where $J$ is the total angular momentum quantum number, which is a constant of motion in the cases considered, and the $x$ quantization axis is chosen to be the direction of polarization in the initial product state. For macroscopic polarization ($J\gg1$) and if the system stays in the vicinity of the spin coherent state ${|{J, m_J^x=J}\rangle}$, the atomic oscillator picture is especially efficient and it allows us to directly define the quasi-canonical atomic quadrature operators, $$\begin{gathered}
  \label{eq:standard-XP}
  \hat X \equiv \frac1{\sqrt J} \hat J_y,
  \qquad
  \hat P \equiv \frac1{\sqrt J} \hat J_z,\end{gathered}$$ with $[\hat X, \hat P] = i\hat J_x / J \approx i$.

For an ensemble consisting of atoms with larger angular momenta $F > \frac12$, it is also possible to define the collective spin operator $\hat{\mathbf J} = \sum_{j=1}^N \hat{\mathbf F}^{(j)}$. If each atom is initially prepared in the same spin coherent state, and the Hamilton operator is an element of the operator algebra generated by the components of $\hat{\mathbf J}$, this collective spin again provides a general description of the system. In that case, it is possible to think of each atom as a collection of $2F$ fictitious spin-$\frac12$ particles, and to think of the initial state 