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arXiv:quant-ph/0108015 · Physical Review A

Spatial correlations in hexagons generated via a Kerr nonlinearity

A. Gatti, Stefano Mancini

Published Dec 14, 2001• 8 citations
arXivHTMLPDF

Abstract

We consider hexagonal pattern forming in the cross section of an optical beam produced by a Kerr cavity, and we study the quantum correlations characterizing this structure. By using arguments related to the symmetry broken by the pattern formation, we identify a complete scenario of six-mode entanglement. Five independent phase quadrature combinations connecting the hexagonal modes are shown to exhibit sub-shot-noise fluctuations. By means of a nonlinear quantum calculation technique, quantum correlations among the mode photon numbers are demonstrated and calculated.

DOI
10.1103/physreva.65.013816

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We consider hexagonal pattern forming in the cross section of an optical beam produced by a Kerr cavity, and we study the quantum correlations characterizing this structure. By using arguments related to the symmetry broken by the pattern formation, we identify a complete scenario of six-mode entanglement. Five independent phase quadrature combinations connecting the hexagonal modes are shown to exhibit sub-shot-noise fluctuations. By means of a nonlinear quantum calculation technique, quantum correlations among the mode photon numbers are demonstrated and calculated.
Summary form only given. Transverse optical patterns are capable of allowing noteworthy aspects linked to quantum fluctuations. These issues have been mainly studied in stripe patterns, which present only two modes in the far-field. We address the study of the fluctuations and correlations in a hexagonal pattern in a ring cavity filled with a self-focusing Kerr medium. The dynamics of the electric field is described, in the mean-field approximation, by a Langevin equation for the scaled slowly varying amplitude of the field. We show that the pattern fluctuations can be understood in terms of the neutral modes of the pattern and modes with small damping. These modes are obtained from a stability analysis of the hexagonal pattern. We find strong correlations between the intensity fluctuations of the modes of the pattern, not only among the fundamental harmonics, but also with the higher order ones. We also find strong anticorrelations between the intensity fluctuations of the modes of the pattern and the homogeneous mode which are related to energy conservation.
Abstract The pairwise production of photons in nonlinear optical processes ensures entanglement to occur between two photons. E.g. when the Kerr effect is exploited, the photons are produced in different frequency modes, which are symmetric with respect to the pump frequency. Since these photons are produced into the same spatial mode, the quadrature entanglement can be witnessed only by the use of a frequency selective device which transforms the adjacent frequency modes into two different spatial modes. We use a Mach‐Zehnder interferometer with a large path length difference to separate symmetric frequency modes located 10.25 MHz from the carrier. We measure correlations of the quadrature components of 1.6 ± 0.1dB below the shot noise in the amplitude and 1.4 ± 0.1dB in the phase.
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.
The dynamical behavior of Kerr optical frequency combs is very well understood today from the perspective of the semi-classical approximation. In this article, we provide a theoretical understanding of the phenomena of quantum correlations, multimode entanglement and squeezed states of light that can occur in these frequency combs when quantum noise is accounted for. We prove that for all stationary spatio-temporal patterns, the side-modes that are symmetrical relatively to the central (pumped) mode in the frequency domain display quantum correlations that can lead to squeezed states of light under some optimal conditions that are analytically determined. We show that these quantum correlations can persist regardless the dynamical state of the system (rolls or solitons), regardless of the spectral extension of the comb (number sidemodes), and regardless of the dispersion regime (normal or anomalous). We study with particular emphasis the two principal architectures for Kerr comb generation, namely the add-through and add-drop configurations. We show that in both cases, the ratio between out-coupling and total losses plays a key role as it directly determines the efficiency of the squeezing. We also explicitly determine the phase quadratures leading to photon entanglement, and analytically calculate their quantum noise spectra. The relevance of Kerr combs for quantum information systems at optical telecommunication wavelengths is also discussed.
We analyze nonlinear transverse mode coupling in a Kerr medium placed in an optical cavity and its influence on bistability and different kinds of quantum noise reduction. Even for an input beam that is perfectly matched to a cavity mode, the nonlinear coupling produces an excess noise in the fluctuations of the output beam. Intensity squeezing seems to be particularly robust with respect to mode coupling, while quadrature squeezing is more sensitive. However, it is possible to find a mode whose quadrature squeezing is not affected by the coupling. \textcopyright{} 1996 The American Physical Society.
The dynamical behavior of Kerr optical frequency combs is very well understood today from the perspective of the semiclassical approximation. These combs are obtained by pumping an ultrahigh-QQQ whispering-gallery mode resonator with a continuous-wave laser. The long-lifetime photons are trapped within the toruslike eigenmodes of the resonator, where they interact nonlinearly via the Kerr effect. In this article, we use quantum Langevin equations to provide a theoretical understanding of the nonclassical behavior of these combs when pumped below and above threshold. In the configuration where the system is under threshold, the pump field is the unique oscillating mode inside the resonator, and it triggers the phenomenon of spontaneous four-wave mixing, where two photons from the pump are symmetrically up- and down-converted in the Fourier domain. This phenomenon, also referred to as parametric fluorescence, can only be understood and analyzed from a fully quantum perspective as a consequence of the coupling between the field of the central (pumped) mode and the vacuum fluctuations of the various side modes. We analytically calculate the power spectra of the spontaneous emission noise, and we show that these spectra can be either single- or double-peaked depending on the value of the laser frequency, chromatic dispersion, pump power, and spectral distance between the central mode and the side mode of interest. We also calculate as well the overall spontaneous noise power per side mode and propose simplified analytical expressions for some particular cases. In the configuration where the system is pumped above threshold, we investigate the phenomena of quantum correlations and multimode squeezed states of light that can occur in the Kerr frequency combs originating from stimulated four-wave mixing. We show that for all stationary spatiotemporal patterns, the side modes that are symmetrical relative to the pumped mode in the frequency domain display quantum correlations that can lead to squeezed states of light under some optimal conditions that are analytically determined. These quantum correlations can persist regardless the dynamical state of the system (rolls or solitons), regardless of the spectral extension of the comb (number side modes) and regardless of the dispersion regime (normal or anomalous). We also explicitly determine the phase quadratures leading to photon entanglement and analytically calculate their quantum-noise spectra. For both the below- and above-threshold cases, we study with particular emphasis the two principal architectures for Kerr comb generation, namely the add-through and add-drop configurations. It is found that regardless of the configuration, an essential parameter is the ratio between out-coupling and total losses, which plays a key role as it directly determines the efficiency of the detected fluorescence or squeezing spectra. We finally discuss the relevance of Kerr combs for quantum information systems at optical telecommunication wavelengths below and above threshold.