probability  [@nature_DLCZ; @PRL_kimble]. With the creation operators acting on the initial state of atomic ensemble, the $W$ state is written as [@nPhys_msmemory; @Dou_FORD] $\left| W_{1}\right\rangle {\rm{ = }}\frac{1}{{\sqrt N }}\sum\limits_{j = 1}^N {{e^{i\Delta{\vec k}\cdot {{\vec r}_j}}}} \left| {{g_1}{g_2}...{s_j}...{g_N}} \right\rangle $, where $N$ is the number of involving atoms, $\Delta{\vec k}$ is the wave-vector of spin wave, ${\vec r}_j$ is the position information of $j_{th}$ excited atom. This generating process of entanglement is shown in Figure 1b, and the W state can be heralded through the detection of one scattering Stokes photon. $\left| W_{1}\right\rangle$ contains only one excitation shared by all motional atoms illustrated in Figure 1c, where every atom posesses the equal probability being excited with spin up.

It is inevitable that the SRS in the generating process may produce high-order excitations with a comparably low probability, but such terms would change the structure of our desired multipartite entanglement. The entangled ensemble with more than one excitation event can be generally expressed as $\left| W_{2}\right\rangle {\rm{=}}\sqrt{\frac{2}{N(N-1)}}\sum\limits_{i<j}^N {{e^{i\Delta{\vec k}\cdot\left({{\vec r}_j+{\vec r}_i}\right)}}}\left|{{g_1}{g_2}...{s_i}...{s_j}...{g_N}}\right\rangle$, and higher-order events have negligible contributions. To certify and qualify the W state, we need to apply another optical probe pulse to convert the shared single excitation in atomic entanglement 