Eqs  and can be used to express the symmetric collective atomic operators in $\hat H_1$. Consider, for example, that $\hat H_1$ is an arbitrary function of the collective operator $\hat{\mathcal O}$. Then we write $$\begin{gathered}
  \label{eq:O-approx-rot}
  \hat{\mathcal O} \approx N \mathcal O_{00}(t)
  + \sqrt N \sum_{\alpha\ne0}
  \bigl[ \mathcal O_{\alpha0}(t) \hat a_\alpha^\dag (t)
  + \mbox{H.c.} \bigr] \\
  + \sum_{\alpha\beta\ne0}
  \big[\mathcal O_{\alpha\beta}(t)
  - \mathcal O_{00}(t) \delta_{\alpha\beta}\big]
  \hat a_\alpha^\dag(t)   \hat a_\beta(t)\end{gathered}$$ in place of $\hat{\mathcal O}$, where the matrix elements are $$\begin{gathered}
  \label{eq:O-ab-rot}
  \mathcal O_{\alpha\beta} (t)
    = {\langle{\phi_\alpha}|} \hat U_0^{(1)\dag}(t)
  \hat{\mathcal O}^{(1)} \hat U_0^{(1)}(t) {|{\phi_\beta}\rangle}.\end{gathered}$$ Depending on whether ${|{\phi_0(t)}\rangle}$ is an eigenstate of $\hat{\mathcal
  O}^{(1)}$ or not, we may neglect the quadratic terms in Eq. . Often $\hat H_1$ is at most quadratic in the oscillator operators, and the equation of motion reduces to a set of ordinary linear differential equations that can be solved.

Spin squeezing {#sec:spsq}
==============

For ensembles of spin-$\frac12$ particles, spin squeezing necessarily involves correlation among the spins [@Nature.409.63]. When these particles are actually the fictitious spins constituting the real spin-$F$ atoms (e.g., the valence electron and the nucleons), correlations may be both intra- and interatomic. In the former case, the atoms are internally squeezed independently of each other, 