Fluctuation dissipation theorems connect the linear response of a physical system to a perturbation to the steady-state correlation functions. Until now, most of these theorems have been derived for finite-dimensional systems. However, many relevant physical processes are described by systems of infinite dimension in the Gaussian regime. In this work, we find a linear response theory for quantum Gaussian systems subject to time dependent Gaussian channels. In particular, we establish a fluctuation dissipation theorem for the covariance matrix that connects its linear response at any time to the steady state two-time correlations. The theorem covers non-equilibrium scenarios as it does not require the steady state to be at thermal equilibrium. We further show how our results simplify the study of Gaussian systems subject to a time dependent Lindbladian master equation. Finally, we illustrate the usage of our new scheme through some examples. Due to broad generality of the Gaussian formalism, we expect our results to find an application in many physical platforms, such as opto-mechanical systems in the presence of external noise or driven quantum heat devices.
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