Over the last decade, systems of individually-controlled neutral atoms, interacting with each other when excited to Rydberg states, have emerged as a promising platform for quantum simulation of many-body problems, in particular spin systems. Here, we review the techniques underlying quantum gas microscopes and arrays of optical tweezers used in these experiments, explain how the different types of interactions between Rydberg atoms allow a natural mapping onto various quantum spin models, and describe recent results that were obtained with this platform to study quantum many-body physics.
I. MANY-BODY PHYSICS WITH SYNTHETIC MATTERMany-body physics is the field that studies the behavior of ensembles of interacting quantum particles. This is a broad area encompassing almost all condensed matter physics, but also nuclear and high-energy physics. Despite the immense successes obtained over the last decades, many phenomena observed experimentally still do not have a fully satisfactory explanation. At the origin of the difficulty lies the exponential scaling of the size of the Hilbert space with the number of interacting particles. In practice, the best known ab-initio methods allow calculating the evolution of less than 50 particles. To investigate relevant questions involving a much larger number of particles (after all even 1 mg of usual matter contains already 10 18 atoms!), one must rely on approximations, and the art of solving the many-body problem largely relies on mastering them. However, using approximations is not always possible and it may be hard to assess their range of validity. One approach to move forward was suggested by Richard Feynman [1] and consists in building a synthetic quantum system in the lab, implementing a model of interest for which no other way to solve it is known. The model may be an approximate description of a real material, but it can also be a purely abstract one. In this case, its implementation leads to the construction of an artificial many-body system, which becomes an object of study in its own. One appealing feature of this approach is the ability to vary the parameters of the model in ranges inaccessible otherwise, thus providing a way to better understand their respective influence. For example, if one is interested in the influence of interatomic interactions on the phase of a given system, synthetic systems become interesting as they allow varying their strength in a way which is usually impossible in real materials. The approach introduced by Feynman is usually referred to as quantum simulation [2, 3], but it can be viewed more generally as exploring many-body physics with synthetic systems: in the same way chemists design new materials exhibiting interesting properties (such as magnetism, superconductivity. . . ), physicists assemble artificial systems and study their properties, with the hope to observe new phenomena.For a long time, this idea remained theoretical as the experimental control over quantum objects was not advanced enough. The situation changed radically...