  Strong and ultra-strong coupling in "cavity - single atom" system are of
great interest for both fundamental and applied physics. It is considered that
the increase in the coupling strength between a cavity mode and an atom leads,
first, to transition from weak to strong coupling and, second, to ultra-strong
coupling regime. In this letter, we refute this common opinion and demonstrate
that the transitions between the coupling regimes occur in different sequence
for the correlations' functions of different orders. We show that for n-th
order correlations' functions, the transition to the strong coupling regime
requires the coupling strength approximately by $n^{2/3}$ times greater than
the one for first order correlations' functions. In contrast, the transition to
the ultra-strong coupling regime manifests in the dynamics of n-th order
correlations' functions at the less coupling strength than in the dynamics of
first order correlations' functions. As a result, there is the order of
correlations' functions, above which the increase in the coupling strength
leads, first, to the transition from the weak coupling first to the
ultra-strong coupling regime, and second to the strong coupling regime. We
argue that the measurement of high orders correlations' functions makes it
possible to observe the ultra-strong coupling in "cavity mode - single atom"
when the coupling strength is much less than one tenth of the oscillation
frequency.
