The motion of a two-level atom with an internal degree of freedom that interacts with the field of a single-mode standing light wave in a high- Q cavity is treated as a scattering problem. At detunings between atomic and field resonances δ, atomic recoil frequencies α, initial atomic momenta ρ 0 , and numbers of excitations N at which the semiclassical equations have chaotic solutions (in the sense of their exponential sensitivity to small changes in the initial conditions), atomic scattering is fractal with strongly pronounced self-similarity of the dependence of time T at which the atom leaves the cavity at its initial momentum; the fractal dimension of this curve is 1.84. In the chaotic regime, there are two infinite sets of atomic initial momenta for which T =∞ (in the idealized case of the absence of any loss). They correspond to separatrix-like trajectories along which the atom asymptotically approaches certain configuration space points and to the trajectories of infinitely long chaotic wandering of atoms in the cavity. These trajectories make up countable and uncountable fractals, respectively. Correlations between Rabi atomic oscillations and atomic motion can lead to Doppler-Rabi resonances, that is, deep oscillations of the internal energy of the atom, for large detunings δ that satisfy the condition |αρ 0 |≈|δ|.
We report a clear evidence of atomic fractals in the nonlinear motion of a two-level atom in a standing-wave microcavity. Fractal-like structures, typical for chaotic scattering, are numerically found in the dependencies of outgoing positions and momenta of scattered atoms on their ingoing values and in the dependence of exit times of cold atoms on their initial momenta in the generic semiclassical models of cavity QED (1) with atoms in a far-detuned amplitude (phase)-modulated standing wave and (2) with coupled atomic external and internal degrees of freedom. Tiny interplay between all the degrees of freedom in the second model is responsible for trapping atoms even in a very short microcavity. It may lead simultaneously, at least, to two kinds of atomic fractals, a countable fractal (a set of initial momenta generating separatrix-like atomic trajectories) and a seemingly uncountable fractal with a set of momenta generating infinite walkings of atoms inside the cavity.
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∣gg1, 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
∣alphap0∣, is found numerically and described analytically (with
alpha to be the normalized recoil frequency and
p0 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
zout, measured with atoms after transversing a cavity, are so sensitive to small changes in the initial inversion
zin that the probability of detecting any value of
zout 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.
Strong coupling of the internal and external degrees of freedom of a cold atom to each other and to the spatially periodic field of the standing light wave in a high-finesse cavity is responsible for the dynamic instability of the atomic center-of-mass motion. Due to a weak interaction of the internal nonlinear resonances in the standard model of cavity QED, a stochastic layer appears, whose width in the semiclassical approximation is estimated in terms of the main parameters of the system: atomic recoil frequency, mean number of excitations, and detuning from the resonance. As a result, the atomic motion in the absolutely regular potential has the fractal character, with long Lévy flights alternating with small chaotic oscillations in potential wells.
We study dynamics of the atom-photon interaction in cavity quantum electrodynamics, considering a cold two-level atom in a single-mode high-finesse standing-wave cavity as a nonlinear Hamiltonian system with three coupled degrees of freedom: translational, internal atomic, and the field. The system proves to have different types of motion including Lévy flights and chaotic walkings of an atom in a cavity. The corresponding equations of motion for expectation values of the atom and field variables have two characteristic time scales: fast Rabi oscillations of the internal atomic and field quantities and slow translational oscillations of the center of the atom mass. It is shown that the translational motion, related to the atom recoils, is governed by an equation of a parametric nonlinear pendulum with a frequency modulated by the Rabi oscillations. This type of dynamics is chaotic with some width of the stochastic layer that is estimated analytically. The width is fairly small for realistic values of the control parameters, the normalized detuning delta and atomic recoil frequency alpha. We consider the Poincaré sections of the dynamics, compute the Lyapunov exponents, and find a range of the detuning, |delta| less, similar 3, where chaos is prominent. It is demonstrated how the atom-photon dynamics with a given value of alpha depends on the values of delta and initial conditions. Two types of Lévy flights, one corresponding to the ballistic motion of the atom and the other corresponding to small oscillations in a potential well, are found. These flights influence statistical properties of the atom-photon interaction such as distribution of Poincaré recurrences and moments of the atom position x. The simulation shows different regimes of motion, from slightly abnormal diffusion with <x(2)> approximately tau(1.13) at delta=1.2 to a superdiffusion with <x(2)> approximately tau(2.2) at delta=1.92 that corresponds to a superballistic motion of the atom with an acceleration. The obtained results can be used to find new ways to manipulate atoms, to cool and trap them by adjusting the detuning delta.
We treat semiclassical nonlinear dynamics of a two-level atom in the strong standing-wave field created in a high-quality cavity and in a free space with the help of two counterpropagating laser beams. First of all, we consider the dynamics of the strongly coupled atom-field system in an ideal cavity without any losses and discuss briefly nonlinear dynamical effects of random walking of an atom in a fully deterministic standing wave and Hamiltonian dynamical fractals. Then we study how these effects change in a free space when we take into account relaxation and external pumping of the standing wave. We discuss briefly new dissipative effects of synchronization between electronic and mechanical degrees of freedom of atoms, limit cycles of different periods and their bifurcations, and strange atomic attractors. Finally, we treat spontaneous relaxation as a random events in the equations of motion and study what happens with atomic fractality and limit cycles. It is shown that a self-similar structure of the scattering function (at least, on a few first levels) is conserved in the presence of spontaneous jumps. As to limit cycles, we could not resolve limit cycles of different period, but the fact of synchronization of internal and external atomic degrees of freedom is strictly established in the system with spontaneous jumps.
Nonlinear dynamics in the fundamental interaction between a two-level atom with recoil and a quantized radiation field in a high-quality microcavity is studied. We consider the strongly coupled atom-field system as a quantum-classical hybrid with dynamically coupled quantum and classical egrees of freedom. We show that, even in the absence of any other interaction with environment, the coupling of quantum and classical degrees of freedom provides the emergence of classical dynamical chaos from quantum electrodynamics. It manifest itself in the atomic external degree of freedom as a random walking of an atom inside a cavity with a prominent fractal-like behavior and in the quantum atom-filed degrees of freedom as a sensitive dependence of atomic inversion on small variations in initial conditions. It is shown that dependences of variance of quantum entanglement and of the maximum Lyapunov exponent on the detuning of the atom-field resonance correlate strongly. This result provides a quantum-classical correspondence in a closed physical system.