section0em2.5em subsection2.5em2.5em

# Light Shining Through a Thin Wall: Evanescent Hidden Photon Detection

Asher Berlin<EMAIL_ADDRESS>Roni Harnik<EMAIL_ADDRESS>Ryan Janish
<EMAIL_ADDRESS>Theory Division, Fermi National Accelerator Laboratory,
Batavia, IL 60510, USA  Superconducting Quantum Materials and Systems Center
(SQMS), Fermi National Accelerator Laboratory, Batavia, IL 60510, USA

###### Abstract

A kinetically-mixed hidden photon is sourced as an evanescent mode by
electromagnetic fields that oscillate at a frequency smaller than the hidden
photon mass. These evanescent modes fall off exponentially with distance, but
nevertheless yield detectable signals in a photon regeneration experiment if
the electromagnetic barrier is made sufficiently thin. We consider such an
experiment using superconducting cavities at GHz frequencies, proposing
various cavity and mode arrangements that enable unique sensitivity to hidden
photon masses ranging from $10^{-5}\ \text{eV}$ to $10^{-1}\ \text{eV}$.

††preprint: FERMILAB-PUB-23-073-SQMS-T

##  I Introduction

Massive vector fields with feeble couplings to Standard Model (SM) particles
may readily exist and present a compelling target for experimental searches.
Such _hidden photons_ Holdom (1986) have been well-studied both in their own
right and as a possible component of the dark matter sector (see, e.g., Refs.
Fabbrichesi  _et al._ (2020); Alexander  _et al._ (2016); Battaglieri  _et
al._ (2017); Antypas  _et al._ (2022) and references within). A simple and
natural possibility is that the hidden photon couples to the SM through a
kinetic mixing $\epsilon\ll 1$,

$\displaystyle\mathscr{L}=-\frac{1}{4}\,F^{2}-\frac{1}{4}\,F^{\prime\,2}+\frac{1}{2}\,m_{A^{\prime}}^{2}A^{\prime\,2}+\frac{\epsilon}{2}\,FF^{\prime}-jA~{},$
(1)

where $A$ is the SM photon field and $F$ its field strength, $A^{\prime}$ is
the hidden photon field with mass $m_{A^{\prime}}$ and $F^{\prime}$ its field
strength, and $j$ is the electromagnetic (EM) current density. We have
suppressed Lorentz indices and absorbed the EM coupling into $j$ for brevity.

Many experiments have directly searched for or indirectly constrained the
existence of such a kinetically-mixed hidden photon Alexander  _et al._
(2016); Battaglieri  _et al._ (2017); Caputo  _et al._ (2021); Antypas  _et
al._ (2022). Most of these efforts have involved searching for the effects of
hidden photons that are produced _on-shell_ , since a detectable signal often
requires propagation across a considerable distance compared to an
$A^{\prime}$ Compton wavelength.111 Exceptions to this are, e.g., limits
derived from tests of Coulomb’s law Williams  _et al._ (1971); Abel  _et al._
(2008) and atomic spectroscopy Jaeckel and Roy (2010). Experiments of this
form include photon regeneration or “light-shining-through-wall” (LSW)
experiments, such as ALPS Ehret  _et al._ (2010) and CROWS Betz  _et al._
(2013), as well as Dark SRF, which recently set its first limits employing two
ultra-high quality superconducting radio frequency (SRF) cavities Romanenko
_et al._ (2023).

In the field basis of Eq. (1), an LSW setup involves driving an EM field $A$
at a frequency $\omega$ which in turn produces hidden photons $A^{\prime}$.
These hidden photons have energy $\omega$ and propagate with momentum
$k=\sqrt{\omega^{2}-m_{A^{\prime}}^{2}}$ across a barrier opaque to SM
photons, after which they source a SM field $A$ in a shielded detection
region. For an off-shell hidden photon $m_{A^{\prime}}>\omega$, the signal is
evanescent, i.e., it is suppressed by the propagation distance $d$ as
$e^{-|k|d}$. In most cases, $d$ corresponds to the separation between the
production and detection regions, which is usually chosen to be
$d\gtrsim\omega^{-1}$. Thus, in such an arrangement, searching for more
massive hidden photons requires operating EM sources at higher frequencies,
since this increases the maximum mass for which such particles can propagate
on-shell. This is the strategy employed in current experiments such as ALPS
Ehret  _et al._ (2010) and recently proposed millimeter wavelength setups
Miyazaki  _et al._ (2022).

Figure 1: Left: A sketch of the LSthinW setup. A highly excited emitter cavity
is separated from a quiet receiver cavity by a thin wall. The driven emitter
and signal receiver modes are shown in blue. Right: A closeup of the thin
wall. The visible photon field (blue) is suppressed over the short London
penetration depth $\lambda_{\textnormal{L}}$ whereas the evanescent invisible
mode (pink) extends across the hidden photon Compton wavelength $\sim
1/m_{A^{\prime}}$, which can be orders of magnitude larger than the wall
thickness. The receiver mode is resonantly excited to detectable levels even
though it is effectively sourced by the thin region of non-zero invisible
field near the receiver side of the wall.

However, using higher frequency sources is not a fundamental requirement. It
appears as a consequence of requiring long propagation distances in an LSW
experiment, which can be avoided by employing a thin barrier,
$d\ll\omega^{-1}$. We are thus motivated to consider light-shining-through-
_thin_ -wall (LSthinW) setups, which allow for the detection of evanescent
(i.e., virtual or off-shell) hidden photon signals for
$m_{A^{\prime}}\gg\omega$. There is considerable advantage in this approach as
opposed to increasing the frequency, since lower frequency sources are able to
generate a larger density of source photons and employ very high-quality
resonators, such as SRF cavities.

It should be noted that thin barriers can still provide effective EM
shielding. In the radio frequency (RF) regime, $\omega\sim 1\ \text{GHz}\sim
10\ \mu\text{eV}$ and the length scale $\omega^{-1}$ is of order
$10\;\textnormal{cm}$. This is much larger than the penetration depth of RF
fields into (super)conductors, which can be as small as $\sim 50\ \text{nm}$,
allowing for $d\ll\omega^{-1}$. Further, it is crucial to note that in an
evanescent LSthinW setup, it is not required that the photon detector be
located within a Compton wavelength $m_{A^{\prime}}^{-1}$ of the barrier.
Although the virtual hidden photons are restricted to this small region, they
act as a localized source of on-shell photons which readily propagate to a
distant detector. It follows that, apart from the thickness of the barrier,
there is no exponential suppression relative to any other experimental length
scale, such as the size or frequency of a detection cavity. Similar insights
were noted previously in Ref. Hoogeveen (1992) within the context of LSW
searches for electromagnetically-coupled axions.

In this work, we propose a simple LSthinW setup to search for hidden photons
with mass $m_{A^{\prime}}\gg 1\ \mu\text{eV}$ using high-quality SRF cavities.
As shown schematically in Fig. 1, a driven “emitter” cavity sources both SM
and hidden fields at frequency $\omega\sim 1\ \text{GHz}$. A narrow
superconducting barrier separates and shields the emitter from a quiet
“receiver” cavity. The SM field is exponentially attenuated through the
barrier over the London penetration depth $\lambda_{\textnormal{L}}\sim 50\
\text{nm}$, while the hidden field is attenuated over $1/m_{A^{\prime}}\sim 1\
\text{mm}\times(\text{meV}/m_{A^{\prime}})$. Thus, for a barrier of thickness
$10\ \mu\text{m}\lesssim d\lesssim m_{A^{\prime}}^{-1}$, the receiver cavity
is shielded from the large driven fields, while the evanescent hidden photon
field can penetrate into the receiver cavity and excite its resonant modes. As
we show, such an experiment can provide leading sensitivity to hidden photons
in the $10\ \mu\text{eV}-100\ \text{meV}$ mass range, whether or not they
constitute the dark matter.

The rest of this work proceeds as follows. In Sec. II, we introduce the
formalism needed to calculate LSthinW signals. In Sec. III, we specialize to
the evanescent regime and highlight the signal parametrics particular to this
case. We apply this formalism to optimize a simple LSthinW setup in Sec. IV
and then estimate the sensitivity of several concrete searches. We conclude in
Sec. V.

##  II General Formalism

Here we review the classical equations of motion governing SM and hidden
photon fields, the production of hidden photons, and their excitation of EM
cavities. We use some of the formalism of Ref. Graham  _et al._ (2014), which
studied on-shell hidden photons in the far-field limit, but we reformulate
some of the key results to highlight the physics of the off-shell evanescent
case. The results of this section are general, however, and can be applied to
hidden photons of any mass (provided that $\epsilon\ll 1$) and are equivalent
to the results of Ref. Graham  _et al._ (2014). Later in Sec. III, we will
apply these results to the evanescent case.

###  II.1 Classical Field Equations

We begin by obtaining the classical wave equations for the photon and hidden
photon fields in a field basis which diagonalizes the kinetic and mass terms
of Eq. (1). Eq. (1) is converted by means of the field-redefinitions

$\displaystyle A^{\mu}$ $\displaystyle\to
A^{\mu}+\frac{\epsilon}{\sqrt{1-\epsilon^{2}}}\,A^{\prime\,\mu}$ (2a)
$\displaystyle A^{\prime\,\mu}$
$\displaystyle\to\frac{1}{\sqrt{1-\epsilon^{2}}}\,A^{\prime\,\mu}~{},$ (2b)

which yields

$\displaystyle\mathscr{L}$
$\displaystyle=-\frac{1}{4}\,F_{\mu\nu}F^{\mu\nu}-\frac{1}{4}\,F_{\mu\nu}^{\prime}F^{\prime\,\mu\nu}+\frac{1}{2}\,\frac{m_{A^{\prime}}^{2}}{1-\epsilon^{2}}\,A_{\mu}^{\prime}A^{\prime\,\mu}$
$\displaystyle-
j_{\mu}\,\Big{(}A^{\mu}+\frac{\epsilon}{\sqrt{1-\epsilon^{2}}}\,A^{\prime\,\mu}\Big{)}~{}.$
(3)

In this basis, SM charges couple to both the massless and massive fields,
which are themselves uncoupled. Hence, the wave equations follow immediately
from Eq. (II.1) as

$\displaystyle\Big{(}\partial_{t}^{2}-\nabla^{2}\Big{)}\,\vec{A}$
$\displaystyle=\vec{j}$ (4a)
$\displaystyle\Big{(}\partial_{t}^{2}-\nabla^{2}+\frac{m_{A^{\prime}}^{2}}{1-\epsilon^{2}}\Big{)}\,\vec{A}^{\,\prime}$
$\displaystyle=\frac{\epsilon}{\sqrt{1-\epsilon^{2}}}\,\vec{j}~{},$ (4b)

where $\vec{A}$ and $\vec{A}^{{\,\prime}}$ are the SM and hidden vector
potentials, respectively, and $\vec{j}$ is the SM current density. Eq. (4a) is
Maxwell’s wave equation in Lorenz gauge, with the scalar potential $\phi$
determined by $\partial_{t}\phi=-\vec{\nabla}\cdot\vec{A}\,$. The hidden
potentials obey an identical condition,222For massive fields, this is not a
choice of gauge, but instead follows from conservation of charge, i.e.,
$m_{A^{\prime}}^{2}\partial_{\mu}A^{\prime\,\mu}=-\partial_{\mu}j^{\prime\,\mu}$.
and the electric and magnetic fields are determined from the potentials in the
usual manner for both the SM and hidden fields.

As discussed in Ref. Graham  _et al._ (2014), to compute field propagation
across a conducting barrier it is useful to define “visible” and “invisible”
linear combinations of fields that couple to or are completely sequestered
from SM sources, respectively. In such a basis, conducting boundary conditions
are simple to enforce as they apply only to the visible field and do so in the
standard way. The required transformation can be read off of Eq. (II.1). We
define

$\displaystyle A_{\text{vis}}^{\mu}$
$\displaystyle=\sqrt{1-\epsilon^{2}}\,A^{\mu}+\epsilon\,A^{\prime\,\mu}$ (5a)
$\displaystyle A_{\text{inv}}^{\mu}$
$\displaystyle=-\epsilon\,A^{\mu}+\sqrt{1-\epsilon^{2}}\,A^{\prime\,\mu}~{},$
(5b)

and also take $e\to\sqrt{1-\epsilon^{2}}\,e$ such that the visible field
couples to SM charge via the empirical EM coupling constant. Eq. (4) becomes

$\displaystyle\Big{(}\partial_{t}^{2}-\nabla^{2}\Big{)}\,\vec{A}_{\textnormal{vis}}$
$\displaystyle=\vec{j}-\frac{\epsilon\,m_{A^{\prime}}^{2}}{1-\epsilon^{2}}\,\vec{A}^{{\,\prime}}$
(6a)
$\displaystyle\Big{(}\partial_{t}^{2}-\nabla^{2}+\frac{m_{A^{\prime}}^{2}}{1-\epsilon^{2}}\Big{)}\,\vec{A}_{\textnormal{inv}}$
$\displaystyle=-\frac{\epsilon\,m_{A^{\prime}}^{2}}{1-\epsilon^{2}}\,\vec{A}~{}.$
(6b)

Note that we have kept the right-hand-side of these equations written in terms
of the mass-basis fields $\vec{A}$ and $\vec{A}^{\prime}$ for simplicity, but
since these are linear combinations of $\vec{A}_{\textnormal{vis}}$ and
$\vec{A}_{\textnormal{inv}}$, Eq. (6) contains only two undetermined fields.
On the right-hand-side of Eq. (6a), $\vec{A}^{\,\prime}$ enters analogously to
a current that sources $\vec{A}_{\textnormal{vis}}$. Hence, to leading order
in $\epsilon\ll 1$, we define this _effective current_ as

$\vec{j}_{\text{eff}}=-\epsilon\,m_{A^{\prime}}^{2}\,\vec{A}^{\,\prime}~{}.$
(7)

This form of the effective current is equivalent to that presented in Ref.
Graham  _et al._ (2014),
$\partial_{t}\,\vec{j}_{\text{eff}}=\epsilon\,m_{A^{\prime}}^{2}(m_{A^{\prime}}^{2}\vec{E}^{{\,\prime}}-\vec{\nabla}\,\vec{\nabla}\cdot\vec{E}^{{\,\prime}})$.
This follows from the definition of $\vec{E}^{{\,\prime}}$ in terms of its
potentials, as well as Gauss’s law for the hidden field in vacuum
$\vec{\nabla}\cdot\vec{E}^{{\,\prime}}=-m_{A^{\prime}}^{2}\phi^{{\,\prime}}$,
as derived from Eq. (II.1).

###  II.2 Sourcing the Hidden Field

Consider an emitter cavity of volume $V_{\textnormal{em}}$ which contains a
driven, monochromatic SM field at frequency $\omega$. We want to determine the
hidden fields that are produced outside of this cavity. We work here with the
mass-basis fields of Eq. (4), as this will prove to yield a result which is
particularly useful for the evanescent case.

At $\mathcal{O}{(\epsilon^{0})}$, the emitter cavity is described by the
driven massless fields $\vec{E}_{\textnormal{em}}$ and
$\vec{B}_{\textnormal{em}}$ oscillating at frequency $\omega$, which vanish
within the conducting walls of the cavity and obey conductor boundary
conditions333 $\vec{B}_{\textnormal{em}}$ is tangential to the surface and
$\vec{E}_{\textnormal{em}}$ is normal to the surface. on the inner cavity
surface $\partial V_{\textnormal{em}}$. These boundary conditions are
maintained by charges and currents on $\partial V_{\textnormal{em}}$, e.g.,
the tangential magnetic field is supported by a surface current
$\vec{K}_{\text{em}}$ given by

$\displaystyle\vec{K}_{\textnormal{em}}=-\hat{n}\times\vec{B}_{\textnormal{em}}~{},$
(8)

where $\hat{n}$ is the unit normal orientated outward to $\partial
V_{\textnormal{em}}$ and $\vec{B}_{\textnormal{em}}$ is evaluated on $\partial
V_{\textnormal{em}}$. From Eq. (4b), this current sources an
$\vec{A}^{{\,\prime}}$ which to leading order in $\epsilon$ is

$\displaystyle\vec{A}^{{\,\prime}}\simeq\frac{\epsilon}{4\pi}\,e^{i\omega
t}\int_{\partial
V_{\text{em}}}\mathop{}\\!\mathrm{d^{2}}{x^{\prime}}\mathop{}\\!~{}\frac{e^{-ik|\vec{x}-\vec{x}^{\,\prime}|}}{|\vec{x}-\vec{x}^{\,\prime}|}~{}\vec{K}_{\text{em}}(\vec{x}^{\,\prime})~{},$
(9)

where the integration is over the inner surface of the emitter cavity and we
have defined the wavenumber

$k=\begin{cases}\sqrt{\omega^{2}-m_{A^{\prime}}^{2}}&(m_{A^{\prime}}<\omega)\\\
-i\,\sqrt{m_{A^{\prime}}^{2}-\omega^{2}}&(m_{A^{\prime}}>\omega)~{}.\end{cases}$
(10)

In Eq. (9), we have ignored boundary conditions on the hidden field (i.e.,
that $\vec{E}_{\textnormal{vis}}$ and $\vec{B}_{\textnormal{vis}}$ must obey
conductor boundary conditions at the surface of the emitter and reciever
cavities). Enforcing these boundary conditions amounts to incorporating small
screening corrections to the current in the integrand of Eq. (9), including
those generated in the receiver cavity. However, this only corrects
$\vec{A}^{{\,\prime}}$ at $\mathcal{O}{(\epsilon^{3})}$ and may be ignored. To
see this, note that outside of the emitter cavity, Eq. (9) implies that
$E_{\textnormal{vis}}\sim\epsilon
E^{{\,\prime}}\sim\mathcal{O}{(\epsilon^{2}\,K_{\textnormal{em}})}$. Thus,
enforcing boundary conditions perturbs the surface currents by
$\mathcal{O}{(\epsilon^{2})}$. Therefore, to leading order in $\epsilon\ll 1$,
Eq. (9) is valid everywhere outside of the emitter cavity, including in the
interior of the receiver cavity.

###  II.3 Cavity Excitation via Hidden Photons

From Eq. (6a), we may derive the wave equation for visible electric fields. To
leading order in $\epsilon$ this is

$\vec{\nabla}\times\vec{\nabla}\times\vec{E}_{\text{vis}}+\partial_{t}^{2}\vec{E}_{\text{vis}}\simeq-\partial_{t}\big{(}\vec{j}+\vec{j}_{\textnormal{eff}}\big{)}~{}.$
(11)

We can identify $\vec{E}_{\textnormal{vis}}$ with the SM electric field. It
obeys the SM wave equation and conductor boundary conditions, except for the
introduction of a new source term, which from Eqs. (7) and (9) is

$\vec{j}_{\text{eff}}\simeq-\frac{\epsilon^{2}\,m_{A^{\prime}}^{2}}{4\pi}\,e^{i\omega
t}\int_{\partial
V_{\text{em}}}\mathop{}\\!\mathrm{d^{2}}{x^{\prime}}\mathop{}\\!~{}\frac{e^{-ik|\vec{x}-\vec{x}^{\,\prime}|}}{|\vec{x}-\vec{x}^{\,\prime}|}~{}\vec{K}_{\text{em}}(\vec{x}^{\,\prime})~{}.$
(12)

The effective current $\vec{j}_{\text{eff}}$ sources visible fields just as SM
currents do. Suppose that $\vec{j}_{\text{eff}}$ is monochromatic with
frequency $\omega$ near the $p^{\textnormal{th}}$ mode of the receiver cavity,
$\omega\simeq\omega_{p}$. We decompose the visible field in the receiver
cavity in terms of this resonant mode as
$\vec{E}_{\textnormal{vis}}(\vec{x},t)=\mathcal{E}_{p}(t)\,\vec{E}_{p}(\vec{x})$,
where $\mathcal{E}_{p}(t)$ is a complex number which gives the mode’s
excitation amplitude and phase at time $t$, and the mode profile
$\vec{E}_{p}(\vec{x})$ satisfies

$\vec{\nabla}\cdot\vec{E}_{p}=(\nabla^{2}+\omega_{p}^{2})\,\vec{E}_{p}=\vec{E}_{p}\times\hat{n}\Big{|}_{\partial
V_{\text{rec}}}=0~{}.$ (13)

Here, $\omega_{p}$ is the mode’s resonant frequency and $\hat{n}$ is the unit
normal to the surface $\partial V_{\text{rec}}$ of the receiver cavity.
Performing this decomposition in Eq. (11) yields an equation for the time-
evolution of $\mathcal{E}_{p}(t)$,

$\Big{(}\partial_{t}^{2}+\frac{\omega_{p}}{Q}\,\partial_{t}+\omega_{p}^{2}\Big{)}\,\mathcal{E}_{p}\simeq-\,\frac{\int_{\text{rec}}\mathop{}\\!\mathrm{d^{3}}{x}\mathop{}\\!\vec{E}_{p}^{*}\cdot\partial_{t}(\vec{j}+\vec{j}_{\text{eff}})}{\int_{\text{rec}}\mathop{}\\!\mathrm{d^{3}}{x}\mathop{}\\!|\vec{E}_{p}|^{2}}~{},$
(14)

where on the left-hand-side we have inserted a damping term, quantified by the
quality factor $Q$ of the receiver resonant mode. The integrals on the right-
hand-side are performed over the volume of the receiver. Assuming that there
are no SM currents within the inner volume of the receiver cavity, $\vec{j}$
can be dropped from the right-hand-side of Eq. (14) since the conducting
boundary condition implies that current on the inner walls of the cavity is
orthogonal to $\vec{E}_{p}$ at the boundary. On resonance
($\omega=\omega_{p}$), the visible field in the receiver is

$\vec{E}_{\textnormal{vis}}(\vec{x})\simeq-\,\frac{Q}{\omega_{p}}\,\vec{E}_{p}(\vec{x})\,\frac{\int_{\text{rec}}\mathop{}\\!\mathrm{d^{3}}{x^{\prime}}\mathop{}\\!\vec{E}_{p}^{*}(\vec{x}^{\,\prime})\cdot\vec{j}_{\text{eff}}(\vec{x}^{\,\prime})}{\int_{\text{rec}}\mathop{}\\!\mathrm{d^{3}}{x^{\prime}}\mathop{}\\!|\vec{E}_{p}(\vec{x}^{\,\prime})|^{2}}~{},$
(15)

which corresponds to a signal power of

$P_{\text{sig}}=\frac{\omega_{p}}{Q}\int_{\text{em}}\mathop{}\\!\mathrm{d^{3}}{x}\mathop{}\\!|\vec{E}_{\textnormal{vis}}|^{2}\simeq\frac{Q}{\omega_{p}}\,\frac{\big{|}\int_{\text{rec}}\mathop{}\\!\mathrm{d^{3}}{x}\mathop{}\\!\vec{E}_{p}^{*}\cdot\vec{j}_{\text{eff}}\big{|}^{2}}{\int_{\text{rec}}\mathop{}\\!\mathrm{d^{3}}{x}\mathop{}\\!|\vec{E}_{p}|^{2}}~{}.$
(16)

##  III Evanescent Signals

Figure 2: A closeup of the thin wall $\mathcal{S}$ separating the emitter and
receiver cavities. To evaluate $\vec{j}_{\text{eff}}$ at the point $\vec{x}$
in the receiver cavity, we use the fact that the integral over $\mathcal{S}$
in Eq. (12) is heavily weighted near the point $\vec{x}^{\,\prime}_{0}$ in the
$m_{A^{\prime}}\gg\omega$ limit. This yields Eq. (17).

Eqs. (12) and (16) can be numerically evaluated for any choice of experimental
and model parameters to determine the corresponding signal strength. However,
it is useful to also have analytical expressions for simple geometries. This
was done in Ref. Graham  _et al._ (2014) in the limit that the separation $d$
between the emitter and receiver cavities is much greater than the size of
either cavity. In this work we are interested in the opposite limit, where the
cavities are very closely spaced and in the evanescent regime. As we show
below, it is also possible to analytically evaluate $\vec{j}_{\text{eff}}$ and
$P_{\textnormal{sig}}$ in this limit.

To begin, let us take a simple LSthinW setup consisting of two cavity volumes
obtained from a single larger volume partitioned by a thin conducting surface
$\mathcal{S}$ (as shown in Fig. 2), whose thickness $d$ is much smaller than
both the cavity length and the Compton wavelength of the hidden photon, but
much thicker than the EM penetration depth of the partition material. Consider
$\vec{j}_{\text{eff}}(\vec{x})$ at a point $\vec{x}$ fixed in the receiver
cavity, within a distance $m_{A^{\prime}}^{-1}$ of $\mathcal{S}$ and a
distance much greater than $m_{A^{\prime}}^{-1}$ from the external walls of
the receiver cavity (i.e., the top and bottom of Fig. 1). The integrand in the
expression for $j_{\text{eff}}$ in Eq. (12) is exponentially suppressed for
$|\vec{x}^{\,\prime}-\vec{x}|\gg 1/m_{A^{\prime}}$, where $\vec{x}^{\,\prime}$
is a point on $\mathcal{S}$. Thus the $\vec{x}^{\,\prime}$ integral is heavily
weighted near $\vec{x}^{\,\prime}_{0}$, defined as the point on $\mathcal{S}$
that is closest to $\vec{x}$. These coordinates are shown in Fig. 2. We thus
expand $\vec{K}_{\text{em}}(\vec{x}^{\,\prime})$ around
$\vec{x}^{\,\prime}_{0}$ in the integrand of Eq. (12), which is a good
approximation for $\vec{x}$ away from the external edges of the receiver
cavity and provided that $\omega/m_{A^{\prime}}\ll 1$. Then, taking
$\mathcal{S}$ to be approximately planar near $\vec{x}^{\,\prime}_{0}$, we may
extend the integration region to an infinite plane and evaluate Eq. (12) to
find

$\displaystyle\vec{j}_{\text{eff}}(\vec{x})\simeq\,$
$\displaystyle\frac{1}{2}\,\epsilon^{2}\,m_{A^{\prime}}\,e^{i\omega
t}\,e^{-m_{A^{\prime}}|\vec{x}-\vec{x}^{\,\prime}_{0}|}~{}\hat{n}\times\vec{B}_{\text{em}}(\vec{x}^{\,\prime}_{0})$
(17)

up to $\mathcal{O}{(\omega/m_{A^{\prime}})}$, where as in Fig. 2 the
$\hat{n}$-axis is defined along the direction normal to $\mathcal{S}$ pointing
from $\vec{x}^{\,\prime}_{0}$ to $\vec{x}$. For later convenience, we
factorize the spatial dependence of $\vec{j}_{\text{eff}}$ into a
dimensionless function $\hat{j\,}$ defined as,

$\vec{j}_{\text{eff}}(\vec{x})\equiv\epsilon^{2}\,m_{A^{\prime}}\,e^{i\omega
t}\,e^{-m_{A^{\prime}}d}\,\bar{B}_{\text{em}}\,\hat{j\,}(\vec{x})~{},$ (18)

where
$\bar{B}_{\text{em}}=\sqrt{\int_{\text{em}}\mathop{}\\!\mathrm{d^{3}}{x}\mathop{}\\!|\vec{B}_{\text{em}}|^{2}/V_{\text{em}}}$
is the amplitude of the emitter magnetic field RMS-averaged over the emitter
cavity volume $V_{\text{em}}$.

Figure 3: The spatial profile (in cylindrical coordinates) of
$\vec{j}_{\text{eff}}$ outside of a cylindrical emitter cavity of radius $R$
and length $L=R$ for two source modes, $\text{TM}_{010}$ (left panel) and
$\text{TE}_{011}$ (right panel). The hidden photon mass is
$m_{A^{\prime}}=10/R$. Only a portion of the emitter cavity is shown, with its
walls indicated by the thick black line. The emitter electric field profile is
shown in blue, with darker shading indicating larger field magnitude and
arrows indicating direction. The relative magnitude of the resulting effective
current is shown by the red shading and its direction by the red arrows.
Figure 4: Comparison of numerical and analytic evaluations of the
$\phi$-component of the effective current profile $\hat{j\,}(\vec{x})$ (see
Eq. (18)) outside the endcap of a cylindrical emitter cavity operating in the
$\text{TE}_{011}$ mode. The solid curves show the numerically-determined
radial profiles computed from the integral in Eq. (12) (evaluated within a
Compton wavelength of the emitter cavity) for various choices of the hidden
photon mass $m_{A^{\prime}}$. For $m_{A^{\prime}}\gtrsim\omega$, the numerical
results agree well with the analytic result of Eq. (17), shown in dotted
black.

An example of $\vec{j}_{\text{eff}}$ computed numerically from Eq. (12) is
shown in Fig. 3 for two different emitter modes, taking the emitter to be a
right-cylindrical cavity. This displays the features expected from Eq. (17)
(discussed below), which applies near the emitter wall and away from the side-
endcap boundary ($\rho=R$, $z=0$). In Fig. 4, we compare the analytic result
of Eq. (17) directly to a numerical evaluation of Eq. (12) for a particular
emitter field and at various hidden photon masses. For $m_{A^{\prime}}\gtrsim
2\,\omega$, $j_{\text{eff}}$ is well-approximated by Eq. (17) to $\lesssim
10\%$, except within $1/m_{A^{\prime}}$ of the outer radial edge ($\rho=R$).
In this region, ignoring contributions from surface currents along the radial
edge of the emitter cavity is no longer a valid approximation, and the
effective current develops a component along the $\hat{z}$ direction. However,
the contribution of such corrections to the signal power is relatively
suppressed by the small volume of this region. In particular, this radial edge
contributes to $P_{\textnormal{sig}}$ as
$\mathcal{O}{(\omega^{4}/m_{A^{\prime}}^{4})}$, which is sub-dominant for an
optimal choice of modes (see Eq. (20)).

The effective current in Eq. (17) has three key features:

  1. 1)

$j_{\text{eff}}\propto m_{A^{\prime}}$. This follows from Eq. (12), as the
relevant length scale in the integral is $1/m_{A^{\prime}}\ll 1/\omega$.

  2. 2)

$\vec{j}_{\text{eff}}$ is tangential to $\mathcal{S}$. This is because (away
from the external radial edges of the cavity) if $m_{A^{\prime}}\gg\omega$
then the only SM current within a Compton wavelength of the receiver cavity is
that running along $\mathcal{S}$.

  3. 3)

$j_{\text{eff}}(\vec{x})$ only has weight within a distance $1/m_{A^{\prime}}$
of $\mathcal{S}$, as evident by the exponential factor in Eq. (17).

These facts have important implications for the signal power and the selection
of modes. In particular, in the numerator of Eq. (16), the integral of
$\vec{E}_{p}^{*}\cdot\vec{j}_{\text{eff}}$ over the receiver cavity volume
involves only the components of the receiver mode electric field that are
tangential to $\mathcal{S}$ and within a distance $1/m_{A^{\prime}}$ from
$\mathcal{S}$. These electric field components are suppressed by
$\omega/m_{A^{\prime}}\ll 1$ relative to the typical field value, due to the
conducting boundary condition on $\mathcal{S}$. Taken together, this implies
that the overlap of $\hat{j\,}$ with the receiver cavity mode scales as
$\int_{\text{rec}}\mathop{}\\!\mathrm{d^{3}}{x^{\prime}}\mathop{}\\!\vec{E}_{p}^{*}(\vec{x}^{\,\prime})\cdot\hat{j\,}(\vec{x}^{\,\prime})\propto
1/m_{A^{\prime}}^{2}$. We thus choose to define a dimensionless overlap
parameter $\eta$,

$\eta\equiv\frac{m_{A^{\prime}}^{2}}{\omega_{p}^{1/2}}\,\frac{\big{|}\int_{\text{rec}}\mathop{}\\!\mathrm{d^{3}}{x^{\prime}}\mathop{}\\!\vec{E}_{p}^{*}(\vec{x}^{\,\prime})\cdot\hat{j\,}(\vec{x}^{\,\prime})\big{|}}{\sqrt{\int_{\text{rec}}\mathop{}\\!\mathrm{d^{3}}{x^{\prime}}\mathop{}\\!|\vec{E}_{p}(\vec{x}^{\,\prime})|^{2}}}~{},$
(19)

such that in the evanescent limit and for optimal mode choices, $\eta$ is
independent of $m_{A^{\prime}}$ and set only by cavity and mode geometry. For
an optimal configuration, $\eta\gtrsim\mathcal{O}{(1)}$ as discussed in Sec.
IV.2. Using this in Eq. (16), the signal power reduces to

$P_{\text{sig}}\simeq
Q\,\bar{B}_{\text{em}}^{2}\,\frac{\epsilon^{4}\,\eta^{2}}{m_{A^{\prime}}^{2}}\,e^{-2m_{A^{\prime}}d}~{}.$
(20)

##  IV LSthinW Design

A full design study is beyond the scope of this work. Here we highlight the
key requirements, focusing on those unique to the evanescent regime. Some
practical challenges, such as matching the frequencies of the cavities and
improving quality factors, are shared with ongoing on-shell LSW searches
Romanenko  _et al._ (2023). A future LSthinW search can make use of their
improvements.

The most important design consideration is the use of a thin barrier. This is
optimally as thin as possible to increase the upper mass reach, while still
sufficiently thick to provide adequate shielding and to preserve the large
field gradients and quality factors associated with SRF cavities. Below, we
also consider the optimal choice of emitter and detector modes as well as
cavity shape, as this has important qualitative differences from the on-shell
case. Finally, we give some example experimental parameters and discuss the
resulting sensitivity to hidden photons in the $10^{-6}\ \text{eV}-10^{-1}\
\text{eV}$ mass range.

###  IV.1 Thin Barrier

As discussed above, the largest mass that an LSthinW experiment is sensitive
to is dictated by the thickness $d$ of the barrier separating the two cavity
volumes, $m_{A^{\prime}}\sim 1/d$, which is evident by the exponential
suppression of $P_{\text{sig}}$ in Eq. (20). Although thinner barriers enhance
the signal for large masses, the penetration of SM EM fields into
superconductors means that a minimum thickness is required to suppress noise
from the strong driven fields of the emitter cavity leaking into the detection
region. We conservatively estimate this minimum thickness by demanding that
such leakage fields $B_{\text{leak}}\sim
e^{-d/\lambda_{\textnormal{L}}}\bar{B}_{\text{em}}$ be no larger than the
signal field $B_{\text{sig}}\sim
Q\,\epsilon^{2}\bar{B}_{\text{em}}(\omega/m_{A^{\prime}})$, where
$\lambda_{\textnormal{L}}\sim 40\ \text{nm}$ is the London penetration depth
of niobium. This implies a minimum barrier thickness of $d\sim\text{2}\
\mu\text{m}$ for the weakest signals that we consider in this work. In our
reach estimates, we adopt an even more conservative minimum of $d>10\
\mu\text{m}$, corresponding to a maximum hidden photon mass of
$m_{A^{\prime}}\sim 0.02\ \text{eV}$.

The manufacture and use of a $d\sim 10\ \mu\text{m}$ barrier poses additional
challenges. One option would be to utilize $\sim 1\ \mu\text{m}$ commercial
niobium foils goo . However, maintaining high-$Q$ requires post-fabrication
treatments to rid the niobium of material contaminants, and this is not simple
to do for very thin surfaces, as it often requires chemical or
electropolishing etching treatments that remove the outer $\sim 100\
\mu\text{m}$ of material Antoine (2014). The incorporation of such standalone
niobium barriers would be additionally complicated by the fact that they must
resist stress induced by vacuum and EM pressure gradients.

An alternative strategy would be to fabricate a rigid barrier by sputtering a
few microns of niobium onto a low-loss, insulating substrate, such as sapphire
Perpeet  _et al._ (1997). In this case, the high-$Q$ of the receiver cavity
could be maintained by orientating the exposed sapphire towards the emitter
volume and the thin niobium towards the receiver volume. For such an
arrangement, achieving large driven fields in the emitter cavity in the
presence of higher loss ($Q\sim 10^{9}$) sapphire Braginsky  _et al._ (1987);
Tobar  _et al._ (1998); Krupka  _et al._ (1999a, b) requires a larger amount
of power to be driven and dissipated in the emitter cavity. However, for the
largest field strengths and volumes that we consider (see Table 1) this
corresponds to $\lesssim 100\ \text{W}$, which can be readily supplied and
then dissipated through the liquid helium already required to cool the emitter
cavity Powers (2019); Hansen  _et al._ (2022).

Thermal transport through the thin barrier is another concern. Heat
transmission along a standalone $d\sim 10\ \mu\text{m}$ niobium barrier or
through a dielectric substrate is restricted, causing the temperature at the
center of the barrier to be larger than that of the cavity’s external walls
which are cooled by liquid helium. If this temperature peak is too large it
may quench the superconductivity of the barrier. This may be mitigated, for
instance, by incorporating helium cooling channels through the substrate
itself, but we leave a dedicated study of such designs to future work. To
account for this, we consider two representative barrier thicknesses: $d=0.5\
\textnormal{mm}$ (which is similar to the wall thickness of existing SRF
cavities and for which the above challenges are not expected to be a concern
Singer (2014)) and $d=10\ \mu\text{m}$.

###  IV.2 Cavity and Mode Geometry

We consider a simple arrangement of two coaxial right-cylinder cavities of
radius $R$ and length $L$ formed by partitioning a single cylinder of radius
$R$ with a planar barrier parallel to its endcaps. From the structure of the
effective current in Eq. (17), we can ascertain that the optimal receiver mode
has electric field components tangential to the barrier surface. This is the
case for the $\text{TE}_{011}$ mode, which is azimuthal,
$\vec{E}(\vec{x})\propto\,J_{1}(\alpha_{11}\rho/R)\,\sin{(\pi
z/L)}\,\hat{\phi}$, where the radial coordinate $\rho$ spans $[0,R]$, $z$
spans $[0,L]$, and $\alpha_{11}\simeq 3.83$ is the first zero of $J_{1}$. Eq.
(17) along with $\hat{n}=\hat{z}$ implies that to excite this receiver mode,
the magnetic field of the emitter should possess radial components near the
endcap. This is also satisfied by the $\text{TE}_{011}$ mode, since
$\vec{B}(z=L)\propto\,J_{1}(\alpha_{11}\rho/R)\,\hat{\rho}$. Hence, we expect
optimal overlap when both cavities are operated in the $\text{TE}_{011}$ mode.
This is also demonstrated schematically by the right panel of Fig. 3.
Analytically evaluating the overlap parameter of Eq. (19) for this mode
choice, we find

$\displaystyle\lim_{m_{A^{\prime}}\gg\omega}\eta^{2}=\frac{\pi^{5}R^{5}}{L^{2}\big{(}\pi^{2}R^{2}+\alpha_{11}^{2}L^{2}\big{)}^{3/2}}~{},$
(21)

such that $\eta\simeq 1.6$ for $R=L$ and $m_{A^{\prime}}\gg\omega$.

The dimensions of the emitter/receiver cavity also strongly impact the signal
power. In particular, from Eq. (21) we see that the evanescent signal grows
significantly with the aspect ratio $R/L$ as

$\lim_{R\gg L\gg m_{A^{\prime}}^{-1}}\eta=\pi R/L~{}.$ (22)

This is expected in the evanescent limit. Since $j_{\text{eff}}$ is peaked
within one Compton wavelength of the barrier, decreasing the receiver length
increases the fraction of the receiver volume which contains appreciable
effective current. Eqs. (20) and (22) imply that the signal power is
independent of the receiver volume provided that
$m_{A^{\prime}}\gg\omega,R^{-1},L^{-1}$. We thus consider as an example a
search with $R/L=50$. Note that such a geometry causes a suppression of the
on-shell ($m_{A^{\prime}}<\omega$) signal, as evident in Fig. 5, due to
decreased receiver volume and incoherence of the emitted hidden photon field
over the emitter cavity, as now $R\gg 1/\omega\simeq L/\pi$. Thus, such a
geometry is purely optimized for evanescent hidden photons.

While there is a strong overlap for a $\text{TE}_{011}$ emitter to excite a
$\text{TE}_{011}$ receiver in the evanescent limit, this does not generally
hold for matched cavity modes. This is evident in the left panel of Fig. 3,
which shows $\vec{j}_{\text{eff}}$ sourced by a $\text{TM}_{010}$ emitter
mode. In the coaxial region ($\rho<R$, $z>0$), $\vec{j}_{\text{eff}}$ is
primarily radial but the mode has electric fields purely in $\hat{z}$. Thus,
$\vec{j}_{\text{eff}}\cdot\vec{E}_{p}^{*}$ is non-zero only within
$1/m_{A^{\prime}}$ of the outer radial edge of the endcap ($\rho=R$, $z=0$).
As a result, for both cavities in the $\text{TM}_{010}$ mode, $\eta$ is
suppressed by $\mathcal{O}{(\omega/m_{A^{\prime}})}\ll 1$ relative to that of
the $\textnormal{TE}_{011}$ case considered above.444Fig. 3 implies that large
overlap is possible for $\text{TM}_{010}$ using a different cavity
arrangement, such as nested concentric cavities.

It is natural to ask if evanescent hidden photons can be searched for in a
setup minimally modified from existing RF LSW efforts, such as Dark SRF
Romanenko  _et al._ (2023), simply by taking the cavity separation to be much
smaller than the $d\sim\omega^{-1}\sim 10\;\textnormal{cm}$ gap currently
used. The Dark SRF setup can be well-approximated as two cylindrical cavities
operating in the $\text{TM}_{010}$ mode. This was chosen to target the
longitudinal $A^{\prime}$ polarization, which provides enhanced sensitivity in
the on-shell $m_{A^{\prime}}\ll\omega$ regime Graham  _et al._ (2014).
However, as discussed above, this mode configuration has a suppressed overlap
in the evanescent limit. Therefore, in addition to decreasing the cavity
separation, a change in modes is needed for an effective search for hidden
photons heavier than $\sim\text{few}\times\mu\text{eV}$. It is also
interesting to note that the CROWS LSW experiment conducted a search employing
the optimal $\text{TE}_{011}$ configuration discussed here Betz  _et al._
(2013). However, their cavity separation $d\simeq\omega^{-1}$ was too large to
enable enhanced sensitivity to evanescent hidden photons.

###  IV.3 Hidden Photon Sensitivity

The reach of an optimized LSthinW setup is estimated by the signal-to-noise
ratio, $\text{SNR}=P_{\text{sig}}/P_{\text{noise}}$, where the noise power
$P_{\text{noise}}$ is assumed to be dominated by thermal occupation of the
receiver mode. We take $\textnormal{SNR}=5$ to determine the projected
sensitivity. If the phase of the emitter cavity is not actively monitored, the
noise power is given by the Dicke radiometer equation, $P_{\text{noise}}\simeq
T\sqrt{\delta\omega/2\pi t_{\text{int}}}\,,$ where $T$ is the receiver cavity
temperature, $\delta\omega$ is the analysis bandwidth which we take to be the
receiver bandwidth $\omega/Q$, and $t_{\text{int}}$ is the experimental
integration time. Following Ref. Graham  _et al._ (2014), an active monitoring
of the emitter field’s phase allows for enhanced sensitivity, corresponding to
an effective thermal noise power of $P_{\text{noise}}\simeq T/t_{\text{int}}$.
We will consider both possibilities in our estimates below.

LSthinW | $d$ | $R$ | $L$ | $Q$ | $t_{\text{int}}$ | readout
---|---|---|---|---|---|---
I | $0.5\ \text{mm}$ | $10\ \text{cm}$ | $10\ \text{cm}$ | $10^{10}$ | $4\ \text{hr}$ | power
II | $0.5\ \text{mm}$ | $10\ \text{cm}$ | $10\ \text{cm}$ | $10^{12}$ | $1\ \text{yr}$ | phase
III | $10\ \mu\text{m}$ | $15\ \text{cm}$ | $0.3\ \text{cm}$ | $10^{12}$ | $1\ \text{yr}$ | phase
Table 1: Benchmark experimental parameters for the three projections shown in
Fig. 5. For each, the peak magnetic field on the emitter cavity wall is fixed
to $B=100\ \textnormal{mT}$ and the readout temperature is $T=2\
\textnormal{K}$. Each setup assumes cylindrical cavities with radius $R$,
length $L$, and quality factor $Q$. The cavities are assumed to be aligned
along a common axis, separated by a wall of thickness $d$.
$t_{\textnormal{int}}$ is the total integration time. The last column denotes
whether the readout involves only a measurement of power in the receiver, or
additionally employs an active monitoring of the emitter field’s phase. See
Sec. IV for details. Figure 5:  The projected reach of three representative
LSthinW setups employing SRF cavities, as shown in red. All the searches
assume coaxial right-cylindrical cavities, with the emitter cavity driven in
the $\textnormal{TE}_{011}$ mode and the receiver cavity tuned to have a
$\textnormal{TE}_{011}$ mode of matching frequency. LSthinW I (solid red)
assumes readily achievable experimental parameters and employs cavities of
equal aspect ratio ($R=L=10\;\textnormal{cm}$) separated by a barrier of
thickness $d=0.5\ \text{mm}$. LSthinW II (dashed red) and III (dotted red) use
improved experimental parameters, similar to the design goals of the ongoing
Dark SRF experiment (whose projected reach is shown in dotted dark gray Berlin
_et al._ (2022)). LSthinW II and III are complementary searches, using
different cavity geometries to specialize to smaller and larger
$m_{A^{\prime}}$, respectively. LSthinW II uses the same cavity shape and wall
thickness as LSthinW I, whereas LSthinW III employs a pancake-like cavity
($R=15\;\textnormal{cm},\,L=0.3\;\textnormal{cm}$) and a smaller barrier
thickness, $d=10\ \mu\text{m}$. For more details, see Table 1 and Sec. IV.3.
Also shown in shaded gray are existing constraints on kinetically-mixed hidden
photons, taken from the repository Ref. O’Hare (2020). These include limits
derived from the CROWS LSW experiment Betz  _et al._ (2013), the Dark SRF LSW
Pathfinder run Romanenko  _et al._ (2023), XENON1T An  _et al._ (2020), CMB
spectral distortions Mirizzi  _et al._ (2009); Caputo  _et al._ (2020); Garcia
_et al._ (2020), and solar energy loss An  _et al._ (2013); Redondo and
Raffelt (2013).

Operating at low temperature is advantageous, and existing RF LSW searches
optimize their cooling strategy by cooling the receiver much more than the
emitter Romanenko  _et al._ (2023). This may not be feasible for an LSthinW
search, as the use of a thin barrier with no vacuum gap places the emitter and
receiver in thermal contact. The entire system likely needs to be cooled to
the temperature of the receiver cavity, and for this reason we adopt a
conservative readout temperature of $T=2\ \textnormal{K}$.

The projected sensitivities of three distinct LSthinW setups are shown in Fig.
5, assuming coaxial right-cylindrical cavities operated in the
$\text{TE}_{011}$ mode. Experimental parameters for each are given in Table 1.
In all cases we take $T=2\ \textnormal{K}$ and we set the magnetic field in
the emitter to have a peak value along the cavity walls of $100\
\textnormal{mT}$, slightly below the critical field of niobium. For the
“LSthinW I” setup, we consider cavities of aspect ratio $R/L=1$, a relatively
thick separating barrier $d=0.5\ \textnormal{mm}$, and generally conservative
parameters regarding cavity volume, quality factor, and integration time. For
“LSthinW II,” we adopt the same cavity and mode geometry, but optimize the
other parameters to the design goals of the existing Dark SRF collaboration
Romanenko  _et al._ (2023). Our estimates demonstrate that either of these two
setups would enable sensitivity to a large range of unexplored parameter space
for hidden photons of mass $10^{-5}\ \text{eV}<m_{A^{\prime}}<10^{-3}\
\text{eV}$. “LSthinW III” employs parameters optimized for larger hidden
photon masses, $10^{-4}\ \text{eV}<m_{A^{\prime}}<10^{-1}\ \text{eV}$, using
cavities with $R/L=50$ and a thinner barrier of $d=10\ \mu\text{m}$. Also
shown as shaded gray in Fig. 5 are existing limits on the existence of
kinetically-mixed hidden photons.

In Fig. 5, the sensitivity is maximized in each setup when the hidden photon
mass is equal to the cavity frequency,
$\omega=\sqrt{(\pi/L)^{2}+(\alpha_{11}/R)^{2}}\,$. The cavity frequency for
LSthinW III is larger than that of LSthinW I and LSthinW II due to the smaller
cavity length. The scaling $\epsilon\propto m_{A^{\prime}}^{1/2}$ for
$m_{A^{\prime}}>\omega$ follows from Eq. (20). For $m_{A^{\prime}}<\omega$ we
find $\epsilon\propto m_{A^{\prime}}^{-2}$, as expected from Ref. Graham  _et
al._ (2014) for an on-shell search in the “transverse configuration.” These
two power laws suggest that the optimal sensitivity occurs for
$m_{A^{\prime}}\simeq\omega$. But there is additionally a resonant peak at
$m_{A^{\prime}}=\omega$, evident in Fig. 5. At this critical mass, the hidden
photon field is sourced with wavenumber $k=0$, giving maximally constructive
interference in the integral of Eq. (9). The width of this feature is narrower
for cavities of larger aspect ratio $R/L\gg 1$, as $R\gg\omega^{-1}$ implies a
larger phase incoherence over the emitter surface for a small deviation of
$m_{A^{\prime}}$ away from $\omega$.

##  V Discussion

The highest mass accessible to light-shining-through-wall experiments is
dictated not by the frequency of the driven field but by the inverse of the
distance separating the emission and detection regions. Thus, the upper part
of the mass range that can be explored can be significantly enlarged in a
“light-shining-through-_thin_ -wall" (LSthinW) experiment. We have focused on
thin superconducting barriers, since their small penetration depth implies
that only several microns is needed for sufficient shielding. In particular,
such an experiment employing superconducting cavities operating at
$\sim\text{GHz}$ frequencies could have exquisite sensitivity to hidden
photons as heavy as $\sim 0.1\;\textnormal{eV}$. The development of thin
barriers, as opposed to higher-frequency sources, is a natural alternative to
enlarging the mass reach to new particles and has strong synergistic overlap
with other efforts, such as future versions of the ARIADNE experiment Geraci
_et al._ (2018); Fosbinder-Elkins  _et al._ (2017), which will tentatively
utilize $\sim 100\ \mu\text{m}$ niobium shields.

A unique feature of this setup is its ability to produce and detect particles
of mass up to $\sim 0.1\;\textnormal{eV}$, regardless of whether they
constitute the dark matter of our Universe. Indeed, the power of this approach
is highlighted by the fact that most experiments operating in this mass range
are often hindered by the difficulty in operating low-loss resonators and
photon detectors at $\text{meV}\sim\text{THz}$ frequencies Caldwell  _et al._
(2017); Gelmini  _et al._ (2020); Liu  _et al._ (2022); Chiles  _et al._
(2022); Chen  _et al._ (2022); Fan  _et al._ (2022). In fact, our projections
suggest that the reach of an LSthinW experiment may even exceed that of
certain dark matter detectors Fan  _et al._ (2022), even though the latter
benefit from the assumed presence of a hidden photon dark matter background.

In this work, we have discussed various setups in which an LSthinW experiment
using electromagnetic cavities may be realized, but dedicated design efforts
are needed to fully bring this proposal to light. While we have considered
simple vacuum cavities with a conducting barrier, other arrangements may be
advantageous and deserve further study. For example, the mass suppression of
the signal power in Eq. (20) is not fundamental, but results from the
vanishing of the receiver cavity mode near the conducting barrier. It may be
that the use of absorbing shield materials or novel mode structures with more
support near the barrier can avoid this suppression and generate an improved
scaling of $P_{\textnormal{sig}}\propto m_{A^{\prime}}^{0}$. Finally, in
addition to searching for hidden photons, the proposed LSthinW approach may
also be applied to enlarge the mass reach for related experiments searching
for different types of particles, such as electromagnetically-coupled axions
Janish  _et al._ (2019); Gao and Harnik (2021) and millicharged particles
Berlin and Hook (2020). We leave these investigations to future work.

## Acknowledgements

We would like to thank Paddy Fox, Timergali Khabiboulline, Sam Posen, Paul
Riggins, Vladimir Shiltsev, and Slava Yakovlev for valuable conversations.
This material is based upon work supported by the U.S. Department of Energy,
Office of Science, National Quantum Information Science Research Centers,
Superconducting Quantum Materials and Systems Center (SQMS) under contract
number DE-AC02-07CH11359. Fermilab is operated by the Fermi Research Alliance,
LLC under Contract DE-AC02-07CH11359 with the U.S. Department of Energy.

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