compat=1.0.0

# Long-range Ising interactions mediated by $\lambda\phi^{4}$ fields:
probing the renormalisation of sound in crystals of trapped ions

G. Martín-Vázquez  Departamento de Física Teórica, Universidad Complutense,
28040 Madrid, Spain  Nano and Molecular Systems Research Unit, University of
Oulu, 90014 Oulu, Finland  G. Aarts  Department of Physics, Swansea
University, Singleton Campus, SA2 8PP Swansea, United Kingdom  European Centre
for Theoretical Studies in Nuclear Physics and Related Areas (ECT*) and
Fondazione Bruno Kessler Strada delle Tabarelle 286, 38123 Villazzano (TN),
Italy  M. Müller  Institute for Theoretical Nanoelectronics (PGI-2),
Forschungszentrum Jülich, 52428 Jülich, Germany  JARA-Institute for Quantum
Information, RWTH Aachen University, 52056 Aachen, Germany  A. Bermudez
Departamento de Física Teórica, Universidad Complutense, 28040 Madrid, Spain

###### Abstract

The generating functional of a self-interacting scalar quantum field theory
(QFT), which contains all the relevant information about real-time dynamics
and scattering experiments, can be mapped onto a collection of multipartite-
entangled two-level sensors via an interferometric protocol that exploits a
specific set of source functions. Although one typically focuses on impulsive
delta-like sources, as these give direct access to $n$-point Feynman
propagators, we show in this work that using always-on harmonic sources can
simplify substantially the sensing protocol. In a specific regime, the
effective real-time dynamics of the quantum sensors can be described by a
quantum Ising model with long-range couplings, the range and strength of which
contains all the relevant information about the renormalisation of the QFT,
which can now be extracted in the absence of multi-partite entanglement. We
present a detailed analysis of how this sensing protocol can be relevant to
characterise the long-wavelength QFT that describes quantised sound waves of
trapped-ion crystals in the vicinity of a structural phase transition, opening
a new route to characterise the associated renormalisation of sound.

###### Contents

  1. I Introduction
  2. II Klein-Gordon scalar fields coupled to $\mathbb{Z}_{2}$ fields
    1. II.1 Impulsive sensors of the generating functional
    2. II.2 Harmonic sensors and long-range Ising models
  3. III Self-interacting scalar fields coupled to $\mathbb{Z}_{2}$ fields
    1. III.1 Spin path integral and scalar-sigma field theory
    2. III.2 Renormalised long-range quantum Ising models
  4. IV Renormalisation of sound in trapped-ion crystals
    1. IV.1  Harmonic trapped-ion crystals: rigidity, massive Klein-Gordon fields and effective Ising models.
    2. IV.2  Anharmonic trapped-ion crystals: $\lambda\phi^{4}$ fields, sound renormalisation and Ising models.
    3. IV.3 Rigidity and renormalisation-group predictions
    4. IV.4 Parametric drivings to control the shear modulus
  5. V Conclusions and outlook
  6. A Generating functional of Klein-Gordon fields
  7. B Generating functional of interacting $\lambda\phi^{4}$ fields
  8. C Elastic crystals: compressibility, massless Klein-Gordon fields, and effective Ising models.

##  I Introduction

Large-scale fault-tolerant quantum computers have the potential to solve
relevant computational problems in ways that surpass the capabilities of any
classical device Nielsen and Chuang (2010). To achieve this long-term goal,
these devices must process information by taking explicit advantage of the
laws of quantum mechanics, while they also correct for the errors that occur
during the computation due to imperfections and noise. In this way, one
actively battles a possible accumulation of the errors Terhal (2015), as has
already been demonstrated in various small-scale processors Cory  _et al._
(1998); Chiaverini  _et al._ (2004); Reed  _et al._ (2012); Schindler  _et
al._ (2011); Nigg  _et al._ (2014); Kelly  _et al._ (2015); Linke  _et al._
(2017); Stricker  _et al._ (2020); Egan  _et al._ (2021); Erhard  _et al._
(2021), including atomic, molecular and optical (AMO) technologies such as
trapped ions Wineland  _et al._ (1998); Schindler  _et al._ (2013); Bruzewicz
_et al._ (2019).

Recent progress with noisy intermediate-scale quantum technologies Preskill
(2018) has shown that, even if these errors are not actively corrected for,
and can thus accumulate, the level of the overall noise is sufficiently low
that current noisy devices can already solve certain computational tasks that
no classical computer could solve in any feasible amount of time Arute  _et
al._ (2019); Zhong  _et al._ (2020). This has allowed to demonstrate the so-
called quantum advantage/supremacy Harrow and Montanaro (2017). In theoretical
physics, some of the most complicated problems appear in the study of quantum
systems with a large, sometimes even infinite, number of coupled degrees of
freedom. These systems are typically formulated in terms of quantum field
theories (QFTs) Peskin and Schroeder (1995); Fradkin (2013), some of which
pose problems whose solution has remained elusive for decades, in spite of the
availability of large computational resources. With some notable exceptions
Jordan  _et al._ (2012), determining their specific complexity, or proving
rigorously that no classical computer will ever be able to solve them in a
feasible amount of time, are very delicate matters. In any case, given the
limitations of current computational approaches in QFTs, e.g. real time
evolution or finite fermion densities, and their importance in understanding
Nature at its most fundamental level Hands (2001), any possible progress using
alternative methods can be of great value. In this context, we find that the
prospects of using noisy intermediate-scale quantum technologies to address
open questions in QFTs is fascinating Feynman (1982); Cirac and Zoller (2012).
We also find particularly enticing that these quantum technologies may offer
novel ways to experimentally realise and probe paradigmatic QFTs which,
although not finding counterparts in Nature, e.g. reduced dimensionality,
contain key fundamental aspects that shape our understanding of it. We will
emphasise below the results of the present work in this respect.

Although the initial focus of such, so-called, quantum simulators (QSs)
Feynman (1982); Lloyd (1996) has been on condensed-matter models, such as the
bosonic Jaksch  _et al._ (1998); Greiner  _et al._ (2002) and fermionic
Hofstetter  _et al._ (2002); Jördens  _et al._ (2008); Schneider  _et al._
(2008) Hubbard models with cold atoms Bloch  _et al._ (2012); Jaksch and
Zoller (2005), or models of interacting spins Porras and Cirac (2004a);
Friedenauer  _et al._ (2008); Islam  _et al._ (2011); Lanyon  _et al._ (2011)
with trapped ions Blatt and Roos (2012); Monroe  _et al._ (2020); QSs may also
find numerous applications for high-energy physics Bañuls  _et al._ (2020).
Trapped-ion systems have already been used to explore the predictions of the
Dirac equation Lamata  _et al._ (2007); Gerritsma  _et al._ (2010, 2011).
Similarly, cold Fermi gases in hexagonal optical lattices Zhu  _et al._
(2007); Tarruell  _et al._ (2012) serve to realise experimentally the fermion
doubling in lattice discretisations of Dirac QFTs Gattringer and Lang (2010).
One can also explore other discretisations where topology plays a key role
Bermudez  _et al._ (2010), which connects to a very fruitful line of research
in cold-atom QSs Goldman  _et al._ (2016). The characteristic level of
microscopic control of these QSs allows to explore QFTs with additional
interactions, the strength of which can be tuned independently, connecting to
paradigmatic four-Fermi models Cirac  _et al._ (2010); Jünemann  _et al._
(2017); Bermudez  _et al._ (2018); Ziegler  _et al._ (2020) or self-
interacting scalar fields Jordan  _et al._ (2012); Bermudez  _et al._ (2017a);
Klco and Savage (2019). In connection to our most fundamental theories of
Nature Yang and Mills (1954), QSs have also been proposed to target gauge
theories Byrnes and Yamamoto (2006); Büchler  _et al._ (2005) with both
Abelian Weimer  _et al._ (2010) and non-Abelian Tagliacozzo  _et al._ (2013)
groups. Proposals to include fermionic matter coupled to such gauge fields
Zohar  _et al._ (2012); Banerjee  _et al._ (2012), together with initial
experimental progress along these lines Martinez  _et al._ (2016); Görg  _et
al._ (2019); Schweizer  _et al._ (2019); Mil  _et al._ (2020); Surace  _et
al._ (2020); Yang  _et al._ (2020); Kokail  _et al._ (2019); Atas  _et al._
(2021); Klco  _et al._ (2018); Lamm and Lawrence (2018), have made it a very
active and rapidly-evolving research direction Lamm  _et al._ (2019); Kaplan
and Stryker (2020) (see Bañuls  _et al._ (2020) for a recent review) with
interesting connections to topological phases of matter and topological order
Magnifico  _et al._ (2019a, b); González-Cuadra  _et al._ (2020); Borla  _et
al._ (2021).

In this article, we focus on one of the aspects emphasised above: the
possibility of realising paradigmatic QFTs that can be probed in unprecedented
ways. In the context of high-energy physics, one typically probes these QFTs
via scattering experiments where the particles, described as the fundamental
excitations of the fields, collide against each other in large accelerators.
In the context of AMO quantum technologies, there are other techniques to
probe a quantum field, which can be traced back to the origins of quantum
optics and the theory of photo-detection of the electromagnetic field Glauber
(1963). Historically, there have been various models of the photo-detector,
such as harmonic-oscillator probes Mollow (1968), an ensemble of ancillary
atoms with a single groundstate and a continuum of excited levels Scully and
Lamb (1969), or a simple two-level atom or qubit Milonni  _et al._ (1995), all
of which get excited by the absorption of photons from the electromagnetic
field. By placing these probes at different spatial locations, and detecting
delayed photon coincidences, one gets information about the retarded
correlation functions of this gauge field. These correlation functions play a
key role in the quantum theory of optical coherence Glauber (1963), and form
the basis of numerous groundbreaking experiments Brown and Twiss (1956);
Hanbury Brown and Twiss (1956) that have shaped the progress of AMO quantum
technologies Unruh (1976). Similar ancillary probes are referred to as
particle detectors in QFTs on curved spacetimes Birrell and Davies (1999),
such as a quantum particle held in a box that moves with constant acceleration
in the background of a scalar field Unruh (1976). The excitation rate of this
accelerated particle, which is non-zero even when the field is in the vacuum,
probes the so-called Unruh effect Crispino  _et al._ (2008). Modified versions
of this particle detector, such as a two-level atom DeWitt (2010); Unruh and
Wald (1984) or a harmonic oscillator Unruh (1992), are commonly referred to as
Unruh-DeWitt detectors which, if correctly switched on/off, can give
information about the so-called Wightman correlation function of the scalar
field de Ramón  _et al._ (2018).

In the context of contemporary applications of quantum technologies, in
particular quantum sensing Degen  _et al._ (2017), these photon and particle
detectors can be categorised as Rabi-type quantum sensors, as they rely on the
excitation of the probe to gather information about the quantum field. There
is, however, a different kind of quantum sensor where, rather than focusing on
its excitation probability, one monitors its coherence via Ramsey
interferometry Ramsey (1950). Ramsey-type sensors are actually used in high-
precision measurements, and exploited as frequency standards in the so-called
atomic clocks Ludlow  _et al._ (2015), which include both trapped-ion and
cold-atom technologies. In the context of QSs of condensed-matter models,
Ramsey sensors have also been considered as probes for quantum many-body
properties, both in equilibrium Bruderer and Jaksch (2006); Schmidt  _et al._
(2018) to probe equal-time correlation functions Elliott and Johnson (2016);
Streif  _et al._ (2016), and out of equilibrium Levitov  _et al._ (1996);
Lesovik  _et al._ (2006); Bermudez  _et al._ (2013); Dorner  _et al._ (2013);
Mazzola  _et al._ (2013), where noise and fluctuations can give crucial
information about transport phenomena.

In the context of QSs of high-energy physics, to the best of our knowledge,
Ramsey-type probes have been much less studied. In Zohar and Reznik (2013),
these probes were used to characterise the Wilson loop operator of a lattice
gauge theory. In Bermudez  _et al._ (2017a), Ramsey-type probes coupled to
self-interacting Klein-Gordon fields have been explored for trapped-ion QSs.
Although the real Klein-Gordon field Klein (1926); Gordon (1926) is an
archetype QFT Peskin and Schroeder (1995), and the inclusion of self-
interactions, e.g. $\lambda\phi^{4}$ terms, shapes our understanding of
crucial aspects of QFTs such as renormalisation Wilson and Kogut (1974); there
is no fundamental particle in Nature described by this real scalar field. It
is thus quite interesting that QSs have the potential to realise this
paradigmatic QFT in real experiments and, actually, explore different
effective dimensionalities. Moreover, as put forth in Bermudez  _et al._
(2017a), one can devise a Ramsey-type protocol to probe this field theory,
turning key QFT concepts such as the generating functional Peskin and
Schroeder (1995) into experimentally measurable observables. In that work, we
showed how one can devise an interferometric protocol that maps the
information of such a generating functional onto a multi-partite entangled
state of ancillary qubits Bermudez  _et al._ (2017a), which are coupled to the
scalar field via a generalisation of the so-called Schwinger sources Schwinger
(1951a, b). It is interesting that, in contrast to the previous probing
protocols that infer either equal-time, retarded, or Wightman correlation
functions, by switching these sources on/off appropriately in the impulsive
(i.e. anti-adiabatic) regime Bermudez  _et al._ (2017a), these entangled
Ramsey sensors give access to any $n$-point Feynman propagator. Such time-
ordered functions form the basis of current approaches to QFTs, and measuring
them is thus a way of extracting all relevant information about equilibrium or
real-time dynamics of the QFT, providing an alternative to scattering
experiments.

In this article, we modify this sensing protocol and dispense with the
requirement of initialising the qubits in a multi-partite entangled state. We
show that, by abandoning the impulsive regime to focus instead on harmonic
Ising-Schwinger sources, the dynamics can be described in terms of inter-qubit
Ising-type unitaries, which are connected to the underlying generating
functional evaluated at these specific source functions. In a regime in which
these sources are off-resonant with the massive $\lambda\phi^{4}$ QFT, we show
that the qubits evolve under an effective quantum Ising model with long-range
couplings that are fully controlled by a dimensionally-reduced Euclidean
propagator of the self-interacting Klein-Gordon field. Since this propagator
contains relevant information about various aspects of the renormalisation of
the scalar field, our results show that these matters can be directly probed
by monitoring the dynamics of the ancillary qubits which, we remark again,
need not be prepared in a multi-partite entangled state. Additionally, in
contrast to the impulsive protocol Bermudez  _et al._ (2017a), the sources
need not be switched on/off in various combined experiments, but are simply
adiabatically switched on during the whole probing sequence.

We show that these results naturally connect to a long-wavelength description
of the aforementioned QSs of spin models in trapped ions Porras and Cirac
(2004a); Friedenauer  _et al._ (2008); Islam  _et al._ (2011); Monroe  _et
al._ (2020), if one works in the vicinity of a structural phase transition
Fishman  _et al._ (2008). As discussed in detail below, the role of the real
Klein-Gordon field is played by the transverse sound waves of the ions which,
for sufficiently low temperatures, describe the vibrations of the ions around
the equilibrium positions of a linear Coulomb crystal. In contrast to sound
waves in solid-state materials, a detailed long-wavelength theory shows that
these vibrational excitations, the so-called phonons, are massive particles
and thus move inside an effective light-cone determined by the transverse
speed of sound, which would be fully tunable in an experiment. As one modifies
the ratio of the trap frequencies, approaching the aforementioned structural
phase transition, the bare mass of the effective Klein-Gordon field reduces,
whereas the role of quartic non-linearities becomes more important. We show
that, by introducing additional lasers, one can couple the transverse motion
of the ions to the internal atomic structure, which can be described in terms
of qubits, such that the low-energy description of this system is a scalar-
sigma QFT of a self-interacting Klein-Gordon field coupled to a constrained
sigma field that represents the underlying qubits. By exploiting the
connection to high-energy physics, and in particular to the concept of
generating functionals and the renormalisation group, we show that the self-
interacting scalar field mediates long-range Ising interactions between the
qubits due to the exchange of virtual phonons which, by virtue of the
additional quartic self-interactions, can now scatter along the way. This
allows us to predict that, in comparison to the standard phonon-mediated Ising
interactions in harmonic trapped-ion crystals Porras and Cirac (2004a);
Friedenauer  _et al._ (2008); Islam  _et al._ (2011); Monroe  _et al._ (2020),
by approaching the structural phase transition, the intensity and range of the
Ising couplings get contributions from these scattering events of the
virtually-excited bosons, and can be neatly described in terms of the mass and
wavefunction renormalisation of the scalar field. In essence, the role of the
quartic interactions leads to a renormalisation of transverse sound in the ion
crystal, which can be probed by monitoring the real-time dynamics of an
effective long-range Ising model. We believe that the results presented in
this work are at reach of several current trapped-ion experiments.

Since the topic of this article is multidisciplinary in nature, we have made a
special effort to present our results in a way that is accessible to two
communities, those working in lattice field theories for high-energy physics
and those more familiarised with quantum optics and AMO quantum technologies.
We have thus included additional material in appendixes to make this work
self-contained, and to present concepts in a way that will hopefully be
accessible for both communities.

##  II Klein-Gordon scalar fields coupled to $\mathbb{Z}_{2}$ fields

In this section, we use canonical-quantisation techniques to describe two
different sensing protocols for the generating functional of the massive
Klein-Gordon field $\mathsf{Z}_{0}[J]$. In Appendix A, the reader can find
details about the derivation of $\mathsf{Z}_{0}[J]$ for this field theory,
which we now build upon to present an alternative take on the quantum sensing
protocol introduced in Bermudez  _et al._ (2017a). As described in Sec. II.1,
the exact form of the unitary evolution operation sheds light on the need of
multi-partite entangled probes to extract $\mathsf{Z}_{0}[J]$ within this
scheme. This discussion also clarifies, as described in Sec. II.2, that the
mapping of $\mathsf{Z}_{0}[J]$ onto the probes relies on interactions mediated
by the virtual exchange of Klein-Gordon bosons, which in turn suggests a
simpler sensing protocol based on the use of off-resonant harmonic sources, as
introduced in Sec. II.2. By measuring the dynamics of a pair of unentangled
probes to infer the characteristics of an effective long-range Ising
Hamiltonian, one can reconstruct the 2-point propagator and with it, also the
free generating functional $\mathsf{Z}_{0}[J]$ in a simpler manner.

###  II.1 Impulsive sensors of the generating functional

Figure 1:  Scheme of the impulsive sensors for the generating functional: A
real scalar field $\phi(t,\textsf{{x}})$ is coupled locally by Ising-Schwinger
sources of strength $J(t,\textsf{{x}})$ to an ancillary Ising field which, in
principle, also resides at every spacetime point
$\bm{\sigma}(t,\textsf{{x}})$. In practice, the QFT will be regularised on a
lattice of spacing $a$, and the discretised Ising field need only be placed at
a reduced set of spatial locations, which can be interpreted as a multi-
partite sensor. By switching on/off the Ising-Schwinger sources are
impulsively, as discussed in the text, one maps the information of the
generating functional of the field $\mathsf{Z}[J]$ onto the Ising spins (see
the first inset). The full time evolution, including the initialisation of the
Ising spins in a multi-partite entangled state, can be represented as a
quantum circuit (see the second inset).

In the framework of canonical quantisation, the real scalar field evolving
under a Klein-Gordon equation Klein (1926); Gordon (1926) can be described by
the following Hamiltonian

$H_{0}\\!=\\!\\!\int\\!\\!{\rm d}^{d}\\!\mathsf{x}\mathcal{H}_{0},\hskip
5.16663pt\mathcal{H}_{0}=\textstyle\frac{1}{2}\pi^{2}(x)+\textstyle\frac{1}{2}(\bm{\nabla}\phi(x))^{2}+\textstyle\frac{1}{2}{m_{0}^{2}}\phi^{2}(x),$
(1)

where $\phi(x),\pi(x)$ are the field operator and its conjugate momentum, both
defined on a $D=d+1$ Minkowski spacetime $x=(t,\textsf{{x}})$, and satisfying
equal-time bosonic commutation relations
$[\phi(t,\textsf{{x}}),\pi(t,\textsf{{x}}^{\prime})]={\rm
i}\delta^{d}\\!(\textsf{{x}}-\textsf{{x}}^{\prime})$. Here, $m_{0}$ stands for
the bare mass of the scalar bosons, $\bm{\nabla}$ contains only spatial
derivatives, and we use natural units $\hbar=c=1$.

The generating functional $\mathsf{Z}_{0}[J]$ depends on the so-called
Schwinger sources $J(x)$ Schwinger (1951a, b), which are classical fields that
introduce local perturbations in the original QFT (1), modifying the number of
excitations of the field, and thus creating or annihilating scalar bosons (see
App. A). We consider the promotion of these classical sources to quantum-
mechanical degrees of freedom Bermudez  _et al._ (2017a), by introducing an
Ising field $\bm{\sigma}(x)$ expressed in terms of the identity and Pauli
matrices

$\sigma^{0}=\mathbb{I}_{2},\hskip 4.30554pt\sigma^{1}=X,\hskip
4.30554pt\sigma^{2}=Y,\hskip 4.30554pt\sigma^{3}=Z,$ (2)

in the corresponding basis
$\\{\ket{0_{\textsf{{x}}}}=\ket{\uparrow_{\textsf{{x}}}},\ket{1_{\textsf{{x}}}}=\ket{\downarrow_{\textsf{{x}}}}\\}$.
With this notation, the Ising fields can be interpreted as qubits Nielsen and
Chuang (2010) that will serve as two-level quantum sensors located at
different spacetime points (see Fig. 1). Accordingly, the Hilbert space
becomes a tensor product
$\mathpzc{H}=\mathpzc{H}_{\phi}\otimes\mathpzc{H}_{\sigma}$, and one can
exploit the coupling between the scalar and Ising fields to define a
measurement protocol for $\mathsf{Z}_{0}[J]$. This becomes particularly
interesting when including interactions, as the same measurement scheme
applies, but the generating functional now contains non-trivial information
about the interacting fields, such as renormalisation and the underlying fixed
points Wilson and Kogut (1974); Hollowood (2013).

As discussed in Bermudez  _et al._ (2017a), the Schwinger sources are promoted
to $J(x)\to\bm{J}(x)\cdot\bm{\sigma}(x)$, which becomes interesting for
$\bm{J}_{\alpha}(x)=J(x)(\delta_{\alpha,0}-\delta_{\alpha,3})/2$, as the
scalar field then couples to

$\bm{J}(x)\cdot\bm{\sigma}(x)=J(x)P(x),$ (3)

where we have introduced the orthogonal Ising projector

$P(x)=\ket{\downarrow_{\textsf{{x}}}}\\!\\!\bra{\downarrow_{\textsf{{x}}}}=\textstyle\frac{1}{2}(1-Z(x)).$
(4)

The new Hamiltonian density $\tilde{\mathcal{H}}_{0}+\tilde{\mathcal{V}_{J}}$
with these Ising-Schwinger sources is

$\tilde{\mathcal{H}}_{0}=\mathcal{H}_{0}+\delta\epsilon(x)Q(x),\hskip
6.45831pt\tilde{\mathcal{V}_{J}}=-J(x)\phi(x)P(x),$ (5)

where we have introduced the remaining orthogonal projector

$Q(x)=\ket{\uparrow_{\textsf{{x}}}}\\!\\!\bra{\uparrow_{\textsf{{x}}}}=\textstyle\frac{1}{2}(1+Z(x)),$
(6)

and $\delta\epsilon(x)$ is the energy density for the Ising field. Since
$[P(x),Q(y)]=0$, it is straightforward to see that the time-evolution operator
$U(t_{\rm
f},t_{0})=U_{\\!\\!\tilde{\mathcal{H}}_{0}}U_{\tilde{\mathcal{V}}_{J}}$ can be
expressed in terms of two unitaries

$U_{\tilde{\mathcal{H}}_{0}}={\rm e}^{-{\rm i}\int\\!{\rm
d}^{D}x{\mathcal{\tilde{H}}}_{0}},\hskip
8.61108ptU_{\tilde{\mathcal{V}}_{J}}=\\!\mathsf{T}\\!\left\\{{\rm e}^{+{\rm
i}\int\\!{\rm d}^{D}xJ(x)\phi_{H}(x)P(x)}\\!\right\\},$ (7)

in complete analogy to the situation for the standard Schwinger sources in Eq.
(128). Here, $\mathsf{T}\\{\cdot\\}$ is the time-ordering operator, the scalar
fields
$\phi_{H}(x)=(U_{\\!\\!\tilde{\mathcal{H}}_{0}})^{\dagger}\phi(x)U_{\\!\\!\tilde{\mathcal{H}}_{0}}$
evolve in the Heisenberg picture with respect to the unsourced Klein-Gordon
Hamiltonian (1), whereas the Ising projectors do not change since
$U_{\\!\\!\tilde{\mathcal{H}}_{0}}P(x)(U_{\\!\\!\tilde{\mathcal{H}}_{0}})^{\dagger}=P(x)$.
By using the Magnus expansion Magnus (1954); Blanes  _et al._ (2010) described
in Eq. (133) of the appendix, and carrying out the steps in analogy to the
Klein-Gordon QFT with classical sources of Appendix A, we find

$U_{\tilde{\mathcal{V}}_{J}}\\!=U_{N}\hskip 0.86108pt{\rm
e}^{-\frac{1}{2}\\!\\!\\!\bigintssss\\!\\!{\rm
d}^{D}x_{1}\bigintssss\\!\\!{\rm
d}^{D}x_{2}P(x_{1})J(x_{1})\Delta_{m_{0}}\\!(x_{1}-x_{2})J(x_{2})P(x_{2})}\\!\\!.$
(8)

Here, we have introduced the normal-ordered unitary

$U_{N}=:{\rm e}^{-{\rm i}\\!\\!\bigintssss\\!\\!{\rm d}^{D}xJ(x)\phi_{\rm
H}(x)P(x)}\\!:,$ (9)

and the time-ordered Feynman propagator of the scalar field

$\Delta_{m_{0}}\\!(x)=\int_{k}\tilde{\Delta}_{m_{0}}\\!(k){\rm e}^{-{\rm
i}kx},\hskip 8.61108pt\tilde{\Delta}_{m_{0}}\\!(k)=\frac{{\rm
i}}{k^{2}-m_{0}^{2}+{\rm i}\epsilon},$ (10)

where $k=(\omega,\textsf{{k}})$, and we define
$\int_{k}=\int_{\phantom{k}}\\!\\!\\!{\rm d}^{D}k/(2\pi)^{D}$, and
$\epsilon\to 0^{+}$. The structure of equation (8) already suggests that it
may be possible to map all the information of the generating functional (11)
of the massive Klein-Gordon field

$\mathsf{Z}_{0}[J]={\rm e}^{-\frac{1}{2}\\!\\!\bigintssss\\!\\!{\rm
d}^{D}x_{1}\\!\\!\bigintssss\\!\\!{\rm d}^{D}x_{2}\hskip
0.60275ptJ(x_{1})\Delta_{m_{0}}\\!(x_{1}-x_{2})J(x_{2})},$ (11)

onto the dynamics of the qubits by an appropriate protocol. Note that
$\mathsf{Z}_{0}[0]=1$, so we are referring to the free normalised generating
functional, or the full one $\mathsf{Z}[0]=1$, in this article.

The key idea underlying the mapping of the generating-functional onto the
Ising probes, presented in Bermudez  _et al._ (2017a) using a different
approach, is that the full time-evolution $U(t_{\rm
f},t_{0})=U_{\tilde{\mathcal{H}}_{0}}U_{\tilde{\mathcal{V}}_{J}}$ can map this
information into the amplitudes of a specific initial state. One starts by
preparing a GHZ state for the qubits through the quantum circuit displayed in
Fig. 1, which applies concatenated CNOT gates to a product input state, and
should be extended to all the locations of the Ising spins. After these gates
Nielsen and Chuang (2010), and right before the Ising-Schwinger sources are
switched on, the state is $\ket{\psi(t_{0})}=\ket{0}\otimes\ket{\rm GHZ}$,
where $\ket{0}$ is the Klein-Gordon vacuum, and $\ket{\rm
GHZ}=(\Pi_{\textsf{{x}}}\ket{\uparrow_{\textsf{{x}}}}+\Pi_{\textsf{{x}}}\ket{\downarrow_{\textsf{{x}}}})\sqrt{2}$
is a multi-partite entangled state. We now let the scalar and Ising fields
couple through the Ising-Schwinger sources, and obtain the time-evolved state
$\ket{\psi(t_{\rm
f})}=U_{\tilde{\mathcal{H}}_{0}}U_{\tilde{\mathcal{V}}_{J}}\ket{\psi(t_{0})}$,
where the unitaries are described in Eq. (7). Due to trivial action of the
normal-ordered part (9) on the Klein-Gordon vacuum, and the action of the
Ising projectors on the corresponding qubit states, only the second part of
the entangled state evolves non-trivially, yielding

$\ket{\psi(t_{\rm
f})}=\frac{1}{\sqrt{2}}\ket{0}\otimes\left(\\!\Pi_{\textsf{{x}}}\ket{\uparrow_{\textsf{{x}}}}+\mathsf{Z}_{0}[J]{\rm
e}^{{\rm i}\int{\rm
d}^{D}x\delta\epsilon(x)}\Pi_{\textsf{{x}}}\ket{\downarrow_{\textsf{{x}}}}\\!\\!\right).$
(12)

Here, we readily find that the free generating functional (11) appears in the
relative amplitude of the time-evolved spin state, after neglecting an
irrelevant global phase factor that oscillates with the zero-point energy of
the scalar field. The signal can be extracted by measuring the global parities

$\begin{split}\mathsf{P}_{1}[J]=\bra{\psi(t_{\rm
f})}\Pi_{\textsf{{x}}}X\\!(\textsf{{x}})\ket{\psi(t_{\rm f})}={\rm
Re}\\{\mathsf{Z}_{0}[J]{\rm e}^{{\rm i}\int{\rm
d}^{D}x\delta\epsilon(x)}\\},\\\ \mathsf{P}_{2}[J]=\bra{\psi(t_{\rm
f})}\Pi_{\textsf{{x}}}Y\\!(\textsf{{x}})\ket{\psi(t_{\rm f})}={\rm
Im}\\{\mathsf{Z}_{0}[J]{\rm e}^{{\rm i}\int{\rm
d}^{D}x\delta\epsilon(x)}\\}.\end{split}$ (13)

After introducing the generating functional of connected propagators, which in
the non-interacting case simply reads

$\mathsf{W}_{0}[J]=-{\rm i}\log Z_{0}[J]=\frac{{\rm
i}}{2}\\!\\!\bigintssss\\!\\!{\rm
d}^{D}x_{1}\\!\\!\\!\\!\bigintssss\\!\\!\\!\\!{\rm d}^{D}x_{2}\hskip
0.86108ptJ(x_{1})\Delta_{m_{0}}\\!(x_{1}-x_{2})J(x_{2}),$ (14)

one can see that this functional is encoded in the relative phase of the
multi-partite entangled state (12), which is which is the typical outcome of
Ramsey interferometry Ramsey (1950), one of the key methods in quantum sensing
Degen  _et al._ (2017). The global parities are thus a multi-partite
generalization of a pair of Ramsey signals

$\begin{split}\mathsf{P}_{1}[J]=\cos\left(\mathsf{W}_{0}[J]+\int{\rm
d}^{D}x\delta\epsilon(x)\right),\\\
\mathsf{P}_{2}[J]=\sin\left(\mathsf{W}_{0}[J]+\int{\rm
d}^{D}x\delta\epsilon(x)\right).\end{split}$ (15)

Selecting specific timings $(t_{0},t_{\rm f})$ would allow one to reconstruct
the desired generating functional for any particular set of sources $J(x)$
from experimental data. This result teaches us an important lesson, namely
that previously-considered mathematical tools in QFTs, such as the generating
functional, can indeed become observable quantities when combining ideas from
high-energy physics and AMO quantum technologies. will be a generalisation of
Ramsey interferometry that As discussed in Bermudez  _et al._ (2017a), the
qubits need not be attached to every spacetime point when one is only
interested in recovering a specific $n$-point propagator (129). In that case,
it suffices to use $n$ qubits located at the desired spatial locations
$\delta\epsilon(x)=\omega_{0}\sum_{i=1}^{n}\delta^{d}\\!(\textsf{{x}}-\textsf{{x}}_{i})$,
where $\omega_{0}$ is the transition frequency between the two levels, and
switch on and off the sources impulsively (i.e. non-adiabatically with respect
to any other timescale in the problem) at the corresponding lapses of time
$J(x)=\sum_{i}\mathsf{J}_{i}\delta^{D}\\!(x-x_{i})$, as indicated in Fig. 1.
One would then gather the interferometric experimental data
$\mathsf{P}_{1},\mathsf{P}_{2}$ for various combinations of the impulsive
sources Bermudez  _et al._ (2017a), and infer the corresponding functional
derivatives that lead to the desired $n$-point propagator. In the simplest
situation, inferring $\Delta_{m_{0}}\\!(x_{1}-x_{2})$ requires repeating the
interferometric protocol four times using two qubits at each of the spatial
locations of interest $\textsf{{x}}_{1},\textsf{{x}}_{2}$, and switching on
and off the sources at the corresponding times $t_{1},t_{2}$ of the spacetime
points $x_{1},x_{2}$. Together with the requirement of using multi-partite
entangled states for the initialisation of the Ramsey probe, maintaining the
coherence of macroscopically-distinct states, and repeating the whole
experimental sequence to extract the quantum-mechanical statistics, this
sensing protocol turns out to be quite challenging from an experimental
perspective. In the following section, we shall describe a simpler
alternative.

###  II.2 Harmonic sensors and long-range Ising models

The above simple calculation for the free Klein-Gordon field unveils an
interesting fact, which was partially hidden under the general formalism for
interacting fields Bermudez  _et al._ (2017a). Equation (8) shows that the
$\mathsf{Z}_{0}[J]$ mapping relies on unitaries describing pairwise
interactions $U_{\tilde{\mathcal{V}}_{J}}=U_{N}\Pi_{x1,x2}U_{x1,x2}$ with

$U_{x1,x2}={\rm
e}^{-\frac{1}{2}P\\!(x_{1})J(x_{1})\Delta_{m_{0}}\\!(x_{1}-x_{2})J(x_{2})P\\!(x_{2})}.$
(16)

This unitary represents a pairwise coupling between distant qubits, which must
be mediated by the Klein-Gordon bosons acting as fundamental carriers of a
spin-spin interaction. Since the Feynman propagator of the scalar field
appears in Eq. (16), there might be simpler measurement protocols to extract
relevant properties of the QFT without resorting to the full generating-
functional protocol just described.

Figure 2:  Scheme of the harmonic sensors for the generating functional: Two
distant Ising spins at positions $\textsf{{x}}_{1}$ and $\textsf{{x}}_{2}$ are
coupled to the real scalar field $\phi(t,\textsf{{x}})$ locally by harmonic
Ising-Schwinger sources
$J(t,\textsf{{x}}_{i})\propto\sin(\omega_{J}t-\textsf{{k}}_{J}\cdot\textsf{{x}}_{i})$
that is adiabatically switched during the whole probing lapse. As a
consequence, the scalar bosons will mediate an effective Ising interaction
between the spins, which is represented by a double blue line. In the upper
inset, we depict the strength of this spin-spin interactions, which is
effectively controlled by a dimensionally-reduced Euclidean propagator of the
scalar field. In the presence of quartic couplings, this propagator will
include all possible scattering events that can be represented in terms of
Feynman diagrams, as displayed in the inset up to second order of the quartic
interaction strength.

Let us note, before moving on, that the crucial aspect of the above pairwise
unitary is not restricted to the specific form of the Ising-Schwinger sources,
which involve orthogonal projectors (5). The key point is that the coupling
must contain a qubit operator $P(x)\to O(x)$ that, in the Heisenberg picture
with respect to the unsourced Hamiltonian (5), commutes with itself at
different instants of time $[O_{H}(x),O_{H}(y)]=0$. As will become clear when
discussing the particular application for trapped-ion quantum computers
Häffner  _et al._ (2008); Ladd  _et al._ (2010), one can find various schemes
where this operator is any of the Pauli matrices
$O(x)\in\\{X(x),Y(x),Z(x)\\}$. In the following, we will solely focus on the
$Z$-type couplings

$\tilde{\mathcal{V}_{J}}=-J(x)\phi(x)Z(x),$ (17)

leading to Ising $ZZ$ pairwise unitaries

$U_{x1,x2}={\rm
e}^{-\frac{1}{2}Z\\!(x_{1})J(x_{1})\Delta_{m_{0}}\\!(x_{1}-x_{2})J(x_{2})Z\\!(x_{2})},$
(18)

but note that the results will be interchangeable to the other $XX$ or $YY$
Ising interactions.

As shown below, for certain types of sources, these pairwise unitaries can be
expressed in terms of an effective time-independent Hamiltonian, which allow
for a different sensing protocol. In particular, the probes evolve under an
effective Ising model with long-range couplings $H_{\rm eff}$, which controls
completely the non-trivial part of the time evolution

$U(t_{\rm f},t_{0})\approx U_{\tilde{\mathcal{H}}_{0}}U_{\rm eff}={\rm
e}^{-{\rm i}(t_{\rm f}-t_{0})\tilde{H}_{0}}{\rm e}^{-{\rm i}(t_{\rm
f}-t_{0})H_{\rm eff}}.$ (19)

Instead of using non-adiabatic sources that are switched on/off in the
impulsive regime, we consider a harmonic source

$J(x)=\mathsf{J}_{0}\sin(k_{J}x)=\mathsf{J}_{0}\sin\big{(}\omega_{J}t-\textsf{{k}}_{J}\cdot\textsf{{x}}\big{)},$
(20)

where $\mathsf{J}_{0}$ is the coupling strength density, and
$k_{J}=(\omega_{J},\textsf{{k}}_{J})$ is the external $D$-momentum determining
the plane-wave harmonic source. We also consider that the source lies below
the resonance of the Klein-Gordon modes (i.e. $\omega_{J}\lesssim m_{0}$) and,
moreover, its strength is constrained according to

$\mathsf{J}_{0}\ll(\omega_{\textsf{{k}}}-\omega_{J}){\rm
d}^{d}\textsf{{k}}<(\omega_{\textsf{{k}}}+\omega_{J}){\rm d}^{d}\textsf{{k}}.$
(21)

In this regime, the time integrals underlying the spacetime formulation of the
evolution operator (8) can be performed analytically using the forward and
backward parts of the Feynman propagator (135). We find that the normal-
ordered part of Eq. (8) becomes negligible, such that the evolution of the
scalar field and the Ising spins gets effectively decoupled. Whereas the
former evolves under the Klein-Gordon dynamics (1), the spins also experience
the second-order term including the pairwise couplings (18), which leads to

$U_{\rm eff}=\\!{\rm e}^{{\rm i}(t_{\rm
f}-t_{0})\\!\\!\bigintssss\\!\\!\\!{\rm
d}^{d}\\!\textsf{{x}}_{1}\\!\\!\bigintssss\\!\\!\\!{\rm
d}^{d}\\!\textsf{{x}}_{2}\\!\\!\bigintssss_{\textsf{{k}}}\\!\\!\\!\frac{\mathsf{J}_{0}^{2}\cos\left((\textsf{{k}}-\textsf{{k}}_{J})\cdot(\textsf{{x}}_{1}-\textsf{{x}}_{2})\right)}{\omega_{\textsf{{k}}}^{2}-\omega_{J}^{2}}Z(\textsf{{x}}_{1})Z(\textsf{{x}}_{2})}\\!\\!,$
(22)

where $\int_{\textsf{{k}}}=\int\\!{\rm d}^{d}\textsf{{k}}/(2\pi)^{d}$ stands
for the integrals about the spatial momentum. Using Eq. (19), one readily
identifies an Ising Hamiltonian

$H_{\rm eff}=\frac{1}{2}\\!\int\\!\\!{\rm
d}^{d}\\!\textsf{{x}}_{1}\\!\\!\int\\!\\!{\rm
d}^{d}\\!\textsf{{x}}_{2}\mathsf{J}(\textsf{{x}}_{1}-\textsf{{x}}_{2})Z(\textsf{{x}}_{1})Z(\textsf{{x}}_{2}),$
(23)

where the factor of $1/2$ avoids double counting of pairs, and we have
introduced the long-range coupling-strength density

$\mathsf{J}(\textsf{{x}}_{1}-\textsf{{x}}_{2})=-2\mathsf{J}_{0}^{2}\\!\\!\bigintssss_{\textsf{{k}}}\frac{\cos\left((\textsf{{k}}-\textsf{{k}}_{J})\cdot(\textsf{{x}}_{1}-\textsf{{x}}_{2})\right)}{\textsf{{k}}^{2}+(m_{0}^{2}-\omega_{J}^{2})}.$
(24)

It is worth pointing out that, despite starting with a local Lorentz-invariant
field theory in Eqs. (1) and (5), it seems that we have arrived to an action
at a distance between the Ising fields (23) that naively violates causality.
Note, however, that the relevant time-scale of the problem $t\sim
1/\mathsf{J}(\textsf{{x}}_{1}-\textsf{{x}}_{2}){\rm
d}^{2d}\\!\textsf{{x}}\gg|\\!|\textsf{{x}}_{1}-\textsf{{x}}_{2}|\\!|/c=t_{\rm
ret}$, where we have included again the speed of light to identify the
corresponding retardation time $t_{\rm ret}$. Accordingly, the regime of
validity of the effective Ising model (23) assumes that the timescales are
sufficiently large for the bosons to causally connect any pair of qubits.

Let us also point out that, in order to achieve this purely unitary dynamics,
keeping the Ising fields off-resonant is crucial. Otherwise, the scalar bosons
could also act as a source of dissipation, as our field theory could be
understood as a multi-qubit continuum limit of the unbiased spin-boson model
at zero temperature Leggett  _et al._ (1987); Unruh (1995); Palma  _et al._
(1996). Here, the harmonic frequency would play the role of the so-called
quantum tunneling $\Delta_{\rm qt}\approx\omega_{J}$, provided that Eq. (60)
is fulfilled. Neglecting possible collective dissipative effects, the main
source of irreversible dynamics would appear in the form of decoherence with a
dephasing rate $\Gamma_{\rm d}$ that is proportional to the spectral density
$\mathsf{S}(\omega)$ of the Klein-Gordon modes at a slightly-renormalised
tunneling amplitude ${\omega}_{J,r}<\omega_{J}<m_{0}$ Díaz-Camacho  _et al._
(2016). Since there are no modes with frequency $\omega_{\textsf{{k}}}<m_{0}$,
the corresponding spectral density would be zero
$\mathsf{S}({\omega}_{J,r})=0$, such that $\Gamma_{\rm d}=0$, and we can
simply focus on the purely coherent dynamics of Eq. (19).

Coming back to the integral (24), we note that it can be expressed in terms of
the dimensionally-reduced Euclidean propagator of a Gaussian field in $d$
dimensions Wilson and Kogut (1974); Zinn-Justin (2012, 2013), which is defined
through the following Green’s function

$\left(-\bm{\nabla}^{2}+m_{\rm eff}^{2}\right)G_{m_{\rm eff}}^{\rm
E}\\!(\textsf{{x}})=\delta^{d}\\!(\textsf{{x}}),$ (25)

where the effective mass is shifted from the bare mass to

$m_{\rm eff}^{2}=m_{0}^{2}-\omega_{J}^{2}.$ (26)

Since the Compton wavelength in natural units is simply the inverse of the
bare mass $\xi_{0}=1/m_{0}$, we can define the following effective wavelength

$\xi_{\rm eff}=m_{\rm
eff}^{-1}=1/\left(m_{0}^{2}-\omega_{J}^{2}\right)^{1/2},$ (27)

which shall control the range of the mediated interactions.

This Euclidean propagator can be obtained by lowering the dimension $D=d+1\to
d$ of Feynman propagator of Eq. (10), and making a Wick rotation, yielding

$G_{m_{\rm eff}}^{\rm
E}\\!(\textsf{{x}})\\!=\\!\\!\bigintssss_{\textsf{{k}}}\frac{{\rm e}^{{\rm
i}{\textsf{{k}}\cdot\textsf{{x}}}}}{\textsf{{k}}^{2}+m_{\rm
eff}^{2}}=\left(\frac{m_{\rm
eff}}{|\\!|\textsf{{x}}|\\!|}\right)^{\\!\\!\nu}\frac{\mathsf{K}_{\nu}(m_{\rm
eff}|\\!|\textsf{{x}}|\\!|)}{(2\pi)^{\nu+1}},$ (28)

where $\nu=\frac{d}{2}-1$, and $\mathsf{K}_{\nu}(u)$ is the modified Bessel
function of the second kind Fradkin (2021), also called Basset function
Gradshteĭn  _et al._ (1965). We have thus found that the effective spin-spin
couplings mediated by a $D$-dimensional Klein-Gordon field subjected to
harmonic Ising-Schwinger sources are controlled by the dimensionally-reduced
Euclidean propagator via

$\mathsf{J}(\textsf{{x}}_{1}-\textsf{{x}}_{2})=-2\mathsf{J}_{0}^{2}G_{m_{\rm
eff}}^{\rm
E}\\!(\textsf{{x}}_{1}-\textsf{{x}}_{2})\cos\big{(}\textsf{{k}}_{J}\cdot(\textsf{{x}}_{1}-\textsf{{x}}_{2})\big{)}.$
(29)

Let us pause for a moment and analyse this result. First of all, one can check
that the above expressions are dimensionally correct in mass/energy units.
According to Eq. (1), the natural or scaling dimension of the scalar field is
$d_{\phi}=(d-1)/2$, whereas the Ising field is dimensionless $d_{\sigma}=0$.
Therefore, the Euclidean propagator (28) has natural dimension $d_{G_{m}^{\rm
E}}=d-2$, whereas the coupling strength $\mathsf{J}_{0}$ (20) has natural
dimension $d_{\mathsf{J}_{0}}=(d+3)/2$, and one finds that the effective Ising
Hamiltonian (23) has units of energy. We now discuss the implications of the
form of the spin-spin couplings (29) in various dimensions. For the $D=3+1$
Klein-Gordon field, $\nu=1/2$, and the modified Bessel function is
$\mathsf{K}_{1/2}(u)=\sqrt{\pi/2u}{\rm e}^{-u}$, such that the Ising couplings
become

$d=3,\hskip
6.45831pt\mathsf{J}(\textsf{{x}})=-\frac{\mathsf{J}_{0}^{2}}{2\pi}\frac{{\rm
e}^{-\frac{|\\!|\textsf{{x}}|\\!|}{\xi_{\rm
eff}}}}{|\\!|\textsf{{x}}|\\!|}\cos\big{(}\textsf{{k}}_{J}\cdot\textsf{{x}}\big{)}.$
(30)

Hence, we identify a Yukawa-type interaction between the qubits (i.e. screened
Coulomb decay) Yukawa (1955). In fact, the analogy becomes clearer when
writing the spins in terms of fermions via a Jordan-Wigner transformation
Jordan and Wigner (1928), since the Ising-Schwinger source term (5) becomes a
Yukawa coupling between the fermionic charge density and a propagating scalar
field, although the former would be spinless and static rather than that
described by a relativistic Dirac field.

We note that the range of the Yukawa interaction (30) is controlled by the
inverse of the effective mass (26), and can thus be tuned by placing the
frequency of the harmonic source $\omega_{{J}}$ closer or further from the
bare mass $m_{0}$. If $\textsf{{k}}_{J}=\bm{0}$, the spin couplings are always
negative, and would thus describe an Ising-Yukawa ferromagnet. Note that, as
advanced above, the Ising field need not be densely distributed over the whole
Minkowski spacetime, but can be arranged at

$\delta\epsilon(x)=\omega_{0}\sum_{i=1}^{n}\delta^{d}\\!(\textsf{{x}}-\textsf{{x}}_{i}).$
(31)

In this situation, ferromagnetic couplings can also be achieved for a harmonic
source with non-zero momentum if the qubits are arranged in a plane/line
orthogonal to the plane wave
$\textsf{{k}}_{J}\perp(\textsf{{x}}_{i}-\textsf{{x}}_{j})$. Likewise, if the
qubits are contained in a region much smaller than the harmonic wavelength
$\textsf{{k}}_{J}\cdot(\textsf{{x}}_{i}-\textsf{{x}}_{j})\approx 0$, the
interactions are approximately ferromagnetic. Away from these conditions, the
Yukawa-decaying couplings will alternate between ferromagnetic and
antiferromagnetic depending on the qubit distance, which can lead to magnetic
frustration.

For the $D=1+1$ Klein-Gordon field, $\nu=-1/2$, and the modified Bessel
function is $\mathsf{K}_{-1/2}(u)=\mathsf{K}_{1/2}(u)$, such that

$d=1,\hskip 6.45831pt\mathsf{J}(\textsf{{x}})=-\mathsf{J}_{0}^{2}\frac{{\rm
e}^{-\frac{|\textsf{{x}}|}{\xi_{\rm eff}}}}{m_{\rm
eff}}\cos\big{(}\textsf{{k}}_{J,\textsf{x}}\textsf{{x}}\big{)}.$ (32)

The situation is similar to the one discussed above, albeit with
exponentially-decaying spin-spin interactions. Whereas for small distances
$|\\!|\textsf{{x}}_{1}-\textsf{{x}}_{2}|\\!|\ll\xi_{\rm eff}$, we get a
Coulomb-type decay in three-dimensions
$\mathsf{J}(\textsf{{x}}_{1}-\textsf{{x}}_{2})\propto
1/|\\!|\textsf{{x}}_{1}-\textsf{{x}}_{2}|\\!|$, which gets exponentially
screened at larger spatial separations, the one-dimensional case (32) is
always described by exponentially-decaying interactions regardless of the
distance.

Finally, for the $D=2+1$ scalar field, $\nu=0$, and the modified Bessel
function cannot be expressed in terms of elementary functions. In this case,
one finds

$d=2,\hskip
6.45831pt\mathsf{J}(\textsf{{x}})=-\frac{\mathsf{J}_{0}^{2}}{\pi}\mathsf{K}_{0}\\!\left(\frac{|\\!|\textsf{{x}}|\\!|}{\xi_{\rm
eff}}\right)\cos\big{(}\textsf{{k}}_{J}\cdot\textsf{{x}}\big{)}\\!.$ (33)

For small spatial separations
$|\\!|\textsf{{x}}_{1}-\textsf{{x}}_{2}|\\!|\ll\xi_{\rm eff}$, there are log-
decaying couplings
$\mathsf{J}_{2}(\textsf{{x}}_{1}-\textsf{{x}}_{2})\approx-\gamma+\log(2\xi_{\rm
eff}/|\\!|\textsf{{x}}_{1}-\textsf{{x}}_{2}|\\!|)$, where $\gamma\approx
0.577$ is Euler’s constant. Conversely, for long distances
$|\\!|\textsf{{x}}_{1}-\textsf{{x}}_{2}|\\!|\gg\xi_{\rm eff}$, we obtain a
power-law coupling that gets exponentially screened.

Before concluding this section, let us also note that one can include quantum
fluctuations in the effective Ising models by modifying the perturbation in
Eq. (5) to

$\tilde{V}\to\hat{V}=\int{\rm
d}^{d}x\big{(}-J(x)\phi(x)Z(x)-H_{\textsf{t}}(x)X\\!(x)\big{)},$ (34)

where $H_{\textsf{t}}(x)$ is a new coupling-strength density that controls the
amount of quantum fluctuations in the Ising fields. We shall assume that the
Ising fields are distributed in a certain spatial arrangement
$\textsf{{x}}_{i}\in\Lambda_{\rm s}$, such that

$\begin{split}H_{\textsf{t}}\\!(x)&=2h_{\textsf{t}}(x)\cos(\omega_{0}t),\hskip
4.30554pth_{\textsf{t}}(x)=h_{\textsf{t}}\sum_{i=1}^{n}\delta^{d}\\!(\textsf{{x}}-\textsf{{x}}_{i}),\\\
J(x)&=\sum_{i=1}^{n}{J}_{0}\sin\big{(}\omega_{J}t-\textsf{{k}}_{J}\cdot\textsf{{x}}_{i}\big{)}\delta^{d}\\!(\textsf{{x}}-\textsf{{x}}_{i}),\end{split}$
(35)

where the couplings $J_{0},h_{\textsf{t}}$ have now units of mass/energy. The
non-trivial part of the time-evolution operator, considering that
$h_{\textsf{t}}\ll 2\omega_{0}$, can now be described as

$U_{\hat{\mathcal{{V}}}_{J}}=\\!\mathsf{T}\\!\left\\{{\rm e}^{+{\rm
i}\int\\!{\rm
d}t\sum_{i}\big{(}J(t,\textsf{{x}}_{i})\phi_{H}(t,\textsf{{x}}_{i})Z(t,\textsf{{x}}_{i})+h_{\textsf{t}}X(t,\textsf{{x}}_{i})\big{)}}\\!\right\\},$
(36)

We can now repeat the previous procedure to calculate the evolution operator,
but differences will arise as the Magnus expansion is no longer exact at
second order (133). Nonetheless, the additional terms can be neglected if the
coupling constraint (60) is changed so that it encompasses both couplings

$|J_{0}|,|h_{\textsf{t}}|\ll(\omega_{\textsf{{k}}}-\omega_{J}),(\omega_{\textsf{{k}}}+\omega_{J}).$
(37)

We then arrive at a long-range version of the transverse-field quantum Ising
model Pfeuty (1970); Kogut (1979), namely

$H_{\rm eff}=\frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n}J_{ij}\hskip
1.07639ptZ(\textsf{{x}}_{i})Z(\textsf{{x}}_{j})-h_{\textsf{t}}\sum_{i=1}^{n}X(\textsf{{x}}_{i}),$
(38)

where the spin-spin couplings
$J_{ij}=\mathsf{J}(\textsf{{x}}_{i}-\textsf{{x}}_{j})J_{0}^{2}/\mathsf{J}_{0}^{2}$
have units of mass/energy, and are thus controlled by the $D$-dimensional
Euclidean Green’s function (29). In analogy to the nearest-neighbor models
Pfeuty (1970); Kogut (1979), there is a competition between magnetic phases
that break spontaneously the $\mathbb{Z}_{2}$ symmetry
$P_{1}=\Pi_{\textsf{{x}}_{i}}X(\textsf{{x}}_{i})$ that inverts
$Z(\textsf{{x}}_{i})\to P_{1}Z(\textsf{{x}}_{i})P_{1}=-Z(\textsf{{x}}_{i})$;
and paramagnetic phases where all spins point in the direction of the external
transverse field $h_{\textsf{t}}$. Neglecting the possible frustration due to
the periodic alternation between ferro- and antiferro-couplings, we expect
that the exponential screening at large distances will lead to critical
theories in the Ising universality class, which have the same scaling
behaviour as the corresponding nearest-neighbor models.

Let us now switch to the discussion of the simplified quantum sensors of the
massive Klein-Gordon field. Regardless of the phase diagram of the Ising spins
in the thermodynamic limit $n\to\infty$, if one is interested in the
characterisation of the underlying scalar field, a pair of Ising spins can
suffice as sensor qubits $n=2$ (see Fig. 2), providing a much simpler scheme
than the one presented above for the full generating functional. First of all,
we do not require initialising the system in a multi-partite entangled state,
but instead in $\ket{\Psi_{0}}=\ket{0}\otimes\ket{+_{i}}\otimes\ket{+_{j}}$,
where we recall that $\ket{0}$ is the Klein-Gordon vacuum, and
$\ket{+_{i}}=(\ket{\uparrow_{\textsf{{x}}_{i}}}+\ket{\downarrow_{\textsf{{x}}_{i}}})/\sqrt{2}$
is a coherent superposition for each of the Ising probes. Secondly, instead of
measuring the global parities (13), it will suffice to measure a single-qubit
observable in the Pauli basis, e.g. $X(\textsf{{x}}_{i})$, in the time-evolved
state $\ket{\Psi_{\rm f}}=U(t_{\rm f},(t_{\rm
f}-t_{0})/2)X(\textsf{{x}}_{i})X(\textsf{{x}}_{j})U((t_{\rm
f}-t_{0})/2,t_{0})\ket{\Psi_{0}}$. Setting $h_{\textsf{t}}=0$, this time
evolution yields

$\bra{\Psi_{\rm f}}X(\textsf{{x}}_{i})\ket{\Psi_{\rm
f}}=\bra{\Psi_{0}}U_{\tilde{\mathcal{V}}_{J}}^{\dagger}X(\textsf{{x}}_{i})U_{\tilde{\mathcal{V}}_{J}}^{\phantom{\dagger}}\\!\ket{\Psi_{0}}=\cos\left(2J_{ij}(t_{\rm
f}-t_{0})\\!\right)\\!,$ (39)

such that one could extract the spin-spin coupling strength $J_{ij}$ from the
real-time evolution of the transverse magnetisation. Then, the bare mass of
the Klein-Gordon field $m_{0}$ and the bare coupling $J_{0}$ could be inferred
by repeating the same scheme for different frequencies of the harmonic source
$\omega_{J}$, as this should change the range of the interaction and lead to
different oscillation frequencies of the observable. With this information,
one can reconstruct $\mathsf{Z}_{0}[J]$ in Eq. (11).

Note that, in addition to dispensing with the need of preparing large multi-
partite entangled states, the previous evolution is of the spin-echo type Hahn
(1950), and will thus refocus the dephasing caused by external fluctuations
that affect the qubits on a slower timescale than each of the single
experimental runs. In contrast, spin echos or any other dynamical-decoupling
sequence Suter and Álvarez (2016), cannot be incorporated in the generating-
functional sensing scheme of Bermudez  _et al._ (2017a), as it would also
refocus some of the signals that are required to estimate the derivatives. We
conclude that the harmonic scheme is not only simpler than the impulsive one,
but also more robust to noise.

##  III Self-interacting scalar fields coupled to $\mathbb{Z}_{2}$ fields

In this section, we describe the previous sensing protocols in the presence of
self-interactions in the massive Klein-Gordon field, which leads to the so-
called $\lambda\phi^{4}$ QFT, and can be addressed by the use of functional-
integral methods. In Appendix B, we review this functional approach for the
full generating functional of the interacting scalar field $\mathsf{Z}[J]$ in
absence of the Ising spins. This yields a graphical representation of
$\mathsf{Z}[J]$ in terms of Feynman diagrams, where the logarithm of the full
generating functional can be expressed as a series in even powers of the
renormalised source functions weighted by the renormalised connected
propagators at the corresponding order. To extend these results to the full
problem with Ising-Schwinger sources, we describe in Sec. III.1 a spin-path
integral representation of the amplitude of propagation between two arbitrary
states of the Ising and scalar fields, which leads to an effective scalar-
sigma QFT. In Sec. III.2, we show that the $\lambda\phi^{4}$ bosons act as
mediators of Ising-type interactions between $2n$ spins, each of which is
controlled by the renormalised $2n$-point connected propagator of the self-
interacting scalar bosons. For harmonic sources in the specific regime of the
previous section, the leading term leads again to an effective quantum Ising
Hamiltonian with long-range couplings, the range of which now accounts for all
the possible scattering events that the mediating boson can undergo while
propagating between the corresponding pair of Ising spins. This shows that the
sensing scheme of Sec. II, when applied to the full interacting case, can gain
information about the renormalisation effects, and thus probe this QFT in a
novel manner.

###  III.1 Spin path integral and scalar-sigma field theory

As discussed in the previous section, the real Klein-Gordon field is the
simplest model where one can introduce key concepts and techniques to be used
later in more complicated QFTs. For instance, including quartic self-
interactions in the Klein-Gordon field (1) leads to the $\lambda\phi^{4}$
model

$\mathcal{H}=\mathcal{H}_{0}+\mathcal{V}_{\rm int},\hskip
6.45831pt\mathcal{V}_{\rm int}=\frac{\lambda_{0}}{4!}\hskip
0.43057pt\phi^{4}(x),$ (40)

where $\lambda_{0}$ is the bare coupling strength. This model has played a
leading role in the development of QFT. On the one hand, it is a cornerstone
in our understanding of the spontaneous breakdown Coleman and Weinberg (1973);
Jackiw (1974) and restoration Dolan and Jackiw (1974); Weinberg (1974) of
symmetries in QFTs. This model has also been a neat playground to understand
the conceptual implications of the renormalisation of QFTs Wilson and Kogut
(1974); Wilson and Fisher (1972), such as the role of fixed points controlling
the long-wavelength properties, which has found many applications in the
theory of phase transitions and critical phenomena Fisher (1998). Finally, we
note that the $\lambda\phi^{4}$ model has also been a cornerstone in the
constructive approach to QFTs, allowing to prove rigorous results for various
dimensions Glimm and Jaffe (1968, 1973); Feldman and Osterwalder (1976); Glimm
and Jaffe (1974); Lüscher and Weisz (1987); Aizenman (1981); Fröhlich (1982).

In Appendix B, we review the effects of the $\lambda\phi^{4}$ interactions
using the path-integral formalism for the generating functional (162).
Paralleling our approach for the free Klein-Gordon field, we can use this
discussion as a guide to understand how the self-interactions affect the
effective Ising models (38) when considering harmonic Ising-Schwinger sources.
As reviewed in the appendix, to get the path-integral formulation (142) of the
full generating functional $\mathsf{Z}[J]$ (141), one uses the resolution of
the identity in the field/momentum orthonormal basis of the scalar field. To
include the Ising-Schwinger sources and the transverse field (34), one may
complement this basis with that of spin coherent states Radcliffe (1971);
Fradkin (2013), which can be defined through the action of the spin ladder
operators on a fiducial state

$\ket{\bm{\Omega}(x)}={\rm
e}^{\tan\left(\\!\frac{\theta_{s}(x)}{2}\\!\right){\rm e}^{{\rm
i}\phi_{s}(x)}S^{+}\\!(x)}\ket{S,-S}_{x}.$ (41)

Here, $\ket{S,-S}_{x}=\ket{\downarrow}_{x}$, and
$S^{+}\\!(x)=\textstyle\frac{1}{2}(X(x)+{\rm i}Y(x))$ for our original qubits
$S=1/2$ (5), but can be readily generalised to any spin-$S$ representations of
the $\mathfrak{su}(2)$ algebra. In addition, $\bm{\Omega}(x)$ is a unit vector
field constrained to reside on the $S_{2}$ sphere, which can thus be
characterised by the polar $\theta_{s}(x)$ and azimuthal $\phi_{s}(x)$ angles
at each spacetime point.

An important difference with respect to the aforementioned field/momentum
basis is that the coherent states are not orthogonal, but satisfy instead

$\langle\bm{\Omega}(x_{1})|\bm{\Omega}(x_{2})\\!\rangle\\!=\\!{\rm e}^{{\rm
i}S\Phi(\bm{\Omega}(x_{1}),\bm{\Omega}(x_{2}),{\bf
e}_{z})}\\!\\!\left(\\!\\!\frac{1+\bm{\Omega}(x_{1})\\!\cdot\bm{\Omega}(x_{2})}{2}\\!\right)^{\\!\\!\\!\\!S}\\!.$
(42)

Here, $\Phi(\bm{\Omega}(x_{1}),\bm{\Omega}(x_{2}),{\bf e}_{z})$ stands for the
area of the spherical triangle with vertices determined by the tips of the
coherent-state unit vectors and the north pole of the $S_{2}$ sphere. These
scalar products appear after splitting in infinitesimal pieces the
corresponding time-evolution operator (36), where the fields now evolve in the
Heisenberg picture with respect to the unsourced $\lambda\phi^{4}$ QFT (see
App. B). Paralleling the path-integral construction for this scalar field
theory, we introduce the resolution of the identity of the field/momentum and
coherent-state basis at nearby fixed instants of time, and obtain a functional
integral for the $\lambda\phi^{4}$ QFT coupled to Ising fields via Ising-
Schwinger sources. Following Fradkin (2013), for each spatial point, we find
that the amplitude of the above overlap contributes to the kinetic energy of a
non-relativistic particle moving along the trajectory
$\bm{\Omega}(t,\textsf{{x}})$ on the unit sphere. However, when recovering the
continuum-time limit, the mass of this particle vanishes, and the trajectory
only depends on the accumulated phase (42) and the source couplings to
$J(x),H_{\textsf{t}}(x)$. The latter can be obtained through the coherent-
state expectation value of the Ising-Schwinger term (34), ${\rm
d}t\langle\bm{\Omega}(t,\textsf{{x}}_{i})|\hat{V}|\bm{\Omega}(t+{\rm
d}t,\textsf{{x}}_{i})\rangle\approx{\rm
d}t\big{(}J(t,\textsf{{x}}_{i})\phi_{H}(t,\textsf{{x}}_{i})2S{\Omega}_{z}(t,\textsf{{x}}_{i})+h_{\textsf{t}}(t,\textsf{{x}}_{i})2S{\Omega}_{x}(t,\textsf{{x}}_{i})\big{)}+\mathcal{O}({\rm
d}t^{2})$. In addition, the accumulated phase (42) for the whole time
evolution can be expressed as the integral of a Berry connection along the
trajectory $\bm{\Omega}(t,\textsf{{x}})$, which is equivalent to the Aharonov-
Bohm phase acquired by a test particle moving in the background gauge field
generated by a monopole of charge $S$ situated at the origin of $S_{2}$.
Remarkably, for the Euclidean path integral that arises from the finite-
temperature $T$ partition function of a spin model, these trajectories must be
closed $\bm{\Omega}(0,\textsf{{x}}_{i})=\bm{\Omega}(1/T,\textsf{{x}}_{i})$,
and the integral of the Berry connection can then be expressed as a
topological Berry phase or, equivalently, as a Wess-Zumino $\theta$ term. This
term plays an important role for both Heisenberg ferromagnets Fradkin (2013)
and anti-ferromagnets Haldane (1983); Fradkin and Stone (1988).

In our case, given the form of the coupling (34), it suffices to parametrise
the dynamics of the spins using coherent states along a great circle of
$S_{2}$, i.e. setting $\phi_{s}(x)=0$ in Eq. (42). Accordingly, the Berry
connection vanishes, and there is no accumulated phase. The amplitude of
propagation of the full field theory between an arbitrary pair of states in
this basis finds the following path-integral representation

$\langle\\{\\!\varphi(x),\bm{\Omega}(x)\\}|U_{\hat{\mathcal{V}}_{J}}|\\{\varphi(y),\bm{\Omega}(y)\\}\rangle\\!=\\!\\!\int_{\rm
b.c.}\\!\\!\\!\\!\\!\\!\\!\\!{\rm
D}[{\varphi,\bm{\Omega}}]\delta(\bm{\Omega}^{2}-1){\rm e}^{{\rm i}S}.$ (43)

Here, we have introduced the action

$S=\int{\rm d}^{D}x\big{(}\mathcal{L}_{0}-\mathcal{V}_{\rm
int}(\varphi)-\hat{\mathcal{V}}_{J}(\varphi,\bm{\Omega})\big{)},$ (44)

where we get the Klein-Gordon Lagrangian (143) with the $\lambda\phi^{4}$
interactions (40), and an additional spin-scalar coupling

$\hat{\mathcal{V}}_{J}=-J(x)2S\varphi(x)\Omega_{z}(x)-h_{\textsf{t}}(x)2S\Omega_{x}(x)$
(45)

Note that the spin-scalar path integral (50) has an integration measure with a
constraint that enforces the vector fields to lie on $S_{2}$, and one
integrates over all possible fields consistent with (b.c.), namely the
boundary conditions associated to the initial/final spin configurations of the
asymptotic states.

In the following, we implement the constraint on the vector fields explicitly
by making the following substitution on the symmetry-breaking and transverse
components

$\sigma(x)={\Omega}_{z}(x),\hskip
5.16663pt\pi(x)=\sqrt{1-{\Omega}_{z}^{2}(x)}=\\!\sum_{\ell=0}^{\infty}g_{\ell}\sigma^{2\ell}\\!(x),$
(46)

where $g_{\ell}=(-1)^{\ell}\binom{\textstyle\frac{1}{2}}{\ell}$, and
$\binom{\alpha}{\ell}=\alpha(\alpha-1)\cdots(\alpha-\ell+1)/\ell!$ is the
generalised binomial coefficient. Upon substitution, the action (44) can be
rewritten as a scalar-sigma model

$S=\int{\rm d}^{D}x\big{(}\mathcal{L}_{0}-\mathcal{V}_{\rm
int}(\varphi,\sigma)+J_{s}(x)\sigma(x)\varphi(x)\big{)},$ (47)

with a non-linear part that includes the self-interactions of the $\varphi$
field, but also those of the $\sigma$ field

$\mathcal{V}_{\rm
int}(\varphi,\sigma)=\frac{\lambda_{0}}{4!}\varphi^{4}(x)+\\!\sum_{\ell}h_{\ell}(x)\sigma^{2\ell}\\!(x).$
(48)

Here, we have defined the couplings

$J_{s}(x)=2SJ(x),\hskip 8.61108pth_{\ell}(x)=2Sg_{\ell}h_{\textsf{t}}(x).$
(49)

This QFT partially resembles a non-linear sigma model Gell-Mann and Lévy
(1960), but note that, in this case, the $\sigma$ fields are non-propagating.
Non-trivial effects come from the coupling to the interacting scalar bosons,
and hence the name scalar-sigma model.

This coupled QFT has a global $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry
where any of the fields gets inverted. In presence of the Ising-Schwinger
source term, this symmetry is reduced to $\mathbb{Z}_{2}$ acting
simultaneously on both fields
$(\varphi(x),\sigma(x))\to(-\varphi(x),-\sigma(x))$. Note that spontaneous
symmetry breaking (SSB) can take place in any dimension $d$ by the appearance
of a vacuum expectation value in any of the fields. This is the field-
theoretic interpretation of the SSB in the effective quantum Ising model (38),
which would lead to the ferromagnetic ordered mentioned in the previous
sections. In that case, SSB could only occur in the $\sigma$ sector, as the
scalar sector is described by free Klein-Gordon fields that do not support a
non-zero vacuum expectation value. Even if the generic $\lambda_{0}\phi^{4}$
field theory can support this scalar SSB channel, we shall not explore it in
this paper for the reasons outlined in Sec. IV.2, where we connect to trapped-
ion experiments.

Figure 3:  Scheme of the 4-spin interactions for the harmonic sensors: Four
distant Ising spins at positions $\textsf{{x}}_{1},\cdot,\textsf{{x}}_{4}$ are
coupled to the real scalar field $\phi(t,\textsf{{x}})$ locally by harmonic
Ising-Schwinger sources
$J(t,\textsf{{x}}_{i})\propto\sin(\omega_{J}t-\textsf{{k}}_{J}\cdot\textsf{{x}}_{i})$
. As a result, there are 4-spin Ising interactions mediated by the scalar
bosons, the strength of which will be related to a connected 4-point
propagator. In the upper inset, we show that in the presence of quartic
couplings, these interactions will include all possible scattering events that
involve two incoming and two outgoing virtual bosons, together with internal
loops due to the self-interaction vertices.

###  III.2 Renormalised long-range quantum Ising models

In this subsection, we start from the scalar-sigma field theory (47), and
exploit functional methods to find an effective description of the unitary
dynamics that governs the model of scalar $\lambda\phi^{4}$ fields coupled to
the Ising spins.

We note that the action of the scalar-sigma model (47) is formally equivalent
to the standard functional description of the self-interacting Klein-Gordon
field in Eq. (142) of Appendix B, provided that one substitutes the sources
$J(x)\to J_{s}(x)\sigma(x)$. If we are interested in a situation where, as
customarily, the scalar field evolves in time from the vacuum into the vacuum,
we can then follow a similar approach as described in Appendix B to express
the amplitude of propagation
$\mathcal{T}_{\bm{\Omega}(y)\to\bm{\Omega}(x)}=\langle
0,\\{\\!\bm{\Omega}(x)\\}|U_{\hat{\mathcal{V}}_{J}}|0,\\{\bm{\Omega}(y)\\}\rangle\\!$
as follows

$\mathcal{T}_{\bm{\Omega}(y)\to\bm{\Omega}(x)}=\\!\\!\int_{\rm
b.c.}\\!\\!\\!\\!\\!\\!\\!\\!{\rm D}{\sigma}\frac{{\rm e}^{-{\rm
i}\\!\\!\bigintssss\\!\\!{\rm d}^{D}x\mathcal{V}_{\rm int}\left(\\!-{\rm
i}\delta_{J_{s}(x)\sigma(x)},\sigma(x)\\!\right)}{\mathsf{Z}}_{0}[J_{s}\sigma]}{{\rm
e}^{-{\rm i}\\!\\!\bigintssss\\!\\!{\rm d}^{D}x\mathcal{V}_{\rm
int}\left(\\!-{\rm
i}\delta_{J_{s}(x)\sigma(x)},\sigma(x)\\!\right)}{\mathsf{Z}}_{0}[J_{s}\sigma]\big{|}_{0}\\!\\!\\!\\!},$
(50)

where we use a short-hand notation for the functional derivatives
$\delta_{J_{s}(x)\sigma(x)}=\delta/\delta J_{s}(x)\sigma(x)$, and (b.c.) now
only refers to the initial and final configurations of the $\sigma$ field,
which are determined by the asymptotic spin states of the transition
amplitude. Note that, in this expression, the free generating functional
${\mathsf{Z}}_{0}[J]\to{\mathsf{Z}}_{0}[J_{s}\sigma]$ has the same form as Eq.
(11), but now describes the mediated couplings between $\sigma$ fields. In
addition, the self-interaction potential must include the $\sigma$-field non-
linearities (48).

This expression can be treated using perturbation theory, and admits a
description in terms of Feynman diagrams. Note that, in principle, the new
interaction term (48) not only includes the scalar vertex with four legs, but
also all possible $\sigma$ vertices involving $2n$ legs. These additional
vertices could in principle combine into a wider landscape of Feynman diagrams
as that of the pure $\lambda\phi^{4}$ theory discussed in Appendix B,
including additional insertions of the $\sigma$ vertices. However, the
constraints in Eq. (37), which allowed us to neglect the terms of the Magnus
expansion beyond second order (133) stemming from higher-order nested
commutators, are equivalent to considering only Feynman diagrams with the
$\sigma$ vertices at tree level, i.e. neglecting the combination of the scalar
and sigma scattering processes in the diagrams. Accordingly, the diagrammatic
description of this transition amplitude (50) is

$\int_{\rm b.c.}\\!\\!\\!\\!\\!\\!{\rm
D}{\sigma}\bigg{(}1\begin{picture}(18.0,0.0)\put(1.25,0.0){$+\frac{{\rm
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This equation should be read as follows: the Ising-Schwinger source terms,
proportional to the $\sigma$ fields, are depicted by arrows
${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\bm{\Uparrow}}}=J_{s}(x)\sigma(x)$
to represent the underlying spins at different spacetime locations $x$. The
blobs
${\color[rgb]{0.644679,0.000000,0.065473}\definecolor[named]{pgfstrokecolor}{rgb}{0.644679,0.000000,0.065473}\bm{\bullet}}=\lambda_{0}$
stand for interaction vertices with the bare quartic coupling. Solid lines
that join an arrow and a blob should be translated into
${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\bm{\times}}}\bm{\\!\\!\\!\\!\\!-\\!\\!\\!\\!-\\!\\!\\!\\!{\color[rgb]{0.644679,0.000000,0.065473}\definecolor[named]{pgfstrokecolor}{rgb}{0.644679,0.000000,0.065473}{\bullet}}}=\Delta_{m_{0}}(x-z)$,
and thus involve the free Feynman propagator of the scalar field (10) from the
point $x$ of the $\sigma$ field to the self-interaction vertex at $z$.
Likewise, solid lines connected to the same blob stand for interaction loops
that should be translated for
$\bigcirc{\color[rgb]{0.644679,0.000000,0.065473}\definecolor[named]{pgfstrokecolor}{rgb}{0.644679,0.000000,0.065473}\bullet}\hskip
8.61108pt=\Delta_{m_{0}}(0)$, while those connecting two distant blobs must be
substituted by the scalar-field propagator between the corresponding spacetime
points
$\bm{{\color[rgb]{0.644679,0.000000,0.065473}\definecolor[named]{pgfstrokecolor}{rgb}{0.644679,0.000000,0.065473}\bullet}\\!\\!\\!-\\!\\!\\!\\!-\\!\\!\\!\\!{\color[rgb]{0.644679,0.000000,0.065473}\definecolor[named]{pgfstrokecolor}{rgb}{0.644679,0.000000,0.065473}\bullet}}=\Delta_{m_{0}}(z_{1}-z_{2})$.
For each of the above diagrams, we should integrate over all possible
spacetime locations of the sources $\int\\!{\rm d}^{D}x_{i}$, and those of the
intermediate interaction vertices $\int\\!{\rm d}^{D}z_{i}$.

Note that the arrows in Eq. (51) are drawn in different directions to
emphasise that, contrary to the standard Schwinger sources, the dynamics of
which is fixed externally, the new Ising-Schwinger sources involve $\sigma$
fields with their own dynamics. In particular, from the perspective of Sec.
II.2 on the free Klein-Gordon field coupled to Ising spins, we know that there
are boson-mediated processes where the spins, when initialised in a particular
basis, can be flipped dynamically. Additionally, if the transverse field is
non-zero, there will be quantum fluctuations that will also flip the spins in
a different basis. Our choice of alternating directions of the arrows
${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\bm{\Uparrow}}},{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{{\Rightarrow}}},{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{\bm{\Downarrow}}},{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}{{\Leftarrow}}},$
thus tries to depict this inherent spin dynamics. Additionally, in Eq. (51),
the free generating functional should be substituted for
$\mathsf{Z}_{0}[J_{s}\sigma]\to\tilde{Z}_{0}[J_{s}\sigma,h_{\ell}]$, which now
includes the Ising-Schwinger sources and the tree-level $\sigma$ vertices

$\tilde{Z}_{0}\\!={\rm e}^{-\frac{1}{2}\\!\\!\bigintssss\\!\\!\\!{\rm
d}^{D}x_{1}\\!\\!\bigintssss\\!\\!\\!{\rm
d}^{D}x_{2}{J}_{s}(x_{1}\\!)\sigma(x_{1}\\!)\Delta_{{m}_{0}}\\!(x_{1}-x_{2})\sigma(x_{2}\\!){J}_{s}(x_{2}\\!)+{\rm
i}\\!\\!\bigintssss\\!\\!\\!\\!{\rm
d}^{D}x\sum_{\ell}\\!h_{\ell}\sigma^{2\ell}\\!(x)}\\!\\!.$ (52)

The graphical representation in terms of Feynman diagrams and $\tilde{Z}_{0}$
in Eq. (51) then has a clear interpretation. Distant Ising spins can virtually
excite the bosonic field via the source coupling, and effectively interact by
a long-range coupling mediated by the exchange of a virtual scalar boson (52).
Additionally, these spins are also subjected to quantum fluctuations captured
by the non-linearities of the $\sigma$ vertices (52). The Feynman diagrams
(51) show that this virtual exchange process is not entirely captured by the
free terms (52), as the scalar field also has self-interactions, and the
mediating boson may get scattered through various processes as it propagates
between the spins, as depicted in the upper inset of Fig. 2.

In addition to modifying the interactions between pairs of spins (52), self-
interactions introduce further possibilities where $2n$-spins also get
effectively coupled. In the fifth diagram of Eq. (51), we show that a 4-spin
interaction can occur by the action of two Ising-Schwinger sources that
virtually excite a pair of bosons, which then scatter into another pair of
bosons that carry the interactions to a different pair of spins. Note that
these two-boson mediated processes also allow for further scattering of the
carriers due to the self-interaction of the scalar field, as described by the
following diagrams of Eq. (51) and depicted in the upper inset of Fig. 3.

We can turn this qualitative explanation of the mediated interactions into a
quantitative analysis by handling the perturbative series in analogy to our
exposition of Appendix B. From now onwards, for our spin $S=1/2$ case, we will
make no distinction between the bare and spin sources since $J_{s}(x)=J(x)$.
The expressions below can be, in any case, readily generalised to arbitrary
$S$. The amplitude of propagation between two different $\sigma$-field
configurations can be written in a condensed form that takes into account all
the aforementioned scattering processes of the bosonic carriers. In summary,
these scattering events lead to additive and multiplicative renormalisations
of the bare mass $m_{0}\to{m}_{r}$, and the bare coupling strength
$\lambda_{0}\to\lambda_{r}$. As discussed in Appendix B, using the power
series of the self-energy
$\mathsf{\Sigma}(k)=\mathsf{\Sigma}(0)+k^{2}\partial_{k^{2}}\mathsf{\Sigma}(k)|_{k^{2}=0}+\cdots$
to second order in the coupling strength, one can obtain the so-called tadpole
and sunrise contributions to
$\Sigma_{m_{0},\lambda_{0}}(0)=\mathsf{\Sigma}^{(1,{\rm
td})}_{m_{0},\lambda_{0}}+\mathsf{\Sigma}^{(2,{\rm
td})}_{m_{0},\lambda_{0}}+\mathsf{\Sigma}^{(2,{\rm sr})}_{m_{0},\lambda_{0}}$,
namely

$\begin{split}\mathsf{\Sigma}^{(1,{\rm
td})}_{m_{0},\lambda_{0}}&=\frac{\lambda_{0}}{2}\\!\\!\bigintssss_{k_{1}}\\!\\!\tilde{\Delta}_{m_{0}}\\!(k_{1}),\\\
\mathsf{\Sigma}^{(2,{\rm td})}_{m_{0},\lambda_{0}}&=-{\rm
i}\frac{\lambda_{0}^{2}}{4}\\!\\!\bigintssss_{k_{1}}\\!\\!\bigintssss_{k_{2}}\\!\\!\tilde{\Delta}_{m_{0}}^{2}\\!(k_{1})\tilde{\Delta}_{m_{0}}\\!(k_{2}),\\\
\mathsf{\Sigma}^{(2,{\rm sr})}_{m_{0},\lambda_{0}}&=-{\rm
i}\frac{\lambda_{0}^{2}}{6}\\!\\!\bigintssss_{k_{1}}\\!\\!\bigintssss_{k_{2}}\\!\\!\tilde{\Delta}_{m_{0}}\\!(k_{1})\tilde{\Delta}_{m_{0}}\\!(k_{2})\tilde{\Delta}_{m_{0}}\\!(k_{1}+k_{2}),\end{split}$
(53)

as well as the sunrise contribution to the wavefunction renormalisation
$z_{m_{0},\lambda_{0}}^{\\!-1}=1-\partial_{k^{2}}\mathsf{\Sigma}(k)|_{k^{2}=0}$,
namely

$\left.\frac{\partial\mathsf{\Sigma}^{(2,{\rm sr)}}}{\partial
k^{2}}\right|_{0}=\frac{\lambda_{0}^{2}}{6}\\!\\!\bigintssss_{k_{1}}\\!\\!\bigintssss_{k_{2}}\\!\\!\tilde{\Delta}_{m_{0}}\\!(k_{1})\tilde{\Delta}_{m_{0}}\\!(k_{2})\tilde{\Delta}^{2}_{m_{0}}\\!(k_{1}+k_{2}).$
(54)

Accordingly, to leading order, the renormalised mass reads

$m_{r}^{2}=\left(m_{0}^{2}\hskip
0.86108pt+\mathsf{\Sigma}_{m_{0},\lambda_{0}}(0)\right)z_{m_{0},\lambda_{0}},$
(55)

whereas the source functions become

${J}_{r}(x)=J(x)\sqrt{z_{m_{0},\lambda_{0}}}.$ (56)

Finally, the interaction strength also gets renormalised due to the last
diagram of Eq. (51), leading to

${\lambda}_{r}=\left(\lambda_{0}-{\rm
i}\frac{3{\lambda}_{0}^{2}}{2}\\!\\!\bigintssss_{k_{1}}\\!\\!\tilde{\Delta}^{2}_{m_{0}}\\!(k_{1})\right)z_{m_{0},\lambda_{0}}^{2}.$
(57)

With these renormalisations, the amplitude of propagation in Eq. (51) can thus
be compactly rewritten as

$\mathcal{T}_{\bm{\Omega}(y)\to\bm{\Omega}(x)}\\!\\!=\\!\\!\\!\int_{\rm
b.c.}\\!\\!\\!\\!\\!\\!\\!\\!{\rm D}{\sigma}{\rm
e}^{-\frac{1}{2}\\!\\!\bigintssss\\!\\!{\rm
d}^{2D}x{J}_{r}(x_{1})\sigma(x_{1})\Delta_{{m}_{r}}\\!(x_{1}-x_{2})\sigma(x_{2}){J}_{r}(x_{2})-\frac{{\rm
i}}{4!}\\!\\!\bigintssss\\!\\!{\rm
d}^{4D}x{J}_{r}(x_{1}){J}_{r}(x_{2})\sigma(x_{1})\sigma(x_{2}){G}^{(4,{\rm
c})}_{{m}_{r},{\lambda}_{r}}\\!\\!(x_{1}\cdots
x_{4})\sigma(x_{3})\sigma(x_{4}){J}_{r}(x_{3}){J}_{r}(x_{4})+{\rm
i}\\!\\!\bigintssss\\!\\!\\!\\!{\rm
d}^{D}x\sum_{\ell}\\!h_{\ell}\sigma^{2\ell}\\!(x)}\\!\\!,$ (58)

where we readily identify the quadratic and quartic long-range interactions of
the $\sigma$ fields mediated by the scalar bosons. Note that, as discussed in
Appendix B, one could consider higher-order terms in the external sources,
which would introduce $2n$-spin interactions with $n\geq 3$.

We now argue that, among all these possible mediated interactions for harmonic
Ising-Schwinger sources (20), all but the 2-spin couplings will be negligible.
The source amplitude gets renormalised ${\mathsf{J}}_{0}\to{\mathsf{J}}_{0,r}$
via Eq. (56) yielding

${J}_{r}(x)={\mathsf{J}}_{0,r}\sin(k_{J}x)={\mathsf{J}}_{0,r}\sin\big{(}\omega_{J}t-\textsf{{k}}_{J}\cdot\textsf{{x}}\big{)},$
(59)

To estimate the strength of the $2n$-spin interactions, we recall our
discussion of Sec. II.2, and the importance of tuning the sources below
resonance
$\omega_{J}\lesssim{m}_{r}\lesssim{\omega}_{\textsf{{k},r}}=(\textsf{{k}}^{2}+{m}_{r}^{2})^{1/2}$
to avoid dissipative processes where propagating bosons instead of virtual
ones get excited. As discussed there, the source couplings should also be
constrained by Eq. (60) to effectively decouple the dynamics of the Ising and
scalar fields. This allowed us to get an effective unitary evolution where the
only non-trivial part was encoded in the Ising Hamiltonian (23). The
renormalised version of such a constraint (60) reads

${\mathsf{J}}_{0,r}\ll({\omega}_{\textsf{{k}},r}-\omega_{J}){\rm
d}^{d}\textsf{{k}}<({\omega}_{\textsf{{k}},r}+\omega_{J}){\rm
d}^{d}\textsf{{k}},$ (60)

which guarantees that the back-action of the Ising spins onto the
$\lambda\phi^{4}$ field is negligible and that, if initialised in the vacuum
(or any other state), the scalar field will remain in such a vacuum during the
whole evolution. The discussion of Sec. II.2 then proceeds by performing
explicitly the time-integrals involving the harmonic sources in the forward
and backward directions of the Feynman propagator. As a result of these
integrations and the coupling constraint (60), one can neglect rapidly
oscillating terms, and describe the evolution unitary by a time-independent
effective Hamiltonian (23). We see in Eq. (24) that the long-range spin-spin
coupling scales with
$\mathsf{J}_{0}\cdot\mathsf{J}_{0}/(\omega_{\textsf{{k}}}-\omega_{J})(\omega_{\textsf{{k}}}+\omega_{J})$,
which is already small according to the constraint (60). hence, the dynamical
effects of these interactions can only be observed by letting the system
evolve for sufficiently long times. For the current renormalised equations
(58), we get a similar behaviour, and one can check that the additional
4-$\sigma$ term in Eq. (58) also contributes to the time-independent
Hamiltonian, but this time with 4-spin interactions that scale with terms like
${\mathsf{J}}_{0,r}\cdot{\mathsf{J}}_{0,r}\cdot{\mathsf{J}}_{0,r}\cdot{\mathsf{J}}_{0,r}/({\omega}_{\textsf{{k}}_{1},r}\pm\omega_{J})({\omega}_{\textsf{{k}}_{2},r}\pm\omega_{J})({\omega}_{\textsf{{k}}_{3},r}\pm\omega_{J})({\omega}_{\textsf{{k}}_{4},r}\pm\omega_{J})$.
These terms should be integrated over in momentum space with a total momentum
conservation, and will lead to spin interactions between 4 distant spins with
some specific long-range coupling. In any case, Eq. (60) shows that these
4-spin interactions will be negligible in comparison to the 2-spin terms, and
more so if one considers higher-order $2n$-spin interactions.

Once we have arrived at this point, the last step is to convert the leading
contribution to the amplitude of propagation from the path-integral
formulation (58) onto a canonical time-evolution operator, reversing the steps
of the path-integral construction. Altogether, when considering that the Ising
spins only occupy a certain spatial arrangement (35), we arrive at the final
result: the full time evolution can be represented by the two concatenated
unitaries in Eq. (19), where the non-trivial dynamics of the Ising spins is
controlled by a renormalised long-range quantum Ising Hamiltonian

${H}_{{\rm eff},r}=\frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n}{J}^{r}_{ij}\hskip
1.07639ptZ(\textsf{{x}}_{i})Z(\textsf{{x}}_{j})-h_{\textsf{t}}\sum_{i=1}^{n}X(\textsf{{x}}_{i}).$
(61)

Here, the spin-spin couplings have a range that is controlled by the
dimensionally-reduced Euclidean propagator

${J}^{r}_{ij}=-2{J}_{0,r}G_{{m}_{{\rm eff},r}}^{\rm
E}\\!(\textsf{{x}}_{i}-\textsf{{x}}_{j})\cos\big{(}\textsf{{k}}_{J}\cdot(\textsf{{x}}_{i}-\textsf{{x}}_{j})\big{)}{J}_{0,r},$
(62)

with an effective renormalised mass given by

${m}_{{\rm eff},r}^{2}={m}_{r}^{2}-\omega_{J}^{2}.$ (63)

Therefore, tuning the frequency $\omega_{J}$ of the harmonic source (59)
closer to, or further from, the renormalised mass $m_{r}$ of the scalar
bosons, one will be able to control the range of the Ising interactions, and
extract the information about the renormalisation of the scalar field using
the sensing method of Sec. II.2. Let us recall again that this mass gets
additive and multiplicative renormalisations (55) from all the intermediate
scattering events of the bosons mediating the Ising interaction. Let us
finally note that the spin-spin coupling strength scales with the renormalised
source amplitudes ${J}_{0,r}$, and that these terms also get a multiplicative
renormalisation described in Eq. (56). Typically, for the decoupled
$\lambda\phi^{4}$ QFT, such source renormalisations do not have any physical
consequence, as one is ultimately interested in the renormalised propagators.
The generating functional in these cases is a mathematical tool, and it
suffices to take functional derivatives with respect to these new sources, or
directly assume that the sources couple to the renormalised fields. On the
contrary, for the current spin-scalar model, the renormalisation of the
sources has an impact on the strength of the spin-spin couplings, which is
relevant as these control the time-scale of the spin dynamics.

Considering Eq. (61) and our previous discussions, the new sensing protocol
that monitors the real-time oscillations of a pair of Ising spins coupled to
the $\lambda\phi^{4}$ QFT, will allow us to recover the renormalised
parameters of the field theory. Once we have them, one can reconstruct the
generating functional in Eq. (162), in this case to second-order in the
sources

$\mathsf{Z}[{J}_{r}]={\rm e}^{-\frac{1}{2}\\!\\!\bigintssss\\!\\!{\rm
d}^{D}x_{1}\\!\\!\bigintssss\\!\\!{\rm
d}^{D}x_{2}{J}_{r}(x_{1})\Delta_{{m}_{r}}\\!(x_{1}-x_{2}){J}_{r}(x_{2})}.$
(64)

We note that, if the timescales of the ever-present experimental noise on the
qubits is sufficiently low, it may also be possible to devise schemes that
sense the effect of the 4-spin (generally 2$n$-spin) interactions, gaining
access to higher-order propagators and the renormalised quartic coupling.

We close this section by noting that we have so far not mentioned any ultra-
violet (UV) cutoff of the QFT, although it is implicit in the discretised
drawings of Figs. 1-3. Such a cutoff is not required for the results presented
in Sec. II for the effective Ising models mediated by free Klein-Gordon
bosons. In this case, the only UV diverging quantity is the zero-point
energies of the fields. However, as soon as $\lambda\phi^{4}$ interactions are
included, a different kind of UV divergences appear, as we see that the
renormalisations involve certain Feynman diagrams, such as the tadpole term in
Eq. (53), which includes the propagator at infinitesimally short distances
${\Delta}_{m_{0}}(0)$, which displays such UV divergences. The QFT thus needs
to be regularised by the introduction of a cutoff

$|k^{\mu}|\leq\Lambda_{\rm c}.$ (65)

As a consequence, the renormalised parameters (55)-(57) will depend on the
cutoff scale. The central result of the renormalisation group is that the bare
coupling constants of the QFT, $m_{0},\lambda_{0},J_{0}$ in this case, will
flow with the cutoff scale and give rise to physical renormalised quantities
${m_{r}},{\lambda}_{r},{J}_{0,r}$ that no longer depend on the arbitrary
cutoff. As we will discuss below, in certain situations, this UV cutoff is not
arbitrary but fixed by the physical system at hand (e.g. physical lattice). In
this case, the renormalisation group allows us to extract the universal long-
wavelength properties, and find a predictive field theory valid at low
energies Shankar (1994). We will discuss this in more detail in the following
section.

##  IV Renormalisation of sound in trapped-ion crystals

In this section, we present a detailed discussion on how the previous results
can be applied to a long-wavelength description of crystals of atomic ions
confined in radio-frequency traps. This description can be formalised within
the framework of the theory of elasticity for the quantised sound waves of
crystals, which is reviewed in Appendix C, where we also comment on the main
obstacles for a solid-state implementation of the Ising-Schwinger sensing
scheme introduced in this work. We show in Sec. IV.1 that the transverse sound
waves of harmonic trapped-ion crystals subjected to state-dependent forces
offer a neat realisation of the massive Klein-Gordon field coupled to Ising
spins. This allows us to apply the results of Sec. II.2, connecting the
predicted long-range Ising models with previous works on phonon-mediated spin-
spin interactions between trapped ions. This clarifies the distance dependence
of the latter, and shows that our theory is a valid long-wavelength
description of trapped-ion crystals under state-dependent forces. In Sec.
IV.2, we abandon the harmonic limit by approaching a structural phase
transition of the ion crystal, where non-linear effects lead to scattering of
the quanta related to the sound waves, namely the phonons, which requires us
to use the renormalisation predictions of Sec. III.2. In particular, in Sec.
IV.3, we use an explicit integration of the renormalisation-group flow
equations in the limit of small quantum fluctuations to predict the
renormalisation of the range of the Ising spin-spin interactions. We conclude
in Sec. IV.4 by showing that additional parametric modulations of the trap
frequencies can give a knob to control the amount of quantum fluctuations,
exploring regimes that go beyond the previous perturbative predictions, where
the results of the proposed QS would be very interesting.

###  IV.1  Harmonic trapped-ion crystals: rigidity, massive Klein-Gordon
fields and effective Ising models.

In this section, we discuss the realisation of the previous ideas using
crystals of trapped atomic ions Bermudez  _et al._ (2017a). Clouds of singly-
ionised atoms can be confined in finite regions of space for hours/days using
storage rings or Penning and Paul traps Ghosh (1995). These traps are held
inside ultra-high vacuum chambers, and additionally shielded from external
fluctuating fields, such that one can explore quantum many-body properties of
the ion cloud in a pristine and controlled environment. In this way, one can
observe the plasma-like behaviour of large thermal clouds Dubin and O’Neil
(1999), or focus on ions in medium- to small-sized clouds. In the latter
context, crucial progress in our understanding of laser cooling during the
past decades Wineland and Itano (1979); Itano and Wineland (1982); Wineland
_et al._ (1987a); Cirac  _et al._ (1992); Morigi  _et al._ (2000) has allowed
the development of various techniques Eschner  _et al._ (2003) to reach ever-
lower temperatures. These ultra-cold atomic ions crystalize as a result of the
competition between trapping and Coulomb forces Diedrich  _et al._ (1987);
Wineland  _et al._ (1987b); Raizen  _et al._ (1992); Gilbert  _et al._ (1988);
Birkl  _et al._ (1992), leading to the so-called Coulomb clusters/crystals
Dubin and O’Neil (1999).

In the following, we shall focus on linear Paul traps, a type of radio-
frequency traps that confine the ions in stable linear configurations along
the trap symmetry axis Raizen  _et al._ (1992). These kind of traps have been
exploited, among other things, for the manipulation of individual quantum
systems Leibfried  _et al._ (2003a), the development of frequency standards
based on optical clocks Ludlow  _et al._ (2015), and the demonstration of
various quantum computing algorithms Wineland  _et al._ (1998); Schindler  _et
al._ (2013); Bruzewicz  _et al._ (2019). In this configuration, the
microscopic description of a single trapped ion Berkeland  _et al._ (1998), or
its extension to a collection of them Bermudez  _et al._ (2017b), shows how
slow secular vibrations with a typical timescale set by the inverse of the
trap frequencies $\\{\omega_{\alpha}\\}_{\alpha=\textsf{x,y,z}}$ get decoupled
from a fast driven motion synchronous with the external radio-frequency field
$\Omega_{\rm rf}\gg\omega_{\alpha}$.

For the ion crystal, the long-wavelength description of the collective secular
dynamics James (1998) resembles the elasto-dynamical theory of phonons in a 1D
solid-state crystal, as reviewed in Appendix C, albeit with important
differences that we now stress. On the one hand, while strict 1D crystals are
thermodynamically unstable, the ion chains are dynamically-driven
inhomogeneous clusters of ions, such that there is no conflict with the
spontaneous breakdown of translational symmetry in static situations Hohenberg
(1967); Mermin and Wagner (1966); Coleman (1973). However, due to the
separation of timescales mentioned above, the much slower secular motion can
be discussed using static equilibrium tools. Secondly, whereas 1D crystals can
only support compressional longitudinal waves (163), ions in a Coulomb chain
can vibrate in both longitudinal and transverse directions with respect to the
trap axis. One of the key advantages of trapped-ion crystals with respect to
the solid state is that these different vibrations can be selectively cooled
or excited by controlling both the frequency and the propagation direction of
additional laser beams. This contrasts the typical situation in solids, where
a macroscopic thermal strain excites all modes simultaneously. The last key
difference with respect to the elasto-dynamical description of sound waves in
solids of Appendix C is that, contrary to the screened inter-atomic potentials
of solids, which are in accordance with the form of the coarse-grained stress
forces, trapped-ion crystals are subjected to long-range Coulomb potentials.
These long-range forces will lead to important differences in the context of
the present work. Additionally, trapped ions allow one to tune the analogue of
the bulk and shear moduli $K_{e},\mu_{r}$, which characterise the stiffness
and rigidity of the crystal to external strain and, in turn, determine the
effective speeds of sound as discussed in the appendix in Eqs. (163) and
(166). More importantly, trapped-ion crystals allow to control microscopically
the importance of non-elastic corrections, providing a neat route to explore
renormalised Ising interactions mediated by $\lambda\phi^{4}$ fields, as
discussed in the following subsection.

Figure 4:  Trapped-ion chain and effective Ising couplings: (a) Equilibrium
positions for a chain of $N=50$ 171Yb+ ions confined in a Paul trap with
secular frequencies $\omega_{\mathsf{x}}/2\pi=0.1$MHz, and
$\omega_{\mathsf{z}}/2\pi=3.75$MHz. The microscopic length-scale corresponds
to $\ell=12.7\mu$m in this case, while the minimum distance at the bulk of the
ion chain is $a=4.4\mu$m. (b) The blue alternating arrows depict the lowest
vibrational mode in the transverse branch, the zigzag mode.

Let us start by describing in detail the first differences, which can already
be understood by inspecting the trapped-ion chain in the secular harmonic
approximation Morigi and Fishman (2004a, b), and the realisation of phonon-
mediated Ising-type interactions thereof Porras and Cirac (2004a); Deng  _et
al._ (2005), making contact with Sec. II. In the secular approximation, there
is a competition between the harmonic trapping potential and the Coulomb
repulsion such that, for
$\omega_{\textsf{x}}\ll\omega_{\textsf{y}},\omega_{\textsf{z}}$, the $N$
trapped ions arrange in equilibrium positions
$\textsf{{x}}_{i}=\textsf{x}_{i}{\bf e}_{\textsf{x}}$ along the trap symmetry
axis (see Fig. 4 (a)). The position of each ion can thus be expressed as
$\bm{r}_{i}(t)={\textsf{x}}_{i}{\bf e}_{x}+\sum_{\alpha}u_{i,\alpha}(t){\bf
e}_{\alpha}$, where $u_{i,\alpha}(t)$ represents the small local vibrations
around equilibrium of the $i$-th ion along the $\alpha$-th axis, which can be
considered as a discretised version of the displacement field
$u_{\alpha}(t,\textsf{{x}})$ introduced in the coarse-grained elasticity
theory (163). In a harmonic approximation, one expands the overall potential
to second order, and finds that the dynamics of the ion chain can be described
by a Hamiltonian of local harmonic oscillators with long-range couplings

$H_{h}\\!=\\!\sum_{i,\alpha}\\!\\!\left(\\!\frac{1}{2}m_{a}(\partial_{t}u_{i,\alpha})^{2}+\frac{1}{2}\kappa^{\alpha}u^{2}_{i,\alpha}+\frac{1}{4}\\!\sum_{j\neq
i}\kappa_{i,j}^{\alpha}\big{(}u_{i,\alpha}-u_{j,\alpha}\big{)}^{\\!2}\\!\right)\\!.$
(66)

Here, $m_{a}$ is the mass of the atomic ions, and

$\kappa_{i,j}^{\mathsf{z}}=-\frac{e^{2}}{4\pi\epsilon_{0}}\frac{1}{|\textsf{x}_{i}-\textsf{x}_{j}|^{3}}=\kappa_{i,j}^{\mathsf{y}}=-\textstyle\frac{1}{2}\kappa_{i,j}^{\mathsf{x}},$
(67)

can be understood as interatomic spring constants with a dipolar decay due to
the net charge of the ions. In addition,

$\kappa^{\alpha}=m_{a}\omega_{\alpha}^{2}$ (68)

is an effective spring constant leading to an elastic local force that aims at
restoring the equilibrium, i.e. $\bm{u}_{i}=\bm{0}$.

Let us note that the elasticity theory of Appendix C relies on the short-range
character of the interatomic forces. In solid state, this is typically
modelled by restricting the interatomic springs to nearest neighbours along
the chain $\kappa_{i,j}^{\textsf{x}}\to\kappa_{i,i+1}^{\textsf{x}}$, and
setting $\kappa^{\textsf{x}}=0$ as there is no additional trapping potential
Goodstein (2017); Ashcroft (2016). Note that, in order to describe a stable
crystalline configuration, the nearest-neighbour couplings must be positive in
this case $\kappa_{i,i+1}^{\textsf{x}}>0$. For the Coulomb chain (66), the
situation is richer, as the non-zero local spring constants
$\kappa^{\alpha}>0$ allow for stable configurations with both attractive and
repulsive elastic couplings $\kappa^{\alpha}_{i,j}\gtrless 0$. The
attractive/repulsive character can be understood by considering that the ion
excursions from equilibrium define distant electric dipoles
$\bm{P}_{i}=e\bm{u}_{i},\bm{P}_{\\!j}=e\bm{u}_{j}$, and that these interact by
repelling (attracting) each other when aligned orthogonal (parallel) to the
axis of separation, here the x-axis, yielding $\kappa_{i,j}^{\textsf{y
}},\kappa_{i,j}^{\textsf{z}}>0$ ($\kappa_{i,j}^{\textsf{x}}<0$). This has a
direct consequence for the collective vibrational branches, as the lowest
longitudinal vibration will correspond to all dipoles lying in parallel, which
requires the ions to vibrate in phase and leads to the so-called center-of-
mass mode James (1998). On the other hand, the lowest transverse vibrations
correspond to alternating dipoles along the y,z axis, such that the ions
vibrate out of phase and lead to the so-called zigzag mode (see Fig. 4 (b)).
For reasons that will become clear below, we will restrict to the transverse
vibrations along the z axis in a situation where the corresponding trapping
potential has been partially relaxed
$\omega_{\textsf{x}}<\omega_{\textsf{z}}\ll\omega_{\textsf{y}}$.

We can now address the connection of this microscopic theory (66) to a Klein-
Gordon field theory (1) at long wavelengths or, equivalently, low energies. As
noted above, the lowest-energy transverse vibration will correspond to out-of-
phase displacement of the ions, which can be associated to a quasi-momentum
$\textsf{k}_{s}=\pi/a$. Here, $a$ is a characteristic microscopic length-scale
of the Coulomb chain James (1998) that will be proportional to the combination
of microscopic parameters

$\ell=(e^{2}/4\pi\epsilon_{0}m_{a}\omega_{\textsf{x}}^{2})^{1/3}.$ (69)

Although we could carry on in the most generic situation Bermudez  _et al._
(2017a), in order to make a more direct contact with the scalar QFT of the
previous sections, we shall assume that the ions are homogeneously distributed
in the chain $\textsf{x}_{i}=ia$ with inter-ion distance, i.e. lattice
spacing, $a$ set by the minimum distance between the bulk ions (see Fig. 4
(a)). This sets directly the aforementioned microscopic length scale, which
turns out to be $a\lesssim\ell$. The homogeneous approximation is valid for
the bulk of the ion crystal, neglecting the differences that arise due to
inhomogeneities as one approaches the chain edges Dubin and O’Neil (1999),
which is consistent with the spirit of our long-wavelength theory. We will
thus apply periodic boundary conditions, and discuss numerically the
inhomogeneous chain at the end.

As is customary in other long-wavelength descriptions of condensed-matter
problems Affleck (1990); Shankar , one proceeds by separating fast and slow
variations of the degrees of freedom of interest Bermudez  _et al._ (2017a);
Shimshoni  _et al._ (2011); Bermudez and Plenio (2012), here the transverse
displacement

$u_{j,\textsf{z}}(t)=a{\rm e}^{-{\rm
i}\textsf{k}_{s}\textsf{x}_{j}}\tilde{\phi}\\!(t,\mathsf{x}),\hskip
8.61108pt\partial_{t}u_{j,\textsf{z}}(t)=\frac{1}{m_{a}}{\rm e}^{-{\rm
i}\textsf{k}_{s}\textsf{x}_{j}}\tilde{\pi}(t,\mathsf{x}),$ (70)

where $\tilde{\phi}(t,{\textsf{x}})$ is the slowly-varying component that will
play the role of the scalar field in the continuum
${\textsf{x}_{j}}\to{\textsf{x}}$, and $\tilde{\pi}(t,\mathsf{x})$ is its
conjugate momentum. These operators are defined such that one recovers the
correct equal-time commutation relations
$[\tilde{\phi}\\!(t,\mathsf{x}_{i}),\tilde{\pi}(t,\mathsf{x}_{j})]={\rm
i}\hbar\delta_{i,j}/a\to{\rm
i}\hbar\delta(\textsf{{x}}-\textsf{{x}}^{\prime})$ in the continuum limit,
corresponding to the canonical commutator of the scalar field defined below
Eq. (1) for natural units. Note, however, that we need to work in SI units as
we have not yet identified the quantity that will play the role of the speed
of light. Accordingly, we must rescale the fields through a canonical
transformation to achieve the correct scaling dimensions of the Klein-Gordon
model, which for $D=1+1$ dimensions is

${\phi}(t,\mathsf{x})=\frac{1}{\sqrt{m_{a}a}}\tilde{\phi}(t,\mathsf{x}),\hskip
8.61108pt{\pi}(t,\mathsf{x})=\sqrt{m_{a}a}\tilde{\pi}(t,\mathsf{x}).$ (71)

Since the field varies slowly, one proceeds by performing a gradient expansion
to capture the long-wavelength physics

${\phi}(t,{\textsf{x}_{j}})\approx{\phi}(t,{\textsf{x}_{i}})+({\mathsf{x}_{j}-\mathsf{x}_{i}})\hat{\nabla}_{\\!\\!\mathsf{x}_{i}}{\phi}(t,{\textsf{x}_{i}})+\cdots,$
(72)

where we have introduced the discretized derivative

$\hat{\nabla}_{\\!\mathsf{x}_{i}}{\phi}(t,{\textsf{x}_{i}})=\frac{{\phi}(t,{\textsf{x}_{i+1}})-{\phi}(t,{\textsf{x}_{i}})}{a}.$
(73)

As a consequence of this gradient expansion, the long-range terms (66)
effectively reduce to nearest-neighbour couplings, leading to a clear
similarity with the solid-state harmonic crystal. As will become clear below,
the long-range couplings should decay sufficiently fast such that the
corresponding series obtained by summing over all neighbours converge, and the
truncated gradient expansion is thus meaningful. This procedure improves upon
the so-called phonon-like approximation Morigi and Fishman (2004b), where one
truncates the dipolar tail to nearest neighbours without a previous gradient
expansion, and thus yields an incorrect speed of sound. Although there exist
other methods to obtain a more accurate dispersion relation at long
wavelengths Morigi and Fishman (2004a, b), the current gradient expansion is
likely the simplest and most-economic one. Substituting the expressions
(70)-(72) in the Coulomb-chain Hamiltonian (66), we find that the vibrations
along the transverse $\mathsf{z}$-axis can be described by

$H_{h}\\!=\\!a\\!\\!\sum_{\mathsf{x}\in\Lambda_{s}}\\!\\!\frac{1}{2}\\!\\!\left(\frac{c_{\mathsf{t}}^{2}{\pi}^{2}\\!(t,\mathsf{x})}{K^{2}\hbar^{2}}+K^{2}\hbar^{2}\\!({\hat{\nabla}_{\\!\\!\mathsf{x}}{\phi}\\!(t,\mathsf{x}))^{2}}+{m}_{0}^{2}c_{\mathsf{t}}^{2}K^{2}\\!{\phi}^{2}\\!(t,\mathsf{x})\\!\\!\right)\\!\\!,$
(74)

where $\Lambda_{s}$ stands for the positions of the ions in the chain, and we
have introduced the parameters

$c_{\textsf{t}}=\omega_{\textsf{x}}a\sqrt{\frac{\ell^{3}}{a^{3}}\log 2},\hskip
6.45831ptm_{0}=\frac{\hbar\omega_{\textsf{zz}}}{c_{\textsf{t}}^{2}},\hskip
6.45831ptK=\frac{m_{a}ac_{\textsf{t}}}{\hbar},$ (75)

where the zigzag mode frequency reads

$\omega_{\textsf{zz}}^{2}=\omega_{\textsf{z}}^{2}-\frac{7}{2}\zeta(3)\omega_{\textsf{x}}^{2}\frac{\ell^{3}}{a^{3}},$
(76)

and we have made use of the Riemann zeta function

$\zeta\\!(z)=\sum_{r=1}^{\infty}\frac{1}{r^{z}}.$ (77)

The physical interpretation of these parameters is now given. In the continuum
limit, one gets $a\sum_{\mathsf{x}\in\Lambda_{s}}\to\int\\!{\rm
d}{\mathsf{x}}$, such that the term inside the parenthesis of Eq. (74) can be
identified with a Hamiltonian density. After a canonical rescaling
$\phi(x)\to\phi(x)/{K}$, $\pi(x)\to{K}\pi(x)$, this Hamiltonian density can be
derived from $\mathcal{L}_{\rm
KG}=\textstyle\frac{1}{2}(\hbar^{2}\partial_{\mu}\phi\partial^{\mu}\phi-
m_{0}^{2}c_{\textsf{t}}^{2}\phi^{2})$, which is the Klein-Gordon Lagrangian
density in SI units. Hence, one clearly sees that $c_{\textsf{t}}$ plays the
role of an effective speed of light, and $m_{0}$ of an effective bare mass. To
understand the role of the dimensionless parameter $K$, we need to use a
different normalisation of the fields (71), such that the canonical algebra is
$[{\phi}\\!(t,\mathsf{x}_{i}),{\pi}(t,\mathsf{x}_{j})]={\rm
i}\delta_{i,j}/a\to{\rm i}\delta(\textsf{{x}}-\textsf{{x}}^{\prime})$, the
scalar field is dimensionless, and its conjugate momentum has units of inverse
length. This is achieved by the transformation
$\phi(x)\to\phi(x)({\frac{{c_{\textsf{t}}}}{K\hbar}})^{1/2}$,
$\pi(x)\to\hbar\pi(x)({\frac{K\hbar}{{c_{\textsf{t}}}}})^{1/2}$ which, in the
massless limit $m_{0}=0$, leads to the Hamiltonian density $\mathcal{H}_{\rm
cb}=\frac{\hbar
c_{\textsf{t}}}{2}\big{(}\frac{1}{K}\pi^{2}+K(\nabla\phi)^{2}\big{)}$. Note
that, in this case, this Hamiltonian density corresponds to the field theory
of a conformal boson with Tomogana-Luttinger parameter $K$ Fradkin (2013).
This parameter appears in the treatment of strongly-correlated 1D models
Haldane (1981a); Giamarchi (2010), where various gapless phases of matter
arising from very different microscopic models Luther and Peschel (1975);
Emery (1976); Haldane (1980) share fundamental properties with the groundstate
of an interacting-fermion model Tomonaga (1950); Luttinger (1963). This leads
to the concept of the Tomonaga-Luttinger liquid Haldane (1981b) where,
regardless of the nature of the fundamental constituents, the relevant low-
energy excitations are bosonic particle-hole pairs with a long-wavelength
description in terms of this conformal boson QFT with additional local
interactions: the so-called sine-Gordon model Coleman (1975). Due to the very
nature of these interactions, the boson field can be compactified on a circle
with a radius that depends on the Tomonaga-Luttinger parameter $R=1/\sqrt{4\pi
K}$ which, in turn, controls the power-law decay of two-point functions of the
original microscopic degrees of freedom Senechal in (2004). In the present
situation, the additional local interactions in the trapped-ion crystal shall
not be those of the sine-Gordon model Coleman (1975), such that there is no
reason to compactify the scalar field. Moreover, as the original degrees of
freedom are already a discretised version of the bosonic field, the Luttinger
parameter will not appear in the two-point functions. In fact, one can readily
see that this parameter does not modify the Heisenberg equations of motion of
the scalar field, and will thus not appear in the dispersion relation. In the
following subsection, we shall see how it plays an important role in the
presence of $\lambda\phi^{4}$ interactions, controlling the amount of quantum
fluctuations. However, prior to that, let us give some additional insight on
the derivation and physical origin of the microscopic expressions for
$c_{\mathsf{t}},{m}_{0}$, and $K$.

The effective speed of light $c_{\mathsf{t}}$ can also be derived from a
coarse-grained elasto-dynamical theory of the ion crystal. As discussed around
Eq. (166) of the Appendix, deformations with shear strain lead to sound waves
that do not alter the length of chain, but describe shape deformations. The
speed of such transverse sound waves can be expressed in terms of the crystal
density $\rho$ and the shear modulus $\mu_{r}$ through

$c_{\mathsf{t}}=\sqrt{\frac{\mu_{r}}{\rho}}.$ (78)

In this case, the shear modulus measures the rigidity of the ion chain to
changes to its linear shape, as opposed to the compression/expansion that
would be described by the bulk modulus and lead to longitudinal sound waves.
Within this gradient expansion, the shear modulus reads

$\mu_{r}=\frac{1}{2a}\sum_{j\neq
i}\kappa^{\textsf{z}}_{i,j}|\textsf{x}_{j}-\textsf{x}_{i}|^{2}\cos(\mathsf{k}_{s}|\textsf{x}_{j}-\textsf{x}_{i}|)=m_{a}\omega_{\textsf{x}}^{2}a\frac{\ell^{3}}{a^{3}}\log
2$ (79)

where, in the last equality, we have taken the thermodynamic limit
$N\to\infty$, substituted $k_{s}|\textsf{x}_{j}-\textsf{x}_{i}|=\pi|j-i|$, and
summed the corresponding series $\sum_{r=1}^{\infty}(-1)^{r}/r=-\log 2$ .
Within this homogeneous approximation, the mass density of the ion chain is
$\rho=m_{a}/a$, and the transverse speed of sound

$c_{\textsf{t}}=\sqrt{\frac{e^{2}}{4\pi\epsilon_{0}}\frac{\log 2}{m_{a}a}},$
(80)

which leads directly to the corresponding expression in Eq. (75), upon using
the length scale in Eq. (69).

Let us now derive the expression for the bare mass or, equivalently, the
Compton wavelength $\xi_{0}$ associated to these massive scalar bosons. By
multiplying the first and third terms in Eq. (74) and, likewise, in the
trapped-ion Hamiltonian (66), we find the following expression

$\xi_{0}^{\\!-2}=\frac{m_{0}^{2}c_{\textsf{t}}^{2}}{\hbar^{2}}\\!=\\!\frac{1}{c_{\textsf{t}}^{2}}\\!\\!\left(\\!{\\!\omega_{\textsf{z}}^{2}-\\!\sum_{j\neq
i}\\!\frac{2\kappa_{j,i}^{\mathsf{z}}(-1)^{j-i}}{m_{a}}\sin^{2}\\!\\!\left(\textstyle\frac{1}{2}\mathsf{k}_{s}|\textsf{x}_{j}-\textsf{x}_{i}|\right)}\\!\\!\right)\\!.$
(81)

Once again, taking the thermodynamic limit in the homogeneous bulk for the ion
chain, we find

$\xi_{0}=a\frac{\omega_{\textsf{x}}}{\omega_{\textsf{zz}}}\sqrt{\frac{\ell^{3}}{a^{3}}\log
2},$ (82)

where the zigzag mode frequency (76) is obtained by summing the series
$\sum_{r=1}^{\infty}(1-(-1)^{r})/r^{3}=7\zeta(3)/4$. This Compton wavelength
(82) will be a key quantity below, and leads directly to the bare mass of Eq.
(75).

Let us now give a physical interpretation of the Tomonaga-Luttinger parameter.
First of all, the conformal boson is a self-dual QFT under the transformation
$\pi(x)\leftrightarrow\phi(x)$, $K\leftrightarrow 1/K$, such that the
definition of the Tomonaga-Luttinger parameter as $K$ or $K^{-1}$ depends on
convention. We follow the choice in Bermudez  _et al._ (2017a); Fradkin
(2013), although we note that it is also customary to take the inverse
convention in condensed matter Giamarchi (2010); Shankar . Also note that the
normalization of the fields may differ, and one often finds definitions $K\to
K/2\pi$ von Delft and Schoeller (1998). With these different choices, one can
relate the Tomonaga-Luttinger parameter to the sound speed and the thermal
compressibility of a 1D harmonic chain Emery in (1993); Gonzalez in (1995).
With our current convention, and considering the transverse nature of the
trapped-ion displacements, this relation would involve the rigidity modulus
instead

$c_{\textsf{t}}K=\frac{\mu_{r}a^{2}}{\hbar},$ (83)

which leads directly to the corresponding parameter of Eq. (75). Being
proportional to the shear modulus, the Tomonaga-Luttinger parameter is thus a
measure of the rigidity of the ion chain to shear deformations. As will be
discussed in more detail in the following section, given the current
microscopic interpretation in terms of shear rigidity, it is not surprising
that the Tomonaga-Luttinger parameter gets renormalised by additional
anharmonicities. Indeed, prior to the canonical rescaling mentioned below Eq.
(74), the coarse-grained Lagrangian of the ion chain would be
$\mathcal{L}_{\rm
KG}=\frac{K^{2}}{2}(\hbar^{2}\partial_{\mu}\phi\partial^{\mu}\phi-
m_{0}^{2}c_{\textsf{t}}^{2}\phi^{2})$, which can be readily identified with
our discussion of the wavefunction renormalisation for the $\lambda\phi^{4}$
theory in the Appendix below Eq. (155). The difference here is that, for the
trapped-ion chain, the Tomonaga-Luttinger parameter can be interpreted as a
wavefunction renormalisation $K^{2}\leftrightarrow z_{\phi}^{-1}$ that occurs
already in the free theory, and is a consequence of the underlying rigidity of
the ion chain. As noted above, from this perspective, it is not surprising
that this rigidity gets renormalised by anharmonicities, which gives a
physical interpretation of the field-theoretic multiplicative renormalisations
(56) due to the sunrise Feynman diagram. Indeed, such ’non-interacting
multiplicative renormalisations’ are directly taken care of by the parameter
definition of the coarse-grained theory (74).

Let us now present an alternative account of these results that will be useful
later. Since the harmonic trapped-ion Hamiltonian (66) is quadratic, one can
derive an exact description of the collective vibrational branches in the
homogeneous limit by a simple Fourier transform Ashcroft (2016). The
collective-mode frequencies for transverse $\mathsf{z}$ vibrations are

$\omega^{2}({\textsf{k}})={\omega_{\mathsf{z}}^{2}+\sum_{j\neq
i}\frac{2}{m_{a}}\kappa_{j,i}^{\mathsf{z}}\sin^{2}\left(\textstyle\frac{1}{2}\mathsf{k}|\textsf{x}_{j}-\textsf{x}_{i}|\right)},$
(84)

where the momentum belongs to the first Brillouin zone ${\rm
BZ}=\\{\textsf{k}=\frac{2\pi}{Na}n\hskip 0.86108pt:\hskip
0.86108ptn\in\mathbb{Z}_{N}\\}$. In the thermodynamic limit, the exact
dispersion relation is expressed in terms of the Riemann zeta function and the
series representation of the polylogarithm

${\rm Li}_{n}(z)=\sum_{r=1}^{\infty}\frac{z^{r}}{r^{n}},$ (85)

where $z\in\mathbb{C}$ and the radius of convergence is $|z|\leq 1$ Gradshteĭn
_et al._ (1965). In the present case, the polylogarithms are evaluated at the
$N$ roots of unity to yiled the dispersion relation

$\omega^{2}({\textsf{k}})=\omega_{\mathsf{z}}^{2}+\omega_{\textsf{x}}^{2}\frac{\ell^{3}}{a^{3}}\\!\left(2\zeta\\!(3)-{\rm
Li}_{3}\\!\big{(}{\rm e}^{{\rm i}\textsf{k}a}\big{)}-{\rm
Li}_{3}\\!\big{(}{\rm e}^{-{\rm i}\textsf{k}a}\big{)}\right).$ (86)

On can now Taylor expand the polylogarithms around the lowest-energy mode for
the transverse vibrations $\textsf{k}a\approx\pi+\delta\textsf{k}a$ for
$|\delta\textsf{k}|\ll\pi/a$, leading to

$\omega^{2}({\delta\textsf{k}})\\!\approx\omega_{\mathsf{z}}^{2}-\omega_{\textsf{x}}^{2}\frac{\ell^{3}}{a^{3}}\\!\\!\\!\left(\\!\frac{7}{2}\zeta(3)-(\delta\textsf{k}a)^{2}\log
2\\!\\!\right)\\!\\!=\\!\frac{{m}_{0}^{2}c_{\textsf{t}}^{4}}{\hbar^{2}}+c_{\textsf{t}}^{2}\delta{\textsf{k}}^{2}\\!,$
(87)

where one obtains the same expressions for the effective speed of light (80)
and Compton wavelength (82). As remarked above, the dispersion relation gives
us no information about the Tomonaga-Luttinger parameter, which can be either
extracted from the gradient expansion, or using the above relationship
$c_{\textsf{t}}K=\mu_{r}a^{2}/\hbar=c_{\textsf{t}}^{2}m_{a}a/\hbar$. It is
interesting to note that, while the longitudinal vibrational modes for a
harmonic crystal with dipolar couplings do not lead to a well-defined sound
speed $\omega_{\delta\textsf{k}}^{2}\approx|\textsf{k}|^{2}\log|\textsf{k}|$
Ashcroft (2016), the transverse sound speed of the ion crystal is perfectly
valid (103) due to the repulsive nature of the harmonic couplings (66) and the
out-of-phase oscillations of neighbouring ions in the lowest-energy mode. This
is the underlying reason for our choice to focus on the transverse vibrations
of the ions along the z axis.

We have seen that the transverse displacement of each ion can be described by
a scalar quantum field discretised in a lattice formed by the positions of the
ions in the Coulomb crystal. In addition, at each of these positions, we have
all the electronic degrees of freedom of each ion, among which we shall select
only a pair of energy levels
$\\{\ket{0_{\textsf{{x}}}}=\ket{\uparrow_{\textsf{{x}}}},\ket{1_{\textsf{{x}}}}=\ket{\downarrow_{\textsf{{x}}}}\\}$.
These will typically correspond to long-lived levels in the ground- and/or
metastable-state manifolds. The specific choices lead to the so-called
hyperfine, Zeeman, optical, or fine-structure qubits Bruzewicz  _et al._
(2019), where we remind that $\omega_{0}$ is the transition frequency between
the two corresponding levels. Using the discretised version of the orthogonal
projectors introduced around (5),
$Q(t,\mathsf{x})=\ket{\uparrow_{\textsf{x}}}\bra{{\uparrow_{\textsf{x}}}}=1-P(t,\mathsf{x})$,
the dynamics of the internal and motional degrees of freedom is described by

$H_{0}=H_{h}+a\\!\sum_{\mathsf{x}\in\Lambda_{s}}\delta\epsilon(\mathsf{x})Q(t,\mathsf{x}),\hskip
5.38193pt\delta\epsilon(\mathsf{x})=\frac{\hbar\omega_{0}}{a}.$ (88)

where the vibrational Hamiltonian in the long-wavelength harmonic
approximation corresponds to Eq. (74). According to our long-wavelength
description based on spin path integrals, this system will then correspond to
$S=\textstyle\frac{1}{2}$ scalar-sigma model.

The coupling of the qubits to the vibrational degrees of freedom is realised
by shining electromagnetic radiation into the ion crystal, which stems from
additional laser or microwave sources, again depending on the chosen qubit
type Bruzewicz  _et al._ (2019). In order to achieve the form of the Ising-
Schwinger sources, either in the $Z(x)$ basis (17) or any other Pauli basis,
one typically works in the Lamb-Dicke regime of resolved sidebands Leibfried
_et al._ (2003a). Here, one selects the propagation direction and frequency of
the radiation to couple selectively to the desired vibrational branch by
creating/annihilating a single phonon. The desired Ising-Schwinger sources can
be achieved by using a Mølmer-Sørensen Sørensen and Mølmer (1999) or Leibfried
Leibfried  _et al._ (2003b) scheme, both of which generate a so-called state-
dependent force that underlies the realization of multi-qubit entangling gates
for quantum computation Soderberg and Monroe (2010). For concreteness,
following the discussion in the first part of this manuscript, we focus on
$Z$-dependent forces and hyperfine/Zeeman qubits, but remark that similar
expressions can be found for other Pauli operators depending on the qubit
choice. Following the notation introduced in Bermudez  _et al._ (2017a), but
sticking to the use of fields in SI units, the corresponding spin-phonon
coupling can be rewritten as

$\hat{V}=a\\!\sum_{\mathsf{x}\in\Lambda_{s}}\cos(\textsf{k}_{s}\mathsf{x})\frac{g}{a}\sin(\Delta\omega_{L}t-\Delta\textsf{{k}}_{L}\cdot\textsf{{x}})Z(t,\textsf{x})\phi(t,\textsf{x}).$
(89)

Here, $\Delta\textsf{{k}}_{L}$ ($\Delta\omega_{L}$) is the wavevector
(frequency) of the beat note arising from a pair of interfering laser beams,
and $g=\hbar\Omega_{L}\Delta\textsf{{k}}_{L}\cdot{\bf
e}_{\textsf{z}}\sqrt{m_{a}a^{3}}$ is the coupling strength between the qubits
and the scalar field, which is controlled by the two-photon Rabi frequency
$\Omega_{L}$ Leibfried  _et al._ (2003b). By a direct comparison with the
harmonic Ising-Schwinger sources introduced in Eq. (35), we directly find that
the harmonic dependence of the source is fully tunable via the laser’s beat
note

$\omega_{J}=\Delta\omega_{L},\hskip
4.30554pt\textsf{{k}}_{J}=\Delta\textsf{{k}}_{L}.$ (90)

Let us now discuss the role of the Tomonaga-Luttinger parameter. As already
discussed above, in order to arrive at the corresponding expression of the
Klein-Gordon QFT, we should use a canonical transformation of the fields
$\phi(x)\to\phi(x)/{K}$, $\pi(x)\to{K}\pi(x)$. For a pure scalar theory, this
rescaling has no effect on the dynamics, such that the Tomonaga-Luttinger
parameter does not have any dynamical manifestation. However, as soon as we
couple the scalar field to the Ising spins, the situation changes. Since the
Tomonaga-Luttinger parameter is a measure of the harmonic internal forces that
quantify the shear stress of the crystal, it does not appear directly in the
bare coupling strength of the Ising-Schwinger sources, which can be understood
as external sources that lead to shear strain. Note that, under the
aforementioned canonical transformation of the fields, the Ising-Schwinger
sources will get effectively rescaled by the Tomonaga-Luttinger parameter,
which can be interpreted as Hook’s law in disguise (i.e. strain proportional
to stress) Landau and Lifshitz (2008); Thorne and Blandford (2017).
Considering the additional alternation due to the oscillating nature of the
lowest-energy $\textsf{k}_{s}=\pi/a$ vibrational mode, one finds that the
amplitude of Ising-Schwinger sources is

$J_{0}\to J_{0}(\mathsf{x})=-J_{0}\cos(\textsf{k}_{s}\mathsf{x}),\hskip
4.30554ptJ_{0}=\frac{g}{K}.$ (91)

Hence, the Tomonaga-Luttinger parameter will appear in the strength of the
boson-mediated Ising interactions (29). This will be particularly important in
presence of non-linearities for the sound waves. As commented in the paragraph
below (83), the Tomonaga-Luttinger parameter can be understood as a non-
interacting wavefunction renormalisation, and one will get further
renormalisations when including interactions, modifying the strength of the
spin-spin couplings.

Finally, to conclude with the discussion about the trapped-ion realisation,
the additional transverse field $H_{\textsf{t}}(x)$ in Eq. (35) can be
obtained by adding another source of radiation, either from additional laser
beams, or from an external microwave source, tuned in resonance with the
carrier transition instead of the vibrational sidebands Leibfried  _et al._
(2003a). Altogether, the time-evolution unitary (36) would correspond to

$\hat{V}=\\!\sum_{\mathsf{x}\in\Lambda_{s}}{J_{0}(\mathsf{x})}\sin(\Delta\omega_{L}t-\Delta\textsf{{k}}_{L}\cdot\textsf{{x}})K\phi(t,\textsf{x})Z(t,\textsf{x})-{h}_{\textsf{t}}X(t,\textsf{x}),$
(92)

where the transverse field
$h_{\textsf{t}}=\hbar\Omega_{0}/2\ll\hbar\omega_{0}$ depends on the carrier
Rabi frequency $\Omega_{0}$.

According to the generic results of Sec. II, the source term (92) will lead to
Ising interactions mediated by the effective massive Klein-Gordon fields,
which can be written as

$H_{\rm eff}=\frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n}J_{ij}\hskip
1.07639ptZ(\textsf{{x}}_{i})Z(\textsf{{x}}_{j})-h_{\textsf{t}}\sum_{i=1}^{n}X(\textsf{{x}}_{i}).$
(93)

Here, the Ising couplings (29) must be updated to account for the spatial
modulation of the source couplings (91), yielding

$J_{ij}=-2J_{0}(\mathsf{x}_{i})G_{m_{\rm eff}}^{\rm
E}\\!(\textsf{{x}}_{i}-\textsf{{x}}_{j})\cos\big{(}\textsf{{k}}_{J,\textsf{x}}(\textsf{{x}}_{i}-\textsf{{x}}_{j})\big{)}J_{0}(\mathsf{x}_{j}).$
(94)

We recall that the decay of the interactions with the distance is controlled
by the dimensionally-reduced Euclidean propagator, which corresponds to $D=1$
dimensions (28) in this case, with an effective mass (26) that depends on the
frequency of the harmonic sources. Note, however, that all the expressions in
Secs. II-III use natural units. In the context of trapped ions, one can
translate them into SI units by exchanging the mass parameter in favour of an
effective Compton wavelength $m_{\rm eff}^{2}\to\xi_{\rm
eff}^{-2}=\xi_{0}^{-2}-\Delta\omega_{L}^{2}/c_{\textsf{t}}^{2}$, such that

$\xi_{\rm
eff}=\frac{c_{\textsf{t}}}{\sqrt{{\omega_{\textsf{zz}}^{2}-\Delta\omega_{L}^{2}}}},$
(95)

where we recall that the beat-note is always red-detuned with respect to the
transverse vibrational branch
$\Delta\omega_{L}\lesssim\omega_{\textsf{zz}}=\mathord{\rm
min}\\{\omega(\textsf{k}),\textsf{k}\in[0,2\pi/a)\\}$. In addition, to be
consistent with the SI units, the Euclidean propagator in Eq. (29) should be
rescaled as $G_{m_{\rm eff}}^{\rm E}\\!(\textsf{{x}}_{1}-\textsf{{x}}_{2})\to
G_{m_{\rm eff}}^{\rm E}\\!(\textsf{{x}}_{1}-\textsf{{x}}_{2})/\hbar^{2}$.