Topological band structure engineering of a one-dimensional optical lattice system with resonant shaking

Floquet engineering is a novel way to generate new properties out of a system. We study the topological band structure of a resonantly shaken one-dimensional optical lattice system, where the lattice potential is periodically modulated to couple the two lowest bloch bands. In a two-band approximation, we numerically show that degenerate edge states appear under a certain driving condition and the corresponding topological phase is protected by the chiral symmetry of the periodically driven system. The system's micromotion is characterized with oscillating Zak phases which are quantized only when the chiral symmetry condition is explicitly satisfied. We also describe the topological charge pumping effect, slowly modulating the driving parameters around a critical point, and investigate its adiabaticity with increasing modulation frequency. Finally, we discuss the experimental feasibility of realizing a flat band in the shaken lattice system.