  The realization of spin-orbit-coupled ultracold gases has driven a wide range
of researches and is typically based on the rotating wave approximation (RWA).
By neglecting the counter-rotating terms, RWA characterizes a single
near-resonant spin-orbit (SO) coupling in a two-level system. Here, we propose
and experimentally realize a new scheme for achieving a pair of two-dimensional
(2D) SO couplings for ultracold fermions beyond RWA. This work not only
realizes the first anomalous Floquet topological Fermi gas beyond RWA, but also
significantly improves the lifetime of the 2D-SO-coupled Fermi gas. Based on
pump-probe quench measurements, we observe a deterministic phase relation
between two sets of SO couplings, which is characteristic for our beyond-RWA
scheme and enables the two SO couplings to be simultaneously tuned to the
optimum 2D configurations. We observe intriguing band topology by measuring
two-ring band-inversion surfaces, quantitatively consistent with a Floquet
topological Fermi gas in the regime of high Chern numbers. Our study can open
an avenue to explore exotic SO physics and anomalous topological states based
on long-lived SO-coupled ultracold fermions.
