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Relaxation time for monitoring the quantumness of an intense cavity field

Recently it was shown that the quantum behavior of an intense cavity field can be revealed by measuring the steady atomic correlations between two ideal atoms, which interact with the same leaking cavity mode. Considering a weak atom-field coupling regime and large average number of photons in the cavity mode ($\bar{n}$), one expects that a semiclassical theory could explain the whole dynamics of the system. However, this system presents the generation of correlations between the atoms, which is a signature of the quantumness of the cavity field even in the limit of $\bar{n} \gg 1$ [Phys. Rev. Lett. \textbf{107}, 153601 (2011)]. Here, we extend this result by investigating the relaxation time for such a system. We have shown that the relaxation time of the system varies proportionally to $\bar{n}$ for a coherent driving, but it is inversely proportional to $\bar{n}$ for an incoherent pumping. Thus, the time required to observe the manifestation of the quantum aspects of a cavity field on the atomic correlations diverges as $\bar{n}$ tends to macroscopic values due to a coherent driving, while it goes to zero for incoherent pumping. For a coherent driving, we can also see that this system presents metastability, i.e., firstly the atomic system reaches a quasi-stationary state which last for a long time interval, but eventually it reaches the real steady state. We have also discussed the effects of small atomic decay. In this case, the steady correlations between the atoms disappear for long times, but the intense cavity field is still able to generate atomic correlations at intermediate times. Then, considering a real scenario, we would be able to monitor the quantumness of a cavity field in a certain time interval.