  We introduce and theoretically analyze a scheme to prepare and detect
non-Gaussian quantum states of an optically levitated particle via the
interaction with a light pulse that generates cubic and inverted potentials. We
show that this allows operating on short time- and lengthscales, which
significantly reduces the demands on decoherence rates in such experiments.
Specifically, our scheme predicts the observation of interference of
nanoparticles with a mass above $10^8$ atomic mass units delocalised over
several nanometers, on timescales of milliseconds, when operated at vacuum
levels around $10^{-10}$~mbar and at room temperature. We discuss the prospect
of using this approach for coherently splitting the wavepacket of massive
dielectric objects using neither projective measurements nor an internal level
structure.
