  The fidelity of the gate operation and the coherence time of neutral atoms
trapped in an optical dipole trap are figures of merit for the applications.
The motion of the trapped atom is one of the key factors which influence the
gate fidelity and coherence time. The motion has been considered as a classical
oscillator in analyzing the influence. Here we treat the motion of the atom as
a quantum oscillator. The population on the vibrational states of the atom are
considered in analyzing the gate fidelity and decoherence. We show that the
fidelity of a coherent rotation gate is dramatically limited by the temperature
of a thermally trapped atom. We also show that the dephasing between the two
hyperfine states due to the thermal motion of the atom could rephase naturally
if the differential frequency shift is stable and the vibrational states do not
change. The decoherence due to the fluctuations of the trap laser intensity is
also discussed. Both the gate fidelity and coherence time can be dramatically
enhanced by cooling the atom into vibrational ground states and/or by using a
blue-detuned trap. More importantly, we propose a "magic" trapping condition by
preparing the atom into specific vibrational states.
