Anderson attractors in active arrays

In dissipationless linear media, spatial disorder induces Anderson localization of matter, light, and sound waves. The addition of nonlinearity causes interaction between the eigenmodes, which results in a slow wave diffusion. We go beyond the dissipationless limit of Anderson arrays and consider nonlinear disordered systems that are subjected to the dissipative losses and energy pumping. We show that the Anderson modes of the disordered Ginsburg-Landau lattice possess specific excitation thresholds with respect to the pumping strength. When pumping is increased above the threshold for the band-edge modes, the lattice dynamics yields an attractor in the form of a stable multi-peak pattern. The Anderson attractor is the result of a joint action by the pumping-induced mode excitation, nonlinearity-induced mode interactions, and dissipative stabilization. The regimes of Anderson attractors can be potentially realized with polariton condensates lattices, active waveguide or cavity-QED arrays.

Scientific RepoRts | 5:13263 | DOi: 10.1038/srep13263 interactions between the modes get suppressed in the strong localization and vanishing dissipation limit, although with significant deviations found beyond 33 .
A new room for dissipation effects was created by the recent progress in experimental manipulations with exciton-polariton condensates [34][35][36][37][38] . A condensate can be considered as an active system balancing between excitation (by a pumping source) and decay (due to the continuous light emission). Further on, one can arrange 1D arrays of condensate centers by synthesizing spatial inhomogeneities 35,[39][40][41] or by rotating ring-shaped optical potentials and switching to the co-moving frame 42,43 . Spatial interaction appears due to polariton diffraction and diffusion and, importantly, would include both Josephson and dissipative terms (the former typically prevails). The resulting collective dynamics is a blend of excitation and lasing effects and can be modeled with Ginzburg-Landau type equations (GLE) 44 . In this framework, dissipative effects act as internal decay mechanisms and their influence on the center dynamics is accounted by additional imaginary terms in the model equations 36,38,45,46 . The recent pioneering theoretical and experimental studies have already demonstrated a rich nonlinear dynamics of traveling and immobile gap solitons in periodic 1D condensate center arrays 40,47 , and further stretched to spatially quasiperiodic structures to uncover the fractal energy spectrum 41 .
Altogether, these advances naturally lead to the question of Anderson localization in active arrays, where pumping and dissipation join the old players, nonlinearity and disorder. Some collective phenomena in such systems are well studied, for example, synchronization 48 and oscillation death [49][50][51] . However, most of the related studies address lattices that crumble into a set of uncoupled oscillators in the linear conservative limit.
In this Report we demonstrate and study Anderson attractors in 1D active arrays, as described by a disordered 22 version of the discrete complex GLE 52 . We find that the increase of the pumping strength leads to the formation of a stationary multipeak pattern formed by a set of excited and interacting Anderson modes. We determine the transition from the regime of Anderson attractors to delocalized collective oscillations upon the increase of pumping. Both excitation and delocalization thresholds scale with the strength of the dissipative coupling and increase with the increase of disorder. Finally, we show that the increase of pumping beyond the delocalization threshold leads to a multi-mode chaos followed by cluster synchronization.

Results
We consider a one-dimensional disordered discrete Ginsburg-Landau equation, a generalization of the original Anderson lattice equations 22  3 . These equations contain both the linear and nonlinear terms that account for dissipation and pumping. Nonlinear terms are responsible for the mode interaction. However, due to the exponential localization of the eigenstates, interactions are confined to localization volume V loc (λ) ≈ 3.3ξ λ 54 .
We start the analysis of Eq. (1) by considering the net norm Z = Σ |z l | 2 . The dynamics of the norm is given by It follows that the zero solution ≡ z 0 l is globally stable for all α ≤ 0. It also suggests that homogeneous in-phase solutions z l + 1 ≈ z l are more energetically favorable than anti-phase ones, z l + 1 ≈ − z l . To study stability of the zero solution, we assign increments p v to the small-amplitude Anderson modes, , and substitute them into Eq. (1). Linearization gives The zero solution is stable when max p v < 0. This quantity depends only on the strength W and particular realization {Δ l } of the disorder, and also on the ratio between incoherent pumping rate and dissipative coupling, α α η = / . Irrespective of the strength and particular realization of disorder, the scaled excitation threshold As the Anderson modes have finite localization lengths for finite disorder strength W and, hence, inside localization volume we have ( ) there is a finite excitation threshold α > 0 for finite W. Figure 1 presents the results of numerical simulations for a particular realization of disorder. Profiles for different values of α were obtained as independent attractor solutions, by setting the system into an initial random low-energy state |z l (0)|≪ 1 and letting it evolve until the corresponding amplitude profile is stabilized. (We observed single-attractor regimes in all performed numerical tests, although multistability is not excluded, in principle.) The key feature of the attractor patterns is the multipeak structure, well pronounced above a certain threshold (e.g. Fig. 1). The positions of the peaks remain unaffected by the further increase of the pumping strength. Zooming into a l single peak, we find that it extends over many sites, top right panel of Fig. 1. By going into the reciprocal Anderson space, we find that the excitation is well-localized at a single Anderson mode, bottom right panel of Fig. 1. This observation supports the conjecture that the attractor peaks are produced through excitation of Anderson modes.
Mode specific excitation conditions can be further analyzed by using the linearized version of equations (4), In the weak disorder limit, W ≪ 1, the localization length of the modes that are far from the band edges is large, ξ λ ≫ 1. Since within the localization volume ( ) , the terms with ν ν , J 1 can be neglected. It follows immediately that the rescaled excitation threshold of v-th Anderson mode can be approximated well by its eigenvalue, This also means that the modes closer the lower band edge will be excited first. However, the localization length of such modes can substantially decrease, potentially, up to ξ λ ~ 1, so that corrections to Eq. (8) due to ν ν , J 1 terms might become significant. The instability threshold can be estimated more accurately by using Eq. , we obtain: Finally, by substituting the localization length ξ 0 ≈ 8 W −2/3 for the modes with λ v ≈ 0 55 in V loc ≈ 3.3ξ λ , we arrive at: 4 3 Note, that this approach is also valid in the strong disorder limit, W ≫ 1, when all Anderson modes are essentially single-site excitations: substituting V loc = 1 in (10)  It follows, that they tend to the limiting value α = ⁎ 2 as W→ ∞. To test the analytical results, we calculate mode excitation thresholds α ν ⁎ according to (7) and plot them as a function of the numerically calculated eigenvalues λ v , Fig. 2. The obtained statistical dependencies corroborate approximation (9) for the modes far from the band edges, especially well in the limit of weak disorder. The values of minimal excitation thresholds correspond to λ v ≈ 0, and the estimate (11) is in a good agreement with numerical results, see inset of Fig. 2. By approximating the dependence around its dip by α α λ − ∝ Δ ⁎ 2 and taking into account the finiteness of the density of Anderson states at λ = 0, we get that the density of excited states scales α α ∝ − ⁎ . By getting over the oscillation threshold α ⁎ one would not immediately excite all modes near the band edge. These modes are well-localized and their interaction with other modes is exponentially weak. In addition, next-neighbor mode interaction remains significantly damped since mode eigenvalues differ substantially due to the level repulsion.
As a result, Anderson modes from the vicinity of the band edge arise in a one-by-one manner as the pumping rate exceeds thresholds α α > ν ⁎ . Mode amplitudes saturate because of the nonlinear dissipation and amplitude asymptotic values can be estimated, by using Eq. (4), as: As the pumping strength increases further, the set of excited modes becomes dense and mode interaction starts contributing to the formation of the system attractor. Multi-mode nonlinear dynamics has two well-known trademarks: chaos and synchronization 48 . Both appear in our model system, see Fig. 3. By gradually increasing the pumping strength, we first observe a transition from the Anderson attractors to the regime of delocalized oscillations, Fig. 3 (middle panel). The delocalized regime is characterized by irregular spatio-temporal patterns. In terms of the localized modes, this is a well-developed mode chaos. When the pumping is increased further, we observe formation of synchronization clusters with the typical size of the Anderson localization length. We can estimate the transition to delocalized oscillations by assuming that it happens when the sum of the localization volumes of the excited modes becomes of the order of the system size, loc . An average localization volume that measures the ratio of effectively excited sites is , where the non-excited modes are formally assigned V loc = 0. By using expression (9) for the mode excitation  In the strong disorder limit the mode excitation thresholds (12) converge to α = ⁎ 2, which, therefore, also approximates the onset of delocalized oscillations, α ≈ ⁎⁎ 2. For a numerical test we average |z l | 2 over observation time and calculate the participation number (a quantity commonly used to estimate the number of effectively excited sites) normalized by the system size: Since the maximally possible P = 1 requires a uniform distribution of |z l |, we use P = 1/2 as the threshold value to indicate localization-delocalization transition. The left panel of Fig. 4 presents the results obtained by averaging over ten disorder realizations. For weak dissipative coupling η ≪ 1, the scaled curves α ( ) P fall closely to each other, in accord to the theoretical prediction, Eq. (15). It also estimates the numerical thresholds reasonably well, e.g. compare α ≈ .  (15) with the scaling α η α ( , , ) = ( , ) P W P W are no longer valid, and the actual delocalization threshold is significantly different from (15). In this limit one cannot neglect the last term in Eq. (8) which is responsible for dissipative interaction between the modes.
In order to quantify the transition to the mode chaos regime, we calculate the largest Lyapunov exponent as a function of the pumping strength, Fig. 4 (right panel). Comparing the exponents, obtained for different values of dissipative coupling constant η, with the results presented in Fig. 4, we confirm that the transition to delocalized oscillations is a precursor of the mode chaos. Remarkably, a further increase of the pumping above α ≈ 1 leads to the drop of the largest Lyapunov exponents to zero thus marking the transition back to regular dynamics. This transition is weakly dependent of η and corresponds to the emergence of synchronized clusters 48 , see Fig. 3 (bottom panel).

Discussion
Anderson localization in active disordered systems is a combined effect produced by the energy pumping, dissipation and nonlinearity. It results in the formation of the Anderson attractor consisting of many localized weakly-interacting modes. We have found that the pumping excitation thresholds for the Anderson modes are mode-specific and those with lowest values correspond to the modes located near the lower band edge. Sequential excitation of Anderson modes by tuned pumping leads to the transition from Anderson attractors to the mode chaos and attractor patterns in the form of delocalized oscillations. These results pose a broad range of theoretical challenges, as studying Anderson attractors in higher dimensions, which allow for a mobility edge or criticality, in other types of localizing potentials, and their counterparts in open quantum systems. It would also be of interest to consider non-uniform dissipation, e.g. absorbing boundaries only. For the experimental perspective, lattices of exciton-polariton condensates and active waveguide arrays are most promising candidates for the realization of Anderson attractors. The recent study of another localizing-quasiperiodically modulated-1D polariton condensate arrays has paved a way 41 , and the on-chip random lasing in the Anderson regime is, probably, the first already present example 32 . Other candidates (although on the model level at the moment) are cavity-QED arrays with the cavities filled up with two-level atoms or qubits, where the dynamics the mean-field states in the adjoint cavities can be described by using GLE-type equation 56,57 and plasmonic nanostructures 58 . Finally, Anderson attractor regimes can be generalized to the systems of coupled disordered Josephson junction arrays, marked by the recent rise of interest to dissipative response effects 59 .