  Heating to high-lying states strongly limits the experimental observation of
driving induced non-equilibrium phenomena, particularly when the drive has a
broad spectrum. Here we show that, for entire families of structured random
drives known as random multipolar drives, particle excitation to higher bands
can be well controlled even away from a high-frequency driving regime. This
opens a window for observing drive-induced phenomena in a long-lived prethermal
regime in the lowest band.
