---
abstract: 'We show that an ensemble of identical $d$-level atoms can be efficiently described by $d-1$ collective oscillator degrees of freedom in the vicinity of a product state with all atoms in the same, but otherwise arbitrary single-particle state. We apply our description to two different kinds of spin squeezing: (i) when each spin-$F$ atom is individually squeezed without creating interatomic entanglement and (ii) when a particular collective atomic oscillator mode is squeezed via quantum non-demolition (QND) measurement and feedback. When combined in sequence, the order of the two methods is relevant in the final degree of squeezing. We also discuss the role of the two kinds of squeezing when multi-sublevel atoms are used as quantum memories for light.'
author:
- 'Z. Kurucz'
- 'K. M[ø]{}lmer'
bibliography:
- 'qmem-refs.bib'
date: 'Dec. 20, 2009'
title: 'Multilevel Holstein-Primakoff approximation and its application to atomic spin squeezing and ensemble quantum memories'
---

Introduction {#sec:introduction}
============

Large ensembles of identical particles are excellent candidates for light-matter interfaces, quantum memories, repeaters for long distance quantum communication, and registers for quantum computing. This is not only due to the fact that in these systems certain collective quantum degrees of freedom efficiently couple to the electromagnetic 