Non-Markovian two-time correlation functions for optomechanical systems

The two-time correlation function (TTCF) is a fundamental tool in precision detection and quantum measurement. In this work, we present a theoretical study of the dynamical behavior of a cavity optomechanical system. Employing the stochastic Schrödinger equation formalism, we systematically investigate both the long-time steady-state and time-dependent two-time correlation functions under different environmental conditions. Our numerical results reveal that the steady-state TTCFs exhibit qualitatively distinct features in Markovian and non-Markovian environments. Moreover, we show that finite-time TTCFs provide a more sensitive and detailed characterization of environmental properties compared to conventional spectral-function-based approaches. Importantly, we demonstrate that the stability of the system is jointly determined by the eigenfrequency of the system and the environmental memory parameter. These results establish a unified framework for the analysis of two-time correlation functions in non-Markovian regimes, offering deeper physical insight into the role of environmental effects in quantum measurement and system stability.