\hat\Sigma_{\alpha\gamma}, \hat\Sigma_{\gamma\beta} \big]$, vanish.

Any pure state of a spin-$\frac12$ particle is a spin coherent state, i.e., a spin up state along a suitably chosen axis, say $x$. The corresponding ensemble state ${|{J, m_J^x=J}\rangle}$ serves as the natural reference state: it is the ground state of the effective atomic oscillator in the Holstein-Primakoff approximation. With a general single-atom basis vector ${|{\phi_0}\rangle}$, we take the product state $\bigotimes^N {|{\phi_0}\rangle}$ as the reference state and introduce $d-1$ independent collective atomic oscillator modes with creation and annihilation operators that redistribute the atomic populations between ${|{\phi_0}\rangle}$ and the other basis states ${|{\phi_\alpha}\rangle}$, $$\begin{gathered}
  \hat a_\alpha^\dag
  {|{n_0, m_\alpha, \ldots}\rangle}
  \equiv \sqrt{m+1} \;
  {|{(n-1)_0, (m+1)_\alpha, \ldots}\rangle},
  \label{eq:a-step}
  \\
  \hat a_\alpha^\dag \hat a_\alpha
  {|{n_0, m_\alpha, \ldots}\rangle}
  = m \; {|{n_0, m_\alpha, \ldots}\rangle},
  \label{eq:a-number}\end{gathered}$$ with $\alpha = 1, \ldots, d-1$. The reference state itself corresponds to the multi-mode vacuum state of the oscillators, while the excitation number states coincide with the symmetric states . In analogy with the Holstein-Primakoff representation, directly comparing Eq.  with and Eq.  with , respectively, the generalized collective spin operators can be expressed as follows, $$\begin{gathered}
  \label{eq:Sigma_a0}
  \hat\Sigma_{\alpha\beta} = \hat a_\alpha^\dag \hat a_\beta,
  \qquad
  \hat\Sigma_{\alpha0} = \hat a_\alpha^\dag \sqrt{\textstyle N -
    \sum_{\beta\ne0} \hat a_\beta^\dag 