Shmuel Fishman

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.

Analog computation with dynamical systems

A b s t r a c t Physical systems exhibit various levels of complexity: their long term dynamics may converge to fixed points or exhibit complex chaotic behavior. This paper presents a theory that enables to interpret natural processes as special purpose analog computers. Since physical systems are naturally described in continuous time, a definition of computational complexity for continuous time systems is required. In analogy with the classical discrete theory we develop fundamentals of computational complexity for dynamical systems, discrete or continuous in time, on the basis of an intrinsic time scale of the system. Dissipative dynamical systems are classified into the computational complexity classes Pd, Co-RPd, NPd and EXP,t, corresponding to their standard counterparts, according to the complexity of their long term behavior. The complexity of chaotic attractors relative to regular ones leads to the conjecture Pa :fi NPj. Continuous time flows have been proven useful in solving various practical problems. Our theory provides the tools for an algorithmic analysis of such flows. As an example we analyze the continuous Hopfield network.

Effective Hamiltonians for periodically driven systems

The dynamics of classical and quantum systems, which are driven by a high-frequency

Excitation of molecular rotation by periodic microwave pulses. A testing ground for Anderson localization

(\ensuremath{\omega})

Exercise stress testing, myocardial perfusion imaging and stress echocardiography for detecting restenosis after successful percutaneous transluminal coronary angioplasty: a review of performance

field, is investigated. For classical systems, the motion is separated into a slow part and a fast part. The motion for the slow part is computed perturbatively in powers of

A Theory of Complexity for Continuous Time Systems

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Quantum resonances and decoherence for δ-kicked atoms

to the order

Structural phase transitions in low-dimensional ion crystals

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Time independent description of rapidly oscillating potentials.

and the corresponding time independent Hamiltonian is calculated. Such an effective Hamiltonian for the corresponding quantum problem is computed to the order

Quantum zigzag transition in ion chains.

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