Semiclassical interaction of moving two-level atoms with a cavity field: from integrability to Hamiltonian chaos

The dynamics of an ensemble of two-level atoms moving through a single-mode lossless cavity is investigated in the semiclassical and rotating-wave approximations. The dynamical system for the expectation values of the atomic and field observables is considered as a perturbation to one of the following integrable versions: (i) a model with atoms moving through a spatially inhomogeneous resonant field, and (ii) a model with atoms interacting with a nonresonant eigenmode which is assumed to be homogeneous on the cavity size. We find the general exact solutions for both the models and show that they contain special solutions describing a coherent effect of population and radiation trapping. Using the Melnikov method, we prove analytically transverse intersections of stable and unstable manifolds of a hyperbolic fixed point under a small modulation of the vacuum Rabi frequency. These intersections are believed to provide the Smale horseshoe mechanism of Hamiltonian chaos. The analytical results are accompanied with direct computation of topographical maps of maximal Lyapunov exponents that give a representative image of regularity and chaos in the atom-field system in different ranges of its control parameters--the frequency detuning, the number, and the velocity of atoms.