state to $F'=2,3,4$ in the excited state[@Cs_Data], $l_{i}(\omega )$ is the normalization lineshape. For more precisely fitting, the normalization lineshape $l_{i}(\omega )$ should be considered as Vigot lineshape [@absorption_model]. More details about the theoretical absorption model and experimental fitting are in the supplementary notes. According to the fitting coefficients, the number density of atoms in the $^{133}$Cs cell is nearly $1.21\times 10^{18} m^{-3}$.

To determine how many atoms participating in the interaction, we need to know the volume of interaction area illuminated by light in the creating and certifying process, which can be evaluated with the volume of Gaussian light within the cell. The light beam used to generate multipartite entanglement is Gaussian beam, whose amplitude of electric field has the spatial dependence in following form $$\label{eq6}
|E(r)|=\frac{E_{0}}{\sqrt{1+\frac{z^{2}}{z_{w}^{2}}}}e^{-\frac{x^{2}+y^{2}}{W_{w}^{2}(1+\frac{z^{2}}{z_{w}^{2}})}}$$ where $z_{w}$ is Rayleigh length of laser beam, $W_{w}$ is the beam waist. Here, we only define the effective area of interaction by the amplitude decrease to ${1}/{10}$ of central magnitude in the Gaussian beam. Actually, the witness applied to certify our $W$ state doesn’t require equal amplitude for the entanglement state in $\left| W_{1}\right\rangle$  [@NC_million-atoms], therefore more atoms can be taken into calculations owing to the extension of Gaussian beam’s intensity. 