In this work, the quantum entanglement and quantum linearity in a trapped ionphonon system are theoretically explained in terms of the second-order terms of the LambDicke parameter, η. For the trapped ionphonon system, the optimum times are 2 and 10 and the optimum probability amplitudes of ions are 0.4 and 0.58, respectively. With the optimum times and amplitudes, it is shown that as η grows, the quality of the entanglement in the trapped ionphonon system increases. In particular, we have demonstrated the quantum linearity due to second-order terms of η.
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