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We investigate the dynamics of a cold trapped ion coupled to the quantizedfield inside a high-finesse cavity. We have used an approach for preparing theSC states of motion of ion. This approach, based on unitary transformating theHamiltonian, allows its exact diagonalization without performing the Lamb-Dickeaproximation. We show that is possible to generate a SC states having rathersimple initial state preparation, e.g., the vacuum sate for both cavity fieldand the ion motion.

Preparation of Schr\"odinger cat states with cold ions in a cavity beyond the Lamb-Dicke limit