  Non-Gaussian states (NGSs) with higher-order correlation properties have
wide-range applications in quantum information processing. However, the
generation of such states with high quality still faces practical challenges.
Here, we propose a protocol to faithfully generate two types of mechanical
NGSs, i.e., Schr\"{o}dinger cat states and Fock states, in open optomechanical
systems, even when the cooperativity is smaller than one
($g^2/\kappa\gamma<1$). In contrast to the usual scheme, a short squeezed field
is pumped to rapidly entangle with a mechanical resonator via a
beam-splitter-like optomechanical interaction, effectively reducing the
mechanical decoherence. Furthermore, by performing an additional amplifier and
a following multi photon subtraction on the entangled optical field, one can
selectively obtain the high-fidelity mechanical cat and Fock states. This
protocol is robust to various imperfections, allowing it to be implemented with
state-of-the-art experimental systems with close to unit fidelity. Moreover, it
can be extended to generate a four-component cat state and provide
possibilities for future quantum applications of NGSs.
