  The Floquet engineering opens the way to create new topological states
without counterparts in static systems. Here, we report the experimental
realization and characterization of new anomalous topological states with
high-precision Floquet engineering for ultracold atoms trapped in a shaking
optical Raman lattice. The Floquet band topology is manipulated by tuning the
driving-induced band crossings referred to as band inversion surfaces (BISs),
whose configurations fully characterize the topology of the underlying states.
We uncover various exotic anomalous topological states by measuring the
configurations of BISs which correspond to the bulk Floquet topology. In
particular, we identify an unprecedented anomalous Floquet valley-Hall state
that possesses anomalous helicallike edge modes protected by valleys and a
chiral state with high Chern number.
