An Effective Scaling Framework for Non-Adiabatic Mode Dynamics

Abstract: This study proposes an effective theoretical framework for non-adiabatic parametric excitation in structured media, incorporating a nonlinear frequency regulator U as a stabilizing mechanism. We introduce the non-adiabaticity parameter as a time-local diagnostic for driven non-stationary systems and analyze its competition with nonlinear spectral detuning through the scaling ratio. The principal physical result is that strongly nonlinear oscillatory systems can exhibit saturation of non-adiabatic parametric amplification: when the nonlinear regulator becomes sufficiently strong, exponential mode growth is dynamically suppressed and the excitation evolves toward a bounded low-occupancy regime. Using numerical verification in an expanded 100-level bosonic Fock basis, we demonstrate a crossover from hyperbolic amplification dynamics toward an effectively bounded response associated with spectral blockade and suppression of higher-order mode occupation. These results suggest that nonlinear spectral stabilization may represent a general mechanism for finite-amplitude non-adiabatic dynamics in driven structured media.