  Kicked atoms under a constant Stark or gravity field are investigated for
experimental setups with cold and ultra cold atoms. The parametric stability of
the quantum dynamics is studied using the fidelity. In the case of a quantum
resonance, it is shown that the behavior of the fidelity depends on arithmetic
properties of the gravity parameter. Close to a quantum resonance, the long
time asymptotics of the fidelity is studied by means of a {\em
pseudo-classical} approximation first introduced by Fishman {\em et al.} [J.
Stat. Phys. {\bf 110}, 911 (2003)]. The long-time decay of fidelity arises from
the tunneling out of pseudo-classical stable islands, and a simple ansatz is
proposed which satisfactorily reproduces the main features observed in
numerical simulations.
