# Rogue waves in quantum lattices with correlated disorder

A. R. C. Buarque  Instituto de Física, Universidade Federal de Alagoas,
57072-900 Maceió, AL, Brazil  Secretaria de Estado da Educação de Alagoas,
57055-055, Maceió, AL, Brazil  W. S. Dias  Instituto de Física, Universidade
Federal de Alagoas, 57072-900 Maceió, AL, Brazil  G. M. A. Almeida  Instituto
de Física, Universidade Federal de Alagoas, 57072-900 Maceió, AL, Brazil  M.
L. Lyra  Instituto de Física, Universidade Federal de Alagoas, 57072-900
Maceió, AL, Brazil  F. A. B. F. de Moura<EMAIL_ADDRESS>Instituto de
Física, Universidade Federal de Alagoas, 57072-900 Maceió, AL, Brazil

###### Abstract

We investigate the outbreak of anomalous quantum wavefunction 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,
fuelling the occurrence of extreme events of much higher amplitudes, specially
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.

##  I Introduction

Rare and unpredictable events carrying huge impact are widespread in nature,
from stock markets to physical sciences. Outliers may change the course of
things more often than we are prone to think, sometimes leading to hazardous
consequences. One example is the emergence of rogue waves in the ocean. The
famous Draupner wave recorded in 1995 at a gas platform in Norway was twice as
big as the significant wave height of the area. This happened to be the first
scientific evidence of a rogue wave [1] and led to a burst of interest in the
field as studies began to suggest that these extreme events would occur more
frequently than assumed from ordinary Gaussian statistics [2]. About a decade
later, Solli et al. introduced rogue waves in optics based on observations
made on fibre supercontinuum generation in the presence of noise [3].

The analogy drawn between oceanic rogue waves and extreme instabilities in
optics associated to long-tailed statistics set the stage for a number of
theories aimed to explain the physical mechanisms behind rogue-wave
generation. Much of the effort has been directed toward establishing whether
and which linear or nonlinear processes play the biggest roles [4, 5, 6, 7, 8,
9, 10, 11]. Oceanic rogue waves, for instance, may result from various
mechanisms in action, such as constructive interference of random fields,
modulational instability, and soliton modes, depending on sea and wind
conditions [12, 13]. Even though there is no definitive consensus on that
matter, neither robust ways to predict where and when rogue waves will occur,
noise and randomness seem to be key ingredients for their occurrence [14, 11,
15] (deterministic rogue waves are also discussed in Refs. [4, 5]).

Some degree of disorder is paramount for generating rogue waves via linear
mechanisms in particular [14, 11, 16, 17, 15, 18, 19, 20, 21, 22, 23, 24, 25].
Very recently, this was addressed in the context of quantum walks [25].
Therein, the authors primarily sought to explore the (hitherto elusive)
relationship between Anderson localization and rogue wave manifestation. They
reported a minimal disorder threshold $\propto N^{-1/2}$ ($N$ being the number
of sites) above which rogue waves were created, whereas intermediate disorder
levels would maximize the chances of seeing one due to a proper balance
between trapping and mobility (see also Ref. [21]). Such results, besides
bringing the rich subject of rogue waves up to the realm of quantum transport,
add important elements to the issue of the actual role played by randomness.

With those ideas in mind, here we set out to track the dynamics of rogue waves
on a single quantum particle propagating in a lattice featuring correlated
disorder. Anderson localization theory settles that all single-particle
eigenstates are exponentially localized for any amount of uncorrelated
disorder in one and two dimensions [26]. This can be violated, however, when
the disorder displays intrinsic correlations [27, 28, 29]. Scale-free
correlations, for instance, are known to support a metal-insulator transition
with sharp mobility edges [27].

In this article, our goal is to investigate the development of sudden,
anomalous quantum amplitudes due to the interplay between localized and
extended states. Indeed, long-range correlated fluctuations in the random
input phases was recently shown to produce rogue waves way above the threshold
in an experiment on linear light diffraction in 1D [22].

We look at a particular kind of correlated disorder in which a single
parameter is able to control the typical correlation length and, in turn, the
local disorder strength. The latter is found to be a crucial factor underlying
the generation of the rogue waves for it sets up the right amount of
wavefunction mobility. This boosts not only the number of occurrences, but
also the average rogue wave amplitude.

##  II Model

We consider a single quantum particle propagating in a one-dimensional array
described by the tight-binding Hamiltonian

$\displaystyle H=J\sum_{n=1}^{N}(|n\rangle\langle n+1|+|n+1\rangle\langle
n|)+\sum_{n}^{N}\varepsilon_{n}|n\rangle\langle n|,$ (1)

where $J$ is the nearest-neighbor hopping strength, $\varepsilon_{n}$ is the
on-site potential, and states $|n\rangle$ represent the location of the
particle and span the whole Hilbert space. As such, an arbitrary quantum state
can be written as $|\Psi\rangle=\sum_{n}\psi_{n}|n\rangle$, with the
normalization condition $\sum_{n}|\psi_{n}|^{2}=1$. The time evolution of the
wave function $\psi_{n}$ is given by the Schrödinger equation

$\displaystyle
i\frac{d}{dt}\psi_{n}=\psi_{n+1}+\psi_{n-1}+\varepsilon_{n}\psi_{n},$ (2)

where $\hbar=J=1$ without loss of generality.

Here we go beyond the standard case of uncorrelated disorder and consider that
the local potentials $\varepsilon_{n}$ are embedded with correlations, as
given by

$\displaystyle\varepsilon_{n}=\sum_{m}\frac{Z_{m}}{(1+d_{n,m}/A)^{2}},$ (3)

where $Z_{m}$ is a random number $\in[-1,1]$, $d_{n,m}$ is the Euclidean
distance between sites $n$ and $m$, and $A$ controls the correlation length of
the series. We impose periodic boundary conditions, $|N+1\rangle=|1\rangle$,
such that each site $n$ can be identified through angular and Cartesian
coordinates, $\theta_{n}=(2\pi/N)n$ and
$(x_{n}=R\sin{\theta_{n}},y_{n}=R\cos{\theta_{n}})$, respectively, rendering
$d_{n,m}=\sqrt{(x_{n}-x_{m})^{2}+(y_{n}-y_{m})^{2}}$. The disordered sequence
is further normalized to have zero mean and unit variance.

To see how the above disorder configuration play out with the correlation
parameter $A$, in Fig. 1(a) we plot the autocorrelation function
$C(r)=\mathrm{cov}(\varepsilon_{i},\varepsilon_{i+r})=\sum_{i=1}^{N-r}\varepsilon_{i}\varepsilon_{i+r}/(N-r)$.
Upon increasing $A$, $C(r)$ exhibits a slower decay, as expected from Eq. 3.
We are then able to set an effective correlation length $L_{c}$ by fitting
$C(r)\propto e^{-r/L_{c}}$. Figure 1(b) shows that $L_{c}\propto A$ thereby
affirming what the latter stands for.

Figure 1: (a) Autocorrelation function
$C(r)=\mathrm{cov}(\varepsilon_{i},\varepsilon_{i+r})$ versus $r$, for many
values of the correlation parameter $A$, averaged over 100 independent
samples. (b) Effective correlation length $L_{c}\propto A$.

##  III Results

Figure 2: Snapshots of the space-time evolution of the particle wavefunction
probability highlighting typical rogue wave events for different correlation
strengths (a) $A=1$, (b) $A=25$, (c) $A=50$, (d) $A=100$. 2D graphs display
spatial and temporal cross-sections for the largest amplitudes. Time is
expressed in units of $J^{-1}$.

We are ready to search for the occurrence of rogue waves and address the role
of the correlated disorder. In all simulations bellow, the set of equations
given in Eq. 2 is numerically solved by employing a high-order Taylor
expansion of the time evolution operator:

$U(\Delta t)=\exp(iH\Delta t)=1+\sum_{l=1}^{n_{0}}\frac{(iH\Delta
t)^{l}}{l!},$ (4)

with time step $\Delta t=0.01J^{-1}$ and $n_{0}=20$, which is enough to
produce smooth outcomes and keep the norm conserved during the whole time
interval. In order to avoid ambiguity between a rogue wave event and trapping
of the wavefunction due to Anderson localization we set
$\psi_{n}(t=0)=1/\sqrt{N}$ for all $n$ [25].

Figure 2 shows typical scenarios of rogue waves, as told by the probability
amplitude of the particle wavefunction $|\psi_{n}(t)|^{2}$, for different
values of the correlation strength $A$. For weak correlations the evolution is
characterized by sparse, low-amplitude waves that eventually add up to produce
the rogue events, as shown in Fig. 2(a) around $t=3000/J$. As the correlation
strength is increased, the background becomes more inhomogeneous – a
fundamental trait in the generation of rogue waves via linear processes [11].
There sets in distinct amplitude domains in space, indicating that $A$ is
pushing for weaker local fluctuations in the disorder distribution (to be
addressed in a moment). As a consequence, rogue waves of exceptionally higher
amplitudes are likely to occur [cf. Fig. 2(d)].

We will not be using here any particular rogue-wave criteria. One measure
employed in various contexts is the significant wave height, commonly defined
as twice the average of largest one third of values in a data set [12]. An
event is thus considered a rogue wave whenever it beats that level.
Considering our initial state and the results obtained in Ref. [25], that
deals with a similar class of problem, such threshold would be of the order of
$1/N$. As our following analysis is built on extreme-value statistics, the
data is heavily loaded with amplitudes well above that mark.

Let us now obtain one of the limiting extreme-value distributions, according
to the Fischer-Tippett-Gnedenko theorem [30]. In order to do so, we pick the
maximum wavefunction amplitude in space at each time step for several
independent realizations of the disorder. The stacked outcomes are seen in the
probability density functions (PDFs) shown in Fig. 3 for different degrees of
correlation $A$. Note that this effective correlation length indeed brings
forth higher amplitudes. However, there is a threshold value for $A$ above
which the right tail of the distribution begins to deflate (rare, extreme
rogue waves can still develop). To learn more about this non-monotonic
relationship between the rogue-wave maximum amplitudes and the correlation
parameter $A$, we portray in Fig. 4 the maximum amplitudes $|\psi|_{max}^{2}$
allowed at a probability level just above $10^{-7}$ (see Fig. 3). This is done
in order to avoid extremely rare outcomes. The scaling with $N$ in Fig. 4 is
employed so we can filter out finite-size effects and focus on the role of the
correlation degree only. Compared with what one would obtain from uncorrelated
series of the potential $\varepsilon_{n}$ (cf. dashed curve in Fig. 4), the
rogue waves can reach almost twice as large amplitudes when supported by the
correlated disorder.

We also find that all the skewed PDFs shown in Fig. 3 belong to the Gumbel
class of extreme value distributions, fitted by $P(x)\propto\exp[-(\alpha
x+\beta\exp(-\alpha x))]$. This is expected since the tail of the parent
function $p(|\psi|^{2})$ decays exponentially (slower than that of a Gaussian
distribution, which is another signature of rogue waves). Similar behavior is
found in quantum walks featuring uncorrelated disorder [25], meaning that the
wavefunction fluctuations are well described by processes involving
independent and identically distributed random variables.

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
$t=15000/J$ with $\Delta t=0.01/J$) for 500 independent samples of a
disordered chain with $N=100$ sites when $A=1,25,50,100$. Right panels show
the fitted Gumbel distribution (red curve) of the form
$P(x)\propto\exp[-(\alpha x+\beta\exp(-\alpha x))]$.  Figure 4: Maximum
wavefunction probability amplitude $|\psi|_{max}^{2}$ scaled with $N$
(excluding exceptionally rare events) for a range of $A/N$ values. Figure 5:
Mean number of sites actively involved in the generation of rogue waves
$\Delta\overline{n}_{RW}$ versus correlation strength $A$, averaged over 500
independent realizations of disorder, for times up to $t=15000/J$ and $N=100$.
Note that the values of $A$ that give highest $\Delta\overline{n}_{RW}$ also
leads to outbreak of much intense rogue waves.

Figure 6: (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.

The regularity of rogue wave events occurring on systems featuring static
disorder can be predicted to some extent based on the energy resonance
conditions across the lattice. Either very weak or very strong levels of
disorder should suppress the onset of anomalous wavefunction fluctuations.
Some studies report that an optimal balance between localization and mobility
can maximize their chances to happen [21, 25]. Here, this balance is
effectively imposed by the correlation level $A$. To show there is indeed a
coordinated dynamics taking place, let us track how many sites, for a given
disordered sample, are mostly involved in the those extreme events. Defining
$\pi(n)$ as the probability of a rogue occurring at site $n$, we compute the
mean number of participating sites as
$\Delta\overline{n}_{RW}=1/[\sum_{n}\pi(n)^{2}]$, which ranges from $1$ (fully
biased) to $N$ (equally distributed). Figure 5 displays this quantity against
the correlation parameter $A$. Again, the non-monotonic behavior is clear and
indicates that intermediate values of $A$ implies in more sites actively
participating in the generation of rogue waves. In turn, waves of higher
amplitudes becomes more likely (cf. Figs. 3 and 4).

Last, we complement the above discussion by taking a look back at the
potential series given by Eq. (3). Figure 6(a) shows how it typically sets up
along the lattice. For $A=1$, the series features a white noise (almost
uncorrelated) profile. Then, at $A=25$ local fluctuations are drastically
reduced. As we further increase the effective correlation length, a rougher
landscape is obtained, but still carrying a predominant harmonic component. To
put all that together, let us compute the local disorder strength in terms of
the local standard deviation
$\sigma_{L_{0}}=\left(\sum_{k=1}^{M}\sigma_{k,L_{0}}\right)/M$ within a
segment with $L_{0}$ sites, where

$\sigma_{k,L_{0}}=\sqrt{\sum_{n=(k-1)L_{0}+1}^{kL_{0}}\frac{\varepsilon_{n}^{2}}{L_{0}}-\left(\sum_{n=(k-1)L_{0}+1}^{kL_{0}}\frac{\varepsilon_{n}}{L_{0}}\right)^{2}}$
(5)

Results are shown in Fig. 6(b), where we are able to confirm once for all that
intermediate values of $A$ makes for minimum local fluctuations, which is
consistent with the smooth potential landscape seen in Fig.6(a).

##  IV Concluding remarks

We explored the intrinsic role played by correlated disorder on the emergence
of rogue waves in a simple quantum tight-binding model. The amplitude of the
particle wave function was found to exhibit strong anomalous fluctuations
inherently unpredictable in time and space. An approach based on extreme-value
statistics revealed that such fluctuations follow a Gumbel distribution,
belonging to the same class as those reported in a quantum walk featuring
uncorrelated disorder [25].

We learned that intermediate values of the correlation parameter $A$, acting
here as an effective correlation length, brings down the local disorder
strength. This properly enhances the mobility of the wavefunction (by spanning
eigenstates with larger localization lengths) and so the number of sites on
which rogue-wave events take place, what in turn amplifies their
characteristic amplitude (almost twice as large when compared to uncorrelated
scenarios [21, 22, 25].

While disorder must be present to promote random fluctuations of the
wavefunction, the underlying single-particle eigenstates must be wide enough
in order to allow the linear superposition of the many components needed to
produce localized waves in space and time. A general result we can establish
is that rogue waves are expected to occur more often in the regime of weak
disorder. Our findings are general to a class of 1D discrete disordered
systems and bring about another perspective on their dynamics as well as
reveal fundamental aspects of the role of randomness in the generation of
rogue waves.

##  V Acknowledgments

This work was supported by CAPES, CNPq, and FAPEAL (Alagoas State agency).

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