Complete Lewis–Riesenfeld Theory of Gaussian Entanglement in Coupled Position-Dependent-Mass Oscillators with Microscopic Tunneling and Dynamical Phase Accrual

We develop a non-perturbative treatment of entanglement dynamics in two coupled harmonic oscillators with position-dependent mass (PDM) and arbitrary Hermitian operator ordering. A time-dependent symplectic transformation diagonalizes the coupled Hamiltonian at every instant (verified symplectic and diagonal to machine precision at all sampled t and μ), reducing the dynamics, up to a small non-adiabatic term, to two Ermakov–Pinney equations with closed-form Gaussian solutions. The formalism is validated against two independent methods, truncated Fock-space propagation and direct Heisenberg-picture integration of the linear response matrix, yielding a maximum relative error of 0.339% on the covariance matrix in both cases (mean Heisenberg-picture error 0.113%; the error on the logarithmic negativity itself reaches 3.5% at t = T), which shows that the Gaussian/Lewis–Riesenfeld reduction reproduces the second moments of the full quantum dynamics to within a fraction of a percent. The bilinear coupling μ is shown to be a threshold-free entanglement resource: the logarithmic negativity at t = T rises monotonically from EN = 0 at μ = 0 to EN = 0.158 at μ = 0.55, saturating as the mixing angle approaches, starting from the decoupled regime (μ = 0), but never reaches the maximal 45° hybridization. A microscopic double-well tunneling model, derived from Wannier-state overlap integrals, yields an effective coupling μeff = μ(1 + g\(√C_T\) ) that scales with the tunneling coherence \(√C_T\) rather than linearly with CT (the squared coherence), enhancing EN by 72.4% relative to the zero-tunneling baseline at strong tunneling (CT = 0.91). Self-consistent PDM renormalization mildly suppresses entanglement by 7.8% at ε = 0.5, via a 9.8% effective-mass increase, while weak local dissipation, treated with the full Gaussian master equation, reduces the final entanglement only moderately (by 8.6% at κ = 0.2) and the purity by about 1%.