# A Nuclear Interferometer for Ultra-Light Dark Matter Detection

Hannah Banks  CERN, Theoretical Physics Department, Geneva, Switzerland
DAMTP, University of Cambridge, Wilberforce Road, Cambridge, UK  Elina Fuchs
CERN, Theoretical Physics Department, Geneva, Switzerland  Institute of
Theoretical Physics, Leibniz Universität Hannover, Appelstr. 2, 30167
Hannover, Germany  Physikalisch-Technische Bundesanstalt (PTB), Bundesallee
100, 38116 Braunschweig, Germany  Matthew McCullough  CERN, Theoretical
Physics Department, Geneva, Switzerland

###### Abstract

Abstract

We propose the nuclear interferometer - a single-photon interferometry
experiment based upon the thorium-229 nuclear clock transition - as a novel
detector for ultra-light dark matter (ULDM). Thanks to the enhanced
sensitivity of this transition to the variation of fundamental constants, we
find that possible realisations of such an experiment deploying either single
ions or clouds of atoms have the potential to be complementary to advanced
very-long-baseline terrestrial clock atom interferometers in the search for
ultra-light dark matter with scalar couplings to photons. Nuclear
interferometry may also offer an unparalleled window to new physics coupling
to the QCD sector via quarks or gluons, with a discovery reach that could
enhance existing and proposed experiments over a range of frequencies in the
direction of well-motivated parameter space.

††preprint: CERN-TH-2024-104

##  I Introduction

The hunt for dark matter (DM) is entering a new era. Despite a panoply of
evidence for its existence, very little is known about the nature or
properties of DM on the particle level. At present, our only real handle on
this elusive substance comes from observations of its gravitational
interactions over astrophysical and cosmological scales, leaving its
microphysics largely unconstrained. Even under the assumption that the energy
density attributed to DM is predominantly harboured by a single species, the
mass in question could viably range from $\sim 10^{-22}$ eV bosons up to $\sim
10^{19}$ GeV mass primordial black holes [1, 2]. Deciphering the fundamental
constituents of DM and its particle-like properties remains of upmost priority
in contemporary particle physics.

Over the last few decades, non-gravitational interactions of weak-scale
particles have been relentlessly pursued as part of a broad experimental
campaign encompassing direct, indirect and collider based searches, all to no
avail [3]. Whilst there remain a number of interesting and well-motivated
avenues in the vicinity of this parameter space yet to explore, the idea that
DM instead comprises sub-eV ultra-light bosonic particles which have remained
hidden thus far due to extremely feeble couplings to the Standard Model (SM),
has gained significant traction in recent years [4, 5]. A number of highly
motivated DM candidates fall into this category including the QCD axion [6, 7]
(see Ref. [8] for a review) and other pseudo-scalar axion-like particles
(ALPs), vector dark photons [9, 10, 11] and scalars such as moduli [12, 13,
14, 15], dilatons [16, 17], Higgs portal dark matter [18] and the relaxion
[19, 20].

The high occupation numbers required for such a light species to saturate the
relic local dark matter density renders its behaviour over galactic scales
akin to that of a coherent, classical wave, giving rise to a host of novel
phenomenological signatures which are not shared by heavier ‘particle-like’
candidates [21, 22]. Direct searches for ultra-light DM (ULDM) thus typically
rely upon different detection strategies to the traditional experiments
conceived for weak-scale candidates which seek impulses resulting from
particle-like interactions. Recent advances in quantum sensing technologies
have been transformative in this respect, opening up a number of exciting new
possibilities to probe new light weakly coupled states which lie within this
‘precision’ frontier [23, 24, 25, 26, 27].

ULDM could conceivably couple to the SM through a number of different
‘portals’, each of which has a distinct set of phenomenological consequences
and in turn, experimental signatures. In this proposal we focus specifically
on scalar ULDM with linear couplings to the SM. These couplings cause
fundamental constants, and thus physical properties such as atomic energy
levels and the distance scales which they govern, to oscillate in time at a
frequency set by the DM mass, see e.g. Refs. [28, 29, 30, 31, 24]. This effect
can be searched for in a variety of different ways including accelerometers
[32], optical cavities [33], mechanical resonators [34, 35] and by comparing
the frequency of two different atomic clocks [29, 31, 36, 37]. More recently,
atom interferometers have been identified to offer the potential to probe
ULDM-induced oscillations of atomic transition frequencies in a complementary
region of parameter space to the aforementioned tests [38, 39]. In its most
basic form, an atom interferometer constitutes an experiment which measures
the phase difference between two spatially delocalised quantum superpositions
of clouds of atoms [40]. Such experiments can be thought of analogously to
optical interferometers but with the traditional beam-splitters and mirror
elements realised via a sequence of controlled duration laser pulses which act
to divide, reflect and recombine the atomic wave-packets by driving
transitions between internal energy states. The presence of ULDM can be
diagnosed by the existence of an oscillating phase difference between a pair
of spatially separated single-photon111Here, ‘single-photon’ refers
specifically to a class of interferometers in which the desired transition is
driven by a single laser frequency. This contrasts with interferometers based
on Raman transitions which involve two distinct optical fields (see e.g. Ref.
[41]). atom interferometers which are resonantly interrogated with common
laser pulses [39]. Such multi-atom interferometer configurations are termed
single-photon atom gradiometers and have the key advantage in that noise
arising from fluctuations and jitters in the laser phase is common to both
instruments and thus cancels in the differential measurement [42]. In order to
achieve sensitivity to ULDM, such proposals look to both long
($\mathcal{O}$(km)) baselines and large-momentum-transfer (LMT) atom
optics222Such techniques (see e.g. Ref. [43]) effect an increase of the
physical baseline by employing an increased number of laser pulses to augment
the momentum transfer between the two arms of each atom interferometer and in
turn the phase difference accumulated at the end of the sequence. [44] (see
Sec. II.2) to increase the recorded phase difference. To sustain these
practices it is essential for the pair of internal states on which the atom
interferometer is based to be sufficiently long-lived such that spontaneous
emission, which would otherwise exponentially degrade the sensitivity due to
loss of coherence, is not significant.333For a discussion of some of the
potential implications of spontaneous decay on long-baseline atom
interferometry in space, see Refs. [45, 46]. For the very long baselines and
high number of sequential pulses demanded by ULDM searches (see e.g. [47,
48]), ultra-narrow transitions such as those deployed in optical lattice
clocks are therefore typically taken advantage of. For this reason such atom
interferometer realisations have earned the title of ‘large-momentum-transfer
clock atom interferometry’ [49].

A number of prototype long-baseline gradiometers including AION-10 [50],
MAGIS-100 [47] and VLBAI [51] are currently in development, and will serve as
essential technological readiness indicators to longer (up to km-scale)
instruments envisioned to offer sensitivity to oscillating ULDM as well as
mid-frequency (0.1-10 Hz) gravitational waves by the mid 2030’s. A
comprehensive overview of the potential capabilities of long-baseline
interferometers in addition to their current status of development can be
found in Ref. [49].

In tandem, the discovery of an excited isomeric state in 229Th [52, 53], has
led to proposals for a ‘nuclear clock’ [54, 55, 56] based on transitions
between the ground and the 229mTh excited isomer state. Arising due to
accidental cancellations between contributions from strong and electromagnetic
interactions [57, 58, 59], the small (8.3 eV [60]) gap in energy between the
ground and excited states lies in the optical range [61] and offers an
enhanced sensitivity to the variation of fundamental constants at the level of
several orders of magnitude [57, 62, 63] over typical optical clock
transitions in atoms, ions and highly charged ions [64]. Frequency comparisons
between nuclear and atomic optical clocks could offer a highly sensitive probe
of ULDM in the near future [65, 66, 67, 29, 68]. Due to rapid internal
conversion, the excited isomeric state in electrically neutral 229Th is short-
lived with a half-life of $\sim 7\,\mu$s [69]. Proposed realisations of the
thorium nuclear clock for deployment in such experiments therefore look to
either ionic forms [55] or to a thorium-doped lattice [70] in which internal
conversion is forbidden and the isomeric state becomes long-lived with a
radiative lifetime of $\sim 10^{4}$ s [65].

Whilst the nuclear clock is yet to be realised experimentally, rapid progress
towards this goal has been made in recent years and especially months. The
production of 229Th at ISOLDE at CERN via the decay chain of 229Fr as a new
pathway [71] refined the knowledge of the nuclear transition energy from the
percent to the per-mill level. Subsequently, based on the development of a new
tunable VUV laser [72], the first laser excitation of the 229Th nucleus was
achieved at PTB [73] in Th-doped crystals [74]. While the resolution of the
transition frequency here is limited by the laser linewidth to 7 GHz [73],
corresponding to a relative uncertainty of $\mathcal{O}(10^{-6})$, this
precision has since further been improved to 3 GHz at UCLA [75] and, by means
of a VUV frequency comb at JILA, to 300 kHz [76], suppressing the relative
uncertainty to $\mathcal{O}(10^{-10})$. With such developments rapidly paving
the way towards an operational nuclear clock, the time to investigate other
ways in which this unique transition could be exploited to probe fundamental
physics is ripe.

In this work we propose to amalgamate the 229Th nuclear clock transition with
the established principles of single-photon matter-wave interferometry to form
what we shall refer to as a ‘nuclear interferometer’. We present two different
possible realisations of this idea - one based on single 229Th ions and the
other on clouds of neutral 229Th atoms. Whilst both present additional
experimental challenges beyond those of the optical clock transitions of
conventional atom interferometry-based ULDM search proposals, we find that for
certain experimental configurations, the enhanced sensitivity of the ground-
to-excited state transition to the variation of fundamental constants in these
thorium systems is able to overcome the loss in sensitivity that respectively
results from the low ion flux (i.e large quantum projection noise) and the
short lifetime of the excited state in neutral form, to offer access to new
regions of ULDM parameter space.

We begin in Sec. II.1 by introducing the ULDM model that we will adopt as an
example in this work, before computing the signal that it would imprint in a
generic nuclear interferometry experiment in Sec. II.2. We consider potential
noise sources in Sec. II.3 and use them to derive an expression for the reach
of a general free-fall nuclear interferometer in terms of a handful of tunable
parameters which characterise the experimental operating configurations. In
Sec. III we explore two different ways, making use of single-ion clocks and
clouds of atoms respectively, in which such a nuclear interferometer could be
realised experimentally, and discuss how the aforementioned principles apply
to each case. Forecasts of the sensitivity of both realisations to scalar ULDM
with linear couplings to SM operators for a variety of experimental
configurations that may become feasible in the future are presented in Sec. IV
and compared to bounds from existing experiments and the predicted
sensitivities of competing proposals. Some concluding remarks are offered in
Sec. V.

##  II The Concept

###  II.1 Ultra-light scalar Dark Matter

To motivate this proposal we consider an ultra-light singlet scalar field,
$\phi$, which couples linearly444Note that quadratic couplings also lead to
the variation of fundamental constants [77], and become relevant if linear
couplings are not present, forbidden for instance by a symmetry. We leave
consideration of such couplings to future work. to the SM by way of non-
derivative operators. We assume that this field comprises an $\mathcal{O}(1)$
fraction of the local DM abundance. Working in units in which $\hbar=c=1$ and
following the notational conventions555In this paramaterisation, the
coefficients and normalisation of each of the terms in the Lagrangian are
specifically chosen such that the dimensionless couplings are all RG-invariant
[78].  of Refs. [78, 29] in parameterising the dimensionless couplings
relative to Newton’s gravitational constant $G_{N}$, the leading-order low
energy Lagrangian, applicable to scales $\gtrsim$ 1 GeV, can be expressed in
terms of a relatively minimal set of operators as

$\begin{split}\mathcal{L}_{\phi}=-\kappa\phi\Biggr{[}\frac{d_{e}}{4e^{2}}F_{\mu\nu}F^{\mu\nu}-\frac{d_{g}\beta_{3}}{2g_{3}}G^{A}_{\mu\nu}G^{A\mu\nu}-d_{m_{e}}m_{e}\bar{e}e\\\
-\sum_{i=u,d}\left(d_{m_{i}}+\gamma_{m_{i}}d_{g}\right)m_{i}\bar{\psi}_{i}\psi_{i}\Biggr{]}~{},\end{split}$
(1)

where $F_{\mu\nu}$, $G_{\mu\nu}$ are respectively the photon and gluon field
strength tensors and $e$, $\psi_{i}$ are the fermion fields of the first
generation. Here, $\kappa=\sqrt{4\pi G_{N}}$, $\beta_{3}(g_{3})$ is the QCD
beta function, $g_{3}$ is the strong coupling constant and the
$\gamma_{m_{i}}$, $i=\\{u,d\\}$ are the anomalous dimensions of the $u$ and
$d$ quarks.

These effective operators feature in well-motivated models of ultra-light dark
matter. For one example, in the presence of additional sources of CP violation
beyond those included within the SM, both the QCD axion, and other pseudo-
scalar axion-like particles possess CP-even scalar couplings in addition to
their CP-odd pseudo-scalar interactions. In particular, the QCD axion can
attain a non-zero scalar coupling to quarks [79, 80, 81], which is bounded
from below by the CP violation in the SM, and from above by measurements of
the neutron electric dipole moment.

Owing to the high occupation numbers and small velocities of DM in the Milky
Way, $\phi$ can be well approximated by a non-relativistic temporally and
spatially oscillating classical plane wave

$\phi(t,\mathbf{x})=\phi_{0}\cos[m_{\phi}(t-\mathbf{v\cdot
x})+\theta]+\mathcal{O}(|\mathbf{v}|^{2})~{},$ (2)

with an amplitude $\phi_{0}\simeq\sqrt{2\rho_{\textnormal{DM}}}/m_{\phi}$
which is set by the requirement that the field saturates the local DM energy
density, $\rho_{\textnormal{DM}}=0.3$ GeV cm-3 [82]. $\theta$ denotes an
arbitrary phase. The field is temporally coherent over a coherence time
$\tau_{c}=2\pi/(m_{\phi}\sigma_{v}^{2})$ and spatially coherent over a
coherence length $\lambda_{c}=2\pi/(m_{\phi}\sigma_{v})$ [83] where
$\sigma_{v}\sim\sqrt{\langle(\mathbf{v}-\langle\mathbf{v}\rangle)^{2}\rangle}\sim
5\times 10^{-4}$ (in natural units) [84] is the DM virial velocity.

###  II.2 The signal

The couplings in Eq. 1 cause the fine structure constant $\alpha$, the strong
coupling constant $\alpha_{s}$ and the fermion masses to vary in time.
Explicitly,

$\displaystyle m_{\psi}$
$\displaystyle=m_{\psi}\left[1+d_{m_{\psi}}\kappa\phi(t,\mathbf{x})\right]$
(3) $\displaystyle\alpha$
$\displaystyle=\alpha\left[1+d_{e}\kappa\phi(t,\mathbf{x})\right]$ (4)
$\displaystyle\alpha_{s}$
$\displaystyle=\alpha_{s}\left[1+d_{g}\kappa\phi(t,\mathbf{x})\right]$ (5)

where $\psi$ represents a generic fermion with mass $m_{\psi}$.

These variations induce oscillations in the nuclear energy and length scales
and, in turn, the thorium nuclear transition energy, $\omega_{N}=8.3$ eV [60].
$\omega_{N}$ is a function, $g$, of both $\alpha$ and the ratio
$X_{q}=m_{q}/\Lambda_{\textnormal{QCD}}$, i.e.

$\omega_{N}=g(\alpha,X_{q})~{}~{}.$ (6)

Here, the QCD scale $\Lambda_{\textnormal{QCD}}$, is defined to be the Landau
pole of the running of the strong coupling constant as a function of an energy
scale $\mu$, $\alpha_{s}(\mu)\sim 1/\ln(\Lambda_{\textnormal{QCD}}/\mu)$, and
is thus related to $\alpha_{s}$ via dimensional transmutation.

As such, in the presence of the scalar DM background $\omega_{N}$ oscillates
as

$\omega_{N}=\omega_{N}+\Delta\omega_{N}~{},$ (7)

where [62, 63]

$\displaystyle\Delta\omega_{N}$
$\displaystyle=\omega_{N}\left(\frac{\partial\ln
g}{\partial\ln\alpha}\frac{\Delta\alpha}{\alpha}+\frac{\partial\ln
g}{\partial\ln X_{q}}\frac{\Delta X_{q}}{X_{q}}\right)$ (8)
$\displaystyle\approx\omega_{N}\left(10^{4}d_{e}+10^{5}(d_{\hat{m}}-d_{g})\right)\phi(t,\mathbf{x})~{}.$
(9)

In this expression $d_{\hat{m}}$ is defined to be the coupling to the
symmetric combination of quark masses:

$d_{\hat{m}}\equiv\frac{d_{m_{d}}m_{d}+d_{m_{u}}m_{u}}{m_{d}+m_{u}}~{}.$ (10)

We now investigate how these oscillations would manifest in a future nuclear
interferometer experiment. Fig. 1 shows a space-time diagram of the
interferometer sequence proposed for a vertical single-photon atom gradiometer
composed of a pair of interferometers situated at positions $z_{1}$ and
$z_{2}$ respectively. This scheme deploys large-momentum-transfer (LMT)
techniques [44] in order to increase the momentum transfer between the two
arms of each interferometer and in turn the sensitivity. In practice this
involves realising the beam-splitter with $n>1$ laser pulses which each impart
a momentum $\hbar\mathbf{k}$ where $\mathbf{k}$ denotes the photon wavevector,
to one of the interferometer arms.

Figure 1: A schematic spacetime diagram of a vertical single-photon atom
gradiometer with $n=4$ large-momentum-transfer atom optics. The experiment
comprises a pair of single-photon atom interferometers based on single-photon
transitions between a ground (blue) and excited (red) state. The wavy lines
denote laser pulses fired from opposite ends of the baseline which are used to
divide, direct and recombine the atomic states to yield an interference
pattern. Atom-laser interactions are shown as black circles.

The phase difference between the two arms of each isolated interferometer at
the end of the interferometer sequence receives contributions from multiple
sources. In the absence of new physics, the phase accumulated on each of the
two arms is expected to cancel due to the symmetry of the interferometric
sequence. It suffices therefore to only track the ULDM-induced phase
accumulated by the excited state relative to the ground state over each arm of
the interferometer.

For a segment of the trajectory between times $t_{0}$ and $t_{1}$ during which
the system is in the nuclear excited state, the ULDM-induced phase that is
accrued (relative to the ground state) is

$\Phi_{t_{0}}^{t_{1}}\equiv\overline{\Delta\omega_{N}}\int_{t_{0}}^{t_{1}}\textnormal{d}t\
\cos(m_{\phi}t+\theta)~{},$ (11)

where we have defined

$\overline{\Delta\omega_{N}}=\omega_{N}\kappa\phi_{0}(10^{4}d_{e}+10^{5}(d_{\hat{m}}-d_{g}))~{},$
(12)

to be the magnitude of $\Delta\omega_{N}$.

The total phase difference recorded by a single interferometer follows from
subtracting the total phase (relative to the ground state) that is accumulated
on each of the two arms during propagation in the excited state.

The gradiometer signal, $\Phi_{s}$, then corresponds to the difference between
the phase difference recorded by the interferometer located at position
$z_{1}$ and that recorded by the interferometer at position $z_{2}$. For the
interferometric sequence depicted in Fig. 1, in the limit that the launch
velocity in the two interferometers is the same, the gradiometer signal can be
shown (see Ref. [85] for a derivation) to be

$\begin{split}\Phi_{s}\approx\frac{\Delta
z}{L}\left(\left[\Phi^{T+L}_{T-(n-1)L}-\Phi_{0}^{nL}\right]\right.-\\\
\left.\left[\Phi^{2T+L}_{2T-(n-1)L}-\Phi_{T}^{T+nL}\right]\right)~{},\end{split}$
(13)

where the interrogation time $T$ is half the time between the two beam-
splitter sequences, and $\Delta z=|z_{1}-z_{2}|$ is the vertical separation of
the two interferometers.666For very long-baseline gradiometers such as those
considered in Ref. [39], the approximation $\Delta z/L\sim 1$ is often
adopted. In more compact gradiometers including some of those considered in
this work, this factor can be an important correction and needs to be
explicitly considered. $n$ refers to the number of large-momentum-transfer
(LMT) kicks applied, i.e. the total number of pulses used to realise each
beam-splitter sequence.

The signal amplitude $\overline{\Phi}_{s}$ for a single phase measurement,
defined according to

$\overline{\Phi}_{s}\equiv\left(\frac{1}{\pi}\int_{0}^{2\pi}\textnormal{d}\theta\hskip
5.0pt\Phi_{s}^{2}\right)^{1/2}~{},$ (14)

can then be shown to be [85]

$\begin{split}\overline{\Phi}_{s}=8\frac{\overline{\Delta\omega_{N}}}{m_{\phi}}\frac{\Delta
z}{L}\sin\left[\frac{m_{\phi}(T-(n-1)L)}{2}\right]\\\
\times\sin\left[\frac{m_{\phi}nL}{2}\right]\sin\left[\frac{m_{\phi}T}{2}\right]~{}.\end{split}$
(15)

In order to isolate the dependence of this signal on the scalar couplings we
introduce the notation

$\overline{\Phi}_{s}=d_{\phi}\overline{\Phi}_{R}~{},$ (16)

where

$d_{\phi}=d_{e}+10(d_{\hat{m}}-d_{g})~{}.$ (17)

###  II.3 Experimental Sensitivity

We now quantify the reach of a potential nuclear interferometer as an ULDM
detector. In practice, over an experimental campaign of duration
$T_{\textnormal{int}}$, a total of $N$ measurements of the gradiometer phase
difference will be performed at a regular sampling interval $\Delta t$
=$T_{\textnormal{int}}/N$, corresponding to a constant shot repetition rate.
Such measurements will contain, in addition to any possible ULDM-induced
signal, contributions from background noise sources.

Assuming that the sensitivity is limited by the quantum projection noise (or
shot-noise) of the interrogated species777Reaching shot-noise limited
detection is a key design goal for atomic sensors and has already been
demonstrated experimentally in atom interferometry - see e.g. Refs. [86, 87].
with spectral density $S_{\textnormal{n}}$, the minimum scalar coupling
$d_{\phi}^{*}$ that could be detected at a signal-to-noise ratio SNR following
a measurement period $T_{\textnormal{int}}$ is

$d_{\phi}^{*}\simeq\frac{\sqrt{\textnormal{SNR}}}{\overline{\Phi}_{R}}\times\sqrt{\frac{S_{\textnormal{n}}}{T_{\textnormal{int}}}\times\textnormal{max}\left(1,\sqrt{\frac{T_{\textnormal{int}}}{\tau_{c}}}\right)}~{}.$
(18)

Further details regarding the origin of this expression are given in Appendix
A.

Inserting Eq. 15 and Eq. 16 into Eq. 18, one finds that the experimental
sensitivity depends on the parameters $T,$ $L$, $n$ which are set in the
experimental design, in addition to the shot-noise spectral density
$S_{\textnormal{n}}$ and the campaign duration $T_{\textnormal{int}}$. For a
gradiometer comprising two identical interferometers, the shot-noise can be
described by the white noise spectrum [88, 89, 85]

$S_{\textnormal{n}}=\frac{2\Delta t}{C^{2}N_{\textnormal{s}}}~{},$ (19)

where $N_{\textnormal{s}}$ is the number of atoms (ions) per shot and the
interferometer contrast, $C\leq 1$, characterises the amplitude of the
oscillations of the probability of being in the ground and excited states.

The values that these parameters can take depend both on the physical nature
of the source (i.e. whether one is working with atoms or ions) and on the
properties of the transition, including, most importantly for the scenarios to
be considered in this work, its width. In clock interferometry, for which
spontaneous decay of the excited state can be neglected, $S_{\textnormal{n}}$
is independent of the choice of $T,$ $n$ and $L$ for a given set of starting
conditions (i.e. the number of species per shot). Given the increase in the
signal phase difference with $n,$ $L$ and $T$, such experiments look to
maximise these parameters. Due to the geometric requirement that the
interrogated clouds remain within the baseline for the duration of the
interferometric sequence, it is important to realise that these parameters
cannot be extremised independently. Any given choice of $L$ restricts the
maximum LMT order $n$ and interrogation time $T$ that can be implemented
without the trajectories of the interrogated species exceeding the baseline.

The picture is more nuanced if one considers operating on a broad transition
for which spontaneous decay becomes important relative to the proposed
interferometer scales. Here, the increase in the phase shift that comes with
increasing $L$ and $n$ is accompanied by a degradation of the interferometer
contrast and the number of species in the cloud at the point of detection
relative to their input design values due to the enhanced degree of
spontaneous decay. Precisely which of these two parameters is degraded depends
on where in the interferometric cycle the decay occurs and how far the decayed
species have travelled away from the cloud before the end of the sequence.
Those that have left the cloud contribute to a reduction in
$N_{\textnormal{s}}$ whereas those which have lost coherence with the cloud
due to decay but have not had time to travel significantly, form a diffuse
background reducing $C$ [90]. Both of these effects manifest as an exponential
increase in the shot-noise spectrum with the product $nL$ which, as we shall
see, ultimately limits both the baseline and the LMT order that can be
usefully deployed in such experiments.

##  III Possible Realisations

We now discuss two distinct ways in which the principles outlined in the
previous sections could be applied to perform interferometry on the ground-
excited isomeric state transition in 229Th - the first involving single-ion
‘clocks’, and the second clouds of neutral atoms. Both scenarios present a
distinct set of experimental challenges from the sources traditionally
considered for long-baseline interferometry and yet, as will be shown, may
offer the potential to open up new phenomenological territory. Differing both
in their physical nature and the properties of the ground-excited state
transition, these two realisations demand starkly different operating
configurations (as characterised by the parameters identified in Sec. II.3)
which we now outline.

###  III.1 Single Ions

As previously discussed, conventional ULDM searches using atom interferometry
propose to make use of ultra-narrow clock transitions which, thanks to long
excited state lifetimes, allow for the use of long baselines and large inter-
arm momentum transfers. Although the excited isomeric state lifetime for
neutral 229Th is only $\sim 10^{-5}$ s due to internal conversion, when
ionised, this decay channel becomes inaccessible and the (now radiative decay
limited) lifetime is of order $\sim 10^{4}$ s, falling within the remits of
clock interferometry. We thus begin our exploration of the potential usage of
229Th in free-fall matter-wave interferometry by considering working with an
ionic888From the perspective of the interferometry sequence, the precise
charge of the ion is unimportant and we thus leave this unspecified. form.

Unlike for a neutral species which could (providing sufficient cooling is
achieved to prevent significant Doppler expansion of the cloud) potentially be
launched in clouds comprising up to $10^{8}-10^{10}$ simultaneously
interrogated atoms, with ions electrostatic repulsion prohibits more than a
single ion per shot (i.e. $N_{\textnormal{s}}$ = 1). Whilst, at least when
exploited for fundamental physics, atom interferometers typically seek to
maximise the number of atoms per shot in an effort to minimise quantum
projection noise, experiments working with single atoms in free-fall have
already been both conceived and performed [91].

In addition to deploying a high number of atoms per shot, the sensitivity
forecasts for long-baseline atom interferometers also make use of multiplexing
(see e.g. Ref. [49]). Exploiting the Doppler shift to the transition frequency
experienced by clouds of atoms moving at different velocities, this technique
allows the time interval between successive shot launches to be reduced below
the interferometer period (2T) with multiple independent interferometers
concurrently in operation. Since (in typical Mach-Zehnder interferometry
schemes) the atoms are launched upwards and subsequently fall under gravity,
the trajectories of atoms in successive clouds cross. The atoms, which can be
treated ballistically, are effectively non-interacting such that this
trajectory overlap can occur without disrupting the generation of interference
at detection. The same can not be said for ions due to electrostatic
repulsion. This imposes a lower limit on the time interval between shots:
$\Delta t\geq 2T$. We note that whilst this condition applies to
interferometry configurations in which the ions are launched upwards and
latterly fall under gravity as in this proposal, if instead a configuration
could be utilised in which they only travelled in one direction (i.e. if
dropped from rest with the primary laser pulses fired vertically downwards)
such that they were detected in a different region from launch, it could be
possible to achieve a higher effective shot flux due to the possibility of now
applying multiplexing. In this case the experimental parameters need to be
carefully selected in order to ensure that there are no intersections between
the trajectories of each of the arms of the two interferometers. In order for
such linear, non-intersecting trajectories to simultaneously satisfy the
geometric constraints of the baseline, smaller $T$, $n$ and $\Delta z$ than
the conventional set-up are likely to be necessary, somewhat suppressing the
potential sensitivity gain associated with a higher ion flux. We leave a more
detailed exploration of the potential benefit of such scenarios to future
work.

In order to mitigate the inherently large quantum projection noise associated
with working with single ions and thus optimise the sensitivity to ULDM of
such an experiment, it is favourable to deploy both high LMT orders and long
baselines. We emphasise that these practices are only possible thanks to the
narrow transition linewidth in ionic form. In estimating the potential of ion-
based nuclear interferometry, we consider the possibility of both terrestrial
and space-based experiments, taking guidance from the configurations forecast
for the analogous future long-baseline clock atom interferometry experiments
based on 87Sr (e.g. AION [50]) in our selection of the relevant experimental
parameters.

For the terrestrial setup, we therefore consider a 1 km baseline with $n$ =
2500 LMT optics. We set the interferometer separation $\Delta z$, to its
maximum value

$\Delta z_{\textnormal{max}}\sim L-v_{n}T~{},$ (20)

where

$v_{n}=n\omega_{N}/m_{\textnormal{Th}}~{},$ (21)

is the velocity of the faster half of the wavepacket at the end of the
beamsplitter sequence relative to the slower half and $m_{\textnormal{Th}}$
denotes the mass of the thorium ion. We select $T$ such that $\Delta
z_{\textnormal{max}}\sim L$.

We model our space-based instrument on the design considered in Refs. [92, 42,
47, 93, 94] comprising two satellites in medium Earth orbit separated by a
distance of 4.4 $\times 10^{7}$ m along a single line-of-sight. In design A we
take the more conservative assumption that the region over which the ions are
interrogated is confined to the interior of the satellites. This constrains
the maximum wavepacket separation during the interferometric sequence,
$v_{n}T$, to be less than $\sim$ 1 m [47, 92]. In contrast, if the
interrogation region could be located outside the satellite (as in the designs
of Ref. [95]), wave-packet separations on the order of $\sim 100$ m could be
achieved. Whilst these inequalities are linear in both $T$ and $n$, we note
that due to the dependence of the ion flux and thus shot-noise spectral
density on $T$, it is preferable to deploy larger $n$. We select $n$ and $T$
to satisfy these inequalities, accounting for the fact that $n$ should be even
and that the total duration of the interferometer sequence should exceed the
total time taken for the (sequentially fired) laser pulses to traverse the
baseline, i.e. $2T>(4n-1)L/c$ , where $c$ is the speed of light. Whilst the
parameters used in our forecasts are representative of the reach that could be
attained in such experiments we note that further optimisation could be
performed if one had a particular signal or region of parameter space in mind.

For each of the three designs we set $S_{\textnormal{n}}$ to be consistent
with Eq. 19, accounting for the fact that one must have both
$N_{\textnormal{s}}=1$ and $\Delta t\geq 2T$.

Setup |  $L$ [m] |  $T$ [s] | $n$ |  $\sqrt{S}_{\textnormal{n}}$ [Hz-1/2] |  $\Delta z$ [m]
---|---|---|---|---|---
Terrestrial | 1000 | 1.2 | 2500 | 2.3 | 965
Space-based (A) | $4.4\times 10^{7}$ | 5 | 16 | 4.7 | $4.4\times 10^{7}$
Space-based (B) | $4.4\times 10^{7}$ | 70 | 120 | 17.6 | $4.4\times 10^{7}$
Table 1: The experimental parameters characterising the single-ion nuclear
interferometers considered in this work. In the space-based configuration A
the interrogation region is confined to be within the satellites, whereas in
configuration B it is exterior.

###  III.2 Neutral Atoms

One could alternatively conceive a nuclear interferometer based upon the
ground - isomeric excited state transition in neutral 229Th atoms. Whilst this
allows for significantly greater effective flux rates and thus much lower
quantum projection noise, there are additional complexities relating to the
short lifetime of the excited state which need to carefully considered.
Compared to clock interferometry, in the case of a broad transition the
experimental sensitivity is negatively impacted by spontaneous decay occurring
both during periods of free propagation in the excited state and during atom-
laser interactions. Each of these effects can be accounted for by the
introduction of an appropriate correction factor as we shall now detail.

Within an order $n$ single-photon LMT interferometer sequence, the ensembles
of atoms are required to spend a total of $n$ periods in the excited state,
each lasting for a time $\sim 2L/c$ set by the time for light to travel across
the baseline and back. The probability of an atom remaining coherently within
the cloud at the end of the sequence is thus (reverting to physical units)
[46]

$P_{A}=e^{\frac{-2nL}{c\tau}}~{},$ (22)

where $\tau\sim 7\,\mu\textnormal{s}/\ln(2)\sim 10\,\mu$s is the (mean)
lifetime of the neutral thorium isomeric excited state [69].

To avoid a significant deterioration of the sensitivity attributed to a given
initial number of atoms, the condition

$nL\leq\frac{c\tau}{2}\sim 1500~{}\textnormal{m}~{},$ (23)

should therefore be satisfied. Unlike single-ions, neutral 229Th is thus
restricted to terrestrial scale interferometry experiments.

It is also important to account for the impact of spontaneous decay on the
(finite duration) atom-laser interactions. This manifests as a reduction in
the efficiency of the so-called $\pi$-pulses which are deployed to invert the
populations of the ground and excited states. The proposed LMT interferometry
scheme depicted in Fig. 1 relies upon a number of $\pi$-pulses which are
Doppler tuned to selectively interact with a single arm of the interferometer.
The velocity resolution of the pulse i.e. the FWHM of the atom velocities
which will interact, is

$\delta
v=\frac{\Gamma_{a}}{\omega_{N}}\left(1+\frac{I}{I_{\textnormal{sat}}}\right)^{1/2}~{},$
(24)

where $\Gamma_{a}=\tau^{-1}$ is the natural linewidth of the ground-excited
isomeric transition in atomic 229Th and the ratio of the applied laser
intensity to the transition saturation intensity $I/I_{\textnormal{sat}}$,
captures the effect of power broadening. For the laser to successfully
interact with a single arm of the interferometer, the velocity separation of
the two arms at the time the pulse is applied, $\Delta v$, must exceed the
velocity resolution. Depending on the location of the pulse within the
interferometer sequence, the velocity separation between the two arms is

$\Delta v=\frac{\omega_{N}l}{m_{\textnormal{Th}}}~{},$ (25)

where $l$ takes integer values between 1 and $n$ and represents the difference
in the number of times that each of the arms has interacted with the laser. In
this instance $m_{\textnormal{Th}}$ now denotes the mass of a thorium atom.

The requirement

$\Delta v>\delta v~{},$ (26)

thus implies a maximum laser intensity

$\frac{I}{I_{\textnormal{sat}}}\Bigg{|}_{\textnormal{max},l}=\left(\frac{\omega_{N}^{2}l}{m_{\textnormal{Th}}}\right)^{2}-1~{}.$
(27)

Given that the Rabi frequency, $\Omega$, which sets the duration of the
$\pi$-pulse according to

$t_{\pi}=\frac{\pi}{\Omega}~{},$ (28)

is related to $I/I_{\textnormal{sat}}$ by

$\frac{\Omega^{2}}{\Gamma_{a}^{2}}=\frac{I}{2I_{\textnormal{sat}}}~{},$ (29)

Eq. 27 thus implies that there is a minimum $\pi$-pulse duration for each $l$,
which, for the transition in question is sizeable relative to the excited
state lifetime. It thus becomes important to account for the impact of
spontaneous decay over the duration of the $\pi$-pulse. This can be formally
achieved by explicitly including spontaneous decay in the optical Bloch
equations which describe the behaviour of a 2-level system interacting with an
electromagnetic field [96, 97]. The result, which we detail explicitly in
Appendix B, is a reduction in the probability that an applied $\pi$-pulse
results in population inversion from its ideal value of unity according to the
value of the ratio $\Omega/\Gamma_{a}$.

The interferometer sequence depicted in Fig. 1 involves (4$n$-3) $\pi$-pulses
in total, all but 1 of which are intended to selectively interact with a
single interferometer arm. Of these, there are 2 pulses which address an $l=1$
velocity separation, 2 of which act on an $l=n$ velocity separation and a
further 4 for each $l\in\\{2,...,n-1\\}$. In estimating the total degradation
of the sensitivity due to $\pi$-pulse inefficiencies over the course of the
interferometer sequence, we assume that the laser intensity used for each
$\pi$-pulse takes its maximum value such that each pulse has as short a
duration and thus high as efficiency as possible. In particular we note that
for $l=1$ and $l=2$ (the only pulse types required for an $n=2$ LMT sequence),
the minimum pulse durations are achieved at intensity ratios
$I/I_{\textnormal{sat}}$ $\sim 25$ and $\sim 105$ respectively. We
additionally assume that the intensity used for the central $\pi$-pulse which
is designed to simultaneously interact with both arms is sufficiently high
such that its duration is negligible with respect to $\tau$. The total
probability of having undergone all the desired transitions at the end of the
interferometer sequence is then:

$P_{B}=P_{1}^{2}P_{n}^{2}\prod_{k=2}^{n-1}P_{k}^{4}~{},$ (30)

where $P_{k}$ denotes the maximum efficiency of a $\pi$-pulse that addresses a
single interferometer arm at an $l=k$ velocity separation, obtained by
evaluating Eq. 39 at the maximum ratio of $\Omega/\Gamma_{a}$ according to
Eqs. 27 and 29.

Given that the reduction in $\pi$-pulse efficiency is sizeable (e.g.
$P_{1}\sim 0.73$), it is preferable to deploy as few LMT kicks as possible and
we thus only consider $n=2$.999Note that it is necessary to have $n>1$ in
order that the atoms predominantly propagate in the ground state allowing for
interrogation times that exceed the excited state lifetime.

As a result of these two effects, the shot-noise spectrum increases to

$S_{\textnormal{n}}=\frac{\tilde{S}_{\textnormal{n}}}{P_{A}P_{B}}=\frac{e^{\frac{2nL}{c\tau}}\tilde{S}_{\textnormal{n}}}{P_{B}}~{},$
(31)

where $\widetilde{S}_{\textnormal{n}}$ denotes the noise spectrum for an
identical experiment subject to the same initial number of atoms per shot but
in the absence of spontaneous decay.

A crucial feature of LMT interferometry is that the atoms propagate in the
ground state for the majority of the sequence meaning that it is possible to
make use of interrogation times which greatly exceed the excited state
lifetime. This has been experimentally demonstrated on table-top scales in
Ref. [43] where LMT interferometry was performed on the 689 nm
intercombination line of 87Sr.101010Note that the interferometer sequence
deployed in Ref. [43] differs from that proposed in this work, in that every
$\pi$-pulse is designed to address both arms of the interferometer
simultaneously. This is realised by making use of sufficiently high laser
intensities such that Eq. 26 is not satisfied.  Whilst the short excited state
lifetime does not therefore impose a direct upper limit on the duration of the
interrogation sequence, the interrogation time does inherit a constraint
indirectly from the geometric requirement that the atom trajectories remain
within the baseline. This condition implies that for any given choice of $L$
and $n$ there is a maximum $T$ that can be utilised. In practice however, if
looking to work in the regime $\Delta z\sim L$, it is necessary to deploy a
value of $T$ substantially smaller than this maximum i.e. there is a trade-off
between $T$ and $\Delta z$ which must be accounted for in parameter selection.

In our sensitivity forecasts for this scenario we consider three distinct
operating configurations, characterised by the parameters $T,$ $L$ and $n$,
the interferometer separation $\Delta z$ and the intrinsic phase noise
$\sqrt{\tilde{S}_{\textnormal{n}}}$ that arises from the number of atoms
launched per shot, representative of an ‘initial’, ‘intermediate’ and more
‘advanced’ experimental design. Whilst we specify our configurations using
$\sqrt{\tilde{S}_{\textnormal{n}}}$, we emphasise that it is
$S_{\textnormal{n}}$, as defined in Eq. 31, which actually enters the
sensitivity calculations in order to account for the additional loss of
coherence/atom number due to spontaneous decay. Some justification of these
values, which we detail in Tab. 2, is in order. As explained above, due to the
substantial impact of spontaneous decay on the $\pi$-pulse efficiency we are
restricted to an $n=2$ LMT order. According to Eq. 23, the greatest
sensitivity will then be reached at a baseline of $\sim$ 750 m which we thus
deploy for our advanced configuration. In order to take advantage of existing
and planned experimental infrastructure, for the initial and intermediate
configurations we specifically choose values for $L$ that match the baselines
that are, or will be, deployed in the prototype iterations of proposed very-
long-baseline clock interferometers based on the 87Sr clock transition.

For $n=2$, the maximum interferometer separation that can be achieved
(assuming equal launch velocities) for a given $L$ and $T$ without the atom
trajectories exceeding the baseline is approximately111111We emphasise that
this approximate expression is valid only for low ($\mathcal{O}(1)$) $n$.

$\Delta z_{\textnormal{max}}\sim L-\frac{1}{2}gT^{2}~{}.$ (32)

Once again we set $\Delta z$ to $\Delta z_{\textnormal{max}}$ and choose $T$
such that $\Delta z_{\textnormal{max}}\sim L$. We explicitly include the
$\Delta z/L$ factor in our sensitivity calculations, noting that this factor
becomes increasingly important the shorter the baseline.

As per Eq. 19, in the shot-noise limit the phase sensitivity is essentially
controlled at the design level by the time-averaged atomic flux
($N_{\textnormal{s}}/\Delta t$). Given that atom interferometry relies upon
the ensemble of atoms being sufficiently cold such that the fraction of atoms
that are addressed with each laser pulse is not significantly degraded by the
thermal expansion of the cloud over the course of the interferometer sequence,
it is laser cooling techniques which ultimately limit the achievable flux
rates. As discussed in detail in Refs. [46, 98], the percentage of atoms which
interact with each laser pulse follows from a convolution of the spatial and
velocity distributions of the ensemble of atoms with the laser pulse Fourier
transform and spatial profile of the beam. In our forecasts we assume that the
laser beam waist in use is sufficient to address every atom in every
interaction. Since the clouds of atoms expand over time as a result of their
thermal velocity, the initial temperature to which the atoms are cooled thus
needs to be sufficiently low such that the cloud size remains appropriate for
the beam waist in use. A greater understanding of the cooling temperatures and
laser properties that may be required to achieve this could be developed
through a more thorough analysis of the convolution integral. Assuming the
usage of a similar beam waist, the requirements on the 3D momentum
distribution of an ensemble of thorium atoms are likely to be of similar order
of magnitude to those of more typical atom interferometry sources such as
strontium. We thus inform our forecasts of potential fluxes with the initial
and more ambitions projections discussed in the context of conventional atom
interferometers (see e.g. Ref. [49]).

Whilst it is possible to produce clouds of $10^{8}-10^{10}$ atoms with current
techniques [99], the challenge in atom interferometry is to achieve the
requisite level of cooling sufficiently rapidly so as to support high shot
repetition rates and thus fluxes [95]. The spectral densities of
$\sqrt{\tilde{S}_{\textnormal{n}}}$ = $10^{-4},$ $10^{-5}$ (Hz)-1/2 projected
for future long-baseline interferometers such as AION could (theoretically) be
realised with $10^{8}$ and $10^{10}$ uncorrelated atoms per shot at a 1 Hz
shot repetition rate, however are in practice expected to be achieved with
fewer atoms and the deployment of squeezing techniques in which the source
atoms are entangled [100]. In adopting these flux rates for our forecasts we
are thus implicitly assuming the development of the requisite laser cooling
and squeezing techniques for 229Th. As is the case for conventional atom
interferometers, our projections should therefore be interpreted as
indications of the reach that could be achieved in the future should the
necessary advancements in cold atom technologies be realised. On comparing our
forecasts to those of conventional atom interferometry proposals however, it
is important to keep in mind that we have not accounted for any increased
experimental challenge that may be associated with cooling and squeezing 229Th
as opposed to 87Sr, and therefore that the timescales over which these
technologies may be developed are likely to differ.

Setup |  $L$ [m] |  $T$ [s] | $n$ |  $\sqrt{\tilde{S}_{\textnormal{n}}}$ [Hz-1/2] |  $\Delta z$ [m]
---|---|---|---|---|---
Initial | 10 | 0.6 | 2 | 10-4 | 8.2
Intermediate | 100 | 1.8 | 2 | 10-5 | 84
Advanced | 750 | 3.1 | 2 | $0.3\times 10^{-5}$ | 702
Table 2: The parameters characterising the atom-based nuclear interferometer
configurations deployed in the sensitivity forecasts presented in this work.
$\sqrt{\tilde{S}_{\textnormal{n}}}$ refers to the ‘intrinsic’ phase noise set
by the shot repetition rate and the number of atoms launched per shot. These
values are corrected prior to entering our sensitivity estimates to account
for the impact of spontaneous decay as detailed in the main text.

A further issue which one needs to remain mindful of is that in neutral 229Th,
the nuclear excitation energy lies above the first ionisation
potential.121212Note that this is not the case in ionic form. Whilst a first-
principles evaluation of the photo-ionisation and photo-excitation rates of
this system lies beyond the scope of this work, we provide an estimation as to
the potential implications of this in Appendix C. We find that whilst photo-
ionisation is likely to make working with unpolarised atoms infeasible, if the
clouds are polarised it may be possible via a careful choice of linear laser
polarisation to greatly suppress the photo-ionisation rate by forbidding the
dominant photo-ionisation channels. The projections shown in Sec. IV assume
the implementation of such a practice such that atom loss due to ionisation is
negligible.

##  IV Discussion

In Figs. 2, 3 and 4, we plot the couplings $d_{e}$, $d_{g}$ and $d_{\hat{m}}$
that could be detected at a signal-to-noise ratio (SNR) of 1 with the single-
ion interferometer designs specified in Tab. 1 following a measurement period
of $T_{\textnormal{int}}=10^{8}$ s, assuming that the coupling in question
dominates $d_{\phi}$. In Figs. 5, 6 and 7, we present the reach of the neutral
atomic interferometer configurations of Tab. 2, once again taking
$T_{\textnormal{int}}=10^{8}$ s and assuming that the clouds of atoms have
been appropriately polarised such that photo-ionisation is negligible (see
Appendix C). For clarity, our plots show the power-averaged envelope of the
experimental response using the approximation $\sin x$ = min($x$,
$1/\sqrt{2}$) as in Ref. [39].

Figure 2: Shot-noise limited projected sensitivity of the single-ion nuclear
interferometer configurations detailed in Tab. 1 to ULDM with a linear scalar
coupling to photons. The curves show the minimum coupling $d_{e}$ that could
be detected at a SNR = 1 following an experimental campaign of duration
$T_{\textnormal{int}}=10^{8}$s. Also shown are existing bounds and forecast
sensitivities of other experiments as detailed in the main text. The parameter
space motivated from a naturalness perspective as described in the text is
shown in red.  Figure 3: Same as Fig. 2 but for ULDM with a linear scalar
coupling to gluons.  Figure 4: Same as Fig. 2 but for ULDM with a linear
scalar coupling to quarks.

For context, also shown in these figures are the 95% confidence level (CL)
limits on the considered couplings from tests of the Equivalence Principle
(EP) both with Eöt-Wash torsion balance experiments [101, 102] and MICROSCOPE
[103, 104], as computed in Ref. [105]. We also display the limits from a
hyperfine frequency comparison of rubidium and caesium (Rb/Cs) atomic
microwave clocks [106], in addition to the estimated reach of a frequency
comparison between the 229Th nuclear clock and an atomic optical clock [29,
5]. Upon the recommendation of Ref. [83], the Rb/Cs limits derived in Ref.
[106] have been rescaled by a factor of 3 in order to properly account for the
stochastic nature of the scalar ULDM field and enable consistent comparison
with competing bounds and proposals.

In the case of the coupling to photons $d_{e}$, we also show limits from the
AURIGA detector [107], the frequency comparison of the electric octupole and
the electric quadrupole clock transitions in single-ion ytterbium (Yb+ E3/E2),
the comparison of the Yb+ E3 transition to the strontium $S\to P$ clock
transition (Yb+ E3/Sr) [37], the comparison of a strontium optical clock with
a silicon cavity (Sr/Si) [108] and the combined constraints of several optical
clock comparisons (Al+/Hg+, Al+/Yb, Yb/Sr) from the Boulder atomic clock
optical network [36]. Forecasts for the sensitivities of proposed compact
mechanical resonators based on superfluid helium, sapphire cylinders and
sapphire pillars [34] are also shown, as are the prospective sensitivities of
very-long-baseline atom interferometers operating on the 87Sr atomic clock
transition. We show forecasts for terrestrial configurations (TVLBAIs) with 10
m, 100 m and 1 km scale baselines in addition to a potential space-based
realisation (SB-VLBAI), assuming the experimental parameters listed in Tab. 3.
Further details on the calculation of these estimates, including a
justification of the parameters used, can be found in Appendix D.

In the case of the couplings to quarks and gluons, we show bounds from the
comparison of a Yb lattice clock with a caesium fountain microwave clock
(Yb/Cs) [109], the comparison of strontium and caesium clocks (Sr/Cs) [110],
the comparison of a hydrogen maser with a silicon cavity [108] and the
comparison of the E3/E2 transitions in Yb+ [111]. Whilst the first three of
these comparisons involve a microwave hyperfine transition, in the latter, the
sensitivity to $d_{g}$ and $d_{\hat{m}}$ derives from ULDM-induced
oscillations of the nuclear charge radius. Also shown are projections for the
comparison of a molecular SrOH clock with an atomic optical clock, assuming a
total data accumulation period of 1 year [112].

We emphasise that whilst a variety of other experiments have probed, or have
been proposed to probe each of these couplings, for clarity and efficiency, we
have shown here just a representative selection of those which yield the
strongest bounds or greatest prospective sensitivities in the parameter space
of relevance. A more detailed overview of the landscape of competing
experiments can be found (e.g. for $d_{e}$) in Ref. [5].

The red shading in each of these figures delineates the parameter space which
is motivated from a naturalness perspective. We derive this following Ref.
[39]. The central argument here is that these scalar interactions break any
shift symmetry which could keep the scalar light, such that a mass-squared
contribution is expected from within the UV theory, scaling as the square of
the coupling. Motivated by the scales already explored by the LHC, we
conservatively take the cutoff of the IR (infrared/ low-energy) theory at
which these UV-contributions are predominantly generated to be $10$ TeV and
determine the parameter space in which these generic UV contributions are
smaller than the scalar mass, thus avoiding the requirement of fine-tuning of
different UV contributions to arrive at a given scalar mass, given a certain
scalar coupling.

Figure 5: Shot-noise limited projected sensitivity of the nuclear
interferometer configurations deploying neutral atoms detailed in Tab. 2 to
ULDM with a linear scalar coupling to photons. The curves show the minimum
coupling $d_{e}$ that could be detected at a SNR = 1 following an experimental
campaign of duration $T_{\textnormal{int}}=10^{8}$ s and assuming that photo-
ionisation has been successfully suppressed. Also shown are existing bounds
and forecast sensitivities of other experiments as detailed in the main text.
The parameter space motivated from a naturalness perspective as described in
the text is shown in red.  Figure 6: Same as Fig. 5 but for ULDM with a linear
scalar coupling to gluons.  Figure 7: Same as Fig. 5 but for ULDM with a
linear scalar coupling to quarks.

It is important to realise that whilst the bounds from tests of the EP are
independent of the abundance of $\phi$, the other experiments shown on these
plots, including our forecasts for the various nuclear interferometer
realisations, implicitly assume $\phi$ to comprise the totality of the relic
local DM abundance. At low masses $m_{\phi}<2\times 10^{-20}$ eV, this
scenario has been ruled out by examination of small-scale cosmological
structure in the Lyman-$\alpha$ forest [113].

As a further point of note, we emphasise that our forecasts assume the
experiment to be shot-noise limited at all frequencies. In practice, for
terrestrial experiments, gravitational gradient noise (GGN) arising from mass
fluctuations around the experiment [114, 115] is thought to dominate at sub-Hz
frequencies [116, 117, 118]. The modelling and mitigation of GGN at
terrestrial atom interferometers constitutes an active and involved area of
research (see e.g. Ref. [119]) which is beyond the scope of this work. As
such, we do not attempt to take the impact of GGN into account here but
caution the reader that below $\sim$1 Hz, our estimations for the terrestrial
nuclear interferometer configurations (and indeed those of the TVLBAIs) should
be read as extrapolations of the reach that could be attained if GGN could be
sufficiently mitigated such that the experiment remains shot-noise limited. As
demonstrated in Ref. [119], the use of multiple interferometers positioned
along the baseline could be promising in this respect, with the potential to
reduce the impact of GGN in ULDM searches by up to two orders of magnitude.

Together, these figures illustrate the discovery reach of nuclear
interferometry, with the potential to open windows to large regions of
currently uncharted phenomenological territory, thus facilitating a broader
exploration of the dark matter landscape. As a result of their differing
operating conditions, the ionic and atomic nuclear interferometer realisations
proposed in this work naturally target contrasting regions of parameter space
and are thus complementary in their potential physics gain. Whilst the single-
ion nuclear interferometer may, if operated in space, be able to achieve
significantly greater peak sensitivities than an instrument deploying clouds
of neutral 229Th, the experimental parameters required for this latter
realisation naturally favour higher mass particles with a significantly
shallower degradation of sensitivity moving in the direction of increasing
ULDM mass.

In Figs. 2 and 5, we respectively show the reach of a single-ion and a neutral
atom nuclear interferometer to ULDM coupling to photons, whose future
experimental landscape is expected to be most strongly constrained by long-
baseline 87Sr atomic clock interferometers and nuclear-atomic optical clock
comparisons. Due to the intrinsically limited ion flux, we find that
terrestrial single-ion nuclear interferometers are unable to fully benefit
from the increased sensitivity of the nuclear clock transition to the
variation of fundamental constants, and are likely to be out-performed by
conventional 87Sr-based terrestrial interferometers, provided that the target
atomic fluxes (as listed in Tab. 3) can be reached. Much higher sensitivities
could be achieved in space-based experiments with the substantial increase in
baseline afforded by such realisations compensating for the high shot-noise
associated with working with single ions.

In contrast, we find that the $\sim$ 4 order of magnitude increase in
sensitivity of the ground-excited isomeric state transition to the variation
of $d_{e}$ compared to standard atomic optical clocks is sufficient to
overcome challenges associated with the short excited state lifetime in
neutral 229Th, rendering terrestrial atom-based nuclear interferometers
competitive with their 87Sr clock analogues in addition to improving their
reach at higher masses. Beyond that, such enhancements can be achieved with
both significantly less demanding LMT requirements and more modest baselines
(as is indeed required due to the short excited isomeric lifetime), than are
necessary to deploy in conventional atom interferometry.

The potential of nuclear interferometry to advance existing experimental
proposals is substantially greater in searches for new physics coupling to the
QCD sector via the couplings $d_{g}$ and $d_{\hat{m}}$, as displayed in Figs.
3, 4, 6 and 7. These couplings enter the frequencies of typical atomic optical
clock transitions via the ratio of the electron mass to the nucleus mass
($m_{e}/m_{\textnormal{nuc}}$) in the reduced mass of the system, rendering
their contribution suppressed by approximately 5 orders of magnitude with
respect to that of $d_{e}$. As such, the sensitivity of conventional atomic
clock interferometers to such effects is not appreciable and for consistency
with literature is not shown in these figures. We note that in Ref. [111] it
was shown that in heavy-nucleus optical clocks such as 171Yb+, the transition
frequency becomes sensitive to the nuclear-charge radius, leading to an
enhanced sensitivity to the couplings $d_{g}$ and $d_{\hat{m}}$ by $\sim 2$
orders of magnitude. Such systems could in principle be deployed in clock atom
interferometry experiments as recently suggested in Ref. [120] where
oscillations of the nuclear charge radius in a space-based atom interferometry
experiment using a (neutral) Yb source were considered. We emphasise that the
intrinsic sensitivity of this transition to variations of $d_{g}$ and
$d_{\hat{m}}$ is still many orders of magnitude below that of the nuclear
clock however.

We highlight that the deployment of the nuclear clock in an interferometry-
based search for ULDM as proposed in this work is complementary to its usage
in nuclear-atomic optical clock frequency comparison experiments. Indeed, as
can be seen in the plots, the nuclear interferometer pushes the potential
parameter space coverage closer to, and in the case of $d_{\hat{m}}$ into,
more theoretically motivated regions where the couplings can be deemed
technically natural.

The entirety of the discussion up to this point has been based on operating
the interferometer in ‘broadband’ mode. In principle, one could alternatively
operate in ‘resonant’ mode [92] in which the interferometer space-time diagram
comprises $Q$ closed diamonds each of duration $2T$. This results in a
$Q$-factor sensitivity enhancement to the mass $m_{\phi}$ = $\pi/T$ over a
mass range $\pi/(QT)$ relative to a broadband search with the same
experimental parameters. Such schemes are, in the context of conventional atom
interferometers, thought to be of particular benefit to space-based designs in
which the atom interrogation is confined to the interior of the satellites,
greatly restricting the LMT order $n$ that can be employed [47, 92]. With the
on-resonance sensitivity scaling linearly with both $Q$ and $n$ in resonant
mode (see Ref. [85]), it is possible to exploit the parameter $Q$ whilst
maintaining a sufficiently low $n$ so as to adhere to the geometric
constraints in order to reach considerably greater sensitivities than could be
achieved with broadband operation. In Ref. [39], it was shown that
systematically scanning through a band of resonant masses over the course of
the experimental campaign by altering $Q$ and $T$ could extend the reach of
space-based 87Sr interferometers to masses in this band relative to the
equivalent broadband search. Given that resonant mode operation involves an
increased number of atom-laser interactions, for the atom-based nuclear
interferometer the low $\pi$-pulse efficiencies mean such schemes are unlikely
to be of any benefit. In contrast, for ion-based interferometers we do expect
a scanned resonant search to yield increased sensitivities, particularly to
masses at the higher end of the band. Due to the dependence of the shot-noise
on the total duration of the interferometer sequence, which in resonant mode
is increased by a factor $Q$ to $2QT$, the relative enhancement over the
analogous broadband search will not be as significant as that which may be
achieved for 87Sr atom interferometers however.

Though not studied in this work, we highlight that the nuclear interferometer
is not limited to investigating ULDM with scalar couplings. In particular, the
pseudo-scalar coupling of axions and axion-like particles to gluons induces
oscillations in the thorium transition energy at quadratic order in the axion
field [121] and would thus generate a signal in a nuclear interferometer. We
leave a more detailed treatment of this scenario to future work.

Whilst the reach of the proposals presented here remains limited by the high
shot-noise and the short lifetime of the excited state respectively, this work
demonstrates that there is nonetheless the potential, provided experimental
challenges can be overcome in the future, to access new, previously uncharted,
phenomenological territory with the nuclear interferometer. Not only does this
serve as testament to the unique and exciting role that 229Th may play in our
future explorations of new physics, but it motivates a closer investigation of
its potential within the field of long-baseline interferometry. For example,
in the case of single-ions, one might explore different interferometer
geometries to enable the use of greater ion fluxes, and in neutral atoms,
alternative interferometry schemes deploying floquet atom optics [122] could
be considered as a means to achieve higher $\pi$-pulse efficiencies.

##  V Conclusions

With an extraordinarily broad landscape of viable candidates which give rise
to a diverse range of phenomenological signatures, new ideas for the detection
of DM are required in order to maximise experimental coverage of the vast
available parameter space. In this work we propose two possible realisations
of a single-photon interferometry experiment based on the nuclear clock
transition in 229Th as a probe of ULDM with scalar couplings to the SM, making
use of single-ions and clouds of neutral atoms respectively. Although working
with either source type presents a unique set of experimental challenges, we
find that the enhanced sensitivity of the nuclear clock transition to the
variation of fundamental constants offers the means to access large regions of
uncharted parameter space over a wide range of frequencies.

In particular, nuclear interferometry could offer a unique opportunity to
probe scalar couplings of ULDM to gluons and quarks, with a sensitivity
exceeding existing and proposed experiments over a range of frequencies.
Employing the nuclear clock in this way complements its usage in nuclear-
atomic optical clock frequency comparisons, with the peak sensitivity moving
towards more theoretically motivated regions of parameter space. Although by
no-means fully optimised, this proposal highlights the important role which
229Th could play in future searches for new physics, motivating a broader
investigation into its potential capabilities both within interferometry and
beyond.

###### Acknowledgements.

This work is supported by the CERN Quantum Technology Initiative. We are
incredibly grateful to Leondardo Badurina and Thomas Hird for discussions
regarding the challenges of using sources with short excited state lifetime in
atom interferometry. We also thank Dennis Schlippert, Marianna Safranova and
Gilad Perez for invaluable feedback on early versions of this manuscript,
Ekkehard Peik and Michael Doser for insight on the nuclear clock and John
Carlton for discussions on the AION experiment. We thank Nathaniel Sherrill
and Abhishek Banerjee for details on the analyses of atomic clock comparison
experiments. HB acknowledges partial support from the STFC HEP Theory
Consolidated grants ST/T000694/1 and ST/X000664/1 and thanks other members of
the Cambridge Pheno Working Group for useful discussions. EF acknowledges
funding by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under Germany’s Excellence Strategy – EXC-2123 QuantumFrontiers –
390837967.

##  Appendix A Sensitivity to ULDM Couplings

Here we briefly outline how the experimental sensitivity given in Eq. 18
follows from Eq. 15, the signal amplitude for a single measurement.

The experimental sensitivity takes into account a total of $N$ phase
measurements made continuously at a regular sampling rate $1/\Delta t$ over
the experiment run time $T_{\textnormal{int}}$. We denote these measurements
as $\\{\Phi_{m}\\}$, where $m$ labels the phase recorded from the
interferometer sequence starting at time $m\Delta t$.

A discrete set of measurements made at different times can be conveniently
characterised by its power spectral density (PSD), which contains information
on both the amplitude and the frequency spread of the data. The PSD can be
constructed via the Fourier transform of the measured signal

$\tilde{\Phi}_{k}=\sum_{m=0}^{N-1}\Phi_{m}\textnormal{exp}\left(-\frac{2\pi
imk}{N}\right)~{},$ (33)

where $k$ takes integer values between 0 and $N-1$.

Using this definition, the PSD, $S_{k}$, can be expressed as

$S_{k}=\frac{(\Delta
t)^{2}}{T_{\textnormal{int}}}\left|\tilde{\Phi}_{k}\right|^{2}~{}.$ (34)

Defining the angular frequency as $\omega=2\pi k/T_{\textnormal{int}}$, in the
limit of shot-noise domination, the SNR at a frequency $\omega_{0}$ is then
given by

$\textnormal{SNR}=\frac{S_{\textnormal{s}}(\omega_{0})}{S_{\textnormal{n}}(\omega_{0})}~{},$
(35)

where $S_{\textnormal{s}}(\omega)$ denotes the PSD of the ULDM induced signal
and $S_{\textnormal{n}}(\omega)$ denotes the shot-noise PSD as given by Eq.
19.

In the limit $T_{\textnormal{int}}<\tau_{c}$, the ULDM signal is dominated by
a single frequency such that the signal PSD takes the form

$S_{s}(\omega)\sim\begin{cases}T_{\textnormal{int}}|\overline{\Phi}_{s}|^{2},&\omega=\omega_{0}\\\
0,&\omega\neq\omega_{0}\end{cases}~{}.$ (36)

The experimental sensitivity to these couplings in this regime is thus

$d_{\phi}\simeq\frac{\sqrt{\textnormal{SNR}}}{\overline{\Phi}_{R}}\times\sqrt{\frac{S_{\textnormal{n}}}{T_{\textnormal{int}}}}~{}.$
(37)

If instead $T_{\textnormal{int}}>\tau_{c}$, the signal PSD has a finite
profile in frequency space determined by the DM velocity distribution. In this
instance, the experimental sensitivity can be computed using Bartlett’s method
[85, 123, 124], leading to

$d_{\phi}\simeq\frac{\sqrt{\textnormal{SNR}}}{\overline{\Phi}_{R}}\times\sqrt{\frac{S_{\textnormal{n}}}{\sqrt{\tau_{c}T_{\textnormal{int}}}}}~{}.$
(38)

Combining these two results leads directly to Eq. 18.

##  Appendix B $\pi$-pulse efficiencies

The behaviour of a 2-level atomic or nuclear system coherently interacting
with an external electromagnetic field (i.e. a laser) over time can be
described by the optical Bloch equations (see e.g. Refs. [96, 97]). These
equations model the time evolution of a density operator $\hat{\rho}$ which
encodes the occupation probabilities of the ground and excited states. When
the field is on resonance with the transition, in the absence of spontaneous
decay, the population probabilities of the ground and excited state oscillate
at the so-called Rabi frequency, $\Omega$, such that after a time
$t_{\pi}=\pi/\Omega$, a system initialised in the ground state has a 100$\%$
probability of having transitioned to the excited state. Pulses of this
duration are commonly referred to as $\pi$-pulses, and as discussed in the
main text, play a crucial role in the coherent transfer of populations in
matter-wave interferometry.

To estimate the $\pi$-pulse efficiency in the presence of spontaneous decay,
one can explicitly introduce spontaneous decay to the optical Bloch equations
and solve for the probability of being in the excited state as a function of
time assuming that the system is initialised in the ground state and that the
electromagnetic field (i.e. the laser) is resonant with the transition. This
calculation, which can be performed analytically, is sometimes known as
Torrey’s solution [125] and is derived in full in Refs. [96, 126]. For a
comprehensive discussion of these equations in the context of nuclear
transitions, we refer the reader to Ref. [127].

Evaluating the excited state probability at $t_{\pi}=\pi/\Omega$ yields

$\begin{split}&P\hskip
5.0pt\Big{|}_{t=t_{\pi}}=\frac{\Omega^{2}}{2\Omega^{2}+\Gamma_{a}^{2}}\hskip
5.0pt\times\\\
&\left[1-e^{-\frac{3\Gamma_{a}\pi}{4\Omega}}\left[\cos{\left(\frac{\Omega_{\Gamma}}{\Omega}\pi\right)}+\frac{3\Gamma_{a}}{4\Omega_{\Gamma}}\sin{\left(\frac{\Omega_{\Gamma}}{\Omega}\pi\right)}\right]\right]~{},\end{split}$
(39)

where

$\Omega_{\Gamma}=\Omega\sqrt{1-\left(\frac{\Gamma_{a}}{4\Omega}\right)^{2}}~{}.$
(40)

In quoting this result we have implicitly taken the limit that the bandwidth
of the exciting laser, $\Gamma_{l}$, is significantly less than the natural
width of the transition $\Gamma_{a}\sim 16$ kHz, such that the rate of
decoherence in the optical Bloch equations:
$\tilde{\Gamma}=(\Gamma_{a}+\Gamma_{l}$)/2 is entirely due to spontaneous
decay i.e. $\tilde{\Gamma}\sim\Gamma_{a}/2$. It is this regime which maximises
the $\pi$-pulse efficiency.

We take Eq. 39 \- i.e. the probability that a pulse of duration $t_{\pi}$
results in a population inversion - to be the $\pi$-pulse efficiency that we
use in making our sensitivity forecasts and emphasise that it is entirely
governed by the ratio $\Omega/\Gamma_{a}$.

##  Appendix C Photo-ionisation vs. photo-excitation of neutral 229Th atoms

The first ionisation potential of neutral 229Th is 6.3 eV - lower than the 8.3
eV nuclear isomeric excitation energy. With the interferometry schemes
proposed in this work reliant on the successful completion of a precise
sequence of coherent, laser-induced nuclear excitations and de-excitations, it
is necessary to explore the extent to which photo-ionisation would compete
with these processes, depleting the number of atoms at the point of detection
and reducing the sensitivity. By comparing estimations of the photo-ionisation
and nuclear photo-excitation cross-sections of atomic 229Th, we offer a rough
assessment of this here before exploring a possible means of mitigation.

From the ground electronic configuration, a laser resonant with the isomeric
transition is sufficiently energetic to ionise the atom via removal of
electrons from either the 6d or 7s electronic shells. At leading order, these
processes proceed via allowed (E1) electric dipole transitions with a rate
that is governed by the scalar product of the laser polarisation with the
effective electric dipole moment of the transition (i.e. the promotion of the
bound electron to the continuum) in question [128]. In an ensemble of
unpolarised atoms, the electric dipole transition moment of a given process is
randomly orientated in different atoms. In this instance, the photo-ionisation
rate follows from the average over all possible orientations of the transition
dipole moment relative to the laser polarisation. If the sample is polarised
however, these moments align in a direction fixed by the electromagnetic field
used to polarise the ensemble. As demonstrated in Ref. [129], the measured
rate of that particular photo-ionisation process then drops to zero if the
laser polarisation is chosen to be orthogonal to the direction of the
transition moment.

The cross-sections for E1 photo-ionisation of the 6d and 7s electrons are
computed numerically in Ref. [130] (assuming the atoms to be unpolarised)
using the flexible atomic code (FAC) based upon the relativistic configuration
interaction method [131]. At an energy of $\sim$ 8 eV these cross-sections are
of order 10-17 cm2 [130].

In the low saturation limit of moderate laser intensities and a laser
bandwidth that exceeds the natural linewidth of the transition, the resonance
nuclear photo-excitation cross-section is given by [127]

$\sigma_{\textnormal{ex.}}\sim\frac{\lambda_{0}^{2}}{2\pi}\frac{\Gamma_{\gamma}}{\Gamma_{l}}\sim\left(3.6\times
10^{-20}\right)\times\frac{\Gamma_{\gamma}}{\Gamma_{l}}~{}\textnormal{cm}^{2},$
(41)

where $\Gamma_{\gamma}\sim 10^{-9}\hskip 5.0pt\Gamma_{a}$ is the radiative
decay rate and $\lambda_{0}$ is the transition wavelength. Setting
$\Gamma_{l}$ to $\Gamma_{a}$, its minimum value consistent with the regime for
which this equation is valid, gives $\sigma_{\textnormal{ex.}}\sim 10^{-20}$
cm2, approximately three orders of magnitude below the photo-ionisation rate.
Whilst, as stated in Appendix B, our sensitivity forecasts assume the laser
bandwidth to be narrow relative to the natural linewidth to reduce atom loss
from spontaneous decay, here we are just interested in an order of magnitude
estimate of the nuclear excitation rate in order to assess the extent to which
photo-ionisation is likely to be a problem. Given that the nuclear excitation
rate should be greatest in the low saturation limit, using this result
provides a conservative assessment of the potential impact of photo-ionisation
on the interferometry sequence and should thus be sufficient here.

As such, we conclude that performing interferometry with unpolarised clouds of
229Th atoms is unlikely to be viable due to a significant depletion of the
number of atoms remaining coherently in the cloud at detection. To perform
nuclear interferometry with neutral 229Th atoms, we suggest that it is
therefore necessary for them to be polarised such that the transition electric
dipole moments of different atoms become aligned. By then making use of a
linearly polarised laser with a polarisation carefully chosen to be orthogonal
to the transition dipole moments of the two E1 photo-ionisation processes, it
could therefore be possible to forbid photo-ionisation at leading order.
Whilst photo-ionisation could still proceed via higher order (e.g. first
forbidden magnetic dipole M1) processes, these rates are, according to the
computations in Ref. [130], approximately 8 orders of magnitude smaller than
the E1 processes and are thus negligible with respect to estimations of the
nuclear photo-excitation rate. The forecasts presented in this work neglect
photo-ionisation, assuming it has been suppressed to negligible levels by
successful adoption of such a practice.

##  Appendix D AION Prospective Sensitivity

The sensitivity curves labelled TVLBAI-10, TVLBAI-100 and TVLBAI-km displayed
in Fig. 5 refer specifically to terrestrial atom interferometers operating on
the 698 nm 5s2 ${}^{1}S_{0}\leftrightarrow$ 5s5p ${}^{3}P_{0}$ clock
transition in 87Sr assuming the experimental parameters detailed in Tab. 3.
These choices match those that may be implemented in future vertical atom
gradiometers such as AION. In particular, the TVLBAI-km parameters correspond
to those used in the plots presented by the terrestrial very-long-baseline
atom interferometry community in Ref. [49]. The TVLBAI-10 and TVLBAI-100
parameter sets match configurations considered by the AION collaboration in
Refs. [85] and [50] respectively. The parameters chosen for the SB-VLBAI
configuration are appropriate for a broadband search at AEDGE with the
interrogation region situated exterior to the satellites. In this respect the
values of $n$ and $T$ are selected to achieve a maximum cloud separation of
$\sim$ 100 m whilst paying heed to the constraints on the maximum number of
laser pulses (1000) and the maximum interferometer sequence duration (300 s)
that could be implemented at AEDGE as stated in Ref. [94]. For each of these
forecasts, a total campaign duration of 108 s is assumed.

Setup |  $L$ (m) |  $T$ (s) | $n$ |  $\sqrt{S_{\textnormal{n}}}$ (Hz-1/2) |  $\Delta z$ (m)
---|---|---|---|---|---
TVLBAI-10 | 10 | 0.3 | 1000 | 10-4 | 7.8
TVLBAI-100 | 100 | 1.4 | 1000 | 10-4 | 90
TVLBAI-km | 1000 | 1.7 | 2500 | $10^{-5}$ | 970
SB-VLBAI | 4.4 $\times 10^{7}$ | 100 | 150 | $10^{-5}$ | 4.4$\times 10^{7}$
Table 3: The parameters used in the 87Sr source terrestrial and space-based
very-long-baseline atom interferometer sensitivity curves displayed in Fig. 2
and Fig. 5. These values correspond to configurations considered by proposed
long-baseline atom interferometry experiments as detailed in the text.

The projections are calculated using Eq. 15 but with
$\overline{\Delta\omega}_{N}$ replaced by

$\overline{\Delta\omega}_{A}=\omega_{A}\kappa\phi_{0}\left(2+\xi_{A}\right)d_{e}~{},$
(42)

where $\omega_{A}$ = 1.78 eV [132] and $\xi_{A}\sim 0.06$ [133].

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