Supplemental Material for Amplified opto-mechanical transduction of virtual radiation pressure

In this supplemental material we present the details of the hybrid opto-mechanical structure considered in the main text. The system consists of an electromagnetic mode ultrastrongly coupled to a matter degree of freedom (via dipole interaction) and to a mechanical oscillator (via radiation pressure). In this way, the mechanical mode can be used as a probe of the dressed structure of the light-matter system, i.e., as a transducer of virtual radiation pressure. To amplify the signal, we consider a modulation of the opto-mechanical interaction at the mechanical frequency. We model the matter as either a spin (in the low-energy limit) or a bosonic mode and find a unified effective master equation (with different parameters) which describe them. We use these results to calculate bounds on the minimal amount of resources necessary to resolve virtual radiation-pressure effects when probing the mechanical quadratures. We finally present an analysis of experimental feasibility in an electro-mechanical setting.