  We theoretically and numerically study the quantum dynamics of two degenerate
optical parametric oscillators with mutual injections. The cavity mode in the
optical coupling path between the two oscillator facets is explicitly
considered. Stochastic equations for the oscillators and mutual injection path
based on the positive $P$ representation are derived. The system of two
gradually pumped oscillators with out-of-phase mutual injections is simulated,
and its quantum state is investigated. When the incoherent loss of the
oscillators other than the mutual injections is small, the squeezed quadratic
amplitudes $\hat{p}$ in the oscillators are positively correlated near the
oscillation threshold. It indicates finite quantum correlation, estimated via
Gaussian quantum discord, and the entanglement between the intracavity
subharmonic fields. When the loss in the injection path is low, each oscillator
around the phase transition point forms macroscopic superposition even under a
small pump noise. It suggests that the squeezed field stored in the low-loss
injection path weakens the decoherence in the oscillators.
