  The generation of entangled states that display negative values of the Wigner
function in the quantum phase space is a challenging task, particularly elusive
for massive, and possibly macroscopic, systems such as mechanical resonators.
In this work, we propose two schemes based on reservoir engineering for
generating Wigner-negative entangled states unconditionally. We consider two
non-interacting mechanical resonators that are radiation-pressure coupled to
either one or two common cavity fields; the optomechanical coupling with the
field(s) features both a linear and quadratic part in the mechanical
displacement and the cavity is driven at multiple frequencies. We show
analytically that both schemes stabilize a Wigner-negative entangled state that
combines the entanglement of a two-mode squeezed vacuum with a cubic
nonlinearity, which we dub cubic-phase entangled (CPE) state. We then perform
extensive numerical simulations to test the robustness of Wigner-negative
entanglement attained by approximate CPE states stabilized in the presence of
thermal decoherence.
