More than 30 years ago, Richard Feynman outlined the visionary concept of a quantum simulator for carrying out complex physics calculations. Today, his dream has become a reality in laboratories around the world. In my talk I will focus on the remarkable opportunities offered by ultracold quantum gases trapped in optical lattices to address fundamental physics questions ranging from condensed matter physics over statistical physics to high energy physcis with table-top experiment.

For example, I will show how it has now become possible to image and control quantum matter with single atom sensitivity and single site resolution, thereby allowing one to directly image individual quantum fluctuations of a many-body system. Such ultrahigh resolution and sensitivity have also enabled us to detect ‘Higgs’ type excitations occurring at 24 orders of magnitude lower energy scales than in high energy physics experiments. I will also show, how recent experiments with cold gases in optical lattices have enabled to realise and probe artificial magnetic fields that lie at the heart of topological energy bands in a solid. Using a novel ‘Aharonov-Bohm’ type interferometer that acts within the momentum space, we are now able to fully determine experimentally the geometric structure of an energy band.