Integrated random processes exhibiting long tails, finite moments and 1/f spectra

A dynamical model based on a continuous addition of colored shot noises is presented. The resulting process is colored and non-Gaussian. A general expression for the characteristic function of the process is obtained, which, after a scaling assumption, takes on a form that is the basis of the results derived in the rest of the paper. One of these is an expansion for the cumulants, which are all finite, subject to mild conditions on the functions defining the process. This is in contrast with the Levy distribution -which can be obtained from our model in certain limits- which has no finite moments. The evaluation of the power spectrum and the form of the probability density function in the tails of the distribution shows that the model exhibits a 1/f spectrum and long tails in a natural way. A careful analysis of the characteristic function shows that it may be separated into a part representing a Levy processes together with another part representing the deviation of our model from the Levy process. This allows our process to be viewed as a generalization of the Levy process which has finite moments.Comment: Revtex (aps), 15 pages, no figures. Submitted to Phys. Rev.

Fractional transport equations for Levy stable processes

The influence functional method of Feynman and Vernon is used to obtain a quantum master equation for a Brownian system subjected to a Levy stable random force. The corresponding classical transport equations for the Wigner function are then derived, both in the limit of weak and strong friction. These are fractional extensions of the Klein-Kramers and the Smoluchowski equations. It is shown that the fractional character acquired by the position in the Smoluchowski equation follows from the fractional character of the momentum in the Klein-Kramers equation. Connections among fractional transport equations recently proposed are clarified.Comment: 4 page

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.

Exotic quantum dark states

We extend studies of velocity selective coherent population trapping to atoms having a $J=1 \rightarrow J=0$ transition. When placed in a two-dimensional laser field these atoms are optically pumped into different velocity selective nonabsorbing states. Each of these distinct energy eigenstates exhibits a unique entanglement between its internal and external degrees of freedom. We use a graphical method that makes easier the description of these states. We confirm our predictions experimentally