  Generating macroscopic non-classical quantum states is a long-standing
challenge in physics. Anharmonic dynamics is an essential ingredient to
generate these states, but for large mechanical systems, the effect of the
anharmonicity tends to become negligible compared to decoherence. As a possible
solution to this challenge, we propose to use a motional squeezed state as a
resource to effectively enhance the anharmonicity. We analyze the production of
negativity in the Wigner distribution of a quantum anharmonic resonator
initially in a squeezed state. We find that initial squeezing enhances the rate
at which negativity is generated. We also analyze the effect of two common
sources of decoherence, namely energy damping and dephasing, and find that the
detrimental effects of energy damping are suppressed by strong squeezing. In
the limit of large squeezing, which is needed for state-of-the-art systems, we
find good approximations for the Wigner function. Our analysis is significant
for current experiments attempting to prepare macroscopic mechanical systems in
genuine quantum states. We provide an overview of several experimental
platforms featuring nonlinear behaviors and low levels of decoherence. In
particular, we discuss the feasibility of our proposal with carbon nanotubes
and levitated nanoparticles.
