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.

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.

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.

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.

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.

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.

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%).

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 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.

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.

Evaluation of the phase-plane ECG as a technique for detecting acute coronary occlusion.

OBJECTIVE: To evaluate the phase plane (PP) ECG as a method for detecting acute coronary occlusion (ACO). BACKGROUND: Balloon inflation in a coronary artery during PTCA produces acute myocardial ischemia. The sensitivity of the standard ECG for detecting ACO is approximately 50%, depending on the number of leads recorded. METHODS: The standard ECG signals of 18 patients (91 leads), undergoing PTCA were sampled and converted to digital data, prior to, and during acute coronary occlusion. PPs were constructed by projecting the ECG signals and their first derivatives onto a two-dimensional plane. Standard ECG signals and PPs, prior to ACO, were compared to their respective recordings and PPs during ACO. RESULTS: Using the standard ECG analysis, the acute occlusion was detected in 39 of 91 leads (43%), and in 15 of 18 patients (83%), whereas using the PP analysis it was detected in 82 of 91 leads (90%), and in all 18 patients (100%) (P<0.001, for leads). The median number of leads per patient demonstrating standard ECG changes was 2.0, whereas for the PP analysis it was 5.5 (P<0.001). The specificity of the PP method was 83.5%. CONCLUSIONS: The sensitivity of the PP method for detecting ACO during PTCA was superior to that of the standard ECG analysis. A smaller lead system is required to detect changes of ACO, during PTCA, when the PP method is used. The PP method is simple, low-priced, and may serve to detect acute myocardial ischemia in a number of clinical settings.

Dynamics of an ion chain in a harmonic potential.

Cold ions in anisotropic harmonic potentials can form ion chains of various sizes. Here, the density of ions is not uniform, and thus the eigenmodes are not phononic-like waves. We study chains of N>>1 ions and evaluate analytically the long-wavelength modes and the density of states in the short-wavelength limit. These results reproduce with good approximation the dynamics of chains consisting of dozens of ions. Moreover, they allow one to determine the critical transverse frequency required for the stability of the linear structure, which is found to be in agreement with results obtained by different theoretical methods [Phys. Rev. Lett. 71, 2753 (1993)]] and by numerical simulations [Phys. Rev. Lett. 70, 818 (1993)]]. We introduce and explore the thermodynamic limit for the ion chain. The thermodynamic functions are found to exhibit deviations from extensivity.

Localized perturbations of integrable quantum billiards.

The statistics of energy levels of a rectangular billiard that is perturbed by a strong localized potential are studied analytically and numerically, when this perturbation is at the center or at a typical position. Different results are found for these two types of position. If the scatterer is at the center, the symmetry leads to additional contributions, some of which are related to the angular dependence of the potential. The limit of the delta-like scatterer is obtained explicitly. The form factor, which is the Fourier transform of the energy-energy correlation function, is calculated analytically, in the framework of the semiclassical geometrical theory of diffraction, and numerically. Contributions of classical orbits that are nondiagonal are calculated and are found to be essential.

Decoherence as a probe of coherent quantum dynamics.

The effect of decoherence, induced by spontaneous emission, on the dynamics of cold atoms periodically kicked by an optical lattice is experimentally and theoretically studied. Ideally, the mean energy growth is essentially unaffected by weak decoherence, but the resonant momentum distributions are fundamentally altered. It is shown that experiments are inevitably sensitive to certain nontrivial features of these distributions, in a way that explains the puzzle of the observed enhancement of resonances by decoherence [Phys. Rev. Lett. 87, 074102 (2001)]. This clarifies both the nature of the coherent evolution, and the way in which decoherence disrupts it.

The correlation dimension of rat hearts in an experimentally controlled environment.

The electric response of several isolated rat hearts in a controlled environment was studied experimentally. The correlation dimension D(2) was estimated and was found to be between 4 and 6.5 when the response was nearly periodic. The variation of D(2) with the concentration of calcium was studied and a general trend of its increase with increasing concentration was found. Two types of ventricular fibrillation (VF) were observed, one that corresponds to a stochastic signal where D(2) is unbounded and the other to a low dimensional dynamical system with 3.5

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.

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.

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.