Eigenmodal analysis of Anderson localization: Applications to photonic lattices and Bose–Einstein condensates

Abstract

We present the eigenmodal analysis techniques enhanced towards calculations of optical and non-interacting Bose-Einstein condensate (BEC) modes formed by random potentials and localized by Anderson effect. The results are compared with the published measurements and verified additionally by the convergence criterion. In 2-D BECs captured in circular areas, the randomness shows edge localization of the high-order Tamm-modes. To avoid strong diffusive effect, which is typical for BECs trapped by speckle potentials, a 3-D-lattice potential with increased step magnitudes is proposed, and the BECs in these lattices are simulated and plotted.2 the analytical results are available only for 1-dimensional (1-D) systems [14], and the higher dimensional ones are mostly studied by numerical methods.In 1983, the evidence of Anderson's prediction was numerically found by the transfer-matrix (TM) method in [15]. The authors applied their method to the disordered electrons in 1-D system.An eigenmodal analysis of transversally-disordered electron waveguides is published in [16], [17].Recently, scientists have spent much effort to study the localization effect of light experimentally.In addition to the experiments, Schwartz and his co-authors [3] proposed a theoretical model and solved it numerically by split step Fourier method [18]. Moreover, the localization of BECs is also a hot topic related to Anderson's original contribution. The phenomenon was realized experimentally, but not much theoretical work has been done to describe these results. In [12], the authors gave a simple equation to exponentially fit the measurements of cold atoms. In [11], the authors solved the Gross-Pitaevskii equation by sum-rules approach [19] for the frequency behavior.Despite a vast literature on the subject, there is no efficient, fast, and general method to provide numerical study of the Anderson localization in different applications. In this paper, an approach is proposed to study this effect for photonic and matter waves. It can be generalized to all kinds of physics which are in compliance with the wave phenomena. The method arises from the analysis of eigen-mode propagation of EM field in waveguides. The method is partly similar to the 1-D analysis in [16], [17], but it uses an iterative algorithm to solve the resulting Hamiltonian matrix instead of using cascaded circuit matrices, equivalent circuit approach and TM model [20].The paper is organized as follows. In the Section 2, the proposed method of eigenmodal analysis is briefly discussed. To demonstrate its application possibilities, the method is employed for 2-D photonic lattices in the Section 3, and for BECs in speckle potentials in the Section 4.1.

Nonlinear dynamics of dipoles in microwave electric field of a nanocoaxial tubular reactor

“…Small size of reactors allows applying high-gradient electric fields to the reactor walls at low voltage, and these fields may affect, for instance, the process of folding and unfolding of protein globules [13]- [17]. Trapping molecules or nanoparticles at room temperature in liquids is more challenging in comparison with confining ultracold atoms [18], [19], and combination of laser light with electrophoretic forces allows trapping nanoparticles in an adaptive manner in nanoor microtraps [20], [21].…”