  Periodically-driven quantum systems are currently explored in view of
realizing novel many-body phases of matter. This approach is particularly
promising in gases of ultracold atoms, where sophisticated shaking protocols
can be realized and inter-particle interactions are well controlled. The
combination of interactions and time-periodic driving, however, often leads to
uncontrollable heating and instabilities, potentially preventing practical
applications of Floquet-engineering in large many-body quantum systems. In this
work, we experimentally identify the existence of parametric instabilities in
weakly-interacting Bose-Einstein condensates in strongly-driven optical
lattices through momentum-resolved measurements. Parametric instabilities can
trigger the destruction of weakly-interacting Bose-Einstein condensates through
the rapid growth of collective excitations, in particular in systems with weak
harmonic confinement transverse to the lattice axis.
