Quantum simulation - Engineering and understanding quantum systems atom- by-atom

The computational resources required to describe the full state of a quantum many-body system scale exponentially with the number of constituents. This severely limits our ability to explore and understand the fascinating phenomena of quantum systems using classical algorithms. Quantum simulation offers a potential route to overcome these limitations. The idea is to build a well-controlled quantum system in the lab, which represents the problem of interest and whose properties can be studied by performing measurements. In this talk I will introduce quantum simulators based on neutral atoms that are confined in optical arrays using laser beams. State-of-the-art experiments now reach several thousand particles, while maintaining control on the level of single atoms. I will show how these systems can be used to study topological phases of matter and how we can use them to address fundamental questions about the thermalization of isolated quantum systems. In the end I will provide a brief outlook on new directions in the field based on the unique properties of alkaline-earth(- like) atoms.