We show how to create quantum squeezed states of magnons and phonons in a cavity magnomechanical system. The magnons are embodied by a collective motion of a large number of spins in a macroscopic ferrimagnet, and couple to cavity microwave photons and phonons (vibrational modes of the ferrimagnet) via the magnetic dipole interaction and magnetostrictive interaction, respectively. The cavity is driven by a weak squeezed vacuum field generated by a flux-driven Josephson parametric amplifier, which is essential to get squeezed states of the magnons and phonons. We show that the magnons can be prepared in a squeezed state via the cavity-magnon beamsplitter interaction, and by further driving the magnon mode with a strong red-detuned microwave field, the phonons are squeezed. We show optimal parameter regimes for obtaining large squeezing of the magnons and phonons, which are robust against temperature and could be realized with experimentally reachable parameters.
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