where 1/T 1 = k +→D , and we assume that for the pure decoherence rate it is half of the population transfer rate between |+⟩ and |-⟩ states (which is indeed valid under the Markovian approximation and this can be seen from the Lindblad master equations ). Under the large N limit (or collective strong coupling limit), √ N g c ≫ γ, the spectral density J ν ( √ N g c ) ∼ 1/ √ N g c (c.f. Eq. 5), and we find the following fundamental scalings