Quantum chaos and fractals with atoms in cavities

We study the coupled translational, electronic, and field dynamics of the combined system "a two-level atom + a single-mode quantized field + a standing-wave ideal cavity". We derive Hamilton -- Schr\"odinger equations for probability amplitudes and averaged position and momentum of a point-like atom interacting with the quantized field in a standing-wave cavity. They constitute, in general, an infinite-dimensional set of equations with an infinite number of integrals of motion which may be reduced to a dynamical system with four degrees of freedom if the quantized field is supposed to be initially prepared in a Fock state. This system is found to produce semiquantum chaos with positive values of the maximal Lyapunov exponent. At large values of detuning $|\delta|\gg 1$, the Rabi atomic oscillations are usually shallow, and the dynamics is found to be almost regular. The Doppler -- Rabi resonance, deep Rabi oscillations that may occur at any large value of $|\delta|$ to be equal to $|\alpha p_0|$, is found numerically and described analytically (with $\alpha$ to be the normalized recoil frequency and $p_0$ the initial atomic momentum). Two gedanken experiments are proposed to detect manifestations of semiquantum chaos in real experiments. In the chaotic regime values of the population inversion $z_{out}$, measured with atoms after transversing a cavity, are so sensitive to small changes in the initial inversion $z_{in}$ that the probability of detecting any value of $z_{out}$ in the admissible interval becomes almost unity in a short time. Chaotic wandering of a two-level atom in a quantized Fock field is shown to be fractal. Fractal-like structures, typical for chaotic scattering, are numerically found in the dependence of the time of exit of atoms from the cavity on their initial momenta.