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Physical Review A

Quantum theory of cross-phase-modulational instability: Twin-beam correlations in a χ(3) process

T. A. B. Kennedy

Published Aug 1, 1991• 20 citations
Publisher

Abstract

We consider the quantum correlations between fields generated through cross-phase-modulational instability in an optical fiber. Two examples of this instability are addressed, one relating to the linear-polarization eigenstates in a high-birefringence fiber, and the other to two-color propagation in a fiber. The degrees of second-order coherence of the quantum noise sidebands generated in this ${\mathrm{\ensuremath{\chi}}}^{(3)}$ process are calculated, and the violation of a classical Cauchy-Schwarz inequality and intensity-difference squeezing is reported for light transmitted by a passive filter. Around 50% intensity-difference squeezing is possible in the case of the two-color instability. This illustrates the quantum-statistical nature of the sideband correlations and how passive processing may in some circumstances help accentuate quantum correlations.

DOI
10.1103/physreva.44.2113

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We consider the quantum correlations between fields generated through cross-phase-modulational instability in an optical fiber. Two examples of this instability are addressed, one relating to the linear-polarization eigenstates in a high-birefringence fiber, and the other to two-color propagation in a fiber. The degrees of second-order coherence of the quantum noise sidebands generated in this \ensuremathχ(3){\mathrm{\ensuremath{\chi}}}^{(3)}\ensuremathχ(3) process are calculated, and the violation of a classical Cauchy-Schwarz inequality and intensity-difference squeezing is reported for light transmitted by a passive filter. Around 50% intensity-difference squeezing is possible in the case of the two-color instability. This illustrates the quantum-statistical nature of the sideband correlations and how passive processing may in some circumstances help accentuate quantum correlations.
The propagation of a quasimonochromatic beam in an optical fiber may be unstable owing to phase-matched four-wave mixing. The phase matching may arise in a variety of ways, and the resulting instabilities are commonly referred to as modulational instabilities. Here I consider the cross-phase-modulational (XPM) instability arising in a high birefringence optical fiber in the normal dispersion (visible) regime when the pump is polarized at 45° to the principal axes of birefringence.1,2 Orthogonally polarized sidebands are generated along these axes. Phase matching arises from a balance between dispersion and the group velocity mismatch of the sidebands and dictates that the up-shifted sideband travels on the fast axis. The sidebands result in terahertz modulation of the pump and are interesting from the point of view of optical communications. From a quantum-statistical perspective, the quantum correlations, which develop as the sidebands are generated from quantum noise, are the focus of attention here.
The propagation of a quantized radiation field in a nonlinear optical fiber is investigated by using a quantum nonlinear Schrödinger equation with Raman scattering contributions.1 Wideband photon pair correlations generated via the optical Kerr effect and the role of group velocity dispersion and Raman scattering will be discussed. Specifically, the quantum correlations of up-shifted and downshifted sidebands in the modulational instability at terahertz frequency detuning from the pump lie outside the GAWBS region, but they are influenced by the low frequency region of the Raman gain curve, which introduces an asymmetry between the sidebands. The effects of this asymmetry on photon pair correlations and twin beams will be discussed.
Evolution of the vacuum fluctuations in the two-wave mixing of the optical fields propagating in a birefringent fiber is studied. The two-wave mixing in the birefringent fiber was suggested as a possible scheme for the entangled-state generation. Our treatment in studying the entangled-state generation uses depleted pump approximation and enables one to trace the influence of the input conditions of classical optical fields on the evolution of vacuum squeezing. We report the periodical modulation of the vacuum squeezing when the input relative phase of coherent waves varies. The measure of nonclassical correlations imposed on the generated light is calculated.
Squeezing is a fundamental quantum optical phenomenon - induced by various nonlinear optical processes - in which a special state of the electromagnetic field is generated, for which quantum noise fluctuations are reduced below the shot noise level for certain frequency ranges. Two qualitatively different kinds of squeezing are currently known to take place in optical fibers. The first is the four-wave mixing (FWM) induced quadrature squeezing, in which an intense continuous wave (CW) pump initiates a spontaneous emission of signal and idler waves from vacuum fluctuations at symmetric detunings from the pump. The noise components at the signal and idler frequencies become correlated via the fiber nonlinearity, leading to a reduction of the noise power spectrum below the shot noise level. All the waves participating to the FWM-induced squeezing are monochromatic waves. The second kind of squeezing is the so called self-phase modulation (SPM) induced squeezing, in which the spectral components of an intense short pulse in a coherent state become correlated via the SPM during propagation. We demonstrate theoretically that a third kind of squeezing is possible in optical fibers, in which the noise spectrum of the quasi-monochromatic resonant radiation (RR) emitted by optical solitons when subject to perturbations shows evidence of photon correlation and a reduction of noise below the shot-noise level.
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.
Squeezed light, in which the quantum noise in one amplitude component is reduced below the vacuum level, has been generated using parametric processes. The starting point described here is the nonlinear Schrodinger equation which models light propagating through optical fibers. When such a dispersive nonlinear medium with a third-order susceptibility is pumped with an optical beam, squeezing can result. It is found that the degree of squeezing oscillates as a function of fiber length in the normal dispersion regime, whereas in the anomalous dispersion regime, it grows exponentially with fiber length. Examination of squeezing near the fiber’s zero dispersion wavelength shows that odd-order dispersive terms have no effect on squeezing. However, small values of higher-order even terms have significant effects on squeezing. In practice, high-frequency components of the squeezing spectrum are difficult to obtain. However, it is shown that a cross-correlation technique may be used to explore the frequency dependence of the squeezing spectrum.