  Periodic driving has emerged as a powerful tool in the quest to engineer new
and exotic quantum phases. While driven many-body systems are generically
expected to absorb energy indefinitely and reach an infinite-temperature state,
the rate of heating can be exponentially suppressed when the drive frequency is
large compared to the local energy scales of the system -- leading to
long-lived 'prethermal' regimes. In this work, we experimentally study a
bosonic cloud of ultracold atoms in a driven optical lattice and identify such
a prethermal regime in the Bose-Hubbard model. By measuring the energy
absorption of the cloud as the driving frequency is increased, we observe an
exponential-in-frequency reduction of the heating rate persisting over more
than 2 orders of magnitude. The tunability of the lattice potentials allows us
to explore one- and two-dimensional systems in a range of different interacting
regimes. Alongside the exponential decrease, the dependence of the heating rate
on the frequency displays features characteristic of the phase diagram of the
Bose-Hubbard model, whose understanding is additionally supported by numerical
simulations in one dimension. Our results show experimental evidence of the
phenomenon of Floquet prethermalization, and provide insight into the
characterization of heating for driven bosonic systems.
