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arXiv:quant-ph/0202167 · Physical Review A

Quantum properties of transverse pattern formation in second-harmonic generation

Morten Bache, Pierre Scotto, Roberta Zambrini, Maxi San Miguel, Mark E. Saffman

Published Jul 19, 2002• 20 citations
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Abstract

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.

DOI
10.1103/physreva.66.013809

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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 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.
The mutual influence of second-harmonic generation and nondegenerate downconversion inside the same cavity is studied. The threshold behavior and the stability of a semiclassical solution with all four modes oscillating resonantly is investigated. It is shown that under certain conditions the self-pulsing instability that occurs in second-harmonic generation is absent in the combined system. The squeezing spectra for the quadrature-phase amplitudes in the second harmonic and the fundamental, as well as the photon statistics in the downconverted modes, are derived analytically. As one of the various nonclassical features of this system, the second harmonic is found to exhibit excellent noise suppression at very large frequencies, with the frequency component of best squeezing being tunable by the pump parameter. The intricate interaction of the two nonlinear processes can give rise to twin beams of correlated photons, with sub-Poissonian statistics in each beam. These results are contrasted with the results for each nonlinear process by itself.
A model is proposed where two ${\ensuremath{\chi}}^{(2)}$ nonlinear waveguides are contained in a cavity suited for second-harmonic generation. The evanescent wave coupling between the waveguides is considered as weak, and the interplay between this coupling and the nonlinear interaction within the waveguides gives rise to quantum violations of the standard quantum limit. Pronounced twin-beam behavior is observed as almost complete noise suppression in the difference as well as the sum of the waveguide intensities close to bistable transitions, self-pulsing instabilities as well as the transition from symmetric to asymmetric states. The correlations imply strong correlations and anticorrelations induced by the coupling. The violations of the standard quantum limit are particularly strong when two instabilities are competing. The results are based on the full quantum Langevin equations derived from the boson operator Hamiltonian and including cavity damping effects. The intensity correlations of the output fields are calculated semianalytically using a linearized version of the Langevin equations derived through the positive-P representation. Confirmation of the analytical results are obtained by numerical simulations of the nonlinear Langevin equations derived using the truncated Wigner representation.
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.