Anderson localization and wave absorption

Experimental signatures of classical wave localization in the absence and in the presence of attenuation are analyzed. The different regimes of the attenuation, reflection, and transmission coefficients for the diffusive and localized regimes are discussed. Apparent contradictory results presented previously by John and Anderson on the renormalization of absorption by localization are reconciled and shown to apply to different situations.     

Anderson localization for one- and quasi-one-dimensional systems

We prove almost-sure exponential localization of all the eigenfunctions and nondegeneracy of the spectrum for random discrete Schrödinger operators on one- and quasi-one-dimensional lattices. This paper provides a much simpler proof of these results than previous approaches and extends to a much wider class of systems; we remark in particular that the singular continuous spectrum observed in some quasiperiodic systems disappears under arbitrarily small local perturbations of the potential. Our results allow us to prove that, e.g., for strong disorder, the smallest positive Lyapunov exponent of some products of random matrices does not vanish as the size of the matrices increases to infinity.     

A statistical test of Anderson localization

Numerical demonstrations of localization in random systems are difficult to obtain and interpret because of statistical fluctuations in the electron probability density. This difficulty can be avoided through the use of correlation functions defined in terms of the electron probability density. The fluctuations can then be eliminated by averaging over a large number of Anderson Hamiltonians. The resulting averaged correlation functions clearly show that electrons are exponentially localized. The localization demonstrated here is sufficient to insure a zero dc conductivity in the limit of large systems.     

Quantum correlations in two-particle Anderson localization

We predict quantum correlations between noninteracting particles evolving simultaneously in a disordered medium. While the particle density follows the single-particle dynamics and exhibits Anderson localization, the two-particle correlation develops unique features that depend on the quantum statistics of the particles and their initial separation. On short time scales, the localization of one particle becomes dependent on whether or not the other particle is localized. On long time scales, the localized particles show oscillatory correlations within the localization length. These effects can be observed in Anderson localization of nonclassical light and ultracold atoms.     

Anderson localization from the replica formalism

We study Anderson localization in quasi-one-dimensional disordered wires within the framework of the replica sigma model. Applying a semiclassical approach (geodesic action plus Gaussian fluctuations) recently introduced within the context of supersymmetry by Lamacraft, Simons, and Zirnbauer, we compute the exact density of transmission matrix eigenvalues of superconducting wires (of symmetry class CI.) For the unitary class of metallic systems (class A) we are able to obtain the density function, save for its large transmission tail.     

Anderson localization in nonlocal nonlinear media

We theoretically and numerically investigate the effect of focusing and defocusing nonlinearities on Anderson localization in highly nonlocal media. A perturbative approach is developed to solve the nonlocal nonlinear Schr?dinger equation in the presence of a random potential, showing that nonlocality stabilizes Anderson states.     

Boundary-induced Anderson localization in photonic lattices

We analyze numerically localization of light in linear square waveguide arrays restricted in one dimension (“ribbons”), whose boundaries are disordered in propagation constant and/or coupling. We find that the disordered boundary induces a localization tendency in the bulk even for relatively wide ribbons.     

A self-consistent treatment of Anderson localization

Anderson localization in two-dimensional systems is discussed with the use of Feynman graph method. A self-consistent treatment of diffusion propagator leads to an expression for dynamical conductivity which vanishes at zero frequency while exhibiting well known logarithmic dependence at higher frequency. It is applied to three-dimensional systems, too, and a discussion is given on mobility edge.     

Self-induced transparency and the Anderson localization of light

Self-induced transparency pulses propagating in a random medium embedded in a two-level system can transfer energy to localized Anderson states. This allows the onset of two-level laserlike action.     

Kolmogorov turbulence, Anderson localization and KAM integrability

A modified Shastry-Sutherland model with fourth-nearest-neighbor dimerization is studied with strong coupling perturbation theory. One-triplet excitations are calculated up to the second order. Bose-Einstein condensation of one-triplet excitations may induce different ordered states, i.e. collinear antiferromagnetic state, Néel state or incommensurate state when the gap closes by changing the physical parameters or by an external magnetic field. Two-triplet bound states in the S = 1 and S = 2 channels are shown to exist around the symmetric lines of K _x = ±K _y and K _x ± K _y = ±??. For ferromagnetic dimer-dimer interaction, the energies of the bound states in the S = 1 (S = 2) channel lie above (below) the two-triplet continuum. By calculating the existence region, the degeneracy, the symmetry and the dispersion relation, a whole image and main features of the bound states are obtained. Experimental results of (CuX)LaNb₂O₇ with X = Cl, Br are interpreted.     

Anderson localization for Bernoulli and other singular potentials

We prove exponential localization in the Anderson model under very weak assumptions on the potential distribution. In one dimension we allow any measure which is not concentrated on a single point and possesses some finite moment. In particular this solves the longstanding problem of localization for Bernoulli potentials (i.e., potentials that take only two values). In dimensions greater than one we prove localization at high disorder for potentials with Hölder continuous distributions and for bounded potentials whose distribution is a convex combination of a Hölder continuous distribution with high disorder and an arbitrary distribution. These include potentials with singular distributions.We also show that for certain Bernoulli potentials in one dimension the integrated density of states has a nontrivial singular component.Partially supported by NSF grant DMS 85-03695Partially supported by NSF grant DMS 83-01889Partially supported by G.N.F.M. C.N.R.     

Short-distance wavefunction statistics in one-dimensional Anderson localization

We investigate the short-distance statistics of the local density of states in long one-dimensional disordered systems, which display Anderson localization. It is shown that the probability distribution function can be recovered from the long-distance wavefunction statistics, if one also uses parameters that are irrelevant from the perspective of two-parameter scaling theory.Received: 10 July 2003, Published online: 15 October 2003PACS: 72.15.Rn Localization effects (Anderson or weak localization) - 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion - 42.25.Dd Wave propagation in random media - 73.20.Fz Weak or Anderson localization     

Emergence of Anderson localization in plasmonic waveguides

The propagation of surface plasmon polaritons in dielectric loaded waveguides with randomly placed scatterers is studied using both numerical simulations and a simplified transfer matrix framework. Despite the importance of losses in this system, we find fingerprints of the localized behavior of one-dimensional disordered systems. Furthermore, losses amplify the impact of the necklace states on the transport properties for systems not much larger than the localization length. The system presented here also offers the possibility to use localization effects for engineering purposes by means of deliberately introduced disorder.