  Strong light fields have created spectacular opportunities to tailor novel
functionalities of solids. Floquet-Bloch states can form under periodic driving
of electrons and enable exotic quantum phases. On subcycle time scales,
lightwaves can simultaneously drive intraband currents and interband
transitions, which enable high-harmonic generation (HHG) and pave the way
towards ultrafast electronics. Yet, the interplay of intra- and interband
excitations as well as their relation with Floquet physics have been key open
questions as dynamical aspects of Floquet states have remained elusive. Here we
provide this pivotal link by pioneering the ultrafast buildup of Floquet-Bloch
bands with time- and angle-resolved photoemission spectroscopy. We drive
surface states on a topological insulator with mid-infrared fields - strong
enough for HHG - and directly monitor the transient band structure with
subcycle time resolution. Starting with strong intraband currents, we observe
how Floquet sidebands emerge within a single optical cycle; intraband
acceleration simultaneously proceeds in multiple sidebands until high-energy
electrons scatter into bulk states and dissipation destroys the Floquet bands.
Quantum nonequilibrium calculations explain the simultaneous occurrence of
Floquet states with intra- and interband dynamics. Our joint experiment-theory
study opens up a direct time-domain view of Floquet physics and explores the
fundamental frontiers of ultrafast band-structure engineering.
