beam coordinate system, the ion motion is then given by

$$\begin{aligned}
r_{x}(t) &=& \frac{1}{\sqrt{2}} \left[ R_{X}(t) -\sin{\alpha} \,R_{Y}(t) -\cos{\alpha} \,R_{Z}(t) \right], \\
r_{y}(t) &=& \frac{1}{\sqrt{2}} \left[ R_{X}(t) +\sin{\alpha} \,R_{Y}(t) +\cos{\alpha} \,R_{Z}(t) \right], \\
r_{z}(t) &=& -\cos{\alpha} \,R_{Y}(t) +\sin{\alpha} \,R_{Z}(t).\end{aligned}$$

This geometry is chosen to make the projection of each beam direction on the trap $Z$ axis equal in magnitude ($\cos{\alpha}/\sqrt{2} = \sin{\alpha}$). In the rest frame of the ion, the phase of a laser beam traveling along the $\mathbf{u}_l$ axis is then given by $\exp{\{-i[\omega_j t - k_jr_l(t) - \phi_{jl}]\}}$ ($j=\mathrm{c},\mathrm{r}$). The phase $\phi_{jl}$ is not relevant.

Figure \[fig:ionmotion\] illustrates the effect of ion motion using numerical parameters typical for an endcap trap: $q_Z=-2q_{X,Y}=0.4$, $\omega_{\text{sec},Z}=2\omega_{\text{sec},X,Y}=2\pi\times 2$MHz, and $\Omega=2\pi\times 12$MHz. The solid (black) curve is the scattering rate for a stationary ion, $T=0$, equal to the dashed (red) curve in Fig. \[fig:pol-mod\] (note the different horizontal axis), and shows oscillations at the frequency $\omega_\mathrm{EOM}$ only. At finite temperatures, there are additional oscillations at the trap frequencies $\omega_{\text{sec},X,Y}$ and $\omega_{\text{sec},Z}$ and at the drive frequency $\Omega$, and the time dependence of the scattering rate is rather complex. The oscillation amplitudes increase with increasing temperature; see the dashed (red) and dash-dotted (green) curves 