True and apparent motion of optomechanical resonators, with applications to feedback cooling of gravitational wave detector test masses

Modern optomechanical systems employ increasingly sophisticated quantum-mechanical states of light to probe and manipulate mechanical motion. Squeezed states are now used routinely to enhance the sensitivity of gravitational-wave interferometers to small external forces, and they are also used in feedback-based trapping and damping experiments on the same interferometers to enhance the achievable cooling of fluctuations in the differential test mass mode (arXiv:2102.12665). In this latter context, an accurate accounting of the true test mass motion, incorporating all sources of loss, the effect of feedback control, and the influence of classical force and sensing noises, is paramount. We work within the two-photon formalism to provide such an accounting, which extends a previously described decomposition of the quantum-mechanical noise of the light field (arxiv:2105.12052). This decomposition provides insight, rooted in physically motivated parameters, into the optimal squeezed state and feedback control configuration that should be employed to achieve the lowest fluctuations. We apply this formalism to feedback damping experiments in current and possible future gravitational-wave interferometers -- LIGO A+, LIGO Voyager, Cosmic Explorer (CE), and CE Voyager -- and discuss how these multi-degree-of-freedom systems might be compared to a single degree-of-freedom oscillator. We find that, for the oscillator definition used most commonly in the literature so far, occupation numbers below 1 are possible in these interferometers over a frequency range comparable to the bandwidth of the trapped and cooled oscillator. We also discuss several technical issues in cooling experiments with gravitational-wave detectors