as a symmetric product state of the total of $2FN$ fictitious spin-$\frac12$ particles, which evolves in the manifold of states of the large collective spin with $J=NF$. The collective spin components constituting the Hamiltonian are invariant under the exchange of any two fictitious spin-$\frac12$ particles irrespective of whether they belong to the same atom or to different atoms. The collective spin description is, however, incomplete in general. For example, the atoms may be individually prepared in arbitrary internal superposition states $\sum_{m} c_m|F,m_F\rangle$ corresponding to correlations (entanglement) among their own fictitious spin-$\frac12$ constituents. Even if the atoms are thus all prepared in the same state, the $2FN$ fictitious spins are not equivalent: those pertaining to the same atom share correlations in the initial atomic product state, while the ones that pertain to different atoms do not.

We will now introduce a generalization of the Holstein-Primakoff representation that is capable of treating not only the total angular momentum operators but any permutation invariant sum of single-particle operators of an ensemble of $d$-level systems.

Generalized spin operators and collective atomic oscillators {#sec:spinops}
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Let us consider an ensemble of $N$ identical $d$-level systems, and let $\{{|{\phi_\alpha}\rangle}\}_{\alpha=0}^{d-1}$ denote an arbitrary orthonormal single-particle basis. Throughout 