Topological Floquet Engineering in One-Dimensional Optical Lattices through Simultaneous Amplitude and Phase Modulations

We present a numerical study on the topological properties of a three-band system in one-dimensional optical lattices subjected to periodic driving. We explore scenarios involving simultaneous modulations of the amplitude and phase of the optical lattice potential, and observe the emergence of topologically nontrivial bands when the modulation frequency resonates with the s-d band transition, contingent on the relative phase of the amplitude and phase modulations. Calculations of the Zak phase and the entanglement spectrum of the Floquet bands illustrate the topologically nontrivial characteristics. We show that parity-time reversal symmetry is the protective symmetry governing the system's topology. Finally, we discuss an experimental scheme to realize this three-band model.