Floquet approach to $\mathbb{Z}_{2}$ lattice gauge theories with ultracold atoms in optical lattices

$\mathbb{Z}_2$ lattice gauge theories (LGTs) are of high interest in condensed matter physics and topological quantum computation. The investigation of strongly-interacting regimes, however, is especially challenging and in general difficult to access with conventional numerical methods. Here, we take a first step using analog quantum simulations and present an approach to realize $\mathbb{Z}_2$ LGTs. We use a two-component mixture of ultracold bosonic atoms with strong on-site interactions in an optical two-site potential together with resonant periodic driving. For particular driving parameters, the effective Floquet Hamiltonian exhibits $\mathbb{Z}_2$ symmetry. We study the dynamics of the system for different initial states and find that it is well described by a full time-dependent description. Moreover, it is non-trivial due to the imposed gauge constraints and in agreement with predictions from the ideal $\mathbb{Z}_2$ LGT. We reveal challenges that arise due to symmetry-breaking terms, which may be relevant for any experimental implementation, and outline potential pathways to overcome them. The results provide important insights for studies of LGTs based on Floquet techniques and the two-site model constitutes a minimal instance for extended LGTs coupled to matter.