\hat a_\beta},\end{gathered}$$ furthermore, using Eq. , $$\begin{gathered}
  \label{eq:Sigma_00}
  \hat\Sigma_{00} =
  N - \sum_{\alpha\ne0} \hat a_\alpha^\dag \hat a_\alpha.\end{gathered}$$

Linearization around the reference state: Holstein-Primakoff expansion of collective operators {#sec:linarization}
----------------------------------------------------------------------------------------------

In the vicinity of the reference state $\bigotimes^N {|{\phi_0}\rangle}$, i.e., for small number of excitations of the atomic oscillators, only the population $\hat\Sigma_{00}$ has macroscopic expectation value, and the spin operators can be approximated as $$\begin{gathered}
  \label{eq:HP-approx}
  \hat\Sigma_{\alpha\beta} = \hat a_\alpha^\dag \hat a_\beta
  \;\; \ll \;\;
  \hat\Sigma_{\alpha0} \approx \sqrt N \hat a_\alpha^\dag
  \;\; \ll \;\;
  \hat\Sigma_{00} \approx N.\end{gathered}$$ This allows us to simplify collective operators that are permutation invariant sums of single-particle operators, or belong to the algebra generated by such operators.

Since $\{ {|{\phi_\alpha}\rangle} {\langle{\phi_\beta}|}\}_{\alpha\beta}$ forms an orthonormal single-particle operator basis, for any Hermitian single-particle operator $\hat{\mathcal O}^{(1)}$ the corresponding $N$-body operator can be expanded as $$\begin{gathered}
  \label{eq:O-expanded}
  \hat{\mathcal O} \equiv \sum_{j=1}^N \hat{\mathcal O}^{(j)}
  = \sum_{\alpha\beta}
  \mathcal O_{\alpha\beta}
  \hat \Sigma_{\alpha\beta},\end{gathered}$$ where we used the shorthand $\mathcal O_{\alpha\beta} =
{\langle{\phi_\alpha}|} \hat{\mathcal O}^{(1)} {|{\phi_\beta}\rangle}$. Now we distinguish between two cases. If ${|{\phi_0}\rangle}$ is an eigenstate of $\hat{\mathcal O}^{(1)}$, then $\mathcal O_{\alpha0} = 0$ for all $\alpha\ne0$ and the collective operator becomes a c-number plus a correction that is quadratic in the creation 