Chaos and flights in the atom-photon interaction in cavity QED

We study dynamics of the atom-photon interaction in cavity quantum electrodynamics (QED), 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 L\'{e}vy flights and chaotic walkings of an atom in a cavity. 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 $\alpha$. 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 L\'{e}vy flights, one corresponding to the ballistic motion of the atom and another one corresponding to small oscillations in a potential well, are found. These flights influence statistical properties of the atom-photon interaction such as distribution of Poincar\'{e} recurrences and moments of the atom position $x$. The simulation shows different regimes of motion, from slightly abnormal diffusion with $\sim\tau^{1.13}$ at $\delta =1.2$ to a superdiffusion with $\sim \tau^{2.2}$ 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 $\delta$.Comment: 16 pages, 7 figures. To be published in Phys. Rev.

A Modified Particle Swarm Optimization Algorithm for the Best Low Multilinear Rank Approximation of Higher-Order Tensors

Abstract. The multilinear rank of a tensor is one of the possible gener-alizations for the concept of matrix rank. In this paper, we are interested in finding the best low multilinear rank approximation of a given ten-sor. This problem has been formulated as an optimization problem over the Grassmann manifold [14] and it has been shown that the objec-tive function presents multiple minima [15]. In order to investigate the landscape of this cost function, we propose an adaptation of the Parti-cle Swarm Optimization algorithm (PSO). The Guaranteed Convergence PSO, proposed by van den Bergh in [23], is modified, including a gradi-ent component, so as to search for optimal solutions over the Grassmann manifold. The operations involved in the PSO algorithm are redefined using concepts of differential geometry. We present some preliminary nu-merical experiments and we discuss the ability of the proposed method to address the multimodal aspects of the studied problem