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Journal of the Optical Society of America B

Nonlinear dynamics and quantum noise for competing χ^(2) nonlinearities

Monika A. M. Marte

Published Nov 1, 1995• 15 citations
Publisher

Abstract

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.

DOI
10.1364/josab.12.002296

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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.
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We consider the process of second-harmonic generation (SHG) in a cavity that is resonant for both the fundamental- and second-harmonic modes. It has been previously shown that such a system displays a phase instability leading to self-pulsing and that one gets important intensity squeezing on both the fundamental- and second-harmonic mode for pump intensities close to but below, this instability threshold. Such a squeezing has been recently observed. We have analytically calculated the intensity-noise spectrum of the frequency-doubled field above such an instability threshold, using a semiclassical input-output technique for the treatment of quantum fluctuations.1 Such a method, introduced to deal with fluctuations around a stable solution, has been generalized to the self-pulsing domain. As expected, the instability brings excess noise to the output beams, especially around the zero-frequency, the self-oscillation frequency, and its harmonics. Remarkably, some regions exist in which where the noise increase is small; squeezing still occurs in this frequency range far in the instability domain.
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The interaction between two cavity resonant nonlinear processes in an intracavity ${\mathrm{\ensuremath{\chi}}}^{(2)}$ medium is studied, namely frequency doubling of a pump field of frequency ${\mathrm{\ensuremath{\omega}}}_{1}$ and subsequent down-conversion of the second harmonic to a pair of photons of frequencies ${\mathrm{\ensuremath{\omega}}}_{1}$\ifmmode\pm\else\textpm\fi{}\ensuremath{\Delta}. A closed set of semiclassical equations of motion for appropriately chosen quantities of the full system is derived and an explicit stationary solution is given. Its features and stability are discussed, the most striking being a stable plateau in the second harmonic, i.e., the field of frequency ${\mathrm{\ensuremath{\omega}}}_{2}$=2${\mathrm{\ensuremath{\omega}}}_{1}$ becomes independent of the pump field scrE1{\mathit{scrE}}_{1}scrE1​, provided scrE1{\mathit{scrE}}_{1}scrE1​ is larger than a certain threshold value. It is demonstrated that undamped self-pulsing can only prevail, if the ratio of the rates of second harmonic generation over and parametric down-conversion exceeds a value which depends solely on the relative damping rates of the second harmonic and fundamental, which yields a tentative theoretical explanation of the experimentally observed quenching of the self-pulsing.
We analyze the evolution of quantum noise in both the fundamental and the harmonic fields that are undergoing traveling-wave interaction in a second-order [χ(2)] nonlinear medium. Assuming perfect phase matching between the fundamental and the harmonic fields and arbitrary input boundary conditions, the behavior of quantum noise in the propagating fields is studied by linearization of the nonlinear-operator equations around the mean-field values. We first consider the degenerate case that is applicable to type-I phase-matching geometries, obtaining expressions for squeezing in both the fundamental and the harmonic fields. We then analyze the polarization-nondegenerate case that applies to type-II phase-matching geometries. In the special case, when the two orthogonally polarized fundamental inputs are of equal amplitude, we obtain analytical results and show that the type-II phase-matched second-harmonic-generation process can be configured to generate sub-Poissonian light in both polarization components of the fundamental field. Finally, we numerically solve the linearized quadrature-operator equations along with the nonlinear mean-field equations for the general case of type-II phase matching. For both type-I and type-II processes, we find that whenever the fundamental field experiences deamplification it is associated with amplitude squeezing and phase desqueezing. If, in contrast, the fundamental field experiences amplification, then it is accompanied by amplitude desqueezing, but with squeezing in the phase quadrature. The harmonic field is amplitude squeezed if the input boundary condition leads to harmonic conversion and is phase squeezed if the input boundary condition leads to parametric amplification.
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.