Floquet engineering is an important tool for the engineering of novel band structures with interesting properties that go beyond those offered by static systems. Recently, Floquet systems have enabled the generation of Bloch bands with non-trivial topological properties, such as the Hofstadter and Haldane model. This led, among others, to direct measurements of momentum-resolved Berry curvatures and the development of novel techniques to determine the Chern-number of the artificially generated energy bands.

Besides this success studies of many-body phases in driven systems remain experimentally challenging in particular due to the interplay between periodic driving and interactions. In driven, time-periodic systems, energy conservation is relaxed due to the absorption and emission of energy quanta from the drive, and any ergodic system is expected to eventually heat up to infinite temperatures.

In this talk, I briefly review recent experimental advances in the generation of topological band structures in the non-interacting regime using Floquet engineering and present first studies of interacting atoms in driven 1D lattices. In particular, I will present experimental results obtained with bosonic atoms in driven 1D lattices that directly reveal the existence of parametric instabilities that lead to a depletion of the condensate. Our results point out ways to overcome these limitations in future experiments.

In the last part of my talk I will present recent results, where we have used a combination of periodic modulation and strong Hubbard interactions to realize a minimal building block of Z2 lattice gauge theories. We engineer a minimal coupling between matter and gauge fields using two different internal states of bosonic Rb atoms. The obtained lattice model displays local Z2 gauge symmetry, which we study experimentally in a double-well potential – the building block of extended lattice models.
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