  In this paper, we analyze the harmonically driven Jaynes-Cummings and
Lipkin-Meshkov-Glick models using both numerical integration of time-dependent
Hamiltonians and Floquet theory. For a separation of time-scales between the
drive and intrinsic Rabi oscillations in the former model, the driving results
in an effective periodic reversal of time. The corresponding Floquet
Hamiltonian is a Wannier-Stark model, which can be analytically solved. Despite
the chaotic nature of the driven Lipkin-Meshkov-Glick model, moderate system
sizes can display qualitatively different behaviors under varying system
parameters. Ergodicity arises in systems that are neither adiabatic nor
diabatic, owing to repeated multi-level Landau-Zener transitions. Chaotic
behavior, observed in slow driving, manifests as random jumps in the
magnetization, suggesting potential utility as a random number generator.
Furthermore, we discuss both models in terms of what we call Floquet Fock state
lattices.
