  A model is proposed where two $\chi^{(2)}$ 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 classical limit. These violations are particularly
strong when two instabilities are competing, where twin-beam behavior is found
as almost complete noise suppression in the difference of the fundamental
intensities. Moreover, close to bistable transitions perfect twin-beam
correlations are seen in the sum of the fundamental intensities, and also the
self-pulsing instability as well as the transition from symmetric to asymmetric
states display nonclassical twin-beam correlations of both fundamental and
second-harmonic intensities. The results are based on the full quantum Langevin
equations derived from the Hamiltonian and including cavity damping effects.
The intensity correlations of the output fields are calculated
semi-analytically 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.
