angular momentum quantum numbers $F'$ and associated $|M'| \leq F'$. In a large ensemble of such atoms, the collective spin picture is less useful, and we will proceed directly to an effective oscillator description of the symmetric collective states of the system and of the system dynamics. Various methods allow the control of single particle states, and the atoms in a large ensemble can be prepared in essentially any superposition of the hyperfine ground states [@PhysRevLett.99.163002]. Therefore, it is a particular purpose of our work to develop a theory of collective states in the vicinity of a product state of arbitrary single atom states, and to determine the interplay between the collective and single particle properties of such samples.

In Sec. \[sec:HP-gen\], we review the usual Holstein-Primakoff approximation, describing a collection of spin-$\frac12$ particles by a single harmonic oscillator degree of freedom, and we generalize this description to multilevel systems expanded around product states of arbitrary single particle state vectors. We discuss the representation of collective operators, which are a sum of single particle operators over the entire ensemble, and we demonstrate that in the vicinity of product states, such operators can be associated with oscillator quadrature operators in a 