A Bogoliubov-ratio framework for quantum-information diagnostics of time-dependent two-mode Boson Hamiltonian

We present a compact and unified framework for quantum-information diagnostics of time-dependent two-mode bosonic systems based on the Bogoliubov ratio $\lambda_k(\eta) \equiv \beta_k(\eta)/\alpha_k(\eta)$. For a general time-dependent quadratic two-mode Hamiltonian, the state dynamics is exactly reduced to a single complex Riccati equation for $\lambda_k$. Upon tracing out one partner mode, the spectrum of the one-mode reduced density matrix is determined entirely by the squared magnitude $q_k(\eta) = \vert{}\lambda_k(\eta)\vert{}^2$. Consequently, we could construct the explicit, model-independent formula for the for the reduced-state purity, linear entropy, R\'enyi-2 entropy, and von Neumann entropy without reconstructing and diagonalizing the reduced density matrix on a model-by-model basis using coupled squeezing parameters ($r_k, \phi_k$). We demonstrate the utility of this framework in two distinct non-stationary setups: primordial cosmological perturbations and a chirped-pulse nondegenerate optical parametric amplifier. In the cosmological context, our formulation clarifies how background-induced phase rotation and frequency softening regulate squeezing growth and state mixedness; in the optical domain, it captures the delayed onset, suppression of squeezing accumulation, and late-time entropy saturation induced by finite pump duration and frequency chirp. By cleanly factorizing model-dependent driving protocols from universal information-theoretic metrics, this framework offers an efficient, standardized diagnostic tool for a broad class of parametrically driven quadratic bosonic systems.