\label{eq:H-int-sq}
 \hat H = g_1 \hat s_z(0) \hat J_x
 + 2g_2 \bigl[\hat s_x(0) \hat V
 + \hat s_y(0) \hat T \bigr],\end{gathered}$$ with the Hermitian collective operators and $$\begin{gathered}
 \label{eq:twist-tilted}
 \hat V \equiv \sum_{j=1}^N  \frac1{2}
 \big( \hat F_{x+}^{(j)2} + \hat F_{x-}^{(j)2} \big)
 = \sum_{j=1}^N \big( \hat F_y^{(j)2} - \hat F_z^{(j)2} \big).\end{gathered}$$

We start from the $x$-polarized spin coherent state and take the single-particle angular momentum eigenbasis ${|{\phi_\alpha}\rangle} = {|{F, m_F^x =  F-\alpha}\rangle}$. The operator $\hat T$ in Eq.  is responsible for squeezing $\hat F_z^{(j)}$ (and anti-squeezing $\hat F_y^{(j)}$) of each atom, and we can select it by setting $\hat s_y = \phi/2$ to have macroscopic expectation value, while the other two light operators, $\hat s_x$ and $\hat s_z$, can be completely neglected. This corresponds to equally strong right and left circularly polarized light components, i.e., a strong linearly polarized field (rotating at twice the Larmor frequency in the laboratory frame).

Under the action of $\hat H_0 = g_2 \phi \hat T$, the initial spin coherent state evolves into the product state $\bigotimes^N [\hat U_0^{(1)} (t) {|{\phi_0}\rangle}]$ in which every atom is internally squeezed, while the atomic oscillators expanded around this rotating reference state are in their vacuum states. Therefore, the statistical 