The Floquet approach to periodically time-dependent quantum systems, providing an analog of the Bloch states for crystal lattices, has recently met with renewed interest. In first part of this colloquium I will give a brief introduction to this framework, using multiphoton transitions of a periodically driven "particle in a box" as a guiding example. The second part will be devoted to Floquet time crystals, which have been heralded as a new paradigm in quantum statistical mechanics. Employing the Bose-Hubbard dimer as a test case which can be investigated numerically without approximation, I will argue that such time crystals appear quite naturally in periodically driven many-body systems. The analysis sheds light on the validity of the mean-field approximation under conditions such that the time-dependent nonlinear Gross-Pitaevskii equation furnishes chaotic solutions, indicating that even fine details customarily associated with chaotic Hamiltonian systems can be resolved by many-body Schrödinger dynamics.
Jürgen Schnack