Single-mode nonclassicality criteria via Holstein-Primakoff transformation

We study different representations of a many-particle system. We reveal the stunning beauty of the connections among the nonclassicalities of a many-particle, a single-mode and a two-mode systems. First, we demonstrate that (separable) atomic coherent states~(ACSs), of a many-particle system, mimic the coherent states of single-mode light. This enables the utilization of a many-particle entanglement~(MPE) criterion as a single-mode nonclassicality~(SMNc) condition via Holstein-Primakoff transformation. As an example, we show that spin-squeezing criterion, for MPE, can be transformed into the quadrature-squeezing, a nonclassicality condition for a single-mode system. Second, we demonstrate a similar connection between the two-mode separable states and separable ACSs of a many-particle system. The second connection makes us be able to obtain a single-mode nonclassicality condition from a two-mode entanglement criterion, using the many-particle system as a bridge. As an example, we obtain Mandel's Q-parameter from the two-mode entanglement criterion of Hillery{\&}Zubairy.