Fractal atom-photon dynamics in a cavity

Nonlinear dynamics in the fundamental interaction between a two-level atom with recoil and a quantized radiation field in a high-quality cavity is studied. We consider the strongly coupled atom-field system as a quantum-classical hybrid with dynamically coupled quantum and classical degrees of freedom. We show that, even in the absence of any other interaction with environment, the interaction of the purely quantum atom-field system with the external atomic degree of freedom provides the emergence of classical dynamical chaos from quantum electrodynamics. Atomic fractals with self-similar intermittency of smooth and unresolved structures are found in the exit-time scattering function. Tiny interplay between all the degrees of freedom is responsible for dynamical trapping of atoms even in a very short microcavity. Gedanken experiments are proposed to detect manifestations of atomic fractals in cavity quantum electrodynamics.Comment: Lecture notes in NATO Advanced Study Institute Summer School on Chaotic Dynamics and Transport (Aug. 18-30, 2003, Cargese, Corsica, France

Nonadiabatic quantum chaos in atom optics

Coherent dynamics of atomic matter waves in a standing-wave laser field is studied. In the dressed-state picture, wave packets of ballistic two-level atoms propagate simultaneously in two optical potentials. The probability to make a transition from one potential to another one is maximal when centroids of wave packets cross the field nodes and is given by a simple formula with the single exponent, the Landau--Zener parameter $\kappa$. If $\kappa \gg 1$, the motion is essentially adiabatic. If $\kappa \ll 1$, it is (almost) resonant and periodic. If $\kappa \simeq 1$, atom makes nonadiabatic transitions with a splitting of its wave packet at each node and strong complexification of the wave function as compared to the two other cases. This effect is referred as nonadiabatic quantum chaos. Proliferation of wave packets at $\kappa \simeq 1$ is shown to be connected closely with chaotic center-of-mass motion in the semiclassical theory of point-like atoms with positive values of the maximal Lyapunov exponent. The quantum-classical correspondence established is justified by the fact that the Landau--Zener parameter $\kappa$ specifies the regime of the semiclassical dynamical chaos in the map simulating chaotic center-of-mass motion. Manifestations of nonadiabatic quantum chaos are found in the behavior of the momentum and position probabilities.Comment: arXiv admin note: substantial text overlap with arXiv:1201.032

Genealogical tree of Russian schools on Nonlinear Dynamics

One of the most prominent feature of research in Russia and the former Soviet Union is so-called scientific schools. It is a collaboration of researchers with a common scientific background working, as a rule, together in a specific city or even at an institution. The genealogical tree of scientific schools on nonlinear dynamics in Russia and the former Soviet Union is grown. We use these terminology in a broad sense including theory of dynamical systems and chaos and its applications in nonlinear physics. In most cases we connect two persons if one was an advisor of the Doctoral thesis of another one. It is an analogue of the Candidate of Science thesis in Russia. If the person had no official advisor or we don't know exactly who was an advisor, we fix that person who was known to be an informal teacher and has influenced on him/her very much

Nonlinear control of chaotic walking of atoms in an optical lattice

Centre-of-mass atomic motion in an optical lattice near the resonance is shown to be a chaotic walking due to the interplay between coherent internal atomic dynamics and spontaneous emission. Statistical properties of chaotic atomic motion can be controlled by the single parameter, the detuning between the atomic transition frequency and the laser frequency. We derive a Fokker-Planck equation in the energetic space to describe the atomic transport near the resonance and demonstrate numerically how to manipulate the atomic motion varying the detuning.Comment: 6 pages, 4 figure

Synchronization of internal and external degrees of freedom of atoms in a standing laser wave

We consider dissipative dynamics of atoms in a strong standing laser wave and find a nonlinear dynamical effect of synchronization between center-of-mass motion and internal Rabi oscillations. The synchronization manifests itself in the phase space as limit cycles which may have different periods and riddled basins of attraction. The effect can be detected in the fluorescence spectra of atoms as equidistant sideband frequencies with the space between adjacent peaks to be inversely proportional to the value of the period of the respective limit cycle. With increasing the intensity of the laser field, we observe numerically cascades of bifurcations that eventually end up in settling a strange chaotic attractor. A broadband noise is shown to destroy a fine structure of the bifurcation scenario, but prominent features of period-1 and period-3 limit cycles survive under a weak noise. The character of the atomic motion is analyzed with the help of the friction force whose zeroes are attractor or repellor points in the velocity space. We find ranges of the laser parameters where the atomic motion resembles a random but deterministic walking of atoms erratically jumping between different wells of the optical potential. Such a random walking is shown to be fractal in the sense that the measured characteristic of the motion, time of exit of atoms from a given space of the standing wave, is a complicated function that has a self-similar structure with singularities on a Cantor set of values of one of the control parameters.Comment: Submitted to Phys. Rev.

Theory of dissipative chaotic atomic transport in an optical lattice

We study dissipative transport of spontaneously emitting atoms in a 1D standing-wave laser field in the regimes where the underlying deterministic Hamiltonian dynamics is regular and chaotic. A Monte Carlo stochastic wavefunction method is applied to simulate semiclassically the atomic dynamics with coupled internal and translational degrees of freedom. It is shown in numerical experiments and confirmed analytically that chaotic atomic transport can take the form either of ballistic motion or a random walking with specific statistical properties. The character of spatial and momentum diffusion in the ballistic atomic transport is shown to change abruptly in the atom-laser detuning regime where the Hamiltonian dynamics is irregular in the sense of dynamical chaos. We find a clear correlation between the behavior of the momentum diffusion coefficient and Hamiltonian chaos probability which is a manifestation of chaoticity of the fundamental atom-light interaction in the diffusive-like dissipative atomic transport. We propose to measure a linear extent of atomic clouds in a 1D optical lattice and predict that, beginning with those values of the mean cloud's momentum for which the probability of Hamiltonian chaos is close to 1, the linear extent of the corresponding clouds should increase sharply. A sensitive dependence of statistical characteristics of dissipative transport on the values of the detuning allows to manipulate the atomic transport by changing the laser frequency

Atomic Fractals in Cavity QED

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.Comment: 4 pages, 4 figures, new experimentally feasible scheme of generating atomic fractals added, submitted to Phys. Rev. Letter

Chaotic walking and fractal scattering of atoms in a tilted optical lattice

Chaotic walking of cold atoms in a tilted optical lattice, created by two counter propagating running waves with an additional external field, is demonstrated theoretically and numerically in the semiclassical and Hamiltonian approximations. The effect consists in random-like changing the direction of atomic motion in a rigid lattice under the influence of a constant force due to a specific behavior of the atomic dipole-moment component that changes abruptly in a random-like manner while atoms cross standing-wave nodes. Chaotic walking generates a fractal-like scattering of atoms that manifests itself in a self-similar structure of the scattering function in the atom-field detuning, position and momentum spaces. The probability distribution function of the scattering time is shown to decay in a non-exponential way with a power-law tail

Chaotic mixing and fractals in a geophysical jet current

We model Lagrangian lateral mixing and transport of passive scalars in meandering oceanic jet currents by two-dimensional advection equations with a kinematic stream function with a time-dependent amplitude of a meander imposed. The advection in such a model is known to be chaotic in a wide range of the meander's characteristics. We study chaotic transport in a stochastic layer and show that it is anomalous. The geometry of mixing is examined and shown to be fractal-like. The scattering characteristics (trapping time of advected particles and the number of their rotations around elliptical points) are found to have a hierarchical fractal structure as functions of initial particle's positions. A correspondence between the evolution of material lines in the flow and elements of the fractal is established

Nonlinear coherent dynamics of an atom in an optical lattice

We consider a simple model of lossless interaction between a two-level single atom and a standing-wave single-mode laser field which creates a one-dimensional optical lattice. Internal dynamics of the atom is governed by the laser field which is treated to be classical with a large number of photons. Center-of-mass classical atomic motion is governed by the optical potential and the internal atomic degree of freedom. The resulting Hamilton-Schr\"odinger equations of motion are a five-dimensional nonlinear dynamical system with two integrals of motion. The main focus of the paper is chaotic atomic motion that may be quantified strictly by positive values of the maximal Lyapunov exponent. It is shown that atom, depending on the value of its total energy, can either oscillate chaotically in a well of the optical potential or fly ballistically with weak chaotic oscillations of its momentum or wander in the optical lattice changing the direction of motion in a chaotic way. In the regime of chaotic wandering atomic motion is shown to have fractal properties. We find a useful tool to visualize complicated atomic motion -- Poincar\'e mapping of atomic trajectories in an effective three-dimensional phase space onto planes of atomic internal variables and momentum. We find common features with typical non-hyperbolic Hamiltonian systems -- chains of resonant islands of different sizes embedded in a stochastic sea, stochastic layers, bifurcations, and so on. The phenomenon of sticking of atomic trajectories to boundaries of regular islands, that should have a great influence to atomic transport in optical lattices, is found and demonstrated numerically