generalized Holstein-Primakoff approximation. In Sec. \[sec:spsq\], we turn to a special discussion of spin squeezing, and we point out the formal distinction between the effect of spin squeezing within each $F > \tfrac12$ hyperfine angular momentum manifold and squeezing of collective spin degrees of freedom. We also show that, when applied sequentially, the order of the two kinds of squeezing is relevant. In Sec. \[sec:qmem\], we discuss the use of samples of multi-sublevel atoms as quantum memories for light and the role of internal spin squeezing in such atoms. Sec. \[sec:conclusions\] concludes the paper.

Multilevel Holstein-Primakoff approximation {#sec:HP-gen}
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A convenient way to describe an ensemble of spin-$\frac12$ particles or two-state atoms which are homogeneously coupled to external perturbations is in terms of the collective spin operator $\hat{\mathbf J} = \sum_{j=1}^N \hat{\mathbf s}^{(j)}$. If the initial state of the system is a product state with all members occupying the same pure state, the dynamically accessible Hilbert space is characterized by a single angular momentum ladder. It is then straightforward to introduce oscillator-like creation and annihilation operators in the Holstein-Primakoff representation, $$\begin{gathered}
  \label{eq:standard-HP}
  \hat J_{x-} \equiv \hat a^\dag \sqrt{2J-\hat a^\dag \hat a},
  \qquad
  \hat J_x \equiv J- \hat a^\dag \hat a,\end{gathered}$$ 