  We examine the validity of the harmonic approximation, where the
radio-frequency ion trap is treated as a harmonic trap, in the problem
regarding the controlled collision of a trapped atom and a single trapped ion.
This is equivalent to studying the effect of the micromotion since this motion
must be neglected for the trapped ion to be considered as a harmonic
oscillator. By applying the transformation of Cook and Shankland we find that
the micromotion can be represented by two periodically oscillating operators.
In order to investigate the effect of the micromotion on the dynamics of a
trapped atom-ion system, we calculate (i) the coupling strengths of the
micromotion operators by numerical integration and (ii) the quasienergies of
the system by applying the Floquet formalism, a useful framework for studying
periodic systems. It turns out that the micromotion is not negligible when the
distance between the atom and the ion traps is shorter than a characteristic
distance. Within this range the energy diagram of the system changes remarkably
when the micromotion is taken into account, which leads to undesirable
consequences for applications that are based on an adiabatic process of the
trapped atom-ion system. We suggest a simple scheme for bypassing the
micromotion effect in order to successfully implement a quantum controlled
phase gate proposed previously and create an atom-ion macromolecule. The
methods presented here are not restricted to trapped atom-ion systems and can
be readily applied to studying the micromotion effect in any system involving a
single trapped ion.
