11institutetext: Theoretical Physics Department, CERN, 1 Esplanade des
Particules, CH-1211 Geneva 23, Switzerland22institutetext: Instituto de Física
Corpuscular (IFIC), Universitat de València-CSIC, Parc Científic UV, C/
Catedrático José Beltrán 2, E-46980 Paterna, Spain33institutetext: Department
of Physics & IPAP & Lab for Dark Universe, Yonsei University, Seoul 03722,
Korea

# Symmetries and Selection Rules: Optimising Axion Haloscopes for
Gravitational Wave Searches

Valerie Domcke, 2 Camilo Garcia-Cely, 1,3 Sung Mook Lee, 1 Nicholas L. Rodd
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In the presence of electromagnetic fields, both axions and gravitational waves
(GWs) induce oscillating magnetic fields: a potentially detectable fingerprint
of their presence. We demonstrate that the response is largely dictated by the
symmetries of the instruments used to search for it. Focussing on low mass
axion haloscopes, we derive selection rules that determine the parametric
sensitivity of different detector geometries to axions and GWs, and which
further reveal how to optimise the experimental geometry to maximise both
signals. The formalism allows us to forecast the optimal sensitivity to GWs in
the range of 100 kHz to 100 MHz for instruments such as ABRACADABRA, BASE,
ADMX SLIC, SHAFT, WISPLC, and DMRadio.

††preprint: CERN-TH-2023-093

##  1 Introduction

Gravitational wave (GW) experiments have begun to probe the GW spectrum over a
vast range, from the Gigaparsec wavelengths probed by the CMB Planck:2018jri
to thousands of kilometers, covered by current ground-based interferometers
which operate in the 100 Hz range aLIGO:2020wna , yielding fundamental
insights into cosmology, astrophysics, and particle physics. Reaching even
higher frequencies poses a significant experimental challenge, but would
represent a unique opportunity to probe possible extensions of the Standard
Models of particle physics and cosmology Aggarwal:2020olq . A cosmological
source of GWs produced at a temperature $T_{*}$ could generate a stochastic GW
background at frequencies of $f\gtrsim 1\,{\rm kHz}~{}(T_{*}/10^{10}\,{\rm
GeV})$, and leave a signature from modifications to the Standard Model at the
highest temperatures. Unfortunately, probing relics of a possible high-
temperature phase of the early Universe is fundamentally challenging.
Experimental sensitivity to GWs can be expressed in terms of the strain $h$.
As the energy density in GWs scales as $\rho\sim h^{2}f^{2}M_{\rm Pl}^{2}$, at
higher frequencies even greater reach in terms of $h$ is required to reach
energy densities below the current bounds on the total energy in radiation in
the early Universe derived from BBN and CMB observations. Instead, exotic
astrophysical events sourcing transient signals appear to be a more promising
medium-term target Franciolini:2022htd ; Domcke:2022rgu . For example, the
merger of two equal mass objects yields GWs at $f\sim 1\,{\rm
kHz}~{}(M_{\odot}/m)$, so that sources such as primordial black holes with
$m\ll M_{\odot}$ could populate the high frequency landscape.

In recent years, significant progress has been made in understanding the
sensitivity of electromagnetic GW detectors in this frequency regime. In a
flat spacetime perturbed by a gravitational wave,
$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$, the usual expressions for
electrodynamics in flat spacetime receive corrections of the schematic form
$\sim hF^{2}$, yielding a graviton-two-photon vertex. As long appreciated,
this interaction can lead to photon-GW mixing Gertsenshtein ; Boccaletti ;
Raffelt:1987im . More generally, however, a GW in the presence of an
electromagnetic background will induce an electromagnetic response, in close
analogy to the signal from axion (scalar) dark-matter arising from the
coupling $aF\tilde{F}$ ($\varphi F^{2}$). Exploiting the considerable
experimental efforts to search for an electromagnetic response from wave-like
dark matter, it has been shown that these same instruments can be used as GW
telescopes Ejlli:2019bqj ; Berlin:2021txa ; Domcke:2022rgu ; Herman:2020wao ;
Gao:2023gph . Largely motivated by the QCD axion, dark matter searches focus
on signals of a MHz or above, and are therefore naturally suited to look for
high-frequency GWs.

In this paper, we will continue the study of the sensitivity of axion
haloscopes to GWs, with a particular focus on instruments operating in the
“low-mass” magnetoquasistatic regime, and sensitivity in the MHz-GHz window;
experiments already operating in this range include ABRACADABRA Kahn:2016aff ;
Ouellet:2018beu ; Salemi:2021gck , ADMX SLIC Crisosto:2019fcj , BASE
Devlin:2021fpq , SHAFT Gramolin:2020ict , and WISPLC Zhang:2021bpa . These
devices feature a strong static magnetic field which in the presence of an
axion – or a GW – sources a small, oscillating induced magnetic field which is
captured by a suitably placed pickup loop. Many of the existing instruments
are effectively prototypes, with a sensitivity that can be improved
significantly by increasing the volume of the magnetic field, and by reading
out the magnetic flux induced in the pickup loop resonantly. By combining both
of these improvements, the goal of the DMRadio collaboration is to reach the
QCD axion prediction for ${\rm neV}\lesssim m_{a}\lesssim\mu{\rm eV}$
DMRadio:2022jfv ; DMRadio:2022pkf ; DMRadio:2023igr . In view of the expected
progress, it is timely to consider how synergies in axion and GW searches can
be optimally exploited, in particular in view of different detector geometries
currently proposed for axion searches.

Reference Domcke:2022rgu  first proposed the use of low-mass axion haloscopes
as GW detectors, and demonstrated that a toroidal magnetic field – as employed
by ABRACADABRA, SHAFT, and the upcoming DMRadio-50L – could detect a passing
GW. Here, we generalise that analysis to additional detector geometries, with
a particular focus on the solenoidal magnetic field used by ADMX SLIC, BASE,
WISPLC and which as been moreover proposed for DMRadio-m3. We provide
analytical expressions for the effective current which the GW sources, the
resulting induced magnetic field, as well as for resulting magnetic flux for
various pickup loop geometries. Armed with these results, we will bootstrap
the expected sensitivities to GW signals from axion searches. A further
improvement over Ref. Domcke:2022rgu  is a careful treatment of the different
timescales involved, in particular the potentially short duration of the GW
signal.

Whilst the GW sensitivity for a solenoidal magnetic field is a practical
result, as for the toroidal magnetic field, the calculation remains involved,
and ultimately it becomes inefficient to compute the GW interaction with all
possible magnetic field geometries. Motivated by this, we derive a series of
symmetry based selection rules, which determine the parametric sensitivity to
a GW signal depending upon the symmetries of the experimental magnetic field
and the pickup loop used to read out the signal. From these results, we will
demonstrate that configurations with a high degree of symmetry can kill the
leading order sensitivity to a GW, even though they may be desirable to
maximise the axion sensitivity. An analogue of this was already observed in
Ref. Domcke:2022rgu , where it was shown that if the flux from a toroidal
magnetic field is read out through a circular pickup loop, then the leading
order GW sensitivity, expected at ${\cal O}[(\omega L)^{2}]$, vanishes, while
sensitivity at ${\cal O}[(\omega L)^{3}]$ remains. Here $\omega$ is the
angular frequency of the GW, $L$ is a characteristic length scale for the
experiment, and in the magnetoquasistatic regime of interest for low-mass
axion haloscopes, $\omega L\ll 1$. We show that if both the external magnetic
field and the pickup loop have cylindrical symmetry, i.e. if they are
invariant under azimuthal rotations and reflections in the $z$ coordinate, any
orientation of the pickup loop which is sensitive to the axion suffers from a
cancellation of the leading order term ($\propto(\omega L)^{2}$) for the GW
signal. This symmetry is commonly exhibited by axion haloscopes, which make
use of solenoidal or toroidal magnetic fields. To recover the dominant
scaling, the cylindrical symmetry must be broken, for instance through the
placement or geometry of the pickup loop. The latter can be most easily
achieved by modifying the pickup loop to span only a fraction of the azimuthal
angle, with the optimal GW sensitivity obtained when the cylindrical symmetry
for the pickup loop is maximally broken (for instance, with a figure-8
configuration as in Ref. Domcke:2022rgu ).

For existing experiments, as the largest axion signal is obtained for
detectors with full cylindrical symmetry, this explains the observation in
Ref. Domcke:2022rgu  that the optimal axion and GW sensitivities cannot be
simultaneously obtained for a haloscope based on a toroidal magnetic field,
and furthermore demonstrates that this conclusion is generic. Nevertheless, we
find that modifying the pickup loop geometry (or including several different
pickup loops) allows one to obtain sensitivity to both the axion and GW
signal, in a manner that at worst reduces the axion sensitivity by an ${\cal
O}(1)$ amount.111This same approach would also allow for discrimination
between a GW and axion signal. Of course, we note that there are many ways to
distinguish these signals, the most important being that in the accessible
parameter space the GW signal will be transient, whereas that from dark matter
is persistent. We illustrate the power of symmetry arguments by determining
the leading power in $(\omega L)$ sensitivity for a range of different
detector geometries without explicit computation, in view of determining the
optimal geometries for GW searches. For the most relevant cases, we provide
the computation to confirm our results.

At the outset, we can already provide an intuitive argument as to why
cancellations in highly symmetric detectors might be expected. To do so,
rather than contrasting GW and axion electrodynamics, as we will in the
remainder of the paper, let us consider a simpler comparison: a scalar versus
a pseudoscalar. In particular, consider first the induced magnetic field
arising from the interaction of a toroidal magnet, ${\bf B}=B_{0}\hat{\bf
e}_{\phi}$, with a pseudoscalar via the interaction, $gaF\tilde{F}$. If we
consider the induced magnetic field in the $z$ direction at the center of the
toroid, as measured by the ABRACADABRA collaboration, we find $B^{a}_{z}\sim
g(\partial a)B_{0}L$. The consistent transformation of this result under
parity, which can be confirmed directly, is critically reliant on the
pseudoscalar nature of the axion. Indeed, if we ask what the induced field
would be for a scalar interaction, $g\varphi F^{2}$, there is no expression we
can write consistent with parity and the cylindrical symmetry of the
instrument. An explicit computation confirms that $B^{\varphi}_{z}=0$. This
argument can be formalised into symmetry based selection rules which determine
the geometries that are sensitive to scalar versus pseudoscalar coupling –
indeed, there are configurations where $B_{a}=0$ whilst $B_{\varphi}\neq 0$ –
and we undertake that exercise in App. B. The general lesson, however, is that
highly symmetric detectors impose symmetry constraints on the induced fields
that can be so restrictive that the measurable signal vanishes. This is true
also for GWs, and we will determine an appropriate set of selection rules to
determine the interplay between signals and geometry.

We can actually determine an additional general lesson by comparing the scalar
and pseudoscalar interaction. As is well known, the axion interaction
generates an effective current proportional to
$\partial_{\nu}(a\tilde{F}^{\nu\mu})=(\partial_{\nu}a)\tilde{F}^{\nu\mu}$, so
that the interactions depends only on a derivative of the axion, as expected
for a pseudo-goldstone boson. The equivalent expression for a scalar is
$\partial_{\nu}(\varphi F^{\nu\mu})=(\partial_{\nu}\varphi)F^{\nu\mu}-\varphi
j^{\mu}$, where $j^{\mu}=\partial_{\nu}F^{\mu\nu}$ is the current that
generates the leading order fields in the laboratory. Accordingly, for the
scalar there is an additional contribution to the effective current localised
at the boundary of the magnetic volume, which turns out to be generic: it will
be present also for the GW, although it has so far been overlooked in the
literature. This contribution can be interpreted as an effective current at
the boundary of the magnetic volume, determined by the component of the
effective magnetisation vector (introduced in Ref. Domcke:2022rgu  for GWs)
parallel to the boundary surface.

In the remainder of this paper we will flesh out these ideas for the GW
signal, and we organise our discussion as follows. Section 2 lays out the
theoretical framework for our work, reviewing the relevant aspects of
electrodynamics in a spacetime perturbed by a GW. Several points, such as a
discussion of the symmetry properties of the induced magnetic field and
response matrix formalism are presented here for the first time in this
context. This sets the stage for deriving the GW sensitivity of axion
haloscopes with solenoidal magnetic field configurations in Sec. 3. The
results are then generalised in Sec. 4 where we derive symmetry principles
which allow us to determine the parametric scaling of the GW sensitivity for
various detector geometries without explicit computation. The symmetry
arguments will then enable us to draw general conclusions about the optimal
strategy for axion and GW searches in axion haloscopes.

Many details of our analyses are deferred to appendices. Appendix A reviews
Maxwell’s equation in curved space time. Within it, we provide a careful
derivation of the main equations governing the interaction of a GW with a
background electromagnetism (EM) field, the derivation of the effective
surface current, and an explanation of why the GW effects we consider scale at
lowest order as $(\omega L)^{2}$. In App. B we study scalar and axion
electrodynamics, with a focus on sharpening an analogy to the GW case. We will
explain how our GW selection rules extend to these spin-0 waves, and the
consequences for various detector geometries. In App. C we summarise the
symmetry properties of the cylindrical magnetic field configurations employed
by axion haloscopes, and demonstrate that they can be decomposed into a
solenoidal and toroidal component. Appendix D expands our discussion of the
response matrix formalism used to describe the detector response to a passing
GW. In App. E we summarise the explicit analytical expressions for all
components of the effective current induced by a GW, up to order $(\omega
L)^{3}$ and for both toroidal and solenoidal external magnetic field
configurations. These expressions may be used as input for full detector
simulations, or for detailed numerical calculations of the relevant GW
effects. In App. F we discuss in detail the bootstrapping of axion search
results to establish GW sensitivity, carefully taking into account the
different time scales involved in the possible signals and detectors. The
appendix further discusses details of several possible sources for high-
frequency GWs. Finally, App. G is dedicated to the possibility of using an
external electric instead of magnetic field for GW detection, and demonstrates
how our symmetry arguments extend to this case.

##  2 Gravitational Wave Electrodynamics

To begin with, we review the general formalism used to compute the magnetic
flux induced by a GW passing through a lumped-element circuit axion haloscope.
We will review the discussion of Ref. Domcke:2022rgu  (see also Ref.
Berlin:2021txa ), pointing out an additional contribution to the induced
magnetic flux due to effective surface currents at the boundary of the
magnetic volume, which was previously overlooked. We then extend this approach
to account for the transformation properties and symmetries of the various
quantities, in particular the induced magnetic field, under rotations and
reflections. The axion haloscopes targeting the magneto-quasistatic regime
($m_{a}\lesssim\mu{\rm eV}$) generally have a high degree of cylindrical
symmetry, and we will study the impact of this on the GW signal
systematically. Doing so will allow us to develop a systematic approach to the
geometries of the external background magnetic field and pickup loop, and
resolve fundamental questions such as determining the optimal geometry for GW
and axion searches.

###  2.1 Proper detector frame

Throughout this paper we will work in the proper detector frame,222This is in
contrast to the transverse traceless (TT) frame, in which coordinate distances
are set by the geodesics of free-falling test masses, and a rigid instrument
and experimental magnetic field no longer have a simple description. in which
coordinate distances to the origin match the proper distance, and thus
coincide with those measured by ideal rigid rulers. As a consequence of this,
up to non-inertial forces such as those associated with the rotation of the
Earth (which can be neglected at high frequencies Ni:1978zz ; Maggiore:2007ulw
), the effect of GWs is simply given by a small Newtonian force proportional
to their amplitude. Throughout this paper, we will assume that these GW forces
do not mechanically deform the experimental setup, in particular the static
electromagnetic fields applied in the experiment remain static in the presence
of a GW. Critically, this implies that in the proper detector frame the
experimentally generated magnetic field coincides with that of flat spacetime,
see App. A for details. This assumption is in particular valid for GW
frequencies below the mechanical resonance frequencies of the setup. At
frequencies around and above the lowest mechanical resonance $\omega_{0}\sim
v_{s}/L$, with $v_{s}$ denoting the speed of sound in the material, the
Newtonian GW force is no longer negligible Berlin:2023grv ; Bringmann:2023gba
. We expect this to impact part of the parameter space relevant for the
experimental setups discussed here, and we leave a quantitative analysis to
future work. Interestingly, in the case of microwave cavities, it was
demonstrated that this effect can enhance the GW sensitivity Berlin:2023grv .

Expanding the metric as $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ with
$\eta_{\mu\nu}$ denoting the flat metric with sign convention $(-+++)$, in the
proper detector frame the GW at the position ${\bf r}$ can be expressed as
Berlin:2021txa ; Domcke:2022rgu ,333 This expression is equivalent to Eq. (S5)
in Ref. Domcke:2022rgu , as can be shown using the completeness relation
$\hat{k}_{i}\hat{k}_{j}+\hat{U}_{i}\hat{U}_{j}+\hat{V}_{i}\hat{V}_{j}=\delta_{ij}$
in the third line of Eq. (1).

$\displaystyle h_{00}$ $\displaystyle=\omega^{2}e^{-\mathrm{i}\mkern
1.0mu\omega t}F({\bf k}\cdot{\bf
r})\,r_{m}r_{n}\sum_{A=+,\times}h^{A}e^{A}_{mn}(\hat{\bf k}),$ (1)
$\displaystyle h_{0i}$
$\displaystyle=\frac{1}{2}\omega^{2}e^{-\mathrm{i}\mkern 1.0mu\omega t}[F({\bf
k}\cdot{\bf r})-\mathrm{i}\mkern 1.0muF^{\prime}({\bf k}\cdot{\bf
r})][\hat{\bf k}\cdot{\bf
r}~{}r_{m}\delta_{ni}-r_{m}r_{n}\hat{k}_{i}]\sum_{A=+,\times}h^{A}e^{A}_{mn}(\hat{\bf
k}),$ $\displaystyle h_{ij}$ $\displaystyle=-\mathrm{i}\mkern
1.0mu\omega^{2}e^{-\mathrm{i}\mkern 1.0mu\omega t}F^{\prime}({\bf k}\cdot{\bf
r})[|{\bf
r}|^{2}\delta_{im}\delta_{jn}+r_{m}r_{n}\delta_{ij}-r_{n}r_{j}\delta_{im}-r_{m}r_{i}\delta_{jn}]\sum_{A=+,\times}h^{A}e^{A}_{mn}(\hat{\bf
k}),$

where $F(\xi)=[e^{\mathrm{i}\mkern 1.0mu\xi}-1-\mathrm{i}\mkern
1.0mu\xi]/\xi^{2}$, $h^{+,\times}$ denotes the amplitude of the two GW
polarisations, and the polarisation tensor $e_{ij}^{+,\times}(\hat{\bf k})$
for a given direction of GW propagation $\hat{\bf k}=\sin\theta_{h}\,\hat{\bf
e}^{\phi_{h}}_{\rho}+\cos\theta_{h}\,\hat{\bf e}_{z}$ can be defined as

$\displaystyle
e_{ij}^{+}=\frac{1}{\sqrt{2}}\left[\hat{U}_{i}\hat{U}_{j}-\hat{V}_{i}\hat{V}_{j}\right]\\!,$
$\displaystyle
e_{ij}^{\times}=\frac{1}{\sqrt{2}}\left[\hat{U}_{i}\hat{V}_{j}+\hat{V}_{i}\hat{U}_{j}\right]\\!,$
$\displaystyle\hat{\bf V}=\hat{\bf e}_{\phi}^{\phi_{h}},$
$\displaystyle\hat{\bf U}=\hat{\bf V}\times\hat{\bf k}.$ (2)

Here $\theta_{h}$ and $\phi_{h}$ denote the azimuthal and polar angle of the
GW, and $\hat{\bf e}^{\phi_{h}}_{\rho}$, $\hat{\bf e}_{\phi}^{\phi_{h}}$ and
$\hat{\bf e}_{z}$ denote unit vectors in the radial, angular and vertical
direction for a polar angle $\phi_{h}$, with both coordinate systems defined
with origin at the center of the experiment. In particular, note that $h_{\mu
i}r_{i}=0$, and consequently
$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}$ for
$dx^{\mu}=(0,dr\,\hat{{\bf r}})$. From this we see that coordinate distances
to the origin coincide with the corresponding proper distance, a defining
characteristic of the proper detector frame Manasse:1963zz .

In this work, we will limit ourselves to the regime of $\omega L\ll 1$, as
appropriate over most of the range covered by lumped-element circuit
instruments.444For the result when taking $\omega L\sim 1$ for the axion
induced signal in these instruments, see Ref. Benabou:2022qpv . We can
therefore treat $\omega L$ as a perturbative parameter, and will do so often,
for instance it will be implicit in our use of the Biot-Savart law and used
throughout our discussion of the implications of the symmetry transformations.
Further, as $F(\xi)=-\tfrac{1}{2}+{\cal O}(\xi)$, it follows from Eq. (1) that
$h_{\mu\nu}$ in the proper detector frame has a leading order contribution at
${\cal O}[(\omega L)^{2}]$. The absence of any contribution at ${\cal
O}[\omega L]$ is a consequence of working in a freely falling reference frame
assumed to be rigid, as we demonstrate in App. A.4. An immediate implication
of this scaling is that for a GW incident on an electromagnetic field that is
static in the proper detector frame, the leading order electromagnetic
response induced will scale as ${\cal O}[(\omega L)^{2}]$.555By gauge
invariance, the same must also be true in the TT frame, where $h\propto
e^{-\mathrm{i}\mkern 1.0mu\omega t}$ and therefore has contributions at all
orders in $\omega L$. This implies there must be a detailed cancellation of
the linear frequency contribution, and the need to keep track of this
highlights the advantage of working in the proper detector frame. For
additional discussion, see Ref. Berlin:2021txa . This demonstrates that the
optimal observables for the GW one can construct will also be at ${\cal
O}[(\omega L)^{2}]$.

As outlined in the introduction, one of the primary goals of the present work
is to understand the role symmetry plays in the GW interactions. In
particular, we will be studying the interaction between a GW and detectors
with a high degree of symmetry. Existing axion instruments tend to have full
cylindrical symmetry, that is, invariance under rotations about $\hat{\bf
e}_{z}$ with an angle $\varphi$, $R_{z}(\varphi)$, and arbitrary reflections,
$P_{\alpha}$, with $\alpha=x,y,z$. Therefore, it is worthwhile already to
characterise the transformation of the GW polarisations and proper detector
frame components when these transformations are applied to the position and
incident direction at which we evaluate these quantities. To begin with,

$e_{ij}^{A}(P_{\alpha}\hat{\bf
k})=\sigma\,[P_{\alpha}]_{ik}\,e_{kl}^{A}(\hat{\bf
k})\,[P_{\alpha}]_{lj},\hskip 14.22636pte_{ij}^{A}(R_{z}\hat{\bf
k})=[R_{z}]_{ik}\,e_{kl}^{A}(\hat{\bf k})\,[R_{z}]_{lj}.$ (3)

Here, $[P_{\alpha}]_{ij}$ is a $3\times 3$ matrix corresponding to the
reflection of the $\alpha$-component, and $[R_{z}]_{ij}$ is similarly the
matrix describing the rotation about $\hat{\bf e}_{z}$.666We emphasise that
$e_{ij}^{A}$ transforms under general rotations as a tensor only up to gauge
transformations Weinberg:1964ew . From the definitions in Eq. (2), however,
the polarisation tensors are true tensors under rotations about $\hat{\bf
e}_{z}$. We have further introduced $\sigma=+1$ ($\sigma=-1$) for the $A=+$
($A=\times$) polarisation, which keeps track of their different
transformations under the reflections. From these results, we conclude

$\displaystyle h_{00}(P_{\alpha}{\bf r},P_{\alpha}{\bf k})=\sigma\,h_{00}({\bf
r,k}),\hskip 14.22636pth_{0i}(P_{\alpha}{\bf r},P_{\alpha}{\bf
k})=\sigma\,[P_{\alpha}]_{ij}\,h_{0j}({\bf r,k}),$ (4) $\displaystyle\hskip
56.9055pth_{ij}(P_{\alpha}{\bf r},P_{\alpha}{\bf
k})=\sigma\,[P_{\alpha}]_{ik}\,h_{kl}({\bf r,k})\,[P_{\alpha}^{T}]_{lj},$

and $h_{\mu\nu}$ transforms as a regular tensor under rotations about the
$z$-axis.

###  2.2 Effective current induced by GWs

The interaction of GWs with electromagnetic fields can be effectively
described as an additional current augmenting Maxwell’s equations in a flat
spacetime. Specifically,

$\partial_{\nu}F^{\mu\nu}=j^{\mu}+j_{\rm eff}^{\mu},\hskip
14.22636pt\partial_{\nu}F_{\alpha\beta}+\partial_{\alpha}F_{\beta\nu}+\partial_{\beta}F_{\nu\alpha}=0,$
(5)

where $j^{\mu}$ is the electromagnetic current in the absence of the GW (i.e.
the ordinary currents in flat spacetime) whereas the effective current can be
written as

$j_{\rm
eff}^{\mu}\equiv\partial_{\nu}\left(-\frac{1}{2}hF^{\mu\nu}+F^{\mu\alpha}{h^{\nu}}_{\alpha}-F^{\nu\alpha}{h^{\mu}}_{\alpha}\right)\\!,$
(6)

with $h\equiv{h^{\mu}}_{\mu}$. This is derived in the App. A (see also Refs.
Herman:2020wao ; Berlin:2021txa ; Domcke:2022rgu ), where we also discuss why
the second equation in Eq. (5) – the homogeneous Maxwell’s equations – are
unaffected by the presence of the GW. Throughout this paper we will be working
to linear order in $h$, so that $F^{\mu\nu}$ as it appears on the right-hand
side of this equation contains only the background fields. In further analogy
to EM, one can define Domcke:2022rgu

$\displaystyle P_{i}$ $\displaystyle\equiv-
h_{ij}E_{j}+\frac{1}{2}hE_{i}+h_{00}E_{i}-\epsilon_{ijk}h_{0j}B_{k},$ (7)
$\displaystyle M_{i}$ $\displaystyle\equiv-
h_{ij}B_{j}-\frac{1}{2}hB_{i}+h_{jj}B_{i}+\epsilon_{ijk}h_{0j}E_{k},$

so that

$j_{\rm eff}^{\mu}=\left(-\nabla\cdot{\bf P},\,\nabla\times{\bf
M}+\partial_{t}{\bf P}\right)\\!.$ (8)

This final formulation is reminiscent of polarisation and magnetisation
vectors for EM in a medium. Hence, the task of calculating the electromagnetic
fields induced by a GW is equivalent to performing standard EM calculations in
such media. If we consider the leading order effect in ${\cal O}[(\omega
L)^{2}]$, we can already see a simplification when the external field is
purely magnetic: the spatial part of the current will be generated only by
${\bf M}$, and be sourced by $h_{00}$ and $h_{ij}$ but not $h_{0i}$, as the
time derivative acting on ${\bf P}$ ensures it will be higher order.

Up to this point, we have not specified the geometry of the background fields
or the pickup loop that will be used to measure the induced fields. The
majority of the axion haloscopes in consideration exploit solely an external
magnetic field with full cylindrical symmetry. For such magnetic field
configurations, as shown in App. C, it is possible to decompose the background
magnetic field into a solenoidal and toroidal piece, ${\bf B}({\bf r})={\bf
B}^{{(s)}}({\bf r})+{\bf B}^{{(t)}}({\bf r})$. We will further assume the
field depends only on $\rho$, namely $|{\bf B}({\bf r})|=B(\rho)$ for an
unspecified function $B$. The benefit of this decomposition is that ${\bf
B}^{{(s)}}({\bf r})\propto\hat{\bf e}_{z}$ and ${\bf B}^{{(t)}}({\bf
r})\propto\hat{\bf e}_{\phi}$, and each component has a well-defined set of
transformations under reflections, which we keep track of through a parameter
$\eta_{\alpha}$, defined through ${\bf B}(P_{\alpha}{\bf
r})=\eta_{\alpha}P_{\alpha}{\bf B}({\bf r})$. In addition to the partial
reflections, we will track the transformation under the full parity
transformation $P=P_{x}P_{y}P_{z}$, ${\bf B}(P{\bf r})=\eta P{\bf B}({\bf
r})$, where consistency requires $\eta=\eta_{x}\eta_{y}\eta_{z}$.

For each pair of magnetic field configuration and spatial reflection, explicit
values of $\eta_{\alpha}$ are summarised in Tab. 1. Each detector
configuration is usually associated uniquely with either ${\bf B}^{{(s)}}$ or
${\bf B}^{{(t)}}$. This is certainly true for the existing and planned axion
haloscopes we consider. Accordingly, we will suppress the superscripts $(s)$
or $(t)$ moving forward. Combining the transformation properties of the
magnetic field with Eq. (4), the spatial part of the induced current – which
fully determines the induced magnetic field – then obeys

${\bf j}_{\rm eff}\left(P_{\alpha}{\bf r},\,P_{\alpha}{\bf
k}\right)=-\sigma\eta_{\alpha}P_{\alpha}\,{\bf j}_{\rm eff}({\bf r},{\bf k}),$
(9)

and ${\bf j}_{\rm eff}\left(R_{z}{\bf r},R_{z}{\bf k}\right)=R_{z}\,{\bf
j}_{\rm eff}$. We emphasise this transformation holds for both the ${\bf M}$
and ${\bf P}$ contributions to ${\bf j}_{\rm eff}$ separately.

|  ${\bf B}^{{(s)}}\propto\hat{\bf e}_{z}$ (solenoid) |  ${\bf B}^{{(t)}}\propto\hat{\bf e}_{\phi}$ (toroid) | $\hat{\bf n}^{\prime}\propto\hat{\bf e}_{z}$ | $\hat{\bf n}^{\prime}\propto\hat{\bf e}_{\phi}$ | $\hat{\bf n}^{\prime}\propto\hat{\bf e}_{\rho}$
---|---|---|---|---|---
$P_{x}$ | $+1$ | $-1$ | $+1$ | $-1$ | $+1$
$P_{y}$ | $+1$ | $-1$ | $+1$ | $-1$ | $+1$
$P_{z}$ | $-1$ | $+1$ | $-1$ | $+1$ | $+1$
$P$ | $-1$ | $+1$ | $-1$ | $+1$ | $+1$
Table 1: The transformation of a solenoidal and toroidal magnetic field
configuration as well as various pickup loop orientations under reflections.
For the magnetic field, the values in the table determine $\eta_{\alpha}$ as
defined in ${\bf B}^{(s,t)}(P_{\alpha}{\bf r})=\eta_{\alpha}P_{\alpha}{\bf
B}^{(s,t)}({\bf r})$, or $\eta$ in the case of the complete parity
transformation $P=P_{x}P_{y}P_{z}$. The transformation of the normal vector to
the pickup loop, $\hat{\bf n}^{\prime}$, are collected in the final three
columns, with $\hat{\bf n}^{\prime}(P_{\alpha}{\bf
r}^{\prime})=\kappa_{\alpha}P_{\alpha}\hat{\bf n}^{\prime}({\bf r}^{\prime})$,
and $\kappa=\kappa_{x}\kappa_{y}\kappa_{z}$. See text for details.

Before we move on to consider the fields generated by ${\bf j}_{\rm eff}$, we
note that from the above discussion we can see the presence of a boundary
contribution that has previously been overlooked (for instance, in Ref.
Domcke:2022rgu ). Recall that at the interface of two bodies with different
values of the magnetisation vector ${\bf M}$, Maxwell’s equations predict a
surface current proportional to $\hat{\bf n}\times\Delta{\bf M}$, where
$\hat{\bf n}$ is the unit vector normal to the surface. For the external
magnetic fields considered in this work, the GW effective magnetisation ${\bf
M}$ in Eq. (7) sharply drops to zero at cylindrical surfaces, where the
external magnetic fields vanish. If this occurs at a radius $\rho=R$, the GW
generates an effective surface current given by ${\bf j}_{\rm
eff}^{(t,s)}=\pm\delta(\rho-R)\,\hat{\bf e}_{\rho}\times{\bf M}$, which must
be accounted for. As already noted, such a contribution does not occur for an
axion, but does for a scalar coupled to EM. (For the axion, such a
contribution will not occur as $\Delta{\bf M}\propto\Delta{\bf
E}\propto\hat{\bf n}$ at any surface. Explicit calculations are provided in
App. B.)

###  2.3 The induced magnetic field

The effective current induced by the GW will source an induced magnetic field,
${\bf B}_{h}$, which is determined by the Biot-Savart law,777This approach is
valid only to ${\cal O}[(\omega L)^{3}]$, see Ref. Domcke:2022rgu  for a
discussion.

${\bf B}_{h}({\bf r}^{\prime},{\bf k})=\frac{1}{4\pi}\int_{V_{B}}d^{3}{\bf
r}~{}\frac{{\bf j}_{\rm eff}({\bf r},{\bf k})\times({\bf r}^{\prime}-{\bf
r})}{|{\bf r}^{\prime}-{\bf r}|^{3}},$ (10)

where $V_{B}$ denotes the detector volume filled by the external magnetic
field. Here and throughout, we use ${\bf r}^{\prime}$ to indicate the position
where we evaluate the induced field, and the pickup loop inserted to measure
that field will integrate over this variable, whereas ${\bf r}$ is where we
evaluate the effective current. Under the assumption that the integration
region $V_{B}$ is invariant under $P_{\alpha}$ – which it is for the
cylindrically symmetric detectors we consider – the Biot-Savart law Eq. (10)
together with Eq. (9) implies the transformation of the induced magnetic field
as,

${\bf B}_{h}(P_{\alpha}{\bf r}^{\prime},P_{\alpha}{\bf
k})=\sigma\eta_{\alpha}P_{\alpha}\,{\bf B}_{h}({\bf r}^{\prime},{\bf k}),$
(11)

and correspondingly ${\bf B}_{h}(R_{z}{\bf r}^{\prime},R_{z}{\bf
k})=R_{z}\,{\bf B}_{h}({\bf r}^{\prime},{\bf k})$. The result in Eq. (11) will
be a key tool in studying the implications of detector symmetry for the
associated GW signal, which we consider in detail in Sec. 4.

We now have all the ingredients to compute the effect of GWs on the observable
used in low-mass axion haloscopes, namely the induced magnetic flux through a
suitably placed pickup loop,

$\Phi_{h}=\int_{A_{\ell}}d^{2}{\bf r}^{\prime}\,{\bf B}_{h}({\bf
r}^{\prime})\cdot\hat{\bf n}^{\prime}({\bf r}^{\prime}),$ (12)

with $A_{\ell}$ the surface enclosed by the pickup loop, and $\hat{\bf
n}^{\prime}({\bf r}^{\prime})$ the unit normal vector to that surface. Our
symmetry arguments will also depend on the transformation of the pickup loop
under reflections, which in direct analogy to the transformation of the
magnetic field we will trace using

$\hat{\bf n}^{\prime}(P_{\alpha}{\bf
r}^{\prime})=\kappa_{\alpha}P_{\alpha}\hat{\bf n}^{\prime}({\bf r}^{\prime}),$
(13)

with the different possible values collected in Tab. 1. In Sec. 3, we will
provide explicit expression of $\Phi_{h}$ for solenoidal geometries and review
how to set constraints on GW signals by recasting the results of axion
experiments.

Rather than employing Eqs. (10) and (12), one can also work directly with the
vector potential,

${\bf A}_{h}({\bf r}^{\prime})=\frac{1}{4\pi}\int d^{3}{\bf r}\,\frac{{\bf
j}_{\rm eff}({\bf r})}{|{\bf r}^{\prime}-{\bf r}|},\hskip
14.22636pt\Phi_{h}=\int_{\ell}d{\bf r}^{\prime}\cdot{\bf A}_{h}({\bf
r}^{\prime}),$ (14)

where $\ell$ is the closed curve describing the pickup loop. This approach
reduces the dimension of the flux integration by one, thereby simplifying
several analytical calculations. Nonetheless, the symmetry based arguments are
often more intuitive when expressed in terms of the magnetic field. We will
use both formalisms as needed.

Moreover, we introduce the response matrix, $D^{mn}({\bf k})$, to study the
dependence of the flux on the polarisation. The response matrix exploits the
observation that the flux in Eq. (12) is a linear functional of the GW, which
in turn is linear in the polarisation tensors, $e^{A}_{mn}$. Hence, there must
exist a matrix $D^{mn}({\bf k})$ such that

$\Phi_{h}=e^{-\mathrm{i}\mkern 1.0mu\omega t}D^{mn}({\bf
k})\sum_{A}h^{A}e^{A}_{mn}(\hat{{\bf k}}).$ (15)

A more detailed discussion of the response matrix and explicit expressions for
$D^{mn}({\bf k})$ are provided in App. D.

Using the response matrix, for a given GW wave vector ${\bf k}$ and
polarisation $A$, we can construct pattern functions $D^{mn}({\bf
k})e^{A}_{mn}$ which encode the angular response of a detector, i.e. describe
the antenna pattern relevant to determine the magnitude of the induced
magnetic flux. This is analogous to the formalism introduced for the response
of interferometers to GWs, see Ref. Maggiore:2007ulw . We emphasise that Eq.
(15) results from the fact that the magnetic flux is linear in the metric
perturbation associated with GWs.

Although more details can be found in App. D, let us here provide two remarks
on the general properties of $D^{ij}$. First, as we see from the form of Eq.
(1), the lowest order frequency contribution to $D^{ij}({\bf k})$ occurs at
${\cal O}[(\omega L)^{2}]$.888For interferometers, the observable is the GW
strain and the calculation can be performed in the TT frame, where the antenna
pattern function at leading order in frequency depends only on the direction
of the GW, ${\hat{\bf k}}$. This is a generic consequence of the use of proper
detector frame. Secondly, the matrices $D^{ij}$ are not unique, but
$D^{ij}({\bf k})\rightarrow D^{ij}({\bf k})+c^{i}k^{j}+c^{j}k^{i}$ with
constants $c^{i,j}$ gives rise to the same magnetic flux since the
polarisation tensors are transverse with respect to ${\bf k}$.

##  3 The GW Sensitivity of Solenoidal Detector Geometries

Having established the general framework of how a GW interacts with an
experimental magnetic field, we now put it to use for the explicit case of
solenoidal instrument. Previous work, see Ref. Domcke:2022rgu , focused on a
toroidal geometry for the external magnetic field, which is used for the axion
searches performed by ABRACADABRA Salemi:2021gck  and SHAFT Gramolin:2020ict .
This is modelled as

${\bf
B}=B_{\max}\frac{R}{\rho}\left[\Theta(R+a-\rho)-\Theta(R-\rho)\right]\hat{\bf
e}_{\phi},$ (16)

where $\Theta(x)$ is the Heaviside step function, which ensures the external
magnetic field only exists on $R<\rho<R+a$. However, another magnetic field
geometry that is being pursued by axion haloscopes is a soleneoidal field,

${\bf B}=B_{0}\Theta(R-\rho)\hat{\bf e}_{z}.$ (17)

Among the detectors making use of a solenoidal field, we will first focus on
instruments where the induced flux is read out through a vertical pickup loop
as depicted in Fig. 1 and implemented in the ADMX SLIC Crisosto:2019fcj  and
BASE Devlin:2021fpq  experiments. Moreover, the planned WISPLC Zhang:2021bpa
and DMRadio-$\text{m}^{3}$ DMRadio:2022pkf ; DMRadio:2023igr  experiments are
also planning to implement a related configuration.999Let us briefly comment
on several of the differences between these experiments. ADMX SLIC experiment
has a single rectangular pickup loop at a fixed polar angle $\phi_{\ell}$. The
BASE experiment relies instead on many such pickup loops placed symmetrically
in the horizontal plane, whereas DMRadio-m3 will use a full toroidal sheath.
For the WISPLC experiment, the current design features a pickup loop at fixed
$\phi_{\ell}$, but which is located outside the region of the external
magnetic fields. (Although other future instruments will use a toroidal
magnetic field, for instance DMRadio-50L.) Therefore, as a straightforward
generalisation of Ref. Domcke:2022rgu , we first calculate the expected
magnetic flux from the incoming GW for a solenoidal magnetic field and
different locations of the pickup loop. Armed with an understanding of how the
GW interacts with a detector for two explicit cases, in the next section we
will then generalise our discussion for general geometries.

Before proceeding, we note that in the main text we will only consider the
interaction between GW and laboratory magnetic fields. The rationale for this
is that axion haloscopes exclusively make use of magnetic fields, as larger
energy densities can be built up in magnetic than electric fields. Further,
the axion interaction with a magnetic field is controlled by $\partial_{t}a$,
which for dark matter is much larger than $\nabla a$, which the electric field
couples to (for a review, see App. B). As the GW is both relativistic and
couples differently than the axion, this final consideration does not apply,
and therefore for completeness we briefly discuss the interaction with an
electric field in App. G.

Figure 1: A cartoon depiction of the geometry of a solenoidal detector that
can be used to search for axions or a passing GW. We show a solenoidal
magnetic field (gray) with a rectangular vertical pickup loop (red). The
solenoid has a height $H$ and radius $R$, and throughout we will generally
assume that $H$ is parametrically larger than all other scales. The pickup
loop is located at an angle $\phi_{\ell}$, and spans over coordinates
$\rho\in[r_{1},r_{2}]$ and $z\in[-l/2,l/2]$, giving it an area
$l(r_{2}-r_{1})$. Finally, in green we depict the direction of the incident
GW, which has a wave vector ${\bf k}$, and comes in at an angle in spherical
coordinates of $\theta_{h}$ and $\phi_{h}$.

###  3.1 The GW signal for a solenoidal magnetic field

Consider first the flux $\Phi_{h}(r)$ caught by a rectangular pickup loop at
fixed polar angle $\phi_{\ell}$, radially ranging from $[0,r]$ with height
$l$, and positioned symmetrically about $z=0$. This scenario is equivalent to
that depicted in Fig. 1 with $r_{1}=0$ and $r_{2}=r$. We can then derive the
equivalent result for an arbitrary width with $r_{1}<r_{2}\leq R$ from
$\Phi_{h}(r_{2})-\Phi_{h}(r_{1})$. Further, we will consider the case where
$r\leq R$, with $R$ the radius of the detector, and $r\geq R$ separately.

For the solenoidal magnetic field in Eq. (17), we can calculate the effective
current, induced magnetic fields, and flux using the formalism of Sec. 2. As
already discussed, we will study the problem perturbatively in $\omega L\ll
1$. Complete expressions for all components of the current to order ${\cal
O}[(\omega L)^{3}]$ are provided in App. E. Here, we state the results for the
flux, which we write as a series in $\omega$ as
$\Phi_{h}=\Phi_{h}^{(2)}+\Phi_{h}^{(3)}+\cdots$, with $\Phi_{h}^{(n)}$
denoting the flux at ${\cal O}[(\omega L)^{n}]$. Explicitly, when $r\leq R$ we
have

$\Phi_{h}^{(2)}=\frac{e^{-\mathrm{i}\mkern 1.0mu\omega
t}}{144\sqrt{2}}\omega^{2}B_{0}lr\left(30R^{2}-13r^{2}\right)s_{\theta_{h}}\left(h^{+}c_{\theta_{h}}s_{\phi_{h}-\phi_{\ell}}+h^{\times}c_{\phi_{h}-\phi_{\ell}}\right)\\!,$
(18)

and

$\displaystyle\Phi_{h}^{(3)}=$ $\displaystyle-\frac{\mathrm{i}\mkern
1.0mue^{-\mathrm{i}\mkern 1.0mu\omega
t}}{2304\sqrt{2}}\omega^{3}B_{0}lr^{2}\Big{[}h^{+}c_{\theta_{h}}s_{2(\phi_{h}-\phi_{\ell})}\\{3l^{2}\\!-\\!2r^{2}\\!+\\!57R^{2}\\!+\\!(l^{2}\\!-\\!22r^{2}\\!+\\!27R^{2})c_{2\theta_{h}}\\}$
(19)
$\displaystyle+2h^{\times}\\{(l^{2}\\!+\\!2r^{2}\\!+\\!18R^{2}\\!+\\![l^{2}\\!-\\!14r^{2}\\!+\\!24R^{2}]c_{2\theta_{h}})c_{2(\phi_{h}-\phi_{\ell})}\\!+\\!6(5r^{2}\\!-\\!12R^{2})s_{\theta_{h}}^{2}\\}\Big{]},$

where we employ the shorthands $c_{x}\equiv\cos x$ and $s_{x}\equiv\sin x$.
The factor of $\mathrm{i}\mkern 1.0mu$ in $\Phi_{h}^{(3)}$ indicates that this
contribution enters with a $\pi/2$ phase shift in time as compared to
$\Phi_{h}^{(2)}$. If the loop instead is placed outside the solenoidal
magnetic field, extending over $\rho^{\prime}\in[r_{1},r_{2}]$ with
$r_{2}>r_{1}>R$, then the leading order flux is

$\Phi_{h}^{(2)}=\frac{5e^{-\mathrm{i}\mkern 1.0mu\omega
t}}{48\sqrt{2}}\omega^{2}B_{0}lR^{4}\,\frac{r_{2}-r_{1}}{r_{1}r_{2}}s_{\theta_{h}}\left(h^{+}c_{\theta_{h}}s_{\phi_{h}-\phi_{\ell}}+h^{\times}c_{\phi_{h}-\phi_{\ell}}\right)\\!,$
(20)

and

$\displaystyle\Phi_{h}^{(3)}=$ $\displaystyle\frac{\mathrm{i}\mkern
1.0mue^{-\mathrm{i}\mkern 1.0mu\omega
t}}{2304\sqrt{2}}\omega^{3}B_{0}\frac{lR^{4}}{(r_{1}r_{2})^{2}}\Bigg{[}2h^{\times}\Bigg{\\{}-72(r_{1}r_{2})^{2}\ln\frac{r_{1}}{r_{2}}s_{\theta_{h}}^{2}$
(21) $\displaystyle\hskip
34.14322pt+c_{2(\phi_{h}-\phi_{\ell})}\left((r_{2}^{2}-r_{1}^{2})(l^{2}+2R^{2}+(l^{2}-8R^{2})c_{2\theta_{h}})+24(r_{1}r_{2})^{2}\ln\frac{r_{1}}{r_{2}}\right)\Bigg{\\}}$
$\displaystyle\hskip
34.14322pt+h^{+}c_{\theta_{h}}s_{2(\phi_{h}-\phi_{\ell})}\Bigg{\\{}(3l^{2}+R^{2})(r_{2}^{2}-r_{1}^{2})+60(r_{1}r_{2})^{2}\ln\frac{r_{1}}{r_{2}}$
$\displaystyle\hskip
119.50148pt-\left((13R^{2}-l^{2})(r_{2}^{2}-r_{1}^{2})+12(r_{1}r_{2})^{2}\ln\frac{r_{1}}{r_{2}}\right)c_{2\theta_{h}}\Bigg{\\}}\Bigg{]}.$

These are the appropriate results one would use for WISPLC.

Consider the symmetry of the above expressions. Firstly, observe that
$\Phi_{h}^{(2)}\neq 0$, $\Phi_{h}^{(3)}\neq 0$, and in fact both polarisations
contribute at each order. However, with a single pickup loop at
$\phi=\phi_{\ell}$, the considered configuration manifestly breaks azimuthal
symmetry. If we were to restore this symmetry – through an array of pickup
loops arranged symmetrically in $\phi$ as for BASE, or with a coaxial
arrangement as for DMRadio-m3 – then all trigonometric functions with the
argument $\phi_{h}-\phi_{\ell}$ will vanish, and we would conclude
$\Phi_{h}^{(2)}=0$ and $\Phi_{h}^{(3)}\propto h^{\times}$, the $h^{+}$
contribution also vanishing. Cylindrical symmetry hence induces a cancellation
of the leading order sensitivity. This same conclusion was reached for a
toroidal magnetic field in Ref. Domcke:2022rgu . There, it was shown that
breaking the azimuthal symmetry by using a non-circular pickup loop restored
sensitivity at ${\cal O}[(\omega L)^{2}]$, exactly as we find in Eqs. (18) and
(20). In particular, Ref. Domcke:2022rgu  showed that by using a magnetometer
or figure-8 configuration of the pickup loop, the GW flux was given by

$\Phi_{8}^{{(t)}}=\frac{e^{-\mathrm{i}\mkern 1.0mu\omega
t}}{3\sqrt{2}}\omega^{2}B_{\max}r^{3}R\ln(1+a/R)s_{\theta_{h}}\left(h^{\times}s_{\phi_{h}}-h^{+}c_{\theta_{h}}c_{\phi_{h}}\right)\\!.$
(22)

We can imagine implementing a similar pickup loop configuration in the
solenoidal case. For instance, we can take Eq. (18) and combine the
contribution from $\phi_{\ell}\in[0,\pi)$ with a $\pi$ phase shift to that
from $\phi_{\ell}\in[\pi,2\pi)$. (For BASE, we can conceive of implementing
this by changing the orientation of the winding of the loops on, for instance,
$0\leq\phi_{\ell}<\pi$.) Doing so, we obtain,101010This is an
oversimplification. One cannot simply add the contribution from a set of
differentially spaced loops, as this neglects the mutual inductances between
the loops. Instead, one would need to model the full sheath and compute the
induced current density across across it. We thank Joshua Foster for
discussions on this point.

$\Phi_{8}^{{(s)}}=\frac{e^{-\mathrm{i}\mkern 1.0mu\omega
t}}{36\sqrt{2}}\omega^{2}B_{0}lr\left(30R^{2}-13r^{2}\right)s_{\theta_{h}}\left(h^{\times}s_{\phi_{h}}-h^{+}c_{\theta_{h}}c_{\phi_{h}}\right)\\!.$
(23)

The results in Eqs. (22) and (23) are very similar – if we take
$B_{0}=B_{\max}$ and set all spatial scales to $L$, the two only differ by a
factor of $\simeq 2$ – suggesting that the change in geometry has only a minor
impact. In Sec. 4, we will explain the origin of these cancellations in the
instruments with full cylindrical symmetry in terms of the symmetry
transformations of the various quantities introduced in Sec. 2, and in
particular explain how the appearance or absence of different terms can be
understood without an explicit calculation. Such arguments will allow us to
determine optimised detector geometries that maximise the GW sensitivity.

We emphasise that the analytic results above all assume the height of the
solenoidal magnet, $H$, is parametrically larger than all other scales. That
we had to assume this is not unique to the GW signal, the equivalent axion
flux (discussed in the next subsection) also only has an analytic form for
parametrically large $H$. In Fig. 2 we compare the leading order flux from the
analytic result in Eq. (18), to the exact result determined numerically, where
we use the currents in App. E, and with the bands denoting the uncertainty in
the integration.111111As a point of caution, we note that the analytic results
must be treated carefully as there can be branch cuts in the integration. In
particular, we compute the non-trivial part of the flux, captured by
$I^{(2)}_{+,\times}$ in

$\Phi_{h}^{(2)}=e^{-\mathrm{i}\mkern 1.0mu\omega
t}\omega^{2}B_{0}\left(h^{+}I^{(2)}_{+}+h^{\times}I^{(2)}_{\times}\right)\\!.$
(24)

We take all dimensions to match those of ADMX SLIC, and show results for the
physical $H$ and $H\to\infty$. Good agreement is observed in the large $H$
limit, whereas for finite $H$ the analytic results overestimate the amplitude
of the flux. This can be understood from considering the integral over $z$
that appears in Eq. (10): schematically, we have
$\int_{-H/2}^{H/2}dz\,[A+(z-z^{\prime})^{2}]^{-3/2}=2/A-4/H^{2}+{\cal
O}(H^{-4})$, so that the first term neglected will suppress the flux
amplitude.

Figure 2: A comparison between the analytic flux in Eq. (18) (dashed) to the
exact result determined numerically (solid band), for the finite (left) and
large $H$ (right) cases. In detail, we show the non-trivial part of flux
specified in Eq. (24). As expected, agreement is observed when $H\to\infty$,
whereas an offset is seen for finite $H$, see text for a discussion. For the
geometry (see Fig. 1), we adopt the various parameters used by ADMX SLIC
Crisosto:2019fcj : $r_{1}=0$, $r_{2}=7.62~{}{\rm cm}$, $l=31.25~{}{\rm cm}$,
$R=8.55~{}{\rm cm}$, and when finite $H=40~{}{\rm cm}$. We take
$\phi_{\ell}=0$, $\phi_{h}=\pi/4$, and show results as a function of
$\theta_{h}$.

###  3.2 Strain sensitivity from recasting axion limits and projections

Combining the above results, we now determine the GW sensitivity of axion
haloscopes making use of a solenoidal external magnetic field. In detail, we
recast the constraints on the axion photon coupling $g_{a\gamma\gamma}$ from
BASE Devlin:2021fpq  and ADMX SLIC Crisosto:2019fcj , as well as the future
instruments WISPLC Zhang:2021bpa  and DMRadio-m3 DMRadio:2022jfv , as expected
sensitivities to the amplitude of GWs. All of these instruments are designed
with the goal of detecting a coherently oscillating axion dark matter
background field, which takes the schematic form

$a(t)=\frac{\sqrt{2\rho_{\scriptscriptstyle{\rm DM}}}}{m_{a}}\sin(m_{a}t),$
(25)

with $\rho_{\scriptscriptstyle{\rm DM}}$ the local dark-matter density, and
$m_{a}$ the unknown dark-matter mass. In the presence of a magnetic field, the
axion background generates an effective current ${\bf j}_{\rm
eff}=g_{a\gamma\gamma}(\partial_{t}a){\bf B}$. For the configuration in Fig. 1
the current induces the following magnetic flux,

$\Phi_{a}=\frac{1}{4}g_{a\gamma\gamma}(\partial_{t}a)B_{0}l(r_{2}^{2}-r_{1}^{2})+{\cal
O}(H^{-2}),$ (26)

where the result for finite $H$ can again be determined numerically.

Our goal is to derive sensitivity to the GW strain, $h$, by reinterpreting
results on $g_{a\gamma\gamma}$, established assuming a signal flux as in Eq.
(26) (modified for the specific experimental configuration). To do so, we
compare $\Phi_{a}$ and $\Phi_{h}$, but further we account for the fact that in
all expressions derived so far the axion and GW are treated as persistent
monochromatic waves, when this is not the case for either one. The dark-matter
axion is indeed persistent, but has a coherence time of $\tau_{a}=2\pi
Q_{a}/m_{a}$, with a quality factor of $Q_{a}\sim 10^{6}$, indicating a highly
coherent signal. The GW signal on the other hand is model dependent (specific
examples of superradiance and primordial black hole (PBH) mergers are
discussed in App. F), but can be described as lasting for a duration $T_{h}$
with coherence time scale $\tau_{h}$, centered at a frequency $f$, so that the
signal has a quality factor $Q_{h}=\tau_{h}f$. For resonant instruments, one
must also account for the quality factor and coherence time of the instrument,
given by $Q_{r}$ and $\tau_{r}=Q_{r}/f$. The ultimate strain sensitivity is
determined from considering the interplay of each of these scales, together
with the experimental run time, an analysis of which we provide in App. F. The
end result is that rather than simply matching the GW and axion fluxes, we
instead arrive at

$\Phi_{h}(h^{+},h^{\times};\,\phi_{h},\theta_{h})={\cal
R}_{c}\,\Phi_{a}(g_{a\gamma\gamma}),$ (27)

where the coherence ratio ${\cal R}_{c}$ accounts for the difference in
coherence between the signals. As defined, ${\cal R}_{c}>1$ implies the GW
signal is harder to detect than a naive matching of the flux would suggest, as
for a fixed $\Phi_{a}$ we want to probe the smallest $h\propto\Phi_{h}$ values
possible.

Here, we will restrict our attention to a single case, where we take
$T_{h}=\tau_{h}$, and imagine a resonant instrument that spends a time
$T_{m}\gg\tau_{a},\,\tau_{h},\,\tau_{r}$ scanning each axion mass. In this
case, the coherence ratio can be computed to be

${\cal
R}_{c}=\left(\frac{T_{m}}{\tau_{h}}\right)^{1/4}\left(\frac{Q_{a}}{Q_{h}}\right)^{1/4}\left\\{\begin{array}[]{lcc}1&&Q_{r}<Q_{a},\,Q_{h},\\\
(Q_{a}/Q_{r})^{1/4}&&Q_{a}<Q_{r}<Q_{h},\\\ Q_{r}/Q_{h}&&Q_{h}<Q_{r}<Q_{a},\\\
(Q_{a}/Q_{r})^{1/4}Q_{r}/Q_{h}&&\textrm{otherwise}.\end{array}\right.$ (28)

This expression is derived in App. F, however, let us briefly describe the
physical origin of each term. The first factor encodes the suppression that
arises as the GW signal does not persist for the full time the instrument
scans this frequency. The quarter scaling follows our assumption that $T_{m}$
is the largest scale, implying that the sensitivities have entered the
asymptotic scaling regime consistent with the Dicke radiometer equation
Dicke:1946glx  (see also Ref. Dror:2022xpi ). The $(Q_{a}/Q_{h})^{1/4}$ factor
arises as signals that are more coherent are easier to discover, and was used
in Refs. Dror:2021nyr ; Domcke:2022rgu . More coherent signals are narrower in
the frequency domain, and therefore can generally be teased out over a smaller
amount of background. In addition, for $Q_{a}<Q_{r}$ there is a slight penalty
to the axion signal of $(Q_{a}/Q_{r})^{1/4}$ as the full axion signal is not
resolved by the instrument. Finally, there is a strong penalty of
$Q_{r}/Q_{h}$ that applies to the GW signal whenever $Q_{r}>Q_{h}$. When this
occurs, the GW fails to fully ring up the resonance response of the
instrument, which strongly decreases the power it deposits, explaining the
linear scaling of this factor, as opposed to the quarter scaling of all
others. Using Eq. (28) – and its generalisations in App. F – for any given
instrument, we can translate GW signals defined by their amplitude $h$,
duration $T_{h}$, and coherence time $\tau_{h}$, to an effective signal
strength $h/{\cal R}_{c}$. We can then compare $h/{\cal R}_{c}$ to the
equivalent plane wave sensitivity shown in Fig. 13, which is derived simply
from matching the flux; explicitly, Eq. (27) assuming ${\cal R}_{c}=1$.

Figure 3: Recast GW sensitivity for low-mass axion haloscopes. On the left, we
show the existing results from ABRA Salemi:2021gck  (the sensitivity to SHAFT
Gramolin:2020ict  is similar), BASE Devlin:2021fpq , and ADMX SLIC
Crisosto:2019fcj . ADMX SLIC already achieves sensitivity to the leading
${\cal O}[(\omega L)^{2}]$ GW contribution, whereas the cylindrical symmetry
of ABRA and BASE lead that to cancel. Accordingly, we also show those
instruments sensitivity to a figure-8 pickup loop configuration. On the right
we show the future reach to WISPLC Zhang:2021bpa  and the three components of
the DMRadio program DMRadio:2022pkf ; DMRadio:2022jfv : 50L, m3, and GUT. We
assume a figure-8 readout for all three DMRadio results, the toroidal result
in Eq. (22) for 50L and GUT, and the solenoidal analogue in Eq. (23) for m3.
In all cases, the sensitivity is determined from Eq. (27) with ${\cal
R}_{c}=1$.131313For comparison, Fig. 1 in Ref. Domcke:2022rgu  employed ${\cal
R}_{c}=\left(Q_{a}/Q_{h}\right)^{1/4}=10^{3/4}$ for their adopted
$Q_{a}=10^{6}$ and $Q_{h}=10^{3}$. As explained in App. F, this is only
appropriate when comparing persistent signals (which PBHs are not) and in the
case where $Q_{a}$ is not smaller than both $Q_{r}$ and $Q_{h}$. Accordingly,
to determine the detectability of any signal ${\cal R}_{c}$ must be included
in the signal prediction, which we do for a putative PBH inspiral or axion
superradiance signal. See text for details.

For the results in Fig. 13, the frequency range is fixed by the corresponding
axion mass range, and therefore falls into the MHz band. The left figure
demonstrates the estimated sensitivity of three existing instruments: ABRA,
BASE, and ADMX SLIC (the reach for SHAFT is comparable to ABRA, see Ref.
Domcke:2022rgu ). Note that BASE and ADMX SLIC perform a resonant search
strategy for the axion, and therefore at present have a deeper sensitivity,
but narrower frequency coverage than ABRA which completed a broadband search.
For ABRA and BASE, the cylindrical symmetry of these instruments suppresses
the leading order flux, for reasons we demonstrate in the next section. For
this reason we also show the sensitivity the instruments could obtain if they
implemented a figure-8 style geometry, as given in Eqs. (22) and (23). On the
right we show the sensitivity for future instruments, in particular DMRadio
and WISPLC. For DMRadio, we show the projected reach of the 50L, m3, and GUT
variants of these instruments, in each case assuming they have adopted a
figure-8 style readout. We assumed a solenoidal magnet for m3, but toroidal
for 50L and GUT. Beyond this, the only difference to these DMRadio projections
and those derived in Ref. Domcke:2022rgu  is the use of the $H\to\infty$
results and the treatment of the coherence ratio, which here we take as ${\cal
R}_{c}=1$, leaving it for the signal predictions.141414In detail, for 50L we
took $r=a=R=0.11\,{\rm m}$, for GUT $r=a=R=0.64\,{\rm m}$, and for m3,
$R=0.8\,{\rm m}$, $r_{1}=0.34\,{\rm m}$, $r_{2}=0.64\,{\rm m}$, $l=1.4\,{\rm
m}$. For WISPLC, we used $R=r_{1}=0.063\,{\rm m}$, $r_{2}=0.19\,{\rm m}$,
$l=0.57\,{\rm m}$.  For WISPLC, we have repurposed their anticipated axion
sensitivity with a resonant detection strategy, and assumed a single pickup
loop employing Eq. (18) (more precisely, WISPLC has two individual pickup
loops Zhang:2021bpa ). For these cases, we chose the angular direction of GW
and relative pickup loop direction which maximise the GW sensitivity. In
computing all fluxes, we have adopted the $H\to\infty$ results; cf. Fig. 2.
Further, as the GW signal depends on the incident direction, in each case we
have simply taken one of the two polarisations, and chosen the incident
direction to maximise the signal, although performing an angular average
instead would only minimally impact the results.

For comparison, on the right we also draw the expected effective signals
$h/{\cal R}_{c}$ coming from superradiance or PBH binaries as benchmarks for
various parameter choices. Details are given in App. F and effective signals
with different parameters for superradiance and PBHs are given in Figs. 4 and
5 respectively. Example values of ${\cal R}_{c}$ for different instruments and
two different benchmark signals are provided in Tab. 2, with $f_{*}$ denoting
a suitable detector reference frequency. The first benchmark (B1) is a GW
signal of duration $1\,{\rm s}$, with a coherence limited only by its finite
duration. The second (B2) is an example of a persistent highly coherent
signal. In both cases, we assume the frequency spectrum of the signal to be
centered around the detectors resonant frequency. These two benchmarks are
loosely inspired by the properties of GWs from primordial black hole mergers
and from superradiance, respectively. Further discussion of each is provided
in App. F. The advantage of these simplified signals is to facilitate
comparisons across different detector concepts for astrophysical signals. This
is an alternative to the comparison of sensitivity curves in terms of noise
spectral densities (see e.g. Ref. Berlin:2023grv ) which requires detailed
knowledge of the relevant noise sources. A further alternative is the use of
the characteristic strain, $h_{c}\sim Q_{h}^{1/2}h$, as advocated e.g. in Ref.
Aggarwal:2020olq . However, in realistic cases the timescales mentioned above
enter into the estimation of the characteristic strain, and hence again
detailed knowledge of signal and detector properties becomes necessary for
comparisons. Of course, the method used here of defining simplified benchmarks
for comparisons also has its drawbacks. In particular, we do not cover
searches for a stochastic GW background. The coherence ratio ${\cal R}_{c}$
can also be computed for more realistic signals. In the right panel of Fig.
13, we present two benchmark GW signals arising from either superradiance or
PBH inspirals for illustration with the details of each signal being discussed
in App. F. For those examples we adopt the following simplified experimental
setups: we have chosen $Q_{r}=10^{4}$ and $T_{m}=1\,{\rm ks}$ for
superradiance while $Q_{r}=10^{4}$ and $T_{m}=1\,{\rm ms}$ for PBH. In the PBH
case, the expected signal depends on the PBH masses and the curve presented in
the figure corresponds to the maximal signals for a given frequency.

Figure 13 demonstrates that axion haloscopes can place competitive bounds on
GWs in this frequency range. Due to the challenges mentioned above in
comparing different sensitivity estimates, we refrain here from including
other detector concepts in the this figure. This is, however, a very active
field and other concepts such as bulk accoustic wave devices Goryachev:2014yra
, levitated sensors Aggarwal:2020umq , interferometers Holometer:2016qoh ,
other electromagnetic GW detectors Ringwald:2020ist ; Berlin:2021txa ;
Berlin:2023grv ; Bringmann:2023gba  and indirect detection methods
Fujita:2020rdx ; Domcke:2020yzq ; Liu:2023mll  have reached, or are expected
to reach, similar sensitivities. As evident from Fig. 13, a further increase
in sensitivity is needed to reach possible astrophysical (or cosmological)
signals. Our work should be seen as part of the quest of paving a possible
path towards this.

| $Q_{r}$ | $T_{m}$ | $f_{*}$ | ${\cal R}_{c}^{({\rm B}1)}$ | ${\cal R}_{c}^{({\rm B}2)}$
---|---|---|---|---|---
ADMX SLIC Crisosto:2019fcj  | $3\times 10^{3}$ | 320 s151515Technically the ADMX SLIC scan strategy involved averaging 10,000 individual 32 ms scans at each frequency, which we here treat as a single 320 s scan. | 50 MHz | 1.6 | 0.1
BASE Devlin:2021fpq  | $4\times 10^{4}$ | 1 min | 0.7 MHz | 3.0 | 0.39
WISPLC Zhang:2021bpa  | $10^{4}$ | 1 min | (30 kHz, 5 MHz) | (6.7, 1.9) | (0.86, 0.24)
DMRadio DMRadio:2022pkf  | $2\times 10^{7}$ | (8 mins, 60 ns) | (100 kHz, 30 MHz) | (787, 1) | (0.18, 1)
Table 2: Coherence ratio factor ${\cal R}_{c}$ for the different experiments
considered in Fig. 13 for two different benchmark signals: $T_{h}=\tau_{h}=1$
s, $Q_{h}=f_{*}\tau_{h}$ (B1) and $T_{h}\gg T_{m}$, $Q_{h}=10^{10}$ (B2). For
example, the best estimated sensitivity of ADMX SLIC when taking ${\cal
R}_{c}=1$ is $h\simeq 1.7\times 10^{-16}$, so that the sensitivity to the
signals B1 (B2) would be $\simeq 2.8\times 10^{-16}$ ($\simeq 1.7\times
10^{-17}$). The $T_{m}$ values for DMRadio follow from our simple model for
the instruments scan strategy in App. F.

##  4 Selection Rules for General Detector Geometries

In Sec. 3 we studied in detail the interaction of a GW with a solenoidal
magnetic field, adding to the existing results where the wave interacts with a
toroidal field Domcke:2022rgu . In this section we seek to generalise these
results with a symmetry based study of a broader class of magnetic fields and
pickup loops. We will consider detectors with both solenoidal and toroidal
magnetic fields (using Eqs. (16) and (17)), but in each case we will consider
all possible directions for the pickup loops: designed to measure the induced
magnetic field in the $\hat{\bf e}_{z}$, $\hat{\bf e}_{\phi}$, and $\hat{\bf
e}_{\rho}$ directions.

A central goal of our analysis is to identify, by symmetry alone, what is the
leading power in $(\omega L)$ of the GW flux, $\Phi_{h}$, for a given
detector. To do so, we will derive three selection rules that hold for the
interaction of a GW with a cylindrical instrument, that allow us to study the
more general case and identify promising detector geometries without explicit
calculation. The results are catalogued in Tab. 3 supplemented by the outcome
of explicit computations, but let us briefly summarise the key findings. The
leading contribution we expect to $\Phi_{h}$ is at ${\cal O}[(\omega L)^{2}]$,
however, we have already seen for the BASE experiment and for ABRA that the
flux at this order vanishes. These are two examples of a general result:
instruments with full cylindrical symmetry (of both the magnetic field and
pickup loop) designed to search for axions have a leading power sensitivity of
at most ${\cal O}[(\omega L)^{3}]$. This is a consequence of two observations.
Firstly, as we will demonstrate, detectors with azimuthal symmetry are only
sensitive to one of the two GW polarisations $h^{+}$ or $h^{\times}$. They are
also only sensitive to either a scalar or axion, as under parity the scalar
transforms as $h^{+}$, whereas the axion transforms as $h^{\times}$ (see App.
B). Secondly, azimuthal symmetry enforces that only the $h^{+}$ contribution
can enter at ${\cal O}[(\omega L)^{2}]$. Consequently, instruments with full
cylindrical symmetry which can detect scalars coupled to EM may also detect
$h^{+}$ GW flux at ${\cal O}[(\omega L)^{2}]$, but no leading order
contribution can appear for axion experiments which employ full cylindrical
symmetry to enhance the axion signal. If the cylindrical symmetry of the
pickup loop is broken, the leading power sensitivity to the GW can be
restored, at the cost of an ${\cal O}(1)$ factor to the axion signal.

###  4.1 Three selection rules for the interaction of a GW with a cylindrical
detector

Let us now derive three general results regarding the form of $\Phi_{h}$ when
we have a magnetic field and pickup loop with full cylindrical symmetry.

#### Selection Rule 1:

For an instrument with azimuthal symmetry, $\Phi_{h}\propto h^{+}$ at ${\cal
O}[(\omega L)^{2}]$.161616We emphasise that this statement is coordinate
dependent, and holds for the definition of $h^{+}$ adopted in Eq. (2). In
general, one can convert $h^{+}$ into $h^{\times}$ by a $\pi/4$ rotation
around the propagation direction of the GW. The coordinate independent
statement is that for geometries with azimuthal symmetry, only a single
polarisation appears at leading order.

Proof: At ${\cal O}[(\omega L)^{2}]$, the effective current ${\bf j}_{\rm
eff}$ receives a contribution only from ${\bf M}$ (as $\partial_{t}{\bf
P}\propto{\cal O}[(\omega L)^{3}]$), which itself depends on $h_{00}$ and
$h_{ij}$, defined in Eq. (1). Moreover, as both $F({\bf k}\cdot{\bf r})$ and
$F^{\prime}({\bf k}\cdot{\bf r})$ are constant at leading order, $h_{00}$ and
$h_{ij}$ depend only on the GW direction through the polarisation tensors in
Eq. (2), to which the induced magnetic field and the flux are proportional.
Equivalently, the leading order response matrix in Eq. (15) is independent of
the GW direction, explicitly $D^{mn}({\bf k})=D^{mn}_{(2)}+{\cal O}[(\omega
L)^{3}]$, where $D^{mn}_{(2)}$ depends on $\omega^{2}$ but not $\hat{\bf k}$.
These results hold in general. We now invoke azimuthal symmetry, which implies
that the result cannot depend on the azimuthal angle associated with the
direction of the GW, so that $\Phi_{h}(\hat{\bf
k})=\Phi_{h}(R_{z}(\varphi)\hat{\bf k})$ for an arbitrary angle $\varphi$.
Combined with the leading order response matrix, we find

$\displaystyle\Phi_{h}(\hat{\bf k})$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}d\varphi~{}\Phi_{h}(R_{z}(\varphi)\hat{\bf
k})=e^{-\mathrm{i}\mkern 1.0mu\omega
t}D_{(2)}^{mn}\int_{0}^{2\pi}d\varphi\,\sum_{A}h^{A}e_{mn}^{A}(R_{z}(\varphi)\hat{\bf
k})+{\cal O}[(\omega L)^{3}]$ (29) $\displaystyle=\frac{e^{-\mathrm{i}\mkern
1.0mu\omega
t}}{2\sqrt{2}}\,h^{+}\sin^{2}\theta_{h}\,D^{mn}_{(2)}\begin{pmatrix}-1&0&0\\\
0&-1&0\\\ 0&0&2\end{pmatrix}_{mn}+{\cal O}[(\omega L)^{3}].$

By explicit computation, the $h^{\times}$ contribution has vanished,
completing the proof. As a corollary, we note that at leading order an
azimuthally symmetric detector can only depend on the incident GW direction
through $\sin^{2}\theta_{h}$.

#### Selection Rule 2:

For an instrument with azimuthal symmetry, the flux is proportional to either
$h^{+}$ or $h^{\times}$, but not both. This holds to all orders in $(\omega
L)$.171717We reiterate that we derive all fluxes from the Biot-Savart law,
which is valid only to ${\cal O}[(\omega L)^{3}]$ (cf. footnote 7).

Proof: Consider a GW of wave vector $\bf k$ incident on the detector. By the
azimuthal symmetry, we can rotate $\bf k$ into the $xz$-plane without loss of
generality. The configuration is now invariant under a $P_{y}$ reflection, in
particular $P_{y}{\bf k}={\bf k}$. If the pickup loop is azimuthally
symmetric, the flux will receive a contribution at ${\bf r}^{\prime}$ and
$P_{y}{\bf r}^{\prime}$. We now compare the contribution at these points, by
evaluating the magnetic field transverse to the pickup loop, which has unit
normal vector $\hat{\bf n}^{\prime}({\bf r}^{\prime})$.

$\hat{\bf n}^{\prime}(P_{y}{\bf r}^{\prime})\cdot{\bf B}_{h}(P_{y}{\bf
r}^{\prime},{\bf k})=[\kappa_{y}P_{y}\hat{\bf n}^{\prime}({\bf
r}^{\prime})]\cdot[\sigma\eta_{y}\,P_{y}{\bf B}_{h}({\bf r}^{\prime},{\bf
k})]=\sigma\kappa_{y}\eta_{y}\,\hat{\bf n}^{\prime}({\bf r}^{\prime})\cdot{\bf
B}_{h}({\bf r}^{\prime},{\bf k}).$ (30)

Here we used the transformation properties of ${\bf B}_{h}$ and $\hat{\bf
n}^{\prime}$ given in Eqs. (11) and (13). The various values of $\eta_{y}$ and
$\kappa_{y}$ are summarised in Tab. 1, and recall $h^{+}$ and $h^{\times}$
transform with $\sigma=\pm 1$. If $\sigma\kappa_{y}\eta_{y}=-1$, then the
contributions of the flux at positions ${\bf r}^{\prime}$ and $P_{y}{\bf
r}^{\prime}$ cancel, and hence the total flux vanishes when integrated over an
azimuthally symmetric pickup loop. This uniquely selects the GW polarisation
$\sigma$ which can be measured in a detector with azimuthal symmetry,
completing the proof.

In Tab. 3, we summarise which polarisation survives for which geometry from
this argument. Together with the first selection rule, this enables us to
identify geometries which are potentially suitable to pick up the ${\cal
O}[(\omega L)^{2}]$ component of the induced flux. Let us work through several
explicit examples, each assumed to be azimuthally symmetric, referring to the
table for visualisations.

  * $\circ$

Toroidal magnet with a horizontal pickup loop. Consider first a configuration
with ${\bf B}_{0}\propto\hat{\bf e}_{\phi}$ and $\hat{\bf
n}^{\prime}\propto\hat{\bf e}_{z}$, as for instance used by ABRA. For these
choices, the flux in Eq. (30) transforms with
$\sigma\kappa_{y}\eta_{y}=-\sigma$. Accordingly, by selection rule 2 the
$h^{+}$ polarisation ($\sigma=+1$) cannot contribute, only $h^{\times}$ will
($\sigma=-1$). Concretely, for ABRA this implies that any azimuthally
symmetric pickup loop will receive no contribution proportional to $h^{+}$ for
all possible positions $z$ of the pickup loop. Selection rule 1 further
implies that the leading order contribution can only occur at ${\cal
O}[(\omega L)^{3}]$. Both of these results were observed by explicit
calculation in Ref. Domcke:2022rgu .

  * $\circ$

Solenoidal magnet with an array of vertical pickup loops. Inspired by BASE we
next consider a setup with ${\bf B}_{0}\propto\hat{\bf e}_{z}$ and $\hat{\bf
n}^{\prime}\propto\hat{\bf e}_{\phi}$, so that in Eq. (30)
$\sigma\kappa_{y}\eta_{y}=-\sigma$, and only $h^{\times}$ can contribute.
Explicitly, the flux generated by $h^{+}$ from a pickup loop at $\phi_{\ell}$
will exactly cancel the flux the plus polarisation generates in a pickup loop
at $-\phi_{\ell}$. Thus, even though the geometry has changed significantly,
our polarisation selection rule applies identically to the ABRA-type
configuration. This explains the cancellation for azimuthally symmetric
solenoidal detectors observed in Sec. 3.

Although the leading order GW flux vanished in the two instances above, using
our selection rules we can straightforwardly conceive of azimuthally symmetric
geometries where the cancellation does not occur, we simply require
$\kappa_{y}\eta_{y}=+1$. One instance is the following example.

  * $\circ$

Solenoidal magnet with a horizontal pickup loop. Taking ${\bf
B}_{0}\propto\hat{\bf n}^{\prime}\propto\hat{\bf e}_{z}$, we have
$\sigma\kappa_{y}\eta_{y}=+\sigma$ in Eq. (30). As a result, the $h^{\times}$
contributions to the flux vanishes to all orders in $(\omega L)$ whereas the
$h^{+}$ contribution survives, and with it a contribution to the flux of order
$(\omega L)^{2}$. In this case, one might worry about the feasibility of
separating the tiny induced field from the large background magnetic field.
Any detection strategy would exploit the AC nature of the GW flux, as opposed
to the (ideally) DC static field, and potentially also the angular dependence
of the GW. (For further discussion, see also Ref. Bloch:2023uis .)

The two selection rules derived so far allow us to understand an important
consequence for the detection of GWs with axion haloscopes. In particular, by
selection rule 2, the flux in an azimuthally symmetric detector sensitive to
the $h^{\times}$ or $\sigma=-1$ component has no dependence on $h^{+}$. But
then by selection rule 1, such an instrument will not have the optimal
sensitivity to the GW, since the ${\cal O}[(\omega L)^{2}]$ contribution will
vanish. This is an important observation for axion haloscopes, because as
shown in App. B, the pseudoscalar axion field transforms with $\sigma=-1$, and
therefore to be sensitive to the axion one is forced into a configuration
where the leading GW flux vanishes. The only way of evading this conclusion is
to break the azimuthal symmetry. One could break this maximally by introducing
a figure-8 configuration, as done in Ref. Domcke:2022rgu . This would revive a
contribution from $h^{+}$ at ${\cal O}[(\omega L)^{2}]$, however as the axion
induced magnetic field ${\bf B}_{a}$ has no angular dependence, its
contribution will vanish. Hence, to detect both axion and GW, one could use a
pickup loop with an opening angle smaller than $2\pi$, which avoids a complete
cancellations. For this purpose, we present the results for $\Phi_{h}^{(2)}$
with a pickup loop with an arbitrary angle in the next subsection. Further
discussion of these points is provided in App. B, where we also demonstrate
that a scalar, $\varphi$, which couples as $\varphi F^{2}$ transforms with
$\sigma=+1$, and so can also be understood through our selection rules.

#### Selection Rule 3:

For an instrument with full cylindrical symmetry, $\Phi_{h}$ will contain only
even or odd powers of $\omega$.

Proof: If the instrument has full cylindrical symmetry, then the flux will
receive a contribution from ${\bf r}^{\prime}$ and $P{\bf r}^{\prime}=-{\bf
r}^{\prime}$, where again $P=P_{x}P_{y}P_{z}$ is the complete parity
transformation. Let us thus consider the property of the induced magnetic flux
under ${\bf r}^{\prime}\rightarrow P{\bf r}^{\prime}$. We start by writing the
GW in the proper detector frame as power series in $\omega$,
$h_{\mu\nu}=\sum_{n=2}^{\infty}h^{(n)}_{\mu\nu}$ with
$h_{\mu\nu}^{(n)}\propto\omega^{n}$. From Eq. (1), we can see the components
transform as follows,

$\displaystyle h_{00}^{(n)}(P{\bf r},{\bf k})=(-1)^{n}\,h_{00}^{(n)}({\bf
r,k}),\hskip 14.22636pth^{(n)}_{0i}(P{\bf r},{\bf
k})=(-1)^{n}\,P_{ij}\,h^{(n)}_{0j}({\bf r,k}),$ (31) $\displaystyle\hskip
68.28644pth^{(n)}_{ij}(P{\bf r},{\bf k})=(-1)^{n}\,P_{ik}\,h_{kl}^{(n)}({\bf
r,k})\,P_{lj}^{T}.$

This is identical to the transformations studied in Eq. (4), but with
$\sigma\to(-1)^{n}$. Either by proceeding through the same steps as used to
derive Eq. (11) or by exploiting the above analogy, we find,

${\bf B}_{h}^{(n)}(P{\bf r}^{\prime},\hat{\bf k})=\eta(-1)^{n}P{\bf
B}_{h}^{(n)}({\bf r}^{\prime},\hat{\bf k}),$ (32)

where we have decomposed the induced field as ${\bf
B}_{h}=\sum_{n=2}^{\infty}{\bf B}_{h}^{(n)}$, with ${\bf
B}_{h}^{(n)}\propto\omega^{n}$. Consequently, the flux contribution from the
point $P{\bf r}^{\prime}$ will be determined by

${\bf n}^{\prime}(P{\bf r}^{\prime})\cdot{\bf B}_{h}(P{\bf r}^{\prime},{\bf
k})=\kappa\eta\sum_{n=2}^{\infty}(-1)^{n}[{\bf n}^{\prime}({\bf
r}^{\prime})\cdot{\bf B}_{h}^{(n)}({\bf r}^{\prime},\hat{\bf k})].$ (33)

The values of $\eta$ and $\kappa$ were given in Tab. 1. Clearly
$\kappa\eta=\pm 1$. If $\kappa\eta=+1$, then for odd $n$, the flux from $P{\bf
r}^{\prime}$ and ${\bf r}^{\prime}$ will cancel, whereas for $\kappa\eta=-1$ a
similar conclusion is reached for even $n$, completing the proof. (Note this
result is independent of $\sigma$.)

The combination of our three selection rules show that instruments with full
cylindrical symmetry have a highly restricted form of the induced GW flux. In
Tab. 3, we apply these selection rules for all possible pickup loop
orientations, and for both solenoidal and toroidal external fields, always
assuming full cylindrical symmetry. (Similar results can be derived for a
scalar and axion, which we study in the appendix and summarise in Tab. 4.) In
the first line of each cell, we denote the surviving polarisation ($h^{+}$ or
$h^{\times}$), if even or odd powers of $\omega L$ contribute, and what is the
leading order contribution. The second line provides the explicit leading
order flux $\Phi_{h}$. The flux is computed assuming a parametrically large
$H$, and for all cases we took $\rho^{\prime}\in[0,r]$, with $r\leq R$. For an
array of pickup loops or a continuous pickup surface (which are required for
cylindrical symmetry in the case of a vertical pickup loop) we integrate the
induced flux over the azimuthal angle $\phi_{\ell}$. In the absence of
detailed information on the detector setup and resulting inductance, this
serves as a proxy to estimate the total flux, and most importantly, will drop
out when recasting axion search results (see Sec. 3.2) since the same factor
appears for the axion case. (See also the discussion in footnote 10.)

The result for a horizontal pickup loop in a toroidal magnetic field was
previously presented in Ref. Domcke:2022rgu , our new results corrects this
expression by a factor $1/3$ which is due taking into account the contribution
from the effective surface current previously overlooked. For the ${\cal
O}[(\omega L)^{2}]$ results in Ref. Domcke:2022rgu , obtained from the use of
a figure-8 pickup loop to break the azimuthal symmetry, the surface current
contribution vanishes, leaving them unchanged. Such cases are considered
explicitly in the next subsection.

For each case in the table, the selection rules determine the leading order
contribution to $\Phi_{h}$ without any explicit calculation being required,
which achieves one of the central goals of this work. For example, take a
solenoidal magnetic field with a radial pickup loop ($\hat{\bf
n}^{\prime}\propto\hat{\bf e}_{\rho}$). As $\sigma\kappa_{y}\eta_{y}=\sigma$
in Eq. (30), only $h^{+}$ can contribute from selection rule 2. However, as
$\kappa\eta=-1$ in Eq. (33), only odd powers of $\omega$ contribute by
selection rule 3, and therefore the leading order contribution is at ${\cal
O}[(\omega L)^{3}]$. If, however, the loop was moved up or down in the
vertical direction, breaking the cylindrical symmetry, selection rule 3 would
no longer hold, and we would have a contribution at ${\cal O}[(\omega L)^{2}]$
as allowed by selection rule 1. Explicitly, if we place the loop at
$z^{\prime}\in[-H/2,-H/2+l]$, and then expand the flux assuming for simplicity
$H\gg l\gg r,a,R$, we find

$\Phi_{h}=\frac{e^{-\mathrm{i}\mkern 1.0mu\omega
t}}{3\sqrt{2}}h^{+}\omega^{2}B_{0}\pi r^{2}l^{2}\sin^{2}{\theta_{h}},$ (34)

so that a leading order contribution has been resurrected. Consider also the
case of a toroidal magnet with a radial pickup loop. Now
$\sigma\kappa_{y}\eta_{y}=-\sigma$ and $\kappa\eta=+1$, so that only
$h^{\times}$ and even orders of $\omega L$ will contribute. But by selection
rule 1, for such a configuration the leading order contribution cannot occur
until ${\cal O}[(\omega L)^{4}]$, where we already expect corrections from our
use of the Biot-Savart law. Again, placing the loop at the vertical bottom (or
top) of the instrument would parametrically enhance the flux to

$\Phi_{h}=-\frac{\mathrm{i}\mkern 1.0mue^{-\mathrm{i}\mkern 1.0mu\omega
t}}{96\sqrt{2}}h^{\times}\omega^{3}B_{\max}\pi
r^{2}aR(a+2R)\sin^{2}\theta_{h}.$ (35)

|  Solenoid: ${\bf B}_{0}\propto\hat{\bf e}_{z}$ ($\eta_{y}=+1$, $\eta=-1$) |  Toroid: ${\bf B}_{0}\propto\hat{\bf e}_{\phi}$ ($\eta_{y}=-1$, $\eta=+1$) |
---|---|---|---
$\hat{\bf n}^{\prime}\propto\hat{\bf e}_{z}$ ($\kappa_{y}=+1$, $\kappa=-1$) |  $h^{+}$, $n$ even $\Rightarrow$ ${\cal O}[(\omega L)^{2}]$ $\Phi_{h}=\frac{e^{-\mathrm{i}\mkern 1.0mu\omega t}}{48\sqrt{2}}h^{+}\omega^{2}B_{0}s_{\theta_{h}}^{2}\pi r^{2}\left(11r^{2}+14R^{2}+16R^{2}\ln\frac{R}{H}\right)$ |  $h^{\times}$, $n$ odd $\Rightarrow$ ${\cal O}[(\omega L)^{3}]$ $\Phi_{h}=\frac{\mathrm{i}\mkern 1.0mue^{-\mathrm{i}\mkern 1.0mu\omega t}}{48\sqrt{2}}h^{\times}\omega^{3}B_{\max}\pi r^{2}aR(a+2R)s^{2}_{\theta_{h}}$ |
$\hat{\bf n}^{\prime}\propto\hat{\bf e}_{\phi}$ ($\kappa_{y}=-1$, $\kappa=+1$) |  $h^{\times}$, $n$ odd $\Rightarrow$ ${\cal O}[(\omega L)^{3}]$ $\Phi_{h}=\frac{\mathrm{i}\mkern 1.0mue^{-\mathrm{i}\mkern 1.0mu\omega t}}{96\sqrt{2}}h^{\times}\omega^{3}B_{0}\pi r^{2}l(12R^{2}-5r^{2})s^{2}_{\theta_{h}}$ |  $h^{+}$, $n$ even $\Rightarrow$ ${\cal O}[(\omega L)^{2}]$ $\Phi_{h}=\frac{3e^{-i\omega t}}{4\sqrt{2}}h^{+}\omega^{2}B_{\max}\frac{\pi r^{2}aRl(a+2R)}{H^{2}}s_{\theta_{h}}^{2}$ |
$\hat{\bf n}^{\prime}\propto\hat{\bf e}_{\rho}$ ($\kappa_{y}=+1,\kappa=+1$) |  $h^{+}$, $n$ odd $\Rightarrow$ ${\cal O}[(\omega L)^{3}]$ $\Phi_{h}=\frac{\mathrm{i}\mkern 1.0mue^{-\mathrm{i}\mkern 1.0mu\omega t}}{96\sqrt{2}}h^{+}B_{0}\omega^{3}c_{\theta_{h}}s_{\theta_{h}}^{2}$ $\times\pi r^{2}l\left(3l^{2}-22(r^{2}+2R^{2})-36R^{2}\ln\frac{R}{H}\right)$ |  $h^{\times}$, $n$ even $\Rightarrow$ ${\cal O}[(\omega L)^{4}]$ $\Phi_{h}=\frac{e^{-\mathrm{i}\mkern 1.0mu\omega t}}{32\sqrt{2}}h^{\times}\omega^{4}B_{\max}\pi r^{2}aRl(a+2R)c_{\theta_{h}}s_{\theta_{h}}^{2}$ |
Table 3: Summary table of sensitivity to GWs for cylindrical detector
geometries depending on the direction of the external magnetic ${\bf B}$ field
and the orientation of the pickup loop ${\hat{\bf n}^{\prime}}$.
$\eta_{\alpha}$ and $\kappa_{\beta}$ are respectively defined in Eqs. (11) and
(13). In each cell, the first line indicates the surviving GW polarisation,
the selection of odd or even powers of $(\omega L)^{n}$, and the leading power
of the induced flux. The second line gives the explicit result for the leading
order of the induced flux, and finally we provide a cartoon representation of
the instrument. See text for detail.

###  4.2 Increased GW sensitivity for pickup loops that break azimuthal
symmetry

All three selection rules above followed from the full azimuthal symmetry of
the detector. When broken, the restrictions the rules impose are lifted, and
in many geometries this allows for a parametric improvement in the GW flux.
While the breaking can occur at the level of the magnetic field or pickup
loop, the latter is far more practical. For instance, as discussed in Ref.
Domcke:2022rgu , an instrument could use multiple pickup loops, one for the
axion, and another for the GW.

With such a possibility in mind, here we compute the leading ${\cal O}[(\omega
L)^{2}]$ flux for various geometries in the case where the detector has a
pickup loop that spans an opening angle $\phi_{\ell}\in[\phi_{1},\phi_{2}]$
for a horizontal or radial readout, or a set of loops that span a fraction
$[\phi_{1},\phi_{2}]$ of a toroid in the case of the vertical loop. As it can
be readily restored, we fix $\phi_{h}=0$ throughout for ease of notation, and
assume $\rho^{\prime}\in[0,r]$ for the radial expanse of the loop. We again
employ the shorthand $c_{x}=\cos x$ and $s_{x}=\sin x$.

For a solenoidal magnetic field, the three results determined by the
orientation of the pickup loop $\hat{\bf n}^{\prime}$ are as follows,

$\displaystyle\hat{\bf e}_{z}\\!:$ $\displaystyle\hskip
7.11317pt\Phi_{h}=\frac{e^{-\mathrm{i}\mkern 1.0mu\omega
t}}{768\sqrt{2}}\omega^{2}B_{0}r^{2}\Bigg{[}12h^{\times}r^{2}c_{\theta_{h}}(c_{2\phi_{2}}-c_{2\phi_{1}})-3h^{+}r^{2}(3+c_{2\theta_{h}})(s_{2\phi_{2}}-s_{2\phi_{1}})$
(36) $\displaystyle\hskip
109.54327pt+8h^{+}(\phi_{2}-\phi_{1})\left(11r^{2}+14R^{2}+16R^{2}\ln\frac{R}{H}\right)s_{\theta_{h}}^{2}\Bigg{]},$
$\displaystyle\hat{\bf e}_{\phi}\\!:$ $\displaystyle\hskip
7.11317pt\Phi_{h}=\frac{e^{-\mathrm{i}\mkern 1.0mu\omega
t}}{144\sqrt{2}}\omega^{2}B_{0}rl\left(30R^{2}-13r^{2}\right)s_{\theta_{h}}\left[h^{+}c_{\theta_{h}}(c_{\phi_{2}}-c_{\phi_{1}})+h^{\times}(s_{\phi_{2}}-s_{\phi_{1}})\right]\\!,$
$\displaystyle\hat{\bf e}_{\rho}\\!:$ $\displaystyle\hskip
7.11317pt\Phi_{h}=\frac{5e^{-\mathrm{i}\mkern 1.0mu\omega
t}}{48\sqrt{2}}\omega^{2}B_{0}rl\left(2R^{2}-3r^{2}\right)s_{\theta_{h}}\left[h^{+}c_{\theta_{h}}(s_{\phi_{2}}-s_{\phi_{1}})-h^{\times}(c_{\phi_{2}}-c_{\phi_{1}})\right]\\!.$

In each case we only state the result to ${\cal O}[(\omega L)^{2}]$ and
leading order in $1/H$. Observe that when $\phi_{1}=0$ and $\phi_{2}=2\pi$,
only the result for $\hat{\bf n}^{\prime}\propto\hat{\bf e}_{z}$ survives,
consistent with Tab. 3.

For a toroid, the analogous results are as follows (again working only to
${\cal O}[(\omega L)^{2}]$),

$\displaystyle\hat{\bf e}_{z}\\!:$ $\displaystyle\hskip
7.11317pt\Phi_{h}=\frac{e^{-\mathrm{i}\mkern 1.0mu\omega
t}}{12\sqrt{2}}\omega^{2}B_{\max}r^{3}R\ln\left(1+\frac{a}{R}\right)s_{\theta_{h}}\left[h^{+}c_{\theta_{h}}(c_{\phi_{2}}-c_{\phi_{1}})+h^{\times}(s_{\phi_{2}}-s_{\phi_{1}})\right]\\!,$
(37) $\displaystyle\hat{\bf e}_{\phi}\\!:$ $\displaystyle\hskip
7.11317pt\Phi_{h}=-\frac{e^{-\mathrm{i}\mkern 1.0mu\omega
t}}{32\sqrt{2}}\omega^{2}B_{\max}r^{2}Rl\ln\left(1+\frac{a}{R}\right)\left[4h^{\times}c_{\theta_{h}}(c_{2\phi_{2}}-c_{2\phi_{1}})-h^{+}(3+c_{2\theta_{h}})(s_{2\phi_{2}}-s_{2\phi_{1}})\right]$
$\displaystyle\hskip 30.15985pt+\frac{3e^{-\mathrm{i}\mkern 1.0mu\omega
t}}{8\sqrt{2}}h^{+}\omega^{2}B_{\max}(\phi_{2}-\phi_{1})\frac{r^{2}aRl(a+2R)}{H^{2}}s_{\theta_{h}}^{2},$
$\displaystyle\hat{\bf e}_{\rho}\\!:$ $\displaystyle\hskip
7.11317pt\Phi_{h}=-\frac{e^{-\mathrm{i}\mkern 1.0mu\omega
t}}{16\sqrt{2}}\omega^{2}B_{\max}r^{2}Rl\ln\left(1+\frac{a}{R}\right)\left[h^{+}(3+c_{2\theta_{h}})(c_{2\phi_{2}}-c_{2\phi_{1}})+4h^{\times}c_{\theta_{h}}(s_{2\phi_{2}}-s_{2\phi_{1}})\right]\\!.$

Again taking $\phi_{1}=0$ and $\phi_{2}=2\pi$, there will only be a
contribution $\hat{\bf n}^{\prime}\propto\hat{\bf e}_{\phi}$, reproducing Tab.
3. That contribution only appears at ${\cal O}(1/H^{2})$, and this is the only
non-leading contribution in $H\to\infty$ we included.

##  5 Discussion

Both axions and GWs induce effective polarisation and magnetisation terms in
Maxwell’s equations. While the formalism for exploiting this effect to search
for the axion has been in place for four decades Sikivie:1983ip , the GW
analogue and its synergies with axion searches remains nascent. Our work
expands our understanding of this latter case, and by focussing on lumped-
element circuits for axion detection (such as ABRADACABRA, SHAFT, BASE, ADMX
SLIC, WISPLC and the DMRadio program), we estimate their sensitivity to
current and future high-frequency GW searches, as shown in Fig. 13.

We also expand the theoretical foundations of the interaction of GWs with
instruments operating in the magnetoquasistatic regime, extending the earlier
results of Ref. Domcke:2022rgu  in a number of ways. Most importantly, we have
developed a symmetry based formalism that largely fixes the form of the
leading GW signal in various instruments. We considered external magnetic
fields with a cylindrical symmetry – toroidal or solenoidal fields – as used
in all ongoing and planned axion haloscopes. We derived selection rules for
the signal strength, which, based on symmetry alone, fix the leading power
sensitivity in $\omega L$, and hence parametrically determine the GW strain
sensitivity without calculation. This allows one to immediately determine the
impact of different geometries for the external magnetic (or electric) field
and the pickup loop on the achievable GW strain sensitivity. As summarised in
Tab. 3, highly symmetric detectors place strong restrictions on the form of
the induced flux as a direct consequence of the tensor nature of the GW. These
arguments can be extended to a scalar or pseudoscalar (axion) coupled to EM,
as we show in App. B. Taken together, we observe that in optimising the
sensitivity to axions, existing instruments can often parametrically suppress
the GW signal. Fortunately, however, the observed cancellation can quite
easily be remedied by minimally breaking the instruments cylindrical symmetry,
for instance by changing the position or shape of the pickup loop. We
demonstrated this for different detector geometries, obtaining a parametric
increase for the GW sensitivity.

Our work provides several technical improvements on the formalism and initial
studies of Ref. Domcke:2022rgu . First, we include the contribution from
effective surface currents induced by the GW, arising due to the change in
effective magnetisation at the boundary of the static magnetic fields. These
effects are generically of the same order as the effects obtained from the
interaction of the GW with the magnetic field itself, and for instance modify
some of the toroidal results of Ref. Domcke:2022rgu , relevant for
ABRACADABRA, SHAFT, and DMRadio-50L. Second, we give a thorough discussion and
prescription of how to compute sensitivities for transient signals, focussing
on resonant detectors. This is particularly important for high-frequency GWs,
since the duration of the expected signals can be much shorter than the
observation time. We include the effect of finite coherence time and finite
duration of the signal, the scanning strategy and the quality factor of the
instrument. Our prescription is based on bootstrapping the axion search
results, allowing an immediate recasting of existing and upcoming axion
searches in terms of GW searches. Third, we introduce linear response matrices
describing the detector response to the GW signal. With this new formalism, we
recover the results of Ref. Domcke:2022rgu , but the alternative approach
played a key role in revealing the symmetry properties of the detectors,
facilitating the derivation of the selection rules mentioned above. With all
this at hand, we provide analytical results for the effective current induced
by a GW up to ${\cal O}[(\omega L)^{3}]$ for a solenoidal magnetic field
configuration.

Much work remains. The symmetry based arguments introduced here can be
deployed for the full set of signals axion haloscopes could detect, including,
for instance, dark photons. Such arguments can help determine the full physics
reach of the future axion dark-matter program, see Ref. Adams:2022pbo . A
dedicated GW search will require a targeted data analysis strategy as well as
a detailed detector simulation. While this is beyond the scope of the current
paper, any such analysis can draw on the tools we have provided here. A
further open question retains to the impact of the mechanical response of the
detector to GWs, which may become relevant once the GW frequency lies above
the lowest mechanical resonance mode, as discussed in Refs. Berlin:2023grv ;
Bringmann:2023gba . We leave this to future work. The achievable strain
sensitivities (see Fig. 13) we obtain by bootstrapping the axion searches
still lie above any expected signals from astrophysical or cosmological
sources. Nevertheless, the sensitivities are competitive with other
experiments and proposals in this frequency regime. We aim with this work to
join the worldwide effort of paving the way towards high-frequency GW
detection.

###### Acknowledgements.

We thank Tael Coren, Jack Devlin, Sebastian Ellis, Joshua Foster, Jai-chan
Hwang, Joachim Kopp, Stefan Ulmer, Jonathan Ouellet, Yotam Soreq, and Zhongyue
Zhang for very helpful discussions, and Francesco Muia for comments on a draft
version of this work. CGC is supported by a Ramón y Cajal contract with Ref.
RYC2020-029248-I. The work of SML is supported in part by the Hyundai Motor
Chung Mong-Koo Foundation Scholarship, and funded by the Korea-CERN
Theoretical Physics Collaboration and Developing Young High-Energy Theorists
fellowship program (NRF-2012K1A3A2A0105178151).

##  Appendix A Additional Details of GW Electrodynamics

In this first appendix, we provide a derivation of the key elements of GW
electrodynamics required for the discussion in the main text. In the first two
subsections we will provide a detailed derivation of Eqs. (5) and (6), which
formed the foundation of the physical effect explored in the main text. After
this, we describe the origin of the effective surface current contributions
that had been missed in previous analyses, and lastly we explain why GW
effects start at ${\cal O}[(\omega L)^{2}]$ in the proper detector frame.

###  A.1 External currents

The external static fields, $f_{\mu\nu}$, upon which GWs interact to generate
electromagnetic effects satisfy Maxwell’s equations in flat spacetime

$\partial_{\nu}f^{\mu\nu}=\left[j^{\mu}\right]_{\text{FLAT}},\hskip
14.22636pt\partial_{\nu}f_{\alpha\beta}+\partial_{\alpha}f_{\beta\nu}+\partial_{\beta}f_{\nu\alpha}=0,$
(38)

where $\left[j^{\mu}\right]_{\text{FLAT}}$ is the external electromagnetic
current sourcing the external fields. For a system of electrons following
trajectories described by $x_{n}^{\mu}(u)$, with $n$ indexing the various
particles, the current is given by Weinberg:2008zzc

$\left[j^{\mu}\right]_{\text{FLAT}}=\int
du\sum_{n}e\,\delta^{(4)}\left(x_{n}(u)-x\right)\frac{dx_{n}^{\mu}(u)}{du}=\sum_{n}e\,\delta^{(3)}\left({\bf
x}_{n}(t)-{\bf x}\right)\frac{dx^{\mu}_{n}(t)}{dt}.$ (39)

For example, solenoidal and toroidal static configurations in which the
electric charges are confined to cylindrical surfaces of constant radius lead
to $\left[j^{\mu}\right]_{\text{FLAT}}\propto\delta(\rho-R)$. For the case of
a solenoid infinitely extended in the $z$-direction

${\bf j}_{\text{FLAT}}^{(s)}=B_{0}\delta(\rho-R)\,\hat{{\bf e}}_{\phi}.$ (40)

Similarly, for the toroidal configurations in the limit of infinite height

${\bf
j}^{(t)}_{\text{FLAT}}=B_{\max}\frac{R}{\rho}\left[\delta\left(\rho-R\right)-\delta\left(\rho-(R+a)\right)\right]\hat{\bf
e}_{z},$ (41)

because there are two cylindrical surfaces.

Observe also that charge conservation directly follows from Eq. (39)

$\displaystyle{\bf\nabla}\cdot{\bf j}_{\text{FLAT}}=$
$\displaystyle\sum_{n}e\,{\bf\nabla}\cdot\left[\delta^{(3)}\left({\bf
x}_{n}(t)-{\bf x}\right)\right]\frac{dx^{\mu}_{n}(t)}{dt}$ (42)
$\displaystyle=$ $\displaystyle-\frac{\partial}{\partial
t}\sum_{n}e\,\delta^{(3)}\left({\bf x}_{n}(t)-{\bf x}\right)$ $\displaystyle=$
$\displaystyle-\partial_{t}j_{\text{FLAT}}^{0}.$

Let us now discuss how the external currents are modified by the presence of
GWs in the proper detector (PD) frame. A generalisation of Eq. (39) to an
arbitrary spacetime can be found by noticing that
$\delta^{(4)}\left(x_{n}(u)-x\right)/\sqrt{-g}$ transforms as a
scalar.181818Recall, it is the spacetime volume $d^{4}x\sqrt{-g}$ rather than
$d^{4}x$ that transforms as a scalar. The current associated with a set of
charges following spacetime trajectories $x_{n}^{\mu}(u)$ is thus given by
Weinberg:2008zzc

$j^{\mu}=\int
du\sum_{n}e\,\frac{1}{\sqrt{-g}}\delta^{(4)}\left(x_{n}(u)-x\right)\frac{dx_{n}^{\mu}(u)}{du}=\frac{1}{\sqrt{-g}}\sum_{n}e\,\delta^{(3)}\left({\bf
x}_{n}(t)-{\bf x}\right)\frac{dx_{n}^{\mu}(t)}{dt}.$ (43)

A calculation similar to that in Eq. (42) shows that
$\partial_{\mu}\left(\sqrt{-g}\,j^{\mu}\right)=0$. This is equivalent to
$\nabla_{\mu}j^{\mu}=0$, which can be proven employing the properties of
Christoffel symbols.

In the proper detector frame, the effect of a GW on the charged particles is
described by a Newtonian force. As stated above, in this work we assume that
the experimental apparatus is rigid, or more precisely, that such a Newtonian
force does not alter the trajectories of the particles. In particular, this
implies that in the proper detector frame we can use the same $x_{n}^{\mu}(t)$
that led to Eqs. (40) and (41). We conclude that

$\left[\sqrt{-g}\,j^{\mu}\right]_{\text{PD}}=\left[j^{\mu}\right]_{\text{FLAT}}.$
(44)

###  A.2 Maxwell’s equations in the spacetime of a gravitational wave

Maxwell’s Equations in an arbitrary spacetime read

$\nabla_{\nu}\left(g^{\alpha\mu}F_{\alpha\beta}\,g^{\beta\nu}\right)=j^{\mu},\hskip
14.22636pt\nabla_{\nu}F_{\alpha\beta}+\nabla_{\alpha}F_{\beta\nu}+\nabla_{\beta}F_{\nu\alpha}=0.$
(45)

where the external current is defined by Eq. (43). Due to the properties of
the Christoffel symbols and the fact the electromagnetic tensor is
antisymmetric, these equations can be cast as landau1975classical

$\partial_{\nu}\left(\sqrt{-g}\,g^{\alpha\mu}F_{\alpha\beta}\,g^{\beta\nu}\right)=\sqrt{-g}\,j^{\mu},\hskip
14.22636pt\partial_{\nu}F_{\alpha\beta}+\partial_{\alpha}F_{\beta\nu}+\partial_{\beta}F_{\nu\alpha}=0.$
(46)

When considering a passing GW in the proper detector frame, the equations are
equivalent to Maxwell’s equations in a flat spacetime with the GW effects
described by an effective current. To see this, we note that the expression in
parenthesis to first order in $h_{\mu\nu}=g_{\mu\nu}-\eta_{\mu\nu}$ is given
by

$\sqrt{-g}\,g^{\alpha\mu}F_{\alpha\beta}\,g^{\beta\nu}=\left(1+\frac{h}{2}\right)F^{\mu\nu}-F^{\mu\alpha}{h^{\nu}}_{\alpha}+F^{\nu\alpha}{h^{\mu}}_{\alpha}+{\cal
O}(h^{2}),$ (47)

where we employ $\sqrt{-g}=1+h/2+{\cal O}(h^{2})$. This motivates us to define
the following current effective current,

$j_{\rm
eff}^{\mu}\equiv\partial_{\nu}\left(-\frac{1}{2}h\,F^{\mu\nu}+F^{\mu\alpha}{h^{\nu}}_{\alpha}-F^{\nu\alpha}{h^{\mu}}_{\alpha}\right)\\!,$
(48)

which, together with Eqs. (46) and (44), leads to the following form for
Maxwell’s equations

$\partial_{\nu}F^{\mu\nu}=\left[j^{\mu}\right]_{\text{FLAT}}+j_{\rm
eff}^{\mu},\hskip
14.22636pt\partial_{\nu}F_{\alpha\beta}+\partial_{\alpha}F_{\beta\nu}+\partial_{\beta}F_{\nu\alpha}=0.$
(49)

This completes our justification of Eqs. (5) and (6).

We combine Maxwell’s equations in the presence of a GW with their flat space
analogues in Eq. (38) to isolate the induced fields, defined by

$F^{h}_{\,\mu\nu}\equiv-f_{\mu\nu}+F_{\mu\nu}.$ (50)

In particular, these fields satisfy the following equations

$\partial_{\nu}F^{h\,\mu\nu}=j^{\mu}_{\rm eff},\hskip
14.22636pt\partial_{\nu}F^{h}_{\alpha\beta}+\partial_{\alpha}F^{h}_{\beta\nu}+\partial_{\beta}F^{h}_{\nu\alpha}=0.$
(51)

Alternatively, one may also write these equations in terms of a contravariant
electromagnetic field strength tensor,
$g^{\alpha\mu}F_{\alpha\beta}\,g^{\beta\nu}-f^{\mu\nu}$, in which case the
second set of equations acquire a source term, as shown in Refs.
Sokolov:2022dej ; Hwang:2023jhs . In the absence of external currents, this
leads to an ambiguity in the definition of the electromagnetic field. This is
the well-known duality of Maxwell’s equations in the absence of charges and
currents. However, in contrast to the claims of those references, such an
ambiguity does not arise here because of the proper detector external current.
Furthermore, the derivation presented here shows that it is not necessary to
define the electric and magnetic field vectors in curved spacetime in order to
describe the effect of GWs propagating in external electromagnetic fields, as
one can always work at the level of $F^{\mu\nu}$. For simplicity, in the main
text and in the rest of the manuscript, we drop the FLAT subscript from the
current in Eq. (49).

###  A.3 Effective surface currents

Comparing the magnetic fields in Eqs. (16) and (17) with the currents in Eqs.
(40) and (41), one can note that the currents close to the surface $\rho=R$
are given by ${\bf j}=\delta(\rho-R)\,\hat{\bf n}\times{\bf B}$, with
$\hat{\bf n}=-\hat{\bf e}_{\rho}$ for the solenoid and $\hat{\bf n}=\hat{\bf
e}_{\rho}$ for the toroid. This can also be written as

${\bf j}=\delta(\rho-R)\,{\bf K},\hskip 14.22636pt\text{with}\hskip
14.22636pt{\bf K}=\hat{\bf n}\times{\bf B},$ (52)

where $\bf K$ is often called the surface current density. The result in Eq.
(52) is a particular example of a phenomenon that takes place whenever there
is discontinuity in the magnetic field across a surface. It is possible to
prove (see, for instance, Ref. jackson_classical_1999 ) that at the interface
of two bodies with different values of ${\bf H}={\bf B}-\bf M_{\rm eff}$,
Maxwell’s equations predict the existence of a surface current density,

${\bf K}=\hat{\bf n}\times\left({\bf H}_{2}-{\bf H}_{1}\right)\\!,$ (53)

where $\hat{\bf n}$ is the unit vector normal to the surface from medium 1 to
2. Note that Eq. (52) is a particular case of Eq. (53) when the magnetic field
vanishes on one side of the surface $\rho=R$, and ${\bf M}_{\rm eff}=0$
everywhere, i.e. there is no GW or (pseudo-)scalar field.

In the presence of GWs or a (pseudo-)scalar field, the magnetisation does not
vanish. For the toroid and solenoid cases of Eqs. (16) and (17), in addition
to Eq. (52), there is an effective current on the surface given by

${\bf j}_{S,{\rm eff}}=\delta(\rho-R)\,\hat{\bf n}\times{\bf M}_{\rm
eff}\big{|}_{\rho=R}.$ (54)

For instance, as we show in App. B, a scalar coupled to electromagnetism will
generate ${\bf M}_{\rm eff}=-g_{\varphi\gamma\gamma}\varphi{\bf B}_{0}$, and
therefore ${\bf j}_{S,{\rm eff}}=-g_{\varphi\gamma\gamma}\varphi\,{\bf j}$.
For axions, however, ${\bf M}_{\rm eff}=g_{a\gamma\gamma}a{\bf E}$ and ${\bf
j}_{S,{\rm eff}}=0$, as the tangential component of the electric field must be
continuous across any boundary jackson_classical_1999 . This will be found by
an explicit calculation in App. B. To obtain the effective surface for GWs,
Eq. (54) can be cast as $\hat{\bf u}\cdot{\bf j}_{S,{\rm
eff}}=\pm\delta(\rho-R)\,(\hat{\bf u}\times\hat{\bf e}_{\rho})\cdot{\bf
M}_{\rm eff}\big{|}_{\rho=R}$, with $\hat{\bf u}$ an arbitrary unit vector, or
in components

$\displaystyle j_{S,{\rm eff},\phi}$ $\displaystyle=$
$\displaystyle\mp\,\delta(\rho-R)(\hat{\bf
e}_{z})_{i}\left(-h_{ij}B_{j}-\frac{1}{2}hB_{i}+h_{jj}B_{i}\right)\\!,$ (55)
$\displaystyle j_{S,{\rm eff},z}$ $\displaystyle=$
$\displaystyle\pm\,\delta(\rho-R)(\hat{\bf
e}_{\phi})_{i}\left(-h_{ij}B_{j}-\frac{1}{2}hB_{i}+h_{jj}B_{i}\right)\\!,$
(56)

where we assume the system is interacting purely with a magnetic field.

###  A.4 $(\omega L)$ power counting in the proper detector frame

The proper detector frame is the closest analogue to the inertial reference
frame of the laboratory, and therefore allows for a simple description of the
experimentally generated electromagnetic fields. As Eq. (1) demonstrates, in
the proper detector frame the leading order frequency contribution to $h$
occurs at ${\cal O}[(\omega L)^{2}]$. Accordingly, when the GW interacts with
static electromagnetic fields, the leading order gauge invariant contribution
to the induced magnetic field and measurable magnetic flux will also be
$(\omega L)^{2}$, as seen explicitly in, for example, Refs. Berlin:2021txa ;
Domcke:2022rgu . In this appendix we explain the physical origin of this
scaling, and in particular justify the absence of any contribution at ${\cal
O}(\omega L)$.

The starting point is that the proper detector frame corresponds to Fermi
normal coordinates FortiniGualdi ; Marzlin:1994ia ; Rakhmanov:2014noa , which
are freely falling locally inertial coordinates defined along a geodesic,
$x_{0}$. In effect, Fermi normal coordinates are the extension of Riemmann
normal coordinates to an entire worldline. In terms of the specified geodesic,
we can evaluate the metric at an arbitrary spacetime point as follows,

$g_{\mu\nu}(x)=g_{\mu\nu}(x_{0})+(x-x_{0})^{\alpha}\partial_{\alpha}g_{\mu\nu}(x_{0})+(x-x_{0})^{\alpha}(x-x_{0})^{\beta}\partial_{\alpha}\partial_{\beta}g_{\mu\nu}(x_{0})+\ldots.$
(57)

To wit, we can take $x_{0}$ to represent the worldline of the center of our
detector, and then $x$ could represent an arbitrary point in the detector
where we wish to evaluate the impact of the GW. This implies that
parametrically $(x-x_{0})\sim L$, where as throughout the main text $L$ is a
characteristic length scale of the instrument. Now, Fermi normal coordinates
are locally flat, and therefore we have $g_{\mu\nu}(x_{0})=\eta_{\mu\nu}$.
Performing the usual linear decomposition of the metric for a GW propagating
in flat space, $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$, we can therefore
identify the contribution from the GW with the derivative terms in Eq. (57).
Assuming there is no backreaction from the instrument on the GW, and in
addition that the detector can be treated as rigid, if we have a monochromatic
incident source of frequency $\omega$, then
$\partial^{n}g_{\mu\nu}(x_{0})\sim\omega^{n}g_{\mu\nu}(x_{0})$. In this
language, the absence of an ${\cal O}(\omega)$ term in the description of the
GW is reduced to explaining why the single derivative contribution to Eq. (57)
must vanish. This is straightforward: Fermi normal coordinates are a locally
inertial reference frame, so that all Christoffel symbols vanish along
$x_{0}$, and therefore $\partial_{\alpha}g_{\mu\nu}(x_{0})=0$. Accordingly,
the GW in Fermi normal coordinates has a leading contribution at ${\cal
O}(\omega^{2})$. The final ingredient is to transform from a freely falling
frame to the non-inertial frame of a laboratory on the surface of the Earth,
which define the proper detector frame. However, it can be shown that this
transformation introduces contributions only at significantly lower
frequencies than we consider here Ni:1978zz , and therefore does not impact
the above argument.

##  Appendix B Scalar and Axion Electrodynamics

In this appendix, we expand on our discussion of scalar and axion
electrodynamics. In particular, we will compare the well studied pseudoscalar
axion interaction
$-\tfrac{1}{4}g_{a\gamma\gamma}aF^{\mu\nu}\tilde{F}_{\mu\nu}$,191919We define
the dual field strength tensor as
$\tilde{F}^{\mu\nu}\equiv\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}$,
where $\epsilon^{\mu\nu\rho\sigma}$ is the ordinary totally antisymmetric
symbol (with $\epsilon^{0123}=1$). to the scalar equivalent202020We remain
agnostic as to the UV details of this scenario, for a discussion, see e.g.
Refs. Arvanitaki:2014faa ; Banerjee:2022sqg . Our focus is simply to study how
such a coupling would differ from the conventional axion interaction.

${\cal L}\supset-\frac{1}{4}g_{\varphi\gamma\gamma}\varphi
F^{\mu\nu}F_{\mu\nu}.$ (58)

In the main text, we used this as a motivating example, as it involved several
of the features we explored for GW signals, explicitly the importance of the
symmetry of the detector and further the additional contributions we receive
from the boundaries of the detector. Here we will provide a more complete
discussion of the specific differences between axion and scalar
electrodynamics, and then determine the flux scalar dark matter could induce
in various lumped-element circuit instruments.

In direct analogy to the axion, a scalar field that couples as in Eq. (58)
will modify Maxwell’s equations. If we work perturbatively in the coupling
$g$, writing $F^{\mu\nu}=F^{\mu\nu}_{0}+F^{\mu\nu}_{a/\varphi}+{\cal
O}(g^{2})$, we have the equations of motion for the induced fields,

$\displaystyle\partial_{\nu}F^{\mu\nu}_{\varphi}$
$\displaystyle=\partial_{\nu}(g_{\varphi\gamma\gamma}\varphi
F_{0}^{\nu\mu})=g_{\varphi\gamma\gamma}(\partial_{\nu}\varphi)F_{0}^{\nu\mu}-g_{\varphi\gamma\gamma}\varphi
j^{\mu},$ (59) $\displaystyle\partial_{\nu}F^{\mu\nu}_{a}$
$\displaystyle=\partial_{\nu}(g_{a\gamma\gamma}a\tilde{F}_{0}^{\nu\mu})=g_{a\gamma\gamma}(\partial_{\nu}a)\tilde{F}_{0}^{\nu\mu},$

The final results follow from $\partial_{\nu}\tilde{F}^{\mu\nu}=0$ for the
axion, and $\partial_{\nu}F_{0}^{\mu\nu}=j^{\mu}$ for the scalar. The presence
of a coupling to the current, which also must be included for the GW, is a
novelty that does not arise for the axion. For instance, this interaction will
give rise to an oscillating contribution to the fields generated by $j^{\mu}$,
as explored in Ref. Bloch:2023uis . (There have also been discussions of using
axion haloscope inspired instruments to detect $g_{\varphi\gamma\gamma}$, see
Refs. Flambaum:2022zuq ; McAllister:2022ibe .)

For both cases, Eq. (59) demonstrates that we can define an effective
magnetisation and polarisation tensor as in Ref. Domcke:2022rgu . In
particular, we have ${\cal
M}_{\varphi}^{\mu\nu}=g_{\varphi\gamma\gamma}\varphi F^{\mu\nu}$ and ${\cal
M}_{a}^{\mu\nu}=g_{a\gamma\gamma}a\tilde{F}^{\mu\nu}$, which yields explicit
polarisation and magnetisation vectors,

${\bf P}_{\varphi}=g_{\varphi\gamma\gamma}\varphi{\bf E},\hskip 14.22636pt{\bf
M}_{\varphi}=-g_{\varphi\gamma\gamma}\varphi{\bf B},\hskip 14.22636pt{\bf
P}_{a}=g_{a\gamma\gamma}a{\bf B},\hskip 14.22636pt{\bf
M}_{a}=g_{a\gamma\gamma}a{\bf E},$ (60)

and, therefore, the following inhomogeneous equations for the induced fields,

$\displaystyle\nabla\cdot{\bf E}_{\varphi}$
$\displaystyle=-g_{\varphi\gamma\gamma}{\bf E}\cdot\nabla\varphi-
g_{\varphi\gamma\gamma}\varphi\rho,$ (61) $\displaystyle\nabla\cdot{\bf
E}_{a}$ $\displaystyle=-g_{a\gamma\gamma}{\bf B}\cdot\nabla a,$
$\displaystyle\nabla\times{\bf B}_{\varphi}$ $\displaystyle=\partial_{t}{\bf
E}_{\varphi}-g_{\varphi\gamma\gamma}(\nabla\varphi)\times{\bf
B}+g_{\varphi\gamma\gamma}(\partial_{t}\varphi){\bf
E}-g_{\varphi\gamma\gamma}\varphi{\bf j},$ $\displaystyle\nabla\times{\bf
B}_{a}$ $\displaystyle=\partial_{t}{\bf E}_{a}\hskip
1.42271pt+g_{a\gamma\gamma}(\nabla a)\times{\bf E}\hskip
2.84544pt+g_{a\gamma\gamma}(\partial_{t}a){\bf B}.$

From these, we can determine the induced fields a scalar or axion would
generate for various laboratory field configurations.

We now specialise to the situation where the background electric field
vanishes, as is commonly employed in experiments. The effective currents that
will then source magnetic fields are determined from Eq. (61) as ${\bf
j}^{a}_{\rm eff}=g_{a\gamma\gamma}(\partial_{t}a){\bf B}$ and ${\bf
j}^{\varphi}_{\rm eff}=-g_{\varphi\gamma\gamma}\nabla\times(\varphi{\bf B})$.
From here, in analogy to Eq. (9), we find the following transformation
properties for these currents

${\bf j}_{\rm eff}^{a}(P_{\alpha}{\bf r},P_{\alpha}{\bf
k})=\eta_{\alpha}P_{\alpha}\,{\bf j}_{\rm eff}^{a}({\bf r},{\bf k}),\hskip
14.22636pt{\bf j}_{\rm eff}^{\varphi}(P_{\alpha}{\bf r},P_{\alpha}{\bf
k})=-\eta_{\alpha}P_{\alpha}\,{\bf j}_{\rm eff}^{\varphi}({\bf r},{\bf k}).$
(62)

In the language we introduced for gravitational waves, we see that the axion
and scalar transform with $\sigma=-1$ and $+1$ respectively, being the spin-0
counterparts of $h^{\times}$ and $h^{+}$. According to selection rule 2
derived in Sec. 4, for a detector with cylindrical symmetry, there is only
sensitivity to either $h^{\times}$ and or $h^{+}$.212121We emphasise once more
that this statement assumes the polarisations being defined as in Eq. (2). The
proof of the selection rule only used Eq. (11) which followed from Eq. (9),
that is directly analogous to the transformations in Eq. (62). Accordingly
axions and scalars must also obey selection rule 2, demonstrating that when
full azimuthal symmetry is in place an instrument can only be sensitive to one
of the two scalar waves. This is shown explicitly in Tab. 4, which summarises
our symmetry based results for those geometries that are sensitive to an axion
and scalar. This is the spin-0 analogue to Tab. 3. Next we will expand upon
these claims by presenting the explicit results for several geometries (these
same three cases were considered for the GW in Sec. 4).

|  Solenoid: ${\bf B}_{0}\propto\hat{\bf e}_{z}$ |  Toroid: ${\bf B}_{0}\propto\hat{\bf e}_{\phi}$
---|---|---
$\hat{\bf n}^{\prime}\propto\hat{\bf e}_{z}$ |  scalar $\Phi_{a}\equiv 0$, $\Phi_{\varphi}\neq 0$ |  axion (ABRA) $\Phi_{a}\neq 0$, $\Phi_{\varphi}\equiv 0$
$\hat{\bf n}^{\prime}\propto\hat{\bf e}_{\phi}$ |  axion (BASE) $\Phi_{a}\neq 0$, $\Phi_{\varphi}=0$ |  scalar $\Phi_{a}\equiv 0$, $\Phi_{\varphi}\neq 0$
$\hat{\bf n}^{\prime}\propto\hat{\bf e}_{\rho}$ |  scalar $\Phi_{a}\equiv 0$, $\Phi_{\varphi}=0$ |  axion $\Phi_{a}=0$, $\Phi_{\varphi}=0$
Table 4: A summary of the sensitivity of various detector geometries to an
axion or scalar coupled to electromagnetism. Columns specify the laboratory
magnetic field configuration and rows specify the component of the induced
magnetic field which is measured. As manifest in Eq. (62) there is a direct
analogy to these results and those for the GW given in Tab. 3. The magnetic
fluxes stated are derived under the assumption of full cylindrical symmetry.
Cases where we write $\Phi=0$ implies that the cylindrical symmetry has killed
these results, as we had either $B^{a,\varphi}\propto z^{\prime}$, or for the
scalar something of the form
$B^{a,\varphi}\propto\sin(\phi^{\prime}-\phi_{\varphi})$. On the other hand,
where we write $\Phi\equiv 0$ sensitivity can only be recovered by modifying
${\bf B}$, as the vanishing flux originates arises from $B^{a,\varphi}=0$.

#### Toroidal magnet with a horizontal pickup loop.

We first study an instrument with ${\bf B}_{0}\propto\hat{\bf e}_{\phi}$ and
where $\hat{\bf n}^{\prime}\propto\hat{\bf e}_{z}$. An explicit example of
such a geometry is ABRACADABRA. The magnetic field in this case was already
provided in Eq. (16), and again the toroid has inner and outer radii given by
$R$ and $R+a$, and a height $H$ which we take to be parametrically larger than
both. The induced field in the $z$ direction – the one the pickup loop will
measure – for each case can be computed as,

$\displaystyle B^{\varphi}_{z}({\bf r}^{\prime})$ $\displaystyle=0,$ (63)
$\displaystyle B^{a}_{z}({\bf r}^{\prime})$
$\displaystyle=g_{a\gamma\gamma}(\partial_{t}a)B_{0}R\left[\ln\left(1+\frac{a}{R}\right)-\frac{a(a+2R)}{H^{2}}\right]\\!.$

As in the main text, ${\bf
r}^{\prime}=(\rho^{\prime},\,\phi^{\prime},\,z^{\prime})$ is the cylindrical
coordinate system where the field is measured and integrated over by the
pickup loop. The axion contribution here is only stated to ${\cal O}(H^{-2})$,
whereas the scalar result is exact: there is no scalar induced magnetic field
in the $z$ direction. This results from the azimuthal symmetry of the toroid,
a toroid with a wedge in $\phi$ removed will have a non-zero
$B^{\varphi}_{z}({\bf r}^{\prime})$. Regardless, the existing and planned
toroidal axion instruments operating in this range, such as ABRACADABRA,
SHAFT, or DMRadio-50L would have an exactly vanishing sensitivity to a scalar
dark-matter signal, as they would for the $h^{+}$ polarisation of a GW, both
consistent with selection rule 2.

#### Solenoidal magnet with an array of vertical pickup loops.

Next we consider the primary configuration studied in the main text, where
${\bf B}_{0}\propto\hat{\bf e}_{z}$ and $\hat{\bf n}^{\prime}\propto\hat{\bf
e}_{\phi}$, as pursued by, for example, the BASE collaboration. We will
compute the azimuthal component of the magnetic field, as measured by a
vertical pickup loop, and again assume that the height of the solenoid is
parametrically larger than the other scales. Adopting the magnetic field in
Eq. (17), to ${\cal O}(H^{-2})$ we have222222To facilitate comparison with the
axion, in the scalar case we used the fact that the phase of the dark-matter
wave will be $\sim m(t-{\bf v}\cdot{\bf x})$ to write $\nabla\varphi=-{\bf
v}(\partial_{t}\varphi)$.

$\displaystyle B^{\varphi}_{\phi}({\bf r}^{\prime})$
$\displaystyle=2g_{\varphi\gamma\gamma}(\partial_{t}\varphi)vB_{0}z^{\prime}\frac{R^{2}}{H^{2}}\sin\theta_{\varphi}\sin(\phi^{\prime}-\phi_{\varphi}),$
(64) $\displaystyle B^{a}_{\phi}({\bf r}^{\prime})$
$\displaystyle=\frac{1}{2}g_{a\gamma\gamma}(\partial_{t}a)B_{0}\rho^{\prime}\left[1-\frac{2R^{2}}{H^{2}}\right]\\!,$

with $(\theta_{\varphi},\,\phi_{\varphi})$ the coordinates of the scalar
field’s velocity on the celestial sphere. If we compare the two results, we
see that if the experiment measures the magnetic flux within a pickup loop
symmetric in $z^{\prime}$ – as done in ADMX SLIC or BASE – then the axion flux
will grow proportional to the height of the loop, whereas the scalar flux will
exactly vanish as $B^{\varphi}_{\phi}({\bf r}^{\prime})\propto z^{\prime}$.
Further, for any loop in a plane of constant $\phi^{\prime}$, if we wrap it in
a full circle in $\phi^{\prime}$ as DMRadio-m3 plans, then again while the
axion flux increases the scalar flux will vanish, consistent with selection
rule 2. It is straightforward to confirm that these results persist to all
orders in $H$. In other words, solenoidal axion instruments will generically
have exactly vanishing sensitivity to a scalar dark-matter signal, even though
$B^{\varphi}_{\phi}({\bf r}^{\prime})\neq 0$.

As discussed in the main body, the key difference between these scalar and
pseudoscalar axion results can be understood on basic symmetry grounds. In
particular, consistent with Tab. 1, the azimuthal component of the magnetic
field will flip sign under parity. For the axion interaction, this sign flip
is produced by the pseudoscalar field itself, whereas for the scalar, the flip
is generated by the $z^{\prime}$ in Eq. (64), as $Pz^{\prime}=-z^{\prime}$.
(In the axion case, the equivalent dimensions were made up by $\rho^{\prime}$,
which is invariant under parity.) Nevertheless, in spite of these differences,
there are pickup loops that can be designed which would have sensitivity to
both the scalar and axion. This is not the case for a background toroidal
field. Similar arguments can be used to understand the remaining results in
Tab. 4.

#### Solenoidal magnet with a horizontal pickup loop.

Finally, we consider a configuration which is optimally sensitive to a scalar
and minimally sensitive to the axion. From selection rule 2, we simply need a
configuration sensitive to the $\sigma=+1$ contribution, or the $h^{+}$
component for a GW. From Tab. 3, one possibility is a solenoidal magnetic
field ${\bf B}_{0}\propto\hat{\bf e}_{z}$, with a pickup loop that reads out
the $z$ component of the induced field, $\hat{\bf n}^{\prime}\propto\hat{\bf
e}_{z}$. For such a configuration, we have

$\displaystyle B^{\varphi}_{z}({\bf r}^{\prime})$
$\displaystyle=-g_{\varphi\gamma\gamma}\varphi
B_{0}+\frac{1}{2}g_{\varphi\gamma\gamma}(\partial_{t}\varphi)vB_{0}\rho^{\prime}\sin\theta_{\varphi}\cos(\phi^{\prime}-\phi_{\varphi}),$
(65) $\displaystyle B^{a}_{z}({\bf r}^{\prime})$ $\displaystyle=0.$

The scalar result is stated to leading order in $1/H$, whereas the axion
result exactly vanishes.

Observe that in Eq. (65) the first term for the scalar is proportional to
$\varphi$ rather than a derivative of the field, and arises from the unique
current term for the scalar already visible in Eq. (59). As discussed, similar
terms arise for the GW. In the scalar case, these contributions are
particularly simple. For instance, as $-g_{\varphi\gamma\gamma}\varphi{\bf
j}\propto{\bf j}$, this contribution must be directly proportional to the
background magnetic field, which is also produced by ${\bf j}$, or ${\bf
B}^{\varphi}\propto{\bf B}_{0}$. This implies that in Tab. 4 such a
contribution can only be measured when $\hat{\bf n}^{\prime}\propto{\bf
B}_{0}$, which only occurs for two of the six cases in the table. This leaves
one case where a scalar contribution is expected by selection rule 2, but
where there is no contribution from the surface current: a solenoidal magnet
with $\hat{\bf n}^{\prime}\propto\hat{\bf e}_{\rho}$ for the pickup loop. In
this case, the magnetic field is generated purely from
$(\nabla\varphi)\times{\bf B}_{0}$, and therefore must be proportional to
$\sin(\phi^{\prime}-\phi_{\varphi})$ or $\cos(\phi^{\prime}-\phi_{\varphi})$.
When the pickup loop has azimuthal symmetry, however, such terms vanish,
explaining why we write $\Phi_{\varphi}=0$ in the table, although it could be
recovered by using a pickup loop that violates the rotational symmetry.

##  Appendix C Parity Properties of External Magnetic Fields

In this appendix, we will demonstrate that it is possible to decompose static
cylindrically symmetric magnetic fields – those invariant under azimuthal and
$z$-reflection symmetries – as a sum of a toroidal and a solenoidal piece,
which as we will show take the form ${\bf B}^{(t)}\propto\hat{\bf e}_{\phi}$
and ${\bf B}^{(s)}\propto\hat{\bf e}_{z}$, respectively. In the idealised
treatment of static laboratory magnetic fields, we usually imagine them
dropping sharply to zero beyond the boundary of a well-defined region.
According to the discussion in App. A.3, a current flows on the surface of
such a boundary, which sources the magnetic field inside. Azimuthal symmetry
dictates that the current takes the form ${\bf
j}\propto\delta(\rho-\rho_{S}(z))$. Together with ${\bf\nabla}\cdot{\bf j}=0$,
this can be used to show that current can always be cast as ${\bf j}={\bf
j}^{(t)}+{\bf j}^{(s)}$, with

$\displaystyle{\bf j}^{(t)}$ $\displaystyle=\frac{{\cal
C}^{(t)}}{\rho}\delta(\rho-\rho_{S}(z))\left([\partial_{z}\rho_{S}(z)]\hat{\bf
e}_{\rho}+\hat{\bf e}_{z}\right)\\!,$ (66) $\displaystyle{\bf j}^{(s)}$
$\displaystyle={\cal C}^{(s)}(z)\delta(\rho-\rho_{S}(z))\,\hat{\bf e}_{\phi},$

where ${\cal C}^{(s)}(z)$ is a function of $z$ while ${\cal C}^{(t)}$ is a
constant. In addition, $z$-reflection symmetry implies that

${\cal C}^{(s)}(-z)={\cal C}^{(s)}(z),\hskip
14.22636pt\rho_{S}(-z)=\rho_{S}(z).$ (67)

Observe that the two terms in Eq. (66) are conserved separately. As a result,
we can similarly decompose the magnetic field sourced by ${\bf j}$ as ${\bf
B}={\bf B}^{(t)}+{\bf B}^{(s)}$, with ${\bf\nabla}\times{\bf B}^{(t)}={\bf
j}^{(t)}$ and ${\bf\nabla}\times{\bf B}^{(s)}={\bf j}^{(s)}$. We will further
discuss each case below, and justify that ${\bf B}^{(t)}\propto\hat{\bf
e}_{\phi}$ and ${\bf B}^{(s)}\propto\hat{\bf e}_{z}$. From this we conclude
that toroidal and solenoidal magnetic fields exhaust all the realistic
configurations invariant under cylindrical and $z$-reflection symmetries, and
further this justifies the symmetry transformations encoded in ${\bf
B}(P_{\alpha}{\bf r})=\eta_{\alpha}P_{\alpha}{\bf B}({\bf r})$ and Tab. 1.

#### Toroidal fields.

As we now show, the magnetic field arising from the toroidal current takes a
simple form, regardless of the function $\rho_{S}(z)$. Due to the cylindrical
symmetry, the magnetic field sourced by ${\bf j}^{(t)}$ points in the
azimuthal direction, ${\bf B}^{(t)}=B(\rho,z)\hat{\bf e}_{\phi}$. For a fixed
$z$, the circulation of the magnetic field along a circular path is $2\pi\rho
B(\rho,z)$. According to Ampère’s law, this must equal $\int d\rho\int
d\phi\,\rho\,j^{(t)}_{z}=2\pi{\cal C}^{(t)}$. Hence

${\bf B}^{(t)}(\rho,z)=\frac{{\cal
C}^{(t)}}{\rho}\Theta(\rho-\rho_{S}(z))\,\hat{\bf e}_{\phi}.$ (68)

The $1/\rho$ dependence of the magnetic field can be alternatively derived by
considering an arbitrary field of the form ${\bf B}=B(\rho,z)\,\hat{\bf
e}_{\phi}$, and then demanding that ${\bf\nabla}\cdot{\bf
B}={\bf\nabla}\times{\bf B}=0$, which holds sufficiently far from the surface.

For the realistic toroidal fields produced in the lab, the current flows up
and then down again, so the total field is the sum of two pieces. Furthermore,
we can usually neglect the $z$-dependence of $\rho_{S}(z)$. In that case, the
general result in Eq. (68) reduces to Eq. (16).

#### Solenoidal fields.

The solenoidal configuration does not lead to an expression as simple as Eq.
(68). Nonetheless, Eq. (67) shows that ${\cal C}^{{(s)}}(z)$ and $\rho_{S}(z)$
are even, and hence $\partial_{z}{\cal
C}^{{(s)}}(0)=\partial_{z}\rho_{S}(0)=0$. Since axion experiments often
measure the induced magnetic flux away from the $z$-boundaries of the external
magnetic field, to a reasonable approximation we can take $\rho_{S}$ and
$\partial_{z}{\cal C}^{{(s)}}$ as constants. In that case ${\bf j}^{(s)}$
reduces to Eq. (40) and the magnetic field takes the form given in Eq. (17).
In general, however, this need not hold, and for instance ${\bf B}^{{(s)}}$
can develop a contribution $\propto\hat{\bf e}_{\rho}$.

##  Appendix D Response Matrix

In this appendix, we derive the response matrix introduced in Eq. (15), and
provide an explicit example. Let us first note that the effective current is a
linear functional of the GW, which implies that there exists a tensor ${\cal
J}^{i}_{mn}({\bf r},{\bf k})$ such that

$j_{\rm eff}^{i}=e^{-\mathrm{i}\mkern 1.0mu\omega t}{\cal J}^{i}_{mn}({\bf
r},{\bf k})\sum_{A}h^{A}e^{A}_{mn}(\hat{\bf k}).$ (69)

Following Eq. (14), this gives Eq. (15)

$\Phi_{h}=e^{-\mathrm{i}\mkern 1.0mu\omega t}D^{mn}({\bf
k})\sum_{A}h^{A}e^{A}_{mn}(\hat{{\bf k}}),\hskip 14.22636pt\text{with}\hskip
14.22636ptD^{mn}({\bf
k})=\int_{\ell}dr^{\prime}_{i}\int_{V_{B}}\frac{d^{3}{\bf r}}{4\pi}\frac{{\cal
J}^{i}_{mn}}{\left|{\bf r}-{\bf r}^{\,\prime}\right|}.$ (70)

As an explicit example, for the solenoidal magnetic field in Eq. (17), we find

$\displaystyle{\cal J}^{\rho}$ $\displaystyle=\frac{1}{12}e^{-\mathrm{i}\mkern
1.0mu\omega t}\omega^{2}B_{0}\Theta(R-\rho)\begin{pmatrix}\rho s_{2\phi}&-\rho
c_{2\phi}&4zs_{\phi}\\\ -\rho c_{2\phi}&-\rho s_{2\phi}&-4zc_{\phi}\\\
4zs_{\phi}&-4zc_{\phi}&0\end{pmatrix}\\!,$ (71) $\displaystyle{\cal J}^{\phi}$
$\displaystyle=\frac{1}{12}e^{-\mathrm{i}\mkern 1.0mu\omega
t}\omega^{2}B_{0}\Theta(R-\rho)\begin{pmatrix}\rho(3+c_{2\phi})&\rho
s_{2\phi}&4zc_{\phi}\\\ \rho s_{2\phi}&\rho(3-c_{2\phi})&4zs_{\phi}\\\
4zc_{\phi}&4zs_{\phi}&-4\rho\end{pmatrix}$
$\displaystyle\;\;-\frac{1}{12}e^{-\mathrm{i}\mkern 1.0mu\omega
t}\omega^{2}B_{0}\delta(R-\rho)\begin{pmatrix}z^{2}+2\rho^{2}+\rho^{2}c_{2\phi}&\rho^{2}s_{2\phi}&4z\rho
c_{\phi}\\\ \rho^{2}s_{2\phi}&z^{2}+2\rho^{2}-\rho^{2}c_{2\phi}&4z\rho
s_{\phi}\\\ 4z\rho c_{\phi}&4z\rho s_{\phi}&5z^{2}-\rho^{2}\end{pmatrix}\\!,$
$\displaystyle{\cal J}^{z}$ $\displaystyle=\frac{1}{4}e^{-\mathrm{i}\mkern
1.0mu\omega t}\omega^{2}B_{0}\Theta(R-\rho)\begin{pmatrix}0&0&-\rho
s_{\phi}\\\ 0&0&\rho c_{\phi}\\\ -\rho s_{\phi}&\rho c_{\phi}&0\end{pmatrix}$
$\displaystyle\;\;-\frac{1}{12}e^{-\mathrm{i}\mkern 1.0mu\omega
t}\omega^{2}B_{0}\delta(R-\rho)\begin{pmatrix}z\rho s_{2\phi}&-z\rho
c_{2\phi}&-\rho^{2}s_{\phi}\\\ -z\rho c_{2\phi}&-2z\rho
c_{\phi}s_{\phi}&\rho^{2}c_{\phi}\\\
-\rho^{2}s_{\phi}&\rho^{2}c_{\phi}&0\end{pmatrix}\\!,$

at ${\cal O}[(\omega L)^{2}]$. From here, considering the specific loop
geometry described in Fig. 1 with $r_{2}=r$ and $r_{1}=0$, the response matrix
is

$D^{mn}({\bf k})=\frac{e^{-\mathrm{i}\mkern 1.0mu\omega
t}}{288}\omega^{2}B_{0}rl\left(13r^{2}-30R^{2}\right)\begin{pmatrix}0&0&-s_{\phi_{\ell}}\\\
0&0&c_{\phi_{\ell}}\\\ -s_{\phi_{\ell}}&c_{\phi_{\ell}}&0\end{pmatrix}\\!.$
(72)

Note that as shown in Eq. (29), for a cylindrical symmetric setup, only the
diagonal part of the response matrix matters, and therefore the zeros in the
response function are consistent with the fact that we do not have any
contribution at ${\cal O}[(\omega L)^{2}]$ for the vertical loop.

##  Appendix E Explicit Expressions for the Effective Current

In this appendix, we provide the analytic expressions for the current induced
by a GW to ${\cal O}[(\omega L)^{3}]$ for both a solenoidal and toroidal
external magnetic field as in Eq. (17) and Eq. (16). Together with Eq. (10) or
Eq. (14), these are used to calculate the induced magnetic field and flux
which can then be measured by a pickup loop. To slightly simplify the
expressions that follow, throughout this appendix we have taken $\phi_{h}=0$,
but this can be restored immediately by sending $\phi\to\phi-\phi_{h}$.

#### Solenoidal magnet.

The three components of the current ${\bf j}=(j_{\rho},\,j_{\phi},\,j_{z})$
are given as follows, where in each case organise the results by polarisation
and power in $\omega$,

$\displaystyle j_{\rho}e^{\mathrm{i}\mkern 1.0mu\omega t}=\,$ $\displaystyle