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In this paper, we propose that a simple model, in which fermions coupled to adynamical lattice gauge field, can be engineered via the Floquet approach. Themodel possess both a independent Maxwell term and the local $Z_{2}$ gaugesymmetry. Our proposal relies on a species dependent optical lattice, and canbe achieved in one, two or three dimension. By an unitary transformation, thismodel can be mapped into a non-interacting composite Fermion system withfluctuating back ground charge. With the help of this composite Fermionpicture, two characteristic observations are predicted. One is theradio-frequency spectroscopy, which exhibits no dispersion in all parameterregimes. Second is the the dynamical localization, which depends on thestructure of the initial states.

Floquet Engineering of a Dynamical $Z_{2}$ Lattice Gauge Field with Ultracold Atoms