= Paul trap micromotion
{c}
{title2=$u_i\simeq X_i[1-(q_i/2)\cos\Omega t]$}
In the small-parameter regime, the slow coordinate $X_i$ describing <Paul trap secular motion> approximately obeys a <harmonic oscillator equation> at <angular frequency> $\omega_i$. Its superposed fast motion is Paul trap micromotion. Substituting the leading fast displacement from the <RF-averaged trapping potential> gives the displayed expression in our phase convention. A small positional ripple can have substantial <velocity>, because differentiating it brings down the large <angular frequency> $\Omega$. Precision <spectroscopy> must therefore address both <Paul trap secular motion> and <Paul trap micromotion>; merely obtaining a bounded trajectory does not eliminate <Doppler shifts>.
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