Non-Markovian two-time correlation functions for optomechanical systems

We extend a stochastic Schrödinger equation framework for calculating non-Markovian two-time correlation functions and spectral responses to cavity optomechanical systems. Conventional spectral analysis relies on the quantum regression theorem, whose standard same-generator form is not generally justified for finite-memory environments. In this paper, we consider a linearized optomechanical system in which the mechanical mode interacts with a structured phononic bath and the optical cavity experiences Markovian dissipation. By constructing two-time correlations directly from paired non-Markovian stochastic trajectories, we obtain finite-time spectral functions without invoking regression assumptions. Numerical simulations reveal that environmental memory generates long-lived correlations and qualitatively distinct finite-window spectral features, including non-Lorentzian line shapes and multiple sideband structures relative to the short-memory regime. This work extends the non-Markovian TTCF method to a continuous-variable mixed-bath optomechanical system and provides a framework for studying dynamical correlations in structured environments.