ion and including the oscillations in the phases of the laser fields.

The equations of motion of a single ion in a quadrupole trap are given by the Mathieu equation and the first-order solution in the trap coordinate system $XYZ = R_XR_YR_Z$ is [@Berkeland1998a] $$R_i(t) \approx \left[ R_{0i} + R_{\text{sec},i} \cos{(\omega_{\text{sec},i} t + \theta_{\text{sec},i})} \right] \left( 1+ \frac{q_i}{2} \cos{\Omega t} \right).$$ Here $R_{\text{sec},i}$ is the secular motion amplitude, which can be obtained from the temperature of the ion, $E_{\mathrm{K}i} = k_\mathrm{B} T_i \approx \frac{1}{2} m R_{\text{sec},i}^2 \omega_{\text{sec},i}^2$ [@Berkeland1998a], $\omega_{\text{sec},i}$ is the trap frequency, and $\theta_{\text{sec},i}$ is the phase, determined by the initial position and momentum of the ion. The trap drive frequency is $\Omega$ and $q_i$ is a parameter that depends on the trap geometry, drive voltage, and ion mass. $R_{0i}$ is the displacement of the average position of the ion that is caused by static electric fields and that results in excess micromotion.

We consider a geometry, where the endcap trap $Z$ axis is at an angle $\alpha = 35.3^\circ$ to the horizontal plane. The possible beam directions are vertical ($z$) and horizontal at $\pm 45^\circ$ to the horizontal projection of the trap axis ($x$ and $y$). In the 