  Quantum Floquet engineering seeks to externally control systems by means of
quantum fields. However, to faithfully capture the physics at arbitrary
coupling, a gauge-invariant description of light-matter interaction is
required, which makes the Hamiltonian highly nonlinear in the photonic
operators. Here we provide a non-perturbative truncation scheme, which is valid
for arbitrary coupling strength. With this framework, we investigate the role
of light-matter correlations, which are absent in systems described by
semiclassical Floquet engineering. We find that even in the high-frequency
regime, their importance can be crucial, in particular for the topological
properties of the system. As an example we show that in an SSH chain coupled to
a cavity, light-matter correlations break chiral symmetry, strongly affecting
the robustness of its edge states. In addition, we show how light-matter
correlations are imprinted in the photonic spectral function, and discuss their
relation with the topology of the photonic bands.
