  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.
