Experimental engineering of Floquet topological phases in a one-dimensional optical lattice

Periodic driving enables realization of topological phases without static counterparts. We experimentally realize and detect a one-dimensional anomalous Floquet topological phase in an optical lattice, using multi-frequency control to manipulate the relative sign structure of the gap windings $(W_0,W_\pi)$ associated with the $0$ and $\pi$ quasienergy gaps. We develop a lattice-depth modulation scheme that induces staggered nearest-neighbor $s$-$p$ orbital couplings and realize a minimal nontrivial Floquet topology under single-tone driving. Introducing a second tone, its relative phase controls the effective coupling signs in the $0$ and $\pi$ gaps, thereby tuning the corresponding windings to add and produce a high-winding phase or to cancel while retaining nontrivial gap indices. We read out $(W_0,W_\pi)$ with a band-inversion-surface (BIS)-resolved Ramsey protocol assisted by lattice-position shaking, which measures relative Floquet phases on the BISs. Controlled quenches further confirm phase-dependent band modifications even at quasimomenta far from resonance. These results establish multi-frequency control with a tunable relative phase as a quantitative route to engineering anomalous Floquet topology, and demonstrate phase-coherent coexistence of distinct drive modalities.