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We show that an ensemble of identical d-level atoms can be efficientlydescribed by d-1 collective oscillator degrees of freedom in the vicinity of aproduct state with all atoms in the same, but otherwise arbitrarysingle-particle state. We apply our description to two different kinds of spinsqueezing: (i) when each spin-F atom is individually squeezed without creatinginteratomic entanglement and (ii) when a particular collective atomicoscillator mode is squeezed via quantum non-demolition (QND) measurement andfeedback. When combined in sequence, the order of the two methods is relevantin the final degree of squeezing. We also discuss the role of the two kinds ofsqueezing when multi-sublevel atoms are used as quantum memories for light.

Multilevel Holstein-Primakoff approximation and its application to atomic spin squeezing and ensemble quantum memories