Gravitational wave signal and noise response of an optically levitated sensor in a Fabry-P\'erot cavity

Optically levitated sensors inside a Fabry-P\'erot cavity have been proposed for high-frequency gravitational-wave (GW) detection, though their configuration for gravitational wave sensitivity exhibits counterintuitive features. We provide a new detailed general relativistic derivation of the interaction between a gravitational wave and a levitated object in an optical cavity, demonstrating gauge independence of the observable response. We find a strong asymmetric dependence of the strain signal on trap position, maximized when the sensor is located near the input mirror, in agreement with previous results. A key new result of this work is the consequence of this asymmetry on the noise coupling: the coupling of input-mirror displacements to the strain signal can be highly suppressed relative to that of end-mirror displacements and common-mode mirror motion. These results clarify the physical origin of the gravitational wave interaction with such a sensor and establish crucial design principles for optical levitation based high-frequency GW detectors.