Non-Gaussian states (NGSs) with higher-order correlation properties have
wide-range applications in quantum information processing. However, the
preparation of such states with high quality still faces practical challenges.
Here, we propose a protocol to rapidly generate two types of mechanical NGSs,
Schr\"{o}dinger cat states and Fock states, in dissipative optomechanical
systems, even when the cooperativity is smaller than one
(g2/κγ<1). In contrast to the usual scheme of directly applying
non-Gaussian operations on the entangled optical mode, we show that an
additional phase-sensitive amplifier can accelerate the generation and also
precisely control the type of NGSs. Then, a principally deterministic
multi-photon subtraction induced by the Rydberg-blockade effect is adopted to
produce large-sized NGSs. The protocol can be implemented with state-of-the-art
experimental systems with close to unit fidelity. Moreover, it can also be
extended to generate a four-component cat state and provide new possibilities
for future quantum applications of NGSs.Comment: 7 pages, 4 figure