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Abstract
We numerically investigate the transport of discrete breather-like states (DBSs) in a nonlinear optical lattice with weak disorder. We find that the DBS’s center of mass experiences a transient anomalous diffusion before its localization. This diffusive process is shown to represent the property of weak ergodicity breaking.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Discrete breathers (DBs), the spatially localized and time-periodic stable excitations, emerge as a prototype class of nonlinear waves in lattice systems [1,2]. They were first observed experimentally through light propagation in 1D waveguide arrays [3], and are currently studied in many areas such as Bose-Einstein condensates (BECs) [4] and crystal physics [5]. The mobile DBs can transport energy in a lattice, where they may need to overcome the so-called Peierls-Nabarro barriers [6]. Moreover, interaction of the moving DBs with ordered lattice impurity leads to various phenomena, including reflection, transmission, or trapping of these DBs [7].
Disorder, on the other hand, can suppress the diffusive transport of wave packets, known as the famous Anderson localization [8–10]. In 2007, the first experiment of Anderson localization was successfully performed by Schwartz et al. and Lahini et al. in disordered photonic lattices [11,12], where they also show the localization can be enhanced under self-focusing nonlinearity. Nevertheless, numerical simulations in 1D tight-binding systems reveal that, Anderson localization can be destroyed by weak nonlinearity, giving rise to a subdiffusive spreading of the wave packets [13,14]. For large nonlinearities, self-trapping of a fraction of the wave packets occurs [14–16], which was related to the emergence of breather-like excitations.
In this paper, facing an aspect of the interaction between disorder and strong nonlinearity, we study the transport of discrete breather-like states (DBSs) in a disordered optical lattice. These localized states oscillate with time that may not be strictly periodic due to the presence of disorder. Related work can be traced back to 1986, when the celebrated Gordon-Haus effect was established for amplified solitons in optical fiber transmission [17]. The Brownian soliton motion (normal diffusion of solitons) was predicted in light-induced random photonic lattices [18], and was also observed recently in BECs with impurities [19]. Besides, the so-called Anderson localization of solitons were reported in random media as well [20–22]. Here, we will numerically show that, the DBSs, analogous to classical particles, experience a transient anomalous diffusion with the property of non-ergodicity, in a 1D disordered nonlinear optical lattice.
2. Model and method
The evolution of optical waves in a 1D lattice of coupled waveguides along the propagation coordinate $z$ is described by the following discrete nonlinear Schrödinger (NLS) equation in its dimensionless form
Equation (1) was integrated by using a fourth-order Runge-Kutta scheme with periodic boundary conditions. The lattice size where $n\in [-N,N]$ was large enough with $N=500$ to make sure that the DBSs are far away from the boundaries. The discretization of the propagation distance was $\Delta z=0.01$, and the total distance was computed up to $L=10^{5}$. Our statistical results as below were obtained by averaging over $N_r=1024$ realizations, and the DBSs remained localized and robust at all times for each realization. To double check our numerical results, we varied $\Delta z$ by orders of magnitude, even changed the integration scheme by a second-order Besse method [26], and found the DBS behaviors kept the same.
3. Results and analysis
Figure 1(a) shows a typical evolution of the DBS for a specific realization of disorder, where the DBS is scattered by the disorder to experience transverse displacements, and may change its moving direction at certain spots of strong impurities. In order to study their particle-like motions, we consider random walks of the DBS’s center of mass $x(z)$ with respect to $z$, defined as $x(z)=\sum _n n|\psi _n|^{2}/P$. Here $x(z)$ physically represents the dependence of the DBS center coordinate on the propagation distance. In Fig. 1(b), $x(z)$ is plotted as functions of $z$ for $2^{10}=1024$ independent realizations of disorder, which appears to display the features of diffusion.
We use two basic quantities to characterize these diffusive motions [27]: one is the ensemble-averaged mean squared displacement (eMSD), and the other one is the time-averaged mean squared displacement (tMSD), where the coordinate $z$ plays a role of the time $t$ in our context. The eMSD is defined as
For classical particles, the eMSD can display either a Brownian motion $\langle x^{2}(z) \rangle \thicksim z$, or an anomalous diffusion $\langle x^{2}(z) \rangle \thicksim z^{\alpha }$ with $\alpha \neq 1$ for a certain scale of $z$. Anomalous diffusion is an important subject in complex systems, and has been studied in various occasions [27–29]. In Fig. 2(a), we plot the eMSD $\langle x^{2}(z)\rangle$ as a function of $z$ on a log-log scale for $5$ orders of magnitude, with the same representative parameters in Fig. 1. The eMSD seems to increase monotonically with $z$ exceeding $z=3000$, corresponding to a diffusive process, and then reaches a plateau (localization of DBSs) for the longer $z$. The diffusion covers several intervals with different scaling exponents $\alpha$ for each interval, as seen in Table 1. We can apparently find a transient anomalous diffusion, that includes an initial stage with $\alpha \approx 3.3$, an intermediate stage with $\alpha \approx 1.5$, and a final stage with $\alpha \approx 0.8$.
Fig. 1. (a) Evolution of the DBS in the lattice for a specific realization of disorder; (b) Center of mass of the DBSs $x(z)$ for 1024 independent realizations of disorder. The parameters are $\mu =0.50$, $\sigma =0.002$, and $\nu =1$.
Fig. 2. (a) The eMSD $\langle x^{2}(z)\rangle$ as a function of $z$; (b) The tMSDs $\overline {\delta ^{2}(\Delta )}$ (red thin curves) for individual trajectories as functions of $\Delta$, and the trajectory-averaged tMSD $\left \langle \overline {\delta ^{2}(\Delta )} \right \rangle$ is denoted by the black bold curve. All these quantities are evaluated from $N_r=1024$ independent realizations. The parameters are the same as in Fig. 1 ($L=10^{5}$).
Table 1. Diffusive property of the DBS’s center of mass. For different intervals of $z$, $\alpha$ and $\alpha '$ are obtained by respectively fitting the eMSD and tMSD curves to $\langle x^{2}(z)\rangle \thicksim z^{\alpha }$ and $\left \langle \overline {\delta ^{2}(\Delta )}\right \rangle \thicksim \Delta ^{\alpha '}$. The $R^{2}>0.997$ is kept for every fit.
On the other hand, from the viewpoint of single particle tracking, the tMSD of an individual trajectory is given by [27]
However, comparison of the eMSD and tMSDs clearly reveals the inequality
for different intervals of the diffusive process ($\Delta \ll L$), with the identification $z\leftrightarrow \Delta$. For the particle-like motion, this inequivalence indicates occurrence of weak ergodicity breaking, meaning that the information obtained in the time average is different from that given by the ensemble average [27,30,31]. This should be important since it reminds us to be careful when we draw some conclusions from a single-trajectory measurement of DBSs with long duration. For example, regarding the interval $z\in [400,1000]$, the tMSD gives rise to the seemingly subdiffusive behavior with $\alpha '\approx 0.9$, as opposed to the eMSD-suggested superdiffusive transport with $\alpha \approx 1.5$. In principle, such a statement of non-ergodicity should be made in the limit of $L\rightarrow \infty$, however, the computational domain $L=10^{5}$ is far larger than the diffusive domain of $z\lesssim 3000$, and the disparity between the eMSD and the tMSDs seems to be kept for longer $L$.To further understand the scatter of the tMSDs $\overline {\delta ^{2}(\Delta )}$ between individual trajectories, we can investigate the distributions of these tMSDs by using the probability density function (PDF) $\phi (\xi )$ of the following normalized ratio
Fig. 3. Scatter distribution $\phi (\xi )$ for the DBS’s diffusive process. Panels (a)-(c) are extracted at $\Delta =20$, $\Delta =450$, and $\Delta =1250$, corresponding to the diffusion intervals with different exponents $\alpha$. The parameters are the same as in Fig. 2.
For longer propagation distance ($z\gtrsim 5000$), the DBSs become trapped, displaying the transverse Anderson-like localization of DBSs. In general, increasing the strength of the disorder may shorten the transient diffusive process, resulting the DBS motions confined in a narrower spatial region.
Since the nonlinear localized states, such as solitons, may be used as the information carriers in optical communication systems, the studies on how their propagations are influenced by noise and random inhomogeneities become necessary. One example is the soliton Gordon-Haus jitter for which the distributed noise leads the variance of soliton displacement to satisfy $\langle x^{2}\rangle \thicksim z^{3}$ [17]. In this work our simulation of the DBS propagation in 1D disordered waveguide lattices reveals the existence of intervals where the displacement variance grows in a slower way, $\langle x^{2}\rangle \thicksim z^{\alpha }$ with $\alpha <3$. This suppressed transport may be realized in types of experimental setup as in [3,12]. On the other hand, we should be careful in interpreting the average of DBS trajectories since the time averages and the ensemble averages give different scaling due to ergodicity breaking for the diffusive process. The non-ergodic optical behaviors can also be found such as for Lévy flight of light in disordered media [27,33], blinking nanoscale light emitters [34], and explosions of dissipative optical solitons [35].
4. Conclusions
In summary, we numerically simulated transport of discrete breather-like states in a disordered nonlinear optical lattice, where the center of mass of the breather-like state represents a transient anomalous diffusion. Related properties were studied to suggest non-ergodicity of such the diffusive process, and binding of the DBSs by disorder was considered to be the mechanism for the diffusion transition. Our study might benefit understanding of the interaction between disorder and strong nonlinearity in optical community.
Funding
Fundamental Research Funds for the Central Universities; Beihang University (Zhuoyue Talent Program); National Natural Science Foundation of China (11902016).
Acknowledgments
The referee is appreciated for his/her valuable comments.
Disclosures
The authors declare no conflicts of interest.
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