Rogue waves in quantum lattices with correlated disorder

We investigate the outbreak of anomalous quantum wave-function amplitudes in a one-dimensional tight-binding lattice featuring correlated diagonal disorder. Such rogue-wave-like behavior is fostered by a competition between localization and mobility. The effective correlation length of the disorder is ultimately responsible for bringing the local disorder strength to a minimum, fueling the occurrence of extreme events of much higher amplitudes, especially when compared to the case of uncorrelated disorder. Our findings are valid for a class of discrete one-dimensional systems and reveal profound aspects of the role of randomness in rogue-wave generation.

Figure 1

(a) Autocorrelation function $C\left(r\right)=\mathrm{cov}({\varepsilon }_i,{\varepsilon }_{i+r})$ vs $r$ for many values of the correlation parameter $A$, averaged over 100 independent samples. (b) Effective correlation length $L_c\propto A$.

Figure 2

Snapshots of the space-time evolution of the particle wave-function probability highlighting typical rogue wave events for different correlation strengths: (a) $A=1$, (b) $A=25$, (c) $A=50$, and (d) $A=100$. Two-dimensional graphs display spatial and temporal cross sections for the largest amplitudes. Time is expressed in units of $J^{-1}$.

Figure 3

Extreme-value distributions generated by sorting up the maximum values of $|{\psi }_n{|}^2$ at each time step during the whole evolution (up to $t=15000/\text{J}$ with $\mathrm{\Delta }t=0.01/\text{J}$) for 500 independent samples of a disordered chain with $N=100$ sites when $A=1,25,50,100$. Bottom panels show the fitted Gumbel distribution (red curve) of the form $P\left(x\right)\propto exp[-(\alpha x+\beta exp(-\alpha x\left))\right]$.

Figure 4

Maximum wave-function probability amplitude ${\left|\psi \right|}_{\max }^2$ scaled with $N$ (excluding exceptionally rare events) for a range of $A/N$ values.

Figure 5

Distribution of ${\left|\psi \right|}^2$ accumulated from an ensemble of ${10}^3$ independent realizations of disorder for a chain with $N=100$ sites, each evolving up to $t=15000/\text{J}$. Correlation parameters are $A=1$ (circles), $A=30$ (triangles), and $A=90$ (diamonds), respectively.

Figure 6

Mean number of sites actively involved in the generation of rogue waves $\mathrm{\Delta }{\overline{n}}_{RW}$ vs correlation strength $A$, averaged over 500 independent realizations of disorder, for times up to $t=15000/\text{J}$ and $N=100$. Note that the values of $A$ that give the highest $\mathrm{\Delta }{\overline{n}}_{RW}$ also lead to outbreaks of very intense rogue waves.

Figure 7

(a) Series of the on-site potential ${\varepsilon }_n$ considering $A=1,25,50$, and 100. (b) Local standard deviation ${\sigma }_{L_0}$ as a measure of the local disorder strength. Weaker local fluctuations are obtained at intermediate values of $A$, the same range that renders better chances of observing rogue waves of higher amplitudes.