# Dynamics of quantum discommensurations in the Frenkel-Kontorova chain

Oksana Chelpanova  Institut für Physik, Johannes Gutenberg-Universität Mainz,
D-55099 Mainz, Deutschland  Shane P. Kelly  Mani L. Bhaumik Institute for
Theoretical Physics, Department of Physics and Astronomy, University of
California at Los Angeles, Los Angeles, CA 90095  Ferdinand Schmidt-Kaler
Institut für Physik, Johannes Gutenberg-Universität Mainz, D-55099 Mainz,
Deutschland  Giovanna Morigi  Theoretische Physik, Universität des Saarlandes,
D-66123 Saarbrücken, Deutschland  Jamir Marino  Institut für Physik, Johannes
Gutenberg-Universität Mainz, D-55099 Mainz, Deutschland

###### Abstract

The ability for real-time control of topological defects can open up prospects
for dynamical manipulation of macroscopic properties of solids. A sub-category
of these defects, formed by particle dislocations, can be effectively
described using the Frenkel-Kontorova chain, which characterizes the dynamics
of these particles in a periodic lattice potential. This model is known to
host solitons, which are the topological defects of the system and are linked
to structural transitions in the chain. This work addresses three key
questions: Firstly, we investigate how imperfections present in concrete
implementations of the model affect the properties of topological defects.
Secondly, we explore how solitons can be injected after the rapid change in
lattice potential or nucleated due to quantum fluctuations. Finally, we
analyze the propagation and scattering of solitons, examining the role of
quantum fluctuations and imperfections in influencing these processes.
Furthermore, we address the experimental implementation of the Frenkel-
Kontorova model. Focusing on the trapped ion quantum simulator, we set the
stage for controllable dynamics of topological excitations and their
observation in this platform.

††preprint: APS/123-QED

##  I Introduction

In recent years, there has been a growing interest in the physics of
incommensurate systems and the role of topological defects in altering the
macroscopic properties of physical systems. This has been particularly
pronounced in the domain of nano-physics, where the commensurate-
incommensurate (C-IC) transition is regarded as a pinning-to-sliding
transition occurring between two surfaces, making it a prominent item in the
study of nano-friction Bonetti  _et al._ (2021); Hormann  _et al._ (2023); Li
_et al._ (2022); Timm  _et al._ (2021a); Dong  _et al._ (2023); Zheng (2023);
Sar  _et al._ (2023); Braun and Naumovets (2006); Timm  _et al._ (2021b). In
materials science, the C-IC transition sparked curiosity in the late 1970s
after experimental work by Naumovets and Fedorous Naumovets and Fedorus
(1977). In their work, different interference peaks were reported depending on
the concentration of atoms absorbed by the surface of the substrate material
(adatoms). A number of studies reported Aristov and Luther (2002); Pokrovsky
and Talapov (1984); Bohlein  _et al._ (2012) that when the concentration of
the adatoms was small, they formed a commensurate structure with the surface,
while for higher concentrations, the long-range Van der Waals force induced an
incommensurate configuration. The emergent lattice modulations have also been
shown to alter the properties of magnetic material. For instance, it has been
shown that IC-C transition in the perovskite $R\text{MnO}_{3}$ is accompanied
by the emergence of ferroelectric order at low temperatures Goto  _et al._
(2004). In general, the C-IC transition has been shown to induce noteworthy
modifications in the solid’s macroscopic properties, piquing significant
interest and setting the stage for future research Coullet (1986); Schuster
_et al._ (1996); Vanossi  _et al._ (2013); Woods  _et al._ (2014); Ahmed  _et
al._ (2023); Bak (1982); Aristov  _et al._ (2002); Zhang  _et al._ (2023);
Menu  _et al._ (2023).

In recent studies, the C-IC transitions have been examined from the
perspective of its implementation in quantum simulators, including driven
Bose-Einstein condensates Büchler  _et al._ (2003); Kasper  _et al._ (2020);
Wybo  _et al._ (2023); Mivehvar  _et al._ (2021); Neuenhahn  _et al._ (2012),
arrays of Rydberg atoms Muñoz  _et al._ (2020); Maceira  _et al._ (2022);
Chepiga and Mila (2021); Zhang  _et al._ (2024), and trapped ion chains Landa
_et al._ (2020); Gangloff  _et al._ (2020); Vanossi  _et al._ (2020); Pyka
_et al._ (2013). What sets these quantum simulators apart is their unique
capability for dynamic parameter control, allowing researchers not only to
investigate the equilibrium C-IC transition but also to actively drive it in
real time by adjusting the simulator’s parameters. This approach offers the
potential to dynamically alter the macroscopic properties of the system and
manipulate its spectrum in real time. On the other hand, quantum fluctuations
and finite-size effects, which are distinctive characteristics of quantum
simulators, can pave the way for exotic phenomena that have not been observed
on a macroscopic scale in traditional condensed matter systems Ejtemaee and
Haljan (2013); De Chiara  _et al._ (2010); Ulm  _et al._ (2013); Chelpanova
_et al._ (2023).

In this paper, we study the discommensurations in such simulators,
concentrating on the parameter regime accessible with the trapped ion
simulator. We consider a one-dimensional chain containing up to a hundred
charged particles within a periodic lattice potential captured by the Frenkel-
Kontorova model (FKM) Frenkel and Kontorova (1938); Frank  _et al._ (1949);
Braun and Kivshar (2004). This enables us to explore various effects and
imperfections, such as discreetness and finite-size effects, as well as
quantum fluctuations, and highlight the key distinctions between the C-IC
transition in the quantum simulator and the C-IC transition in classical
systems.

Furthermore, we demonstrate how topological defects, holes, or double-filled
sites (kinks and anti-kinks of the model) can be dynamically injected into the
system by adjusting the amplitude of the lattice potential. Also, we show how
quantum fluctuations can lead to nucleation of topological defect, resulting
in a state that is a quantum superposition of commensurate and incommensurate
chains.

We delve further into the impact of simulator imperfections on the system’s
dynamics. To this end, we study the scattering of solitons in the model and
show how finite-size effects, along with the quantum fluctuations, alter such
scattering, comparing with the ideal scenario.

This paper is an extension of our previous work Chelpanova  _et al._ (2023),
in which we solely studied the injection of the kink through an abrupt change
in lattice periodicity.

### Summary of results

The paper is dedicated to exploring three main directions listed below.

  * •

Impact of imperfections in the quantum simulation of solitons. In Sec. II, we
summarize the role of discreteness, finite size, and quantum fluctuations on
soliton dynamics. In particular, we review the conditions under which the
long-wavelength limit of the FKM is applicable. We discuss features that break
the integrability of the model, namely the corrections to the continuum limit
due to the fact that the system is effectively discrete, which we will refer
to as “discreteness effects”. Moreover, we will discuss finite-size effects.
We then summarize different limits of the FKM in Sec. II.6 and discuss
possible implementation of the dynamics we predict in a trapped ion simulator
in Section V.

  * •

Real-time control of the platform and solitons injection. In Sections III, we
show how solitons can be injected into the system from its boundaries in a
controlled fashion. We consider driving dynamics by adjusting the amplitude of
the lattice potential. Interestingly, in this case, we also distinguish the
generation of a bounded pair of kink and anti-kink, which is not observed when
the lattice periodicity is instead varied, cf. Ref. Chelpanova  _et al._
(2023).

  * •

Scattering of solitons as a test of integrability breaking in the FKM. In Sec.
IV, we revisit kink-anti-kink scattering in the presence of quantum
fluctuations and accounting for discreteness effects and discuss the
difference from solitons scattering in the continuum limit of the model.
Finally, we study a specific case of the scattering of the kink against the
boundary of the Frenkel-Kontorova chain. We distinguish different outcomes of
such scattering, namely, reflection and mirroring of the soliton or injection
of multiple solitons. We provide explanations for these outcomes based on
energy conservation arguments.

Figure 1: Equilibrium properties of the FKM. (a) The equilibrium phase diagram
in terms of the lattice potential amplitude $m_{K}$ and the misfit parameter
$\delta$. The number of solitons $Q$ is encoded in the colormap. (b) Phase
$\phi_{n}$ for the commensurate configuration. (c) Sketch of particle
positions on top of the lattice potential in the commensurate phase. (d) Phase
$\phi_{n}$ for the incommensurate configuration. Each kink describes the local
distribution of the empty lattice site. (e) Sketch of particle positions on
top of the lattice potential in the incommensurate phase. (f) Sketch of the
spectrum of the FKM in the incommensurate phase.

##  II Preliminaries

This section is dedicated to giving a comprehensive introduction to the FKM.
To highlight the main properties, we compare the predictions of this model
with the ones of its continuous integrable counterpart. We discuss what breaks
the integrability in the FKM and how such integrability breaking affects the
dynamics of the system. Furthermore, we explore the influence of boundary
conditions and finite-size effects, demonstrating how these elements can be
leveraged to dynamically control topological defects within the system.
Additionally, we briefly discuss an efficient approach for incorporating
quantum effects into the dynamics through semi-classical methods. At the end
of the section, we summarize various parameter regimes that offer insights
into the diverse physics encapsulated by the FKM. Readers already familiar
with these concepts can proceed to the following section.

###  II.1 The Frenkel-Kontorova model

We consider a one-dimensional chain comprising $N$ charged particles. Due to
the confinement, the particles are at uniform inter-particle spacing $a_{0}$.
In addition, the particles also interact with a substrate lattice potential
with periodicity $a_{s}$. The displacement between the particle positions
$x_{n}$ and the closest minimum of the external lattice, $na_{s}$, is
expressed in terms of a phase: $\phi_{n}=2\pi(x_{n}-na_{s})/a_{s}-\pi$, where
$n=1,\ldots,N.$ The phase $\phi_{n}$ takes value $\pi$ when the ions’
equilibrium positions coincide with a lattice minima. The dynamics of this
phase are captured by the Frenkel-Kontorova model Frenkel and Kontorova
(1938); Frank  _et al._ (1949)

$H=\sum_{n}\frac{p_{n}^{2}}{2}+{m_{K}^{2}}(1+\cos\phi_{n})+\sum_{r=\pm
1}\frac{(\phi_{n+r}-\phi_{n}-2\pi r\delta)^{2}}{2}$ (1)

where $p_{n}=\dot{\phi}_{n}$. Here, the first term describes the system’s
kinetic energy, the second term models the external periodic lattice
potential, and the last term accounts for the Coulomb repulsion between
particles, here truncated to the nearest-neighbor terms. The parameter
$m_{K}^{2}$ captures the amplitude of the periodic lattice potential, while
the misfit parameter $\delta$ measures the discommensuration of the system and
is defined as a relative difference between two length scales of the model,
$\delta=(a_{0}-a_{s})/a_{s}.$ In Eq. (1), we set all variables and parameters
to be dimensionless, and, if necessary, rescaled in such a way that
$|\delta|<0.5$. A detailed derivation of the model (1) along with the
rescaling can be found in Sec. V.

The equations of motion that govern the dynamics of the finite-size FKM read

$\displaystyle\ddot{\phi}_{1}={m_{K}^{2}}\sin\phi_{1}+\left(\phi_{2}-\phi_{1}-{2\pi}\delta\right)$
(2)
$\displaystyle\ddot{\phi}_{n=2:N-2}={m_{K}^{2}}\sin\phi_{n}+\left(\phi_{n+1}-2\phi_{n}+\phi_{n+1}\right)$
$\displaystyle\ddot{\phi}_{N}={m_{K}^{2}}\sin\phi_{N}+\left(\phi_{N-1}-\phi_{N}+{2\pi}\delta\right).$

Note that here, for particles in the bulk discommensuration effects from the
left $+2\pi\delta$ and from the right $-2\pi\delta$ neighboring particles
cancel each other, and thus, the misfit parameter does not directly affect the
dynamics of this part of the chain. At the same time, the boundary sites are
sensitive to the value of the misfit parameter $\delta.$

By varying $\delta$, the system can undergo an equilibrium transition from the
topologically trivial commensurate state to the incommensurate configuration
characterized by the presence of solitons, see Fig. 1(a). When
$\delta<\delta_{c}\approx 2m_{K}/\pi^{2},$ the external periodic potential
dominates over inter-particle interaction, and particles are pinned by the
substrate lattice, i.e., each particle sits in a minimum of the potential and
$\phi_{n}=\pi.$ Figure 1(b) displays the corresponding phase configuration,
indicating that all particles are situated at the lattice minima positions,
while Figure 1(c) illustrates the typical distribution of particles within the
lattice in the commensurate state.

In contrast, for $\delta>\delta_{c},$ inter-particle interactions destabilize
the commensurate structure, leading to particles rearrangement and the
formation of dislocations, see Fig. 1(e). These dislocations correspond to the
local distribution of either holes ($\delta>0$) or excess particles
($\delta<0$) across the lattice and can be mapped to the solitons of the FKM,
which are called respectively kinks or anti-kinks [see Fig.1(d)]. Each kink
(anti-kink) corresponds to the ‘step’ in the phase $\phi_{n},$ namely, the
phase value changes by $2\pi$ ($-2\pi$) over a few lattice sites Landa  _et
al._ (2020); Bak (1982); Braun and Kivshar (2004); Pruttivarasin  _et al._
(2011); García-Mata  _et al._ (2007); Bak (1982). The number of solitons

$Q=\frac{1}{2\pi}\left(\phi_{N}-\phi_{1}\right)$ (3)

is a quantity that allows us to distinguish between the commensurate ($Q=0$)
and the incommensurate ($Q\neq 0$) phases Aristov and Luther (2002) by
counting the total phase accumulated in units of $2\pi$. For instance, the
incommensurate configuration in Fig. 1(d,e) has three ‘steps’ in $\phi_{n}$
and corresponds to $Q=3.$ In the incommensurate phase, $Q$ is controlled by
$\delta.$

When the length of the kink is significantly larger than the lattice spacing,
the discrete charge distribution can be approximated by a continuum. The
continuum limit of the FKM is the so-called Pokrovsky and Talapov
modelPokrovsky and Talapov (1984). Importantly, the Pokrovsky and Talapov
model is integrable, and its analytical solutions are often employed as a
benchmark for the FKM. In the following section, we summarize predictions of
the Pokrovsky-Talapov model that are relevant to this study.

###  II.2 Long-wavelength limit of the Frenkel-Kontorova model

When the size of the discommensuration region is much larger than inter-
particle spacing, the phase function $\phi_{n}$ can be replaced with the field
$\phi(x)$, and the finite differences in (1) can be replaced by partial
derivatives in space. The effective model can be then described by the
Pokrovsky-Talapov Hamiltonian, which reads $H=\int\mathrm{d}xh,$ where the
Hamiltonian density is given by

$h=\frac{1}{2}p^{2}+\frac{1}{2}\left(\partial_{x}\phi\right)^{2}+\frac{m_{K}^{2}}{\beta^{2}}(1+\cos\beta\phi)-\frac{\mathcal{H}\beta}{2\pi}\partial_{x}\phi,$
(4)

where $p=\dot{\phi}$, $\beta$ calls the interaction constant, and parameter
$\mathcal{H}$ sets the density of topological defects. The interplay between
kinetic terms and non-linearities in the first three terms enables soliton
solutions, while the last term controls the density of these solitons within
the chain.

The evolution of the field $\phi(x)$ according to Eq. (4) follows the sine-
Gordon equation Malard (2013)

$\partial_{t}^{2}\phi=\partial_{x}^{2}\phi+\frac{m_{K}^{2}}{\beta}\sin\beta\phi.$
(5)

The possible quasi-particle solutions of the model (5) are determined by the
interaction constant $\beta$ Ruelle (2009); Del Vecchio  _et al._ (2023). From
mapping between models (1) and (4) one can find $\beta=1.$ This specific
choice enables kink (anti-kink) solutions. Additionally, the model can host a
high-energy breather solution, which is the bounded kink-anti-kink pair that
oscillates back and forth around a common center.

The important feature of the chain governed by the Pokrovsky-Talapov model is
that the density of the defects in it can be controlled by adjusting the
parameter $\mathcal{H}$, cf. Refs. Aristov and Luther (2002); Kasper  _et al._
(2020). The last term in Eq. (4) bears the analogy with the chemical potential
in statistical physics Huang (2009). However, instead of controlling the
number of particles in a macro-canonical ensemble, it controls the number of
topological defects in the chain.

In particular, the so-called ‘chemical potential’ $\mathcal{H}$ sets the
boundary conditions where the equilibrium solution seeks to minimize the total
energy, cf. Refs. Aristov and Luther (2002); Kasper  _et al._ (2020);
Lazarides  _et al._ (2009),

$E=\int_{0}^{L}dx\left(\frac{1}{2}\left(\partial_{x}\phi\right)^{2}+\frac{m_{K}^{2}}{\beta^{2}}(1+\cos\beta\phi)\right)-\mathcal{H}Q_{c}.$
(6)

Here, we denote $Q_{c}=\beta/(2\pi)(\phi(L)-\phi(0))$. This parameter
quantifies the change in the field $\phi$ over the length of the chain,
counting the number of kinks present within it. Adjusting the ‘chemical
potential’ $\mathcal{H}$ makes it possible to steer the system across various
meta-stable minima within the energy landscape. Each of these minima
corresponds to distinct soliton configurations within the chain Lazarides  _et
al._ (2009).

It has been shown Aristov and Luther (2002); Kasper  _et al._ (2020) that in
equilibrium, the ground state of the Pokrovsky-Talapov model reads

$\phi(x)=\left\\{\begin{aligned} &\pi/\beta,&\quad\mathcal{H}<M_{s},\\\
&\frac{2}{\beta}\operatorname{am}\left(\dfrac{xm_{K}}{k},k^{2}\right),&\quad{\mathcal{H}\geq
M_{s}}.\end{aligned}\right.$ (7)

Here, the parameter $k$ depends on $m_{K}$ and $\mathcal{H}$,
$M_{s}=8m_{K}/\beta^{2}$ is the soliton mass, and $\operatorname{am}()$
denotes the Jacobi amplitude function Abramowitz  _et al._ (1988). If
$\mathcal{H}<M_{s}$, the energy of the incommensurate configuration exceeds
that of the commensurate chain. In this case, in the ground state, the
particles organize themselves in the equidistant configuration with
$\phi(x)=\pi/\beta$. On the other hand, when $\mathcal{H}\geq M_{s}$, the
ground state contains a finite density of discommensurations. In the ground
state, the field adopts a staircase configuration, where each step has a
height $2\pi/\beta$ and corresponds to a kink. The length of a single kink,
$l_{m}\propto 1/m_{K}$, represents the size of the region in which $\phi$
changes by $2\pi/\beta$. Conversely, the distance between kinks is controlled
by the chemical potential $l_{\delta}\propto 1/\mathcal{H}$. Note that these
results qualitatively describe the corresponding commensurate-incommensurate
transition in the FKM mentioned above, where the correspondence between
chemical potential and the misfit parameter is given by
$\mathcal{H}=4\pi^{2}\delta/\beta^{2}$.

Depending on the resulting ground state, the model (4) can host various types
of excitations. The excitation spectrum of the model (4) can be derived by
expanding field $\phi(x)$ on top of the ground state configuration
$\phi^{(0)}(x)$ in powers of fluctuations $\eta(x)$ as
$\phi(x)=\phi^{(0)}(x)+\eta(x).$ In the lowest order, which is valid in the
low energy limit, this expansion leads to the quadratic Hamiltonian Aristov
and Luther (2002)

$\mathcal{H}_{\eta}=\frac{1}{2}\left(\partial_{t}\eta\right)^{2}+\frac{1}{2}\left(\partial_{x}\eta\right)^{2}-\frac{1}{2}m_{K}^{2}\left(\cos\beta\phi^{(0)}\right)\eta^{2}+\ldots$
(8)

The equations of motion governing the dynamics of the excitations read

$\left(-\partial_{x}^{2}-m_{K}^{2}\cos\beta\phi^{(0)}\right)\eta_{i}(x)=\omega_{i}^{2}\eta_{i}(x).$
(9)

Here, the index $i$ labels the $i$-th mode. In the commensurate phase, the
spectrum exhibits a gap proportional to $m_{K}$. Following the terminology in
Ref. Aristov and Luther (2002), which is inherited from the solid state
physics Ashcroft and Mermin (2001), we refer to this branch of the spectrum as
the optical modes. The optical modes govern individual excitations of
particles around their equilibrium positions, with frequencies proportional to
$m_{K}$.

In the incommensurate phase, the spectrum contains two branches. The first
branch is formed from the hybridization of the bound states related to
individual $Q_{c}$ kinks in the chain. These $Q_{c}$ modes correspond to
gapless acoustic excitations that control the propagation of $Q_{c}$ kinks
along the chain Aristov and Luther (2002); Kutnjak-Urbanc and Žekš (1995). The
second branch describes individual oscillations of the rest of the particles
around their equilibrium positions.

The integrable Pokrovsky-Talapov model (4) provides valuable insights and
foundation for developing an intuitive understanding of the commensurate-
incommensurate transition and the characteristics of solitons. Nevertheless,
in realistic implementations, numerous factors can disrupt integrability and
significantly alter the chain’s behavior. In the following subsections, we
summarize the impact of finite-size effects, the chain’s discreteness, and
quantum corrections on the system’s dynamics. We also establish the conditions
under which these effects are negligible, and the continuous description
remains valid.

###  II.3  Discreteness effects

In this section, we summarize the role of discreteness effects and determine a
parameter regime within which the continuum model (4) can provide accurate
predictions for soliton dynamics. The FKM is a discrete lattice model, with
the amount of ‘discreteness’ controlled by the amplitude of the lattice
potential, $m_{K}.$ Depending on $m_{K}$, the dynamics (propagation) of the
kinks may exhibit very different features: In the limit $m_{K}\to 0$, solitons
can propagate ballistically through the lattice, resembling the behavior
commonly observed in integrable models. On the other hand, it has been
reported that in the regime where the continuum approximation becomes invalid
($m_{K}\propto 1$), solitons do not propagate ‘frictionlessly’ through the
lattice. Instead, they slow down during dynamics or even become immovable
after short periods of time Braun and Kivshar (2004); Peyrard and Kruskal
(1984); Askari  _et al._ (2020); Willis  _et al._ (1986). As we show in Sec.
IV.1, the discreteness effects become important in scattering processes since
kinematic characteristics of solitons before collision are crucial for the
outcome of the scattering.

The role of discreteness effects on the continuous model can be understood
from a symmetry point of view: In the continuous model, the system possesses
continuous translation invariance, which is broken in the incommensurate phase
and restored in the commensurate phase. The breaking of this symmetry results
in the appearance of the gapless Goldstone mode in the excitation spectrum,
the acoustic mode mentioned earlier, which governs the propagation of kinks
through the lattice Aristov and Luther (2002); Kasper  _et al._ (2020). In the
discrete model, instead, continuous translation invariance is reduced to a
discrete one, and its breaking in the incommensurate phase is associated with
the appearance of a gaped Peierls-Nabarro mode in the spectrum Braun and
Kivshar (2004), cf. Fig. 1(f). The difference in the nature of the acoustic
mode in continuous and discrete regimes leads to distinct dynamical
properties.

We can map the discrete model to the continuum one by substituting
$\phi_{n}\to\phi(x)$. For the finite differences in Eqs. (2), we can employ a
Taylor series expansion, which takes the form

$\phi_{n+1}-2\phi_{n}+\phi_{n-1}\to\frac{\partial^{2}\phi}{\partial
x^{2}}+\frac{2}{4!}\frac{\partial^{4}\phi}{\partial x^{4}}+\ldots.$ (10)

The sine-Gordon model is integrable, while including higher-order derivatives
disrupts this integrability and significantly impacts the model’s properties.
However, these higher-order corrections become negligible by decreasing the
amplitude of the potential $m_{K}.$ Qualitatively, this stems from the fact
that in the $m_{K}\to 0$ limit the length of the kink $l_{m}\propto
1/m_{K}\to\infty$, and the kink appears wide and smooth. Thus, higher-order
derivatives $\partial^{2n}\phi/\partial x^{2n}$ are suppressed compared to the
contribution from the $\partial^{2}\phi/\partial x^{2}$ term. As $m_{K}$
increases, the influence of higher-order terms becomes more significant in
shaping the dynamics; thereby, the model (4) does not approximate FKM
correctly.

To estimate the value of $m_{K}$ at which the continuous model (4) fails to
accurately predict the kinematics of a single kink of the FKM, one can
consider the propagation of a kink across a single lattice site. During this
process, the positions of the particles neighboring the kink change, resulting
in the change of the system’s potential energy along the path,
$V_{\operatorname{pot}}=\sum_{n}m_{K}^{2}(1+\cos\phi_{n})+\sum_{r=\pm
1}(\phi_{n+r}-\phi_{n}-2\pi r\delta)^{2}/{2}$. Thus, to propagate to the
following site, the kink should overcome a Peierls-Nabarro barrier $E_{PN}$
Braun and Kivshar (2004); Ablowitz  _et al._ (2021); Partner  _et al._ (2013);
Willis  _et al._ (1986), which is the difference between the maximal, see Fig.
2(b), and minimal, see Fig. 2(a), potential energy of the system through this
process

$~{}E_{PN}=V_{\operatorname{pot}}(\phi_{\operatorname{kink}}^{\operatorname{unstable}})-V_{\operatorname{pot}}(\phi_{\operatorname{kink}}^{\operatorname{stable}}).$
(11)

The barrier $E_{PN}$ has a clear physical meaning as the minimum kinetic
energy the kink must have to move to the following site. If the kink cannot
overcome this barrier, it gets stuck at the initial site. When the kink
propagates through the lattice, it needs to climb the $E_{PN}$ barrier, which
results in a decrease in its kinetic energy. The released kinetic energy of
the kink is transferred to the lattice vibrations in the form of the optical
modes, and surrounding particles start oscillating around their equilibrium
positions.

Figure 2: The Peierls-Nabarro barrier. (a-b) Two stationary kink
configurations (compare the particles’ positions in the middle of the chain).
The Peierls-Nabarro barrier is determined as the difference in the potential
energies of configurations (b) and (a). (c) The Peierls-Nabarro barrier as a
function of $m_{K}.$

The dependence of the Peierls-Nabarro barrier on $m_{K}$ is plotted in Fig.
2(c). One can recognize that for small $m_{K}$, the kink can freely propagate
through the lattice, while for $m_{K}\gtrsim 0.8$, the Peierls-Nabarro barrier
increases. For $m_{K}>1.1$, a single kink can propagate only a few lattice
sites before it comes to a complete stop.

###  II.4 Finite-size and boundary effects

The finite-size and boundary effects play a crucial role in the ability to
dynamically manipulate topological defects. The finite system size makes it
necessary to treat exactly the boundary condition of the model, which, as
opposed to the continuous model, brings a direct dependence of the dynamics of
the system on the misfit parameter $\delta.$

In the FKM, the boundaries act as a source of topological defects with some
chemical potential, and thus, the corresponding boundary conditions are
different from the ones often considered in the continuum Pokrovsky-Talapov
model Aristov and Luther (2002). Precisely, equations of motion (2) are
directly sensitive to the misfit parameter through the boundary sites,
enabling the possibility to dynamically pump kinks or anti-kinks into the
system from the boundaries, thereby altering the macroscopic properties of the
chain, as we illustrate in Sec. III.

Additionally, the boundary conditions of the finite-size FKM [cf. Eqs. (2)]
result in the dressing (i.e., minor modification) of the number of kinks $Q$.
For small but non-zero values of the misfit parameter, $0<\delta<\delta_{c}$,
the equilibrium value of the phase $\phi_{1,N}$ deviates from $\pi$ by the
quantity, linear in $\delta.$ Due to this fact, the number of kinks $Q$ [cf.
Eq. (3)] also deviates from integer by the value linear in $\delta$ even in
the commensurate phase. This additional soliton density is induced solely by
the boundary conditions scales like $\delta/N$ with the system size, and it
becomes irrelevant only for an infinite system. However, such boundary defects
in finite-size chains can have a pronounced effect on scattering processes,
see Sec. IV.2.

A second finite size effect comprises modulation of the soliton density with
the system size. Following the Pokrovsky-Talapov model, the total length of
one kink equals $l_{m}+l_{\delta}.$ Thus, even if the system size is varied
within $[N,N+l_{m}+l_{\delta})$ in equilibrium, the number of kinks remains
constant, determined by the number of kinks that can fit a given chain, namely
$Q=\lfloor N/(l_{m}+l_{\delta})\rfloor$. The injection of another kink
requires additional energy costs due to the effective repulsion between kinks.
As a result, the kinks density, which is equal to $\rho=Q/N$ varies
periodically with the system size in the range
$\rho=(Q/N,Q/(N+l_{m}+l_{\delta})),$ see Ref. Kasper  _et al._ (2020) for
comprehensive discussion.

Finally, if the system size is smaller than $l_{m}+l_{\delta}$, the form of a
kink will be modified to make it sharper and fit the lattice. This change of
the kink’s shape requires additional energy cost, resulting in a higher
equilibrium critical value of the misfit parameter $\delta_{c}$ than predicted
by the Pokrovsky-Talapov model. In this regime, a field theory (4) fails to
predict system properties. Our analysis, however, is in the regime where at
least a few solitons fit into the system.

###  II.5 Quantum mechanics of the Frenkel-Kontorova model

The quantum simulators of the FKM operate on length and energy scales where
quantum fluctuations can play a significant role and, therefore, should be
taken into account. Quantum fluctuations may destroy ordered phases, generate
new dynamical phases, or give rise to a quantum critical region where the
transition from one phase to another appears as a crossover Sachdev (1999);
Fernandes  _et al._ (2019); Shimshoni  _et al._ (2011a); Gornostyrev  _et al._
(1999); Shimshoni  _et al._ (2011b); Lazarides  _et al._ (2009); Lerose  _et
al._ (2018, 2019); Marino  _et al._ (2022); Berdanier  _et al._ (2019); Wybo
_et al._ (2023). Consequently, we extend the model (1) such that is

$\hat{H}=\sum_{n}\frac{\hat{p}_{n}^{2}}{2}+m_{K}^{2}(1+\cos\hat{\phi}_{n})+\sum_{r=\pm
1}\frac{(\hat{\phi}_{n+r}-\hat{\phi}_{n}-2\pi r\delta)^{2}}{2}.$ (12)

Here we introduce momentum and coordinate operators $\hat{p}_{n},$
$\hat{\phi}_{n}$ instead of classical variables $p_{n},\phi_{n}.$ These
operators satisfy canonical commutation relations

$\left[\hat{\phi}_{n},\hat{p}_{m}\right]=ih_{\operatorname{eff}}\delta_{n,m}.$
(13)

Here, the effective Plank constant is a dimensionless small parameter that
controls the strength of the quantum fluctuations. The value of
$h_{\operatorname{eff}}$ depends on the particular experimental implementation
of the model (1). Details of Eq. (12) and $h_{\operatorname{eff}}$ are found
in Sec. V.

In the limit $h_{\operatorname{eff}}\to 0$, quantum fluctuations can be
considered as a minor correction to the classical effects and thus treated
perturbatively. In this regard, one can build a Bogoliubov-Born-Green-
Kirkwood-Yvon (BBGKY) hierarchy of corrections in powers of
$h_{\operatorname{eff}},$ when higher-order corrections contribute to dynamics
at later timescales Negele and Orland (2018).

In the following, we treat quantum corrections in a semi-classical limit,
where only corrections linear in $h_{\operatorname{eff}}$ are included in the
dynamics. To do so, we follow the truncated Wigner approximation (TWA) and
consider $\hat{\phi}_{n}$ and $\hat{p}_{n}$ as a sum of the classical term and
the small quantum correction Polkovnikov (2010); Lancaster  _et al._ (2010);
Bistritzer and Altman (2007). For simplicity, we initialize our system in the
commensurate phase, for which the ground state can be approximated by the
Gaussian wave function with the Wigner quasi-probability distribution

$W\left(\eta_{q}^{0},n_{q}^{0}\right)=\prod_{q}\exp\left[-{\left|\eta_{q}^{0}\right|^{2}}/{\sigma_{q}}-{\left|\psi_{q}^{0}\right|^{2}}/b_{q}\right],$
(14)

where $\eta^{0}_{q}=\phi^{0}_{q}-\langle\hat{\phi}_{q}\rangle_{0}$ and
$\psi^{0}_{q}=p^{0}_{q}-\langle\hat{p}_{q}\rangle_{0}$ are the phase space
variables relative to the mean value of the operators in the initial state,
while
$\sigma_{q}=\langle\hat{\eta}_{q}^{2}\rangle_{0}=h_{\operatorname{eff}}/\omega_{q}$
and
$b_{q}=\langle\hat{\psi}_{q}^{2}\rangle_{0}=h_{\operatorname{eff}}\omega_{q}$
encode the variance of quantum fluctuations in the initial state. Here,
$\omega_{q}$ is determined by the discrete counterpart of the Eq. (9), cf.
Fig. 1(f). At zeroth order in $h_{\operatorname{eff}}$, the TWA approximation
is simply the mean-field dynamics as in Eq. (2). At first order in
$h_{\operatorname{eff}}$, the dynamics of phase space variables are still
described by Eq. 2, but the initial state is sampled from the positive Wigner
distribution in Eq. (14). Observables at later times are calculated by
averaging over the trajectories resulting from the initial quantum noise.
Taking into account higher-order corrections implies adding quantum noise
during dynamics. Precisely, at randomly distributed time moments, the momentum
$p_{m}$ undergoes additional jumps induced by quantum fluctuations Polkovnikov
(2010); Lancaster  _et al._ (2010).

###  II.6 Summary of regimes of the Frenkel-Kontorova model

Figure 3: Cartoon of possible limits of the FKM depending on the $m_{K}$ and
the strength of $h_{\operatorname{eff}}.$ (I) Range of values of $m_{K}$ in
which the Pokrovsky-Talapov model governs dynamics correctly. Note that for
higher values of $m_{K}$ quantum corrections are required to capture dynamics.
(II) Region of parameters in which quantum fluctuations should be taken into
account to recover the dynamics of the system correctly. (III) For
$m_{K}\gtrsim 0.8$, discreteness effects provoke significant dissipation of
the soliton’s kinetic energy. In this limit, the Pokrovsky-Talapov model is
invalid. (IV) When $1/m_{K}\gtrsim N$, where $N$ is the system size, the
length of a single kink exceeds the system size, resulting in a change in the
critical misfit parameter compared to the Pokrovsky-Talapov model predictions.

The sine-Gordon model is integrable in classical and quantum limits Del
Vecchio  _et al._ (2023). Thus, the sources of the integrability breaking in
the model are discreteness and finite size effects but not a finite
$h_{\operatorname{eff}}$ value. As a result, one can probe different regimes,
integrable or non-integrable, classical or quantum, by varying $m_{K}$ and
$h_{\operatorname{eff}}$ separately.

The illustrative sketch in Fig. 3 provides a concise summary of the diverse
regimes within the FKM that have been discussed thus far. When $m_{K}\to 0,$
one can obtain a regime in which continuous description works relatively well,
and solitons behave like solitons of the integrable sine-Gordon model (I).
When $m_{K}$ increases and Peierls-Nabarro barrier $E_{PN}$ becomes finite
(cf. Sec. II.3), solitons start experiencing dissipation of their kinetic
energy to the lattice vibrations (III). Along with that, by adjusting the
value of the effective Plank constant, one can make the effect of quantum
fluctuations sizable. As elaborated in subsequent sections, these fluctuations
significantly influence the critical region, giving rise to a novel dynamical
regime characterized by a superposition of chains with and without solitons
(II).

Furthermore, another parameter range can be explored when dealing with an
exceptionally weak lattice potential amplitude, $m_{K}\to 0$, where the length
of a single kink surpasses the system’s dimensions (IV). In such instances,
the critical misfit parameter at which the transition occurs becomes sensitive
to the system’s size and deviates from the predictions of the Pokrovsky-
Talapov model.

In the distinct parameter regimes depicted in Fig. 3, the impact of finite
size, discreteness, and quantum effects extends beyond the equilibrium
properties of the system to also influence its dynamics. The following section
provides a more detailed exploration of these effects, focusing on the
system’s dynamical response to abrupt parameter changes.

##  III  Real-time control of the platform and solitons injection

Figure 4:  Sketch of the dynamical phase diagram in terms of the initial
$m_{K}^{\operatorname{init}}$ and final $m_{K}^{\operatorname{fin}}$ values of
$m_{K}$. For some combination of parameters, only the optical mode can be
excited (white central region), solitons can be injected (yellow), or the
breather can be excited (blue).

In this section, we explore structural C-IC transitions dynamically driven by
quenches of the lattice amplitude $m_{K}$. We start with the classical
treatment and discuss different modes that can be excited by varying $m_{K}$.
We show how, in this way, one can inject topological defects into the system,
altering the system’s properties. Then, we explore how quantum fluctuations
modify the dynamics of the systems in proximity to the boundary that separates
the commensurate and incommensurate phases. Finally, we provide insights into
the distinctions between fast quenches and the gradual, adiabatic adjustment
of parameters.

###  III.1 Excitations

We initialize the system in the equilibrium configuration with given initial
$m_{K}^{\operatorname{init}},$ and then evolve it with some final value
$m_{K}^{\text{fin}}$. Throughout this process, we maintain a constant misfit
parameter $\delta$. Various dynamical responses on such quench are summarized
in Fig. 4. Here, we find three possible outcomes of the quenching $m_{K}$ in
the FKM

  1. 1.

The number of kinks remains constant, while small excitations of the particles
around their equilibrium positions are excited as an optical mode; this
corresponds to the white region in Fig. 4;

  2. 2.

Injection of kinks from the boundaries of the chain, accompanied via
excitation of the acoustic modes; this corresponds to the yellow region;

  3. 3.

Excitation of a bounded superposition of the kink and anti-kink (breather);
this corresponds to the blue region.

We explore these regimes in more detail below.

####  III.1.1 Excitation of the optical modes after the quench of $m_{K}$

Figure 5: (a) Time evolution of the $\phi_{n}$ after the quench of $m_{K}.$
For small changes of $m_{K}$, particles start oscillating around their new
equilibrium positions. (b) Sketch of the particle dynamics in the chain. Here
$m_{K}^{\operatorname{init}}=0.1$, $m_{K}^{\operatorname{fin}}=0.15$, and
$\delta=0.01$.

Referring to the equilibrium phase diagram in Figure 1(a), it is evident that
the parameter $m_{K}$ can be adjusted without inducing a transition between
commensurate and incommensurate phases, namely without change of the soliton’s
number $Q$. In real-time dynamics, such quench of $m_{K}$ can be seen as a
tiny perturbation since the energy injected with the quench remains below the
threshold required to inject kink or breather excitations. In this process,
the energy difference between two stable configurations for initial
$m_{K}^{\operatorname{init}}$ and final $m_{K}^{\text{fin}}$ is transferred
into vibration modes, see the white region in Fig. 4.

The quenching of $m_{K}$ is achieved through a rapid change of the lattice
amplitude, and its effects can be comprehended by examining equilibrium
solutions for both the initial and final values of $m_{K}.$ In the
incommensurate phase, $m_{K}$ sets the length of a single kink, with $l_{m}$
being inversely proportional to $m_{K}$ (i.e., $l_{m}\propto 1/m_{K}$).
Consequently, when the lattice amplitude is adjusted, the equilibrium spacing
between kinks also changes. Particles that are initially positioned at their
equilibrium locations before the quench are no longer in equilibrium
afterward, as depicted in Fig. 5(b). These particles, displaced from their
equilibrium positions, commence oscillating around new equilibrium positions
determined by $m_{K}^{\text{fin}}.$ The difference in energy between the state
after the quench and the equilibrium configuration with
$m_{K}^{\operatorname{fin}}$ contributes to the excitation of optical modes.
The frequency of these oscillations is determined by the $m_{K}^{\text{fin}}$,
and the amplitude of the oscillations is set by the amount of the exceeding
energy.

For quenches within the commensurate phase, the displacement of particles in
the bulk in equilibrium for both values of $m_{K}^{\operatorname{init}}$ and
$m_{K}^{\operatorname{fin}}$ read $\phi_{n}=\pi$. Thus, after the quench,
these particles are not displaced from equilibrium and remain at rest.
However, starting from the commensurate phase, the optical mode can still be
excited after the quench of $m_{K}$. This happens because, for any finite
misfit parameter $\delta\neq 0$, the deviation of the phase at the boundaries
is non-zero and depends on $\delta$ and $m_{K}$. Thus, after the quench of
$m_{K}$ optical mode is excited at the boundary sites, and then this
excitation propagates due to the nearest neighbor interaction into the bulk
with the speed of sound $c=1$, see Fig. 5(a).

If $m_{K}$ is varied slightly, only optical modes can be excited in the
system, cf. Fig. 4. As we will show below, the injection of a soliton and the
excitation of a breather require higher energy and more substantial system
perturbation.

####  III.1.2 Injection of kinks accompanied via excitation of the acoustic
mode

Figure 6: (a) Evolution of $\phi_{n}$ after the quench of
$m_{K}^{\operatorname{init}}\to m_{K}^{\operatorname{fin}}\leq m_{K}^{c}$ when
kinks periodically enter the system from both boundaries. (b) Number of kinks
in the system as a function of time. The inset shows the energy of two kinks
in equilibrium as a function of the distance between their centers. Here
$m_{K}^{\operatorname{init}}=0.5$, $m_{K}^{\operatorname{fin}}=0.25$, and
$\delta=0.08$.

According to the equilibrium phase diagram in Fig. 1, the commensurate-
incommensurate transition can occur when keeping the misfit parameter fixed
and decreasing $m_{K}.$ If we decrease the mass of the kink $m_{K}$ below a
certain threshold, $m_{K}^{\operatorname{init}}\to
m_{K}^{\operatorname{fin}}\leq m_{K}^{c}$, the initial energy of the system
can be sufficient to inject a finite density of kinks into the system. In this
case, we lower the amplitude of the lattice potential, causing particles to
redistribute in the chain and form local discommensurations, see the yellow
region in Fig. 4.

The example of dynamics with injection of solitons is illustrated in Fig.
6(a). Starting from the commensurate configuration, kinks enter the chain one
by one via the system boundaries. After a timescale of $\tau\propto N$, the
density of kinks saturates, and then kinks traverse the lattice back and
forth, scattering off each other. Importantly, since the topological charge
within the chain must be conserved, discommensurations can enter or leave the
system solely at the boundaries, which can be considered as a reservoir of
topological defects, see also Refs. Yakimenko  _et al._ (2015); Wright  _et
al._ (2013); Kochetov  _et al._ (2019); Ma  _et al._ (2020); Rajaraman (1987);
Dauxois and Peyrard (2006); Brox  _et al._ (2017).

The number of kinks $Q$ in the system as a function of time is plotted in Fig.
6(b). Just after the quench, $Q$ starts growing in a staircase fashion, and
then it saturates around a certain value. One can recognize that the height of
each ‘step’ in the staircase is equal to two, as the injection of kinks occurs
independently from both the left and right boundaries. The duration of each
plateau (a time interval during which the number of kinks $Q$ remains
constant) is proportional to $l_{\delta}/c$, while the time intervals during
which the charge changes from $Q$ to $Q+2$ are set approximately by $l_{m}/c$.

The staircase structure of $Q(t)$ can be elucidated by analyzing the energies
associated with various equilibrium configurations of the kinks. The inset in
Fig. 6(b) shows the energy of two kinks as a function of the relative
distance, $r$, between their centers. If the kinks overlap, their combined
energy is higher than the energy of the configuration with two infinitely
separated kinks, resulting in effective short-distance repulsion Carretero-
González  _et al._ (2022); Braun and Kivshar (2004). As such, the average
distance between kinks after the quench is set approximately by
$l_{m}+l_{\delta}$ because arranging kinks closer requires additional energy.
Consequently, over longer times, the number of kinks tends to saturate at
approximately $Q\approx 2N/(l_{m}+l_{\delta})$. This quantity is primarily
determined by the total number of solitons that can be accommodated within a
system of a given size $N$. The factor of two arises from the behavior of the
nearest neighbor interaction model, where the left boundary remains unaware of
the injection of kinks through the right boundary until a time of $N/c$ when
the first injected kinks reach the opposite boundaries. Accordingly,
$N/(l_{m}+l_{\delta})$ kinks are injected from the left boundary and an equal
number from the right boundary, independently. This factor two disappears when
particles interact with each other in a long-range fashion. In this case, the
injection of kinks stops once the number of kinks reaches the value
$N/(l_{\delta}+l_{m}),$ cf. Ref. Chelpanova  _et al._ (2023)

Until a time scale of $\tau\propto N/c$, systems with different numbers of
particles in the short-range interacting model evolve with the same staircase
temporal profile, as their dynamics are universally governed by ${\delta}$ and
$m_{K}$. However, in the long-range interacting model, the temporal profiles
of $Q$ can vary depending on the system sizes. This mechanism is identical to
the one reported in Ref. Chelpanova  _et al._ (2023) for the case when a
quench in $\delta$ is performed.

Note that the critical value of $m_{K}^{cd}$ at which transition to
incommensurate phase occurs in dynamics differs from the critical $m_{K}^{c}$
in equilibrium: While in equilibrium, kinks are immovable, the dynamical phase
transition is associated with the excitation of the acoustic mode, which
governs the propagation of kinks. As such, the dynamical phase transition
requires higher energy and happens at the smaller value of $m_{K}.$ We omit
the apex d when discussing dynamical critical values for simplicity in the
rest of the text but recover it again in Sec. VI.

Figure 7: (a) The time evolution of $\phi_{n}$ subsequent to the quench, where
$m_{K}^{\operatorname{init}}$ is significantly less than
$m_{K}^{\operatorname{fin}}$. Dashed red lines indicate the positions of the
breathers. Here $m_{K}^{\operatorname{init}}=0.1$,
$m_{K}^{\operatorname{fin}}=0.7$, and $\delta=0.025$. (b) The shape of a
single kink before (black line) and just after (blue line) the quench. The
gray shading is employed to denote discommensurations corresponding to the
formation of breathers. (c) Sketch of $\phi_{n}$ near the center of the chain.
After the quench of $m_{K}$, two breathers arise in the system. Each kink-
anti-kink pair is characterized by a non-zero velocity $\pm{v}$, and the
breathers move away from the kink.

####  III.1.3 Excitation of a pair of breathers

Within the incommensurate phase, higher values of $m_{K}$ correspond to
smaller solitons density [see Fig. 1(a)]. According to the equilibrium phase
diagram, a substantial increase of lattice amplitude,
$m_{K}^{\operatorname{init}}\ll m_{K}^{\text{fin}},$ can potentially trigger a
transition from the incommensurate to the commensurate phase. However, this
can appear differently when we consider real-time dynamics.

The only way to destroy kinks and enter the commensurate phase is to excite
the acoustic mode and move kinks toward the boundaries, where they can
disappear. However, during a quench with $m_{K}^{\operatorname{init}}\ll
m_{K}^{\text{fin}}$, the lattice amplitude increases, and each particle is
trapped in a deep potential well, escaping from which it would require a high
energy cost. Furthermore, during the quench of $m_{K}$, the solitons do not
acquire a distinct momentum direction. Consequently, there are no preferred
directions for their propagation. Thus, the transition to the commensurate
phase remains unattainable after quenching $m_{K}^{\operatorname{init}}\ll
m_{K}^{\text{fin}}$. Nonetheless, the outcome of this kind of quenches can be
non-trivial, including the generation of highly non-linear types of
excitations. In particular, after increasing $m_{K}$ above a certain value, a
breather can be excited, see the blue region in Fig. 4.

The example of dynamics when a pair of breathers are excited on top of the
kink is shown in Fig. 7(a). After increasing $m_{K}$, the kink in the middle
of the chain remains immovable, while breathers appear on the left and right
sides of the kink. For simplicity, we mark the boundaries of the breathers in
the Figure with red dashed lines. Each breather consists of a bounded kink-
anti-kink pair, which propagates back and forth around a common center. This
is reflected in the periodic temporal oscillation of $\phi_{n}$ around the
breather. One can also recognize that in this particular example, the
breathers slowly drift away from the center of the chain.

Qualitatively, the excitation of breathers can be understood as follows: After
increasing $m_{K}$, the width of the kink, $l_{m}$, shrinks as $l_{m}\propto
1/m_{K}$. Consequently, immediately after the quench, particles near the
center of the kink are compelled to occupy the lattice potential minima to
minimize the energy. This results in the formation of a sharp kink at the
center, see the blue line in Fig. 7(b). The displacements of the particles
located immediately adjacent to the kink result in the creation of defects
characterized by a negative charge. These displaced particles subsequently
acquire momentum, driving them towards the system’s boundaries. These anti-
kink-like defects then engage with the positive-charged dislocations at the
ends of the initial kink and form breathers, illustrated by the gray regions
in Fig. 7(b). These breathers move toward the boundaries as their initial
anti-kinks constituents acquire non-zero momentum after the quench, cf. Fig.
7(c). By increasing the final value of $m_{K}$ one can simultaneously excite a
few breather modes on top of the kink.

###  III.2  Semi-classical treatment

The results discussed so far have been evaluated neglecting quantum
fluctuations. However, for specific implementations, e.g., in trapped ions or
cold atoms systems, quantum effects can start playing a significant role,
affecting long-time dynamics and physics close to the boundary between phases.
In this case, quantum fluctuations in the model can be exponentially enhanced,
inducing non-trivial dynamical responses.

The mechanism of the amplification of the quantum fluctuations in the sine-
Gordon model has been shown by Starobinsky for the re-heating phase in the
inflation theory in cosmology Greene  _et al._ (1999), and it can be briefly
illustrated as follows. In the low energy limit [see Eq. (8)], dynamics of
quantum fluctuations are governed by

$\ddot{\eta}_{q}\approx
m_{K}^{2}\cos\left(\phi^{(0)}(t)\right)\eta_{q}-c^{2}q^{2}\eta_{q},$ (15)

which is valid at the early stages of the dynamics when fluctuations are
small, with variance $\propto\sqrt{h_{\operatorname{eff}}}$. After the quench,
the classical solution $\phi^{(0)}$ performs oscillations governed by the
optical mode. Thus, the dynamics of fluctuations are similar to those of a
harmonic oscillator with a periodically driven mass. Starobinsky has shown
that modes with $|q|\in\left[0,m_{K}\sin\left|\phi^{(0)}/2\right|\right]$ are
exponentially enhanced in this case, with amplification rates
$2\Gamma_{q}\simeq|q|[\sin^{2}\left(\phi^{(0)}/2\right)-q^{2}/m_{K}^{2}]^{1/2}$,
see Greene  _et al._ (1999) for more details. Amplification of these modes
results in effects beyond linear response theory accompanied by the appearance
of visible signatures of quantum effects in physical systems. For instance, in
Ref. Neuenhahn  _et al._ (2012), short-lived quasi-breathers were reported in
the sine-Gordon model as a manifestation of parametric resonance of quantum
fluctuations in cold atoms.

Figure 8: Quench $m_{K}^{\operatorname{init}}\to m_{K}^{f}\gtrsim
m_{K}^{c}>0.$ (a) The phase $\phi_{n}$ after the quench at $t=90$ computed for
different noise realizations within TWA method. The resulting semi-classical
configuration is plotted with the solid black line. Here
$m_{K}^{\operatorname{init}}=0.5$, $m_{K}^{\operatorname{fin}}=0.29$,
$\delta=0.08$, and $h_{\operatorname{eff}}=0.07$. (b) The time dependence of
the charge of the system evaluated in the classical (blue) and semi-classical
limits (red).

The dynamics of the quantum fluctuations are mostly determined by the lattice
amplitude $m_{K}.$ For instance, if $m_{K}^{\operatorname{fin}}=0$, according
to Eq. (15), the dynamics of fluctuations and classical phase $phi_{n}$
decouple, rendering the mean-field theory exact at arbitrary timescales. On
the other hand, for the quench with $m_{K}^{\operatorname{fin}}\to 1$, the
classical part $\phi_{n}^{(0)}$ and the quantum fluctuations $\eta$ in Eq.
(15) are strongly coupled and parametric amplification of fluctuations is
possible. As a result, the larger $m_{K}$ is, the quicker quantum effects
become visible in the dynamics.

An example of the non-trivial manifestation of quantum effects for the quench
$m_{K}^{\operatorname{init}}\to m_{K}^{\operatorname{fin}}\gtrsim m_{K}^{c}>0$
is shown in Fig. 8. Here, we use the TWA method from Sec. II.5 to evaluate
dynamics in the presence of quantum noise. Panel (a) shows the profile of the
phase after the quench. Different lines result from the different initial
conditions in the TWA method, and the black solid lines show the value of the
phase averaged over several noise realizations. One can recognize that after
this quench, some initial conditions result in commensurate configuration
while others evolve towards an incommensurate state. As a result, the number
of kinks averaged over several noise realizations in panel (b) (red line) is
non-integer and deviates from the value evaluated in the classical limit (blue
line). The outcome of the quench is a quantum superposition of the system with
and without the kink, resulting in a state without classical counterpart, as
signaled by the non-integer value of the topological charge $Q$.

Figure 9: (a) The energy difference between the state with a kink and the
ground commensurate state of the system as a function of $m_{K}$. The energy
gap between commensurate and incommensurate configurations closes linearly
with the mass of the kink. (b) The probability of nucleating a kink for a
single trajectory as a function of $m_{K}^{c}-m_{K}^{\operatorname{fin}}$.

To see how enhanced quantum fluctuations modify the dynamics of the Frenkel-
Kontorova chain in the case above, we revisit the quench within the
commensurate phase $m_{K}^{\operatorname{init}}\to
m_{K}^{\operatorname{fin}}\gtrsim m_{K}^{c}>0.$ Classically, this quench keeps
the system within the commensurate phase since injecting a single kink would
require an additional energy cost $\Delta
E=|E\left(m_{K}^{c}\right)-E\left(m_{K}^{\operatorname{fin}}\right)|$, see
Fig. 9(a). Here, we plot the energy barrier the system needs to overcome to
enter the incommensurate phase as a function of $m_{K}$. The barrier $\Delta
E$ is evaluated as the energy difference between the chains with and without a
kink. Adding quantum fluctuations allows the system to tunnel through this
barrier and nucleate a kink at the boundary. The transition probability can be
calculated with the Wentzel-Kramers-Brillouin (WKB) approximation Berry and
Mount (1972), considering the propagation of a ‘particle’ along the $m_{K}$
‘direction’, tunneling through the barrier $\Delta E$. This yields the formula
$P=P_{0}\exp(-\alpha h_{\text{eff
}}^{-1}\int_{m_{K}^{c}}^{m_{K}^{\operatorname{fin}}}\sqrt{|E(m_{K}^{c})-E(m_{K})|}\mathrm{d}m_{K})=P_{0}\exp(-\alpha^{\prime}|m_{K}^{c}-m_{K}^{\operatorname{fin}}|^{3/2})$,
which we fit in good agreement with numerical results, as shown in Fig. 9(b).
This also captures that by performing a quench closer to the boundary between
the two phases $m_{K}^{c}$, more final trajectories will contain a kink as
they need to overcome smaller energy barriers.

The effective Planck constant $h_{\operatorname{eff}}$ determines the width of
the region in $m_{K}^{c}-m_{K}^{\operatorname{fin}}$, where a quantum
superposition takes place. Our numerical results show that
$\log(P/P_{0})\propto 1/\sqrt{h_{\operatorname{eff}}}$, and the effective
range of $m_{K}^{\operatorname{fin}}$, for which nucleation of kinks occurs,
is smaller than the one predicted via the WKB approximation, $\log P\propto
h_{\operatorname{eff}}^{-1}$. This resulting dependence reflects that $m_{K}$
controls not only the dynamics of the classical part of the phase but also the
coupling between the classical phase and quantum fluctuations, see Eq. (15).
In the quenches where kinks are injected (or nucleated) into the system, the
final mass is much smaller than the initial one, resulting in a suppression of
quantum effects. Altogether, this provokes a more complicated dependence of
$P$ on the effective Planck constant. For values of $m_{K}$ that are farther
from the critical ones, the final state remains commensurate, and quantum
noise instead provokes dephasing among trajectories at longer timescales
$\propto 1/h_{\operatorname{eff}}$. The overall result is that the width of a
single kink will be broadened, and the positions of particles will be more
challenging to resolve.

The regime most sensitive to quantum fluctuations is the one involving
breather excitations. Firstly, in the corresponding protocols, the final
$m_{K}^{\operatorname{fin}}$ is several times larger than the initial one
$m_{K}^{\operatorname{init}}$, thus strengthening the coupling between the
classical background $\phi^{(0)}$ and quantum fluctuations $\eta$ in Eq. (15).
Secondly, in the breather regime, $\phi^{(0)}$ oscillates, providing the
mechanism for amplifying the low-energy fluctuating modes discussed above.
Thus, the effects of quantum fluctuations become visible at relatively short
timescales. According to semi-classical approximation, Eq. (15) breaks down,
and different trajectories dephase. This dephasing provokes the decay of the
breather mode, making its detection more complicated.

Fig. 10 shows the evolution of a phase where, after the quench, a pair of
breathers is excited on top of the kink. For classical dynamics [see Fig.
10(a)], breathers are well-localized structures that can survive for
arbitrarily long times. In contrast, for semi-classical dynamics [Fig. 10(b)],
breathers are washed out after a few periods due to the dephasing effects. The
effective lifetime of the breather mode in the presence of quantum
fluctuations can be extended by working in a lower $m_{K}$ parameter regime
and suppressing the interplay between classical and quantum parts of
$\phi_{n}$ [cf. Eq. (15)]. However, this regime requires a larger system size,
as the size of a single kink scales as $l_{m}\propto 1/m_{K}.$

Figure 10: Evolution of the phase after the quench of $m_{K}$ when breather
mode is excited. (a) Dynamics evaluated classically. Two breathers are excited
and propagate from the center to the boundaries. (b) Semi-classical dynamics.
Due to the amplification of quantum noise on top of the oscillating
background, breathers are washed out. Here $h_{\operatorname{eff}}=0.07.$

Finally, we briefly comment on the importance of the quench protocol. If,
instead of the quench, we slowly vary $m_{K}$ – for instance, following an
adiabatic protocol – some of the results discussed so far will be altered. The
main difference is that as the system can be adiabatically transferred between
the ground states of the initial and final Hamiltonian, no oscillatory phases
can be excited. This results in the absence of breathers and optical modes,
thus, better system dynamics robustness against quantum fluctuations. However,
the system is out of its ground state manifold after driving the commensurate-
incommensurate transition, and oscillatory modes are excited, which allows
quantum effects to show up at later timescales.

As shown in this section, quenching $m_{K}$ can lead to various outcomes,
including the excitation of the optical mode, the injection of kinks, and even
the generation of breathers — distinct bounded states composed of kink and
anti-kink. Traditionally, breathers have been shown to emerge in kink and
anti-kink scattering processes, cf. Refs. Braun and Kivshar (2004); Carretero-
González  _et al._ (2022); Kivshar and Malomed (1989). Subsequently, in the
following section, we examine such processes within the framework of the FKM.
Additionally, we explore another intriguing phenomenon: the scattering of a
kink against a boundary. We demonstrate how the final result of this
particular process is contingent on the value of the misfit parameter
$\delta$.

##  IV Scattering processes in the Frenkel-Kontorova model

In the integrable sine-Gordon model, the scattering of solitons is elastic
Braun and Kivshar (2004); Carretero-González  _et al._ (2022); Kivshar and
Malomed (1989), while discreteness of the FKM breaks integrability and results
in the distinct outputs of solitons scatterings Gomes  _et al._ (2018); Gani
_et al._ (2019); Braun and Kivshar (2004); Kivshar and Malomed (1989); Malard
(2013); Braun and Kivshar (1998). In this section, we probe these effects of
integrability breaking on scattering processes. Firstly, we concentrate solely
on discreteness effects and study kink-anti-kink scattering, comparing it to
the analogous process of the sine-Gordon model. Then, we study additionally
the impact of the finite size effects by analyzing the scattering of a kink
against the boundary. We show results for zero and finite effective Planck
constants in both cases.

###  IV.1 Kink-anti-kink scattering

The sine-Gordon model can support solutions in the form of both moving and
immovable solitons, as well as bounded kink-anti-kink pairs, which are also
known as breathers Rajaraman (1987); Dauxois and Peyrard (2006); Braun and
Kivshar (2004). Therefore, if we take a kink and an anti-kink that are
initially separated by a distance $r$ and have initial velocities $\pm v$ [cf.
Fig. 11(a)], we can see that after the scattering, this configuration will
either evolve towards a breather or a kink-anti-kink configuration Ref.
Carretero-González  _et al._ (2022).

The outcome of the scattering can be predicted from energy conservation Gani
_et al._ (2019). Fig. 11(b) depicts the energy of the kink-anti-kink pair $E$
as a function of the distance $r$ between their centers. Here, $E_{\infty}$
denotes the energy of two infinitely separated solitons. The figure shows that
at small distances, $r\lesssim l_{m}+l_{\delta},$ solitons experience an
effective attractive potential. This suggests that if at $t=0$ the separation
between the solitons is smaller than the length of a single kink,
$r<l_{m}+l_{\delta}$, solitons must expend the energy difference,
$E_{\infty}-E_{r},$ to propagate further from the center. Therefore, if the
system’s kinetic energy is less than $E_{\infty}-E_{r},$ solitons remain bound
within the effective attractive potential and form a breather. The yellow
region in Fig. 11(c) indicates the initial conditions for which the system
evolves towards a breather state.

Conversely, if the kinetic energy exceeds $E_{\infty}-E_{r}$ [dark blue region
in Fig. 11(c)], the kink and anti-kink pass each other, with the system’s
final kinetic energy reduced by the value $E_{\infty}-E_{r}.$ If the initial
spacing between the kink and anti-kink is larger than the length of a single
kink, they scatter elastically. In this case, by moving closer to each other,
they first release potential energy $E_{\infty}-E_{r}$ and accelerate. After
passing each other, they pay back this energy difference from the kinetic
energy, and the solitons’ velocities return to their initial values.

Figure 11: Kink-anti-kink scattering. (a) Cartoon of the scattering process.
The initial separation between solitons $r$ is the distance between their
centers. The initial kink (anti-kink) velocity equals $v$ ($-v$). (b) The
effective attractive potential of the kink-anti-kink pair as a function of the
distance between their centers. (c) Sketch of the phase plot for kink-anti-
kink scattering when $m_{K}=0.5\ll 1$, effectively recovering the field theory
prediction. The yellow region represents the initial parameter values for
which the kink-anti-kink pair evolves into the breather after scattering, and
for parameters in the dark blue region, scattering is elastic. (d) The same as
in panel (c) but for $m_{K}=1.1$. The green region indicated velocities for
which kinetic energy is smaller than $E_{PN}$, and the light blue region shows
parameters for which solitons stop before the scattering.

In the discrete FKM, the picture described above remains qualitatively
unchanged for small $m_{K}$, where the Peierls-Nabarro barrier $E_{PN}$ [Eq.
(11)] is negligible, as shown in Fig. 2. However, for $m_{K}\gtrsim 0.8$, the
increase of $E_{PN}$ leads to several interesting modifications to the picture
in Fig. 11(c). The corresponding sketch of the phase plot is depicted in Fig.
11(d).

The first major difference is that if the system’s kinetic energy is less than
$E_{PN}$ [green region in Fig. 11(d)], solitons are unable to propagate
through the chain (cf. Sec. II.3). Such solitons remain confined at their
initial locations, and the kinetic energy of the solitons is transferred, with
the single possible channel, into oscillatory optical modes. The final state
is a localized kink-anti-kink pair with frozen positions of both solitons.

The second major difference occurs when solitons are widely separated at
$t=0$. In this case, solitons start propagating towards each other but
experience deceleration from the Peierls-Nabarro barrier (cf. Sec. II.3). The
kinetic energy the kink loses is proportional to the number of sites through
which it propagates. Consequently, there is a critical propagation distance
after which solitons radiate a major part of their kinetic energy to the
optical mode and cannot propagate further. The range of parameters $(r,v)$,
corresponding to the scenario when solitons lose all their kinetic energy
before they scatter and ultimately form a localized state, are shown via light
blue in Fig. 11(d).

Finally, note that the boundary of the breather phase in the discrete model
[yellow region in Fig. 11(d)] is determined from the condition
$E_{\text{kin}}-(E_{\infty}-E_{r})-E_{\operatorname{sc}}=0$, taking into
account the part of the kinetic energy $E_{\operatorname{sc}}$ that system
loses before the scattering due to the Peierls-Nabarro barrier. In the
continuum limit, the boundary is given simply by
$E_{\text{kin}}-(E_{\infty}-E_{r})=0$.

In finite-size discrete systems, an additional modification of the kink-anti-
kink scattering arises from the interaction between the solitons and the
excited optical modes. To start with, consider the propagation of one of the
solitons through the discrete lattice, namely $m_{K}>0.8$. After the particle
‘jumps’ from site $n_{0}$ to site $n_{0}-1$, this particle oscillates around
the potential minima. Due to the nearest neighbor interaction, this
oscillatory mode propagates through the chain with the speed of sound $c$.
Upon reaching the boundary site $n=1$, this ‘wave’ reflects and continues
propagating in the opposite direction. Consequently, after a time interval
$\Delta t_{\operatorname{opt}}=(n_{0}+N/2)/c$, these optical oscillations
reach the center of the chain where kink-anti-kink scattering takes place. If
the optical mode reaches the center of the chain faster than the corresponding
kink (or anti-kink) that released this optical mode, i.e., $\Delta
t_{\operatorname{opt}}<(N/2-n_{0})/|v|$, one must consider the scattering of
the kink, anti-kink, and the optical mode. Depending on the phase of the
oscillations in an optical mode at the center of the chain during the
scattering, the outcome can vary: either the optical mode provides additional
energy for the solitons to overcome the effective interaction and scatter into
unbounded pair, or the additional energy in the center of the chain is
insufficient, and the kink-anti-kink pair evolves towards a bounded state. As
a result, for small system sizes, the boundary between bounded and unbounded
outcomes of the kink-anti-kink scattering may exhibit non-monotonic behavior
with $r.$

It is also important to account for the effect of quantum fluctuations on
solitons scattering. The first of such effects is the appearance of the
critical region close to the separatrix between the bounded and unbounded
parts in Fig. 11(d). In the semi-classical limit, the initial conditions
$(r,v)$ are defined along with uncertainties in positions and momenta on the
order of $\sqrt{h_{\operatorname{eff}}}$ [cf. Eq. (14)]. Therefore, if the
initial conditions correspond to the critical regime on the phase plot, some
trajectories appear to evolve towards the breather state, while for the rest
of the trajectories, the kink-anti-kink pair scatters without forming a
localized state. As a result, at longer timescales, the averaged over
different noise realizations phase $\phi$ evolves towards a superposition of
unbounded kink-anti-kink states and breathers.

An additional manifestation of quantum fluctuations appears after the
scattering. This effect can be explained by the dephasing between different
realizations of the dynamics that stem from different initial conditions [cf.
Sec. III.2]. Within the bounded region in Fig. 11(d), such dephasing leads to
a finite lifetime for breathers. Here, at late times the phase of breathers in
each particular realization becomes significantly shifted, and after averaging
over all noise realizations, the phase $\phi_{n}$ is constant. Such effects
have also been reported in other systems, cf. Ref. Neuenhahn  _et al._ (2012).
Within the unbounded region, the dephasing and variance in the initial
velocities of soliton between different realizations of scattering can result
in the broadening of (anti-)kink at late times.

###  IV.2  Scattering of the kink against the boundary

Figure 12: (a) Initial state of the system: We prepare the initial
configuration as a sum of the single kink with the velocity $v$ and the center
at position $x_{0}$ (in black) and the commensurate background phase with the
boundary defect. The boundary defect takes the form of the tail of kink
($\delta>0,$ yellow) or anti-kink ($\delta<0$, blue). (b) The dynamical
response of the system to the scattering of the kink against the boundary
defect as a function of kink velocity $v$ and misfit parameter $\delta$. Here,
the final state of the system is monitored immediately after the scattering.
Figure 13: Scattering of a kink against the boundary. (a) Sketch of the
effective interaction between kink and boundary for positive (yellow) and
negative (dark blue) values of misfit parameter $\delta$ experienced by the
kink in close proximity to the boundary. The instance of dynamics where kink
(b) repels from the boundary, (c) leaves the chain through the boundary, (d)
leaves the boundary and the anti-kink is injected instead, (e) leaves the
boundary, and two anti-kinks are injected.

The boundary conditions of the FKM result in a unique form of the phase
$\phi_{n}$ when $\delta\neq 0$. While each particle has two nearest neighbors
in bulk, the edge particles interact only with the one neighbor on the left or
right. Such imbalance in the interaction of edge particles [cf. Eqs. (2)]
results in the modification of the form of the $\phi_{n}$ close to the
boundaries in such a way that it becomes reminiscent of the tail of a single
kink [for $\delta>0$, cf. yellow lines in Fig. 12(a)] or anti-kink [for
$\delta<0$, cf. blue lines in Fig. 12(a)]. In this part of the paper, we
explore the scattering of a single kink against such unique boundary defects.
This process is important because in the finite-size chain, moving solitons
reach boundaries at finite time $t~{}\propto N/c$ and, depending on the
$\delta$, their fate therein can vary.

To concentrate directly on the kink-boundary scattering and neglecting other
kink-(anti-)kink interactions (which induce hybridization of the acoustic
mode), we prepare the initial state as a combination of the commensurate
background and moving kink, $\phi_{n}=\phi_{C}+\phi_{\operatorname{kink}}$.
Here, $\phi_{C}$ represents the commensurate background phase with the
boundary defect. Precisely, $\phi_{C}$ is a ground state solution of Eq. (2)
at a given value of $|\delta|<\delta_{c}$. On the other hand,
$\phi_{\operatorname{kink}}$ represents a kink which we place at a specific
position $x_{0}$ and give it a velocity of $v$ directed towards the left
boundary as shown in Fig 12(a). This setup allows us to study the behavior of
the kink as it moves towards the boundary and scatters against the boundary
defect at different values of $\delta$.

We fix $m_{K}=0.5.$ On one hand, at such amplitude of the lattice potential,
the Peierls Nabarro barrier $E_{PN}$ is negligible [cf. Fig. 2(c)], and
solitons can freely propagate through the lattice without slowing down. In
this way, the velocity of the kink before the scattering is equal to the
initial value of $v$. On the other hand, $m_{K}=0.5$ is large enough to make
the effect of quantum fluctuation pronounced. At this $m_{K}$ the effective
Plank constant $h_{\operatorname{eff}}$ can be varied from
$h_{\operatorname{eff}}\to 0$ to some fixed value, targeting classical and
semi-classical regimes of the model.

For $|\delta|<\delta_{c}$, there are four possible outcomes of the scattering,
depending on the initial velocity of the kink $v$ and the value of the misfit
parameter $\delta$. These outcomes are illustrated in Fig. 12(b) and include

  1. 1.

Repulsion: The kink repels from the boundary and reverses its direction of
propagation.

  2. 2.

Trapping: The kink leaves the chain through the boundary, and optical modes
are excited.

  3. 3.

Reflection with acceleration: The kink exits the system through the boundary
while an anti-kink is injected.

  4. 4.

Pumping (of two anti-kinks): Several anti-kinks are injected into the chain
after the kink leaves the system.

Below, we discuss each scenario separately, starting from the effects for
$\delta\lesssim\delta_{c}$ and decreasing the misfit parameter till
$\delta\gtrsim-\delta_{c}.$

####  IV.2.1 Repulsion

For $\delta>0$, the displacement of the phase on the boundary takes the form
of the tail of the kink [cf. Fig. 12(a)]. The yellow line in Fig. 13(a) shows
the relationship between the system’s potential energy and the kink’s
proximity to this kink-like boundary. When the kink approaches the boundary,
it encounters an effective repulsive potential. If the initial velocity of a
kink is low and the kinetic energy is smaller than the amplitude of this
potential [$E^{1}_{\operatorname{kin}}$, blue dashed line in Fig. 13(c)], the
kink cannot pass the boundary. In this case, as the kink approaches the edge,
it slows down, comes to a halt, and ultimately reverses its direction of
propagation, see Fig. 13(b). We indicate the initial parameters for which the
“Repulsion” scenario takes place via yellow color in Fig. 12(b).

Note that injection of kinks and excitation of the acoustic mode in Sec.
III.1.2 at late times $t>N/c$ undergoes the scenario “Repulsion”. Here, the
injection of anti-kink or ejection of kink requires additional energy cost,
which preserves the total number of solitons.

####  IV.2.2 Trapping

If the kinetic energy of the kink is slightly higher than the effective
repulsion potential [$E^{2}_{\operatorname{kin}}$, the light blue line in Fig.
13(c)], the kink crosses the boundary and disappears. The released kinetic and
potential energy of the kink is transferred into the oscillatory modes, see
dynamics in Fig. 13(b). The initial parameters corresponding to this regime
are shown in green in Fig. 12(b). In this regime, quantum fluctuations may be
significantly enhanced, potentially provoking dephasing effects due to the
excited optical modes (cf. discussion in Sec. III.2).

The effective repulsion is proportional to the $\phi^{2}\propto\delta^{2},$
and the initial kinetic energy of the kink is proportional to the $v^{2}.$
Thus, the critical velocity, for which the boundary repulses the kink, scales
linearly with the misfit parameter, which agrees with the numerical results
[see the linear boundary between the “Trapping” and “Repulsion” regions in
Fig. 12(b)].

####  IV.2.3 Reflection with acceleration

When the misfit parameter is negative, $\delta<0$, the phase $\phi_{n}$ on the
boundary takes the form of the tail of an anti-kink, see Fig. 12(a). The
potential energy of the system, in this case, decreases with the distance
between the kink and the boundary [see dark blue line in Fig. 13(a)]. Here,
kink experiences the effective attractive potential as it comes closer to the
boundary. Thus, as the kink approaches the boundary, it accelerates and exits
from the chain through the edge. The energy released in this process is enough
to draw another soliton into the system. Since $\delta<0$, a new anti-kink is
injected into the chain. After the anti-kink enters the system, it experiences
effective repulsion from the anti-kink-like boundary and accelerates toward
the bulk. Overall, in this scattering process, the kink is replaced with an
anti-kink, the velocity of which is higher than the initial velocity of the
kink, as shown in Fig. 13(d). The kinetic energy increase comes from the
effective attraction between the initial kink and anti-kink-like boundary.
Note that as the effective interaction between the boundary and the kink
scales like $\delta^{2}$, the higher the absolute value of the misfit
parameter, the greater the velocity of the injected anti-kink. The initial
parameters for which the “Reflection” scenario takes place are indicated in
blue in Fig. 12(b).

####  IV.2.4 Pumping

When the value of the misfit parameter is negative and close to the critical
value $|\delta|\lesssim|\delta_{c}|$, two anti-kinks can enter the system
similarly. The speed of the first anti-kink is equal to the speed of sound,
and the remaining energy determines the kinetic energy of the second anti-
kink. The dynamics of the system in this case are illustrated in Fig. 13(e).
The initial parameters for this regime are indicated in dark blue in Fig.
12(b).

Given the finite strength of quantum fluctuations, major modifications to the
picture discussed so far occur at critical regions in Fig. 12. Here, for
parameters that correspond to the boundary between two distinct dynamical
responses, the outcome of the scattering process can be the quantum
superposition of these responses.

##  V Experimental considerations

In this section, we address the experimental implementation of the FKM in the
trapped ion simulator and discuss the limitations affecting the results
discussed thus far.

The trapped ion setup comprises $N$ ions confined within a trap and subjected
to the lattice potential, see Fig. 14. The ions’ motion within the plane is
effectively frozen due to the presence of a strong trapping potential.
Simultaneously, the particles are constrained along the $x$ axis by a weaker
potential. Thus, the kinetic energy of ions is equal to
$E_{k}={M}\sum_{n}\dot{x}_{n}^{2}/2,$ where $M$ is the mass of a single ion,
and by $x_{n}$ and $\dot{x}_{n}$ we denote the position and velocity of $n$-th
ion. The inter-particle spacing, denoted as $a_{0}$, characterizes the average
distance between adjacent particles in the absence of the lattice potential.

The mutual Coulomb repulsion can be approximated by harmonic interaction
$V_{\operatorname{int}}=\sum_{n}{K}\left({x_{n+1}-x_{n}}-a_{0}\right)^{2}/2,$
with the spring constant $K=2q^{2}/\left(4\pi\varepsilon_{0}a_{0}^{3}\right),$
cf. Landa  _et al._ (2020); Braun and Kivshar (2004). Here $q$ is the
electron’s charge, and $\varepsilon_{0}$ is the vacuum’s permittivity. It is
important to note that the harmonic approximation is valid only when the
distance between ions, $x_{n+1}-x_{n}$, is much larger than the local
displacement of ions, $a_{s}\psi_{n}/(2\pi)=x_{n}-na_{s}$. In a typical
situation, higher-order corrections are suppressed by about four orders of
magnitude Marquet  _et al._ (2003).

Figure 14: Sketch of the setup. Two counter-propagating laser beams (red) near
$\lambda=397$nm form lattice potential with the periodicity
$a_{s}=\lambda/(2\cos\gamma)$; $\gamma$ is the angle formed between the laser
beams and the $x$-axis. Ions are localized in the $yz$ plane due to the strong
trapping potential (yellow), while the weaker harmonic confinement along the
$x$ (green) direction and the mutual inter-ion Coulomb repulsion dictate the
positions in the ion crystal.

Furthermore, ions are subjected to the periodic potential
$V_{\operatorname{sub}}={\epsilon}/{2}\sum_{n}[1-\cos(2\pi{x_{n}}/{a_{s}})],$
which is formed by the counter-propagating laser beams. Here $\epsilon$ is the
lattice potential depth, and $a_{s}$ is its period. The periodicity $a_{s}$
can be tuned by changing the relative angle $\gamma$ between the ion axes and
the propagation direction of the laser beams, $a_{s}=\lambda/(2\cos\gamma).$
Here $\lambda$ is the wavelength of the laser, cf. Fig. 14.

The total Hamiltonian reads
$H=E_{k}+V_{\operatorname{int}}+V_{\operatorname{sub}}$. By introducing a
dimensionless $m_{K}^{2}={(2\pi)^{2}\epsilon}/{(2Ka_{s}^{2})}$ and
$\delta={a_{0}}/{a_{s}}-1$, the dynamics of the displacement $\theta_{n}$ of
$n$-th ion from the $n$-th minima of the lattice potential are described by
the following Hamiltonian

$\displaystyle\tilde{H}=$ $\displaystyle\frac{(2\pi)^{2}H}{Ka_{s}^{2}}=$ (16)
$\displaystyle=$
$\displaystyle\sum_{n}\frac{1}{2}\dot{\theta}_{n}^{2}+m_{K}^{2}\left(1+\cos\theta_{n}\right)+$
$\displaystyle+$
$\displaystyle\sum_{r>0}\frac{1}{2|r|^{m+2}}\left((\theta_{n+r}-\theta_{n})-2\pi
r\delta\right)^{2}.$

where $\theta_{n}=\psi_{n}+\pi$ and $\tau=t\sqrt{K/M}$ is a dimensionless
time. In previous experiments von Lindenfels  _et al._ (2019); Linnet  _et
al._ (2012); Gangloff  _et al._ (2020); Cetina  _et al._ (2013), the inter-
particle spacing was much larger than the lattice spacing, $a_{0}/a_{s}\gg 1$,
such that ions are separated by a few dozens of empty lattice sites. To
exclude artificial kinks corresponding to these empty sites, we can redefine
variables in the following way $\phi_{n}\to\theta_{n}-2\pi
n\lfloor\delta\rfloor,$ and $\delta\to\\{\delta\\}$, where
$\lfloor\delta\rfloor$ and $\\{\delta\\}$ denote integer and non-integer part
of the misfit parameter respectively. Finally, by neglecting long-range
interactions for $r>1$, the final Hamiltonian takes the form of (1).

Figure 15: The dependence of the critical temperature at which thermal
fluctuations dominate over quantum ones as a function of the system size.

To include quantum mechanical effects into consideration, we consider quantum
FKM Hu and Li (2000), which can be obtained from (1) by replacing classical
variables with the quantum operators $x_{n}\to\hat{x}_{n}$ and
$p_{n}\to\hat{p}_{n},$ which satisfy canonical commutation relations

$\displaystyle\left[\hat{x}_{n},\hat{p}_{m}\right]$
$\displaystyle=i\hbar\delta_{nm},$ (17)
$\displaystyle\left[\hat{x}_{n},\hat{x}_{m}\right]$ $\displaystyle=0,$
$\displaystyle\left[\hat{p}_{n},\hat{p}_{m}\right]$ $\displaystyle=0.$

To identify the effective Planks constant, $h_{\operatorname{eff}}$, we
rewrite everything in terms of the dimensionless function $\theta$ (or
operator $\hat{\theta}$). The corresponding expressions for momentum in the
dimensionless form read

$\tilde{p}_{n}=\dfrac{d\theta_{n}}{d\tau},\quad\hat{\tilde{p}}_{n}=-i\dfrac{(2\pi)^{2}\hbar}{a_{s}^{2}\sqrt{KM}}\dfrac{d}{d\hat{\theta}_{n}}=-ih_{\operatorname{eff}}\dfrac{d}{d\hat{\theta}_{n}},$
(18)

which sets the expression for the effective Plank constant
$h_{\operatorname{eff}}={(2\pi)^{2}\hbar}/({a_{s}^{2}\sqrt{KM}}).$ Note that
the same expression can be obtained while bringing the action to the
dimensionless form, namely $\int
Hdt/\hbar=\int\tilde{H}d\tau/h_{\operatorname{eff}}$.

We substitute experimental parameters from Ref. von Lindenfels  _et al._
(2019), which are $a_{0}\in[5,15]~{}\mu$m, ${N}=40-100$ ions,
$\epsilon\in[0,2\pi\cdot 2.7]$ MHz, $a_{s}=\lambda/(2\cos\alpha)$ where
$\lambda=397$nm, $0<\alpha<22^{\circ};$ ${M}$ is the mass of 40Ca ions, and
$N\leq 100$. They correspond to the following range for the dimensionless
variables: $0<m_{K}\lesssim 1.1$, which allows us to target both ‘continuous’
and ‘discrete’ regimes from Sec. II.3. The misfit parameter can be varied in
the range $|\delta|<0.5$, and $h_{\operatorname{eff}}<10^{-2}$, where both
genuine classical or genuine quantum mechanical effects may be observed. The
parameter regimes accessible with the trapped ion implementation of the FKM
are sufficient for probing all regions in Fig. 3. Furthermore, by dynamically
changing the amplitude of the lattice potential, one can also potentially
target the experiment regimes where solitons can dynamically injected into the
chain or breathers are excited from Sec. III.

Note that to observe the quantum superposition of states with and without the
kink (see Sec. III.2), one needs to work at a low-temperature regime.
Precisely, the de Broglie wavelength
$\lambda_{dB}^{\operatorname{thermal}}=[(2\pi\hbar^{2})/(Mk_{B}T)]^{1/2}$ must
be smaller than the typical correlation length in the system, which is of the
order of $a_{s}/m_{K}.$ Here $k_{B}$ is the Boltzmann constant and $T$ is the
temperature. This estimation sets the upper bound of the temperature
$T<10^{-9}-10^{-7}K$. The critical temperature scales with the system size,
see Fig. 15. Quantum superposition of chains with many particles requires an
initialization of the system at a very low temperature. The detailed
calculations are given in Appendix A.

Finally, in typical experiments, the spacing between ions $a_{0}$ is set by
competition between the Coulomb repulsion and the external (harmonic) trapping
potential [depicted in green in Fig. 14]. Consequently, the spacing between
ions is denser at the center and wider at the borders, effectively reducing
part of the crystal for which the C-IC transition can be observed (cf. Refs
Morigi and Fishman (2004); Dubin (1997); Kamsap  _et al._ (2017)). Adjusting
the setup’s geometry (the periodicity $a_{s}$, etc.) makes it possible to
separate a region with a few dozen ions where the C-IC region can be observed.
At the same time, the setup with a harmonic confining potential in $x$
direction will be unsuitable for studying a kink’s scattering against the
boundary: The spatial dependence of the inter-particle distance near the
boundary would alter the results and induce a more complicated dependence of
the kink velocity with the distance to the boundary. Alternatively, one might
add optical tweezer potentials to create steep borders Mazzanti  _et al._
(2021); Bond  _et al._ (2022), which would effectively result in the model
studied in the body of the paper. In this way, the harmonic potential would be
fully mitigated, and the number of ions for which the C-IC transition takes
place would largely increase.

##  VI Conclusions and outlook

The FKM, compared to other models that can host solitons, contains a ‘chemical
potential’ term $\delta$, which enables the injection of kinks directly from
the boundary. Without this term, the creation of solitons can be very
challenging Wybo  _et al._ (2023). On the other hand, utilizing the chemical
potential enables a straightforward path toward dynamical manipulation of
topological defects.

The injection of topological defects into the system changes the spectrum of
excitations and opens novel avenues toward the dynamical control of materials.
The propagation of kinks can be used, for instance, to create controllable
transport to reduce nano-friction when considering driven versions of the
model studied here. For instance, it has been recently proposed to use
topological defects in ion crystals for enhancing heat transport Timm  _et
al._ (2023). The versatility of implementing this model in various platforms,
such as cavity QED with the Rydberg atoms Muñoz  _et al._ (2020), cold atoms
Kasper  _et al._ (2020); Wybo  _et al._ (2023, 2022) suggest a rich array of
parameters’ regimes that can be targeted in experiments. One can also explore
different geometries utilizing extra spatial dimensions, and study, for
instance, zigzag transition in trapped ion simulators Shimshoni  _et al._
(2011b, a); Landa  _et al._ (2010); Mielenz  _et al._ (2013); Wybo  _et al._
(2023, 2022).

The trapped ion realization of the FKM can also be viewed as a versatile
platform for modeling the physics of false vacuum decay Coleman (1977).
Starting in the commensurate phase, there is a range of post-quench values of
$m_{K}^{\operatorname{fin}}\in(m_{K}^{cd},m_{K}^{c}]$ for which the system
occupies meta-stable commensurate state, while in equilibrium these masses
$m_{K}$ correspond to stable incommensurate phase. In this respect, after the
quench, the system remains in the false vacuum (commensurate) state, while
quantum fluctuations can make it decay to the stable (incommensurate) state
Lagnese  _et al._ (2021, 2023). Utilizing Ramsay interferometry Baltrusch  _et
al._ (2012, 2011) to measure this metastable state offers a tunable platform
for simulating the impact on dynamics of false vacuum decay, which can have
applications in high-energy physics and cosmology.

## Acknowledgments

We thank R.J. Valencia-Tortora, V. Vuletić, and A. Ziolkowska for useful
discussions. J. M. acknowledges E. Demler and V. Kasper for previous
collaborations on related topics. This project has been supported by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID
429529648 – TRR 306 QuCoLiMa (“Quantum Cooperativity of Light and Matter”),
and by the Dynamics and Topology Centre funded by the State of Rhineland
Palatinate and Topology Centre funded by the State of Rhineland Palatinate.
The authors gratefully acknowledge the computing time granted on the
supercomputer MOGON 2 at Johannes Gutenberg-University Mainz (hpc.uni-
mainz.de).

##  Appendix A Estimation of a critical temperature via coherence resource
theory

In this Appendix, we use the relative entropy of coherence to determine the
temperature range at which quantum fluctuations dominate over thermal ones,
and a quantum superposition of state with and without a kink can be
dynamically generated.

The relative entropy of coherence is a quantifier for the resource theory of
coherence which encodes the amount of superposition a given state, $\rho,$ has
in a certain basis Streltsov  _et al._ (2017); Kelly  _et al._ (2022); Huang
(2009). If the basis has projectors $P_{d}=\ket{d}\bra{d}$, then the relative
entropy of coherence is written as

$C(\rho)=S\left(\sum_{d}P_{d}\rho P_{d}\right)-S\left(\rho\right),$ (19)

where $S(\rho)$ is the von Neumann entropy of the state $\rho$. Importantly,
the state $\sum_{d}P_{d}\rho P_{d}$ has only diagonal entries with weights
equal to the probability $P(d)=\bra{d}\rho\ket{d}$ of a measurement in the
basis giving the outcomes $d$. Thus, we find that $S\left(\sum_{d}P_{d}\rho
P_{d}\right)$ is equal to the Shannon entropy, $H(P(d))$, of the distribution
$P(d)$ Bromiley  _et al._ (2004). The relative entropy of coherence is simply
the difference between the entropy of the classical distribution $P(d)$ and
that of the quantum state $\rho$. If the quantum state is pure, we have
$S(\rho)=0$, and all entropy in $P(d)$ is due to superposition such that
$C(\rho)=H(P(d))$; on the other hand if the state is mixed, some of the
entropy in $P(d)$ is simply due to the mixed nature of the state $\rho$, and
$C(\rho)<H(P(d))$.

In Sec. III.2, we demonstrate a way of generating uncertainty in the number of
solitons, $Q$, present in the system. Since $Q=(\phi_{N}-\phi_{1})/(2\pi)$,
then we should investigate the coherence in a basis such that the observables
$\hat{\phi}_{i}$ are diagonal:
$\hat{\phi}_{j}\ket{\\{\phi_{i}\\}}=\phi_{j}\ket{\\{\phi_{i}\\}}$. After the
system evolves into the statistical mixture of states with and without a
soliton, it will have some distribution $P(\\{\phi_{i}\\})$, for which we will
want to identify how much entropy is due to superposition or a classical
mixture of states: $C(\rho)=H(P(\\{\phi_{i}\\}))-S(\rho)$.

While the Shannon entropy, $H\left(P\left(\\{\phi_{i}\\}\right)\right)$, is
inaccessible in TWA calculations due to a sampling complexity problem, we can
provide a lower bound on it using a data processing inequality Cover and
Thomas (2012) applied to a statistical classifier,
$f\left(\\{\phi_{i}\\}\right)\in(0,1)$ which identifies if the state has zero
or one soliton. The data processing inequality states
$H\left(P\left(\\{\phi_{i}\\}\right)\right)\geq
H\left(P\left(f\left(\\{\phi_{i}\\}\right)\right)\right)\equiv
H\left(P\left(f\right)\right)$ such that

$C(\rho)\geq C_{f}\equiv H\left(P\left(f\right)\right)-S(\rho).$ (20)

When $C_{f}=1$, then the state is an equal superposition of a chain with and
without a soliton, and can only occur when $S(\rho)=0$ (the initial state of
the system is pure and the evolution over time is unitary). While $C_{f}$
drops below zero when initial state thermal fluctuations dominate and
$S(\rho)>1.$

We compute $H\left(P\left(f\right)\right)$ from $P(f=1)=1-P(f=0)$ obtained by
counting the fraction of TWA trajectories with and without a soliton. To
compute $S(\rho)$, we assume the initial state is in a thermal Gibbs state
$\rho_{0}\sim e^{-\beta H}$ and undergoes strictly unitary evolution
$\rho=U_{t}\rho_{0}U^{\dagger}_{t}$ where $U_{t}$ is the unitary generated by
the Hamiltonian in Eq. (1). The entropy of the state $\rho$ is then equivalent
to the thermal entropy and can be evaluated as
$S=-\sum_{q}\rho_{q}\log\rho_{q}$ Huang (2009), where $\rho_{q}={e^{-\beta
H_{q}}}/{Z_{q}}$ is the reduced density matrix for mode $q$, $\beta$ is the
inverse temperature and $Z_{q}$ is a normalization factor. For each mode $q$
(9) we have

$\displaystyle Z_{q}=$ $\displaystyle\operatorname{tr}\exp(-\beta H_{q})=$
(21) $\displaystyle=$ $\displaystyle\operatorname{tr}\exp\left(-\beta
h_{\operatorname{eff}}\omega_{q}\left(n_{q}+\frac{1}{2}\right)\right)=$
$\displaystyle=$ $\displaystyle\frac{e^{\frac{-\beta
h_{\operatorname{eff}}\omega_{q}}{2}}}{1-e^{-\beta
h_{\operatorname{eff}}\omega_{q}}},$

and after evaluating $\rho_{q,n}=e^{-\beta
h_{\operatorname{eff}}\omega_{q}n_{q}}\left(1-e^{-\beta
h_{\operatorname{eff}}\omega_{q}}\right)$ the thermal entropy reads

$S(\rho)=\dfrac{h_{\operatorname{eff}}\omega_{q}}{k_{B}T}\langle
n_{q}\rangle-\log\left(1-e^{-\beta h_{\operatorname{eff}}\omega_{q}}\right)$
(22)

with $\langle n_{q}\rangle=(e^{\beta
h_{\operatorname{eff}}\omega_{q}}-1)^{-1}.$

Let us consider a state that is the symmetric superposition of the
commensurate and incommensurate chains, and thus $H(P(f))\approx 1$. For small
temperatures, the impact of the thermal entropy is negligible, and the
statistical uncertainty is solely due to superposition. Above a certain
temperature scale $T_{c}$, $C_{f}$ drops below zero, indicating the state is
dominated by thermal fluctuations. For $N=10$ ions, the critical temperature
is equal to $T_{c}=0.1\mu$K. Fig. 15 illustrates the scaling of critical
temperature with the system size. Thermal entropy grows linearly with the
system size; thus, at a fixed temperature, more quantum information is
preserved in a system of smaller lengths. The critical temperature can be
increased by enlarging the strength of quantum fluctuations $\propto
h_{\operatorname{eff}}$ at time $t=0$ or by enhancing a driving mass term for
quantum fluctuations $\eta_{q}$ by increasing $m_{K}$, see Eq. (9).

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