## 3 Parametric Heating of Atoms in an Optical Tweezer  
Atoms in optical tweezers will generate electric dipole moments under the action of the oscillating electric field. We can regard these atoms as electric dipoles interacting with the oscillating external field. As shown in Figure 4a when the electric dipole is driven by an electric field with a frequency lower than the resonant frequency of atomic transition, due to the effect of attraction, the atom will be captured in the region with the strongest electric field. Otherwise, when the electric dipole is driven by an electric field with a frequency higher than the resonant frequency, the atom will be excluded from the region with the strongest electric field. In other words, when the frequency of optical tweezer is red-detuning to the atomic transition frequency concerned, the atoms are captured at the bottom of the optical tweezer [21]. When the optical tweezer frequency is blue-detuning to the atomic transition frequency, the atoms will be excluded from the optical tweezer [22]. Using a focused single-spatial-mode Gaussian beam with red detuning can form an optical tweezer [23], with intensity distribution in the axial and radial direction as follows,  
\[I(r,z)=\frac{2P}{\pi w^{2}(z)}\exp\left[-2\frac{r^{2}}{w^{2}(z)}\right] \tag{3}\]  
where \(P\) is the power of the optical tweezer laser and \(w(z)\) is the beam radius at \(z\)[23],  
\[w(z)=w_{0}\sqrt{1+\frac{z^{2}}{z_{R}^{2}}} \tag{4}\]  
for \(w_{0}\) is the beam wrist, \(z_{R}\) is the Rayleigh length, the axial and radial trap frequencies are \(\omega_{a}=\sqrt{\frac{2U}{Mz_{R}^{2}}}\) and \(\omega_{r}=\sqrt{\frac{4U}{Mw_{0}^{2}}}\), and \(U\) is the trap potential depth of the optical tweezer. Figure 4: Scheme of optical tweezer. (**a**) A strongly focused single-spatial mode Gaussian beam of a red-detuned laser can confine atoms. (**b**) The beam waist radius and Rayleigh length of 1064-nm optical tweezer are 2.2 \(\upmu\)m and 11.7 \(\upmu\)m, respectively. The Hamiltonian of single-atom in the optical tweezer is [23]  
\[H=\frac{p^{2}}{2M}+\frac{1}{2}M\omega_{tr}^{2}[1+\varepsilon(t)]x^{2}, \tag{5}\]  
for \(M\) is the mass of the Cs atom, \(\omega_{tr}^{2}=k_{0}/M\) is the mean square of trap frequency, the elastic coefficient \(k_{0}\) is in direct proportion to the light intensity of optical tweezer \(I_{0}\), we can write the light intensity fluctuation as \(\varepsilon(t)=\frac{I(t)-I_{0}}{I_{0}}\). Here we use the first-order perturbation theory to clarify how the light intensity fluctuation acts on the single-atom, and take the heating process as the transformation of the atom from \(|n\rangle\) at \(t=0\) to \(|m\
eq n\rangle\) at \(t=T\). The average transition probability \(R_{m\leftarrow n}\) can express as the power spectral density of the laser intensity fluctuation \(S_{\varepsilon}(\omega)\)[23]  
\[\int_{0}^{\infty}d\omega S_{\varepsilon}(\omega)=\int_{0}^{\infty}dvS_{ \varepsilon}(\
u)=\left\langle\varepsilon^{2}(t)\right\rangle=\varepsilon_{ 0}^{2}, \tag{6}\]  
for \(\varepsilon_{0}\) is the mean square of intensity fluctuation and \(\omega=2\pi\
u\). The average heating rate of atom in the optical tweezer is [23]  
\[\left\langle\dot{E}\right\rangle=\sum_{n}P(n)2\hbar\omega_{tr}(R_{n+2\gets n }-R_{n-2\gets n})=\frac{2}{\pi}\omega_{tr}^{2}S_{\varepsilon}(2\omega_{tr })\langle E\rangle, \tag{7}\]  
where \(P(n,t)\) is the probability of the atom stay at \(|n\rangle\). For \(\left\langle\dot{E}\right\rangle=\Gamma_{\varepsilon}\langle E\rangle\), the constant [23]  
\[\Gamma_{\varepsilon}=\frac{1}{T_{I(\sec)}}=\pi^{2}\
u_{tr}^{2}S_{\varepsilon} (2\
u_{tr}) \tag{8}\]  
Among others, \(\
u_{tr}\) is the trap frequency. According to Equation (8), the heating rate of single-atom in the optical tweezer is related to the trap frequency and its double frequency of the optical tweezer. For a single-atom in the optical tweezer, it will not only couple with the fluctuation resonance to the trap frequency of the tweezer, but also couple with the double-frequency more intensely.