Anderson localization of matter waves in chaotic potentials

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Abstract

In the present paper we study Anderson localization of Bose–Einstein condensate with a weakly positive nonlinearity under the influence of chaotic potentials. We investigate the one-dimensional Gross–Pitaevskii equation numerically, in order to unveil the influence of the parameters that describe the potential and nonlinearity on the localization of the condensate. The results show that matter localization is possible, in certain regions in parameter space. Similarities to chaotically disordered photonic lattices suggest that the results are of direct interest to such photonic lattices as well.

Introduction

In many physical systems the presence of weak disorder cannot be ignored and may change dramatically its properties [1]. A notorious example is the Anderson localization (AL) of non-interacting particles [2], which is the phenomenon of transport suppression due to a destructive interference of the many paths associated with coherent multiple scattering from the modulations of a disordered potential [2]. This effect is a characteristic of wave physics and occurs when the disordered potential presents a weak amplitude, making the localized state with exponentially decaying tails and absence of diffusion [3]. The AL has been experimentally demonstrated in many scenarios such as light in 3D random media [4], [5], in 2D [6] and 1D [7] disordered photonic lattices, and in Bose–Einstein condensate (BEC) of atoms [8], [9]. However, for strong amplitude potentials the system becomes confined and the quantum state cannot escape of the large barriers of the disordered strong potential, presenting a Gaussian shape similar to that of a potential with very high barriers, unlike the exponentially decaying tails characteristic of the AL phenomenon.

In recent years, the study of AL on BEC has appeared as an interesting possibility, mainly because one can investigate the effects of nonlinearity of the weakly interacting cold atoms on the localization. For instance, the recent works of Billy et al. and Roati et al. have achieved experimental realization of AL in a BEC in disordered potential created by a laser speckle [8] and 1D quasiperiodic bichromatic optical lattice [9], respectively. These works triggered a lot of interest, and theoretical and experimental efforts have been developed to investigate AL in the BEC context, focusing on the study of the disordered potential created by bichromatic optical lattice [10], shaken optical lattice [11], cold atom lattice [12], atomic mixture [13], and under the presence of inhomogeneous magnetic fields [14]. Most of the previous studies consider harmonic potential superposed to the disordered potentials. In particular, detailed investigation of AL in cigar-shaped BEC in the presence of 1D random speckle and disordered cold atom lattice was done in Ref. [15].

In the present work, unlike previous works that consider quasiperiodic [16] or random [17] potential as disordered potential, we study AL of cigar-shaped BEC in the presence of 1D chaotic potential. The motivation to study this type of disordered potential comes from the fact that the details of the localization effect can strongly depend on the type of disorder. Although the behavior of the chaotic systems appears to be similar to random systems, the former exhibit a sensitivity to initial conditions that produce completely different final states even over fairly small time scales for small differences in the initial conditions [18]. As far as we know, chaotic potential as disordered potential has not yet been considered, although it is common knowledge that nonlinear systems can exhibit chaotic behavior.

The approach we consider is to study numerically the GPE, using an imaginary-time method for propagation of an initial Gaussian pulse with the chaotic potential modeled by the logistic map in two distinct cases: in the first, we consider a uniform grid with chaotic amplitude potential, and in the second case we consider identical Gaussian spikes distributed chaotically. The conditions for occurrence of AL in the system are obtained using an imaginary-time propagation of the Gaussian function, to get the ground state solution of the disordered potential. In a previous work, some of us have considered distinct route, in which one uses the logistic map to describe chaotic nonlinearities affecting propagation of solitonic solutions of the GPE equation [19]. The main results of [19] show an important distinction between the presence of chaotic and random effects in the cubic nonlinearity: in the chaotic case, the soliton may be destroyed, while in the random case, it may move around, without destruction. This is another source of motivation for the present study, where we investigate how the chaotic effects of the disordered potential contributes to the localization of the condensate.

In order to ease understanding, we organize the paper as follows: in Section 2 we introduce and study both the theoretical model described by the 1D GPE, and the disordered chaotic potentials. In Section 3 we deal with the numerical results and with discussions concerning the chaotic amplitude potential and the chaotic speckle potential. Finally, in Section 4 we present our comments and conclusions.

Section snippets

Theoretical model

We start the investigation with the 1D GPE in its standard dimensionless form, in the presence of cubic nonlinearity, iψt=12ψxx+Vcψ+g|ψ|2ψ, where ψ=ψ(x,t) is the wave-function describing the collective state of the atoms in a BEC or the electric field propagating in a nonlinear crystal (NC). Here we use ψtψ/t and ψxx2ψ/x2, and Vc=Vc(x) represents the chaotic potential. Also, g is the nonlinear parameter; it is a real number that controls the cubic nonlinearity. In a BEC, g is associated

Numerical results

We now focus on the numerical investigation. We start integrating Eq. (1) numerically, using a split-step algorithm based on the Crank–Nicolson method to solve the dispersive term. Here we use the values 0.04 and 0.001 for space and time steps, respectively. Also, we obtain the ground state solution via an imaginary-time method for propagation of an initial Gaussian pulse. Next,we employ a real-time propagation of this new profile and analyze the conditions for occurrence of AL in the system.

Comments and conclusions

In this work we presented a numerical study of AL in the one-dimensional GPE. We considered a BEC of weakly interacting cold atoms in the presence of potentials with chaotic amplitudes and with chaotic spike positions, respectively. The investigation focused on the influence of the nonlinear cubic term as well as the amplitude of the chaotic potential on the localization of matter waves. The main results show that: in the case of potential with chaotic amplitudes, localization is favored when

Acknowledgments

We would like to thank CAPES, CNPq, FUNAPE/GO, and INCT-IQ, for partial financial support.

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