  Parametric instabilities in interacting systems can lead to the appearance of
new structures or patterns. In quantum gases, two-body interactions are
responsible for a variety of instabilities that depend on the characteristics
of both trapping and interactions. We report on the Floquet engineering of such
instabilities, on a Bose-Einstein condensate held in a time-modulated optical
lattice. The modulation triggers a destabilization of the condensate into a
state exhibiting a density modulation with a new spatial periodicity. This new
crystal-like order directly depends on the modulation parameters: the interplay
between the Floquet spectrum and interactions generates narrow and adjustable
instability regions, leading to the growth, from quantum or thermal
fluctuations, of modes with a density modulation non commensurate with the
lattice spacing. This study demonstrates the production of metastable exotic
states of matter through Floquet engineering, and paves the way for further
studies of dissipation in the resulting phase, and of similar phenomena in
other geometries.
