Quantum correlations for a simple kicked system with mixed phase space.

We investigate both the classical and quantum dynamics for a simple kicked system (the standard map) that classically has mixed phase space. For initial conditions in a portion of the chaotic region that is close enough to the regular region, the phenomenon of sticking leads to a power-law decay with time of the classical correlation function of a simple observable. Quantum mechanically, we find the same behavior, but with a smaller exponent. We consider various possible explanations of this phenomenon, and settle on a modification of the Meiss-Ott Markov tree model that takes into account quantum limitations on the flux through a turnstile between regions corresponding to states on the tree. Further work is needed to better understand the quantum behavior.

Universal exponent for transport in mixed Hamiltonian dynamics.

We compute universal distributions for the transition probabilities of a Markov model for transport in the mixed phase space of area-preserving maps and verify that the survival probability distribution for trajectories near an infinite island-around-island hierarchy exhibits, on average, a power-law decay with exponent γ=1.57. This exponent agrees with that found from simulations of the Hénon and Chirikov-Taylor maps. This provides evidence that the Meiss-Ott Markov tree model describes the transport for mixed systems.

Soliton mobility in disordered lattices.

We investigate soliton mobility in the disordered Ablowitz-Ladik (AL) model and the standard nonlinear Schrödinger (NLS) lattice with the help of an effective potential generalizing the Peierls-Nabarro potential. This potential results from a deviation from integrability, which is due to randomness for the AL model, and both randomness and lattice discreteness for the NLS lattice. The statistical properties of such a potential are analyzed, and it is shown how the soliton mobility is affected by its size. The usefulness of this effective potential in studying soliton dynamics is demonstrated numerically. Furthermore, we propose two ways to enhance soliton transport in the presence of disorder: one is to use specific realizations of randomness, and the other is to consider a specific soliton pair.

Diffusion for ensembles of standard maps.

Two types of random evolution processes are studied for ensembles of the standard map with driving parameter K that determines its degree of stochasticity. For one type of process the parameter K is chosen at random from a Gaussian distribution and is then kept fixed, while for the other type it varies from step to step. In addition, noise that can be arbitrarily weak is added. The ensemble average and the average over noise of the diffusion coefficient are calculated for both types of processes. These two types of processes are relevant for two types of experimental situations as explained in the paper. Both types of processes destroy fine details of the dynamics, and the second process is found to be more effective in destroying the fine details. We hope that this work is a step in the efforts for developing a statistical theory for systems with mixed phase space (regular in some parts and chaotic in other parts).

Soliton trapping in a disordered lattice.

In recent years, the competition between randomness and nonlinearity was extensively explored. In the present paper, the dynamics of solitons of the Ablowitz-Ladik model in the presence of a random potential is studied. In the absence of the random potential, it is an integrable model and the solitons are stable. As a result of the random potential, this stability is destroyed. In a certain regime, for short times, particlelike dynamics with constant mass is found; in another regime, particlelike dynamics with varying mass takes place. In particular, an effective potential is found that predicts correctly changes in the direction of motion of the soliton. This potential is a scaling function of time and strength of the potential, leading to a relation between the first time when the soliton changes direction and the strength of the random potential.

Statistics of the island-around-island hierarchy in Hamiltonian phase space.

The phase space of a typical Hamiltonian system contains both chaotic and regular orbits, mixed in a complex, fractal pattern. One oft-studied phenomenon is the algebraic decay of correlations and recurrence time distributions. For area-preserving maps, this has been attributed to the stickiness of boundary circles, which separate chaotic and regular components. Though such dynamics has been extensively studied, a full understanding depends on many fine details that typically are beyond experimental and numerical resolution. This calls for a statistical approach, the subject of the present work. We calculate the statistics of the boundary circle winding numbers, contrasting the distribution of the elements of their continued fractions to that for uniformly selected irrationals. Since phase space transport is of great interest for dynamics, we compute the distributions of fluxes through island chains. Analytical fits show that the "level" and "class" distributions are distinct, and evidence for their universality is given.

Transport in time-dependent random potentials.

The classical dynamics in potentials that are random both in space and time is studied. The potentials are generated by a stationary process. This can be intuitively understood with the help of Chirikov resonances that are central in the theory of chaos, and explored quantitatively in the framework of the Fokker-Planck equation. In particular, a simple expression for the diffusion coefficient was obtained in terms of the average power density of the potential. The resulting anomalous diffusion in velocity is classified into universality classes. The general theory was applied and numerically tested for specific examples relevant for optics and atom optics.

Universality classes of transport in time-dependent random potentials.

The growth of the average kinetic energy of classical particles is studied for potentials that are random both in space and time. Such potentials are relevant for recent experiments in optics and in atom optics. It is found that for small velocities uniform acceleration takes place, and at a later stage fluctuations of the potential are encountered, resulting in a regime of anomalous diffusion. This regime was studied in the framework of the Fokker-Planck approximation. The diffusion coefficient in velocity was expressed in terms of the average power spectral density, which is the Fourier transform of the potential correlation function. This enabled to establish a scaling form for the Fokker-Planck equation and to compute the large and small velocity limits of the diffusion coefficient. A classification of the random potentials into universality classes, characterized by the form of the diffusion coefficient in the limit of large and small velocity, was performed. It was shown that one-dimensional systems exhibit a large variety of universality classes, contrary to systems in higher dimensions, where only one universality class is possible. The relation to Chirikov resonances, which are central in the theory of chaos, was demonstrated. The general theory was applied and numerically tested for specific physically relevant examples.

Effective noise theory for the nonlinear Schrödinger equation with disorder.

For the nonlinear Shrödinger equation with disorder it was found numerically that in some regime of the parameters Anderson localization is destroyed and subdiffusion takes place for a long time interval. It was argued that the nonlinear term acts as random noise. In the present work, the properties of this effective noise are studied numerically. Some assumptions made in earlier work were verified, and fine details were obtained. The dependence of various quantities on the localization length of the linear problem were computed. A scenario for the possible breakdown of the theory for a very long time is outlined.

Quantum chaos of a mixed open system of kicked cold atoms.

The quantum and classical dynamics of particles kicked by a Gaussian attractive potential are studied. Classically, it is an open mixed system (the motion in some parts of the phase space is chaotic, and in some parts it is regular). The fidelity (Loschmidt echo) is found to exhibit oscillations that can be determined from classical considerations but are sensitive to phase space structures that are smaller than Planck's constant. Families of quasienergies are determined from classical phase space structures. Substantial differences between the classical and quantum dynamics are found for time-dependent scattering. It is argued that the system can be experimentally realized by cold atoms kicked by a Gaussian light beam.

Scaling properties of weak chaos in nonlinear disordered lattices.

We study the discrete nonlinear Schrödinger equation with a random potential in one dimension. It is characterized by the length, the strength of the random potential, and the field density that determines the effect of nonlinearity. Following the time evolution of the field and calculating the largest Lyapunov exponent, the probability of the system to be regular is established numerically and found to be a scaling function of the parameters. This property is used to calculate the asymptotic properties of the system in regimes beyond our computational power.

Quantum zigzag transition in ion chains.

A string of trapped ions at zero temperature exhibits a structural phase transition to a zigzag structure, tuned by reducing the transverse trap potential or the interparticle distance. The transition is driven by transverse, short wavelength vibrational modes. We argue that this is a quantum phase transition, which can be experimentally realized and probed. Indeed, by means of a mapping to the Ising model in a transverse field, we estimate the quantum critical point in terms of the system parameters, and find a finite, measurable deviation from the critical point predicted by the classical theory. A measurement procedure is suggested which can probe the effects of quantum fluctuations at criticality. These results can be extended to describe the transverse instability of ultracold polar molecules in a one-dimensional optical lattice.

Spreading for the generalized nonlinear Schrödinger equation with disorder.

The dynamics of an initially localized wave packet is studied for the generalized nonlinear Schrödinger equation with a random potential, where the nonlinear term is beta|psi|ppsi and p is arbitrary. Mainly short times for which the numerical calculations can be performed accurately are considered. Long time calculations are presented as well. In particular, the subdiffusive behavior where the average second moment of the wave packet is of the form m2 approximately t(alpha) is computed. Contrary to former heuristic arguments, no evidence for any critical behavior as function of p is found. The properties of alpha(p) for relatively short times are explored, a scaling property and a maximal value for p approximately 1/2 are found.

Asymptotic localization of stationary states in the nonlinear Schrödinger equation.

The mapping of the nonlinear Schrödinger equation with a random potential on the Fokker-Planck equation is used to calculate the localization length of its stationary states. The asymptotic growth rates of the moments of the wave function and its derivative for the linear Schrödinger equation in a random potential are computed analytically, and resummation is used to obtain the corresponding growth rate for the nonlinear Schrödinger equation and the localization length of the stationary states.

Transport and Anderson localization in disordered two-dimensional photonic lattices.

One of the most interesting phenomena in solid-state physics is Anderson localization, which predicts that an electron may become immobile when placed in a disordered lattice. The origin of localization is interference between multiple scatterings of the electron by random defects in the potential, altering the eigenmodes from being extended (Bloch waves) to exponentially localized. As a result, the material is transformed from a conductor to an insulator. Anderson's work dates back to 1958, yet strong localization has never been observed in atomic crystals, because localization occurs only if the potential (the periodic lattice and the fluctuations superimposed on it) is time-independent. However, in atomic crystals important deviations from the Anderson model always occur, because of thermally excited phonons and electron-electron interactions. Realizing that Anderson localization is a wave phenomenon relying on interference, these concepts were extended to optics. Indeed, both weak and strong localization effects were experimentally demonstrated, traditionally by studying the transmission properties of randomly distributed optical scatterers (typically suspensions or powders of dielectric materials). However, in these studies the potential was fully random, rather than being 'frozen' fluctuations on a periodic potential, as the Anderson model assumes. Here we report the experimental observation of Anderson localization in a perturbed periodic potential: the transverse localization of light caused by random fluctuations on a two-dimensional photonic lattice. We demonstrate how ballistic transport becomes diffusive in the presence of disorder, and that crossover to Anderson localization occurs at a higher level of disorder. Finally, we study how nonlinearities affect Anderson localization. As Anderson localization is a universal phenomenon, the ideas presented here could also be implemented in other systems (for example, matter waves), thereby making it feasible to explore experimentally long-sought fundamental concepts, and bringing up a variety of intriguing questions related to the interplay between disorder and nonlinearity.

Localization length of stationary states in the nonlinear Schrödinger equation.

For the nonlinear Schrödinger equation (NLSE), in the presence of disorder, exponentially localized stationary states are found. We demonstrate analytically that the localization length is typically independent of the strength of the nonlinearity and is identical to the one found for the corresponding linear equation. The analysis makes use of the correspondence between the stationary NLSE and the Langevin equation as well as of the resulting Fokker-Planck equation. The calculations are performed for the "white noise" random potential, and an exact expression for the exponential growth of the eigenstates is obtained analytically. It is argued that the main conclusions are robust.

Scaling and universality of the complexity of analog computation.

We apply a probabilistic approach to study the computational complexity of analog computers which solve linear programming problems. We numerically analyze various ensembles of linear programming problems and obtain, for each of these ensembles, the probability distribution functions of certain quantities which measure the computational complexity, known as the convergence rate, the barrier and the computation time. We find that in the limit of very large problems these probability distributions are universal scaling functions. In other words, the probability distribution function for each of these three quantities becomes, in the limit of large problem size, a function of a single scaling variable, which is a certain composition of the quantity in question and the size of the system. Moreover, various ensembles studied seem to lead essentially to the same scaling functions, which depend only on the variance of the ensemble. These results extend analytical and numerical results obtained recently for the Gaussian ensemble, and support the conjecture that these scaling functions are universal.

Time-independent approximations for periodically driven systems with friction.

The classical dynamics of a particle that is driven by a rapidly oscillating potential (with frequency omega) is studied. The motion is separated into a slow part and a fast part that oscillates around the slow part. The motion of the slow part is found to be described by a time-independent equation that is derived as an expansion in orders of omega(-1) (in this paper terms to the order omega(-3) are calculated explicitly). This time-independent equation is used to calculate the attracting fixed points and their basins of attraction. The results are found to be in excellent agreement with numerical solutions of the original time-dependent problem.

Changes in the slope of the first major deflection of the ECG complex during acute coronary occlusion.

To evaluate the effect of acute coronary occlusion (ACO) during percutaneous coronary intervention (PCI) on the slope of the first major deflection of the QRS complex (the initial QRS slope). Standard ECG signals of 18 patients (89 leads), undergoing PCI were recorded prior to and during ACO. The initial QRS slope was calculated in the baseline state and during ACO. Changes in the standard ECG were detected in 36 of 89 leads (40%). The initial QRS slope during ACO was significantly different from baseline in 74 of 89 leads (83%). The specificity of the change in the slope during ACO was low (29%).

Eigenmodes and thermodynamics of a Coulomb chain in a harmonic potential.

The density of ions trapped in a harmonic potential in one dimension is not uniform. Consequently the eigenmodes are not phononlike waves. We calculate the long-wavelength modes in the continuum limit, and evaluate the density of states in the short-wavelength limit for chains of N>>1 ions. Remarkably, the results that are found analytically in the thermodynamic limit provide a good estimate of the spectrum of excitations of small chains down to few tens of ions. The spectra are used to compute the thermodynamic functions of the chain. Deviations from the extensivity of the thermodynamic quantities are found. An analytic expression for the critical transverse frequency determining the stability of a linear chain is derived.