  Stability and instability of quantum evolution are studied in the interaction
between a two-level atom with photon recoil and a quantized field mode in an
ideal cavity, the basic model of cavity quantum electrodynamics (QED). It is
shown that the Jaynes-Cummings dynamics can be unstable in the regime of
chaotic walking of the atomic center-of-mass in the quantized field of a
standing wave in the absence of any kind of interaction with environment. This
kind of quantum instability manifests itself in strong variations of reduced
quantum purity and entropy, correlating with the respective classical Lyapunov
exponent, and in exponential sensitivity of fidelity of quantum states to small
variations in the atom-field detuning. The connection between quantum
entanglement and fidelity and the center-of-mass motion is clarified
analytically and numerically for a few regimes of that motion. The results are
illustrated with two specific initial field states: the Fock and coherent ones.
Numerical experiments demonstrate various manifestations of the
quantum-classical correspondence, including dynamical chaos and fractals, which
can be, in principle, observed in real experiments with atoms and photons in
high finesse cavities.
