Assembling quantum matter one atom at a time Antoine Browaeys Laboratoire Charles Fabry, Institut d’Optique, CNRS, 2 avenue A. Fresnel, 91127 Palaiseau, France e-mail: antoine.browaeys@institutoptique.fr Over the last twenty years, physicists have learned to manipulate individual quantum objects: atoms, ions, molecules, quantum circuits, electronic spins... It is now possible to build "atom by atom" a synthetic quantum matter. By controlling the interactions between atoms, one can study the properties of these elementary many-body systems: quantum magnetism, transport of excitations, superconductivity... and thus understand more deeply the N-body problem. More recently, it was realized that these quantum systems may find applications in the industry, such as finding the solution of combinatorial optimization problems. This seminar will present an example of a synthetic quantum system, based on lasercooled ensembles of individual atoms trapped in microscopic optical tweezer arrays. By exciting the atoms into Rydberg states, we make them interact, even at distances of more than ten micrometers. In this way, we study the magnetic properties of an ensemble of more than a hundred interacting ½ spins, in a regime in which simulations by usual numerical methods are already very challenging. Some aspects of this research led to the creation of a company, Pasqal.