We show how to generate quadratic and bi-quadratic phonon-photon interactions through a driven three-level ion inside a cavity. With such a system it is possible to squeeze the cavity-field state, the ion motional state or even the entangled phonon-photon state. We present a detailed analysis of the cavity-field squeezing process, distinguishing three different regimes of this amplification mechanism: the subcritical, critical, and supercritical regimes, which depend, apart from the coupling parameters, on the excitation of the vibrational state. As an application of the engineered Hamiltonians, we show how to implement a Fock-state filter for the vibrational mode. New aspects of the technique of adiabatic elimination emerge in this analysis.
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