Title: A cavity-microscope for micrometer-scale control of atom-photon interactions

Abstract: Cavity quantum electrodynamics offers the possibility to observe and control the motion of few or individual atoms, enabling the realization of various quantum technological tasks such as quantum-enhanced metrology or quantum simulation of strongly-correlated matter. A core limitation of these experiments lies in the mode structure of the cavity field, which is hard-coded in the shape and geometry of the mirrors. As a result, most applications of cavity QED trade spatial resolution for enhanced sensitivity. Here, we propose and demonstrate a cavity-microscope device capable of controlling in space and time the coupling between atoms and light in a single-mode high-finesse cavity, reaching a spatial resolution an order-of-magnitude lower than the cavity mode waist. This is achieved through local Floquet engineering of the atomic level structure, imprinting a corresponding atom-field coupling. We illustrate this capability by engineering micrometer-scale coupling, using cavity-assisted atomic measurements and optimization. Our system forms an optical device with a single optical axis and has the same footprint and complexity as a standard Fabry-Perot cavity or confocal lens pair, and can be used for any atomic species. This technique opens a wide range of perspectives from ultra-fast, cavity-enhanced mid-circuit readout to the quantum simulation of fully connected models of quantum matter such as the Sachdev-Ye-Kitaev model.