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

We formulate a compact dynamical representation of quantum-information diagnostics for time-dependent two-mode bosonic systems in terms of the Bogoliubov ratio λ k(η)=β k(η)/α k(η). For vacuum-evolved pure Gaussian pair states generated by a Hermitian quadratic Hamiltonian, λ k obeys a closed complex Riccati equation. After one partner mode is traced out, the reduced-state spectrum is determined by q k=λ k squared, from which the purity, linear entropy, Rényi-2 entropy, and von Neumann entropy follow directly. These Gaussian results are mathematically equivalent to those obtained from the standard density-matrix or covariance-matrix formalism; the advantage of the ratio representation is a compact separation between model-dependent dynamics and universal state diagnostics. Writing λ k= sqrt q k e iθ k further exposes how pair production drives radial growth, whereas frequency rotation and detuning act through the phase and can suppress coherent squeezing accumulation. We also establish an exact extension to non-Gaussian SU(1,1) sectors. For a lowest-weight Fock seed with conserved number difference d, the same ratio equation governs the evolution, while the reduced spectrum becomes a negative-binomial distribution determined by (q k,d). The Gaussian formulas are recovered at d=0. Cosmological perturbations and a chirped-pulse optical parametric amplifier illustrate the common dynamical mechanism and its information-theoretic consequences.