## 2 Simulation of Behaviors of a Single Atom Driven by Mean Force  
Before being loaded into the narrow-line MOT, atoms need to undergo some pre-cooling stages which force the velocity distribution into the capture range of the narrow-line MOT. A typical MOT configuration is shown in Figure 1a and the corresponding time sequence is shown in Figure 1b for strontium bosons on the ground. Atoms are firstly loaded into a blue MOT where the radiation forces are given by the transition of \((5s^{2})^{1}S_{0}-(5s5p)^{1}P_{1}\) which is characterized by a 32-MHz natural linewidth and a 461-nm transition wavelength. The \(R\) value of this broadline is about \(1.0\times 10^{5}\), so the gravity can be neglected at this stage. The second stage laser cooling is carried out in a red MOT based on transition of \((5s^{2})^{1}S_{0}-(5s5p)^{3}P_{1}\) at 689 nm. To make an efficient transfer from blue MOT to red MOT, the 689 nm laser is modulated to be broadband, about 2 MHz, and then changed to be at a single frequency. As mentioned above, the transition \((5s^{2})^{1}S_{0}-(5s5p)^{3}P_{1}\) has a natural linewidth of 7.6 kHz, giving a \(R\) value of about 16. To illustrate the role of the gravity in the single-frequency, red MOT, we model the trajectory of a single \({}^{88}\)Sr atom in one dimension. The isotope of \({}^{88}\)Sr has a simpler energy level configuration than \({}^{87}\)Sr [35,36], allowing us to abstract the physics essence in a most direct way. The dynamics of the atoms in this narrow-line-based MOT are mainly governed by the scattering force and gravitational force, which are given as below according to the semiclassical theory. \[F_{\alpha}(\mathbf{v},\mathbf{x}) = F_{-}(\mathbf{v},\mathbf{x})-F_{+}(\mathbf{v},\mathbf{x})-mg_{\alpha}, \tag{1}\] \[F_{\mp}(\mathbf{v},\mathbf{x}) = \frac{\hbar k\Gamma}{2}\frac{s}{1+s^{\prime}+\frac{4[\Delta\mp k \cdot\mp\mu\frac{\hbar k\Gamma}{2}x]^{2}}{\Gamma^{2}}}, \tag{2}\]  
where subscripted \(\alpha=G\) or \(M\) denotes the case on ground or in \(\mu\)-gEs, respectively. \(\mathbf{v}\) is the velocity of atom, and \(\mathbf{x}=\{x,y,z\}\). \(s=I/I_{s}\) is the saturation parameter where \(I\) is the single beam peak intensity and \(I_{s}\) is the transition saturation intensity. \(s^{\prime}\geq s\) accounts for saturation induced by other four trapping beams. \(\Delta=\omega_{L}-\omega_{A}\) where \(\omega_{L}\) is the laser angular frequency and \(\omega_{A}\) is the angular frequency of atomic resonance. \(\mu=g_{I}\mu_{B}/\hbar\) where \(g_{J}\) is the Lande g-factor of \({}^{3}P_{1}\) state and \(\mu_{B}\) is the Bohr magneton. \(m\) is the mass of atom. \(\mathbf{g}_{G}\) and \(\mathbf{g}_{M}\) are the gravitational accelerations on ground and in \(\mu\)-gEs, respectively. Here, we suppose a value \(\mathbf{g}_{M}=5\times 10^{-4}\times 9.8\) m/s\({}^{2}\). To simulate the trajectory of a single atom in z direction (vertical direction), we numerically solve the differential equations as:  
\[F_{\alpha,z}[z(t),v_{z}(t)] = m\frac{dv_{z}(t)}{dt}, \tag{3}\] \[v_{z}(t) = \frac{dz(t)}{dt}, \tag{4}\]  
with initial condition \(v_{z}(0)=v_{0}\), and \(z(0)=z_{0}\), where \(v_{z}(t)\) is the velocity in z direction and \(z(t)\) denotes the atom position in \(z\) direction. Here, \(t=0\) is the starting point of single-frequency MOT, as shown in Figure 1b. Figure 2a,b show the simulation results with typical values of laser detuning (\(\Delta=-2\pi\times 1.8\,\mathrm{MHz}\)) and laser intensity (\(s=s^{\prime}=250\)). The parameters used in this simulation are \(m=88\times 1.66\times 10^{-27}\) kg, \(\Gamma=2\pi\times 7.6\) kHz, \(k=2\pi/(689\times 10^{-9})\mathrm{m}^{-1}\), \(g_{J}=1.5\), \((dB_{z})/dz=0.1\) T/m, \(g_{G}=9.8\) m/s\({}^{-2}\), \(g_{M}=5\times 10^{-4}\times 9.8\) m/s\({}^{-2}\), \(v_{z}(0)=2\) cm/s, and \(z(0)=0\). Figure 2a compares the mean force between the cases on ground and in a \(\mu\)-gE of a rest atom in different position. Although the difference between these two forces is small, the resultant behaviors are quite different.