Large detuning limit for the multipartite systems interacting with electromagnetic fields

We study the dynamics of the multipartite systems nonresonantly interacting with electromagnetic fields and show that, since the coupling strength $g$ is collectively enhanced by the square root of the number of microparticles involved, the more rigorous large detuning condition for neglecting the rapidly oscillating terms for the effective Hamiltonian should be $\mathrm{\Delta }\gg \sqrt{N}g$, instead of $\mathrm{\Delta }\gg g$ usually used in the literature even in the case of multipartite systems, with $\mathrm{\Delta }$ the detuning and and $N$ the number of microparticles. This result is significant since merely the satisfaction of the original condition will result in the invalidity of the effective Hamiltonian and the errors of the parameters associated with the detuning in the multipartite case.