  We show that the correspondence between quantum and classical mechanics can
be tuned by varying the coupling strength between the cavity modes and an atom
or a molecule. In the acceleration gauge the cavity-matter system is
represented by an effective Hamiltonian with a non-trivial coupling appearing
in the potential, and a renormaized mass. Importantly, the acceleration-gauge
coupling is non-monotonic with the strength cavity-matter interaction. As a
result one obtain effective approximately decoupled field-matter dynamics for
weak and strong interactions. In the weak coupling regime the effective mass is
essentially the original mass. In contrast, the renormalized mass is increased
as the interaction is increased. This results in acceleration gauge dynamics of
atom/molecule with the original Hamiltonian and effective Planck's constant
that is reduced when the interaction is increased. This approach might lead in
particular to the possibility of studying the correspondence of "quantum-chaos"
(quantum stochasticity) with classical chaos, as well as either enhancement or
suppression of tunneling, by varying a controllable physical parameter.
Physical realization of our findings is briefly discussed.
