  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.
