# Synthetic $\mathbb{Z}_{2}$ gauge theories based on parametric excitations of
trapped ions

O. Băzăvan  Department of Physics, University of Oxford, Clarendon Laboratory,
Parks Road, Oxford OX1 3PU, United Kingdom  S. Saner  Department of Physics,
University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, United
Kingdom  E. Tirrito  The Abdus Salam International Centre for Theoretical
Physics (ICTP), Strada Costiera 11, 34151 Trieste, Italy  SISSA, Via Bonomea
265, 34136 Trieste, Italy  G. Araneda  Department of Physics, University of
Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, United Kingdom  R.
Srinivas  Department of Physics, University of Oxford, Clarendon Laboratory,
Parks Road, Oxford OX1 3PU, United Kingdom  A. Bermudez  Instituto de Física
Teorica, UAM-CSIC, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid,
Spain.

###### Abstract

We present a detailed scheme for the implementation of $\mathbb{Z}_{2}$ gauge
theories with dynamical bosonic matter using analog quantum simulators based
on crystals of trapped ions. We introduce a versatile toolbox based on a
state-dependent parametric excitation, which can be implemented using
different interactions that couple the ions’ internal qubit states to their
motion, and induces a tunneling of the vibrational excitations of the crystal
mediated by the trapped-ion qubits. To evaluate the feasibility of this
toolbox, we perform numerical simulations of the considered schemes using
realistic experimental parameters. This building block, when implemented with
a single trapped ion, corresponds to a minimal $\mathbb{Z}_{2}$ gauge theory
on a synthetic link where the qubit resides, playing the role of the gauge
field. The vibrational excitations of the ion along different trap axes mimic
the dynamical matter fields carrying a $\mathbb{Z}_{2}$ charge. We discuss how
to generalise this minimal case to more complex settings by increasing the
number of ions. We describe various possibilities which allow us to move from
a single $\mathbb{Z}_{2}$ plaquette to full $\mathbb{Z}_{2}$ gauge chains. We
present analytical expressions for the gauge-invariant dynamics and
confinement, which are benchmarked using matrix product state simulations.

###### Contents

  1. I Introduction
  2. II Synthetic dimensions: lattice field theories under background gauge fields
    1. II.1  Parametric tunnelling and synthetic dimensions
    2. II.2 Peierls ladders with trapped-ion chains
  3. III Dynamical gauge fields: the $\mathbb{Z}_{2}$ theory on a link
    1. III.1 State-dependent parametric tunnelling
    2. III.2 One-boson sector: Rabi oscillations and matter-gauge-field correlated dynamics
    3. III.3 Two-boson sector: Dark states and entanglement between modes of the matter fields
  4. IV Trapped-ion toolbox: phonons and qubits
    1. IV.1  Scheme I: Analog scheme for the $\mathbb{Z}_{2}$ link
    2. IV.2 Scheme II: Pulsed scheme for the $\mathbb{Z}_{2}$ link
    3. IV.3  $\mathbb{Z}_{2}$ gauge link scheme comparison and other sources of noise
    4. IV.4  Comparison with neutral atoms
  5. V Minimal plaquettes and synthetic dimensional reduction for a $\mathbb{Z}_{2}$ gauge chain
    1. V.1  $\mathbb{Z}_{2}$ plaquette: Wegner-Wilson and ’t Hooft loops for gauge-field entanglement
    2. V.2  $\mathbb{Z}_{2}$ chain: synthetic dimensional reduction
  6. VI Conclusions and Outlook
  7. A Quadrupole light-shift scheme

##  I Introduction

In quantum many-body physics, a major research effort has been devoted to
identifying the guiding principles that explain how a rich variety of complex
collective phenomena can emerge from apparently-simple microscopic models
Anderson (1972). In this context, the invariance of a physical model under a
specific global symmetry and its spontaneous breakdown Landau (1937) have
turned into a key mechanism to understand and characterise many important
phases of matter. In parallel, understanding how these global symmetries can
be gauged, i.e. how they can be upgraded to local symmetries through the
introduction of additional gauge fields Gross (1996), has been of paramount
importance in our efforts to unveil the fundamental laws of nature Yang and
Mills (1954). Since equilibrium states can only display a non-zero expectation
value for a gauge-invariant observable Elitzur (1975), these local symmetries
cannot be spontaneously broken, and one must look for different principles and
characterisation tools to understand possible phases in gauge theories. For
instance, understanding the specific mechanism for the (de)confinement of
quarks Greensite (2020), which is a non-perturbative phenomenon that can be
addressed directly on a lattice Wilson (1974), could be relevant in the search
for other exotic forms of matter Kogut and Stephanov (2003).

Understanding the ordering mechanisms underlying these exotic phases is a very
complicated open problem in the standard model of particle physics Peskin and
Schroeder (1995), which has been partially hindered by current limitations of
numerical lattice approaches in addressing finite-density regimes and real-
time effects Troyer and Wiese (2005); Nagata (2022). One can gain important
insights by looking at lower dimensions and simpler gauge groups, where tools
that could the be ported to the more complex scenarios can be developed and
benchmarked. For example, the study of gauge theories in one spatial dimension
has been an important step to advance our understanding of confinement
Schwinger (1962); Casher  _et al._ (1973); Coleman  _et al._ (1975); ’t Hooft
(1974); Callan  _et al._ (1976). In another example, exploring models with a
discrete gauge group, such as the pure $\mathbb{Z}_{2}$ gauge theory in two
spatial dimensions Wegner (1971), has identified the importance of non-local
order parameters, the so-called Wegner-Wilson loops Wilson (1974); Wegner
(1971), to characterise a confinement-deconfinement phase transition. The
deconfined phase of this model displays an exotic collective ordering, so-
called topological order Wen (2007, 2017), with a ground state that displays
two important features. First, it has a degeneracy that depends on topological
invariants related to the homology of electric- or magnetic-like field lines
on the manifold in which the model is defined. Second, it displays long-range
entanglement in spite of having an energy gap which, typically, would be
associated to an exponential decay of correlations and short-range
entanglement.

Let us now review some aspects of these discrete-group lattice gauge theories
that also make them interesting from a broader perspective. First of all,
discrete gauge theories can arise in long-wavelength descriptions of
condensed-matter models, e.g. high-temperature superconductivity and
frustrated magnetism Baskaran and Anderson (1988); Read and Sachdev (1991). In
the Hamiltonian formulation of lattice gauge theories Kogut and Susskind
(1975), space is discretised, and the matter and gauge fields are represented
by operators defined on the sites and links of the lattice, respectively. One
typically works in Weyl’s temporal gauge, where there is a residual redundancy
associated to the local gauge symmetries. Using Gauss’ law, one restricts the
dynamics to a specific super-selection sector of the full Hilbert space. On
the other hand, from the perspective of effective models in condensed matter
Fradkin (2013), other super-selection possible sectors can arise, which are
described by specific distributions of effective static charges that act like
a background Moessner  _et al._ (2001). Furthermore, one can even design the
Hamiltonian in such a way that Gauss’ law is not a strict constraint, but
rather an energy penalty that becomes effective in the ground state, but can
be violated to create certain excitations Kitaev (2003). This allows to define
both magnetic- and electric-type excitations, which have another
characteristic property of topological order: mutual anyonic statistics.
Understanding the robustness of the ground state under external perturbations
Trebst  _et al._ (2007); Vidal  _et al._ (2009, 2009), as well as the creation
and subsequent dynamics of these excitations, is relevant in the context of
topological quantum error correction Dennis  _et al._ (2002); Terhal (2015).
Some of these perturbations Tupitsyn  _et al._ (2010) can be actually mapped
onto the problem of $\mathbb{Z}_{2}$ gauge fields coupled to matter Fradkin
and Shenker (1979). It is worth mentioning that these $\mathbb{Z}_{2}$ gauge
theories with a soft Gauss’ constraint can also arise in perturbative regimes
of more-standard models of quantum magnetism Kitaev (2006), which are actively
being searched for in certain solid-state materials Takagi  _et al._ (2019);
Motome and Nasu (2020).

Due to all of these cross-disciplinary connections between high-energy
physics, condensed matter, and quantum information, the study of discrete
lattice gauge theories, including higher-dimensional $\mathbb{Z}_{d}$ gauge
fields Horn  _et al._ (1979) and their coupling to distinct types of matter,
has seen a considerable increase of interest in recent years Assaad and Grover
(2016); Gazit  _et al._ (2017, 2018); Smith  _et al._ (2018); Gazit  _et al._
(2020); König  _et al._ (2020); Emonts  _et al._ (2020); Robaina  _et al._
(2021); Borla  _et al._ (2022). There has also been a remarkable progress in
one-dimensional Ercolessi  _et al._ (2018); Magnifico  _et al._ (2019a, b);
Borla  _et al._ (2020); Kebrič  _et al._ (2021); Magnifico  _et al._ (2020);
Surace and Lerose (2021); Aramthottil  _et al._ (2022); Das  _et al._ (2021);
del Pino and Zilberberg (2023); Ge and Nori (2022); Domanti  _et al._ (2023)
and quasi-one-dimensional Sugihara (2005); González-Cuadra  _et al._ (2020);
Nyhegn  _et al._ (2021); Brenig (2022); Pradhan  _et al._ (2022) cases. We
note that, in the limit $d\to\infty$, one can recover the physics of a
continuous gauge group from these discrete gauge theories, such that these
studies connect to the quantum-electrodynamics sector of the standard model in
reduced spacetime dimensions. In this respect, we should also mention the
studies of lattice gauge theories based on the “quantum-link” approach Horn
(1981); Orland and Rohrlich (1990); Chandrasekharan and Wiese (1997). In this
approach, the gauge degrees of freedom belong again to a finite-dimensional
Hilbert space but, this time, one can represent directly the continuous gauge
group by using link operators corresponding to $(2S+1)$-dimensional
representations of the angular momentum, i.e., spins. There has been a
considerable resurgence of interest in these link models during the last
years, both for Abelian and non-Abelian gauge groups Banerjee  _et al._
(2013a); Rico  _et al._ (2014); Kühn  _et al._ (2014); Pichler  _et al._
(2016); Cardarelli  _et al._ (2017); Huang  _et al._ (2019); Silvi  _et al._
(2019); Cardarelli  _et al._ (2020); Banerjee and Sen (2021); Felser  _et al._
(2020); Lerose  _et al._ (2020); Papaefstathiou  _et al._ (2020); Magnifico
_et al._ (2021); Zache  _et al._ (2021); Halimeh  _et al._ (2021); Hashizume
_et al._ (2022); Banerjee  _et al._ (2022); Halimeh  _et al._ (2022).

The focus on discrete gauge theories has been further encouraged by key
advances in the field of quantum simulations Feynman (1982); Cirac and Zoller
(2012); Bloch  _et al._ (2012); Blatt and Roos (2012); Altman  _et al._
(2021). Starting with the pioneering proposals for the digital Weimer  _et
al._ (2010); Tagliacozzo  _et al._ (2013a, b); Glaetzle  _et al._ (2014) and
analog Büchler  _et al._ (2005); Zohar and Reznik (2011); Zohar  _et al._
(2012); Banerjee  _et al._ (2012); Zohar  _et al._ (2013a); Banerjee  _et al._
(2013b); Zohar  _et al._ (2013b) quantum simulation of lattice gauge theories
with ultracold atoms in optical lattices, a considerable effort has been
devoted to push this research topic in various directions (see reviews Wiese
(2013); Zohar  _et al._ (2015); Dalmonte and Montangero (2016); Bañuls  _et
al._ (2020); Bañuls and Cichy (2020); Aidelsburger  _et al._ (2022); Klco  _et
al._ (2022); Bauer  _et al._ (2022)). These advances have lead to the first
experimental quantum simulators for lattice gauge theories Martinez  _et al._
(2016); Dai  _et al._ (2017); Klco  _et al._ (2018); Schweizer  _et al._
(2019); Kokail  _et al._ (2019); Surace  _et al._ (2020); Klco  _et al._
(2020); Mil  _et al._ (2020); Yang  _et al._ (2020); Zhou  _et al._ (2022);
Atas  _et al._ (2021); Bauer  _et al._ (2021); Nguyen  _et al._ (2022);
Ciavarella  _et al._ (2021); A Rahman  _et al._ (2021); Ciavarella and
Chernyshev (2022); Wang  _et al._ (2022); Atas  _et al._ (2022); Farrell  _et
al._ (2022a, b); Frölian  _et al._ (2022); Charles  _et al._ (2023), hinting
at a promising future in the near term. We would like to discuss briefly the
potential of this type of experiments from the perspective of quantum
advantage. Quantum simulators do not suffer from the sign problem underlying
the aforementioned limitations of numerical Monte Carlo approaches Troyer and
Wiese (2005); Nagata (2022), even when dealing with fermionic matter at finite
densities and real-time dynamics. Therefore, they have the potential of
addressing certain problems that have remained elusive for decades. In fact,
solving the sign problem lies in the class of $NP$ (nondeterministic
polynomial time)-hard problems Troyer and Wiese (2005), such that no
polynomial-time classical algorithm is likely to be found. This strengthens
the potential of quantum simulators as a way to go beyond the capabilities of
classical computers and demonstrate quantum advantage in a problem of
practical relevance. Moreover, recent works have shown that digital quantum
simulations for the dynamics of even simpler lattice field theories, such as
the self-interacting scalar field, are provably among the hardest problems
that can be solved efficiently with quantum hardware Jordan  _et al._ (2012,
2011, 2018). Computing the vacuum persistence amplitude, which gives access to
arbitrary Feynman propagators via the generating functional, is a $BQP$
(bounded-error quantum polynomial time)-hard problem. Its solution would allow
one to solve any other problem that is solvable in polynomial time by a
quantum computer. Unless the $BQP$ complexity class reduces to its classical
analog, computing the generating function of a quantum field theory is already
a problem that cannot be addressed efficiently using classical devices, but
requires instead quantum-mechanical ones Bermudez  _et al._ (2017a); Martín-
Vázquez  _et al._ (2022). We note that this complexity-class collapse is very
unlikely, as it would imply that any quantum computation can be efficiently
simulated by a classical computer, which is not believed to be the case within
our current understanding.

Let us close this general introduction by commenting further on the
experimental progress of quantum simulators for gauge theories. Experimental
realisations based on the concatenation of gates, which exploit optimised
encodings for the discretised fields using the underlying symmetries and
Gauss’ law, have already allowed for the implementation of several digital
quantum simulations and variational quantum algorithms of lattice gauge
theories Martinez  _et al._ (2016); Klco  _et al._ (2018); Kokail  _et al._
(2019); Klco  _et al._ (2020); Atas  _et al._ (2021); Bauer  _et al._ (2021);
Nguyen  _et al._ (2022); Ciavarella  _et al._ (2021); A Rahman  _et al._
(2021); Ciavarella and Chernyshev (2022); Atas  _et al._ (2022); Farrell  _et
al._ (2022a, b); Charles  _et al._ (2023). Although less flexible, the
experiments for the analog quantum simulation of gauge fields Dai  _et al._
(2017); Surace  _et al._ (2020); Schweizer  _et al._ (2019); Mil  _et al._
(2020); Yang  _et al._ (2020); Zhou  _et al._ (2022); Wang  _et al._ (2022);
Frölian  _et al._ (2022) are, in principle, more amenable to be scaled to
larger systems even in the presence of noise and experimental imperfections.
This is due to the accumulation of errors. For digital simulators, the errors
grow fast when enlarging the simulated system and extending the simulated
time, since the number of imperfect gates that must be used increases. For the
typical intensive observables of interest, however, the error of analog
simulators does not increase with the simulated time and lattice volume as
fast as in the case of its digital concatenated-gate counterparts. This
difference has been recently assessed quantitatively for condensed-matter
simulators Flannigan  _et al._ (2022), and should hold similarly for gauge
theories.

In this work, we thus focus on analog quantum simulators of $\mathbb{Z}_{2}$
lattice gauge theories coupled to dynamical matter. In previous realisations,
this gauge theory has only been addressed in the single-link cold-atom
experiments Schweizer  _et al._ (2019), and superconducting-qubit arrays Wang
_et al._ (2022). Unfortunately, the later is limited by additional microscopic
terms that explicitly break the gauge symmetry; whereas the former has not
been scaled to larger system sizes yet. In this manuscript, we present in
detail a new toolbox for quantum simulations of $\mathbb{Z}_{2}$ lattice gauge
theories coupled to matter using trapped-ion systems. This toolbox includes
simple building blocks that can be realised with state-of-the-art ion trap
technologies, but also, more scalable schemes that could be implemented upon
further technological developments in the near term. We believe that the
results hereby presented open an interesting direction to extend the
experiments on $\mathbb{Z}_{2}$ lattice gauge theories coupled to matter to
more interesting and challenging scenarios.

Our results are presented as follows. In Sec. II, we describe the underlying
idea of a scheme to induce background gauge fields using a parametric
frequency conversion. We discuss how this scheme can be implemented using the
transverse vibrational modes in a trapped-ion chain, giving rise to a
synthetic Hall ladder for the bosonic modes. In Sec. III, based on the
previous discussion, we show how one can generate a gauge-invariant
Hamiltonian where a $\mathbb{Z}_{2}$ gauge field mediates the tunnelling
between a pair of bosonic modes. In the single-boson sector, where the
tunnelling of the boson is correlated to the stretching/compressing of the
electric field at the link, the dynamics corresponds to detuned Rabi
oscillations. In the two-boson sector, we show that the system gives rise to
bright and dark states, and that the gauge-invariant dynamics leas to mode
entanglement in the matter sector. In Sec. IV, we describe various realistic
schemes for the implementation of this $\mathbb{Z}_{2}$ gauge theory on a link
using a single trapped ion. We discuss a light-shift- or a Mølmer-Sørensen-
type scheme to implement a state-dependent parametric tunnelling between the
phonons along two different trap axes, presenting a thorough numerical
comparison of the ideal and realistic dynamics with current trapped-ion
parameters. We also discuss a different possibility that combines two
orthogonal state-dependent forces, and can be combined in a pulsed
Trotterization to yield the desired gauge-invariant model with realistic
trapped-ion parameters. Finally, we show in Sec. V how one can also realise
other $\mathbb{Z}_{2}$ models by including more ions. In particular, we show
that the center-of-mass modes of a two-ion system can be used to simulate a
gauge-invariant model with two links forming a plaquette. We show that the
superposition principle of the possible encircling paths is that a single
boson can lead to an entangled state for the $\mathbb{Z}_{2}$ gauge fields. In
this section, we also show how a string of ions with dimerised center-of-mass
modes can also be used to implement a $\mathbb{Z}_{2}$ gauge theory on an
entire chain. We present analytical solutions for the confinement dynamics on
the single- and two-boson sector, and also explore the phenomenon of string
breaking in the half-filled sector using Matrix-Product-State simulations. We
present our conclusions in Sec. VI.

##  II Synthetic dimensions: lattice field theories under background gauge
fields

###  II.1  Parametric tunnelling and synthetic dimensions

Using periodic resonant modulations, it is possible to design quantum
simulators on a lattice that has a connectivity different from the original
one. This leads to the concept of synthetic dimensions Boada  _et al._ (2012);
Celi  _et al._ (2014), as recently reviewed in Ozawa and Price (2019), which
have been realised in various experimental platforms Mancini  _et al._ (2015);
Stuhl  _et al._ (2015); Livi  _et al._ (2016); Han  _et al._ (2019); Bernier
_et al._ (2017); An  _et al._ (2017); Fang  _et al._ (2017); Zilberberg  _et
al._ (2018); Lustig  _et al._ (2019); Dutt  _et al._ (2019, 2020); Chalopin
_et al._ (2020). We start this section by discussing parametric excitations
from this perspective. In its original context, a parametric excitation can be
used to induce couplings between different modes of the electromagnetic field,
leading to a well-known technique for frequency conversion and linear
amplification of photons Louisell  _et al._ (1961); Haus and Mullen (1962);
Gordon  _et al._ (1963); Mollow and Glauber (1967); Caves (1982). For
instance, as originally discussed in Louisell  _et al._ (1961), a small
periodic modulation of the dielectric constant of a cavity can lead to
different couplings between the cavity modes that can be controlled by tuning
the modulation frequency to certain resonances. In the particular case of
frequency conversion, this scheme can be understood in terms of a parametric
tunnelling term between two synthetic lattice sites labelled by the
frequencies of the two modes, which is the essence of the schemes for
synthetic dimensions discussed in Boada  _et al._ (2012); Celi  _et al._
(2014). As emphasised in Fang  _et al._ (2012a), this simple parametric
tunnelling already inherits the phase of the drive Louisell  _et al._ (1961),
such that one could design and implement Fang  _et al._ (2013) non-trivial
schemes where this phase mimics the Aharonov-Bohm effect of charged particles
moving under a static background magnetic field Aharonov and Bohm (1959).
These ideas can also be exploited when the modes belong to distant resonators
coupled via intermediate components, such as mixers Fang  _et al._ (2012b) or
tunable inductors Chen  _et al._ (2014). This leads to parametric tunnelling
terms where the phase can be tuned locally, leading to quantum simulators of
quantum Hall-type physics Roushan  _et al._ (2017). Furthermore, periodic
modulations of the mode frequencies with a relative phase difference can also
lead to these synthetic background gauge fields Bermudez  _et al._ (2011);
Hauke  _et al._ (2012), as demonstrated in experiments that exploit Floquet
engineering in optical lattices Aidelsburger  _et al._ (2013); Miyake  _et
al._ (2013); Atala  _et al._ (2014); Aidelsburger  _et al._ (2015),
symmetrically-coupled resonators Estep  _et al._ (2014), and trapped-ion
crystals Kiefer  _et al._ (2019). We should also mention that there are other
schemes for the simulations of static background gauge fields, which do not
exploit periodic modulations, but instead mediate the tunnelling by an
intermediate quantum-mechanical system Koch  _et al._ (2010); Hafezi  _et al._
(2011, 2013); Mittal  _et al._ (2014, 2018); Manovitz  _et al._ (2020);
Shapira  _et al._ (2022).

Given the importance of the idea of parametric tunnelling for the quantum
simulation schemes of dynamical gauge theories presented in the following
sections, we now describe its details. We consider a set of bosonic/fermionic
particles that can be created and annihilated by a set of operators
$a^{\dagger}_{d},a^{\phantom{\dagger}}_{d}$ with the corresponding
commutation/anti-commutation algebra. Here, the index $d\in\mathcal{D}$ labels
a specific degree of freedom of these particles. In the context of quantum
many-body models and lattice field theories, the indexing set $\mathcal{D}$
typically contains the positions of a microscopic lattice in which the
particles can reside, as already mentioned in the introduction. In this
context, the geometry of the lattice determines the kinetic energy of the
microscopic Hamiltonian, which is described by a tunnelling term
$t^{\phantom{\dagger}}_{dd^{\prime}}a^{\dagger}_{d}a^{\phantom{\dagger}}_{d^{\prime}}$,
where $t^{\phantom{\dagger}}_{dd^{\prime}}$ is the hopping matrix element
between a pair of sites labeled by $d\neq d^{\prime}$. Typically, these
tunnelings decay very fast with the distance, and one only considers nearest
neighbours, such that the connectivity of the lattice, i.e. the edges/links of
a graph, gets directly in-built in the tunnelling matrix. Additionally, in
condensed matter, $d\in\mathcal{D}$ can contain other internal degrees of
freedom, e.g. spin of the valence electrons. In the context of synthetic
dimensions, it is these extra degrees of freedom that provide us with a new
means to engineer a synthetic dimension.

The idea of synthetic dimensions is that the effective connectivity of the
tunnelling matrix can be externally designed by introducing additional
periodic drivings. These, in fact, induce new couplings that can be
interpreted as effective edges/links even when the corresponding degrees of
freedom are not related to any Bravais lattice at all. A possible scheme uses
a parametric tunnelling, as illustrated now with a simple example. We consider
two modes $d\in\mathcal{D}=\\{1,2\\}$ of energies $\omega_{d}$ ( $\hbar=1$
henceforth), such that the bare Hamiltonian is

$H_{0}=\omega_{1}a^{{\dagger}}_{1}a^{\phantom{\dagger}}_{1}+\omega_{2}a^{{\dagger}}_{2}a^{\phantom{\dagger}}_{2}.$
(1)

One now adds the following parametric excitation

$V(t)=\Omega_{\rm d}a^{{\dagger}}_{2}a^{\phantom{\dagger}}_{1}\cos(\phi_{\rm
d}-\omega_{\rm d}t)+{\rm H.c.},$ (2)

where $\Omega_{\rm d}$, $\omega_{\rm d}$, and $\phi_{\rm d}$ are the
amplitude, frequency, and phase of the drive, respectively. In the parametric
regime, i.e.,

$\omega_{\rm d}=\omega_{2}-\omega_{1},\hskip 8.61108pt|\Omega_{\rm d}|\ll
4|\omega_{2}-\omega_{1}|,$ (3)

one can show that an effective tunnelling term between both modes is induced
by the drive. Going to the interaction picture with respect to Eq. (1), it is
straightforward to recognise that the resonance condition of Eq. (3) provides
the required energy to bridge the gap between the modes and couple them.
Additionally, when the driving amplitude is constrained by Eq. (3), a
rotating-wave approximation shows that the mode coupling $V_{\rm I}(t)\approx
H_{\rm eff}$ becomes a time-independent effective Hamiltonian with a simple
frequency-conversion term

$H_{\rm
eff}=t^{\phantom{\dagger}}_{1,\textbf{e}_{1}}a^{{\dagger}}_{2}a^{\phantom{\dagger}}_{1}+{\rm
H.c.},\hskip 8.61108pt\text{with}\hskip
4.30554ptt^{\phantom{\dagger}}_{1,\textbf{e}_{1}}=\frac{\Omega_{\rm d}}{2}{\rm
e}^{{\rm i}\phi_{\rm d}}.$ (4)

In the context of synthetic dimensions, one finds that a non-zero tunnelling
has been established, which could be understood as a new connectivity link of
a synthetic lattice. This tunnelling
$t^{\phantom{\dagger}}_{1,\textbf{e}_{1}}$ is labelled by the synthetic
lattice site index 1 from which the particle departs, and the unit link vector
$\textbf{e}_{1}$ that connects it to the lattice site 2, into which the
particle tunnels. In this simple case, accordingly, the synthetic lattice is
just composed of two sites labelled by the indexes of the mode frequencies. We
note that, for a single link, the complex phase of the tunnelling is trivial
and has no dynamical consequences, i.e. it can be readily gauged away by a
local $U(1)$ transformation acting on the modes. However, this parametric
scheme can be generalised to a larger set $\mathcal{D}$, in which the complex
phase of the effective tunnelling (4) may have non-trivial consequences. As
discussed in Fang  _et al._ (2012b); Roushan  _et al._ (2017), one can create
synthetic lattices in a way that, when the particle tunnels around a closed
path $\gamma$, it gains a non-zero phase
$\sum_{\ell\in\gamma}\phi_{\ell}=\Phi_{\rm AB}$ that simulates a synthetic
Aharonov-Bohm phase. Even if the particles have a vanishing charge, their
tunnelling resembles that of a charged particle in an external magnetic field
via the so-called Peierls’ substitution Peierls (1933), which originally
concerned electrons in a narrow-band material subjected to a perpendicular
magnetic field Hofstadter (1976). The quadratic lattice models with Peierls’
phases provide a playground for studying the integer quantum Hall effect and
topological band theory Thouless  _et al._ (1982). In the following section,
we will discuss this point in detail.

###  II.2 Peierls ladders with trapped-ion chains

So far, we have not yet discussed in detail how the parametric term (2) can be
created and controlled in a specific physical system. The parametric scheme
has been implemented in arrays of superconducting circuits Roushan  _et al._
(2017) but, to the best of our knowledge, its realisation in trapped-ion
crystals has not been discussed so far. We now describe how to exploit this
method to build a quantum simulator of a bosonic quantum Hall ladder using the
transverse vibrations of a chain of $N$ trapped ions in a linear Paul trap
Raizen  _et al._ (1992). This will serve as a warm-up for the scheme of
dynamical $\mathbb{Z}_{2}$ gauge fields that is the core of our work, which
will exploit similar concepts with a new twist, and will be covered in the
following section.

Figure 1: Transverse vibrational excitations: Schematic representation of an
ion chain in a linear Paul trap. In the insets, we represent the transverse
local vibrational excitations of a single ion in the chain.

Following Steane (1997); James (1998); Marquet  _et al._ (2003), for a linear
Paul trap with trap frequencies $\omega_{z}\ll\omega_{x},\omega_{y}$, the ions
form a linear chain along the $z$-axis (see Fig. 1). The transverse vibrations
of each ion Porras and Cirac (2004) around its equilibrium position are
described by

$H_{0}=\sum_{d}\omega^{\phantom{\dagger}}_{d}a^{{\dagger}}_{d}a^{\phantom{\dagger}}_{d}+\sum_{d\neq
d^{\prime}}t^{\phantom{\dagger}}_{dd^{\prime}}a^{{\dagger}}_{d}a^{\phantom{\dagger}}_{d^{\prime}}.$
(5)

Here, the labelling index reads $d=(i,\alpha)\in\mathcal{D}$, and the set
$\mathcal{D}$ contains the label for the ions in the chain
$i\in\\{1,\cdots,N\\}$, and the label for the two possible directions of the
vibrations transverse to the chain $\alpha\in\\{x,y\\}$. In addition,
$a^{{\dagger}}_{d},a^{\phantom{\dagger}}_{d}$ are the bosonic creation-
annihilation operators for the corresponding local vibrations around the
equilibrium positions of the ions $\bm{r}_{i}^{0}$, which we have assumed to
be aligned along the null of the radio-frequency (rf) pseudo-potential of the
linear Paul trap, such that excess micromotion can be neglected Berkeland  _et
al._ (1998). Additionally, the modulation frequencies to be introduced below
must be much lower than the rf driving of the trap to also neglect the
intrinsic quantum-mechanical micromotion Bermudez  _et al._ (2017b). As
discussed in Marquet  _et al._ (2003), the expansion of the Coulomb
interaction to second order leading to Eq. (5) does not mix the $x,y$ modes,
and one finds that the tunnelling matrix decay with the inter-ion distance
following a dipolar law

$t_{(i,\alpha)(j,\beta)}=\frac{1}{2m\omega_{\alpha}}\frac{e^{2}}{8\pi\epsilon_{0}|\bm{r}^{0}_{i}-\bm{r}_{j}^{0}|^{3}}\delta_{\alpha,\beta},$
(6)

where $m$ is the ion mass, $\epsilon_{0}$ the vacuum permittivity, and
$\delta_{\alpha,\beta}$ is the Kronecker delta. The on-site energies are
related to the effective trap frequencies $\omega_{\alpha}$ of the time-
averaged pseudo-potential Porras and Cirac (2004) by the following expression

$\omega^{\phantom{\dagger}}_{i,\alpha}=\omega_{\alpha}-\sum_{j\neq
i}\frac{1}{2m\omega_{\alpha}}\frac{e^{2}}{8\pi\epsilon_{0}|\bm{r}^{0}_{i}-\bm{r}_{j}^{0}|^{3}}.$
(7)

Since the Hamiltonian of Eq. (5) has a global $U(1)\times U(1)$ symmetry under
$a^{\phantom{\dagger}}_{d}\mapsto{\rm e}^{{\rm
i}\varphi}a^{\phantom{\dagger}}_{d}$, $a^{\dagger}_{d}\mapsto{\rm e}^{-{\rm
i}\varphi}a^{\dagger}_{d}$, the number of transverse vibrational excitations
along each axis is individually conserved. Although phonons in crystals
typically refer to the excitations of the collective vibrational modes, it is
customary to refer to these local vibrational modes also as phonons in the
trapped-ion community, and we will follow this convention in the rest of the
manuscript. The novelty with respect to the crystal phonons underlying the
transverse sound waves in elastic solids Martín-Vázquez  _et al._ (2022) is
that, since their number is conserved when $|t_{dd^{\prime}}|\ll
2\omega_{\alpha}$ Porras and Cirac (2004), we can thus think of these
transverse phonons as particles localised to each of the ions. Just like
electrons on a solid, the phonons tend to spread over the chain due to the
dipolar tunnelling of Eq. (6). We note that the dynamics of these local
phonons due to the effective tight-biding Hamiltonian of Eq. (5) has been
observed in various trapped-ion experiments Brown  _et al._ (2011); Harlander
_et al._ (2011); Wilson  _et al._ (2014); Ramm  _et al._ (2014); Toyoda  _et
al._ (2015); Debnath  _et al._ (2018); Hakelberg  _et al._ (2019); Kiefer  _et
al._ (2019); Tamura  _et al._ (2020).

Let us now discuss how to exploit parametric excitations to realise synthetic
phonon ladders subjected to an effective background gauge field. There have
been some prior works on trapped-ion parametric drivings which, to the best of
our knowledge, have only been considered in a different context. For instance,
in Heinzen and Wineland (1990), parametric modulations are used to design
cooling and detection methods for the spectroscopy of a single trapped
electron and proton, as well as for squeezing and linear amplification. The
former requires a parametrically-modulated quadrupole potential that couples
two different vibrational directions Cornell  _et al._ (1990), whereas the
latter employs a periodic modulation of the trap frequencies, and can thus be
achieved by applying an additional oscillating potential to the rf trap
electrodes. Parametric modulations can also be obtained optically, exploiting
the cross-beam ac-Stark shift of a pair of far-detuned laser beams Wineland
_et al._ (1998a); Meekhof  _et al._ (1996). Although the parametric
modulations obtained through the electronic equipment have led to larger
amplification in recent experiments Burd  _et al._ (2019), we will stick to
optical ones in this work, as they are more flexible for the generation of
synthetic gauge fields.

Our goal is to interpret the transverse vibrational directions of Fig. 1 as a
new synthetic “dimension”. Note that, however, the $x$ and $y$ directions are
decoupled at this quadratic order (Eq. (6)), such that the Hamiltonian of Eq.
(5) describes two decoupled dipolar chains. We now discuss how a parametric
excitation of the tunnelling can be induced, and how this term can be used to
derive a model with couplings between the two chains, such that the global
symmetry reduces to $U(1)\times U(1)\mapsto U(1)$, and only the total number
of transverse vibrational quanta is conserved. We will see that, from this
perspective, the phonons move in a synthetic two-leg ladder and, moreover,
argue that they can also be subjected to an effective Peierls’ phase mimicking
the microscopic model for charged particles under external magnetic fields. We
consider that each ion is illuminated by a global two-beam laser field, the
beat note of which is far detuned from any electronic transition Wineland  _et
al._ (1998a). The ions, all prepared in the same internal state of the ground-
state manifold 111A similar argument can be made for a qubit encoding formed
by two sub-levels of a metastable manifold., thus experience an ac-Stark shift
that yields the following optical potential

$V(t)=\sum_{n,n^{\prime}=1,2}\sum_{i=1}^{N}\Omega_{n,n^{\prime}}{\rm e}^{{\rm
i}(\bm{k}_{{\rm L},n}-\bm{k}_{{\rm
L},n^{\prime}})\cdot\bm{r}_{i}-(\omega_{{\rm L},n}-\omega_{{\rm
L},n^{\prime}})t}+{\rm H.c.}.$ (8)

Here, $\omega_{{\rm L},n}$($\bm{k}_{{\rm L},n}$) is the frequency (wave-
vector) of each beam $n\in\\{1,2\\}$ of the global laser field, and
$\Omega_{n,n^{\prime}}=-\Omega_{{\rm L},n}^{\phantom{*}}\Omega_{{\rm
L},n^{\prime}}^{*}/4\Delta$ is the ac-Stark shift arising from two-photon
processes. In those processes, a photon is absorbed from the $n$-th beam by
the $i$-th ion with the Rabi frequency $\Omega_{{\rm L},n}$ and a large
detuning $\Delta$, such that the ion is only virtually excited. Subsequently,
the ion is de-excited to the same internal state by emitting a photon onto the
$n^{\prime}$-th beam Wineland  _et al._ (1998a).

In addition to the standard ac-Stark shifts in Eq. (8), i.e. terms with
$n=n^{\prime}$ that contribute with an energy shift $\Delta E_{\rm
ac}=\sum_{n}\Omega_{n,n}$, one also obtains crossed beat note terms that lead
to periodic modulations in space (time) when the laser wave-vectors
(frequencies) are not co-linear (equal). In this case, one defines the beat
note wave-vector and frequency as $\bm{k}_{\rm d}=\bm{k}_{{\rm
L},1}-\bm{k}_{{\rm L},2}$, and $\omega_{\rm d}=\omega_{{\rm L},1}-\omega_{{\rm
L},2}$, respectively. Noting that the ion positions can be expanded in terms
of the local phonon operators via
$\bm{r}^{\phantom{0}}_{i}=\bm{r}^{0}_{i}+\sum_{\alpha}\textbf{e}_{\alpha}\frac{1}{\sqrt{2m\omega_{\alpha}}}(a^{\phantom{\dagger}}_{i,\alpha}+a^{\dagger}_{i,\alpha})$,
where $\textbf{e}_{\alpha}$ is the unit vector in the direction of the
transverse ion vibration $\alpha$, one can substitute and expand the optical
potential (8) in the so-called Lamb-Dicke regime

$\eta_{\alpha}={\bm{k}_{\rm
d}\cdot\textbf{e}_{\alpha}}/{\sqrt{2m\omega_{\alpha}}}\ll 1.$ (9)

The Taylor expansion of Eq. (8) then leads to a sum of terms with all possible
powers of the phonon operators. By choosing the correct beat note frequency,
it is possible to select which of them brings in the leading contribution. In
particular, for

$\omega_{\rm d}=\omega_{y}-\omega_{x},\hskip 8.61108pt|\Omega_{\rm
d}|\ll|\omega_{y}-\omega_{x}|,$ (10)

where we assume that $|\omega_{x}-\omega_{y}|\ll\omega_{x},\omega_{y}$, a
rotating-wave approximation shows that the optical potential contains the
desired parametric excitation of a tunnelling term that generalises Eq. (2) to
an arbitrary number of lattice sites, namely

$V(t)\approx\sum_{i}\Delta E_{\rm ac}+\sum_{i}\Omega_{\rm
d}\cos(\phi_{i}-\omega_{\rm
d}t)a^{\dagger}_{i,y}a^{\phantom{\dagger}}_{i,x}+{\rm H.c.},$ (11)

where we have introduced the parameters

$\Omega_{\rm d}=|\Omega_{1,2}|\eta_{x}\eta_{y},\hskip
8.61108pt\phi_{i}=\bm{k}_{\rm d}\cdot\bm{r}_{i}^{0}+{\arg}(-\Omega_{1,2}).$
(12)

Note that, due to the constraints in Eq. (10), we have neglected other
contributions in the Lamb-Dicke expansion.

Figure 2: Synthetic dimensions in trapped-ion chains: (a) Schematic
representation of a synthetic Peierls ladder. The sites of the upper and lower
legs of the ladder represent the local vibrations of the ions along the $x$
and $y$ transverse directions, respectively. The dipolar tunnelings (6) are
represented by intra-leg links that connect distant ions. The resulting
parametric tunnelings in Eq. (16) are depicted by the vertical inter-leg
links, and correspond to the frequency-conversion process of (b). (c) For a
pair of ions, the effective rectangular plaquette can lead to a net Aharonov-
Bohm phase $\Phi_{\rm AB}$ (17) for a phonon that tunnels along the
corresponding synthetic links. (d) For $\Phi_{\rm AB}=\pi$, there can be
perfect destructive interference for the phonon, which mimics the Aharonov-
Bohm interference of an electron that travels around an infinitely-thin
solenoid.

One can readily see that, in addition to the irrelevant ac-Stark shift $\Delta
E_{\rm ac}$, we have obtained a parametric modulation like Eq. (2) that
involves simultaneously all of the ions in the chain. Repeating the same
arguments as in the simple two-mode case (Eq. (2)), one finds that the
parametric drive can activate the tunnelling of a phonon along the new
synthetic direction (see Fig. 2 (a)), such that the tunnelling matrix of Eq.
(6) becomes $t_{dd^{\prime}}\mapsto\tilde{t}_{dd^{\prime}}$ with

$\tilde{t}_{(i,\alpha)(j,\beta)}={t}_{(i,\alpha)(j,\beta)}+\frac{\Omega_{\rm
d}}{2}{\rm e}^{{\rm
i}\epsilon_{\alpha,\beta}\phi_{j}}\delta_{i,j}(1-\delta_{\alpha,\beta}),$ (13)

Here, in addition to the Kronecker delta, we have used the fully anti-
symmetric tensor defined as $\epsilon_{x,y}=-\epsilon_{y,x}=1$,
$\epsilon_{x,x}=\epsilon_{y,y}=0$. By making the following identification

$\begin{split}a^{\phantom{\dagger}}_{i,x},a^{\dagger}_{i,x}\mapsto
a^{\phantom{\dagger}}_{i\textbf{e}_{1}},a_{i\textbf{e}_{1}}^{\dagger},\hskip
8.61108pta^{\phantom{\dagger}}_{i,y},a^{\dagger}_{i,y}\mapsto
a^{\phantom{\dagger}}_{i\textbf{e}_{1}+\textbf{e}_{2}},a_{i\textbf{e}_{1}+\textbf{e}_{2}}^{\dagger},\end{split}$
(14)

we obtain, in the interaction picture, a tight-biding model for bosons in a
synthetic two-leg ladder

$H_{\rm
eff}=\sum_{\bm{i}}\sum_{\bm{\ell}\in\mathcal{L}\\{\bm{i}\\}}t^{\phantom{\dagger}}_{\bm{i},\bm{\ell}}a^{{\dagger}}_{\bm{i}+\bm{\ell}}a^{\phantom{\dagger}}_{\bm{i}}.$
(15)

Here, a boson at the synthetic lattice site $\bm{i}$ can tunnel horizontally
or vertically to the site $\bm{i}+\bm{\ell}$ along the synthetic links
labelled by $\bm{\ell}\in\mathcal{L}\\{\bm{i}\\}$. The tunnelling amplitudes
read

$\begin{split}t_{i\textbf{e}_{1},\ell\textbf{e}_{1}}&=\tilde{t}_{(i+\ell,x)(i,x)},\hskip
21.52771ptt_{i\textbf{e}_{1},\textbf{e}_{2}}=\frac{\Omega_{\rm d}}{2}{\rm
e}^{+{\rm i}\phi_{i}},\\\
t_{i\textbf{e}_{1}+\textbf{e}_{2},\ell\textbf{e}_{1}}&=\tilde{t}_{(i+\ell)(y;i,y)},\hskip
4.30554ptt_{i\textbf{e}_{1}+\textbf{e}_{2},-\textbf{e}_{2}}=\frac{\Omega_{\rm
d}}{2}{\rm e}^{-{\rm i}\phi_{i}}.\end{split}$ (16)

In comparison to the parametric tunnelling of Eq. (2), which also lead to a
tunnelling strength with a complex phase ( Eq. (4)), we see that the current
scheme leads to a site-dependent phase as a consequence of the spatial
modulation of the optical potential (8), as depicted in Fig. 2(b). This
inhomogeneity can be exploited, as depicted in Fig. 2 (c), to induce an
effective Peierls’ phase, such that the phonons in the synthetic ladder mimic
the dynamics of electrons under a magnetic field. In fact, if the local phonon
tunnels around the smallest rectangular plaquette
$t_{i\textbf{e}_{1},\textbf{e}_{1}}t_{(i+1)\textbf{e}_{1},\textbf{e}_{2}}t_{(i+1)\textbf{e}_{1}+\textbf{e}_{2},-\textbf{e}_{1}}t_{i\textbf{e}_{1}+\textbf{e}_{2},-\textbf{e}_{2}}\propto{\rm
e}^{{\rm i}\Phi_{\rm AB}}$, it gains a net phase that can no longer be gauged
away as in the simple two-mode case of Eq. (4). In fact, this phase is
analogous to the Aharonov-Bohm phase Aharonov and Bohm (1959) for electrons
moving in a plane under a perpendicular magnetic field

$\Phi_{\rm AB}=\bm{k}_{\rm
d}\cdot(\bm{r}^{0}_{i+1}-\bm{r}^{0}_{i})=:2\pi\frac{\Phi_{B}}{\Phi_{0}},$ (17)

where $\Phi_{B}=\int_{\square}{\rm d}\bm{S}\cdot\bm{B}_{\rm bg}$ is the flux
of an effective magnetic field $\bm{B}_{\rm bg}$ across the plaquette
$\square$, and $\Phi_{0}=h/e$ is the quantum of flux. As a consequence of this
flux, which can be controlled by tilting the laser wave-vector with respect to
the ion chain, one could for instance observe Aharonov-Bohm destructive
interference for $\Phi_{\rm AB}=\pi$, in which a single phonon cannot tunnel
two sites apart along a synthetic plaquette (see Fig. 2 (d)). Let us note that
this Aharonov-Bohm interference occurs at the level of phonons, which have a
zero net charge, and thus differs from the interference of charged ions
tunnelling between two different crystalline configurations, neatly observed
in experiments with a real magnetic field Noguchi  _et al._ (2014).

For larger ion crystals, the analogy with a homogeneous magnetic field is
still valid despite the existence of dipolar tunnelings, provided that the
equilibrium positions of the ions are equally spaced. The equal spacing can be
achieved by designing arrays of individual traps with micro-fabricated
surface-electrode traps Chiaverini  _et al._ (2005); Pearson  _et al._ (2006);
Seidelin  _et al._ (2006); Chiaverini and Lybarger (2008); Labaziewicz  _et
al._ (2008); Schmied  _et al._ (2009); Kumph  _et al._ (2011); Welzel  _et
al._ (2011); Sterling  _et al._ (2014); Mielenz  _et al._ (2016); Bruzewicz
_et al._ (2016); Kumph  _et al._ (2016); Jain  _et al._ (2020), by introducing
anharmonic confining potentials in segmented ion traps Lin  _et al._ (2009);
Xie  _et al._ (2017); Johanning (2016); Leung and Brown (2018, 2018); Pagano
_et al._ (2018), or in ring traps Li  _et al._ (2017); Urban  _et al._ (2019).
Let us note that, even if one achieved an homogeneous spacing, the longer-
range nature of the tunnelings imply that the excitations can now enclose
larger plaquettes potentially changing the interference phenomena. In the pi-
flux case, the next-to-nearest neighbour tunnelling leads to plaquettes with
zero flux, which may challenge the existence of the perfect destructive
interference of Fig. 2 (d). Nevertheless, these larger plaquettes are enclosed
at a considerably slower pace, as the tunnelling strengths decay with the cube
of the distance (6). We have numerically observed that an almost perfect
Aharonov-Bohm interference can still occur when considering how a phononic
excitation travels between opposite corners of the synthetic ladder Martínez-
Carracedo (2019).

Let us close this section by highlighting that the effective magnetic field
underlying Eq. (17) is not a true dynamical magnetic field, but rather a fixed
background field. One can indeed push the analogy to the level of the vector
potential using $\bm{B}_{\rm bg}=\bm{\nabla}\times\bm{A}_{\rm bg}$, but the
$U(1)$ gauge field $\bm{A}_{\rm bg}$ would still be a background field, the
dynamics of which can only be fixed externally, and has nothing to do with
Maxwell electrodynamics. In the following section, we describe how this scheme
of parametric tunnelings can be generalised to get closer to this situation,
and be able to explore lattice gauge theories.

##  III Dynamical gauge fields: the $\mathbb{Z}_{2}$ theory on a link

###  III.1 State-dependent parametric tunnelling

Let us start by revisiting the simplest parametric setup of two modes
introduced in Sec. II.1, and discuss how it can be generalised towards the
simulation of the simplest discrete gauge theory: a $\mathbb{Z}_{2}$ gauge
link. At this abstract level, we consider introducing an additional quantum
system composed of $2$ levels, a so-called qubit in quantum information
science Nielsen and Chuang (2000), which is initially decoupled from the
modes. The bare Hamiltonian of the system (Eq. (1)) now reads

$\tilde{H}_{0}=\omega_{1}a^{{\dagger}}_{1}a^{\phantom{\dagger}}_{1}+\omega_{2}a^{{\dagger}}_{2}a^{\phantom{\dagger}}_{2}+\frac{\omega_{0}}{2}\sigma^{z}_{1,{\bf
e}_{1}},$ (18)

where we have introduced the transition frequency between the qubit levels
$\omega_{0}$, and the Pauli matrix $\sigma^{z}_{1,{\bf
e}_{1}}=\ket{\uparrow_{1,{\bf e}_{1}}}\\!\bra{\uparrow_{1,{\bf
e}_{1}}}-\ket{\downarrow_{1,{\bf e}_{1}}}\\!\bra{\downarrow_{1,{\bf e}_{1}}}$.
Here, we are using an apparently convoluted notation for the index of the
qubit, which will be justified below once we interpret the effective model in
the light of a synthetic lattice gauge theory.

The idea now is to consider a generalisation of Eq. (2) that includes two
tones $\tilde{V}(t)=\tilde{V}_{1}+\tilde{V}_{2}$. One of them induces a state-
dependent parametric drive

$\tilde{V}_{1}=\Omega_{\rm d}\sigma^{z}_{1,{\bf
e}_{1}}a^{{\dagger}}_{2}a^{\phantom{\dagger}}_{1}\cos(\phi_{\rm d}-\omega_{\rm
d}t)+{\rm H.c.},$ (19)

which has an amplitude that depends on the state of the qubit. Additionally,
the other tone drives transitions on the qubit

$\tilde{V}_{2}=\tilde{\Omega}_{\rm d}\sigma^{x}_{1,{\bf
e}_{1}}\cos(\tilde{\phi}_{\rm d}-\tilde{\omega}_{{\rm d}}t)+{\rm H.c.},$ (20)

where we have introduced another Pauli matrix $\sigma^{x}_{1,{\bf
e}_{1}}=\ket{\uparrow_{1,{\bf e}_{1}}}\\!\bra{\downarrow_{1,{\bf
e}_{1}}}+\ket{\downarrow_{1,{\bf e}_{1}}}\\!\bra{\uparrow_{1,{\bf e}_{1}}}$.
Considering that the frequency and strength of this additional driving are
constrained by

$\tilde{\omega}_{\rm d}=\omega_{0},\hskip 8.61108pt|\tilde{\Omega}_{\rm d}|\ll
4\omega_{0},$ (21)

we can follow the exact same steps as in Sec. II.1 to show that, after setting
$\phi_{\rm d}=\tilde{\phi}_{\rm d}=0$, the two-tone drive leads to a time-
independent effective Hamiltonian $\tilde{V}_{I}(t)\approx\tilde{H}_{\rm eff}$
that supersedes Eq. (4), and reads

$H_{\rm
eff}=\left(t^{\phantom{\dagger}}_{1,\textbf{e}_{1}}a^{{\dagger}}_{2}\sigma^{z}_{1,{\bf
e}_{1}}a^{\phantom{\dagger}}_{1}+{\rm H.c.}\right)+h\sigma^{x}_{1,{\bf
e}_{1}}$ (22)

where we have introduced the effective couplings

$t^{\phantom{\dagger}}_{1,\textbf{e}_{1}}=\frac{\Omega_{\rm d}}{2},\hskip
4.30554pt\text{and}\hskip 4.30554pth=\frac{\tilde{\Omega}_{\rm d}}{2}.$ (23)

Figure 3: Synthetic $\mathbb{Z}_{2}$ gauge links and Gauss’ law: (a) Schematic
representation of the effective Hamiltonian in Eq. (22). The two modes
labelled by 1,2, which play the role of matter fields, are coupled by a
synthetic tunnelling of strength $t_{1,{\bf e}_{1}}$ that is mediated by a
qubit that plays the role of the gauge field, and effectively sits on the
synthetic link. In addition to the tunnelling, the electric-field term of
strength $h$ drives transitions in the qubit (inset). (b) For a single
particle, Gauss’ law (25) for a distribution of background charges $q_{1}=0$,
$q_{2}=1$, is fulfilled by the two states $\ket{1_{1},-_{1,{\bf
e}_{1}},0_{2}},\ket{0_{1},+_{1,{\bf e}_{1}},1_{2}}$, characterised by the
absence or presence of an electric field attached to the matter particle
sitting on the leftmost or rightmost site. These electric-field states are
represented by arrows parallel (anti-parallel) to the external field $h$, and
the presence (absence) of the corresponding electric-field line is represented
by a thicker (shaded) golden link.

There are two important aspects to highlight. First, we have again managed to
engineer a synthetic link connecting the two modes via the tunneling of
particles. However, in contrast to the previous case (4), the qubit enters in
this process and mediates the tunnelling. In the spirit of synthetic
dimensions, we can say a the qubit effectively sits on the synthetic link (see
Fig. 3(a)). It is for this reason, that the label used for the qubit $(1,{\bf
e}_{1})$ refers to the link that connects the synthetic site $1$ to its
nearest neighbour $2$ via the direction specified by the vector ${\bf e}_{1}$.
This notation has a clear generalisation to larger lattices and different
geometries, and is common in the context of lattice gauge theories Gattringer
and Lang (2010). The second important aspect to remark is that the dynamics
dictated by the Hamiltonian of Eq. (22), considering also the term
proportional to $h$, has a local/gauge $\mathbb{Z}_{2}$ symmetry. This gauge
symmetry is related to the previous $U(1)$ phase rotation of the modes
discussed below Eq. (7) when restricting to a $\pi$ phase. More importantly,
this $\pi$ phase can be chosen locally. We can transform either
$a^{\phantom{\dagger}}_{1},a_{1}^{\dagger}\mapsto-a^{\phantom{\dagger}}_{1},-a_{1}^{\dagger}$,
or
$a^{\phantom{\dagger}}_{2},a_{2}^{\dagger}\mapsto-a^{\phantom{\dagger}}_{2},-a_{2}^{\dagger}$
by a local $\pi$ phase, and retain gauge invariance in the Hamiltonian by
simultaneously inverting the link qubit $\sigma^{z}_{1,{\bf
e}_{1}}\mapsto-\sigma^{z}_{1,{\bf e}_{1}},\sigma^{x}_{1,{\bf
e}_{1}}\mapsto\sigma^{x}_{1,{\bf e}_{1}}$. Accordingly, the qubit can be
interpreted as a $\mathbb{Z}_{2}$ gauge field introduced to gauge the global
$\mathbb{Z}_{2}$ inversion symmetry of the Hamiltonian (4), paralleling the
situation with other groups where gauge fields are introduced in the links of
the lattice and mediate the tunnelling of matter particles Gattringer and Lang
(2010). Accordingly, we see that the analogy with gauge theories is not a mere
notation choice relying on how we have decided to label the qubit and assign
it to a synthetic link, but it instead rests on our scheme for gauging a
symmetry: the engineered tunnelling leads to a discretised version of the
covariant derivative that is required to upgrade a global symmetry into a
local one.

As advanced in the introduction, in Hamiltonian approaches to lattice gauge
theories Kogut and Susskind (1975), one works in Weyl’s temporal gauge, such
that there is a residual redundancy that is dealt with by imposing Gauss’ law.
As usually in the literature Assaad and Grover (2016); Gazit  _et al._ (2017,
2018); Smith  _et al._ (2018); Gazit  _et al._ (2020); König  _et al._ (2020);
Emonts  _et al._ (2020); Robaina  _et al._ (2021); Borla  _et al._ (2022);
Ercolessi  _et al._ (2018); Magnifico  _et al._ (2019a, b); Borla  _et al._
(2020); Kebrič  _et al._ (2021); Magnifico  _et al._ (2020); Surace and Lerose
(2021); Aramthottil  _et al._ (2022); Das  _et al._ (2021); Ge and Nori
(2022); Sugihara (2005); González-Cuadra  _et al._ (2020); Nyhegn  _et al._
(2021); Brenig (2022); Pradhan  _et al._ (2022), one refers to the external
field $h$ as the electric field term, and talks about the Hadamard states
$\ket{\pm_{1,{\bf e}_{1}}}=(\ket{\uparrow_{1,{\bf
e}_{1}}}\pm\ket{\downarrow_{1,{\bf e}_{1}}})/\sqrt{2}$ as the electric-field
basis, such that the $\ket{+_{1,{\bf e}_{1}}}$ state describes an electric
field line connecting two neighbouring sites. The generators of the
aforementioned local symmetries $[H_{\rm eff},G_{1}]=[H_{\rm
eff},G_{2}]=[G_{1},G_{2}]=0$ are thus

$G_{1}={\rm e}^{{\rm i}\pi
a_{1}^{\dagger}a_{1}^{\phantom{\dagger}}}\sigma_{1,{\bf e}_{1}}^{x},\hskip
8.61108ptG_{2}=\sigma_{1,{\bf e}_{1}}^{x}{\rm e}^{{\rm i}\pi
a_{2}^{\dagger}a_{2}^{\phantom{\dagger}}},$ (24)

and Gauss’ law is imposed by fixing a specific super-selection sector that
fulfills

$G_{1}\ket{\Psi_{\rm phys}}={\rm e}^{{\rm i}\pi q_{1}}\ket{\Psi_{\rm
phys}},\hskip 4.30554ptG_{2}\ket{\Psi_{\rm phys}}={\rm e}^{{\rm i}\pi
q_{2}}\ket{\Psi_{\rm phys}}.$ (25)

Here, $q_{i}\in\\{0,1\\}$ are the so-called background charges, and
$\ket{\Psi_{\rm phys}}$ is a generic state of the physical system
corresponding to specific common eigenstates of the generators of the local
symmetry. These generators, which fulfil $G_{i}^{2}=\mathbb{I}$, and thus only
have the two discrete eigenvalues $\pm 1$, can be used to define orthogonal
projectors $P_{\\{q_{i}\\}}=\prod_{j}\frac{1}{{2}}(\mathbb{I}+{\rm e}^{{\rm
i}\pi q_{j}}G_{j})$. Due to the gauge symmetry, the Hamiltonian is block-
diagonal in subspaces with a fixed arrangement of background charges $H_{\rm
eff}=\sum_{\\{q_{i}\\}}P_{\\{q_{i}\\}}H_{\rm eff}P_{\\{q_{i}\\}}$, and the
different super-selection sectors cannot be connected by the gauge-invariant
dynamics.

In Fig. 3(b), we depict the possible states for this physical subspace in the
case of a single particle, where one can see how the link qubit must lie in a
specific electric-field state such that Gauss’ law is fulfilled for the choice
$q_{1}=0,q_{2}=1$. In particular, when a single particle sits in the rightmost
site 2, an electric field line is created by flipping the qubit in the link
that connects it to the leftmost site 1. This is a clear analogy to Maxwell
electrodynamics, where the regions with a net charge act as sinks/sources of
the electric field. In fact, for (1+1)-dimensional quantum electrodynamics,
i.e. the Schwinger model Schwinger (1962); Casher  _et al._ (1973); Coleman
_et al._ (1975), a $U(1)$ positive charge acts as a source of an electric
field that remains constant in space until a negative charge is found, which
acts a sink. This leads to an electric field string connecting the electron-
positron pair Coleman  _et al._ (1975). As we will discuss in more detail
below, a similar effect occurs when the $\mathbb{Z}_{2}$ gauge theory (22) is
extended to larger chains. Here, the particle moves by stretching or
compressing the electric field string, which resembles the Dirac string
construction that attaches an infinitely-thin solenoid carrying magnetic flux
to a magnetic charge, i.e. a magnetic monopole Dirac (1931). The electric
analog is, nonetheless, a 1D effect and, furthermore, it is gauge-invariant
and observable as emphasised below.

Before going to these generalisations, let us discuss simple dynamical
manifestations of this electric field string that can serve as a guide for the
trapped-ion implementation. Although the quantum simulation scheme leading to
Eq. (22) works for both fermionic and bosonic matter, we will thus focus on
the later that will be mapped onto trapped-ion phonons. For two bosonic modes
and a single gauge qubit, the Hilbert space is infinite dimensional
$\mathcal{H}=\mathcal{F}\otimes\mathbb{C}^{2}$, where
$\mathcal{F}=\oplus_{n=0}^{\infty}\mathcal{F}_{n}$, and each subspace
$\mathcal{F}_{n}={\rm span}\\{\ket{n_{1}}\otimes\ket{n_{2}}:n_{1}+n_{2}=n\\}$
contains $n$ bosonic particles in total. These subspaces can be spanned by the
corresponding Fock states
$\ket{n_{i}}=(a_{i}^{\dagger})^{n_{i}}\ket{0_{i}}/{\sqrt{n_{i}!}}$, where
$\ket{0_{i}}$ is the vacuum of the $i$-th mode. As a consequence of the global
$U(1)$ symmetry and Gauss’ law (25), we can reduce the size of the subspace
where the dynamics takes place, and find some neat manifestations of the
correlations between the charge and electric field degrees of freedom. In
particular, we will show how the dynamics of the $\mathbb{Z}_{2}$ link can be
understood in terms of typical phenomena in quantum optics, such as Rabi
oscillations, dark states, and mode entanglement.

###  III.2 One-boson sector: Rabi oscillations and matter-gauge-field
correlated dynamics

When the initial state contains a single particle, we can restrict the Hilbert
space to $\mathcal{F}_{1}\otimes\mathbb{C}^{2}$ using the global $U(1)$
symmetry. Moreover, Gauss’ law (25) for $q_{1}=0,q_{2}=1$ allows us to
restrict the physical states further to a two-dimensional subspace
$\mathcal{V}_{\rm phys}\subset\mathcal{H}$, such that the states have the form
$\ket{\Psi_{\rm phys}(t)}=c_{r}(t)\ket{\rm R}+c_{l}(t)\ket{\rm L}$, where we
have introduced

$\begin{split}\ket{\rm L}=\ket{1_{1}}\otimes\ket{-_{1,{\bf
e}_{1}}}\otimes\ket{0_{2}},\hskip 8.61108pt\ket{\rm
R}=\ket{0_{1}}\otimes\ket{+_{1,{\bf e}_{1}}}\otimes\ket{1_{2}}.\end{split}$
(26)

According to the effective Hamiltonian of Eq. (22), these levels are split in
energy by $2h$, and transitions between them are induced by the gauge-
invariant tunnelling of strength $t_{1,\textbf{e}_{1}}$ (23). The problem thus
reduces to that of Rabi oscillations Rabi (1937) of a driven two-level atom
Allen and Eberly (1987) (see Fig. 4), and has an exact solution
$\bm{c}(t)={\rm e}^{-{\rm i}\Omega_{0}t\bm{n}\cdot\bm{\sigma}}\bm{c}(0)$,
where $\bm{c}(t)=(c_{r}(t),c_{l}(t))^{\rm t}$, and we have introduced the
vector of Pauli matrices $\bm{\sigma}=(\sigma^{x},\sigma^{y},\sigma^{z})$, and
the following quantities

$\Omega_{0}=\sqrt{t_{1,\textbf{e}_{1}}^{2}+h^{2}},\hskip
8.61108pt\bm{n}=\frac{1}{\Omega_{0}}(t_{1,\textbf{e}_{1}},0,h).$ (27)

Assuming that the particle occupies initially the leftmost site
$\ket{\Psi_{\rm phys}(0)}=\ket{\rm L}$, we see that the tunnelling to the
right is accompanied by the build-up of an electric field line across the
gauge link, which is thus attached to the dynamical $\mathbb{Z}_{2}$ charge
carried by the particle. This correlated dynamics can be observed by measuring
the periodic oscillations of the following gauge-invariant observables

$\begin{split}\overline{n}_{2}(t):=\langle
a_{2}^{\dagger}a_{2}^{\phantom{\dagger}}(t)\rangle&=\frac{t_{1,\textbf{e}_{1}}^{2}}{\Omega_{0}^{2}}\sin^{2}(\Omega_{0}t),\\\
\overline{n}_{1}(t):=\langle
a_{1}^{\dagger}a_{1}^{\phantom{\dagger}}(t)\rangle&=1-\frac{t_{1,\textbf{e}_{1}}^{2}}{\Omega_{0}^{2}}\sin^{2}(\Omega_{0}t),\\\
\overline{s}_{x}(t):=\langle\sigma^{x}_{1,{\bf
e}_{1}}(t)\rangle&=\frac{2t_{1,\textbf{e}_{1}}^{2}}{\Omega_{0}^{2}}\sin^{2}(\Omega_{0}t)-1.\\\
\end{split}$ (28)

Figure 4: Effective two-level system for $\mathbb{Z}_{2}$-link tunnelling: In
the super-selection sector (25) with background charges $q_{1}=0,q_{2}=1$, the
physical subspace for a single particle is composed of two states (26)
depicted in Fig. 3 (b). The gauge-invariant Hamiltonian (22) can then be
mapped onto the problem of detuned Rabi oscillations of a two-level atom in
the rotating frame, where the tunnelling plays the role of the Rabi frequency,
and the electric-field term is proportional to the detuning of the Rabi drive.

In Fig. 5 (a), we compare these analytical predictions (28) for $h=0$ to the
numerical simulation for an initial state
$\ket{\Psi(0)}=\ket{1_{1}}\ket{-_{1,\textbf{e}_{1}}}\ket{0_{2}}$. Note that,
for the numerical simulation, we do not restrict the Hilbert space to the
single-particle subspace, nor to the gauge-invariant basis of Eq. (26). We
truncate the maximal number of Fock states in each site to $n_{i}\leq n_{\rm
max}$, and compute the exact dynamics of the $\mathbb{Z}_{2}$-link Hamiltonian
(22) after this truncation, checking that no appreciable changes appear when
increasing $n_{\rm max}$. The lines depicted in this figure represent the
numerical results for matter and gauge observables
$\overline{n}_{1}(t)=\langle
a_{1}^{\dagger}a_{1}^{\phantom{\dagger}}(t)\rangle$,
$\overline{n}_{2}(t)=\langle
a_{2}^{\dagger}a_{2}^{\phantom{\dagger}}(t)\rangle$ and
$\overline{s}_{x}(t)=\langle\sigma^{x}_{1,\textbf{e}_{1}}(t)\rangle$. Fig. 5
(a) also shows the expectation value of the sum of Gauss’ generators (24)
which, according to the specific distribution of external background charges
$q_{1}=0,q_{2}=1$, should vanish exactly at all times, i.e., $\langle
G_{1}(t)+G_{2}(t)\rangle/2=({\rm e}^{{\rm i}\pi q_{1}}+{\rm e}^{{\rm i}\pi
q_{2}})/2=0$. The symbols represent the respective analytical expressions in
Eq. (28). The picture shows a clear agreement of the numerical and exact
solutions, confirming the validity of the picture of the correlated Rabi
flopping in the matter and gauge sectors. As the boson tunnels to the right
$\ket{1_{1},0_{2}}\to\ket{0_{1},1_{2}}$, the electric field line stretches to
comply with Gauss’ law until, right at the exchange duration, the link qubit
gets flipped $\ket{-_{1,\textbf{e}_{1}}}\to\ket{+_{1,\textbf{e}_{1}}}$. This
behaviour is repeated periodically as the boson tunnels back and forth, and is
a direct manifestation of gauge invariance.

Figure 5: $\mathbb{Z}_{2}$-invariant tunnelling and correlated Rabi flopping:
(a) Dynamics of an initial state $\ket{\Psi(0)}=\ket{\rm L}$ characterised by
the gauge invariant observables $\overline{n}_{1}(t)=\langle
a_{1}^{\dagger}a_{1}^{\phantom{\dagger}}(t)\rangle$,
$\overline{n}_{2}(t)=\langle
a_{2}^{\dagger}a_{2}^{\phantom{\dagger}}(t)\rangle$ and
$\overline{s}_{x}(t)=\langle\sigma^{x}_{1,\textbf{e}_{1}}(t)\rangle$, as well
as the averaged expectation value of the local-symmetry generators $\langle
G_{1}(t)+G_{2}(t)\rangle/2$. The symbols correspond to the numerical
evaluation in the full Hilbert space, whereas the lines display the analytical
predictions for $h=0$ (28). (b) Dynamics for the $\mathbb{Z}_{2}$ gauge link
when the electric field $h$ is increased. The symbols correspond to the
numerical simulations, and the lines to the corresponding analytical
expressions (28).

From the two-level scheme in the right panel of Fig. 4, we see that a non-zero
electric field $h>0$ plays the role of a detuning in the Rabi problem Allen
and Eberly (1987). Accordingly, as the electric field gets stronger, i.e.
$h\gg|t_{1,\textbf{e}_{1}}|$, it costs more energy to create an electric field
line, and the particle ceases to tunnel, i.e. the contrast of the Rabi
oscillations between the L/R levels diminishes (see Fig. 5 (b)). It is worth
comparing to the case of Peierls’ phases and static/background gauge fields of
Sec. II.2. There, four modes were required to define a plaquette and get an
effective flux that can lead to Aharonov-Bohm destructive interference, which
inhibits the tunnelling of a single boson between the corners of the synthetic
plaquette (see Fig. 2 (d). In the case of the $\mathbb{Z}_{2}$ gauge model on
a link, only two modes and a gauge qubit are required. The tunnelling of the
boson is inhibited by increasing the energy cost of stretching/compressing the
accompanying electric field line. As discussed in more detail below, for
larger lattices, this electric-field energy penalty is responsible for the
confinement of matter particles in this $\mathbb{Z}_{2}$ gauge theory, a
characteristic feature of this type of discrete gauge theories Assaad and
Grover (2016); Gazit  _et al._ (2017, 2018); Smith  _et al._ (2018); Gazit
_et al._ (2020); König  _et al._ (2020); Emonts  _et al._ (2020); Robaina  _et
al._ (2021); Borla  _et al._ (2022); Ercolessi  _et al._ (2018); Magnifico
_et al._ (2019a, b); Borla  _et al._ (2020); Kebrič  _et al._ (2021);
Magnifico  _et al._ (2020); Surace and Lerose (2021); Aramthottil  _et al._
(2022); Das  _et al._ (2021); Ge and Nori (2022); Sugihara (2005); González-
Cuadra  _et al._ (2020); Nyhegn  _et al._ (2021); Brenig (2022); Pradhan  _et
al._ (2022).

###  III.3 Two-boson sector: Dark states and entanglement between modes of the
matter fields

Let us now move to the two-particle case, and describe how the connection to
well-known effects in quantum optics can be pushed further depending on the
exchange statistics. A pair of fermions can only occupy the state
$\ket{1_{1}}\otimes\ket{+_{1,{\bf e}_{1}}}\otimes\ket{1_{2}}$, and do not
display any dynamics due to the Pauli exclusion principle. On the other hand,
if the particles are bosonic, the dynamics can be non-trivial and lead to
interesting effects such as mode entanglement. Due to the $U(1)$ symmetry and
Gauss’ law (25) for $q_{1}=q_{2}=0$, the physical subspace can now be spanned
by three different states

$\begin{split}\ket{\rm L}&=\ket{2_{1}}\otimes\ket{-_{1,{\bf
e}_{1}}}\otimes\ket{0_{2}},\\\ \ket{\rm C}&=\ket{1_{1}}\otimes\ket{+_{1,{\bf
e}_{1}}}\otimes\ket{1_{2}},\\\ \ket{\rm R}&=\ket{0_{1}}\otimes\ket{-_{1,{\bf
e}_{1}}}\otimes\ket{2_{2}}.\end{split}$ (29)

A pair of charges sitting on the same site have a vanishing net
$\mathbb{Z}_{2}$ charge $1\oplus 1=(1+1){\rm mod}\,2=0$, and cannot act as a
source/sink of electric field. Therefore, the L and R states in Eq. (29) do
not sustain any electric field. On the other hand, when the pair of
$\mathbb{Z}_{2}$ charges occupy the two different sites, Gauss’ law imposes
that an electric field line must be established at the link. Since creating
this electric field costs energy, these three levels are then separated in
energy by $2h$, and the gauge-invariant tunnelling of the Hamiltonian in Eq.
(22) leads to a $\Lambda$-scheme in quantum optics (see Fig. 6).

Figure 6: $\Lambda$-scheme for 2-boson $\mathbb{Z}_{2}$-invariant tunnelling:
In the left panel, we depict the three possible states in Eq. (29) for the
distributions of the $\mathbb{Z}_{2}$ charges and electric field. In the right
panel, we depict the quantum-optical level scheme, in which the gauge-
invariant tunneling couples the $\ket{\rm L}$ and $\ket{\rm R}$ states to the
state $\ket{\rm C}$ with one boson at each site, and a electric-field string
in the link. The electric field $h$ act as a detuning of these transitions,
leading to a $\Lambda$-scheme.

As it is known to occur for three-level atoms Arimondo and Orriols (1976);
Arimondo (1996), one can find the so-called bright $\ket{\rm B}=(\ket{\rm
L}+\ket{\rm R})/\sqrt{2}$ and dark $\ket{\rm D}=(\ket{\rm L}-\ket{\rm
R})/\sqrt{2}$ states, which here correspond to the symmetric and anti-
symmetric super-positions of the doubly-occupied sites at the left and right
sites. In general, the state of the system can be expressed as a superposition
of $\ket{\rm B}$, $\ket{\rm D}$ and $\ket{\rm C}$, namely $\ket{\Psi_{\rm
phys}(t)}=d(t)\ket{\rm D}+c_{b}(t)\ket{\rm B}+c_{c}(t)\ket{\rm C}$. However,
as the dark state decouples completely from the dynamics, its amplitude
evolves by acquiring a simple phase $d(t)={\rm e}^{{\rm i}ht}d(0)$.
Conversely, the amplitudes of the remaining states mix and display periodic
Rabi oscillations $\bm{c}(t)={\rm e}^{-{\rm
i}\tilde{\Omega}_{0}t\tilde{\bm{n}}\cdot\bm{\sigma}}\bm{c}(0)$ where, in this
case $\bm{c}(t)=(c_{b}(t),c_{c}(t))^{\rm t}$, and

$\tilde{\Omega}_{0}=\sqrt{4t_{1,\textbf{e}_{1}}^{2}+h^{2}},\hskip
8.61108pt\tilde{\bm{n}}=\frac{1}{\tilde{\Omega}_{0}}(2t_{1,\textbf{e}_{1}},0,h).$
(30)

We can now discuss a different manifestation of the gauge-invariant dynamics
with respect to the single-particle case (28). Let us consider the initial
state to be $\ket{\Psi_{\rm phys}(0)}=\ket{\rm C}$ with one boson at each
site, and an electric-field line at the link in between. If we look at the
local number of bosons, we do not observe any apparent dynamics
$\overline{n}_{1}(t):=\langle
a_{1}^{\dagger}a_{1}^{\phantom{\dagger}}(t)\rangle=1=\langle
a_{2}^{\dagger}a_{2}^{\phantom{\dagger}}(t)\rangle=:\overline{n}_{2}(t)$.
However, looking into the electric field at the link, we find periodic Rabi
flopping again, i.e.

$\overline{s}_{x}(t):=\langle\sigma^{x}_{1,{\bf
e}_{1}}(t)\rangle=1-\frac{8t_{1,\textbf{e}_{1}}^{2}}{\tilde{\Omega}_{0}^{2}}\sin^{2}(\tilde{\Omega}_{0}t).$
(31)

Since the gauge field cannot have independent oscillations with respect to the
matter particles, there must be a non-trivial dynamics within the matter
sector which, nonetheless, cannot be inferred by looking at the local number
of particles. In this context, it is the interplay of the superposition
principle of quantum mechanics and gauge symmetry, which underlies a neat
dynamical effect. This effect becomes manifest by inspecting the state after a
single exchange period $\Delta t_{\rm ex}=\pi/2\tilde{\Omega}_{0}$ for $h=0$.
After this time, a boson can either tunnel to the left or to the right. In
both cases, the electric field string compresses, since a doubly-occupied site
amounts to a vanishing net $\mathbb{Z}_{2}$ charge, and there is thus no
sink/source of electric field. Accordingly, when the bosons tunnel along
either path respecting gauge invariance, the state ends up with the same link
configuration, namely $\ket{-}_{1,\textbf{e}_{1}}$. Then, according to the
superposition principle, both paths must be added, and the state of the system
at time $t_{e}$ is given by

$\ket{\Psi_{\rm phys}(\Delta t_{\rm
ex})}=\frac{1}{\sqrt{2}}\left(\ket{2_{1},0_{2}}+\ket{0_{1},2_{2}}\right)\otimes\ket{-_{1,{\bf
e}_{1}}}.$ (32)

We see that, as a consequence of the dynamics, mode entanglement Benatti  _et
al._ (2020) has been generated in the matter sector since the state cannot be
written as a separable state $\ket{\Psi_{\rm phys}(\Delta t_{\rm ex})}\neq
P(a_{1}^{\dagger})Q(a_{2}^{\dagger})\ket{0_{1},0_{2}}\otimes\ket{-_{1,{\bf
e}_{1}}}$ for any polynomials $P,Q$. The specific state (Eq. (32)) in the
matter sector is a particular type of NOON states, which have been studied in
the context of metrology Sanders (1989); Bollinger  _et al._ (1996); Boto  _et
al._ (2000); Lee  _et al._ (2002); Giovannetti  _et al._ (2011). Note that
this state cannot be distinguished from the initial state if one only looks at
the local boson numbers $\overline{n}_{1}(t)=\overline{n}_{2}(t)=1$. The non-
trivial dynamics becomes instead manifest via the link field and the quantum-
mechanical mode-mode correlations.

In Fig. 7, we present a comparison of the analytical predictions with the
corresponding numerical results where, once more, we do not restrict to the
basis in Eq. (29), nor to the 2-boson subspace. We initialise the system in
$\ket{\Psi(0)}=\ket{1_{1}}\ket{+_{1,\textbf{e}_{1}}}\ket{1_{2}}$, and
numerically truncate the Hilbert space such, that $n_{i}\leq n_{\rm max}$,
computing numerically the Schrödinger dynamics for the $\mathbb{Z}_{2}$-link
Hamiltonian (22). In Fig. 7(a), we represent these numerical results with
lines for the observables
$\overline{n}_{1}(t),\overline{n}_{2}(t),\overline{s}_{x}(t)$, as well as the
average of the local-symmetry generators $\langle G_{1}(t)+G_{2}(t)\rangle/2$.
Once again, these numerical results agree perfectly with the corresponding
analytical expressions (31), which are represented by the symbols. In Fig.
7(b), we show the fidelity of the system state with respect the NOON state of
Eq. (32), namely $\mathcal{F}_{\rm NOON}(t)=|\bra{\Psi_{\rm phys}(t_{e})}{\rm
e}^{\\-{\rm i}tH_{\rm eff}}\ket{\Psi(0)}|^{2}$ at different evolution times.
We see that this fidelity tends to unity at the periodic exchange periods
$t=mt_{e},\,\,m\in\mathbb{Z}^{+}$. Note that the timescale in the horizontal
axis is the same as the one for the one-particle case in Fig. 5, but the
periodic oscillations are twice as fast. This two-fold speed-up is caused by
the bosonic enhancement due to the presence of two particles in the initial
state, providing a $\sqrt{2}$ factor, and the enhancement due to the bright
state, which brings the additional $\sqrt{2}$ factor. This total speed-up is
the only difference if one compares the dynamics of the bosonic sector with
that of a standard beam splitter leading to the Hong-Ou-Mandel interference
Hong  _et al._ (1987). In fact, in the trapped-ion literature, the bare
tunneling terms between a pair of vibrational modes are commonly referred to
as a beam splitter due to the formal analogy with the optical device that
splits an incoming light mode into the transmitted and reflected modes
Leibfried  _et al._ (2002).

Figure 7: Two-boson $\mathbb{Z}_{2}$-invariant tunneling and entanglement: (a)
Dynamics of an initial state $\ket{\Psi(0)}=\ket{\rm L}$ characterised by the
gauge invariant observables $\overline{n}_{1}(t)=\langle
a_{1}^{\dagger}a_{1}^{\phantom{\dagger}}(t)\rangle$,
$\overline{n}_{2}(t)=\langle
a_{2}^{\dagger}a_{2}^{\phantom{\dagger}}(t)\rangle$ and
$\overline{s}_{x}(t)=\langle\sigma^{x}_{1,\textbf{e}_{1}}(t)\rangle$, as well
as the averaged expectation value of the local-symmetry generators $\langle
G_{1}(t)+G_{2}(t)\rangle/2$. The symbols correspond to the numerical
evaluation in the full Hilbert space, whereas the lines display the analytical
predictions for $h=0$. (b) State fidelity with respect to a mode-entangled
2-boson NOON state (32), which tends to unity at the integer exchange periods
$\Delta t_{\rm ex}$, and $2\Delta t_{\rm ex}$.

##  IV Trapped-ion toolbox: phonons and qubits

We will now discuss two different schemes for implementing the state-dependent
parametric modulation (19) experimentally. The first scheme (I) is based on
trapped-ion analog quantum simulators that generalise straightforwardly from
Eq. (8). The second scheme (II) exploits recent ideas developed in the context
of continuous-variable quantum computing Sutherland and Srinivas (2021a).

Before describing these schemes, we first review the progress of trapped-ion-
based quantum simulations for lattice gauge theories. As discussed in Martinez
_et al._ (2016); Muschik  _et al._ (2017); Davoudi  _et al._ (2020), certain
gauge theories can be mapped exactly onto spin models that represent the
fermionic matter with effective long-range interactions mediated by the gauge
fields. Following these ideas, the $U(1)$ Schwinger model of quantum
electrodynamics in 1+1 dimensions Martinez  _et al._ (2016); Nguyen  _et al._
(2022) and variational quantum eigensolvers Kokail  _et al._ (2019) have been
simulated digitally in recent trapped-ion experiments. As discussed in Davoudi
_et al._ (2020); Paulson  _et al._ (2021), there are theoretical proposals to
generalise this approach to gauge theories in 2+1 dimensions. Although not
considered in the specific context of trapped ions, digital quantum
simulators, and variational eigensolvers have also been recently considered
for $\mathbb{Z}_{2}$ gauge theories Gustafson and Lamm (2021); Lumia  _et al._
(2022); Carena  _et al._ (2022); Mildenberger  _et al._ (2022). Rather than
eliminating the gauge fields as in the cases above Martinez  _et al._ (2016);
Muschik  _et al._ (2017); Davoudi  _et al._ (2020), one could consider the
opposite, and obtain effective models for the gauge fields after eliminating
the matter content Greenberg  _et al._ (2022); Irmejs  _et al._ (2022).

In order to move beyond those specific models, it would be desirable to
simulate matter and gauge fields on the same footing. Trapped-ion schemes for
the quantum-link approach to the Schwinger model have been proposed in Hauke
_et al._ (2013); Andrade  _et al._ (2022). In particular, for the specific
spin-$1/2$ representation of the link operators, the gauge-invariant tunneling
becomes a three-spin interaction. This could be implemented using only the
native two-spin interactions in trapped-ion experiments and by imposing an
additional energetic Gauss penalty Hauke  _et al._ (2013). Alternatively, one
may also generate three-spin couplings Andrade  _et al._ (2022) directly by
exploiting second-order sidebands that use the phonons as carriers of these
interactions Bermudez  _et al._ (2009); Katz  _et al._ (2022a, b). We note
that there have also been other proposals Yang  _et al._ (2016); Davoudi  _et
al._ (2021) to use the motional modes to encode the $U(1)$ gauge field,
whereas the fermionic matter is represented by spin-$1/2$ operators. In this
case, the gauge-invariant tunneling can be achieved via other second-sideband
motional couplings Yang  _et al._ (2016), or by combining digital and analog
ingredients in a “hybrid” approach Davoudi  _et al._ (2021). A different
possibility is to use the collective motional modes to simulate bosonic matter
and reserve the spins to represent the quantum link operators for the gauge
fields Yang  _et al._ (2016). In this way, one can simulate a quantum link
model provided that all the collective vibrational modes can be individually
addressed in frequency space Yang  _et al._ (2016), which can be complicated
by frequency crowding as the number of ions increases. We note that
engineering the collective-motional-mode couplings has also been recently
considered in the context of continuous-variable quantum computing, boson
sampling, and quantum simulation of condensed-matter models Hou  _et al._
(2022); Sutherland and Srinivas (2021b); Katz and Monroe (2022); Chen  _et
al._ (2023). In the following, we present a trapped-ion scheme for the quantum
simulation of $\mathbb{Z}_{2}$ gauge theories based on our previous idea of a
state-dependent parametric tunneling (19), and using motional states along two
different transverse directions, and a pair of electronic states to encode the
particles and the gauge field.

###  IV.1  Scheme I: Analog scheme for the $\mathbb{Z}_{2}$ link

####  IV.1.1 Light-shift-type parametric tunneling

In section II.2, we showed that the effective Peierls’ tunneling of Eq. (13)
between the two local transverse vibrations in an ion chain could be
synthesised by exploiting the optical potential (8) created by a far-detuned
two-beam laser field. There, we considered that all of the ions were
initialised in the same ground-state level. However, depending on the nuclear
spin of the ions, the ground-state manifold can contain a variety of levels
$\\{\ket{s}$}, which can be used to generalise the scheme towards the state-
dependent parametric tunneling of Eq. (19). In general, provided that the two-
beam field is far-detuned from any direct transition, and that its beat note
is also far-detuned from any Raman transition between ground state levels, the
optical light-shift potential of Eq. (8) becomes state-dependent Wineland  _et
al._ (1998a), namely

$V_{1}(t)=\sum_{n,n^{\prime}=1,2}\sum_{i,s\phantom{l}}\Omega_{n,n^{\prime}}^{(s)}\ket{s_{i}}\bra{s_{i}}{\rm
e}^{{\rm i}(\bm{k}_{{\rm L},n}-\bm{k}_{{\rm
L},n^{\prime}})\cdot\bm{r}_{i}-(\omega_{{\rm L},n}-\omega_{{\rm
L},n^{\prime}})t}+{\rm H.c.}.$ (33)

Here, $\Omega_{n,n^{\prime}}^{(s)}$ is the amplitude of the light-shift terms
discussed after Eq. (8), in which the corresponding Rabi frequencies now refer
to the particular ground-state level $\ket{s_{i}}$ of the $i$-th ion involved
in the two virtual transitions. These light shifts then depend on the specific
state and the intensity and polarisation of the laser fields. As discussed in
the context of state-dependent dipole forces Wineland  _et al._ (1998a);
Leibfried  _et al._ (2003a); Wineland  _et al._ (2003), one can focus on a
particular pair of states $s_{1},s_{2}$ and tune the polarisation, detuning
and intensity of the light, such that the corresponding amplitudes for the
crossed beat note terms attain a specific differential value Leibfried  _et
al._ (2003a); Wineland  _et al._ (2003). In the present case, we consider that
this amplitude is the opposite for each of the electronic states. Since they
are used to encode the aforementioned qubit, we will denote them by
$s_{1},s_{2}\to\uparrow,\downarrow$ from now on, thus assuming the
differential crossed terms

$\Omega_{\rm
d}:=\Omega_{1,2}^{(0)}\eta_{x}\eta_{y}=-\Omega_{1,2}^{(1)}\eta_{x}\eta_{y}.$
(34)

One can now follow the same steps as in Sec. III, introducing the local
transverse phonons via the position operators, and performing a Lamb-Dicke
expansion assuming Eq. (9). Using the same set of constraints, i.e. Eq. (10),
we find that

$V_{1}(t)\approx\sum_{i}\Delta E_{\rm ac}\sigma_{i}^{z}+\sum_{i}\Omega_{\rm
d}\sigma_{i}^{z}\cos(\phi_{i}-\omega_{\rm
d}t)a^{\dagger}_{i,y}a^{\phantom{\dagger}}_{i,x}+{\rm H.c.},$ (35)

where all parameters have been defined in Eq. (12). We are already close to
the idealised situation. It may seem that, as in the case of the background
gauge fields of Eq. (11), the scheme already works for an arbitrary number of
ions.

Figure 8: Trapped-ion synthetic $\mathbb{Z}_{2}$ gauge theory on a link:
Schematic representation of the single-ion system that can realise the
$\mathbb{Z}_{2}$ gauge theory on a synthetic link (22). On the left, we depict
an ion vibrating in the transverse $x$ direction, and the inset represents the
state of the corresponding qubit in
$\ket{-_{1}}=(\ket{\uparrow}_{1}-\ket{\downarrow_{1}})/\sqrt{2}$. On the
right, we can see how, as a consequence of the trapped-ion effective
Hamiltonian (36), the vibrational excitation along $x$ is transferred into a
vibrational excitation along $y$, while simultaneously flipping the qubit into
$\ket{+_{1}}=(\ket{\downarrow}_{1}+\ket{\uparrow_{1}})/\sqrt{2}$. This
dynamics, which is fully consistent with the local gauge symmetry, can be
engineered by shining a far-detuned two-beam laser field with wave-vectors
associated to each frequency represented by green and blue arrows, leading to
a beat note along the grey arrow that yields the desired term (36).

However, even if the form of the parametric term of Eq. (35) does in fact
generalise the single-link case of Eq. (19) to the entire ion chain, a simple
counting argument shows that the effective model cannot achieve
$\mathbb{Z}_{2}$ gauge invariance. For a string of $N$ ions, we have $2N$
local motional modes along the transverse directions, which lead to the
synthetic ladder of Fig. 2 (a). There are $2(N-1)+N=3N-2$ links in this
ladder, and each of them requires a gauge qubit to mediate the tunneling while
preserving a local $\mathbb{Z}_{2}$ symmetry. Since we only have $N$ trapped-
ion qubits at our disposal, i.e. one qubit per ion, it is not possible to
build a gauge-invariant model for the synthetic ladder in a straightforward
manner. Instead, we will present in Sec. V a solution to this problem by
introducing a mechanism that we call synthetic dimensional reduction.

Prior to that, let us discuss the minimal case where gauge invariance can be
satisfied in the trapped-ion experiment: a single $\mathbb{Z}_{2}$ link. This
link requires a single ion: one gauge qubit for the link, and two motional
modes for the matter particles, which can be the vibrations along any of the
axes. In Fig. 8, we consider the two transverse modes, and thus restrict Eq.
(35) to a single ion. Following the same steps as in the derivation of Eq.
(22), we move to an interaction picture and neglect rapidly rotating terms
under the conditions of Eq. (10). We obtain a time-independent term that
corresponds to a $\mathbb{Z}_{2}$ gauge-invariant tunneling

$V_{1}(t)\approx\frac{\Omega_{\rm d}}{2}{\rm e}^{{\rm i}\phi_{\rm
d}}a^{{\dagger}}_{{1},y}\sigma^{z\phantom{\dagger}\\!\\!\\!\\!}_{1}a^{\phantom{\dagger}}_{1,x}+{\rm
H.c.},$ (36)

where the microscopic parameters are

$\Omega_{\rm d}=|\Omega_{1,2}|\eta_{x}\eta_{y}\hskip 8.61108pt\text{and}\hskip
8.61108pt\phi_{\rm d}=\bm{k}_{\rm
d}\cdot\bm{r}_{1}^{0}+{\arg}(-\Omega_{1,2}).$ (37)

At this point, the driving phase $\phi_{\rm d}$ is completely irrelevant and
can be set to zero without loss of generality. Identifying the trapped-ion
operators with those of the lattice gauge theory

$\begin{split}a^{\phantom{\dagger}}_{1,x},a^{\dagger}_{1,x}&\mapsto
a^{\phantom{\dagger}}_{1},a_{1}^{\dagger},\\\
a^{\phantom{\dagger}}_{1,y},a^{\dagger}_{1,y}&\mapsto
a^{\phantom{\dagger}}_{2},a_{2}^{\dagger},\\\
\sigma_{1}^{x},\sigma_{1}^{z}\hskip 8.61108pt&\mapsto\sigma_{1,{\bf
e}_{1}}^{x},\sigma_{1,{\bf e}_{1}}^{z},\end{split}$ (38)

we obtain a realisabove-idealisedation of the $\mathbb{Z}_{2}$ gauge-invariant
tunneling on a link (22) using a single trapped ion, such that

$t_{1,{\bf e}_{1}}=\frac{\Omega_{\rm
d}}{2}=\frac{|\Omega_{1,2}|}{2}\eta_{x}\eta_{y}.$ (39)

As explained above, this exploits the qubit as the gauge field, and two
vibrational modes to host the $\mathbb{Z}_{2}$-charged matter. If the above
condition (34) is not satisfied, one can still obtain a state-dependent
tunneling, but this would not have the desired local invariance under the
above $\mathbb{Z}_{2}$ gauge group (25). Nonetheless, such state-dependent
tunneling can be interesting for other purposes in the context of hybrid
discrete-continuous variable quantum information processing, as realised in
recent trapped-ion experiments Gan  _et al._ (2020). In the following
subsection, we will present alternative schemes that do not depend on this
condition for the differential light shift.

For a single $\mathbb{Z}_{2}$ link, any two of the three motional modes can be
used. In the following simulations, with the trapped-ion parameters of the
considered setup, we encode the matter particles in an axial ($z$) and a
transverse ($x$) mode such that we can benefit from the larger Lamb-Dicke
parameter of the axial mode and higher frequency separation between the
motional modes. We model numerically the possible deviations of a realistic
trapped-ion implementation from the above-idealised expressions used in Fig.
5. To quantify these deviations, we numerically solve the trapped-ion
evolution, starting in $\ket{\textrm{L}}$ (26), where we apply the interaction
for a duration $t=\Delta t_{\rm ex}$ such that, by the end of it, the overlap
squared to the desired state $\ket{\textrm{R}}$ is maximised. If we consider
only the idealised tunneling term, Eq. (36), the exchange duration is given by
$\Delta t_{\rm ex}=\pi/2t_{1,{\bf e}_{1}}$. In the simulation of the more
realistic trapped-ion case, there are additional terms neglected in the ideal
case that can change the optimal exchange duration. We thus find $\Delta
t_{\rm ex}$ by maximising the fidelity $\mathcal{F}(t)=|\langle{\rm
R}|{\psi(t)}\rangle|^{2}$ of achieving the desired state $\ket{\textrm{R}}$.
We also calculate the expectation value of the local symmetry generators (24)
for $\braket{G_{1}(t)+G_{2}(t)}/2$ to check if the effective gauge symmetry is
fulfilled. Moreover, when introducing the electric field with magnitude $h$ in
the simulations, we plot the maximum contrast $\mathcal{C}$ in the
oscillations of $\overline{s}_{x}(t)$ (28). Thus, we can evaluate the reduced
tunneling probability caused by the energy penalty for stretching/compressing
the electric field line as one increases $h>0$.

We consider a realistic parametrically-driven trapped-ion system and compare
it to the idealised gauge-invariant Hamiltonian (22). For the simulations
presented below, we perform direct numerical integration of the relevant
Hamiltonians using the QuantumOptics.jl package in Julia Krämer  _et al._
(2018). We consider the tunneling dynamics arising due to spin-dependent light
shifts, described by the full Hamiltonian (33), using experimentally feasible
parameters. In the simulation, we do not assume the Lamb-Dicke expansion and
thus include possible off-resonant carrier excitations as well as other non-
linear terms neglected in Eq. (35). We use a single ion and two of its
motional modes, which map to a $\mathbb{Z}_{2}$ link connecting two bosonic
matter sites (38). We restrict the simulation to the subspace of a two-level
system together with two bosonic modes, each truncated at phonon number
$n_{\rm max}=7$. We consider a ${}^{88}\mathrm{Sr}^{+}$ ion confined in the
setup presented in Refs. Schafer (2018); Thirumalai (2019). The secular
frequency of the axial in-phase mode can be set to $\omega_{z}/2\pi=1.2\,$MHz,
while the radial secular frequency is $\omega_{x}/2\pi=1.9\,$MHz. The qubit
states $\ket{\uparrow_{1}},\ket{\downarrow_{1}}$ can be defined by two ground
state levels of the $5S_{1/2}$ manifold shown in Fig. 9, or by the ground
state $5S_{1/2}$, $m_{j}=-1/2$ and the metastable state $4D_{5/2}$,
$m_{j}=-1/2$, leading to an optical qubit (quadrupole transition) discussed in
the Appendix.

$S_{\frac{1}{2}}$
---
$P_{\frac{3}{2}}$
---
$\delta$
---
$\Delta$
---
Figure 9: Beam configuration for light-shift parametric tunneling in
${}^{88}{\rm Sr}^{+}$: We consider the ground state qubit (cyan lines). The
gauge-invariant tunneling is created via two far-detuned Raman transitions of
detunings $\Delta$ and $\Delta+\delta$ from the auxiliary level
($P_{{3}/{2}}$), depicted by black arrows. These Raman transitions virtually
couple the qubit states in the $S_{{1}/{2}}$ level to an excited state in the
$P_{{3}/{2}}$ level. When the beat note of the two corresponding tones
$\delta$ is on resonance with the difference of two secular trap frequencies,
we attain the desired state-dependent parametric tunneling.

For a ground state qubit, the light shifts are created by a far-detuned
dipole-mediated Raman transition Gan  _et al._ (2020), whereas for an optical
qubit, two off-resonant beams driving a quadrupole transition can be used. In
both cases, the two beams are assumed to be counter-propagating $\bm{k}_{{\rm
L},1}=-\bm{k}_{{\rm L},2}=:\bm{k}$, such that the beat note wave-vector is
$\bm{k}_{\rm d}=2\bm{k}$. Moreover, we assume that the angle between
$\bm{k}_{\rm d}$ and the axial mode ($z$) is $45^{\circ}$, while the angle
with respect to the transverse mode ($x$) is $60^{\circ}$.The two Raman beams
at near $\lambda=402~{}\textrm{nm}$ are detuned by $\Delta/2\pi=10$ THz from
the $S_{1/2}\leftrightarrow P_{3/2}$ transition (see Fig. 9). The Lamb-Dicke
factors (9) of the two motional modes are $\eta_{z}=2\times 0.077$ and
$\eta_{x}=2\times 0.043$. In this system, light shifts of up to
$\Omega_{1,2}/2\pi=1.1$ MHz can be achieved. In the full Hamiltonian (33), we
set the beam detunings to $\delta=\omega_{x}-\omega_{z}$, resulting in a beat
note frequency of $\omega_{\rm d}=\delta$ in Eq. (10).

Figure 10: $\mathbb{Z}_{2}$ gauge link dynamics using Scheme I: We simulate
the $\mathbb{Z}_{2}$ dynamics using the Raman-based light-shift parametric
tunneling and try to replicate Fig. 5 (a). The markers are full numerical
simulations including non-linear terms (33) beyond the desired Lamb-Dicke
expansion, while the continuous lines are analytical predictions in Eq. (28)
using the effective tunneling strength (39) and $h=0$. The adiabatic pulse
shaping sets the minimum pulse duration to the rising and falling edge
($10\text{\,}\mathrm{\SIUnitSymbolMicro s}$ each).

The simulated dynamics at $\Omega_{1,2}/2\pi=$1.1\text{\,}\mathrm{MHz}$$ is
shown in Fig. 10, where the coloured lines represent the analytical
predictions for the various observables in Eq. (28) using the effective
tunneling strength (39). The coloured symbols stand for the full numerical
simulations including non-linear terms (33) beyond the idealised Lamb-Dicke
expansion (35). This figure is thus the trapped-ion analog of Fig. 5 (a)
following Scheme I. We note that, in order to find a better agreement with the
idealised evolution (28), we have incorporated an adiabatic pulse shaping of
the light-matter coupling that restricts the minimal duration of the real-time
dynamics as discussed in the caption of Fig. 10. For this specific choice of
parameters, we see that the exchange duration of the tunneling of the phonon
and the stretching of the electric-field line is about several tens of $\mu$s,
which is sufficiently fast compared to other possible sources of noise such as
heating and dephasing, as discussed in more detail below. Once the viability
of trapped-ion scheme I for the quantum simulation of the $\mathbb{Z}_{2}$
gauge-invariant tunneling has been demonstrated, we can make a more detailed
analysis of the errors, and also consider including the electric field term.

In Fig. 11, we present simulations of the resulting hopping duration $\Delta
t_{\rm ex}$, the fidelity error $1-\mathcal{F}$ and the gauge-invariance
operator $\braket{G_{1}(t)+G_{2}(t)}/2$ as a function of applied light shift
$\Omega_{1,2}$, which can be increased by using higher laser intensities or
lower Raman detunings in order to obtain a faster gauge-invariant dynamics
(39). When comparing the dynamics of the full Hamiltonian to the ideal gauge-
invariant tunneling (39), we observe excellent agreement with $\Delta t_{\rm
ex}=\pi/2t_{1,{\bf e}_{1}}$. The fidelity error and the symmetry operator
$\braket{G_{1}(\Delta t_{\rm ex})+G_{2}(\Delta t_{\rm ex})}/2$ also remain
very low even when including the trapped-ion additional terms such as the off-
resonant carrier, which underlies the adequacy of the considered parameters
for this specific scheme. We can achieve an effective tunneling coupling rate
of up to $7.1$ kHz inferred as $1/(4\Delta t_{\rm ex})$. The desired tunneling
Hamiltonian can also be synthesised for the optical qubit using two detuned
674-nm beams, see Appendix A.

Figure 11: Light-shift $\mathbb{Z}_{2}$ tunneling with Raman couplings:
Numerical simulations of the exchange duration $\Delta t_{\rm ex}$ (upper
panel), state infidelity $1-\mathcal{F}$ (middle panel), and Gauss’ symmetry
operator $\braket{G_{1}+G_{2}}/2$ (lower panel) as a function of coupling
strength $\Omega_{1,2}$. The results are shown for the full Hamiltonian (33)
(black) and the idealized invariant-gauge tunneling (36) (green). The green
solid line in the upper panel is the expected analytic dependence of the
exchange duration on $\Omega_{1,2}$, extracted from Eq. (39). For both the
full and ideal Hamiltonian the expectation value of the symmetry operator is
consistent with $0$, as desired, down to $10^{-3}$.

To achieve large tunneling rates given the achievable $\Omega_{1,2}$ of
current systems, and the low coupling rate of the second-order process
$\propto\eta_{x}\eta_{z}$ (39), one needs to push the parameters to a regime
that violates the $\delta\gg|\Omega_{1,2}|$ requirement. This results in off-
resonant driving of spurious interactions, most prominently direct off-
resonant carrier coupling. As mentioned above we minimise this effect by
employing adiabatic amplitude pulse shaping Roos (2008) with a rise time of
$10\text{\,}\mathrm{\SIUnitSymbolMicro s}$.

$\Delta$
---
Raman
---
$P_{\frac{3}{2}}$
---
quadrupole
---
$S_{\frac{1}{2}}$
---
$D_{\frac{5}{2}}$
---
$\delta$
---
$\delta$
---
$\delta$
---
$\delta$
---
Figure 12:  Mølmer-Sørensen-type parametric drive:. For the ground state qubit
(cyan lines), the Raman scheme which is now near resonant with the qubit
frequency $\omega_{0}+\delta$, we need to introduce a third tone, here
depicted by a blue arrow . For the optical qubit (magenta lines), which is
driven directly via the quadrupole transition, we symmetrically detune the two
tones (blue and red) about the qubit resonance by $\pm\delta$.

####  IV.1.2 Mølmer-Sørensen-type parametric tunneling

In this section, we present an alternative scheme for synthesising the gauge-
invariant tunneling that could enable a technically simpler implementation
when introducing the electric field term, as we will discuss later on.

The tunneling Hamiltonian now arises from a “bichromatic” field that is no
longer far-detuned from the qubit transition but, instead, has two components
symmetrically detuned from the qubit frequency. For the ground state qubit
addressed with a Raman configuration, the ”bichromatic” field can be achieved
by having one of the Raman beams at $\omega_{L}+\omega_{0}$ and the other
counter-propagating Raman beam consisting of two tones at
$\omega_{L}\pm\delta$, i.e. $\bm{k}_{\rm d}=2\bm{k}$ (see Fig. 12). For the
optical qubit, one can obtain a similar light-matter interaction by addressing
it with a single beam consisting of two tones at $\omega_{0}\pm\delta$, i.e.
$\bm{k}_{\rm d}=\bm{k}$ (see Fig. 12). The Rabi frequency $\Omega$ of the
blue- $(+\delta)$ and red-detuned $(-\delta)$ tone must be the same. Instead
of getting the state-dependent light-shift potential of Eq. (33), one now
finds the following term in the interaction picture

$\tilde{V}_{1}(t)=\Omega\cos(\delta
t)\sum_{i}\ket{\uparrow_{i}}\bra{\downarrow_{i}}{\rm e}^{{\rm i}\bm{k}_{\rm
d}\cdot\bm{r}_{i}}+{\rm H.c.}.$ (40)

As in our previous derivation, we expand the ion positions
$\bm{r}^{\phantom{0}}_{i}=\bm{r}^{0}_{1}+\sum_{\alpha}\textbf{e}_{\alpha}\frac{1}{\sqrt{2m\omega_{\alpha}}}(a^{\phantom{\dagger}}_{i,\alpha}+a^{\dagger}_{i,\alpha})$
in the Lamb-Dicke parameters assuming Eq. (9). By focusing again on a single
trapped ion, and choosing the detuning of the beams to be resonant with
$\delta=\omega_{x}-\omega_{z}$, we reach the frequency-conversion requirement
and obtain an effective tunneling that simultaneously flips the qubit

$\tilde{V}_{1}(t)\approx\frac{\Omega_{\rm
d}}{2}{a}_{1,z}^{\dagger}{\sigma}_{1}^{x\phantom{\dagger}\\!\\!\\!\\!}{a}_{1,x}^{\phantom{\dagger}}+{\rm
H.c.},\hskip 4.30554pt\Omega_{\rm d}=\Omega\eta_{x}\eta_{z}.$ (41)

The gauge-invariant tunneling rate (22) then reads

$t_{1,{\bf e}_{1}}=\frac{\Omega_{\rm
d}}{2}=\frac{|\Omega|}{2}\eta_{x}\eta_{z}.$ (42)

which is analogous to the previous case of Eq. (37). We thus obtain a rotated
version of the gauge-invariant tunneling in Eq. (36), the only difference
being that the operators need to be transformed as
$\sigma_{1}^{z}\mapsto\sigma_{1}^{x}$, and
$a_{1,y}^{\phantom{\dagger}},a_{1,y}^{\dagger}\mapsto
a_{1,z}^{\phantom{\dagger}},a_{1,z}^{\dagger}$, which must also be considered
in the mapping to the operators of the lattice gauge theory in Eq. (38).
Accordingly, the generators of the local symmetries now read

$G_{1}={\rm e}^{{\rm i}\pi
a_{1}^{\dagger}a_{1}^{\phantom{\dagger}}}\sigma_{1,{\bf e}_{1}}^{z},\hskip
8.61108ptG_{2}=\sigma_{1,{\bf e}_{1}}^{z}{\rm e}^{{\rm i}\pi
a_{2}^{\dagger}a_{2}^{\phantom{\dagger}}}.$ (43)

This configuration is commonly used to drive two-qubit entangling gates
according to a Mølmer-Sørensen scheme Sørensen and Mølmer (2000). The
difference is that for the tunneling term, we set $\delta$ to the frequency
difference of two motional modes $\omega_{x}-\omega_{z}$, instead of a
motional mode frequency used for the entangling gate, e.g.
$\delta=\omega_{z}$. Then, it is possible to drive higher-order terms in the
Lamb-Dicke expansion that do not lead to a state-dependent force but, instead,
induce the spin-conditioned tunneling of Eq. (41). This is a state-dependent
beam splitter interaction in the Hadamard basis $\ket{\pm_{1}}$. We refer to
this scheme as a Mølmer-Sørensen(MS)-type parametric tunneling.

Figure 13: Mølmer-Sørensen-type $\mathbb{Z}_{2}$ tunneling: with Raman
couplings (left column) and quadrupole couplings (right column). Simulation of
the exchange duration, fidelity error, and symmetry operator
$\braket{G_{1}+G_{2}}/2$ as a function of the coupling strength. The results
are shown for the full Hamiltonian (40) (black) and the idealized invariant-
gauge tunneling (41) (green). The green solid line in the upper panel is the
expected analytic dependence of the exchange duration on $\Omega$, extracted
from Eq. (42). For both the full and ideal Hamiltonian the expectation value
of the symmetry operator is consistent with $0$, as desired, down to
$10^{-3}$.

Once again, we numerically integrate the full dynamics for (40) and compare it
to the idealized tunneling term (41). For the Raman scheme, we assume the same
experimental parameters as in the previous section. For implementing this on
the optical qubit we use the quadrupole $674$ nm transition. The Lamb-Dicke
parameters are $\eta_{z}=0.05$ and $\eta_{x}=0.024$ and consider Rabi-
frequencies of up to $\Omega/2\pi=$1.1\text{\,}\mathrm{MHz}$$. As in the
light-shift scheme, we restrict ourselves to the subspace of a two-level
system and two bosonic modes truncated at $n_{\rm max}=7$ phonons. We obtain a
maximum effective tunneling coupling strength
$$3.22\text{\,}\mathrm{kHz}$=1/(4\Delta t_{\rm ex})$ for the Raman scheme, and
a slower one $$0.30\text{\,}\mathrm{kHz}$=1/(4\Delta t_{\rm ex})$ for the
quadrupole scheme (see Fig. 13). We note that a similar principle has been
recently used to generate single-mode squeezing in reference Katz  _et al._
(2022c), which can then be used for the quantum simulation of spin models with
multi-spin interactions Bermudez  _et al._ (2009); Andrade  _et al._ (2022);
Katz  _et al._ (2022d). We add adiabatic pulse shaping to the simulation as it
enables smooth transitioning into the interaction picture and suppresses off-
resonant (non-commuting) carrier excitations. This effectively reduces the
strength of the tunneling but, importantly, it retains the state dependence.
The non-commuting off-resonant carrier term sets a limit on the achievable
interaction magnitude; this is reflected by the global minimum in tunneling
duration (first row, Fig. 13).

####  IV.1.3 Implementation of the electric-field term

To realise the full $\mathbb{Z}_{2}$ gauge model in Eq. (22), we also need a
term that drives the qubit transition (20) with Rabi frequency
$\tilde{\Omega}_{\rm d}$, which corresponds to the electric field
$h=\tilde{\Omega}_{\rm d}/2$ in Eq. (23). The technique to induce this
additional electric-field term depends on the specific scheme. For the scheme
based on the light-shift potential (Sec. IV.1.1), one needs to add a field
driving the qubit transition resonantly. For an optical qubit, this term would
arise from a resonant laser driving the quadrupole-allowed transition. On the
other hand, if the qubit is encoded in the ground state, this term can be
induced by either a resonant microwave field or a pair of Raman laser beams.
In both cases, trapped-ion experiments routinely work in the regime of Eq.
(21), where the value of $\tilde{\Omega}_{\rm d}$ can be controlled very
precisely by tuning the amplitude of the laser or microwave field Bruzewicz
_et al._ (2019). Note that the resonance condition in Eq. (21) must be
modified to account for the ac-Stark shifts shown in Eq. (35), namely

$\tilde{\omega}_{\rm d}=\omega_{0}+2\Delta E_{\rm ac},\hskip
8.61108pt|\tilde{\Omega}_{\rm d}|\ll 4(\omega_{0}+2E_{\rm ac}).$ (44)

This leads to the desired Hamiltonian

$V_{1}(t)\approx\left(\frac{\Omega_{\rm d}}{2}{\rm e}^{{\rm i}\phi_{\rm
d}}a^{{\dagger}}_{{1},z}\sigma^{z\phantom{\dagger}\\!\\!\\!\\!}_{1}a^{\phantom{\dagger}}_{1,x}+{\rm
H.c.}\right)+\frac{\tilde{\Omega}_{\rm d}}{2}\sigma_{1}^{x},$ (45)

which maps directly onto the desired $\mathbb{Z}_{2}$ gauge model on the link
in Eqs. (22)-(23) with the new term playing the role of the electric field

$h=\frac{\tilde{\Omega}_{\rm d}}{2}.$ (46)

Figure 14: Light-shift $\mathbb{Z}_{2}$ tunneling with Raman couplings in the
presence of a non-zero electric field: we simulate the hopping duration,
contrast $\mathcal{C}$ and symmetry operator $\braket{G_{1}+G_{2}}/2$ as a
function of electric field strength $h$ relative to the coupling strength
$\Omega_{1,2}$. The results are shown for the full Hamiltonian (33) (black),
including the additional carrier driving (20) with a modified resonant
condition (44). We also show the results for the ideal gauge-invariant
Hamiltonian (45) (green). The green solid line is the analytic dependence of
the exchange duration and contrast respectively on $\Omega_{1,2}$, extracted
from Eqs. (27) and the effective couplings (39) and (46). For both the full
and ideal Hamiltonian the expectation value of the symmetry operator is
consistent with $0$, as desired, down to $10^{-3}$. Figure 15: Mølmer-Sørensen
$\mathbb{Z}_{2}$ tunneling in the presence of a non-zero electric field: with
Raman couplings (left column) and quadrupole couplings (right column).
Simulating the exchange duration, contrast, and symmetry operator
$\braket{G_{1}+G_{2}}/2$ as a function of electric field strength $h$ relative
to the coupling strength $\Omega$. The results are shown for the full
Hamiltonian (40) (black), including the additional carrier driven by a simple
shift of the Mølmer-Sørensen detuning (48). We also show the results for the
ideal gauge-invariant Hamiltonian (47) (green). The green solid line in the
upper panel is the expected analytic dependence of the exchange duration on
$\Omega$, extracted from Eqs. (27) and the effective couplings (42) and (48).
For both the full and ideal Hamiltonian the expectation value of the symmetry
operator is consistent with $0$, as desired, down to $10^{-3}$.

We now present our numerical simulations for the light-shift type scheme as a
function of the electric-field strength $h$. In Fig. 14, we present the
exchange duration $\Delta t_{\rm ex}$ defined at maximum state fidelity
$\mathcal{F}$, the maximum contrast $\mathcal{C}$ in $\overline{s}_{x}(t)$,
and the expectation value of the gauge-symmetry generators
$\braket{G_{1}+G_{2}}/2$, all of them as a function of the ratio between the
transverse electric field $h$ and the differential ac-Stark shift amplitude
$\Omega_{1,2}$. While the presence of the non-commuting off-resonant carrier
coupling ($z$ basis) present in the full Hamiltonian reduces the effect of the
transverse term ($x$ basis) in comparison to the ideal case, the gauge
invariance is preserved.

For the Mølmer-Sørensen-type scheme discussed in Sec. IV.1.2, the spin
conditioning of the tunneling occurs in the Hadamard basis $\ket{\pm_{1}}$,
such that the effective electric field must also be rotated with respect to
Eq. (22). We can introduce this term by simply shifting the centre frequency
of the bichromatic laser field (40) relative to the qubit resonance by a
detuning $\delta_{\rm s}$. In a rotating frame, this modifies Eq. (40) by
introducing an additional term, namely

$\tilde{V}_{1}(t)\approx\left(\frac{\Omega_{\rm
d}}{2}{a}_{1,z}^{\dagger}{\sigma}_{1}^{x\phantom{\dagger}\\!\\!\\!\\!}{a}_{1,x}^{\phantom{\dagger}}+{\rm
H.c.}\right)+\frac{\delta_{\rm
s}}{2}{\sigma}_{1}^{z\phantom{\dagger}\\!\\!\\!\\!},$ (47)

which leads to the effective electric-field term

$h=\frac{\delta_{\rm s}}{2}.$ (48)

This is a considerable advantage with respect to the light-shift scheme, as no
additional tones are required to implement the electric field term. The
resulting exchange duration $\Delta t_{\rm ex}$, the maximum contrast
$\mathcal{C}$ in $\overline{s}_{x}(t)$ and gauge-symmetry generators
$\braket{G_{1}+G_{2}}/2$, found through numerical simulations of the
Hamiltonian (47) and are compared to the ideal gauge tunneling (22) (see Fig.
15). As before, the presence of the off-resonant carrier coupling reduces the
effective magnitude of the transverse term, but the gauge invariance is
preserved.

Figure 16: Amplitude-shaped pulses considered in scheme II: (a) Shaped pulse
that could be used for implementing just the tunneling term. (b) Trotterized
pulse sequence for implementing the tunneling term. (c) Trotterized pulse
sequence for implementing the full $\mathbb{Z}_{2}$-link Hamiltonian.

####  IV.1.4 Experimental considerations

Let us now discuss in detail how the experiment would proceed. One of the
advantages of trapped ions is that they also offer a variety of high-precision
techniques for state preparation and readout Bruzewicz  _et al._ (2019). For a
single trapped ion, it is customary to perform optical pumping to the desired
internal state, say $\ket{\uparrow}$. One can then use laser cooling in the
resolved-sideband limit for both vibrational modes, and prepare them very
close to the vibrational ground state. Using a blue-sideband coupling directed
along a particular axis, say the $x$-axis, one can flip the state of the qubit
and, simultaneously, create a Fock state with a single vibrational excitation
in the mode. We note that the initial state of the $\mathbb{Z}_{2}$ gauge
theory would correspond to $\ket{\rm
L}=\ket{1_{1}}\otimes\ket{\downarrow_{1,{\bf e}_{1}}}\otimes\ket{0_{2}}$,
which is the rotated version of the one discussed previously and is thus
directly valid for the Mølmer-Sørensen-type scheme. For the light-shift
scheme, one must apply a Hadamard gate to initialise the system in $\ket{\rm
L}=\ket{1_{1}}\otimes\ket{-_{1,{\bf e}_{1}}}\otimes\ket{0_{2}}$, which can be
accomplished by driving a specific single-qubit rotation.

One can then let the system evolve for a fixed amount of time under the
effective Hamiltonian in either Eq. (45) for the light-shift scheme, or Eq.
(47) for the Mølmer-Sørensen-type scheme. We have shown that the effective
Hamiltonians approximate the ideal Hamiltonian accurately. After this real-
time evolution, the laser fields are switched off. Then, the measurement stage
starts, where one tries to infer the matter-gauge field correlated dynamics in
Eq. (28). In order to do that, one would take advantage of the readout
techniques developed in trapped ions Bruzewicz  _et al._ (2019), which
typically map the information of the desired observable onto the qubit. After
this mapping, the qubit can be projectively measured in the $z$-basis via
state-dependent resonance fluorescence. In order to measure the electric field
operator of the Mølmer-Sørensen-type scheme
$\overline{s}_{z}(t)=\langle\sigma^{z}_{1,{\bf e}_{1}}(t)\rangle$, one can
collect the state-dependent fluorescence. For the light-shift scheme, one
needs to measure $\overline{s}_{x}(t)=\langle\sigma^{x}_{1,{\bf
e}_{1}}(t)\rangle$, which requires an additional single-qubit rotation prior
to the fluorescence measurement. On the other hand, in order to infer the
phonon population $\langle
a_{2}^{\dagger}a_{2}^{\phantom{\dagger}}(t)\rangle$=$\langle
a_{1,z}^{\dagger}a_{1,z}^{\phantom{\dagger}}(t)\rangle$, and observe the
gauge-invariant tunneling, one would need to map the vibrational information
onto the qubit first prior to the fluorescence measurement Leibfried  _et al._
(2003b). All these techniques are well developed, see for example Wineland
_et al._ (1998b).

###  IV.2 Scheme II: Pulsed scheme for the $\mathbb{Z}_{2}$ link

####  IV.2.1 Orthogonal-force parametric tunneling

We now discuss an alternative strategy to realise the $\mathbb{Z}_{2}$ gauge
link based on digital quantum simulation and the concatenation of gates.
First, we focus on a new way of engineering the gauge-invariant tunneling term
using two orthogonal state-dependent forces and, then, explain how it can be
used to experimentally implement the $\mathbb{Z}_{2}$ gauge model in Eq. (22).
As before, we consider the case of a single $\mathbb{Z}_{2}$ link, i.e. one
single ion and two vibrational modes. Following the scheme proposed in Ref.
Sutherland and Srinivas (2021a) for hybrid discrete-continuous variable
approaches in trapped-ion quantum information processing, we consider two
orthogonal state-dependent forces acting on the two vibrational modes with
Lamb-Dicke parameters $\eta_{z}$ and $\eta_{x}$, respectively. We thus start
from two terms like Eq. (40), each of which will be tuned to yield a different
state-dependent force, i.e. $\bm{k}_{{\rm d},1}||{\bf e}_{x}$, and
$\bm{k}_{{\rm d},2}||{\bf e}_{z}$,

$\tilde{V}_{1}(t)=\Omega\cos(\delta
t)\sum_{i}\sum_{n=1,2}\ket{\uparrow_{i}}\bra{\downarrow_{i}}{\rm e}^{{\rm
i}\bm{k}_{{\rm d},n}\cdot\bm{r}_{i}+\phi_{{\rm d},n}}+{\rm H.c.}.$ (49)

In the interaction picture with respect to the qubit frequency, $\omega_{0}$,
motional frequencies, $\omega_{z}$ and $\omega_{x}$, the interaction is
described by

${V}_{1}(t)\approx\eta_{x}\Omega{\sigma}_{1}^{x}{a}_{1,x}{\rm e}^{-{\rm
i}\delta t}+\eta_{z}\Omega{\sigma}_{1}^{y}{a}_{1,z}{\rm e}^{-{\rm i}\delta
t}+\text{H.c.},$ (50)

where $\eta_{\alpha}\Omega$ is the coupling strength to the respective
vibrational mode, and $\delta$ is the detuning away from the sidebands
$\omega_{0}\pm\omega_{x/z}$. These two terms can be derived using similar
steps as before, which only differ on the specific selection of the leading
contribution by the appropriate choice of the laser frequencies. The
interference of these two forces can lead, in the second order, to an
effective state-dependent tunneling. After using a Magnus expansion for the
time-ordered evolution operator $U(t)=\mathcal{T}\\{{\rm exp}\\{-{\rm
i}\int_{0}^{t}{\rm d}sV_{1}(s)\\}\\}$ Magnus (1954), the second-order term
$U(t)\approx{\rm exp}\\{-{\rm i}H_{\rm eff}t\\}$ yields the following
interaction

${H}_{\textrm{eff}}=\frac{\Omega_{\rm
d}}{2}{a}_{1,z}^{\dagger}{\sigma}_{1}^{z}{a}_{1,x}^{\phantom{\dagger}}+{\rm
H.c.},\hskip 8.61108pt\Omega_{\rm d}={\rm
i}\frac{\Omega^{2}}{\delta}\eta_{x}\eta_{z},$ (51)

which maps directly onto the desired gauge-invariant tunneling of Eq. (22)
with a tunneling strength of

$t_{1,{\bf e}_{1}}=\frac{\Omega_{\rm d}}{2}={\rm
i}\frac{\Omega^{2}}{2\delta}\eta_{x}\eta_{z}.$ (52)

In this derivation, we have neglected higher-order contributions in the Magnus
expansion that would lead to errors $\epsilon=O([\eta\Omega/\delta]^{3})$ that
must be kept small (we assumed that $\eta_{x}$ and $\eta_{z}$ are the same
order of magnitude $\eta$). If a fixed error $\epsilon$ in these higher-order
terms is considered, $\delta$ is then linear in $\eta$. Consequently, the
tunneling coupling rate is also linear in $\eta$. This linear dependence is in
contrast to scheme I, where the coupling is quadratic in $\eta$. Additionally,
the first-order term in the Magnus expansion must be accounted for, which
leads to additional state-dependent displacements in the joint phase space of
both vibrational modes. As in the case of trapped-ion entangling gates, these
displacements vanish for specific evolution times corresponding to integer
multiples of $2\pi/\delta$. Hence, the tunneling term (22) can be achieved by
applying the interaction for a duration that is an integer multiple of
$2\pi/\delta$.

####  IV.2.2 Implementation of the electric-field term

For this scheme, the electric field term $h\sigma_{1}^{x}$ in Eq. (22) can be
introduced through Trotterization. For this, we split the interaction time
into segments with durations that are integer multiples of $2\pi/\delta$, and
we alternate between applying the tunneling term and the external field term
$h\sigma_{1}^{x}$, which can be achieved by a simple carrier driving (see Fig
16). In this way, the electric field term can be easily introduced by
interleaving short pulses on resonance with the carrier with periods of small
evolution under the combination of the two orthogonal state-dependent forces.

####  IV.2.3 Experimental considerations

One way to implement the two orthogonal state-dependent forces in a laser
system is by having two sets of Mølmer-Sørensen-style bichromatic fields, see
Sec. IV.1.2. One bichromatic field is symmetrically detuned from the carrier
by $\delta_{1}=\pm(\omega_{z}+\delta)$ and the other by
$\delta_{2}=\pm(\omega_{x}+\delta)$. As described in Sec. IV.1.2, the
bichromatic fields can either couple levels in the ground state via a Raman
transition, or two levels of an optical qubit via a quadrupole transition.
Moreover, for having two orthogonal state-dependent forces, we set the phase
between the two bichromatic beams such that

$\left(\frac{\phi_{+}+\phi_{-}}{2}\right)_{2}=\left(\frac{\phi_{+}+\phi_{-}}{2}\right)_{1}+\frac{\pi}{2},$
(53)

where $\phi_{+}$, $\phi_{-}$ are the phases of the blue and red detuned tones
in each of the bichromatic fields, $1$ and $2$. Applying these four tones
gives rise to the two orthogonal state-dependent forces (50) that are needed
for engineering the tunneling term (51). However, it also leads to spurious
carrier terms that drive off-resonant qubit rotations around axes that are
orthogonal to each of the corresponding state-dependent forces. As mentioned
above, a technique to mitigate the effect of the carrier term is pulse shaping
Roos (2008), which is mainly needed at large coupling strengths, i.e., large
$\Omega$ in Eq. (50). We describe the amplitude shaping of the pulses and
further discuss its effect in our numerical simulations below.

We consider the same experimental apparatus as in Section IV.1, and
investigate the implementation of the gauge-invariant tunneling term of Eqs.
(22)-(23) using the two orthogonal state-dependent forces. For this, we use
the optical qubit and two motional modes with $\eta_{z}=0.05$,
$\omega_{z}/2\pi=1.2$ MHz, and $\eta_{x}=0.024$, $\omega_{x}/2\pi=1.9\,$MHz,
respectively. We choose the detuning of the bichromatic fields from the
respective vibrational mode to be $\delta/2\pi=75\,$kHz, and set
$\Omega/2\pi=0.75/\sqrt{2}~{}\textrm{MHz}$ for each of the four tones. This
should allow us to reach an effective tunneling coupling rate of up to $1.3$
kHz, inferred as $1/(4\Delta t_{\rm ex})$. The numerically simulated dynamics,
truncating the bosonic modes at phonon number $n_{\rm max}=5$, and for the
same observables as those in Fig. 5 (a), are shown in Fig. 17. This figure is
thus the trapped-ion analog of Fig. 5 (a) following Scheme II. This was
simulated as a series of pulses that are ramped on and off with 3.6$\,\mu$s
ramp durations and with the FWHM duration of $2\pi/\delta$, as shown in Fig 16
(b). The same dynamics can be achieved by having a single pulse with the FWHM
duration equal to an integer multiple of $2\pi/\delta$ (see Fig 16 (a)). By
inspecting the figure more closely, we see that there are small deviations
with respect to the idealized Hamiltonian (51) that deserve a more detailed
analysis.

Figure 17:  $\mathbb{Z}_{2}$ gauge link dynamics using scheme II: We simulate
the $\mathbb{Z}_{2}$ dynamics using two orthogonal spin-dependent forces and
try to replicate Fig. 5 (a). The markers are the numerical simulation data
considering the evolution under the two state-dependent forces in Eq. (50),
but also including the additional off-resonant carriers that stem from the
Lamb-Dicke expansion of Eq. (49). The coloured lines are the analytical
predictions in Eq. (28), considering the effective tunneling strength in Eq.
(52) and $h=0$.

In the top panel of Fig. 18, we vary $\Omega$ in each one of the tones and
evaluate how this affects the exchange duration $\Delta t_{\rm ex}$,
calculated by maximising the state fidelity as in the previous section. We
conduct simulations for three cases. Initially, we simulate the interaction in
Eq. (50) that only contains the two orthogonal state-dependent forces. We then
introduce the off-resonant carrier terms, which would arise from the Lamb-
Dicke expansion of Eq. (40) as the leading off-resonant perturbations to the
state-dependent forces (50). Finally, we also consider the amplitude pulse
shaping. When the carrier terms are excluded, and the pulse is applied for a
duration that is an integer multiple of $2\pi/\delta$, the inferred tunneling
duration follows closely the theory from Eq. (51), which is represented by a
green solid line according to $\Delta t_{\rm ex}=\pi/2t_{1,{\bf e}_{1}}$ with
the effective tunneling of Eq. (52). As $\Omega$ is increased, the error term
$O([\eta\Omega/\delta]^{3})$ becomes significant, and small deviations start
to appear. As above, we use the state fidelity $\mathcal{F}$ (the overlap to
the desired state) and $\braket{G_{1}+G_{2}}/2$ (the expectation value of the
symmetry generators) in order to evaluate the quality of the effective
Hamiltonian with respect to Eq. (22). Introducing the carrier terms does not
change the exchange duration $\Delta t_{\rm ex}$, $\mathcal{F}$, or
$\braket{G_{1}+G_{2}}/2$ significantly. However, amplitude shaping the pulses
substantially improves the quality of tunneling as it suppresses the coupling
to higher order terms $O([\eta\Omega/\delta]^{3})$. Introducing the amplitude-
shaping ramp effectively decreases the area of the pulses, which translates
into slightly slower exchange durations (top panel), but also leads to smaller
errors (middle and bottom panels).

Figure 18:  Pulsed-scheme $\mathbb{Z}_{2}$ tunneling with orthogonal forces:
We vary the strength of the tones in the bichromatic field $\Omega$, and
evaluate the exchange duration, the infidelity in obtaining the desired state
$\ket{R}$, and the expectation value of the local symmetry generators. We do
this for three cases: excluding the carrier from the interaction and thus
considering Eq. (50) (green crosses); looking at the full interaction and thus
including the spurious carrier terms (black circles); and, finally,
considering the full interaction while slowly ramping the pulses on and off
(magenta triangles). The green solid line in the upper panel is the expected
analytic dependence of the exchange duration on $\Omega$, extracted from Eq.
(52). For the Hamiltonian including the spurious carrier and adiabatic ramp
the expectation value of the symmetry operator is consistent with $0$, as
desired, down to $10^{-2}$.

Once the viability of trapped-ion Scheme II for the quantum simulation of the
$\mathbb{Z}_{2}$ gauge-invariant tunneling has been demonstrated, we can
consider the errors that would stem from adding the electric field term. In
Fig 19, we present our numerical results, evaluating the decrease in contrast
of the Rabi oscillations between the $\ket{\rm L}$ and $\ket{\rm R}$, and
$\braket{G_{1}+G_{2}}/2$, as one increases the electric field $h$. When the
carrier terms are excluded and we do not use the amplitude-shaped pulses, the
decrease, in contrast, follows Eq. (28), which is represented as a green solid
line in the top panel. We find that, in the realistic experimental situation
with carrier terms present, and using amplitude-shaped pulses, we need
slightly higher values of $h$ to achieve the same contrast. This is mainly due
to the effect of the ramp.

Figure 19:  Pulsed-scheme $\mathbb{Z}_{2}$ tunneling with orthogonal forces in
the presence of a non-zero electric field: We simulate the $\mathbb{Z}_{2}$
dynamics using two orthogonal spin-dependent forces for different magnitudes
of $h$. The value used for $t_{1,e_{1}}$ was calculated as $1/(4\Delta t_{\rm
ex})$ at $h=0$. We obtain dynamics similar to Fig. 5 (b) and infer the maximum
contrast in $\overline{s}_{x}(t)$ and the expectation value of the local
symmetry at the point of maximum contrast. The markers are numerical
simulations, while the continuous lines are analytical predictions for
different values of $h$ Eq. (28). For the Hamiltonian including the spurious
carrier and adiabatic ramp the expectation value of the symmetry operator is
consistent with $0$, as desired, down to $10^{-2}$.

The shortest time step that we can use in order to close the loops in phase
space is $2\pi/\delta$, which was used for the simulations above. Hence, the
Trotter error is given as $ht_{1,e_{1}}(2\pi/\delta)^{2}$ which must be
negligible. In the simulation results presented above, for the highest value
of $h$, the Trotter error was $\approx 8\times 10^{-4}$. Higher values of
$\delta$ reduce the $O([\eta\Omega/\delta]^{3})$ error, and enable finer time
steps in the time scans, as we always want to measure at instants where the
loops in phase space that arise from first-order contributions in the Magnus
expansion are closed, and the leading effect is the state-dependent tunneling.
However, a higher $\delta$ also reduces the effective tunneling rate (51).
This translates into shorter pulses which, when becoming comparable to the
ramp length, do not give the effective desired dynamics anymore. Comparing
scheme II to scheme I in terms of observing the matter-gauge field dynamics,
the state preparation and measurement stages are the same. The difference is
how the effective Hamiltonian in Eq. (22) is experimentally implemented and
that is by using a pulse sequence as the one shown in Fig. 16 (c).

###  IV.3  $\mathbb{Z}_{2}$ gauge link scheme comparison and other sources of
noise

The two schemes presented above outline different viable strategies to
implement the gauge-invariant model (22) using current trapped-ion hardware.
There are, however, several experimental challenges worth highlighting. It is
crucial that the tunneling rates are large compared to any noise process
present in the physical system. The dominating sources are the qubit and
motional decoherence. In the considered experimental setup Schafer (2018);
Thirumalai (2019), the decoherence time of the qubit is the most stringent one
with $T_{2}\approx 2.4\,$ms for the ground state qubit and $T_{2}\approx
5\,$ms for the optical qubit. These numbers could be improved by several
orders of magnitude by using a clock-qubit encoding. However, in this
encoding, the differential dipole light-shift that leads to Eq. (36) would
vanish and, hence, the method proposed in Sec. IV.1.1 would not work 222Light
shifts can be created using quadrupole-Raman transitions Baldwin  _et al._
(2021) but extra care needs to be taken to symmetrise the shift induced on
each qubit state.. An alternative way of increasing the qubit coherence would
be to use magnetic shielding or active magnetic-field stabilisation.

In terms of motional coherence, the heating rate is limiting the coherence
time for the longitudinal motional mode to be ca. 14 ms. The coherence time
for the transverse modes has not been properly characterised in the current
system and it might be further limited by noise in the trap rf drive. However,
actively stabilising the amplitude of the rf drive has shown improvement in
the coherence time Johnson  _et al._ (2016). We note that a similar, yet non-
gauge-invariant tunneling, has been implemented in a trapped-ion experiment
Gan  _et al._ (2020) in the context of continuous-variable quantum computing.
This experiment successfully used the transverse modes and measured coherence
times of 5.0(7) and 7(1) ms for ${}^{171}\textrm{Yb}^{+}$ ions.

Consequently, we can conclude that the exchange timescales of the
$\mathbb{Z}_{2}$ tunneling implementations investigated in this paper using
scheme I in Secs. IV.1.1 ($\approx$40\text{\,}\mathrm{\SIUnitSymbolMicro s}$$,
using Raman beams) and IV.1.2 ($\approx$100\text{\,}\mathrm{\SIUnitSymbolMicro
s}$$, using Raman beams), as well as scheme II in Sec. IV.2