  Experimental platforms based on ultracold atomic gases have significantly
advanced the quantum simulation of complex systems, yet the exploration of
phenomena driven by long-range interactions remains a formidable challenge.
Currently available methods utilizing dipolar quantum gases or multi-mode
cavities allow to implement long-range interactions with a $1/r^3$ character or
with a spatial profile fixed by the mode-structure of the vacuum
electromagnetic field surrounding the atoms, respectively. Here we propose an
experimental scheme employing laser-painted cavity-mediated interactions, which
enables the realization of atom-atom interactions that are fully tunable in
range, shape, and sign. Our approach combines the versatility of cavity quantum
electrodynamics with the precision of laser manipulation, thus providing a
highly flexible platform for simulating and understanding long-range
interactions in quantum many-body systems. Our analytical predictions are
supported by numerical simulations describing the full dynamics of atoms,
laser, and cavity. The latter demonstrate that there is a wide and
experimentally accessible parameter regime where our protocol optimally works.
The methodology not only paves the way for exploring new territories in quantum
simulation but also enhances the understanding of fundamental physics,
potentially leading to the discovery of novel quantum states and phases.
