analog to the quantum correlations found in a nondegenerate optical parametric oscillator.
We have measured sub-shot-noise quantum correlations of spatial fluctuations in the far-field image of the parametric fluorescence created in a type I $\ensuremath{\beta}$-barium-borate nonlinear crystal, either between opposite angular sectors (nondegenerate configuration) or between opposite pixels (degenerate configuration). Imaging is performed at a very low light level (0.20.20.2 photons per pixel) with an electron-multiplying CCD camera, resulting in purely spatial coincidences between single photons when detecting on pixels. Experimental results overcome the standard quantum-limit-shot-noise level without subtraction of the variance of the detection noise. We compare these experimental results with numerical results given by the quantum Green's function method, which is proved to have strong advantages over stochastic simulations.