  High-order quantum nonlinearity is an important prerequisite for the advanced
quantum technology leading to universal quantum processing with large
information capacity of continuous variables. Levitated optomechanics, a field
where motion of dielectric particles is driven by precisely controlled tweezer
beams, is capable of attaining the required nonlinearity via engineered
potential landscapes of mechanical motion. Importantly, to achieve nonlinear
quantum effects, the evolution caused by the free motion of mechanics and
thermal decoherence have to be suppressed. For this purpose, we devise a method
of stroboscopic application of a highly nonlinear potential to a mechanical
oscillator that leads to the motional quantum non-Gaussian states exhibiting
nonclassical negative Wigner function and squeezing of a nonlinear combination
of mechanical quadratures. We test the method numerically by analysing highly
instable cubic potential with relevant experimental parameters of the levitated
optomechanics, prove its feasibility within reach, and propose an experimental
test. The method paves a road for unique experiments instantaneously
transforming a ground state of mechanical oscillators to applicable
nonclassical states by nonlinear optical force.
