Harmonic damped oscillators with feedback. A Langevin study

We consider a system in direct contact with a thermal reservoir and which, if left unperturbed, is well described by a memory-less equilibrium Langevin equation of the second order in the time coordinate. In such conditions, the strength of the noise fluctuations is set by the damping factor, in accordance with the Fluctuation and Dissipation theorem. We study the system when it is subject to a feedback mechanism, by modifying the Langevin equation accordingly. Memory terms now arise in the time evolution, which we study in a non-equilibrium steady state. Two types of feedback schemes are considered, one focusing on time shifts and one on phase shifts, and for both cases we evaluate the power spectrum of the system's fluctuations. Our analysis finds application in feedback cooled oscillators, such as the Gravitational Wave detector AURIGA.