Title: Laser-painted cavity-mediated interactions in a quantum gas

Abstract: 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.