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Optics Express

From quantum to classical images

A. Gatti, Luigi A. Lugiato, Gian‐Luca Oppo, Richard Martin, P. Di Trapani, Audrius Berzanskis

Published Jul 7, 1997• 38 citations
Publisher

Abstract

We display the results of the numerical simulations of a set of Langevin equations, which describe the dynamics of a degenerate optical parametric oscillator in the Wigner representation. The scan of the threshold region shows the gradual transformation of a quantum image into a classical roll pattern. An experiment on parametric down-conversion in lithium triborate shows strikingly similar results in both the near and the far field, displaying qualitatively the classical features of quantum images. c 1997 Optical Society of America.

DOI
10.1364/oe.1.000021

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We display the results of the numerical simulations of a set of Langevin equations, which describe the dynamics of a degenerate optical parametric oscillator in the Wigner representation. The scan of the threshold region shows the gradual transformation of a quantum image into a classical roll pattern. An experiment on parametric down-conversion in lithium triborate shows strikingly similar results in both the near and the far field, displaying qualitatively the classical features of quantum images. c 1997 Optical Society of America.
We illustrate the concept of quantum image in the context of degenerate parametric oscillators. In the case of a cavity with planar mirrors, we show the gradual transformation of a quantum image into a classical image as the threshold region is scanned. Thus the quantum image behaves as a precursor of the roll pattern which appears above threshold. In the far field, the configuration of the spatial correlation function of the intensity and of the quadrature components shows unambiguously the quantum nature of fluctuations that generate the quantum image, leading to effects of quantum noise reduction below the shot noise level, and to the formulation of an Einstein–Podolsky–Rosen paradox.
Quantum images, that is inhomogeneous field distributions purely generated by quantum fluctuations, persist when passing from the degenerate to the non-degenerate case of optical parametric oscillators (OPO). Below the threshold for parametric oscillation where the near-field distributions are homogeneous both in intensity and phase, appropriate spatial correlation functions anticipate the transverse spatial pattern that appears above threshold. In particular, the angular dependence of the far field spatial correlation function is able to reveal the travelling-wave nature of the phase pattern above threshold typical of nondegenerate OPOs. Cross-correlation functions between signal and idler intensities show clear evidence of the non-classical nature of the output light.
A Langevin model is introduced to study quantum fluctuations below the threshold of pattern formation for optical parametric oscillators (OPO's). In particular we compare analytical and numerical results for the OPO with one and two transverse spatial dimensions and in the presence of either plane or spherical cavity mirrors. The far-field structure and the correlation functions of the fluctuating signal field anticipate the onset of a transverse spatial pattern which arises classically even in the presence of Gaussian input beams. Correlation functions also reveal the squeezed nature of the OPO field. Numerical simulations of the Langevin model describe the result of short time measurements and show that close to threshold the near-field signal is a noisy spot pattern evolving on a time scale longer than the inverse decay rate of the resonator. This and other far-field features should be experimentally accessible.
We illustrate the spatial quantum features that emerge in a degenerate optical parametric oscillator below threshold, where the signal field is purely generated by quantum fluctuations. We analyze the behaviour of the spatial correlation function of a generic quadrature component of the signal field in both cases of a cavity with plane mirrors and a cavity with spherical mirrors. The correlation length diverges when the semiclassical threshold for signal field generation is approached. We focus on the novel concept of quantum image, which is analyzed both in the near field and in the far field.
Summary form only given.We study spatiotemporal fluctuations of the signal field in the output from a degenerate optical parametric oscillator using the Langevin model of A. Gatti et al. (1997). The spatial structure of the output signal is calculated for pump values just below the threshold for the formation of patterns with a critical wave vector. We consider the effects of a squeezed input signal, both homogeneous and spatially varying, on the output correlations, which can be measured in both the near- and far-field.
Photon statistics of light produced by a nondegenerate optical parametric oscillator operating below threshold are calculated. Positive-P representation is used to map quantum dynamics onto a set of classical Langevin equations for the complex field amplitudes associated with the annihilation and creation operators for the field produced by the oscillator. Using a Karhunen Loeve expansion of the resulting stochastic processes the moment generating function for photon counting statistics is obtained.