# Quantum nanofriction in trapped ion chains with a topological defect

L. Timm<EMAIL_ADDRESS>Institut für Theoretische Physik,
Leibniz Universität Hannover, Appelstr. 2, 30167 Hannover, Germany  L. A.
Rüffert  Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116
Braunschweig, Germany  H. Weimer  Institut für Theoretische Physik, Leibniz
Universität Hannover, Appelstr. 2, 30167 Hannover, Germany  L. Santos
Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstr. 2,
30167 Hannover, Germany  T. E. Mehlstäubler  Physikalisch-Technische
Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany  Institut für
Quantenoptik, Welfengarten 1, 30167 Hannover

(August 27, 2024)

###### Abstract

Trapped ion systems constitute a well controllable scenario for the study and
emulation of nanofriction, and in particular of Frenkel-Kontorova-like models.
This is in particular the case when a topological defect is created in a
zigzag ion Coulomb crystal, which results in an Aubry transition from free
sliding to pinned phase as a function of the trap aspect ratio. We explore the
quantum effects of the Aubry transition by means of an effective simplified
model, in which the defect is treated like a single quantum particle that
experiences an effective Peierls-Nabarro potential and a position-dependent
mass. We demonstrate the relevance of quantum tunneling in a finite range of
aspect ratios close the critical point, showing that the quantum effects may
be observed in the kink dynamics for sufficiently low temperatures. Finally,
we discuss the requirements to reveal quantum effects at the Aubry transition
in future experiments on trapped ions.

##  I Introduction

Friction is a relevant phenomenon for a wide range of macroscopic and
microscopic systems Vanossi  _et al._ (2013); Gao  _et al._ (2004); Kühner
_et al._ (2007). While its deceleration effect when two macroscopic objects
slide on top of each other is phenomenologically described by the laws of
Amontons and Coulomb Amontons (1699); Coulomb (1821), the physics of
nanofriction on an atomic scale is much more intriguing Mo  _et al._ (2009).
One of the simplest and best-known models on nanofriction is the Frenkel-
Kontorova (FK) model, in which a chain of particles, which interact through a
harmonic potential, is subjected to a sinusoidal corrugation potential
Kontorova and Frenkel (1938); Braun and Kivshar (2004). The model is hence
governed by the interplay between two length scales, the equilibrium distance
of the particles and the period of the corrugation potential. This interplay
leads to an intriguing physics, relevant to disparate fields ranging from
solid-state physics Dienwiebel  _et al._ (2004); Brazda  _et al._ (2018);
Bohlein  _et al._ (2012) to biophysics Englander  _et al._ (1980); Bormuth
_et al._ (2009) and nonlinear physics Chaikin and Lubensky (1995); Lepri
(2003).

Interestingly, if the ratio of the two lengths is incommensurate there exists
a transition, the so-called Aubry transition, from a free sliding phase for
small potential amplitudes to a pinning phase for strong potentials Aubry and
Daeron (1983). For weak corrugation potentials the classical minimal energy
configuration has constant distances given by the inter-particle interaction.
For incommensurate ratios this has the consequence that particles with
arbitrary positions relative to the potential can be found. As a consequence
the chain of particles may freely slide over the corrugation Sacco and
Sokoloff (1978). In contrast, when the potential amplitude is increased above
a critical value the atoms slide towards the minima of the potential getting
pinned by the substrate Sharma  _et al._ (1984); Braiman  _et al._ (1990). The
picture changes when the FK model is treated quantum mechanically, since
particles may travel into a neighboring potential well via tunneling effects.
The behavior of the FK model in the quantum regime has been studied in
numerous works, employing Monte Carlo methods Borgonovi  _et al._ (1989); Hu
and Li (2000); Zhirov  _et al._ (2003) and density matrix renormalization
group calculations Ma  _et al._ (2014a, b). Whereas the classical pinning
phase is characterized by a non-analytic step-wise Hull function due to the
presence of Peierls-Nabarro barriers Bak (1982), quantum tunneling result in
the softening of the Hull function Borgonovi  _et al._ (1990); Berman  _et
al._ (1994, 1997).

Following first proposals García-Mata  _et al._ (2007); Benassi  _et al._
(2011), recent experiments have demonstrated the ability to study nanofriction
in trapped-ion setups. Subsequently, two main approaches to study Aubry
physics have been realized. The first one builds on a one-dimensional chain of
ions confined in a stiff Paul trap Bylinskii  _et al._ (2015, 2016). They are
exposed to a nonlinear substrate potential via the dipole forces of a standing
wave laser. Nanofriction has then been characterized by the forces necessary
to dislocate the particles, revealing the absence of a friction force in the
sliding regime. A different setup emulates Aubry physics in a self-organized
system, that is the two sub-chains of an ion crystal under the presence of a
topological defect, which provides the necessary incommensurability between
the length scales of the subchains Kiethe  _et al._ (2017, 2018). Experiments
on this approach have demonstrated the softening of a motional mode at the
transition point by measuring its resonance frequency. Recent work has
explored the relevance of quantum effects in trapped ion experiments on Aubry
physics Bonetti  _et al._ (2021). It explicitly revealed the splitting of ion
wavefunctions at the Aubry transition when the crystal is cooled close to the
ground state and investigated the effect of incommensurability in the system.

In this paper we provide an alternative approach for the study of quantum
effects in the FK model. In our approach, we derive close to the Aubry
transition an effective collective description, that permits a considerably
simplified analysis of quantum effects.

This approach is generally applicable to FK systems. As an example we focus in
this paper on a trapped ion realizations with a topological defect. We show
that quantum effects are important at the Aubry transition, and discuss how
the quantum regime may be enlarged in experiments. We discuss as well the case
of finite temperature, discussing the requirements for the observation of
quantum effects.

This paper is structured as follows. In Sec. II we briefly review the Aubry
transition in two dimensional ion crystals. In Sec. III we derive the
effective single-particle model for the motion of a topological defect in the
atomic chain. In Sec. IV we quantize the kink motion, and analyze the
characteristics of the eigenstates of the defect. Section V is devoted to
signatures of quantum effects across the Aubry transition and the requirements
for their observation. In Sec. VI we discuss the experimental strategies to
reveal the quantum effects of the defect. Finally, in Sec. VII we summarize
our results.

##  II Aubry transition

In this section, we briefly introduce the Aubry transition in a trapped ion
crystal with a topological defect. We consider $N$ ions in a harmonic trap.
The overall potential experienced by the ions is composed by the trap
potential and the Coulomb interaction

$V(\\{\vec{r}_{i}=(x_{i},z_{i})\\})=\sum_{i}^{N}\left(z_{i}^{2}+\alpha^{2}x_{i}^{2}\right)+\sum_{j\neq
i}\frac{1}{|\vec{r}_{i}-\vec{r}_{j}|},$ (1)

where $(x_{i},z_{i})$ is the position of the $i$–th ion. Hamiltonian (1) is
written in a dimensionless form, employing as energy unit $u=m\omega_{z}l^{2}$
and length unit $l=\left(C/m\omega_{z}^{2}\right)^{1/3}$, where $m$ is the
mass of the ions and $C=e^{2}/4\pi\epsilon_{0}$ is the Coulomb potential
strength with the elementary charge $e$ and the vacuum permittivity
$\epsilon_{0}$. The trap is characterized by the secular frequency of the trap
in the axial ($z$) direction, $\omega_{z}$ and by the aspect ratio between
transversal and axial frequencies $\alpha=\omega_{x}/\omega_{z}>1$. In this
following we fix $\omega_{z}=2\pi\cdot$150\text{\,}\mathrm{kHz}$$.

For a vanishing temperature the ions crystallize in their equilibrium
positions $\vec{r}_{i}^{0}$, which in turn depend on $N$ and $\alpha$. For a
strong transverse confinement the ions are placed in a single chain, i. e.
$x_{i}^{0}=0$, with a non-uniform spacing along $z$ due to the harmonic
confinement. For a decreasing $\alpha$ the crystal undergoes a structural
linear-to-zigzag phase transition at a critical value $\alpha_{\text{ZZ}}$.
Due to the inhomogeneous spacing, the transition occurs first at the trap
center in the region of maximal charge density. This configuration is twofold
degenerate (zigzag/zagzig) due to the mirror symmetry $x\leftrightarrow-x$ of
the potential.

When $\alpha$ is tuned below a value $\alpha_{K}$ the zigzag chain can host a
kink defect, i.e. a domain wall between a zigzag and a zagzig region,
connecting them in a continuous way (see Fig. 1). Crucially, due to the
presence of the defect, the axial distance in the upper and lower subchains
differ. As a result, the kink emulates an incommensurate interface of two
particle chains, whose influence on one another is determined by the Coulomb
repulsion Kiethe  _et al._ (2018). For low values of $\alpha$ the defect is
captured in the trap center due to its repulsion from the crystal boundaries
Partner  _et al._ (2013). When $\alpha$ is increased the soliton undergoes a
Aubry-type transition at $\alpha_{\text{A}}$, because upper particles are
pinned by the periodic Coulomb potential of the lower ones and vice versa. The
Aubry transition results in spontaneous breaking of the mirror symmetry as the
kink equilibrium position shifts away from the trap center Timm  _et al._
(2020).

##  III Effective Potential

Figure 1: (a) Center section of an ion crystal with a topological defect
(blue) and in a defect-free zigzag alignment (grey) for
$\alpha-\alpha_{\text{A}}=-0.014$. The points are connected to guide the eye.
(b) $\phi^{2}=(\delta-\delta^{zz})^{2}$ (see text) for the same choice of
parameters.

In the following, we derive an effective model to describe the kink dynamics.
The presence of the kink results in the deviation of the axial ion distances
from the defect-free case (see Fig. 1). We therefore define the position
variable of the kink $K$ as

$K(\\{\vec{r}_{i}\\})=\frac{1}{X_{0}}\sum_{j}^{N-1}\bar{z}_{j}(\delta_{j}-\delta_{j}^{zz})^{2}.$
(2)

Here $\delta_{j}=z_{j+1}-z_{j}$ is the $z$-distance between neighboring ions
($\delta_{j}>0,\leavevmode\nobreak\ \forall j$, as we assume the ions are
labelled in $z$-ordering), $\delta_{j}^{zz}$ is the respective distance with
no kink present, $\bar{z}_{j}=(z_{j+1}+z_{j})/2$, and
$X_{0}=\sum_{j}(\delta_{j}-\delta_{j}^{zz})^{2}$ is a normalization factor.
The kink position is hence the average axial ion position weighted with the
deviation from the regular zigzag configuration. An example for these
deviations is depicted in Fig. 1 (b).

The dynamics of the kink position is determined by the Peierls-Nabarro (PN)
potential Peierls (1940); Nabarro (1947). To map out the potential landscape
we minimize the potential energy of the system under the constraint
$K(\\{\vec{r}_{i}\\})=X$, for different positions $X$. The minimization yields
ion positions $\vec{r}_{i,C}(X)$ that determine a path in phase space Partner
_et al._ (2013). The PN potential of the solitonic defect is then defined as
the potential energy along that path

$U_{\text{PN}}(X)=V(\\{\vec{r}_{i,C}(X)\\})$ (3)

The PN potential is shaped by two major effects. On one hand, in the
considered parameter regime the defect is repelled by crystal boundaries,
where the transversal distance of the ions decreases. As a result, the
potential is globally confining such that the kink remains trapped in the
central region of the system. On the other hand, the repulsion of the two ion
subchains is responsible for a sinusoidal modulation of the potential due to
the periodic array of the ions in the crystal, see figure 2. For
$\alpha<\alpha_{\text{A}}$ the transverse ion distances are large such that
the repulsion of the subchains is dominated by the repulsion of the
boundaries. Hence, the PN potential of the soliton is practically harmonic
with a global minimum at $X_{0}=0$, which is the classical equilibrium of the
kink, see figure 3 (a). At $\alpha_{\text{A}}$ however, the subchains are
forced so close together such that the increased Coulomb repulsion causes the
PN potential to develop a periodic modulation which leads to several local
minima (see Fig. 2(b)). In particular, in the system center the potential
resembles a double well, as a local maximum emerges at $X=0$ while two local
minima form at $\pm X_{0}$. Classically, the symmetry of the crystal is
spontaneously broken as an infinitesimal perturbation leads to a new
equilibrium position $X_{0}\neq 0$ of the soliton. As we will see in section
IV this is not the case for the quantized kink defect. Increasing $\alpha$
further away from $\alpha_{\text{A}}$ leads to an increase of $X_{0}$ as well
as a quadratic increase of the potential barrier between the two degenerate
local minima $X_{0}$ and $-X_{0}$, see Fig. 3 (c).

Figure 2: (a) Central part of an ion crystal with a kink, composed of the
upper (light blue) and lower (dark blue) subchain, and an equipotential line
of the corrugation potential of the lower sub-chain felt by the upper ions.
(b) Effective PN potential of the kink, calculated as derived in section III,
for $\alpha-\alpha_{\text{A}}=0.191$. The overall confining potential exhibits
periodic modulations due to the influence of the two sub-chains on each other
(see (a)). Close to the critical point only the central two local minima are
of relevance. Figure 3: PN potential (black) as described in section III and
wavefunctions of the ground state (red) and the first excited state (blue),
calculated from the model (5), for $\alpha-\alpha_{\text{A}}=-0.014$ (a),
$0.011$ (b) and $0.031$ (c). The graphs of the wave functions have a vertical
offset for better visibility. The dashed lines indicate the energy of the
shown states, in (b) and (c) the states are quasidegenerate.

The effective mass of the kink along the minimization path in phase space is
given by

$M_{\text{eff}}(X)=m\sum_{i}\left(\frac{d\vec{r}_{i,C}(X)}{dX}\right)^{2}$ (4)

and in general is position dependent. Note that the description of the kink
dynamics by means of the PN potential and the effective mass only considers
configurations which lie on the path in phase space given by
$\vec{r}_{i,C}(X)$, and therefore assumes a good isolation from other degrees
of freedom of the system. A full description of the dynamics of the defect in
the environment of the crystal would demand the inclusion of excitations
orthogonal to the phase space path, which can be done in the frame of a kink
dressing approach Willis  _et al._ (1986); Boesch  _et al._ (1988). However,
molecular dynamics simulations have shown that for low enough temperatures the
kink motion couples close to the Aubry transition only weakly to the
additional modes of the crystal due to the low frequency of the kink motion
Partner  _et al._ (2013); Landa  _et al._ (2010). Hence, we assume in the
following that the kink dynamics is fully described by the PN potential.

##  IV Quantum kink soliton

In this section we compute the classical PN potential $U_{\text{PN}}(X)$ and
its associated $M_{\text{eff}}(X)$ in the way described in section III and
subsequently quantize the kink defect in the PN potential. We write down the
Hamiltonian of the kink as

$\hat{H}_{s}=\hat{P}\frac{1}{2M_{\text{eff}}(\hat{X})}\hat{P}+U(\hat{X}).$ (5)

Note that $M=M_{\text{eff}}(\hat{X})$ does not commute with the momentum
operator $\hat{P}$. We determine the eigenstate wavefunctions $\varphi_{i}(X)$
and the eigenenergies of the defect by exactly diagonalizing the Hamiltonian
(5). We are particularly interested in possible tunneling effects close to the
Aubry transition and the energy scale which is necessary to resolve these
effects.

Due to the double-well shape of the PN potential, above the critical value of
$\alpha$ the ground-state wavefunction develops a double-peak structure in the
pinned phase, as seen in Fig. 3(b). In order to quantify the appearance of a
double-peaked wavefunction, and hence of significant quantum tunneling through
the PN barrier, we determine the Binder cumulant

$B=1-\frac{\braket{(\hat{X}-\braket{\hat{X}})^{4}}}{3\braket{(\hat{X}-\braket{\hat{X}})^{2}}^{2}},$
(6)

which as recently shown for the case of an 1D ion chain, can adequately
characterize the change of the ground-state wavefunction from a single- to a
double-peak form Bonetti  _et al._ (2021). A Gaussian wavefunction is
characterized by $B=0$ whereas a double-peak structure leads to a nonzero
value of $B$ (with the chosen normalization, $0\leq B\leq 2/3$).

Figure 4: Binder cumulant of the ground-state wave function. We show the graph
for different particle numbers $N$. The critical point differs for different
choices of the particle number: $\alpha_{\text{A}}(N=30)=6.409$,
$\alpha_{\text{A}}(N=50)=9.577$ and $\alpha_{\text{A}}(N=70)=12.555$. We only
show the graph of the Binder cumulant up to the quasiclasssical regime (see
text), marked by the vertical grey bars.

In Fig. 4 we present the Binder cumulant of the ground state wavefunction as a
function of the trap aspect ratio $\alpha$ across the Aubry transition for
different $N$. For $\alpha<\alpha_{\text{A}}$ the defect ground state is of
Gaussian shape, resulting in $B=0$, as also seen in figure 3 (a). At
$\alpha=\alpha_{\text{A}}$ the Binder cumulant increases smoothly to a value
of $B=2/3$, indicating non-negligible tunneling effects close to the critical
point. The low energy states consist of a double peak structure in a symmetric
($\varphi_{+}$) or antisymmetric ($\varphi_{-}$) configuration (see Fig. 3
(b)). However, as $\alpha$ is increased the distance of the two local minima
of the PN potential as well as the energy barrier between them grows. As a
result, for a sufficiently large $\alpha$ tunneling becomes negligible and the
states localized in the left ($\varphi_{L}$) and the right ($\varphi_{R}$)
potential well decouple from each other, the ground state enters a quasi-
classical regime. In that regime the Binder cumulant becomes inadequate for
reasons discussed in section V.

For a given $N$ the importance of tunneling effects can be characterized by
the size of an effective Planck’s constant:

$\tilde{\hbar}=\frac{\hbar\omega_{z}}{m\omega_{z}^{2}l^{2}}=\hbar\left(\frac{\omega_{z}}{mC^{2}}\right)^{1/3}$
(7)

comparing the quantum energy scale $\hbar\omega_{z}$ with the classical
counterpart $m\omega_{z}^{2}l^{2}$. Hence, the size of the $\alpha$ window
with considerable tunneling effects can be tuned by different choices of
$\omega_{z}$ and the ion species, and also by the particle number $N$ (see
Fig. 4). Deeper trap potentials or higher number of ions result in shorter ion
distances, which increase quantum effects, whereas a smaller ion mass leads to
a larger spread of the kink wavefunction, enhancing quantum tunneling.

##  V Signatures of quantum effects

Figure 5: Low energy spectrum of the effective kink Hamiltonian across the
classical transition point $\alpha_{\text{A}}$ for $N=30$. The black line
indicates the energy barrier of the PN potential $\Delta$.

As we will discuss, the Binder cumulant alone does not serve as a good measure
to assess the presence of quantum effects, therefore we present two
alternative ways to reveal the quantum nature of the kink. Firstly, we present
the low-energy spectrum of the quantum kink.

In the classical regime the transition is marked by a soft mode having
vanishing frequency at the critical point Kiethe  _et al._ (2017). In Fig. 5
we show the low-energy spectrum resulting from exact diagonalization for
$N$=30 as a function of $\alpha$.

Due to quantum fluctuations the energy gap between the ground state and the
first excited state remains finite at $\alpha_{\text{A}}$. When $\alpha$ is
increased, the eigenenergies of the lowest states become smaller than the PN
barrier
$\Delta=U(0)-\displaystyle{\min_{X}U(X)}\propto(\alpha-\alpha_{A})^{2}$ (see
Fig. 5) but maintain their finite gap, indicating the tunneling regime.

At higher values of $\alpha$ the gap to the first excited state becomes
arbitrarily small, resulting in a quasi-degenerate ground-state. States with
higher energies show the same behavior at subsequently larger values of
$\alpha$, forming pairwise degenerate states. The loss of the energy gaps can
be again associated to the vanishing of tunneling effects, leading to
spatially separated states located at each potential minimum (see Fig. 3).
This is the regime where the spontaneous symmetry breaking of the Aubry
transition occurs, i.e., any source of decoherence in the system will lead to
the spontaneous selection of a linear combination of the degenerate wave
functions that is localized either in the left or the right well. However,
even for such a classical state, the Binder cumulant remains nonzero, hence it
is not a sufficient condition for the appearance of the quantum kink. Note
that the quasi-classical regime occurs when the eigenenergies of the pairs of
states becomes significantly lower than $\Delta$ (see Fig. 5). In contrast,
the quantum kink is characterized by the existence of a finite tunneling gap,
i.e., at sufficently low temperature, spontaneous symmetry breaking can no
longer occur due to the lack of degeneracy. Importantly, the ground state in
the quantum kink regime retains the mirror symmetry of the Hamiltonian, which
means that is an ordered state (as shown by the finite Binder cumulant) beyond
the notion of spontaneous symmetry breaking.

Figure 6: Probability to find the kink at $X<0$ as a function of time for
$N=30$ and $\alpha-\alpha_{\text{A}}=0.011$ for different kink temperatures.
The kink is prepared in a thermal state of the symmetry broken potential
$U(X)-h\cdot X$, where $h/k_{B}T=0.864\leavevmode\nobreak\ \mu m^{-1}$. The
opaque points display a running mean of the dynamics depicted in transparent
points. The black line indicates the absence of tunneling for
$\alpha-\alpha_{\text{A}}=0.031$, even for very small temperatures of
$T=$0.1\text{\,}\mathrm{\SIUnitSymbolMicro K}$$. Inset: Ratio between the
variance of the running mean and variance of the dynamics within the averaging
blocks as a function of the temperature.

Another approach to demonstrate the presence of quantum effects focuses on the
different dynamical behavior resulting from tunneling.

To demonstrate this we initially prepare the kink at one of the potential
wells. This is done as follows. We break the mirror symmetry of the PN
potential by adding a potential gradient $\hat{H}_{g}=-g\hat{X}$, and assume
that the system is in a thermal state of the overall Hamiltonian
$\hat{H}=\hat{H}_{s}+\hat{H}_{g}$. We choose $g=k_{B}T/2X_{0}$ where $X_{0}$
is the position of the potential minimum and $k_{B}$ is the Boltzmann
constant, such that the potential wells are separated by an energy of
$k_{B}T$. At $t=0$ we set $g=0$, and monitor the inter-well dynamics. Figure 6
shows the probability $P_{<}$ to find the kink at $X<0$. When quantum
tunneling is non-negligible $P_{<}$ oscillates coherently with a frequency
given by the energy splitting of the ground state and the first excited state.
When $\alpha$ is tuned into the quasiclassical regime, the kink stays
localized in one of the potential wells due to the inability to tunnel through
the PN barrier. However, for sufficiently large temperatures, the contrast of
the oscillations tends eventually to zero, since the kink may move from one
well to the other by thermally-activated jumps over the PN barrier, reducing
the oscillation amplitude and introducing high frequency oscillations in the
signal.

To demonstrate this behavior, the trap needs to be stabilized in a
configuration which exhibits considerable quantum effects. This demands
working with $\alpha/\alpha_{A}-1<0.002$ (see Fig. 4) Johnson  _et al._
(2016), although the window can be enlarged by increasing the effective
Planck’s constant $\tilde{\hbar}$. Secondly, the temperature of the defect
needs to be well below the energy of the PN barrier in order to preclude
thermally activated jumps into the neighboring potential well. To quantify the
necessary temperature scale, we compute running mean values of $P_{<}$ and
comparing the variance of the averaged values with the variance of the data
within the averaging blocks (see inset of Fig. 6).

The results show that at the chosen $\alpha$ value the kink should be cooled
to $T<$5\text{\,}\mathrm{\SIUnitSymbolMicro K}$$ in order to reveal quantum
tunneling of the kink. Here again, we want to note that the overall energy
scale of the system $E^{3}=mC^{2}\omega_{z}^{2}$ can be tuned by the trap
frequency $\omega_{z}$, the ion mass $m$ and the particle number $N$, which
would influence the necessary temperature scale in turn.

##  VI Experimental accessibility

In this section we discuss possible strategies to observe the implications of
quantum effects in the Aubry transition with trapped ions. Experimentally
reaching the required temperature regime close to the critical point could be
achieved by preparing the system deeper in the pinned phase, where the mode
frequency of the kink is high enough in order to allow for sufficient cooling
while being energetically well isolated from the residual phonon spectrum
Landa  _et al._ (2010). In addition, studies have shown, that heating of short
wavelength motional modes by dc electric field noise is strongly suppressed
Kalincev  _et al._ (2021); Wineland  _et al._ (1998); King  _et al._ (1998),
such that effective cooling of the kink dynamics is possible. The kink in a
crystal of $30$ ions could be prepared at $\alpha-\alpha_{\text{A}}=0.091$
where the kink mode exhibits a frequency of
$2\pi\cdot$65.17\text{\,}\mathrm{kHz}$$ at an axial trapping frequency of
$\omega_{z}=2\pi\cdot$150\text{\,}\mathrm{kHz}$$. At this frequency, the
required temperature of $T<$5\text{\,}\mathrm{\SIUnitSymbolMicro K}$$
corresponds to a mean motional state occupation of $\bar{n}<1.15$. After
Doppler cooling the entire crystal to the Doppler limit of about
$0.5\text{\,}\mathrm{mK}$ cooling can be applied in order to reach a mean
occupation number of around $\bar{n}\lesssim 3$ for the whole crystal Ejtemaee
and Haljan (2017); Joshi  _et al._ (2020), which is close to the desired
temperature regime. Further cooling of the kink mode can be achieved by
incorporating red sideband cooling techniques. After the cooling stages,
quenching $\alpha$ at a rate of
$\dot{\alpha}\approx-$0.1\text{\,}\mathrm{kHz}$$ ensures adiabaticity during
the approach of $\alpha_{\text{A}}$ while the total quench time on the order
of milliseconds is much smaller than the measured lifetime of the defect Pyka
_et al._ (2013). Ultimately, the excitation spectrum of the quantum kink
presented in Fig. 5 can be mapped out in a spectroscopic way.

In an alternative approach aiming to measure the quantum dynamics shown in
Fig. 6 the kink needs to be initialized in one of the potential wells. This
can be accomplished by introducing controllable higher order contributions in
the trap potential that break the symmetry in the $z$ direction Brox  _et al._
(2017). After the initialization and cooling stage the trap potential can be
tuned to be harmonic again in order to trigger the tunneling motion of the
kink through the potential barrier. This motion translates into an oscillation
of the central ions which also takes place on a timescale of tens of
milliseconds (see Fig. 6 ). The oscillation has an amplitude on the order of a
few hundred nanometers, and requires a non-destructive detection without
heating.

##  VII Conclusions

In summary, we have discussed an effective model for the dynamics of a
topological defect in an ion Coulomb crystal. Working with an effective
potential and effective mass of the kink yields a simplified single-particle
description of the relevant physics at the Aubry transition. Our results for
the quantum kink Hamiltonian demonstrate the presence of quantum effects close
to the critical Aubry point, which manifest themselves, in the coherent
oscillation of the kink between the two minima of the PN potential. The
observation of quantum tunneling of the defect demands in typical experiments
$T<$5\text{\,}\mathrm{\SIUnitSymbolMicro K}$$. Sub-Doppler cooling the motion
of the kink in ion Coulomb crystals to reach these low temperatures will be a
challenge for future experiments on quantum Aubry physics based on trapped
ions. Finally, we would like to mention that our effective single-particle
model can be directly applied to other systems that exhibit Aubry physics. In
particular, with an appropriate definition of the kink position variable
$K({\vec{r}_{i}})$, it can be applied to an alternative approach to emulate
nanofriction in a trapped ion setup based on a linear ion string in a
superimposed periodic lattice, as recently investigated in Bylinskii  _et al._
(2015); Bonetti  _et al._ (2021).

###### Acknowledgements.

We thank Jan Kiethe for fruitful discussions. This project has been funded by
the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
Germany’s Excellence Strategy - EXC-2123 QuantumFrontiers - 390837967 and
through CRC 1227 (DQ-mat), project A07. This project 17FUN07 CC4C has received
funding from the EMPIR programme co-financed by the Participating States and
from the European Union’s Horizon 2020 research and innovation programme.

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