radiation field, but also because the desired control is provided by simple, experimentally accessible interaction mechanisms. This is the case, e.g., for large collections of $N$ identical spin-$\frac12$ particles. Such an ensemble can be effectively described using a single collective spin, if the initial state is a pure product state with all particles prepared in the same single-particle state, and if the Hamiltonian describing the system is a sum of identical single-particle operators. Since, in this case, the quantum state of the entire collection remains invariant under permutations of particles, the corresponding restricted Hilbert subspace is equivalent to the states of a large spin-$\tfrac N2$ spin. If, in addition, the dynamics only weakly perturbs the initial state, the collective states explore only a limited range of states, and the pseudospin eigenbasis can be mapped onto an oscillator basis of states, and one may benefit from the simple and well studied properties of harmonic oscillator systems.

In this paper we wish to generalize the oscillator description of two-level systems to multilevel particles. A natural example is the one of atoms having a Zeeman degenerate ground state with total angular momentum $F$ or a hyperfine ground state manifold with a range of 