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

Technical Summary: A Bogoliubov-Ratio Framework for Quantum-Information Diagnostics

Problem Statement
The dynamics of time-dependent two-mode bosonic systems, such as those describing primordial cosmological perturbations or nondegenerate optical parametric amplifiers, are conventionally characterized by a squeezing amplitude $r_k(\eta)$ and a squeezing phase $\phi_k(\eta)$, or equivalently by Bogoliubov coefficients $\alpha_k(\eta)$ and $\beta_k(\eta)$. While these parametrizations are complete, their direct implementation in general time-dependent systems is often cumbersome. The variables $r_k$ and $\phi_k$ satisfy coupled nonlinear differential equations, and the phase $\phi_k$ can undergo rapid oscillations. Furthermore, evaluating information-theoretic quantities (such as purity and entropy) typically requires reconstructing and diagonalizing the reduced density matrix for each specific model or driving protocol. This model-by-model approach obscures the universal information-theoretic structure inherent to normalized two-mode pair states and complicates comparisons between different physical realizations (e.g., cosmological vs. optical).

Methodology
The authors propose a compact and unified framework based on the Bogoliubov ratio defined as $\lambda_k(\eta) \equiv \beta_k(\eta)/\alpha_k(\eta)$, subject to the normalization condition $|\alpha_k|^2 - |\beta_k|^2 = 1$ and the convergence domain $|\lambda_k| < 1$.

1. State Representation: The authors establish that a normalized generalized two-mode squeezed state can be exactly represented in the paired number basis $|n, n\rangle$ using a single complex variable $\lambda_k$:
   $$|\psi_k(\eta)\rangle = \sqrt{1 - |\lambda_k(\eta)|^2} \sum_{n=0}^{\infty} \lambda_k^n(\eta) |n_k, n_{-k}\rangle.$$
   This formulation absorbs state-level deformations (arising from generalized Lanczos coefficients or specific background dynamics) directly into $\lambda_k$.

2. Dynamical Reduction: For a general time-dependent quadratic two-mode Hamiltonian of the form:
   $$\hat{H}_k(\eta) = \omega_k(\eta) (\hat{a}^\dagger_k \hat{a}_k + \hat{a}^\dagger_{-k} \hat{a}_{-k} + 1) + i g_k(\eta) (\hat{a}^\dagger_k \hat{a}^\dagger_{-k} - \hat{a}_k \hat{a}_{-k}),$$
   the Schrödinger equation reduces to a single, closed Riccati-type equation for $\lambda_k$:
   $$\frac{d\lambda_k}{d\eta} = g_k(\eta)(1 - \lambda_k^2) - 2i\omega_k(\eta)\lambda_k.$$
   Here, $\omega_k$ governs the free phase rotation, and $g_k$ controls pair production.

3. Universal Diagnostics: Upon tracing out the partner mode ($-k$), the phase of $\lambda_k$ drops out. The spectrum of the one-mode reduced density matrix is determined entirely by the squared magnitude $q_k(\eta) = |\lambda_k(\eta)|^2$:
   $$p_n(\eta) = [1 - q_k(\eta)] q_k^n(\eta).$$
   Consequently, standard information-theoretic metrics—purity ($\mu_k$), linear entropy ($S_{L,k}$), Rényi-2 entropy ($S_{2,k}$), and von Neumann entropy ($S_{vN,k}$)—are derived as explicit, model-independent functions of $q_k$.

Key Contributions

• Variable Reduction: The framework reduces the description of two-mode squeezing dynamics from two coupled real variables ($r_k, \phi_k$) to a single complex variable ($\lambda_k$) governed by a first-order differential equation.
• Model-Independent Formulas: The authors provide explicit formulas for reduced-state purity and entropies that depend solely on $|\lambda_k|^2$. This eliminates the need to reconstruct or diagonalize the reduced density matrix for every new time-dependent Hamiltonian.
• Separation of Dynamics and Diagnostics: The framework cleanly factorizes the problem into two steps:
  1. Model-dependent dynamics: Solving the Riccati equation for $\lambda_k$ using specific inputs $\omega_k(\eta)$ and $g_k(\eta)$.
  2. Universal diagnostics: Converting $|\lambda_k|^2$ to information measures using fixed, universal formulas.

Results
The framework is demonstrated in two distinct non-stationary setups:

1. Primordial Cosmological Perturbations:

   • The authors apply the framework to modes with effective frequency $\omega_k = \sqrt{k^2 + \alpha/\eta^2}$ and coupling $g_k = -1/\eta$.
   • Numerical solutions show that background-induced phase rotation and frequency softening (controlled by the parameter $\nu$) regulate the growth of squeezing.
   • As the effective frequency softens (larger $\nu$), the competition between phase rotation and pair production shifts, leading to larger $|\lambda_k|$, reduced one-mode purity, and increased entropies.
   • The results clarify how background dynamics suppress coherent squeezing accumulation compared to an ideal resonant benchmark.

2. Chirped-Pulse Nondegenerate Optical Parametric Amplifier:

   • The framework is applied to a system with a Gaussian pump envelope and a time-dependent frequency chirp (detuning).
   • The results capture the delayed onset of correlation production (localized near the pulse center) and the suppression of squeezing accumulation due to the chirp.
   • Unlike the constant-coupling limit where entropies grow indefinitely, the finite pump duration and frequency chirp lead to finite late-time saturation values for purity and entropy.
   • The numerical recovery of the standard $\tanh(\tau)$ limit for constant resonant coupling validates the sign conventions and normalization of the proposed Riccati equation.

Significance and Claims
The paper claims that this Bogoliubov-ratio formulation offers an efficient, standardized diagnostic tool for a broad class of parametrically driven quadratic bosonic systems. Its primary significance lies in:

• Unification: It isolates the universal information content of two-mode pair production from the specific physical mechanisms (cosmological expansion vs. optical pumping) responsible for the evolution.
• Computational Efficiency: By avoiding the reconstruction of reduced density matrices for each model, it streamlines the calculation of quantum information metrics.
• Physical Insight: It provides a clear mechanism to understand how background-induced phase rotation and frequency softening regulate state mixedness and squeezing growth.

The authors note that the framework applies to any system where the evolved state retains a normalized paired Bogoliubov representation generated by a Hermitian quadratic Hamiltonian. They explicitly state that extensions to genuinely open-system dynamics (e.g., Lindblad evolution) or weak non-Gaussian interactions are outside the scope of the current work.