Fig. 3: Displacement of ions in the Y-direction (D_Y) and motional heating in the X-direction.

a, c Typical ions’ fluorescence images at different Y-displacements and the corresponding cooling laser detuning. Larger displacement results in higher cooling laser detuning (greater motional heating). The trap nodal line is defined as position 0. a Cooling with repumping. Inserted images show ions at the trap nodal line (0 μm), positive 210 μm, and 432 μm in the Y-direction, corresponding to 0.01 GHz, 1.09 GHz, and 2.00 GHz, respectively. b Numerical solution (We assume that both \(n\) and \(\alpha\) are 1) of Eq. (4) at different values of detuning and maximum speeds. Theoretical peaks are all shifted by about −5 MHz to overlap with experimental results, and the rightmost data is multiplied by a factor of 0.2. c Cooling without repumping. Inserted images show ions at 133 μm, 370 μm, and 593 μm in the positive Y-direction, corresponding to 0.63 GHz, 1.74 GHz, and 2.80 GHz, respectively. d, e Relationship among Y-displacement, cooling laser detuning, and maximum velocity in X-direction. 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. d Cooling with repumping. Black dots: Cooling laser detuning \((\delta )\) as a function of Y-displacement (D_Y). A linear fit yields 4.63 ± 0.03 MHz μm⁻¹ and −4.65 ± 0.02 MHz μm⁻¹ in different directions. Red squares: Maximum velocity (\({v}_{mX}\)) as a function of displacement (D_Y). The linear fit, \({v}_{{{\rm{mX}}}}={\eta }_{Y}{D}_{Y}\), shows \({\eta }_{Y}=\) 2.06 ± 0.01 (m s⁻¹) μm⁻¹. e Cooling without repumping. Black dots: Cooling laser detuning \((\delta )\) as a function of Y-displacement (D_Y). A linear fit yields 4.64 ± 0.06 MHz μm⁻¹ and −4.58 ± 0.04 MHz μm⁻¹ in different directions. Red squares: Maximum velocity (\({v}_{mX}\)) as a function of displacement (D_Y). The linear fit, \({v}_{{mX}}={{\eta }^{{\prime} }}_{Y}{{{\rm{D}}}}_{{{\rm{Y}}}}\), shows \({{\eta }^{{\prime} }}_{Y}=\) 2.05 ± 0.01 (m s⁻¹) μm⁻¹. f Energy level diagram of Be⁺. F = 1 and F = 2: The hyperfine structure of Be⁺ in the ground state. Red solid line: frequency shift due to the Doppler effect of the ion’s micromotion. Red/Blue dashed line: red/blue detuning during cooling via ion motional heating. The purple area indicates the region where the ²S_1/2 (F = 1) → ²P_3/2 transition occurs within one period. When sufficient micromotion-induced Doppler shift (\(+{\omega }_{{{\rm{micro}}}{{\rm{mo}}}{{\rm{tion}}}}\)) spans the two ground-state energy levels of Be⁺, cooling without repumping is achieved, see also Supplementary Note 1 for more details.