# Phonon-assisted coherent transport of excitations in Rydberg-dressed atom
arrays

Arkadiusz Kosior  Institute for Theoretical Physics, University of Innsbruck,
6020 Innsbruck, Austria  Servaas Kokkelmans  Department of Applied Physics,
Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The
Netherlands  Maciej Lewenstein  Institut de Ciencies Fotoniques, The Barcelona
Institute of Science and Technology, 08860 Castelldefels, Barcelona, Spain
ICREA, Passeig Lluis Companys 23, 08010 Barcelona, Spain  Jakub Zakrzewski
Instytut Fizyki Teoretycznej, Uniwersytet Jagielloński, Łojasiewicza 11,
30-348 Kraków, Poland  Mark Kac Center for Complex Systems Research,
Jagiellonian University, Łojasiewicza 11, 30-348 Kraków, Poland  Marcin
Płodzień  Institut de Ciencies Fotoniques, The Barcelona Institute of Science
and Technology, 08860 Castelldefels, Barcelona, Spain

###### Abstract

Polarons, which arise from the self-trapping interaction between electrons and
lattice distortions in a solid, have been known and extensively investigated
for nearly a century. Nevertheless, the study of polarons continues to be an
active and evolving field, with ongoing advancements in both fundamental
understanding and practical applications. Here, we present a microscopic model
that exhibits a diverse range of dynamic behavior, arising from the intricate
interplay between two excitation-phonon coupling terms. The derivation of the
model is based on an experimentally feasible Rydberg-dressed system with
dipole-dipole interactions, making it a promising candidate for realization in
a Rydberg atoms quantum simulator for excitation dynamics interacting with
optical phonons. Remarkably, our analysis reveals a growing asymmetry in Bloch
oscillations, leading to a macroscopic transport of non-spreading excitations
under a constant force. Finally, we demonstrate the robustness of our findings
against on-site random potential.

##  I Introduction

Polarons are quasi-particles that emerge from the coupling between electrons
(or holes) with ions of a crystalline structure in polarizable materials. The
idea of electron self-trapping due to lattice deformations dates back to
Landau’s seminal 1933 paper [1], but the modern concept of a polaron as an
electron dressed by phonons was formulated in 1946 by Pekar [2], and developed
later by Fröhlich [3], Feynman [4, 5], Holstein [6], and Su, Schrieffer and
Heeger [7, 8, 9]. Since their discovery, polarons have been extensively
investigated, both theoretically and experimentally, not only in the field of
condensed matter physics (for reviews see Refs. [10, 11]), but also in various
chemical and biological contexts, e.g., in protein propagation [12, 13, 14].
In particular, in the modeling of charge migration in DNA molecules, it is
assumed that a localized polaron is formed in the helix near a base due to an
interaction between a charge carrier and a phonon. When a uniform electric
field is applied, the polaron moves at a constant velocity, and a current
flows through the chain [15, 16, 17]. The charge carrier transport takes place
due to coupling between carrier and phonons; in contrast, in the absence of
phonons, an external constant force induces Bloch oscillations [18, 19, 20],
where the mean position of the carrier is constant while its width
periodically changes in time.

Polarons have been studied in many, seemingly different experimental setups,
ranging from ultracold ions [21, 22, 23, 24], polar molecules [25, 26, 27,
28], mobile impurities in Bose and Fermi gases [29, 30, 31], ultracold dipolar
and Rydberg atoms [32, 33, 34, 35, 36, 37, 38, 39, 40], to quantum dots on a
carbon nanotube [41]. Although each of these platforms possesses its unique
strengths and benefits, recently there has been an exceptional outburst of
interest in quantum simulation and computation with Rydberg atoms, which
provide a remarkable level of flexibility for executing quantum operations and
constructing quantum many-body Hamiltonians [42]. While the latter can
contribute to our comprehension of the static properties of many-body systems,
their main benefits are centered around exploring the complex dynamics
displayed by these systems. In particular, in the context of polarons, it has
been demonstrated that the dipole-dipole interactions between distinct
Rydberg-dressed states can result in coherent quantum transport of electronic-
like excitations [32], which can further be coupled to optical phonons [33].
The paradigmatic one-dimensional topological Su-Schrieffer-Heeger (SSH) model
[7] describing the soliton formation in long-chain polyacetylene due to
excitation-phonon coupling, has been realized in Rydberg arrays [43, 44, 45].

In this paper, we continue along this path and present theoretical studies of
an implementation of a microscopic model featuring the interplay of Su-
Schrieffer-Heeger (SSH) and Fröhlich electron-phonon coupling mechanisms
between optical phonons and excitations, under the influence of an external
force and disorder. In particular, we focus on the directional transport of an
excitation interacting with phonons. We indicate an excitation-phonon coupling
regime where the competition between Bloch oscillations and interactions
results in the coherent transport of a well-localized wave packet over a long
distance. We show the robustness of such a coherent transport of well-
localized wave packets to the on-site random potential, indicating that a
relatively strong disorder does not affect significantly the transport
properties. Moreover, for completeness, we consider also excitation coupling
to acoustic phonons.

The paper is divided into three parts. In the first part, Sec. II, we describe
the physical setup and derive the effective Hamiltonian in Rydberg-dressed
atomic arrays. The second part, described in Section III, focuses on the
dynamics of the system under experimentally relevant parameters. In this
section, we observe the macroscopic transport of the center of mass and a
transition between Bloch oscillations and moving polaron regimes. In the third
part, Sec. IV, we comprehensively analyze the previously derived microscopic
model, which exhibits a rich phase diagram due to the interplay of two
different electron-phonon coupling mechanisms. Finally, we compare the
behavior of excitations with acoustic and optical phonons and demonstrate the
robustness of our results.

##  II The model and its Hamiltonian

We consider a one-dimensional chain of $N$ equidistant Rydberg atoms with
lattice constant $x_{0}$ and positions $x_{j}=jx_{0}$, confined in a periodic
trap, implemented either by an optical lattice [46, 47], an optical tweezer
array [48, 49, 50], a Rydberg microtrap [51], or a painted potential [52]. We
assume that the spatial motion of the atoms is suppressed by the strong
confinement of each Rydberg atom in local potential minima. Although the
atomic motion is frozen, it is remarkable that such a Rydberg system can
display highly non-trivial dynamics. In particular, the induced dipole-dipole
interactions between distinct Rydberg-dressed states can lead to the emergence
of coherent quantum transport of electronic-like excitations [32]. In the
following, we first briefly repeat the derivation of the Hamiltonian that
characterizes the dynamics of single excitations [32]. The purpose of this
recap is to modify the setup in order to incorporate nearly arbitrary on-site
potential terms. Next, after introducing phonons into the system [33], we
derive an effective nearest-neighbor Hamiltonian that includes two excitation-
phonon coupling terms, which we comprehensively study in the forthcoming
sections, focusing on the dynamics in the presence of an external constant
field.

###  II.1 Single excitation Hamiltonian in arbitrary potentials

We assume that each Rydberg atom can be initially found in one of the ground
state hyperfine levels, $|g\rangle$ or $|g^{\prime}\rangle$. By applying far-
detuned dressing laser fields, with effective Rabi frequencies $\Omega_{s}$,
$\Omega_{p}$ and detunings $\Delta_{s}$, $\Delta_{p}$ respectively, these two
hyperfine states can be coherently coupled to selected highly excited Rydberg
states, $|s\rangle$ or $|p\rangle$, with principal quantum number $n\gg 1$ and
different angular momenta. Consequently, each atom can be found in one of the
two Rydberg dressed states [53, 32, 54, 55, 56, 57], which are a slight
admixture of Rydberg states to the atomic ground states,

$|0\rangle_{j}\approx|g\rangle_{j}+\alpha_{s}|s\rangle_{j}\quad\mbox{or}\quad|1\rangle_{j}\approx|g^{\prime}\rangle_{j}+\alpha_{p}|p\rangle_{j},$
(1)

with $\alpha_{s/p}=\Omega_{s/p}/[2\Delta_{s/p}]$ and $j$ denoting the position
of an atom. Treating $\alpha_{s}$, $\alpha_{p}$ as perturbation parameters in
van Vleck perturbation theory, Wüster at al. [32] have shown that the dipole-
dipole interaction can exchange the internal states of a neighboring pair,
e.g. $|1\rangle_{1}|0\rangle_{2}\rightarrow|0\rangle_{1}|1\rangle_{2}$. This
process can be viewed as a hopping of an excitation from $j=1$ to $j=2$
lattice site, which conserves the number of excitations.

The perturbation analysis can be extended to a chain of $N$ atoms, where the
effective Hamiltonian in the single excitation manifold (up to the fourth
order in $\alpha_{s}$ and $\alpha_{p}$) reads [32, 33]

$\hat{H}_{0}=\sum_{j}\hat{n}_{j}(E_{2}+E_{4}+A_{j})+\sum_{j,k}A_{jk}\hat{a}_{j}^{\dagger}\hat{a}_{k},$
(2)

where $\hat{a}_{j}$ ($\hat{a}^{\dagger}_{j}$) denote an annihilation
(creation) operator of excitation on site $j$, while

$\displaystyle A_{j}$
$\displaystyle=\hbar\alpha_{s}^{2}\alpha_{p}^{2}\left(\sum_{k\neq
j}\frac{1}{1-\bar{U}_{kj}^{2}}\right)(\Delta_{s}+\Delta_{p}),$ (3a)
$\displaystyle A_{jk}$
$\displaystyle=\hbar\alpha_{s}^{2}\alpha_{p}^{2}\frac{\bar{U}_{jk}}{1-\bar{U}_{jk}^{2}}(\Delta_{s}+\Delta_{p}),$
(3b)

with
$\bar{U}_{jk}=C_{3}/[\hbar\left|x_{i}-x_{j}\right|^{3}(\Delta_{s}+\Delta_{p})]$
and $C_{3}$ quantifying the transition dipole moment between the Rydberg
states, describe perturbative dipole-dipole interactions. Finally, $E_{2}$ and
$E_{4}$ are constant energy shifts of the second and fourth order,
respectively,

$\displaystyle E_{2}/\hbar$
$\displaystyle=(N-1)\alpha_{s}^{2}\Delta_{s}+\alpha_{p}^{2}\Delta_{p},$ (4a)
$\displaystyle E_{4}/\hbar$
$\displaystyle=(N-1)\alpha_{s}^{4}\Delta_{s}+\alpha_{p}^{4}\Delta_{p}$
$\displaystyle+(N-1)\alpha_{s}^{2}\alpha_{p}^{2}(\Delta_{s}+\Delta_{p}).$ (4b)

Although in principle constant energy terms could be always ignored as they do
not contribute to the dynamics of excitations, let us consider now a scenario
where the Rabi frequency $\Omega_{p}$ depends on the atomic position on the
lattice, i.e., we assume that

$\Omega_{p}\rightarrow\Omega_{p}(j)\equiv\Omega_{p}\left[1+\delta\Omega(j)\right],$
(5)

where $\delta\Omega(j)$ is arbitrary, but small correction of the order
$(\alpha_{p/s})^{2}$. With this assumption, and by retaining terms up to the
fourth order, the effective Hamiltonian in Eq. (2) acquires an additional
term, namely

$\hat{H}=\hat{H}_{0}+\hbar\alpha_{p}^{2}\Delta_{p}\sum_{j}\delta\Omega(j)\hat{n}_{j}\;.$
(6)

Because the term proportional to $\alpha_{p}^{2}\delta\Omega(j)$ is of the
same order as $A_{j}$, it can be incorporated into the definition of $A_{j}$
in Eq. (3a). With this simple modification, we have gained a position-
dependent effective potential term that can strongly affect the dynamics of
excitations. Although the potential term can be tailored almost arbitrarily,
from now on we consider one of its simplest forms, i.e., we choose

$\delta\Omega(j)=2\alpha_{s}^{2}\left(Fj+\epsilon_{j}\right).$ (7)

The first term in the parentheses, being linearly proportional to position
$j$, emulates the presence of a constant external field $F$. The second term,
with $\epsilon_{j}$ being a random variable, gives rise to the on-site
potential disorder. Note that both terms lead to localization of the
excitation either due to Stark localization [18] in a constant tilt, $F$, or
Anderson localization [58] due to random $\epsilon_{j}$. As explained in the
next part, the situation is not so straightforward.

###  II.2 Excitation-phonon Hamiltonian

In this part, we relax our previous assumption that the atoms of the array are
completely immobile. Although we still assume that no atom can move through
the lattice, we now let them vibrate in the vicinity of their local
equilibrium points. This will affect, as we shall see, the dynamics of
excitations. We consider now a scenario where an atom in the $j$-th lattice
site and with mass $m$ may oscillate with a frequency $\omega_{0}=\sqrt{k/m}$
inside a local potential well, that can be approximated by a quadratic
potential

$\frac{k}{2}(x-jx_{0})^{2}\equiv\frac{kx_{0}^{2}}{2}(u_{j})^{2},$ (8)

with $k$ being the force constant and where $u_{j}$ denotes dimensionless
distortion from the local equilibrium position. The motion of atoms can be
quantized $u_{j}\rightarrow\hat{u}_{j}$ and described by a simple quantum
harmonic oscillator. This vibrational motion is responsible for the distortion
of an atomic array and can be considered as a phonon. Since the Hamiltonian of
the previous section describing the motion of single excitations strongly
depends on the position of atoms, phonons can propagate through space due to
the coupling to excitations. Before proceeding to derive the effective
Hamiltonian of the system with phonon-excitation coupling, for clarity and
simplicity we assume that

$\alpha\equiv\alpha_{s}=\alpha_{p},\quad\Delta\equiv\Delta_{s}=\Delta_{p}.$
(9)

Moreover, from now on we also fix the time and energy scales and go to the
dimensionless units by dividing all the energy scales by
$2\hbar\alpha^{4}\Delta$.

Although the setup described in Section II.1 admits only dispersionless
optical phonons that correspond to local vibrations of atoms around local
minima, we consider here two different types of phonons. We proceed by writing
the phononic Hamiltonian explicitly in terms of the dimensionless position and
momentum operators $\hat{u}_{j}$, $\hat{p}_{j}$ of local distortions,

$\hat{H}_{\text{ph}}=\sum_{j}\frac{\hat{p}^{2}_{j}}{2m_{\text{eff}}}+\frac{m_{\text{eff}}\omega_{\text{eff}}^{2}}{2}(\hat{u}_{j}-\eta\hat{u}_{j-1})^{2},$
(10)

with the effective dimensionless mass,

$m_{\text{eff}}=2mx_{0}^{2}\alpha^{4}\Delta/\hbar,$ (11)

and the effective oscillator frequency,

$\omega_{\text{eff}}=\omega_{0}/(2\alpha^{4}\Delta),\quad\omega_{0}=\sqrt{k/m},$
(12)

where $\omega_{0}$ is the bare frequency. By changing the parameter $\eta$ in
Eq. (10), diverse phonon types can be achieved. In particular, $\eta=0$
corresponds to the aforementioned local vibrations (i.e., dispersionless
optical phonons), and $\eta=1$ describes acoustic phonons. These two phonon
types are characterized by the dispersion relation

$\epsilon_{q}=\begin{cases}\omega_{\text{eff}},&\mbox{(optical phonons,
$\eta=0$)}\\\ 2\omega_{\text{eff}}\left|\sin(qx_{0}/2)\right|,&\mbox{(acoustic
phonons, $\eta=1$)}\end{cases},$ (13)

which can be readily found by writing the phononic Hamiltonian (10) in terms
of its eigenmodes,

$\hat{H}_{\text{ph}}=\sum_{q}\epsilon_{q}\left(\hat{b}^{\dagger}_{q}\hat{b}_{q}+\frac{1}{2}\right),$
(14)

where $\hat{b}^{\dagger}_{q}$ ($\hat{b}_{q}$) creates (annihilates) the phonon
with quasi-momentum $q$, and are related to the local dimensionless momentum
and position operators $\hat{p}_{i}$, $\hat{u}_{i}$ of distortion by

$\begin{split}\hat{u}_{j}&=\sum_{q}\frac{1}{\sqrt{2N\epsilon_{q}m_{\text{eff}}}}(\hat{b}_{q}+\hat{b}^{\dagger}_{-q})e^{iqjx_{0}},\\\
\hat{p}_{j}&=-i\sum_{q}\sqrt{\frac{\epsilon_{q}m_{\text{eff}}}{2N}}(\hat{b}_{q}-\hat{b}^{\dagger}_{-q})e^{iqjx_{0}}.\end{split}$
(15)

Figure 1: Schematic illustration of all processes in the effective Hamiltonian
$\hat{H}_{\text{eff}}$ in Eq. (16), describing the dynamics of a single
excitation in a one-dimensional array of Rydberg atoms located at
$x_{0}(j+u_{j})$, with $u_{j}$ being a dimensionless distortion from an
equilibrium position. $\hat{H}_{\text{ex}}$ describes a bare hopping (in the
limit $u_{j}\rightarrow 0$) of an excitation at site $j$ to its neighboring
sites with amplitude $J_{0}$ in the presence of a constant force $F$ and on-
site disorder $\epsilon_{j}$, see Eq. (17). The effective hopping and on-site
potential is further modified by the phonon couplings $g_{J}$ and $g_{W}$,
respectively, see Eq. (19).

Having discussed the phononic degrees of freedom, we can now write the fully
effective Hamiltonian governing the motion of single excitations coupled to
phonons. The derivation is straightforward and requires: (i) the expansion of
the position-dependent coefficients [given by Eq. (3)] in the Hamiltonian (6)
of the previous section up to the first order in $\hat{u}_{j}$, and (ii)
dropping the next-to-nearest neighbor contributions [59]. By following these
steps, we obtain the effective excitation-phonon Hamiltonian [cf. Fig. 1],
which consists of four parts, i.e.,

$\hat{H}_{\text{eff}}=\hat{H}_{\text{ph}}+\hat{H}_{\text{ex}}+\hat{H}_{\text{J}}+\hat{H}_{\text{W}},$
(16)

where $\hat{H}_{\text{ph}}$ is the phononic Hamiltonian, Eq. (10),

$\hat{H}_{\text{ex}}=J_{0}(\hat{a}^{\dagger}_{j+1}\hat{a}_{j}+\mbox{H.c.})+\sum_{j}(jF+\epsilon_{j})\hat{a}^{\dagger}_{j}\hat{a}_{j},$
(17)

describes excitations with the hopping amplitude

$J_{0}=\kappa/(1-\kappa^{2}),\quad\kappa=C_{3}/(2\hbar\Delta x_{0}^{3}),$ (18)

experiencing an external constant force $F$, and a local on-site disorder
$\epsilon_{j}$. Finally,

$\displaystyle\hat{H}_{\text{J}}$
$\displaystyle=g_{J}\sum_{j}(\hat{u}_{j+1}-\hat{u}_{j})\hat{a}^{\dagger}_{j+1}\hat{a}_{j}+\mbox{H.c.}\,,$
(19a) $\displaystyle\hat{H}_{\text{W}}$
$\displaystyle=g_{W}\sum_{j}(\hat{u}_{j+1}-\hat{u}_{j-1})\hat{a}^{\dagger}_{j}\hat{a}_{j},$
(19b)

are the notable SSH and Fröhling Hamiltonians [7, 11], respectively, that
correspond to two different mechanisms of excitation-phonon couplings, with
dimensionless coupling parameters

$\displaystyle g_{J}$
$\displaystyle=-3\kappa(1+\kappa^{2})/(\kappa^{2}-1)^{2},$ (20a)
$\displaystyle g_{W}$ $\displaystyle=-6\kappa^{2}/(\kappa^{2}-1)^{2}.$ (20b)

###  II.3 Equations of motion

The full numerical analysis of the polaron dynamics on the many-body level is
one of the most challenging computational tasks, due to the non-conserved
total number of phonons in the system, which prevents it from working in a
restricted, fixed particle-number Hilbert space sector of the phononic degrees
of freedom. Additionally, even without a force $F$ the effective Hamiltonian
of the systems (16) depends, in principle, on many parameters, namely
$J_{0},g_{W},g_{J},\omega_{\rm eff}$ and $m_{\rm eff}$, making the full
analysis of the system even more challenging.

To analyze the dynamical properties of the considered system, in the following
we make the semiclassical approximation by applying the Davydov Ansatz [60,
61, 62, 63, 64, 65, 66], which relies on two fundamental assumptions: (i)
treating phononic oscillations classically and (ii) representing solutions as
separable states without entanglement between quantum-like excitations and
classical-like phonons. In other words, we assume that phonons are in a
coherent state and that the full wave function is a product state of the
excitation and coherent phonons part, as

$|\Psi(t)\rangle=\left(\sum_{j}\psi_{j}(t)\hat{a}_{j}^{\dagger}\right)\otimes\left(e^{-i\sum_{j}\left[u_{j}(t)\hat{p}_{j}-p_{j}(t)\hat{u}_{j}\right]}\right)|\mathtt{vac}\rangle,$
(21)

where $|\psi_{j}(t)|^{2}$ is a probability of finding an excitation at a site
$j$, $u_{j}(t)$ and $p_{j}(t)$ are expectation values of phononic position and
momentum operators. The equation of motion for $\psi_{j}(t)$ and $u_{j}(t)$
can be subsequently derived from a classical conjugate variable Heisenberg
equations of motions using the generalized Ehrenfest theorem, see, for
example, Ref. [14]. By following these steps, we obtain a closed set of
coupled differential equations for the excitation amplitude $\psi_{j}(t)$ and
classical field $u_{j}(t)$. The equations can be written in a concise form, as

$\displaystyle i\dot{\psi_{j}}$
$\displaystyle=J_{j}\psi_{j+1}+J_{j-1}\psi_{j-1}+W_{j}\psi_{j},$ (22a)
$\displaystyle\ddot{u}_{j}$ $\displaystyle=-\omega_{\text{eff}}^{2}\,{\cal
D}[\\{u_{j}\\}]+{\cal S}[\\{\psi_{j}\\}],$ (22b)

where the effective potential experienced by an excitation $W_{j}(t)$, and the
effective hopping amplitude $J_{j}(t)$ are both time-dependent functions due
to the coupling to the gradient of the phononic field $u_{j}(t)$, i.e.

$\begin{split}W_{j}(t)&=jF+\epsilon_{j}+g_{W}[u_{j+1}(t)-u_{j-1}(t)]\\\
J_{j}(t)&=J_{0}+g_{J}[u_{j+1}(t)-u_{j}(t)].\end{split}$ (23)

As such, both $W_{j}(t)$ and $J_{j}(t)$ are responsible for the self-trapping
of an excitation. Similarly, the phononic equation (22b) also depends on the
excitation amplitude $\psi_{i}(t)$ through the ${\cal S}[\\{\psi_{j}\\}]$
operator, given by

$\begin{split}{\cal
S}[\\{\psi_{j}\\}]=-\frac{g_{W}}{m_{\text{eff}}}(|\psi_{j+1}|^{2}-|\psi_{j-1}|^{2})\\\
-\frac{g_{J}}{m_{\text{eff}}}[\psi_{j}^{*}(\psi_{j+1}-\psi_{j-1})+\mbox{c.c.}],\end{split}$
(24)

which acts as a time-dependent source for the phonon field $u_{j}(t)$.
Finally, the phononic dispersion relation, given by Eq. (13), is necessarily
present in the phononic equation through the ${\cal D}[\\{u_{j}\\}]$ operator,

${\cal D}[\\{u_{j}\\}]=\begin{cases}u_{j},&\eta=0,\\\
2u_{j}-u_{j+1}-u_{j-1},&\eta=1,\end{cases}\;,$ (25)

which introduces a crucial difference in the propagation of optical ($\eta=0$)
and acoustic ($\eta=1$) phonons [67], which we investigate in the next
sections.

###  II.4 Analysed observables

Throughout this article we choose the initial conditions
$\psi_{j}(0)=\delta_{j,0}$ and $u_{j}(0)=\dot{u}_{j}(0)=0$ for the equations
of motion, Eq. (22), that correspond to a single excitation on a central
lattice site and initially unperturbed lattice. Without a phonon-coupling and
for $F=0$, these initial conditions simply correspond to a quantum particle
that spreads symmetrically in both lattice directions characterized by a
constant Lieb-Robinson velocity [68], so that its center of mass remains
localized at the initial position. Contrary to the classical case, a quantum
particle on a lattice will not even move in the presence of a constant force
$F$, but instead, it starts to perform Bloch oscillations [69]. The situation
is different in interacting systems, either in a case of particle-particle
interactions [19], which may further lead to disorder-free many-body
localization [70, 71, 72, 73, 74, 75, 76, 77, 78, 79], or in the presence of
phonons, which can induce transient polarons at the end of Bloch oscillation
periods [64, 80] (see also Ref. [81]).

In this study, we investigate how the propagation of a single excitation is
influenced by the two competing phonon-coupling mechanisms under the applied,
constant force. Specifically, we aim at answering the two following questions:
(i) how much does the excitation spread due to the coupling with phonons, and
(ii) does its center of mass move in the presence of the constant force $F$?
In order to respond to these questions we focus on three simple observables
that can be calculated based on the local density measurements. First, we
consider the participation ratio (PR), defined as [82]:

$\mbox{PR}(t)=\bigg{(}\sum_{j}\left|\psi_{j}(t)\right|^{4}\bigg{)}^{-1},$ (26)

where we have assumed a unit normalization of the wavefunction
$\sum_{j}|\psi_{j}|^{2}=1$. The participation ratio PR is equal to $1$ where
excitation is localized on a single lattice site and equals $N$ when is
completely delocalized over the whole lattice. The second observable is the
center of mass position of the wave packet, i.e.,

$x(t)=\sum_{j=-N/2}^{N/2}j\left|\psi_{j}(t)\right|^{2}.$ (27)

Moreover, in some cases, analyzing the ratio of the two quantities mentioned
above can provide valuable insights. We define this ratio, denoted as $\xi$,
as:

$\xi(t)=\frac{|x(t)|}{\mbox{PR}(t)}.$ (28)

$\xi$ is a quantity ranging from 0 to $\xi_{\textrm{max}}=N/2$. The maximum
value $\xi_{\textrm{max}}$ corresponds to a moving, maximally-localized, non-
dispersive solution that has reached the boundary of the system. As such,
$\xi$ can be viewed as an indicative measure for selecting well-localized
solutions moving in one direction.

Finally, it is worth mentioning that it is often not necessary to analyze the
entire time range of the above observables. In fact, to discern various
dynamic behaviors, it is usually sufficient to look at $\mbox{PR}(t)$, $x(t)$
and $\xi(t)$ at the final evolution time $t_{f}\gg 1$. For example, large
$\mbox{PR}(t_{f})$ (relative to the system size $N$) suggests that excitation
is not stable and has delocalized over a lattice.

##  III Polaron dynamics: experimental considerations

In this section we elaborate on the results of the previous sections and study
the dynamics of a Rydberg excitation under the presence of the external force
$F$, solving the equations of motion for a physically relevant range of
parameters. The effective Hamiltonian (16) of the system relies on several
effective, dimensional parameters, including
$m_{\text{eff}}=2mx_{0}^{2}\alpha^{4}\Delta/\hbar$,
$\omega_{\text{eff}}=\omega_{0}/(2\alpha^{4}\Delta)$, as well as $J_{0}$,
$g_{J}$, $g_{W}$, given by Eq. (18) and Eqs. (20). However, it is worth noting
that the latter three parameters are not independent within our setup, and
their values are determined by a single parameter $\kappa=C_{3}/(2\hbar\Delta
x_{0}^{3})$. This provides us with significant flexibility in selecting
appropriate physical parameters for our convenience. We stress, that the
proposed quantum simulator allows for simulating excitation coupled to optical
phonons only.

In the following, we choose the highly excited Rydberg states $|s\rangle$,
$|p\rangle$ of Rubidium-87 with principal quantum number $n=50$ and angular
momentum equal to 0 or $\hbar$, for which $C_{3}=3.224$ GHz$\times\mu m^{-3}$.
We fix the lattice spacing $x_{0}=2\,\mu\mbox{m}$, and the local trap
frequency $\omega_{0}=20$ kHz. In the numerical simulations, we vary the
dimensionless parameter $\kappa$ between $0.80-0.86$, which is equivalent to
the change of the detuning $\Delta\sim 234-252$ MHz, and corresponds to the
dressing parameter $\alpha\sim 0.04$. Importantly, by increasing $\kappa$ we
also increase the phonon coupling strength from around
$g_{J}/m_{\textrm{eff}}\sim g_{W}/m_{\textrm{eff}}\sim-4.5$ to
$g_{J}/m_{\textrm{eff}}\sim g_{W}/m_{\textrm{eff}}\sim-8$. Furthermore, we
remind the reader that in our setup only the optical phonons (i.e.,
dispersionless vibrations) are experimentally relevant and, therefore, in this
section we set $\eta=0$. Finally, we fix the value of the force at $F=0.2$,
and we choose the system size to $N=401$.

Figure 2:  The top panel illustrates the center of mass motion $x(t)$ of an
excitation dashed blue line) under a constant, external force $F=0.2$, and the
corresponding participation ratio $\text{PR}(t)$ (solid green line), evaluated
at the final evolution time $t_{f}$, and plotted as functions of the
dimensionless parameter $\kappa$ (see Eq. (16) and definitions below it). In
the bottom panel, the ratio $\xi=|x|/\text{PR}$ is shown. The peaks in the
plot correspond to parameter regimes where a well-localized excitation is
transferred under the influence of a constant force $F$. All physical
parameters have been chosen with careful consideration of their experimental
relevance, as discussed in the main text. The time evolution range is
$t\in[0,t_{f}]$, where the final time $t_{f}$ is chosen as
$t_{f}=2.1\,T_{B}=4.2\,\pi/F\approx 66$.

In order to characterize the transport properties of an excitation
$\psi_{i}(t)$, in the top panel of Fig. 2 we plot its center of mass position
$x(t)$ and the corresponding participation ratio $\text{PR}(t)$, see Eqs.
(26)-(27) for the respective definitions. In the bottom panel, we additionally
illustrate the ratio $\xi=|x|/\text{PR}$. All these quantities are plotted as
a function of $\kappa$, at a fixed time $t_{f}=2.1\,T_{B}\approx 66$, where
$T_{B}=2\pi/F$ is the Bloch oscillation period. We find that up to $\kappa\sim
0.83$ both $x(t_{f})$ and $\text{PR}(t_{f})$ are small (relative to the system
size $N$) which corresponds to the Bloch oscillation-like dynamics where the
phonon-influence is minimal. In contrast, phonons play important role above
$\kappa\sim 0.83$ where the system dynamics is quite sensitive to the choice
of microscopic parameters. Within the chaotic-like regime, the typical Bloch
oscillation dynamics is completely disrupted, as the majority of solutions
become delocalized across the lattice, leading to large values of
$\text{PR}(t)$. However, amidst this chaotic behavior, we also discover
intervals of stability, characterized by peaks of $\xi(t_{f})$, where a
substantial portion of the wave packet becomes well-localized and exhibits
near-constant velocity motion.

We illustrate those different dynamical behaviours in Fig. 3, where the first
column, i.e., panels (a)-(d), show the time evolution of the excitation
density $|\psi_{j}(t)|^{2}$, while the second column [panels (e)-(h)]
illustrates the corresponding time evolution of the center of mass position
$x(t)$ and the participation ratio PR$(t)$. In the first row ($\kappa=0.8$),
we observe almost perfect Bloch oscillations. However, upon closer
examination, a subtle asymmetry becomes apparent, which is evident by a non-
zero $x(t)$. The asymmetry is enhanced for a higher $\kappa=0.83$, as depicted
in the second row of Fig. 3. Finally, the last two rows of Fig. 3 illustrate
the time evolution of the excitation density in the chaotic-like regime above
$\kappa\sim 0.83$, cf. Fig. 2, where most of the solutions are delocalized
over a lattice, as in Fig. 3(d) for $\kappa=0.86$. In contrast, in Fig. 3(c)
we illustrate a regular behaviour for $\kappa=0.834$, which lies inside one of
the aforementioned stability windows. In this scenario, due to constructive
interference after one Bloch oscillation period, a prominent portion of the
wave function coalesces into a very narrow non-dispersive wave packet that
moves with a nearly constant velocity. Overall, Fig. 3 offers a comprehensive
visual representation of the dynamic phenomena investigated in this section,
shedding light on the varying dynamical behaviors and properties of the system
with increasing phonon interaction.

Figure 3: The panels illustrate the diverse dynamic behaviors observed in our
study. The first column, panels (a)-(d), showcases the time evolution of the
excitation density $|\psi_{j}(t)|^{2}$ (color encoded), while the second
column, panels (e)-(h), illustrates the corresponding temporal changes in the
center of mass position $x(t)$ (dashed blue lines) and the participation ratio
PR$(t)$ (solid green lines). In the first row ($\kappa=0.8$), near-perfect
Bloch oscillations are observed. However, upon closer examination, a subtle
yet discernible asymmetry becomes apparent, as indicated by the non-zero value
of $x(t)$. This asymmetry becomes more pronounced in the second row for a
higher $\kappa$ value of 0.83. The subsequent rows of Fig. 3 provide insights
into the time evolution of the excitation density for the specific cases of a
well-localized wave-function ($\kappa=0.834$) and a spreading wave-function
($\kappa=0.86$). These distinct parameter regimes highlight the contrasting
behavior and spatial characteristics of the excitations. All physical
parameters are the same as in Fig. 2.

##  IV Dynamical phase diagrams of the effective Hamiltonian

Figure 4: Color encoded participation ratio ${\rm PR}(t_{f})$, Eq. (26), at
the final evolution time for a broad range of coupling strengths,
$g_{J}\in[0,45]$ and $g_{W}\in[-20,20]$, in the presence of optical (left
column, $\eta=0$) and acoustic phonons (right column, $\eta=1$). Each row
corresponds to a specific value of the effective mass $m_{\rm eff}$. Panels
(a)-(b) correspond to $m_{\rm eff}=2$, panels (c)-(d) correspond to $m_{\rm
eff}=1$, and panels (e)-(f) correspond to $m_{\rm eff}=0.5$. Across all
panels, we observe a mixture of extended states (warm colors) and well-
localized, non-spreading wave solutions (dark blue colors). Moreover,
decreasing effective mass $m_{\rm eff}$ narrows the delocalized phase. Despite
some differences, both types of phonons exhibit qualitatively similar
behavior, see the discussion in the main text. The remaining parameters used
for this analysis are $\omega_{\rm eff}=10$, $J_{0}=1$.

In the previous sections, we have derived and then analysed a microscopic
Hamiltonian (16), governing the dynamics of an excitation coupled to phonons
through two different mechanisms, i.e., the SSH and Fröhling Hamiltonians, see
Eq. (19). While maintaining a close connection to the experimental platform,
it is important to note that in the considered Rydberg setup, the phonon
coupling strengths $g_{J}$ and $g_{W}$ are not independent. Instead, they can
both be expressed in terms of a single parameter $\kappa$, as demonstrated in
Eq. (20). Consequently, investigating the interplay between these two
competing phonon-coupling mechanisms within the current Rydberg platform
becomes challenging. To address this limitation and explore the complete phase
diagram in a more general context, in this section, we treat $g_{J}$ and
$g_{W}$ as completely independent and fix other parameters. In the initial
phase, as described in Section IV.1, our primary objective is to identify a
stable polaron regime. Specifically, we aim to find a regime in which an
initially localized excitation does not spread during the course of time
evolution. Subsequently, in Section IV.2, we demonstrate the existence of
stable islands where polarons can exhibit non-dispersive motion when subjected
to a constant force, even in the presence of substantial disorder.
Furthermore, in this part, we thoroughly examine the quantitative differences
in dynamics of optical and acoustic phonons. In the following, we set the
system size to $N=401$, and solve the equations of motions in a fixed time
interval $t\in[0,t_{f}=16.5]$. Unless explicitly stated otherwise, we also set
$m_{\rm eff}=0.5$, $\omega_{\rm eff}=10$, and $J_{0}=1$.

Figure 5: The influence of a constant force on the propagation of non-
spreading solutions. Panels (a) and (b) depict the color encoded participation
ratio, PR$(t_{f})$, for optical and acoustic phonons, respectively, showing a
shift in the boundary between extended and localized states due to the applied
force. Panels (c) and (d) display color encoded $\xi(t_{f})$, a measure for
selecting well-localized solutions propagating in a single direction. Stable
transport islands of such solutions are observed, indicated by warm colors.
Panel (c) corresponds to optical phonons, while panel (d) corresponds to
acoustic phonons. The remaining parameters used for this analysis are $F=0.2$,
$m_{\rm eff}=0.5$, $\omega_{\rm eff}=10$, $J_{0}=1$. While comparing with Fig.
4 mind a shifted color scale.

###  IV.1 Polaron formation

In the preceding section, we have already witnessed the emergence of a non-
dispersive, self-trapped polaron through the excitation-phonon coupling.
Building upon this observation, here we independently vary the two coupling
strengths, $g_{J}$, and $g_{W}$, to identify a stable polaron regime. It is
worth noting that the Hamiltonian of the system, as described by Eq. (16), is
invariant under the simultaneous transformation: $u_{j}\rightarrow-u_{j}$,
$g_{J}\rightarrow-g_{J}$, and $g_{W}\rightarrow-g_{W}$. Therefore, without
loss of generality, we can assume $g_{J}\geq 0$.

In Fig. 4, we present a phase diagram of the participation ratio PR calculated
at the final evolution time for a broad range of values: $g_{J}\in[0,45]$ and
$g_{W}\in[-16,20]$. Each panel of Fig. 4 corresponds to distinct values of
$m_{\rm eff}$ and $\eta$, as specified in the figure caption. In terms of the
layout, the left (right) column corresponds to the optical (acoustic) phonons,
and $m_{\rm eff}$ increases from top to bottom. In all panels of Fig. 4, we
observe wide regions with both extended states (warm colors) and well-
localized solutions (dark blue colors), with the latter corresponding to
stable, stationary polarons. We discover a non-trivial dependence of the
participation ratio on both coupling strengths. Moreover, we find
qualitatively similar behavior for both types of phonon, however, the acoustic
phonons exhibit greater dynamic stability. This is evident from the presence
of a chaotic-like region (the light blue dotted area), [compare with Fig. 2
and see the discussion in Sec. III]. Finally, we indicate that a decrease of
effective mass $m_{\rm eff}$ stabilizes the excitation supporting localized
polaron formation.

Figure 6: Transport of an excitation coupled to phonons under an external
force $F=0.2$. Panels a) and c) present color-encoded time-evolution of the
excitation density $|\psi_{i}|^{2}$, while panels b) and d) present color-
encoded time-evolution of the phonon field $|u_{i}|$. The top row corresponds
to optical phonons ($g_{W}=16$, $g_{J}=7$), while the bottom row corresponds
to acoustic phonons ($g_{W}=17,g_{J}=10$).

###  IV.2 Robustness of coherent transport against disorder

Figure 7: Robustness of non-dispersive moving solutions against on-site
disorder, $\epsilon_{j}\leavevmode\nobreak\ \in\leavevmode\nobreak\
[-W/2,W/2]$. Panels (a) and (b) display color encoded time evolution of the
excitation density, $|\psi_{i}|^{2}$, for optical ($g_{W}=16$, $g_{J}=7$) and
acoustic ($g_{W}=17$, $g_{J}=10$) phonons, respectively, disorder strength
$W=0.6$. Panels (c) and (d) show the center of mass position, $x(t_{f})$, and
the participation ratio, $\textrm{PR}(t_{f})$, at the final evolution time
$t_{f}$, plotted as functions of the disorder amplitude $W$ (top lines
(circles) correspond to acoustic phonons, while bottom lines (diamonds) to
optical phonons). Although the center of mass positions decreases, a
substantial macroscopic transport remains visible. The remaining parameters
used for this analysis are $F=0.2$, $m_{\rm eff}=0.5$, $\omega_{\rm eff}=10$,
$J_{0}=1$.

In this paragraph, we focus on the parameters regime, where a well-localized
excitation can be transported over a long distance. Namely, after identifying
stable polaron regimes, we proceed to apply a constant force to investigate
the propagation of non-spreading solutions.

For this analysis, we fix $F=0.2$, $m_{\rm eff}=0.5$ and select the coupling
strengths within the range $g_{J}\in[4,16]$ and $g_{W}\in[8,20]$. These
regions are indicated by a dashed square in the bottom panels of Fig. 4. The
results are presented in Fig. 5. The top row of Fig. 5 illustrates the
participation ratio, PR$(t_{f})$, for both optical [panel (a)] and acoustic
[panel (b)] phonons. In both panels, we observe a shift in the boundary
between extended and localized states due to the presence of the applied
force. However, the prevalence of dark blue colors, indicating localized
regimes, remains evident. The bottom row of Fig. 5 displays $\xi(t_{f})$, as
given by Eq. (28). This quantity serves as a measure for selecting well-
localized solutions propagating in a single direction. We observe stable
transport islands of such solutions, indicated by warm colors. Panel (c)
corresponds to optical phonons, while panel (d) corresponds to acoustic
phonons.

In Fig.6, we present an example of transportation of an excitation coupled to
phonons under an external force without disorder potential $W=0$. The left
column presents the time-evolution of excitation density $|\psi_{i}|^{2}$,
while the right column presents the evolution of the classical phonon field
$|u_{i}|$. Next, in Fig.7, we examine the robustness of the non-dispersive
moving solutions against on-site disorder, $\epsilon_{j}$, as in Eq.(16). The
disorder is introduced by assuming $\epsilon_{j}$ to be a pseudorandom
variable drawn from a uniform distribution in $[-W/2,W/2]$. Panels (a) and (b)
depict the time propagation of excitations for optical and acoustic phonons,
respectively, $W=0.6$. Panel Fig. 7(c) illustrates the center of mass
position, while Fig. 7(d) presents the participation ratio evaluated at the
final evolution time, plotted as functions of the disorder amplitude $W$. The
results are averaged over 200 independent realizations of disorder. Notably,
the participation ratio for both acoustic and optical phonons remains
relatively constant, providing evidence for the robustness of the polaron
self-trapping mechanism, while the center of mass positions takes place on a
significant distance.

##  V Summary and conclusions

In summary, we propose a quantum simulator with Rydbeg-dressed atom arrays for
SSH-Frölich Hamiltonian allowing studies of polaron formation and dynamics.
The interplay between two competing excitation-phonon coupling terms in the
model results in a rich dynamical behavior, which we comprehensively analyze.
In particular, our findings reveal the presence of asymmetry in Bloch
oscillations allowing coherent transport of a well-localized excitation over
long distances. Moreover, we compare the behavior of excitations coupled to
either acoustic or optical phonons and indicate similar qualitative behavior.
Finally, we demonstrate the robustness of phonon-assisted coherent transport
to the on-site random potential.

Our analysis is restricted to weak lattice distortions related to a small
number of phonons per lattice site, however, the proposed quantum simulator
allows the studies of the excitation dynamics in strong distortion limit, as
well as studies of a plethora of different scenarios, such as bi- and many-
polaron dynamics, and investigation of the quantum boomerang effect [83, 84,
85] affected by the presence of phonons, both in a single-particle and many-
body scenario. We believe, that our work opens up new avenues for research in
Rydberg-based quantum simulators.

###### Acknowledgements.

A.K. acknowledges the support of the Austrian Science Fund (FWF) within the
ESPRIT Programme ESP 171-N under the Quantum Austria Funding Initiative. S.K.
acknowledges the Netherlands Organisation for Scientific Research (NWO) under
Grant No. 680.92.18.05, as well as financial support from the Dutch Ministry
of Economic Affairs and Climate Policy (EZK), as part of the Quantum Delta NL
program. ICFO group acknowledges support from: ERC AdG NOQIA; MICIN/AEI
(PGC2018-0910.13039/501100011033, CEX2019-000910-S/10.13039/501100011033, Plan
National FIDEUA PID2019-106901GB-I00, FPI; MICIIN with funding from European
Union NextGenerationEU (PRTR-C17.I1): QUANTERA MAQS PCI2019-111828-2);
MCIN/AEI/10.13039/501100011033 and by the “European Union NextGeneration
EU/PRTR" QUANTERA DYNAMITE PCI2022-132919 (QuantERA II Programme co-funded by
European Union’s Horizon 2020 programme under Grant Agreement No 101017733),
Proyectos de I+D+I “Retos Colaboración” QUSPIN RTC2019-007196-7); Fundació
Cellex; Fundació Mir-Puig; Generalitat de Catalunya (European Social Fund
FEDER and CERCA program, AGAUR Grant No. 2021 SGR 01452, QuantumCAT
U16-011424, co-funded by ERDF Operational Program of Catalonia 2014-2020);
Barcelona Supercomputing Center MareNostrum (FI-2023-1-0013); EU Quantum
Flagship (PASQuanS2.1, 101113690); EU Horizon 2020 FET-OPEN OPTOlogic (Grant
No 899794); EU Horizon Europe Program (Grant Agreement 101080086 — NeQST),
National Science Centre, Poland (Symfonia Grant No. 2016/20/W/ST4/00314); ICFO
Internal “QuantumGaudi” project; European Union’s Horizon 2020 research and
innovation program under the Marie-Skłodowska-Curie grant agreement No
101029393 (STREDCH) and No 847648 (“La Caixa” Junior Leaders fellowships
ID100010434: LCF/BQ/PI19/11690013, LCF/BQ/PI20/11760031, LCF/BQ/PR20/11770012,
LCF/BQ/PR21/11840013). Views and opinions expressed are, however, those of the
author(s) only and do not necessarily reflect those of the European Union,
European Commission, European Climate, Infrastructure and Environment
Executive Agency (CINEA), nor any other granting authority. Neither the
European Union nor any granting authority can be held responsible for them.
The work J.Z. was funded by the National Science Centre, Poland under the OPUS
call within the WEAVE programme 2021/43/I/ST3/01142 as well as via project
2021/03/Y/ST2/00186 within the QuantERA II Programme that has received funding
from the European Union Horizon 2020 research and innovation programme under
Grant agreement No 101017733. A partial support by the Strategic Programme
Excellence Initiative within Priority Research Area (DigiWorld) at
Jagiellonian University is acknowledged. M.P. acknowledges the support of the
Polish National Agency for Academic Exchange, the Bekker programme no:
PPN/BEK/2020/1/00317. Views and opinions expressed are, however, those of the
author(s) only and do not necessarily reflect those of the European Union,
European Commission, European Climate, Infrastructure and Environment
Executive Agency (CINEA), nor any other granting authority. Neither the
European Union nor any granting authority can be held responsible for them.

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