  The use of periodic driving for synthesizing many-body quantum states depends
crucially on the existence of a prethermal regime, which exhibits drive-tunable
properties while forestalling the effects of heating. This motivates the search
for direct experimental probes of the underlying localized nonergodic nature of
the wave function in this metastable regime. We report experiments on a
many-body Floquet system consisting of atoms in an optical lattice subjected to
ultrastrong sign-changing amplitude modulation. Using a double-quench protocol
we measure an inverse participation ratio quantifying the degree of prethermal
localization as a function of tunable drive parameters and interactions. We
obtain a complete prethermal map of the drive-dependent properties of Floquet
matter spanning four square decades of parameter space. Following the full time
evolution, we observe sequential formation of two prethermal plateaux,
interaction-driven ergodicity, and strongly frequency-dependent dynamics of
long-time thermalization. The quantitative characterization of the prethermal
Floquet matter realized in these experiments, along with the demonstration of
control of its properties by variation of drive parameters and interactions,
opens a new frontier for probing far-from-equilibrium quantum statistical
mechanics and new possibilities for dynamical quantum engineering.
