Exact Gaussian Entanglement Dynamics and Initial-State Control in Coupled Parametric Oscillators
We study two bilinearly coupled harmonic oscillators driven by a common time-dependent spring. A fixed normal-mode transformation reduces the pair to two parametric oscillators, each solved exactly by its Lewis-Riesenfeld invariant and Ermakov-Pinney amplitude, including the ladder-operator construction and the Lewis-Riesenfeld phase. For invariant-vacuum inputs the physical two-mode covariance matrix is obtained in closed form, and the logarithmic negativity collapses to $E_{N}=\frac{1}{2} \operatorname{arcosh}(X/2)$, where a single dimensionless invariant $X \geq 2$ measures the relative phase-space deformation of the two normal modes. We give a normalization-explicit account of the Duan--Simon comparison: after local symplectic optimization the Duan test and the positive partial transpose share the entanglement threshold for this pure exchange-symmetric Gaussian family, but the optimized Duan quantity is a witness, not an independent measure, with $D_{\rm opt}/D_{\rm sep}=e^{-E_N}$. We carefully separate a genuinely separable product preparation from the already-entangled ground state of the coupled Hamiltonian, and use only the former for claims of entanglement generation. Under sinusoidal stiffness modulation a separable input shows bounded correlations away from resonance and sustained growth near parametric resonance, and we demonstrate numerically that the initial width and chirp act as two independent, experimentally accessible controls of the entanglement at a chosen target time. An $SU(1,1)$ formulation identifies $2 E_{N}$ with the hyperbolic separation of the two normal-mode squeezing trajectories on the $SU(1,1)/U(1)$ disk and shows that entanglement is governed by both the relative squeezing magnitude and the relative squeezing angle; ; we exhibit regimes in which nearly equal magnitudes still produce strong entanglement through the angle alone.