Quantum conversion between the cavity fields and the center-of-mass motion of ions in a quantized trap

We propose a technique to generate nonclassical vibrational states of the quantized center-of-mass motion of an ion in a harmonic trap, based on the quantum conversion between the quantum cavity field and the quantized center-of-mass motion. It is shown that when an ion trap system interacts with the eigenmode of a single-mode Fabry-Pérot cavity, where the trap is set in, and with an external classical driving electromagnetic field through Raman transitions, the interchange of the quantum features between the quantum cavity field and the quantized trap occurs. This kind of quantum conversion can be used to prepare some nonclassical trap states for the ion trap as well as to measure the quantum statistics of an initial nonclassical vibrational state.