Fig. 2: Displacement of ions in the X-direction (D_X) and motional heating in the Y-direction.

a Typical ion fluorescence images at different X-displacements. The motional length of ions is determined by calculating the number of pixels corresponding to the ion fluorescence. Larger displacement results in longer motional length of the ions (greater motional heating). b Cooling laser detuning at different X-displacements. The detuning of the cooling laser frequency increases slightly when the ions are displaced in the X-direction from 0 to 680 μm. c Schematic of ion motional position as a function of time, \({{\rm{Y}}}\left(t\right)=A\sin \left(\Omega t+{\varphi }_{0}\right)\). d Cooling with repumping. The error bars represent the standard error of 5 individual measurements, and strong linear fitting results in a narrow 95% confidence interval, which is smaller than the data points. Black dots: The length of the ion trajectory (2A) as a function of X-displacement. A linear fit yields 0.214 ± 0.003 (μm) μm⁻¹ and −0.218 ± 0.003 (μm) μm⁻¹ in different directions. Red squares: Maximum velocity \(\left({v}_{{mY}}=\Omega A\right)\) as a function of displacement in the X-direction (D_X). The linear fit, \({v}_{{mY}}={\eta }_{X}{D}_{X}\), shows \({\eta }_{X}=\) 2.12 ± 0.01 (m s⁻¹) μm⁻¹.