  In quantum mechanics, adiabatic elimination is a standard tool that produces
a low-lying reduced Hamiltonian for a relevant subspace of states,
incorporating effects of its coupling to states with much higher energy.
Suppose this powerful elimination approach is applied to quasi-energy states in
periodically-driven systems, a critical question then arises that the violation
of the adiabatic condition caused by driven forces challenges such a presence
of spectral reduction in the non-equilibrium driven system. Here, both
theoretically and experimentally, we newly reported two kinds of driven-induced
eliminations universal in topologically-protected Floquet systems. We named
them "quasi-adiabatic elimination" and "high-frequency-limited elimination", in
terms of different driven frequencies that deny the underlying requirement for
the adiabatic condition. Both two non-adiabatic eliminations are observed in
our recently developed microwave Floquet simulator, a programmable test
platform composed of periodically-bending ultrathin metallic coupled corrugated
waveguides. Through the near-field imaging on our simulator, the mechanisms
between the adiabatic and driven-induced eliminations are revealed, indicating
the ubiquitous spectral decomposition for tailoring and manipulating Floquet
states with quasi-energies. Finally, we hope our findings may open up profound
and applicable possibilities for further developing Floquet engineering in
periodically-driven systems, ranging from condensed matter physics to
photonics.
