## 4 Measurement of Trap Frequency of Optical Tweezer  
The output power of the 1064-nm laser (DBR-1064P, Thorlabs, America) and Ytterbium-doped fiber amplifier (YDFA) system is up to 2 W. The output 1064-nm laser beam passes a lens focal length of 100 mm to form a parallel beam with a diameter of 18-19 mm. Then the parallel beam goes through a lens assembly with a focal length of 36 mm and a numerical aperture of NA = 0.29 to form the optical tweezers. Figure 4b shows the measurement of the beam waist radius and \(M_{z}^{2}\) factor of 1064 nm optical tweezer are \(2.2\pm 0.4\)\(\upmu\)m and \(1.18\pm 0.21\), respectively. The Rayleigh length \(Z_{R}\) is 11.7 \(\upmu\)m.  
As shown in Figure 5, the part in the blue box is used to measuring the trap frequency of the optical tweezer, which is an important parameter of the optical tweezer. We use a function generator (Model DG635 SRS, America) to load modulation signals of different frequencies on AOM 2 (Model 3110-197, Crystal Technology, America) to simulate the intensity fluctuation of the optical tweezer so that the optical tweezer will run with intensity fluctuation at a specific frequency.  
Then, by measuring the transfer efficiency of single-atom between the MOT and the optical tweezer to observe how the intensity fluctuation of different frequency acts on single-atom trapping lifetime and Figure 6 shows a typical atomic transfer signal between the MOT and the optical tweezer. As shown in Figure 6, the transfer efficiency gets extremely low when the modulation frequency is resonant with the trap frequency and its double frequency of the optical tweezer.  
Figure 7 shows the trap frequency and its double frequency on the axial and radial direction. The power of 1064 nm optical tweezer is 48.5 mW, the optical tweezer trap depth is 1.3 mK. The measured axial trap frequency is 4.70 kHz, for double frequency is 9.00 kHz (should be 9.40 kHz), the radial trap frequency is 41.80 kHz, and the double frequency is 80.80 kHz (should be 83.60 kHz). The deviation between the experimental value and the theoretical calculation is mainly because that the suppression effect of the laser intensity fluctuation feedback loop cannot be kept at the same level due to the long duration of the experimental process.  
Figure 5: Experimental setup. Polarization-maintaining (PM) fiber; Multi-mode (MM) fiber; magneto-optical trap (MOT); optical-dipole trap (ODT), otherwise the optical tweezer; optical isolator (OI); acousto-optical modulator (AOM); polarization beam splitter (PBS); single-photon counting module (SPCM); ultra-high vacuum (UHV). The red dash box shows the laser intensity fluctuation feedback loop by controlling the diffraction efficiency of AOM 1 to stabilize the intensity fluctuation of the 1064-nm laser. The blue dash box shows the measurement setup of the trap frequency of optical tweezer. The green dash box shows the single-atom cooling and trapping system, the anti-Helmholtz coils, but the cooling and repumping laser beams of the MOT are not shown here.  
Figure 6: Atomic transfer signal between the MOT and the optical tweezer. The period is 5.5 s, in which the MOT duration is 4.5 s and the optical tweezer last for 1.0 s. The MOT overlaps with the optical tweezer for 25 ms, and the single-atom is transferred between the MOT and the optical tweezer.  
The photon scattering rate of the optical tweezer is [24]  
\[\Gamma_{\rm{SC}}=\frac{3\pi c^{2}\omega_{\rm{L}}^{3}}{2\hbar\omega_{0}^{6}}\bigg{(} \frac{\Gamma}{\omega_{0}^{2}-\omega_{L}^{2}}+\frac{\Gamma}{\omega_{0}^{2}+ \omega_{L}^{2}}\bigg{)}^{2}, \tag{9}\]  
where \(\Gamma\) is the spontaneous decay rate of the atomic transition, \(\omega_{0}\) is the transition frequency, \(\omega_{L}\) is the frequency of the optical tweezer. In a 1064-nm optical tweezer, for Cs 6 S\({}_{1/2}\) (F\({}_{\rm{g}}=4\)) \(-\)6 P\({}_{3/2}\) (F\({}_{\rm{e}}=5\)) transition, the photon scattering rate is 8.3 photons/s.