Matter-wave localization in disordered cold atom lattices

We propose to observe Anderson localization of ultracold atoms in the presence of a random potential made of atoms of another species or spin state and trapped at the nodes of an optical lattice, with a filling factor less than unity. Such systems enable a nearly perfect experimental control of the disorder, while the possibility of modeling the scattering potentials by a set of pointlike ones allows an exact theoretical analysis. This is illustrated by a detailed analysis of the one-dimensional case.     

Disorder-induced localization of light near edges of nonlinear photonic lattices

We investigated the influence of edges and corners on the Anderson localization of light in disordered two-dimensional photonic lattices that are optically induced in nonlinear saturable photorefractive media. A systematic quantitative study of gradual transition from corner to bulk Anderson localization in truncated two-dimensional photonic lattices was carried out. We analyzed numerically the localization at several corners and edges of the square and triangular photonic lattices and compared them with the localization in bulk medium. We found that, for strong disorder, corners and edges effectively suppress Anderson localization, as compared to the bulk, but to a varying degree.     

Anderson Localization in the Induced Disorder System

We propose a coherently prepared three-level atomic medium that can provide a flexible disordered scheme for realizing the Anderson localization.Different disorder levels can be attained by modulating the intensity ratio between the two control beams.Due to the real-time tunability,the localization of the signal beam is observable and controllable.The influences of the induced disorder level,atomic density and the initial waist radius of the signal beam on the Anderson localization in the medium are also discussed.     

Light localization in nonuniformly randomized lattices

We address Anderson localization of light in disordered optical lattices where the disorder strength varies across the transverse direction. Such variation changes the preferred domains where formation of localized eigenmodes is most probable, hence drastically impacting light localization properties. Thus, step-like disorder results in formation of modes with different decay rates at both sides of the interface, while a smoothly varying disorder yields appearance of modes that are extended within weakly disordered domains and rapidly fade away in strongly disordered domains.     

THERMALLY ACTIVATED HOPPING IN Dy1-xPrxBa2Cu3O7-δ SYSTEM

The temperature dependence of resistivity in Dy_1-xPr_xBa₂Cu₃O_7-δ system with x>0.6 was measured. The experimental results show that Pr substitution leads to the localization of mobile holes and such a localization is enhanced with increasing Pr concentration. The gradually enhancing of localization induces Anderson transitions one by one in this system, including the transition from the conduction by excitation of holes to the one by thermal activation hopping between localized states, the so called Anderson transition type-I, and the transition from nearest neighbor hopping (NNH) to variable range hopping (VRH), the Anderson transition type-II, and the Anderson transition type-lI from 3D to 2D.     

Anderson Localization in Dissipative Lattices

Anderson localization predicts that wave spreading in disordered lattices can come to a complete halt, providing a universal mechanism for dynamical localization. In the one-dimensional Hermitian Anderson model with uncorrelated diagonal disorder, there is a one-to-one correspondence between dynamical localization and spectral localization, that is, the exponential localization of all the Hamiltonian eigenfunctions. This correspondence can be broken when dealing with disordered dissipative lattices. When the system exchanges particles with the surrounding environment and random fluctuations of the dissipation are introduced, spectral localization is observed but without dynamical localization. Previous studies consider lattices with mixed conservative (Hamiltonian) and dissipative dynamics and are restricted to a semiclassical analysis. However, Anderson localization in purely dissipative lattices, displaying an entirely Lindbladian dynamics, remains largely unexplored. Here the purely-dissipative Anderson model in the framework of a Lindblad master equation is considered, and it is shown that, akin to the semiclassical models with conservative hopping and random dissipation, one observes dynamical delocalization in spite of strong spectral localization of the Liouvillian superoperator. This result is very distinct from delocalization observed in the Anderson model with dephasing, where dynamical delocalization arises from the delocalization of the stationary state of the Liouvillian.     

Quantum simulation of disordered systems with cold atoms

This paper reviews the physics of quantum disorder in relation with a series of experiments using laser-cooled atoms exposed to “kicks” of a standing wave, realizing a paradigmatic model of quantum chaos, the kicked rotor. This dynamical system can be mapped onto a tight-binding Hamiltonian with pseudo-disorder, formally equivalent to the Anderson model of quantum disorder, with quantum chaos playing the role of disorder. This provides a very good quantum simulator for the Anderson physics.     

Localization and delocalization of a one-dimensional system coupled with the environment

We investigate several models of a one-dimensional chain coupling with surrounding atoms to elucidate disorder-induced delocalization in quantum wires, a peculiar behaviour against common wisdom. We show that the localization length is enhanced by disorder of side sites in the case of strong disorder, but in the case of weak disorder there is a plateau in this dependence. The above behaviour is the conjunct influence of the coupling to the surrounding atoms and the antiresonant effect. We also discuss different effects and their physical origin of different types of disorder in such systems. The numerical results show that coupling with the surrounding atoms can induce either the localization or delocalization effect depending on the values of parameters.     

Anomalous localization in low-dimensional systems with correlated disorder

This review presents a unified view on the problem of Anderson localization in one-dimensional weakly disordered systems with short-range and long-range statistical correlations in random potentials. The following models are analyzed: the models with continuous potentials, the tight-binding models of the Anderson type, and various Kronig–Penney models with different types of perturbations. Main attention is paid to the methods of obtaining the localization length in dependence on the controlling parameters of the models. Specific interest is in an emergence of effective mobility edges due to certain long-range correlations in a disorder. The predictions of the theoretical and numerical analysis are compared to recent experiments on microwave transmission through randomly filled waveguides.     

Some mathematical aspects of Anderson localization: boundary effect,multimodality, and bifurcation

Anderson localization is a famous wave phenomenon that describes the absence of diffusion of waves in a disordered medium. Here we generalize the landscape theory of Anderson localization to general elliptic operators and complex boundary conditions using a probabilistic approach, and further investigate some mathematical aspects of Anderson localization that are rarely discussed before. First, we observe that under the Neumann boundary condition, the low energy quantum states are localized on the boundary of the domain with high probability. We provide a detailed explanation of this phenomenon using the concept of extended subregions and obtain an analytical expression of this probability in the one-dimensional case. Second, we find that the quantum states may be localized in multiple different subregions with high probability in the one-dimensional case and we derive an explicit expression of this probability for various boundary conditions. Finally, we examine a bifurcation phenomenon of the localization subregion as the strength of disorder varies. The critical threshold of bifurcation is analytically computed based on a toy model and the dependence of the critical threshold on model parameters is analyzed.     

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.     

Correlated electrons in the presence of disorder

Several new aspects of the subtle interplay between electronic correlations and disorder are reviewed. First, the dynamical mean-field theory (DMFT) together with the geometrically averaged (“typical”) local density of states is employed to compute the ground state phase diagram of the Anderson-Hubbard model at half-filling. This non-perturbative approach is sensitive to Anderson localization on the one-particle level and hence can detect correlated metallic, Mott insulating and Anderson insulating phases and can also describe the competition between Anderson localization and antiferromagnetism. Second, we investigate the effect of binary alloy disorder on ferromagnetism in materials with f-electrons described by the periodic Anderson model. A drastic enhancement of the Curie temperature T_c caused by an increase of the local f-moments in the presence of disordered conduction electrons is discovered and explained.     

Anderson cross-localization

We report Anderson localization in two-dimensional optical waveguide arrays with disorder in waveguide separation introduced along one axis of the array, in an uncorrelated fashion for each waveguide row. We show that the anisotropic nature of such disorder induces a strong localization along both array axes. The degree of localization in the cross-axis remains weaker than that in the direction in which disorder is introduced. This effect is illustrated both theoretically and experimentally.     

Observation of the gradual transition from one-dimensional to two-dimensional Anderson localization

We study the gradual transition from one-dimensional (1D) to two-dimensional (2D) Anderson localization upon transformation of the dimensionality of disordered waveguide arrays. An effective transition from a 1D to a 2D system is achieved by increasing the number of rows forming the arrays. We observe that, for a given disorder level, Anderson localization becomes weaker with increasing numbers of rows-hence the effective dimension.     

Localization length of wave functions in one-dimensional disordered systems with periodic modulation

The localization length of wave functions in one-dimensional disordered systems with periodic modulation is studied. The role of spatial inhomogeneity in the problem is considered. We calculate the localization length varying with energy for the compositionally modulated systems with either the disorder of Anderson distributions or the randomness of periods. The results show that the non-uniformity of disordered systems leads to some different properties of the localized wave functions.     

Two-dimensional Anderson-Hubbard model in the DMFT + Σ approximation

The density of states, the dynamic (optical) conductivity, and the phase diagram of the paramagnetic two-dimensional Anderson-Hubbard model with strong correlations and disorder are analyzed within the generalized dynamical mean field theory (DMFT + Σ approximation). Strong correlations are accounted by the DMFT, while disorder is taken into account via the appropriate generalization of the self-consistent theory of localization. We consider the two-dimensional system with the rectangular “bare” density of states (DOS). The DMFT effective single-impurity problem is solved by numerical renormalization group (NRG). The “correlated metal,” Mott insulator, and correlated Anderson insulator phases are identified from the evolution of the density of states, optical conductivity, and localization length, demonstrating both Mott-Hubbard and Anderson metal-insulator transitions in two-dimensional systems of finite size, allowing us to construct the complete zero-temperature phase diagram of the paramagnetic Anderson-Hubbard model. The localization length in our approximation is practically independent of the strength of Hubbard correlations. But the divergence of the localization length in a finite-size two-dimensional system at small disorder signifies the existence of an effective Anderson transition.     

Routes towards Anderson-like localization of Bose-Einstein condensates in disordered optical lattices

We investigate, both experimentally and theoretically, possible routes towards Anderson-like localization of Bose-Einstein condensates in disordered potentials. The dependence of this quantum interference effect on the nonlinear interactions and the shape of the disorder potential is investigated. Experiments with an optical lattice and a superimposed disordered potential reveal the lack of Anderson localization. A theoretical analysis shows that this absence is due to the large length scale of the disorder potential as well as its screening by the nonlinear interactions. Further analysis shows that incommensurable superlattices should allow for the observation of the crossover from the nonlinear screening regime to the Anderson localized case within realistic experimental parameters.     

Zero Energy Anomaly in One-Dimensional Anderson Lattice with Exponentially Correlated Weak Diagonal Disorder

We calculated numerically the localization length of one-dimensional Anderson model with correlated diagonal disorder. For zero energy point in the weak disorder limit, we showed that the localization length changes continuously as the correlation of the disorder increases. We found that higher order terms of the correlation must be included into the current perturbation result in order to give the correct localization length, and to connect smoothly the anomaly at zero correlation with the perturbation result for large correlation.     

Zero Energy Anomaly in One-Dimensional Anderson Lattice with Exponentially Correlated Weak Diagonal Disorder

We calculated numerically the localization length of one-dimensional Anderson model with correlated diagonal disorder. For zero energy point in the weak disorder limit, we showed that the localization length changes continuously as the correlation of the disorder increases. We found that higher order terms of the correlation must be included into the current perturbation result in order to give the correct localization length, and to connect smoothly the anomaly at zero correlation with the perturbation result for large correlation.