and annihilation operators, $$\begin{gathered}
  \label{eq:O-approx2}
  \hat{\mathcal O} =  N \mathcal O_{00} + \sum_{\alpha\beta\ne0}
  (\mathcal O_{\alpha\beta} - \mathcal O_{00} \delta_{\alpha\beta})
  \hat a_\alpha^\dag   \hat a_\beta.\end{gathered}$$ For example, taking the spin projection eigenstates ${|{\phi_\alpha}\rangle} = {|{F, m_F^x = F-\alpha}\rangle}$, the longitudinal component of the total angular momentum is $\hat J_x = \sum_\alpha (F-\alpha) \hat{\Sigma}_{\alpha\alpha} = NF - \sum_{\alpha\ne0} \alpha \, \hat a_\alpha^\dag \hat a_\alpha$.

If ${|{\phi_0}\rangle}$ is not an eigenstate of $\hat{\mathcal O}^{(1)}$, we get cross terms between the $\alpha = 0$ and the $\alpha \neq 0$ components, and we obtain the dominant contribution $$\begin{gathered}
  \label{eq:O-approx1}
  \hat{\mathcal O} \approx N \mathcal O_{00} + \sqrt N \sum_{\alpha\ne0}
  \big( \mathcal O_{\alpha0} \hat a_\alpha^\dag +
  \mathcal O_{0\alpha} \hat a_\alpha \big).\end{gathered}$$ Apart from a constant, Eq.  is a linear function of the creation and annihilation operators, hence, it is a linear function of the collective atomic quadrature variables $$\begin{gathered}
  \label{eq:atom-XP}
  \hat X_\alpha \equiv
  (\hat a_\alpha + \hat a_\alpha^\dag) / \sqrt2
  \quad \mbox{and} \quad
  \hat P_\alpha \equiv
  (\hat a_\alpha - \hat a_\alpha^\dag) / i\sqrt2.\end{gathered}$$ Namely, Eq.  can be written as $$\begin{gathered}
  \label{eq:O-approx1b}
  \hat{\mathcal O} \approx N \mathcal O_{00}
  + \sqrt{2N} \sum_{\alpha\ne0} \big(
  {\mathrm{Re}\,}\mathcal O_{\alpha0} \hat X_\alpha +
  {\mathrm{Im}\,}\mathcal O_{\alpha0} \hat P_\alpha \big).  \end{gathered}$$

Let us choose real numbers 