Abstract: | How fast can a quantum system evolve between two states? This question is important not only because of its fundamental significance, but also because of its far-reaching implications for future quantum technologies. It is addressed by the quantum speed limits, which set an ultimate upper bound on the evolution rate. In this colloquium, I will report on our experimental studies testing these limits with single atoms held in an optical lattice, and discuss further theoretical developments we obtained. I will also present the related problem of quantum brachistochrones, where the goal is to find the fastest route to transport an atom between spatially distant states. Here, we show that under certain circumstances the speed limits fail to capture the relevant time scale of the process, and a solution can be found instead by quantum optimal control. |