# Searches for New Particles, Dark Matter,
and Gravitational Waves with SRF Cavities

Asher Berlin  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  Sergey
Belomestnykh  Superconducting Quantum Materials and Systems Center (SQMS),
Fermi National Accelerator Laboratory, Batavia, IL 60510, USA  Applied Physics
and Superconducting Technology Division, Fermi National Accelerator
Laboratory, Batavia, IL 60510, USA  Department of Physics and Astronomy, Stony
Brook University, Stony Brook, NY 11794, USA  Diego Blas  Grup de Física
Teòrica, Departament de Física, Universitat Autònoma de Barcelona, Bellaterra,
08193 Barcelona, Spain  Institut de Fisica d’Altes Energies (IFAE), The
Barcelona Institute of Science and Technology,
Campus UAB, 08193 Bellaterra (Barcelona), Spain  Daniil Frolov
Superconducting Quantum Materials and Systems Center (SQMS), Fermi National
Accelerator Laboratory, Batavia, IL 60510, USA  Anthony J. Brady  Department
of Electrical and Computer Engineering, University of Arizona, Tucson, Arizona
85721, USA  Superconducting Quantum Materials and Systems Center (SQMS), Fermi
National Accelerator Laboratory, Batavia, IL 60510, USA  Caterina Braggio
Dipartimento di Fisica e Astronomia, Padova, Italy  INFN, Sezione di Padova,
Padova, Italy  Superconducting Quantum Materials and Systems Center (SQMS),
Fermi National Accelerator Laboratory, Batavia, IL 60510, USA  Marcela Carena
Theory Division, Fermi National Accelerator Laboratory, Batavia, IL 60510, USA
Department of Physics, University of Chicago, Chicago, Illinois, 60637, USA
Superconducting Quantum Materials and Systems Center (SQMS), Fermi National
Accelerator Laboratory, Batavia, IL 60510, USA  Raphael Cervantes
Superconducting Quantum Materials and Systems Center (SQMS), Fermi National
Accelerator Laboratory, Batavia, IL 60510, USA  Mattia Checchin
Superconducting Quantum Materials and Systems Center (SQMS), Fermi National
Accelerator Laboratory, Batavia, IL 60510, USA  Crispin Contreras-Martinez
Superconducting Quantum Materials and Systems Center (SQMS), Fermi National
Accelerator Laboratory, Batavia, IL 60510, USA  Applied Physics and
Superconducting Technology Division, Fermi National Accelerator Laboratory,
Batavia, IL 60510, USA  Raffaele Tito D’Agnolo  Université Paris Saclay, CEA,
CNRS, Institut de Physique Théorique, 91191, Gif-sur-Yvette, France  Sebastian
A. R. Ellis  Département de Physique Théorique, Université de Genève,
24 quai Ernest Ansermet, 1211 Genève 4, Switzerland  Grigory Eremeev
Superconducting Quantum Materials and Systems Center (SQMS), Fermi National
Accelerator Laboratory, Batavia, IL 60510, USA  Applied Physics and
Superconducting Technology Division, Fermi National Accelerator Laboratory,
Batavia, IL 60510, USA  Christina Gao  Department of Physics, University of
Illinois at Urbana-Champaign, Urbana, IL 61801, USA  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  Bianca Giaccone  Superconducting Quantum
Materials and Systems Center (SQMS), Fermi National Accelerator Laboratory,
Batavia, IL 60510, USA  Anna Grassellino  Superconducting Quantum Materials
and Systems Center (SQMS), Fermi National Accelerator Laboratory, Batavia, IL
60510, USA  Applied Physics and Superconducting Technology Division, Fermi
National Accelerator Laboratory, Batavia, IL 60510, USA  Roni Harnik
<EMAIL_ADDRESS>Superconducting Quantum Materials and Systems Center (SQMS),
Fermi National Accelerator Laboratory, Batavia, IL 60510, USA  Theory
Division, Fermi National Accelerator Laboratory, Batavia, IL 60510, USA
Matthew Hollister  Superconducting Quantum Materials and Systems Center
(SQMS), Fermi National Accelerator Laboratory, Batavia, IL 60510, USA  Applied
Physics and Superconducting Technology Division, Fermi National Accelerator
Laboratory, Batavia, IL 60510, USA  Ryan Janish  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  Yonatan Kahn  Department of Physics,
University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
Superconducting Quantum Materials and Systems Center (SQMS), Fermi National
Accelerator Laboratory, Batavia, IL 60510, USA  Sergey Kazakov
Superconducting Quantum Materials and Systems Center (SQMS), Fermi National
Accelerator Laboratory, Batavia, IL 60510, USA  Applied Physics and
Superconducting Technology Division, Fermi National Accelerator Laboratory,
Batavia, IL 60510, USA  Doga Murat Kurkcuoglu  Fermilab Quantum Institute,
Fermi National Accelerator Laboratory, Batavia, IL 60510, USA  Superconducting
Quantum Materials and Systems Center (SQMS), Fermi National Accelerator
Laboratory, Batavia, IL 60510, USA  Zhen Liu  School of Physics and Astronomy,
University of Minnesota, Minneapolis, MN 55455, USA  Superconducting Quantum
Materials and Systems Center (SQMS), Fermi National Accelerator Laboratory,
Batavia, IL 60510, USA  Andrei Lunin  Superconducting Quantum Materials and
Systems Center (SQMS), Fermi National Accelerator Laboratory, Batavia, IL
60510, USA  Alexander Netepenko  Superconducting Quantum Materials and Systems
Center (SQMS), Fermi National Accelerator Laboratory, Batavia, IL 60510, USA
Applied Physics and Superconducting Technology Division, Fermi National
Accelerator Laboratory, Batavia, IL 60510, USA  Oleksandr Melnychuk
Superconducting Quantum Materials and Systems Center (SQMS), Fermi National
Accelerator Laboratory, Batavia, IL 60510, USA  Roman Pilipenko
Superconducting Quantum Materials and Systems Center (SQMS), Fermi National
Accelerator Laboratory, Batavia, IL 60510, USA  Applied Physics and
Superconducting Technology Division, Fermi National Accelerator Laboratory,
Batavia, IL 60510, USA  Yuriy Pischalnikov  Superconducting Quantum Materials
and Systems Center (SQMS), Fermi National Accelerator Laboratory, Batavia, IL
60510, USA  Applied Physics and Superconducting Technology Division, Fermi
National Accelerator Laboratory, Batavia, IL 60510, USA  Sam Posen
<EMAIL_ADDRESS>Superconducting Quantum Materials and Systems Center (SQMS),
Fermi National Accelerator Laboratory, Batavia, IL 60510, USA  Applied Physics
and Superconducting Technology Division, Fermi National Accelerator
Laboratory, Batavia, IL 60510, USA  Alex Romanenko  Superconducting Quantum
Materials and Systems Center (SQMS), Fermi National Accelerator Laboratory,
Batavia, IL 60510, USA  Applied Physics and Superconducting Technology
Division, Fermi National Accelerator Laboratory, Batavia, IL 60510, USA  Jan
Schütte-Engel  Department of Physics, University of Illinois at Urbana-
Champaign, Urbana, IL 61801, USA  Superconducting Quantum Materials and
Systems Center (SQMS), Fermi National Accelerator Laboratory, Batavia, IL
60510, USA  Changqing Wang  Superconducting Quantum Materials and Systems
Center (SQMS), Fermi National Accelerator Laboratory, Batavia, IL 60510, USA
Vyacheslav Yakovlev  Superconducting Quantum Materials and Systems Center
(SQMS), Fermi National Accelerator Laboratory, Batavia, IL 60510, USA  Applied
Physics and Superconducting Technology Division, Fermi National Accelerator
Laboratory, Batavia, IL 60510, USA  Kevin Zhou  SLAC National Accelerator
Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA  Silvia Zorzetti
Superconducting Quantum Materials and Systems Center (SQMS), Fermi National
Accelerator Laboratory, Batavia, IL 60510, USA  Quntao Zhuang  Department of
Electrical and Computer Engineering, University of Arizona, Tucson, Arizona
85721, USA  Superconducting Quantum Materials and Systems Center (SQMS), Fermi
National Accelerator Laboratory, Batavia, IL 60510, USA  J. C. Wyant College
of Optical Sciences, University of Arizona, Tucson, Arizona 85721, USA

###### Abstract

This is a Snowmass white paper on the utility of existing and future
superconducting cavities to probe fundamental physics. Superconducting radio
frequency (SRF) cavity technology has seen tremendous progress in the past
decades, as a tool for accelerator science. With advances spearheaded by the
SQMS center at Fermilab, they are now being brought to the quantum regime
becoming a tool in quantum science thanks to the high degree of coherence. The
same high quality factor can be leveraged in the search for new physics,
including searches for new particles, dark matter, including the QCD axion,
and gravitational waves. We survey some of the physics opportunities and the
required directions of R&D. Given the already demonstrated integration of SRF
cavities in large accelerator systems, this R&D may enable larger scale
searches by dedicated experiments.

—

_Submitted to the Proceedings of the US Community Study_

_on the Future of Particle Physics (Snowmass 2021)_

—

††preprint:  FERMILAB-PUB-22-150-SQMS-T

###### Contents

  1. I Introduction
    1. I.1 Dark Matter vs. New Particle Searches
  2. II Searches for New Degrees of Freedom
    1. II.1 Dark Photon Searches
    2. II.2 Light-shining-through wall: Dark SRF
    3. II.3 Proposals for Axion Searches
      1. II.3.1 Two cavities with Static Field
      2. II.3.2 Two Cavities with a pump mode
      3. II.3.3 Single-Cavity Axion Search and Euler-Heisenberg
      4. II.3.4 Ongoing R&D for Axion searches
  3. III Dark Matter Searches with Ultra-high Q
    1. III.1 Haloscopes and the Scan Rate
    2. III.2 High-$Q$ Cavities in a Static $B$-field
    3. III.3 Sub-SQL Dark Photon Dark Matter Search with a Highly-Tunable SRF Cavities
    4. III.4 Axion Dark Matter Searches with Up-Conversion
    5. III.5 A Quantum Network of Haloscopes
  4. IV High Frequency Gravitational waves
  5. V Capabilities and Infrastructure
    1. V.1 Enhanced sensitivity by High-Efficient Microwave-Optical transduction
    2. V.2 Cryogenic Platform for Scaled-up Sensing Experiments
  6. VI Outlook and future R&D
  7. VII Acknowledgements

##  I Introduction

Superconducting radio frequency (SRF) cavities are critical components in
particle accelerators. When excited at their resonant frequency, they build up
large-amplitude electric fields while dissipating very little heat. This makes
it possible for them to provide extremely efficient conversion of radio-
frequency power into energy gain for the charged particle beams that pass
through them. But SRF cavities are, in their essence, extremely high quality
electromagnetic resonators, devices that are now of strong active interest for
quantum information science (QIS). For quantum computing, quantum states can
be stored and manipulated in electromagnetic resonators, and superconductors
at millikelvin temperatures are employed to sustain the coherence of the
quantum states for long enough to perform complex computations. For quantum
sensing, SRF cavities can furnish a large volume where very weak signals of
radio-frequency photons can be collected, with only a small fraction of
photons being lost to heat at the cavity walls.

Accelerator scientists have been improving SRF cavities for 50 years, and new
accelerator applications have been enabled as performance has continuously
improved over the years Reece (2016); et al. (2014); Reschke  _et al._ (2017);
Gonnella  _et al._ (2018). The performance improvements were the results of
improved understanding of RF superconductivity and materials science as well
as new SRF cavity processing techniques that were developed to overcome
performance-limiting phenomena, such as field emission, and to enhance
superconducting properties Saito  _et al._ (1990); Bernard  _et al._ (1992);
Kneisel  _et al._ (1993); Visentin  _et al._ (1999); Kneisel (2000);
Grassellino  _et al._ (2013, 2018); Posen  _et al._ (2020). The advances in
cavity performance, and the modern understanding of SRF systems that has been
developed for particle accelerator applications, can also be leveraged for
QIS. In this whitepaper, we focus on quantum sensing applications of SRF
cavities.

###  I.1 Dark Matter vs. New Particle Searches

The search for new physics follows the standard scientific paradigm of
hypothesis testing. In the context of light degrees of freedom that couple to
photons, there are two distinct hypotheses that can be tested:

##### A new particle:

It is well motivated to test the notion that there may be new degrees of
freedom in the Lagrangian (or Hamiltonian) which interacts with Standard Model
(SM) particles. The well-motivated cases that will be best probed by
electromagnetic cavities are new particles that interact with photons, such as
axions or dark photons. Hence, examples of hypotheses that we can test are

$\mbox{Hypothesis:}\qquad\mathcal{L}\supset\mbox{ axion? dark photon?}$ (1)

where $\mathcal{L}$ is the Lagrangian density. Experiments that test this
hypothesis produce and detect the new particle. As such, the experimenter has
control over the frequency of the axion or dark photon field, as well as over
its phase. Light-shining-through-wall (LSW) experiments Okun (1982); Van
Bibber  _et al._ (1987); Hoogeveen and Ziegenhagen (1991) are a canonical
example of axion and dark photon searches in this class.111Collider and fixed-
target experiments are also in this category, but these typically search for
heavier and more strongly-coupled particles. Searches for high-frequency
gravitational waves (GWs), while not strictly speaking a test for new
particles (at least since direct detection of GWs at LIGO Abbott  _et al._
(2021)), are closely related to new particle searches because gravity couples
similarly to the photon as do axions.

##### A dark matter candidate:

Here we test the notion that a new set of degrees of freedom exist, as above,
and in addition these degrees of freedom make up the dark matter (DM) that
dominates our galaxy. Hence, the tested hypothesis is

$\mbox{Hypothesis:}\ \ \left\\{\begin{array}[]{c}\qquad\ \mathcal{L}\ \
\supset\ \mbox{ axion? dark photon?}\\\\[8.0pt] \qquad\mbox{and}\\\\[7.0pt]
\vbox{\includegraphics[width=19.91684pt]{DMhalo.png}}\supset\mbox{ axion? dark
photon?}\end{array}\right.$ (2)

where the DM halo is represented pictorially. As opposed to pure lab searches,
we do not know the frequency of the DM signal nor its phase. Because the DM is
non-relativistic, it is expected to have a velocity of order the galactic
virial velocity $v\sim 10^{-3}\,c$. The frequency $\omega$ of the DM field is
thus set by the DM mass, an unknown parameter, with a small spread
$\Delta\omega$, of order $\Delta\omega/\omega\sim v^{2}\sim 10^{-6}$. The
wave-like DM field can therefore be thought of as a coherent classical source
with an unknown frequency and an effective quality factor of order
$Q_{\mathrm{DM}}\sim 10^{6}$.

We note that these two hypotheses are complementary and can be pursued
somewhat independently. Of course, the DM candidate hypothesis is strictly
stronger than the new particle hypothesis. A new particle search, such as an
axion LSW experiment, is not strictly constrained by an axion DM search,
because the axion need not be the DM. On the other hand, a null result in the
parameter space of a new particle search will necessarily constrain the
parameter space of a DM search. That said, a DM search is likely to have an
advantage in probing the feeble couplings of a new particle because the number
density of DM particles in the regime of interest is typically much larger
than the density of particles that can feasibly be produced in the lab.

##  II Searches for New Degrees of Freedom

Here we summarize opportunities to search for new particles with SRF cavities.
The central characters are dark photons and axions (or axion-like particles),
though we will touch upon other new physics candidates. We will discuss dark
photons and axions in turn.

###  II.1 Dark Photon Searches

The dark photon Holdom (1986) is a well-motivated extension of the SM, with
its allure lying in its simplicity and theoretical completeness. The model
posits that in addition to the photon there is another $U(1)$ gauge field, the
dark photon. The dark photon, unlike its SM counterpart, is massive (which
breaks the new $U(1)$ symmetry). The only renormalizable interaction (and
hence the only relevant interaction at low energies) one can write involving
the dark photon is a kinetic mixing with the SM photon. The Lagrangian is

$\displaystyle\mathcal{L}=\mathcal{L}_{\mathrm{SM}}+\epsilon\,F^{\mu\nu}F^{\prime}_{\mu\nu}+\frac{1}{2}\,m_{A^{\prime}}^{2}\,A^{\prime}_{\mu}A^{\prime\mu}~{},$
(3)

where primes adorn the dark photon field strength and vector potential and
$m_{A^{\prime}}$ is the dark photon mass.

The dark photon is constrained by a host of experiments including collider and
fixed-target setups at relatively high masses, as well as stars and CMB
observations (see, e.g., Ref. Essig  _et al._ (2013) for a review). As a rule,
limits on dark photons become weaker as a power of the dark photon mass over
the frequency it is emitted in a given experiment. This is because the kinetic
mixing term in Eq. (3) can be rotated away by a field redefinition in the
absence of a mass term. It has been shown that limits on the longitudinal
polarization of dark photons are weakened more slowly at low mass and are
hence more constraining An  _et al._ (2013a, b).

LSW experiments have set new limits on dark photons in optical setups, such as
OSQAR Ballou  _et al._ (2015) and ALPS Hoogeveen and Ziegenhagen (1991);
Sikivie  _et al._ (2007); Ehret  _et al._ (2009) with enhanced plans for ALPS
II Bähre  _et al._ (2013), and in RF setups such as CROWS Betz  _et al._
(2013). To date, these experiments have not taken advantage of the enhanced
sensitivity to longitudinal dark photons.

###  II.2 Light-shining-through wall: Dark SRF

We can explore dark photon scenarios using SRF cavities in a setup that
resembles shining light through a cavity wall. The conversion of some of the
photons to dark photons before the wall and conversion back to regular photons
past the wall makes such a detection possible, if dark photons exist at a
hypothesized mass and coupling. Resonant cavities can be used on both sides of
the wall to increase the number of photons on the emitting side and to enhance
the probability of conversion of dark photons to visible ones on the receiver
side. In particular, in an RF cavity the system can be designed to search for
the parametrically enhanced longitudinal coupling of the dark photon Graham
_et al._ (2014). The Dark SRF experiment plans to conduct such a search with
ultra-high quality cavities Harnik (2019); Grassellino  _et al._ (2020);
Collaboration .

Figure 1: The experimental setup for the Dark SRF experiment consisting of two
1.3 GHz cavities Collaboration .  Figure 2:  The projected 95% C.L.
sensitivity on the dark photon parameter space Collaboration  from an upcoming
pathfinder experiment (blue) along with various future SQMS focus targets
(orange, green, and red). Current limits from the cavity experiment CROWS Betz
_et al._ (2013), a Cavendish-type test of Coulomb’s law Williams  _et al._
(1971), and CMB observations are shown as gray shaded regions Mirizzi  _et
al._ (2009a, b); Caputo  _et al._ (2020).

The kinetic mixing term in Eq. (3) effectively allows the SM and dark
electromagnetic fields to mix. As a result, a coherently oscillating SM
electromagnetic field acts as a source of dark photons at the same frequency.
In addition, a coherent dark photon field can resonantly excite an RF cavity
with the appropriate frequency. Given an emitter cavity with quality factor
$Q$ pumped with a power $P_{\mathrm{em}}$ at a frequency $\omega$, the
radiated dark photon field will then deposit power in a nearby receiver cavity

$P_{\mathrm{rec}}=\epsilon^{4}\left(\frac{m_{A^{\prime}}}{\omega}\right)^{4}\left|G\right|^{2}Q^{2}P_{\mathrm{em}}~{},$
(4)

where $G$ is a form factor that is specified in the Appendix, but
heuristically consists of the wavefunction overlap of the dark photon field
with the spatial mode shape in the receiver cavity. Assuming that the noise in
the receiver is thermal, the signal-to-noise ratio (SNR) is given by the
radiometer formula Dicke (1946)

$\mathrm{SNR}=\frac{P_{\mathrm{rec}}}{P_{\mathrm{th}}}\sqrt{\delta\nu\,t_{\mathrm{int}}}=\frac{P_{\mathrm{rec}}}{k_{B}T_{\mathrm{eff}}}\sqrt{\frac{t_{\mathrm{int}}}{\delta\nu}}~{},$
(5)

where $\delta\nu$ is the bandwidth of the analysis, $t_{\mathrm{int}}$ is the
integration time, and $T_{\mathrm{eff}}$ is the effective noise temperature.
The factor of $\sqrt{\delta\nu\,t_{\mathrm{int}}}$ is the root number of
independent measurements.

The experimental setup shown in Fig. 1 has been assembled using two 1.3 GHz
high quality factor $Q_{0}$ SRF cavities and is employed in the current run of
the Dark SRF experiment at SQMS. The top cavity is the emitter cavity and the
bottom cavity is the receiver cavity. This setup is a powerful dark photon
experiment provided that it properly matches frequencies, minimizes amplifier
noise, and reduces cross talk between cavities. Detailed technical information
will be presented in an upcoming work of Dark SRF Collaboration .

In Fig. 2 we show the expected reach of an upcoming Dark SRF pathfinder run as
well as future projections meeting various SQMS targets. The pathfinder run
will be sensitive to regions above the blue solid line. Compared to existing
constraints from the cavity experiment CROWS Betz  _et al._ (2013), a test of
Coulomb’s law Williams  _et al._ (1971), and observations of the CMB Mirizzi
_et al._ (2009a, b); Caputo  _et al._ (2020), this search will provide the
world’s best limit on dark photons in this mass range, improving constraints
on $\epsilon$ by up to two orders of magnitude for dark photon masses between
$10^{-8}$ and $6\times 10^{-6}$ eV. We note that the improvement in $\epsilon$
sensitivity scales as ${\rm SNR}^{\mathrm{1}/4}$, which clearly shows that the
Dark SRF setup is advantageous in many aspects. Main factors contributing to
such improvement are: high $Q$, noise suppression, and optimal placement of
the receiver cavity. The pathfinder setup corresponds to approximately
$\order{1000}$ photons. Enhancing this sensitivity to the few-photon limit in
the receiver cavity requires lower temperatures (achievable with a dilution
fridge) and improved suppression in amplifier noise. One can also improve the
quality factors of both the emitter and receiver cavities to probe deeper into
the dark photon parameter space. Finally, a phase sensitive readout of the
receiver cavity will enable further improvements. These active research areas
will be developed in the coming years and their corresponding projections are
shown as additional colored lines in Fig. 2.

The longitudinal Dark SRF setup is also sensitive to other new physics. In
particular, the high $E$-fields in the emitter cavity can cause Schwinger
pair-production of very light millicharged particles (mCPs) Berlin and Hook
(2020). Upon production in the emitter cavity, these MCPs will then be
accelerated in the oscillating RF fields, and due to the longitudinal
arrangement of the cavities will pass through the receiver cavity. This
process occurs with a period that is set by the emitter cavity frequency and
hence is in resonance with the receiver. Ref. Berlin and Hook (2020) has shown
that this setup can set the strongest lab limits on mCPs for masses below
$10^{-4}$ eV.

###  II.3 Proposals for Axion Searches

Axions and axion-like particles (ALPs) are natural and well-motivated
extensions of the SM. The QCD axion was proposed as a natural solution of the
strong CP problem Peccei and Quinn (1977a, b); Weinberg (1978); Wilczek
(1978). It is a pseudo Nambu-Goldstone boson, arising from the spontaneous
symmetry breaking (SSB) of an approximate global $U(1)$, and thus is naturally
light. The solution to strong CP is possible due to the axion’s coupling to
gluons. Once this coupling is posited, a coupling to photons is a natural
prediction of the model,

$\mathcal{L}\supset\frac{1}{4}\,g_{a\gamma\gamma}\,a\,F\tilde{F}=g_{a\gamma\gamma}\,a\,\vec{E}\cdot\vec{B}\,.$
(6)

The ALP is a generalization of the QCD axion which does not couple to QCD, but
does couple to photons or SM fermions. ALPs are well motivated in their own
right in top-down constructions Svrcek and Witten (2006); Arvanitaki  _et al._
(2010).

The coupling in Eq. (6) is nonlinear. As a result, such an axion can mix with
a photon in the presence of a background electromagnetic field. LSW
experiments are a classic laboratory setup to search for ALPs.222ALP searches
that are not LSW experiments include PVLAS Della Valle  _et al._ (2016), a
search for vacuum birefringence in a magnetic field. A different recently
proposed example is the “dark SPDC” process, in which an optical pump photon
down-converts to a signal and an axion (missing energy) Estrada  _et al._
(2021). In these experiments, a large number of photons is kept in an enclosed
region with a strong constant background magnetic field, corresponding to the
emitter cavity. In the presence of the $g_{a\gamma\gamma}$ interaction,
photons will convert to axions and escape the enclosure. A receiver cavity
with a similarly strong field is set up nearby to detect axions that convert
back to photons. The current best limit from such LSW setups has been achieved
by the OSQAR experiment: $|g_{a\gamma\gamma}|<3.5\times 10^{-8}\
\textrm{GeV}^{-1}$ for $m_{a}<0.3$ eV Ballou  _et al._ (2015). Operating at
optical frequencies with high-finesse cavities, the ALPS experiment took
advantage of resonant production and detection Hoogeveen and Ziegenhagen
(1991); Sikivie  _et al._ (2007) and achieved a limit Ehret  _et al._ (2009)
comparable to that set by OSQAR, with the prospect of an improvement by a
factor $\sim 10^{3}$ in ALPS II Bähre  _et al._ (2013). At microwave
frequencies, the LSW setup has been implemented by the CROWS experiment and
has set a comparable limit Betz  _et al._ (2013).

Like the dark photon case, LSW-type axion searches can benefit from high
quality factors, which warrants the harnessing of advances in SRF technology.
The necessity for a background magnetic field, however presents a challenge,
as high-quality superconductivity does not survive large fields. To overcome
this, several proposals have been put forth, as we discuss below.

####  II.3.1 Two cavities with Static Field

Figure 3:  Schematic depiction of an SRF light-shining-though-wall cavity
search for axions Janish  _et al._ (2019). The conversion region is in
practice a superconducting toroidal magnet and the coupling to the detector
(i.e., receiver) cavity is achieved via a pickup loop in the center of the
toroid.

One technique to utilize both high-$Q$ SRF cavities and large magnetic fields
for a LSW axion search is to sequester the required magnetic fields away from
the production and detection cavities. This was studied in Ref. Janish  _et
al._ (2019) and is depicted schematically in Fig. 3. The experiment consists
of three primary components: two SRF cavities and a conversion region
containing a confined magnetic field. Source modes are driven so that
$\vec{E}\cdot\vec{B}$ is non-vanishing in the emitter cavity, which allows for
the possibility of axion production. This is in contrast to the conventional
RF approach of applying a static magnetic field to the emitter cavity, which
for an SRF cavity would result in magnetic flux vorticies in the cavity walls
that readily dissipate RF currents and spoil the cavity’s large $Q$-factor.
Produced axions will pass through the cavity walls and propagate to the
conversion region. The conversion region is a volume of confined magnetic
field in the form of a superconducing, broken-toroid magnet. In this magnetic
field some axions will convert back to photons. It is essential that these
photons are not confined in the conversion region as is the static magnetic
field, but rather they must propagate to a receiver which couples them into
the receiver cavity. This is achieved by opening gaps in the toroidal
structure of the magnet, for example by constructing it from polodially
wrapped superconducting wire. This forms an effective polarization filter
which confines the toroidal-pointing static field and passes the polodial-
pointing signal fields sourced by axion conversion into photons. With this
arrangement, axion re-conversion will result in a non-zero AC magnetic field
in the center of the toroid. This field may be detected by a pick-up loop and
amplified by coupling into the SRF receiver cavity.

With this approach neither SRF cavity is subject to large magnetic fields and
neither suffers a degradation of $Q$-factor. However, losses in the walls of
the conversion region can result in a decrease of the effective $Q$ of the
entire system. The SNR is given by

$\text{SNR}\sim\frac{g^{4}B_{c}^{4}B_{0}^{2}L_{t}}{\omega^{5}}\,\beta^{2}\,\text{Min}\left[Q,\frac{\omega
L_{t}}{R_{t}}\right]\,\frac{t_{\text{int}}}{T_{\text{sys}}}~{}.$ (7)

$T_{\text{sys}}$ is the system temperature, with noise dominated by either
thermal or quantum fluctuations. $B_{0}$ is the static magnetic field in the
conversion region, and $B_{c}\approx 0.2\,\text{T}$ the SRF cavity’s lower
critical field which limits the power pumped into the source cavity. The
signal power scales linearly with integration time $t_{\text{int}}$ as the
phase of the input signal is known. $\beta\sim 10^{-3}$ is a geometric factor,
accounting for the propagation of the signal between the production,
conversion, and detection regions. $R_{t}$ is the toroid’s surface resistance
and $L_{t}$ its inductance for toroidal currents. For sufficiently small
$L_{t}/R_{t}$, the system’s $Q$ is set by the toroid’s properties. The toroid
surface resistance is estimated in Janish  _et al._ (2019) to be $R_{t}\sim
100\,\text{n}\Omega$, due to the fringing components of the static conversion
field which will lead to flux vorticies in the toroid perpendicular to the
flow of signal current. This resistance is limiting for $Q\geq 10^{10}$. To
take full advantage of a cavity with $Q\sim 10^{12}$, the fringe fields must
be sufficiently suppressed to allow $R_{t}\sim\text{n}\Omega$. The reach in
both cases is given in Fig.4, assuming a cavity with source frequency
$\nu=\text{GHz}$ and integration time $t_{\text{int}}=30\,\text{days}$, taking
$B_{0}=5\ \text{T}$ and $T_{\text{sys}}=0.1\ \text{K}$. $L_{t}$ is
proportional to the toroid radius and is taken to be $L_{t}=125\,\text{nH}$
which corresponds to a $10\,\text{cm}$ toroid. Reaching these sensitivities
will require exquisite control over the receiver cavity mode frequencies, as
it is essential that a resonant mode of this cavity lies within $Q^{-1}$ of
the source frequency. This will likely require continual active control to
keep the receiver cavity tuned to the source.

---
Figure 4:  Sensitivity to axion-photon coupling $g_{a\gamma\gamma}$ of the LSW
search Janish  _et al._ (2019). See text for details. The yellow region is
constrained by solar axion detection Anastassopoulos  _et al._ (2017).

####  II.3.2 Two Cavities with a pump mode

Figure 5: Setup for axion search using two cavities separated by a distance
$d$, with a pump mode with a frequency $\omega_{0}$ in the receiver and two
active modes $\omega_{0,1}$ in the emitter. Axion is produced from the emitter
with a frequency equal to the sum or difference of the two active modes,
$\omega_{\pm}=\omega_{1}\pm\omega_{0}$. The axion may in turn produce a photon
in the receiver with frequencies equal to $\omega_{\pm}\pm\omega_{0}$, in
which the combination that yields $\omega_{1}$ can be resonantly produced.

The LSW proposal above introduces a conversion region in between the
production and detection cavities, so a large static $B$-field can be run
there and turn the produced axions into signal photons. An alternative
approach as shown schematically in Fig. 5, is to replace the static $B$-field
with an oscillatory $B$-field, which can then be directly run inside the
receiver cavity. The oscillatory $B$-field naturally comes about as a mode of
the cavity, which is the pump mode. Given an integration time $t_{\rm int}$,
the SNR is approximately given by Equation (5). with the signal bandwidth
$\Delta\omega_{1}$ is traditionally chosen as $\omega_{1}/Q$ but can be as
small as $t^{-1}_{\rm int}$ Janish  _et al._ (2019). We have assumed that the
dominant noise is from thermal noise in Eq. (5), characterized by the
temperature $T$. Other sources of noise will be discussed later. Since the
axion signal frequency is $\omega_{a}=\omega_{1}\mp\omega_{0}$, we have that
$\omega_{a}\sim\omega_{1}\sim\omega_{0}\sim V^{-1/3}$, where $V$ is the volume
of the cavity. In the limit that the axion is massless, taking
$\Delta\omega_{1}$ to be $t_{\rm int}^{-1}$ yields

$\begin{split}\mbox{SNR}\sim&\frac{Q}{8\omega_{1}}Vg_{a\gamma\gamma}^{2}E^{2}_{\rm
peak}\left(\omega_{a}\frac{\eta^{2}_{01}g_{a\gamma\gamma}VE^{2}_{\rm
peak}}{4\pi d}\right)^{2}\frac{1}{T}t_{\rm int}\\\
\sim&5\,\left(\frac{Q}{10^{10}}\right)\left(\frac{V}{(0.2\rm
m)^{3}}\right)^{3}\left(\frac{g_{a\gamma\gamma}\,\rm GeV}{5\times
10^{-11}}\right)^{4}\left(\frac{E_{\rm peak}}{80\rm MV/m}\right)^{6}\\\
&\quad\times\left(\frac{0.4\rm m}{d}\right)^{2}\left(\frac{\omega_{a}}{\rm
GHz}\right)\left(\frac{t_{\rm int}}{1\rm year}\right)\left(\frac{1.4\rm
K}{T}\right)\end{split}$ (8)

where both the emitter and receiver cavities are to have the same volume $V$
and the 3 active modes are assumed to have the same peak field value $E_{\rm
peak}$.

The SNR above assumed that the only important background is from thermal
fluctuations. In order to reach this level of sensitivity one must mitigate
other sources of background. A particular worry of this multi-mode setup is
that the signal mode lives in the same cavity as a spectator mode which is
being driven to high occupancy. A small leakage of power either from the
driving source or from the spectator mode to the signal mode can easily
dominate over the thermal background. To put this in perspective, an excited
mode with $E_{\mathrm{peak}}\sim 80$ MV/m at GHz frequencies has roughly
$10^{26}$ photons. The thermal background at these frequencies is of order a
few thousand photons. These challenges were already identified in Ref. Sikivie
(2010). In Ref. Gao and Harnik (2021), it is shown that both of these noise
sources can be mitigated by using a pump with high $Q$ and with the pump
frequency well separated from the signal mode frequency. In addition, such
noise sources can be further suppressed by optimizing the cavity geometry and
material science techniques to reduce nonlinearities. As opposed to DM
searches, the signal in a LSW experiment may be turned off by deactivating or
detuning the emitter in order to characterize the noise in the receiver
cavity.

####  II.3.3 Single-Cavity Axion Search and Euler-Heisenberg

---
Figure 6:  Integrating out an axion leads to an effective quartic nonlinearity
in the QED Lagrangian (top). Integrating out the electron leads to a similar
nonlinear Lagrangian, the Euler-Heisenberg Lagrangian, that is present in pure
QED (bottom).

The presence of an axion with a coupling $g_{a\gamma\gamma}$ to
electromagnetic fields induces small deviations from the linearity of
electrodynamics Bernard (1997); Evans and Rafelski (2018), and axion searches
can thus also be understood as precision tests of quantum electrodynamics
(QED). Famously, loop contributions from virtual electrons will also induce
such nonlinearities in pure QED, which are parameterized by the Euler-
Heisenberg (EH) Lagrangian Heisenberg and Euler (1936); Schwinger (1951). The
EH Lagrangian makes a prediction for light-by-light scattering within the SM,
which has never been observed at photon frequencies below the electron mass
$m_{e}=511\ {\rm keV}$ because the effect is highly suppressed at low
energies. Here we will describe a proposed experiment to search for both the
axion-induced and EH nonlinearities using high-$Q$ SRF cavities. The operating
principle Brodin  _et al._ (2001); Eriksson  _et al._ (2004); Bogorad  _et
al._ (2019) is that in an SRF cavity simultaneously pumped at two resonant
frequencies $\omega_{1}$ and $\omega_{2}$, the enormous occupation numbers of
the pumped modes could generate a weak but observable signal at the signal
frequency $\omega_{s}=2\omega_{1}-\omega_{2}$ if the cavity geometry is tuned
to make $\omega_{s}$ a resonant mode as well.

Consider the Feynman diagram in Fig. 6, top left, which represents photon-
photon scattering mediated by an intermediate axion. When $m_{a}$ is large
compared to the energies of the scattering photons, the axion may be
“integrated out” by replacing its propagator as
$(\omega^{2}-m_{a}^{2})^{-1}\to-m_{a}^{-2}$. This leads to a local Lagrangian
of the form shown in Fig. 6, top right, which describes a 4-photon
interaction, or equivalently a cubic nonlinearity in the equations of motion.

This argument has a close analogy with the EH Lagrangian, which is the low-
energy limit of QED when the electron has been integrated out, $\omega\ll
m_{e}$. Feynman diagrams of the type shown in Fig. 6, bottom left, give rise
to a local interaction with the Lagrangian given in Fig. 6, bottom right. The
nonlinearities of the EH Lagrangian may be described in terms of an effective
charge and current density

$\rho_{\rm EH}\\!=\\!-\frac{4\alpha^{2}}{45m_{e}^{4}}\nabla\cdot\mathbf{P}\,;\
\mathbf{J}_{\rm
EH}\\!=\\!\frac{4\alpha^{2}}{45m_{e}^{4}}\\!\left(\nabla\\!\times\\!\mathbf{M}\\!+\\!\frac{\partial\mathbf{P}}{dt}\right),$
(9)

with
$\mathbf{P}=7(\mathbf{E}\cdot\mathbf{B})\mathbf{B}+2(\mathbf{E}^{2}-\mathbf{B}^{2})\mathbf{E}\,$
and
$\mathbf{M}=7(\mathbf{E}\cdot\mathbf{B})\mathbf{E}-2(\mathbf{E}^{2}-\mathbf{B}^{2})\mathbf{B}\,$.
Note that both $\mathbf{P}$ and $\mathbf{M}$ are cubic in the electromagnetic
fields, and will thus source cubic nonlinearities; for example, an oscillating
$E$-field at frequency $\omega$ will source a response field at frequency
$3\omega$.

The axion-induced nonlinearities may also be _non-local_ when the axion is
light, $\omega\gg m_{a}$, in which case a simple Lagrangian description is not
available. The situation for any $m_{a}$ may still be parameterized in terms
of an effective current,

$\rho_{a}=-g_{a\gamma\gamma}\mathbf{B}\cdot\nabla a\,;\
\mathbf{J}_{a}=g_{a\gamma\gamma}\left(\nabla
a\times\mathbf{E}+\mathbf{B}\frac{\partial a}{\partial t}\right),$ (10)

where $a$ is the intermediate axion field determined by integrating a Green’s
function. As we will see below, the form of $a$ will depend on the relative
sizes of $m_{a}$ and the frequencies of the external $E$\- and $B$-fields, but
as with the EH Lagrangian, $\rho_{a}$ and $\mathbf{J}_{a}$ are cubic in the
electromagnetic fields for any $m_{a}$.

A quick estimate may be made for the size of $g_{a\gamma\gamma}$ at which
axion-induced nonlinearities will be of the same order of magnitude as the EH
nonlinearities. Comparing the coefficients of the two local Lagrangians in
Fig. 6, we have that the axion-induced effect dominates for

$\frac{g_{a\gamma\gamma}}{m_{a}}\gtrsim\mathcal{O}(1)\times\frac{\alpha}{m_{e}^{2}}\simeq\frac{10^{-10}\
\mathrm{GeV}^{-1}}{10^{-6}\ \mathrm{eV}}~{}.$ (11)

Since $10^{-6}\ \mathrm{eV}$ corresponds to GHz frequencies, and the strongest
limits on $g_{a\gamma\gamma}$ which are agnostic to whether or not the axion
makes up the cosmological DM of the universe are $g_{a\gamma\gamma}<6.6\times
10^{-11}\,\mathrm{GeV}^{-1}$ from the CAST experiment Anastassopoulos  _et
al._ (2017), Eq. (11) shows that an experiment sensitive to the EH
nonlinearities could also set the strongest bounds on the axion-photon
coupling.

---
Figure 7:  Schematic of the proposed single-cavity experiment Eriksson  _et
al._ (2004); Bogorad  _et al._ (2019). Note that the signal mode $\omega_{s}$
extends through the bulk of the cavity, but the filtering geometry suppresses
the pump fields in the detection region.

Consider an SRF cavity pumped simultaneously with resonant electromagnetic
fields $\mathbf{E}_{1},\mathbf{B}_{1}$ and $\mathbf{E}_{2},\mathbf{B}_{2}$
with frequencies $\omega_{1}$ and $\omega_{2}$ respectively, as illustrated in
Fig. 7. The intermediate axion field can be obtained by integrating the source
$(\mathbf{E}_{1}+\mathbf{B}_{1})\cdot(\mathbf{E}_{2}+\mathbf{B}_{2})$ against
the massive Klein-Gordon Green’s function, yielding an axion effective current
from Eq. (10) containing a frequency component at
$\omega_{s}=2\omega_{1}-\omega_{2}$,

$\mathbf{J}_{a}(x,t)\supset\mathbf{J}_{a}^{(c)}(x)\cos(\omega_{s}t)+\mathbf{J}_{a}^{(s)}(x)\sin(\omega_{s}t)~{}.$
(12)

For general $m_{a}$, both quadratures in $\mathbf{J}_{a}$ are present, but in
the local limit $m_{a}\gg\omega_{s}$, $\mathbf{J}_{a}$ is always in phase with
the pump modes. Let $\hat{\mathbf{E}}_{s}$ be the dimensionless signal mode in
a cavity with volume $V$, with normalization $\int
d^{3}x\,|\hat{\mathbf{E}}_{s}(x)|^{2}=V$, and quality factor $Q_{s}$. The
axion-sourced signal field $\mathbf{E}_{a}$ is then Bogorad  _et al._ (2019)

$\mathbf{E}_{a}(x,t)=\frac{Q_{s}}{\omega_{s}V}\hat{\mathbf{E}}_{s}(x)\int
d^{3}x^{\prime}\,\hat{\mathbf{E}}_{s}(x^{\prime})\cdot\mathbf{J}_{a}(x^{\prime},t)~{},$
(13)

from which we can obtain the total number of photons in the signal field as

$N_{s}=\frac{1}{2\omega_{s}}\left\langle\int
d^{3}x\,|\mathbf{E}_{a}(x,t)|^{2}\right\rangle,$ (14)

where the brackets denote time averaging. Note that all of the above
considerations hold equally well for the EH-sourced signal field, using Eq.
(9) instead for the effective current.

---
Figure 8:  Projected sensitivity of the single-cavity experiment with
different assumptions about the cavity parameters, bandwidth, and measurement
time; see text for details.

We can obtain an estimate of the sensitivity by assuming that thermal noise
dominates the background and using the Dicke radiometer equation for the SNR,

${\rm SNR}=\frac{P_{s}}{T}\sqrt{\frac{t}{B}}\approx\frac{N_{s}}{N_{\rm
th}}\frac{1}{2LQ_{s}}\sqrt{\frac{t}{B}},$ (15)

where $P_{s}$ is the signal power, $t$ is the total measurement time, $B$ is
the signal bandwidth, $L$ is the length of the cavity, and $N_{\rm
th}=T/\omega_{s}$ is the number of thermal photons at the signal frequency
(valid for temperatures $T\gg\omega_{s}$). Setting SNR = 1 and solving for
$g_{a\gamma\gamma}$ for a cylindrical cavity with $a=0.5$ m and $d=1.56$ m,
temperature $T=1.5$ K, peak pump field strength $E_{0}=45$ MV/m, and the mode
combination TE011/TM010/TM020 (with $f_{s}=\omega_{s}/(2\pi)=0.527\ {\rm
GHz}$) yields the projected sensitivities shown in Fig. 7, for various
measurement times and readout schemes. The red curves correspond to
$Q_{s}=3\times 10^{8}$, a measurement time of $t=1$ day, and a bandwidth
$B=\frac{\omega_{s}}{2\pi Q_{s}}=2$ Hz (dark red) or $B=1/t$ (light red); the
latter may be achieved with a phase-sensitive readout such as a lock-in
amplifier, because the phase of the signal is known according to Eq. (12). The
green curve corresponds to $Q_{s}=10^{12}$, $t=20$ days, and $B=1/t$, and
would achieve sensitivity to the EH nonlinearity. In order for $B=1/t$ to be
viable, the frequencies of the cavity, and in particular the mode-matching
condition, must be robust to vibrational noise, possibly with an active
frequency-stabilizing scheme.

For all of the sensitivity curves, the limiting sensitivity to
$g_{a\gamma\gamma}$ is independent of $m_{a}$ for $m_{a}\ll\omega_{s}$ and
degrades linearly with $m_{a}$ for $m_{a}\gg\omega_{s}$. This is one potential
advantage of the single-cavity scheme compared to the two-cavity schemes:
since the signal generation in the single-cavity scheme takes place in the
same spacetime region as the pump region, a heavy off-shell axion does not
have to propagate to be converted back to photons, a fact which causes the
reach of the two-cavity scheme to degrade exponentially at large $m_{a}$. On
the other hand, the single-cavity scheme is far more sensitive to noise
sources which generate nonlinearities in the pump region, so these two schemes
are highly complementary.

####  II.3.4 Ongoing R&D for Axion searches

The previously discussed axion searches will require custom designed SRF
cavities in order to fulfill the stringent requirements on frequency matching
between the pump mode(s) and the signal mode. In addition, multiple issues,
both technical and scientific, need to be addressed in order to maximize the
search sensitivity. A feasibility study is being conducted to evaluate how the
multimode axion searches can be implemented with the necessary sensitivity.

An elliptical 1.3 GHz 9-cell SRF Nb cavity is being used to study the presence
and magnitude of non-linear effects of the Nb surface that could cause the
excitation of frequencies or resonant modes other than the ones intentionally
excited in the cavity. The geometry of this cavity, being multi-cell, is
different from the single cell cavities that will be used in the axion
searches. However, the multi-cell design allows for easy access to multiple
resonant modes, both nearby ($\approx$MHz apart) or well-separated
($\approx$GHz) in frequency. The measurements are being conducted at 2 K in a
liquid helium dewar and include:

  * •

the excitation of one pump mode and the search for excited modes both in the
same resonant pass-band or in different TM or TE pass-bands.

  * •

the excitation of two pump modes and the search for their linear combinations,
focusing in particular on the combinations that would correspond to the axion
signal: $\omega_{3}=2\omega_{1}\pm\omega_{2}$.

In addition to the study of non-linear effects, it is also crucial to identify
the possible sources of spurious excitation in the RF system that could
propagate to the cavity (as for example higher harmonics of the pump modes
generated by the high power amplifier or resonant modes excited in the cavity
by the amplifier noise). Analyzing the sources of noise in the current
experimental system will help guide the choice of appropriate RF instruments
and components to be used in the axion searches in order to maximize their
sensitivity.

To guide the mechanical design of the axion cavities, effort is also being
directed at characterizing the cavity mechanical modes and the microphonics
present in the helium bath, and how these result in the cavity deformation and
frequency shift for each resonant mode of interest. These developments will
als directly benefit the development of the axion DM search discussed below in
Sec. III.4.

##  III Dark Matter Searches with Ultra-high Q

Both axions and dark photons are compelling candidates for DM Preskill  _et
al._ (1983); Dine and Fischler (1983). These candidates fall into a broader
category of wave-like DM, since their wave, rather than particle, nature
provides a more useful description. As light bosonic degrees of freedom,
axions or dark photons can form non-relativistic waves which would carry
energy density, and hence gravitate. Focusing on the axion for concreteness, a
non-relativistic wave is well-approximated as a coherent field oscillating
with a frequency $\omega_{a}\simeq m_{a}$, where $m_{a}$ is the axion mass.
The amplitude, $a_{0}$ of the axion field oscillations is set by the DM energy
density $\rho_{{}_{\text{DM}}}$ via
$a_{0}\sim\sqrt{\rho_{{}_{\text{DM}}}}/m_{a}$. The DM axion wave is expected
to carry momentum of roughly $m_{a}v$ with $v\sim 10^{-3}\ c$ of order the
local virial velocity in our halo. This leads to a small average correction to
the axion oscillation frequency, of order $v^{2}$. This implies that the
effective quality factor of the axion as a monochromatic source is $Q_{a}\sim
10^{6}$. Hence, the axion field is expected to be coherent over roughly a
million periods, by which time the axion picks up a random phase with respect
to some initial state. In addition, the axion momentum implies that there is a
small $\order{v}$ gradient in the axion field, though in many cases it can be
ignored. A similar story follows for dark photon DM, with the notable
exception that the dark photon is a vector field, and hence picks a random
spatial direction which changes over a coherence time.

Ref. | Material |  $f$ (GHz) |  $B_{a}$ (T) |  $T$ (K) | $Q_{0}$
---|---|---|---|---|---
Alesini  _et al._ (2019) | Nb3Sn | 3.9 | 6.0 | 4.2 | $(5.3\pm 0.3)\times 10^{5}$
Golm  _et al._ (2021) | NbTi/Cu | 9.08 | 5 | 4.2 | $2.95\times 10^{5}$
Ahn  _et al._ (2019) | Nb3Sn | 9 | 8 | 4.2 | $6\times 10^{3}$
Vora  _et al._ (2022) | REBCO | 9 | 11.6 | 4.2 | $7\times 10^{4}$
Di Vora  _et al._ (2022) | YBCO | 6.93 | 8.0 | 4.2 | $3.2\times 10^{5}$
Table 1: A list of existing cavities which were operated in strong magnetic
fields and the corresponding quality factors.

It should be noted that both axions and dark photons are attractive candidates
also because the abundance of DM can be explained by a random initial
displacement of the field. For the axion this can occur either during the QCD
phase transition or during inflation Dine and Fischler (1983); Preskill  _et
al._ (1983). For the QCD axion, an initial displacement of order the axion
decay constant motivates searches around a GHz, but lower frequencies can also
arise naturally. Several mechanisms could produce cosmic dark photons, the
simplest being through quantum fluctuations during inflation Graham  _et al._
(2016). These fluctuations seed excitations in the dark photon field,
resulting in the cold DM observed today in the form of coherent oscillation of
this field. The predicted mass from this mechanism is $m_{A^{\prime}}\approx
10\,{\mu\mathrm{eV}}\left(10^{14}\,\mathrm{GeV}/H_{I}\right)^{4}$, where
$H_{I}$ is the Hubble constant during inflation. Measurements of the CMB
tensor to scalar ratio constrain $H_{I}\lesssim 10^{14}$ GeV Ade  _et al._
(2016), which makes the search for $m_{A^{\prime}}\gtrsim 10^{-5}$ eV well-
motivated. Other mechanisms are possible and are described in Refs. Caputo
_et al._ (2021); Arias  _et al._ (2012).

###  III.1 Haloscopes and the Scan Rate

A scheme that has proven successful to reach sensitivity relevant for QCD
axion detection is that of the cavity haloscope. This detector relies on axion
to photon conversion inside RF cavities in a strong magnetic field. Similarly
a DM dark photon can convert to a photon, even in the absence of a magnetic
field. The signal power read out of a haloscope with a cavity of volume $V$, a
loaded quality $Q_{l}$, magnetic field $B$, and coupling $\beta$ is

$P_{\mathrm{sig}}=P_{0}\frac{\beta}{1+\beta}\frac{Q_{l}Q_{\mathrm{DM}}}{Q_{l}+Q_{\mathrm{DM}}}$
(16)

where

$P_{0}=\left\\{\begin{array}[]{cl}g_{a\gamma\gamma}^{2}\frac{\rho_{a}}{m_{a}}B^{2}VC_{a}&\mbox{for
axions}\\\ \epsilon m_{A^{\prime}}\rho_{A^{\prime}}VC_{A^{\prime}}&\mbox{for
dark photons}\end{array}\right.$ (17)

and the $C$’s are form factors, normalized mode ovelaps between the DM field
and the cavity mode, typically of order 0.1.

Figure 9:  Loaded quality factor as a function of the applied magnetic field
for a TESLA and a cigar-shaped Nb3Sn cavity Posen  _et al._ (2022). The inset
focuses on the low-field region. For the cigar-shaped Nb3Sn cavity,
measurements were made both for increasing and for decreasing field.

The axion (dark photon) mass is unknown, so cavity haloscopes must be tunable
to search through $g_{a\gamma\gamma}$ vs. $m_{a}$ ($\epsilon$ vs.
$m_{A^{\prime}}$) parameter space. Thus the scan rate is a key figure of merit
for such experiments. Accounting for thermal and amplifier noise, it was shown
that the scan rate allowed by fixing the SNR can be higher even as the cavity
$Q$ exceeds the quality of the DM source Kim  _et al._ (2020a). It is thus
well motivated to use ultra-high quality cavities in the search for DM, and in
the case of the axion, to develop high coherence cavities that can withstand
large magnetic fields.

###  III.2 High-$Q$ Cavities in a Static $B$-field

Figure 10: Implementing the photon counting technique will increase the scan
rate by a factor $\bar{n}_{SQL}/\bar{n}_{c}$ compared to the standard quantum
limit. The calculated $\bar{n}_{c}$ assumes a 50 mK cavity and that only real
thermal photons contribute to the dark count.

Using large cylindrical copper cavities, axions with mass in the range
$(2.7-4.2)\ \mu\text{eV}$ (650 MHz $-$ 1025 MHz) have been excluded at DFSZ
model sensitivity Bartram  _et al._ (2021). To extend the search to heavier
axions (up to 15 GHz), next generation experiments target the development of
new cavity technologies with higher quality factors to compensate for the poor
scaling of the scan rate with increasing frequency, mostly determined by the
effective volume of the chosen axion-sensitive cavity mode. For copper
cavities employed so far in leading haloscope experiments, the quality factor
is limited by the anomalous skin effect Pippard and Bragg (1947) to $\sim
10^{4}-10^{5}$ at $\sim$ GHz frequencies Alesini  _et al._ (2021). Thus,
radically different approaches need to be considered to realize better
resonator haloscopes.

Figure 11: The projected sensitivity of the proposed search for dark photon
dark matter with SQMS systems assuming a fixed scan rate.

SRF cavities in accelerators typically are operated in small ${\sim}\mu
T$-scale magnetic fields, with $Q_{0}\sim\order{10^{10}}$. In these low
magnetic field environments, niobium is expected to be in the Meissner state,
in which the superconductor acts as a diamagnet. In $T$-scale magnetic fields,
niobium would be normal conducting, but superconductors with higher critical
fields like Nb3Sn and NbTi can still be superconducting, though typically in
the vortex state, in which applied magnetic fields penetrate through the
material in a lattice of quantized flux. This trapped flux will degrade the
$Q_{0}$ of the cavity to some extent, but the cavity may still have a
substantially higher $Q_{0}$ than an equivalent normal conducting cavity.

Experiments exploring the performance of superconducting cavities in $T$-scale
magnetic fields have been successful in producing $Q_{0}$ values in the
mid-$10^{5}$ range at frequencies between 1-10 GHz, as shown in Table 1. These
are promising for improving cavity haloscope-based axion search experiments,
in which the sensitivity scales approximately with $\sqrt{Q_{0}}$ and the scan
rate scales approximately with $Q_{0}$. The performance of a Nb3Sn cavity from
the SQMS collaboration is shown in Fig. 9. This cavity was measured only up to
6 T due to test stand limitations, but may continue to have high $Q_{0}$ up to
even higher magnetic fields, and may have even higher $Q_{0}$ at lower
temperatures than 4.2 K Posen  _et al._ (2022).

Dielectric resonators McAllister  _et al._ (2018); Kim  _et al._ (2020b) are
another effective strategy to obtain high quality factors in multi-Tesla
fields, especially in the high-frequency range. In this type of cavity,
higher-order TM0n0 modes are exploited for detection, and the inserted
dielectric shells or rings coaxial with the cavity axis are properly shaped to
maximize the figure of merit $(C_{0n0}V)^{2}Q_{0}$, and in turn the haloscope
scan rate, in FEM simulations. $C_{0n0}$ is the form factor, that accounts for
the overlap between the wave functions of the axion and the produced photon.
For cylindrical cavities, the TM0n0 mode is conventionally considered since it
yields the largest form factor in a magnetic field, while the higher-order
modes have form factor values that decrease with increasing order. For
instance, $C_{030}=0.05$ is parametrically smaller than $C_{010}=0.69$ in an
empty cavity. However, form factors that are very close to those of the
fundamental mode can be in principle be obtained even for high-order modes if
one dielectric shell is inserted in the cavity McAllister  _et al._ (2018).
Alternatively, in the double-shell configuration, two concentric dielectric
hollow cylinders are employed to suppress the evanescent field at the copper
boundaries, to achieve quality factors sufficiently large to compensate the
loss in form factor Vora  _et al._ (2022). The estimated scan rate of this
latter dielectric resonator exceeds that of a conventional resonator by more
than a factor 4, with measured quality factors exceeding $9\times 10^{6}$ at
more than 8 T magnetic fields.

###  III.3 Sub-SQL Dark Photon Dark Matter Search with a Highly-Tunable SRF
Cavities

SQMS has recently started to design a dark photon dark matter (DPDM) search
that will implement a widely tunable SRF cavity to scan the parameter space
from 5 GHz to 8 GHz. SQMS will also take advantage of its in-house QIS
expertise to reduce the noise floor below the Standard Quantum Limit (SQL) by
using a superconducting qubit to count photons potentially deposited by a dark
photon Dixit  _et al._ (2021a).

Cavity haloscopes have traditionally extracted the DM signal via an antenna
connected to a linear amplifier, such as a Josephson Parametric Amplifier.
Unfortunately, these linear amplifiers contribute to their own noise power,
and their minimum contribution is the standard quantum limit (SQL). SQL noise
increases linearly with frequency, and thus it is necessary to subvert the SQL
to make higher-mass searches feasible. Two experiments have demonstrated sub-
SQL detection by sacrificing phase information: HAYSTAC by implementing vacuum
squeezing Backes  _et al._ (2021) and SQuAD by implementing qubit-based photon
counting Dixit  _et al._ (2021b).

SQMS plans to increase the scan rate by implementing the photon-counting
technique developed in Ref. Dixit  _et al._ (2021b). A transmon qubit will be
situated inside the SRF cavity. The resonant frequency of the qubit will
depend on the number of photons in the cavity, and Ramsey interferometry will
be implemented to extract this resonant frequency. Since this is a quantum
nondemolition measurement, this measurement can be repeated many times to
reduce uncertainties and reject false positives. If the only source of false
positives, i.e., dark count, comes from real thermal photons, then the scan
rate would improve by a factor of $\mathcal{O}(10^{2})$ to
$\mathcal{O}(10^{3})$ compared to SQL-limited detection. The photon occupation
number for thermal photons in the cavity at temperature
$T_{c}=50\,\mathrm{mK}$ is
$\bar{n}_{c}=\big{(}\exp(hf/k_{b}T_{c})-1\big{)}^{-1}$. The occupation number
from the standard quantum limit is $\bar{n}_{SQL}$, and the scan rate
improvement from subverting the SQL limit is $\bar{n}_{SQL}/\bar{n}_{c}$. The
calculated scan rate improvement (assuming very generously that the dark count
is dominated by the finite cavity temperature) is shown in Fig. 10.

The cavity conceptualized in this section will be sensitive to DPDM with
masses between $20.3\,\mathrm{\mu eV}\,(5~{}\mathrm{GHz})$ and
$33\,\mu\mathrm{eV}\,(8~{}\mathrm{GHz})$ with kinetic mixing angles of
$\order{10^{-15}}-\order{10^{-16}}$, as shown in Fig. 11. The form factor
varies from $C=0.05$ to $C=0.225$, assuming dark photons are randomly
polarized.333Because dark photons are often thought to be randomly polarized,
the form factor for a dark photon haloscope, in this case, is 1/3 that of an
axion haloscope. The Nb cavity is expected to have an unloaded quality factor
of $10^{10}$. However, the superconducting qubit is likely to reduce $Q_{L}$
by a few orders of magnitude.

This project is currently at its conceptual and design stage. The projection
is likely to fluctuate by an order of magnitude as this project matures.
Regardless, SQMS is excited to combine SRF cavity technology and qubit-based
photon counting to increase the DM search scan rate by many orders of
magnitude.

---
Figure 12:  Schematic of the proposed up-conversion axion dark matter
experiment. A pump mode photon at frequency $\omega_{0}\sim\text{GHz}$ is
upconverted by axion dark matter into a nearly degenerate photon of frequency
$\omega_{1}=\omega_{0}+m_{a}$. See Ref. Berlin  _et al._ (2020) for additional
details.

###  III.4 Axion Dark Matter Searches with Up-Conversion

To date, most experiments searching for electromagnetically-coupled axion DM
employ _static_ magnetic fields, since these are more easily sourced at large
field strengths. In the presence of an axion DM background, the axion-photon
interaction sources a feeble electromagnetic field oscillating at the axion
mass $m_{a}$, which can be detected with a resonator tuned to the same
frequency. This is the strategy employed by typical haloscope cavity
experiments, which operate at $\sim\text{GHz}$ frequencies, corresponding to
axion masses of $m_{a}\sim\mu\text{eV}$. Since the resonant frequency is
typically controlled by the inverse geometric size of the detector, probing
much lighter axion masses requires prohibitively large cavities. To circumvent
this, LC circuit resonator haloscopes are in the beginning stages of
development to search for much lighter axions, enabled by the fact that their
resonant frequency is not directly dictated by the size of the circuit
Chaudhuri  _et al._ (2015). However, in the limit that the axion’s Compton
wavelength is much larger than the characteristic size of the detector
$L_{\text{det}}$, the signal power is parametrically suppressed by the small
quantity $m_{a}\,L_{\text{det}}\ll 1$.

---
Figure 13:  Projected sensitivity of an up-conversion axion dark matter
experiment. As three representative examples, we show the projected
sensitivity of a 50L prototype, a cubic meter setup, and a futuristic
experiment with a total instrumented volume of $5\ \text{m}^{3}$. Regions
motivated by the QCD axion and ALP dark matter generated by the misalignment
mechanism are shown as orange and blue bands, respectively.

As highlighted in Refs. Berlin  _et al._ (2020); Lasenby (2020); Berlin  _et
al._ (2021a), a heterodyne search employing the oscillating electromagnetic
fields of a resonant cavity completely undoes this unneeded suppression. This
can be understood from the form of the axion-induced effective current
introduced above in Eq. (10), which in the long-wavelength limit is
$\mathbf{J}_{a}\propto\mathbf{B}\,\partial_{t}a$. For a background magnetic
field oscillating at a frequency of $\omega_{0}$, the resulting axion-induced
emf is controlled by the beats of $\omega_{0}$ and $m_{a}$, i.e.,
$\mathcal{E}_{a}\propto\partial_{t}\mathbf{J}_{a}\propto\omega_{0}\pm m_{a}$.
Thus, compared to an experiment employing static magnetic fields, oscillating
the field at $\omega_{0}\sim\text{GHz}$ enhances the signal by
$\text{GHz}/m_{a}\gg 1$ in the low-mass limit.

Figure 14: Illustration of entangled sensor-cavities within a network-volume
of radius $\bar{r}$, taken to be much smaller than the axion coherence length,
$\lambda$. Thanks to this hierarchy of scales, the entire network experiences
a coherent signal that can be combined at the amplitude level. The cylinders
represent the cavities, and the colored lines represent their entanglement.
The solid black lines represent joint processing of the measurement to search
for dark matter.

This enhancement carries over directly to the reach of a proposed heterodyne
search utilizing a single SRF cavity Berlin  _et al._ (2020); Lasenby (2020);
Berlin  _et al._ (2021a). A schematic of the setup is shown in Fig. 12. A
cavity is pumped at a frequency $\omega_{0}\sim\text{GHz}$. The axion field
interacts with the pump mode magnetic field, resonantly driving power into a
signal mode of nearly degenerate frequency $\omega_{1}\simeq\omega_{0}\pm
m_{a}$ and distinct spatial geometry, provided that the pump $B$-field and
signal $E$-field have a large overlap $\mathbf{B}_{0}\cdot\mathbf{E}_{1}$
integrated over the volume of the cavity (which is indeed the case when the
pump and signal modes are, e.g., low-lying TE and TM modes). A scan over
possible axion masses can be achieved by slightly perturbing the geometry of
the cavity, thus modifying the frequency splitting $|\omega_{1}-\omega_{0}|$
to match a wide range of axion masses. Compared to a static-field LC circuit
experiment of comparable size, the SNR of a thermal-noise limited SRF
heterodyne setup is enhanced by

$\frac{\text{SNR}_{\text{SRF}}}{\text{SNR}_{\text{LC}}}\sim\frac{\omega_{1}}{m_{a}}\,\bigg{(}\frac{Q_{\text{SRF}}}{Q_{\text{LC}}}\bigg{)}^{\frac{1}{2}}\bigg{(}\frac{B_{\text{SRF}}}{B_{\text{LC}}}\bigg{)}^{2}\bigg{(}\frac{T_{\text{LC}}}{T_{\text{SRF}}}\bigg{)}^{\frac{1}{2}},$
(18)

where $Q$, $B$, and $T$ are the quality factor, magnetic field strength, and
temperature of the corresponding setup, respectively. Taking representative
experimental parameters in the above expression, the larger quality factor of
SRF cavities approximately compensates for the smaller magnetic fields and
larger temperatures achievable in a superconducting heterodyne setup, leaving
a residual enhancement
$\text{SNR}_{\text{SRF}}/\text{SNR}_{\text{LC}}\sim\mu\text{eV}/m_{a}\gg 1$
that becomes increasingly important at smaller axion masses. Furthermore,
since the frequency of the axion signal $\omega_{1}\sim\omega_{0}+m_{a}$ is
approximatley independent of the axion mass in the long-wavelength limit, GHz
cavities can efficiently couple to extremely slowly oscillating axion fields,
opening up sensitivity to axion masses below a kHz, a region typically
inaccessible with static field resonators.

Figure 15: Scaling of the (normalized) scan-rate with the number of sensors
$M$ in the network (log-log plot). Cavities in the network differ by their
intrinsic linewidths, which fall within the interval $\gamma_{\ell}\in[1,3]$
(normalized by smallest linewidth). Solid lines correspond to squeezed input
at gain $G=4$. Dashed lines correspond to the quantum limited setup (zero
squeezing). For a network of independent sensors, each sensor must be
independently squeezed. Observe quadratic scaling of the scan rate with $M$
for a coherent network of sensors versus linear scaling with $M$ for a network
of independent sensors (i.e., the slope of the former is about twice that of
the latter).

The projected sensitivity to the axion parameter space is shown in Fig. 13 for
three choices of experimental parameters, including noise estimates that are
anchored to previous experimental measurements, such as those performed for
the MAGO gravitational wave experiment Bernard  _et al._ (2001, 2002);
Ballantini  _et al._ (2005). A wider range of experimental parameters is shown
in Refs. Berlin  _et al._ (2020, 2021a). The projection for the 50L prototype
assumes a peak pump-mode magnetic field of $B_{\text{pump}}=0.2\ \text{T}$, an
intrinsic quality factor of $Q=10^{10}$, and no attenuation of mechanical
vibrations. The sensitivity of the cubic meter setup assumes the same magnetic
field, a larger quality factor of $Q=10^{12}$, and 100 dB of vibrational
attentuation. Finally, the projections for the futuristic setup assumes a
total integrated volume of $5\ \text{m}^{3}$, $B_{\text{pump}}=0.4\ \text{T}$,
$Q=10^{12}$, and 130 dB of vibrational attenuation.

These estimates indicate that noise induced by mechanical vibrations are
likely to limit the sensitivity for $m_{a}\lesssim\text{kHz}$, while a
thermal-noise limited setup is potentiallly attainable for
$m_{a}\sim\text{kHz}-\text{GHz}$. In particular, a thermal-noise limited
cubic-meter sized cavity woud be sensitive to the QCD axion for $m_{a}\sim
100\ \text{kHz}-100\ \text{MHz}$. As mechanical noise increases rapidly at
lower frequency splittings, there is strong motivation to consider active or
passive attenuation of vibrations stemming from the immediate environment,
such as the cooling apparatus, as was considered in Ref. Ballantini  _et al._
(2005). Regardless, a wide range of uncharted axion DM parameter space remains
attainable without any active attenuation and for modest experimental
assumptions.

###  III.5 A Quantum Network of Haloscopes

The operation of haloscopes in ultra-coherent cavities and quantum techniques
presents new opportunities to enhance the scan rate. One such example has been
shown by the HAYSTAC experiment Backes  _et al._ (2021). There, the SNR and
the scan rate for axion DM have been enhanced with a quantum metrology
protocol Zheng  _et al._ (2016). In particular, a squeezed vacuum state
injected into an RF cavity and homodyne detection along the squeeze direction
together allow for an improved SNR and a faster scan.

Recently, it was shown that the performance of a quantum network could be
utilized further to improve axion DM searches Brady  _et al._ (2022). In
particular, a network of microwave cavities that is local will experience a
coherent axion signal if the size of the network is less than the axion
coherence length, of order $10^{3}\,m_{a}^{-1}$, as illustrated in Fig. 14.
However, the noise in the network will be incoherent among the network nodes.
One can make use of distributed squeezed states Zhuang  _et al._ (2018) to
exploit the coherent nature of the DM signal. The results of Brady  _et al._
(2022) indicate that combining quantum resources (squeezing) in a distributed-
network setting can allow for a scan that is faster by a factor of the square
number of network nodes in the ideal case. The improvement is enabled by
adding the signal at the amplitude level rather than adding powers in the
classical network case. The improved scan rate is shown in Fig. 15 as compared
with a classical network, both with squeezing and without squeezing (i.e., at
the quantum limit).

##  IV High Frequency Gravitational waves

Figure 16: A gravitational wave (red) interacts with a static external
$B$-field (blue), resonantly exciting a cavity mode (green).

Since gravity couples to all forms of energy, there is an interaction between
gravitational waves (GWs) and photons Gertsenshtein (1962); Zeldovich (1972);
Zel’dovich (1973). A direct consequence of this coupling is that microwave
cavities can also be used to detect high-frequency GWs. As with the static-
field axion searches described above, an incoming GW can be thought of as
sourcing an effective current through its interaction with a background
magnetic field $B_{0}$. This effective current then sources a small signal
field $B_{\rm sig}\sim h\,B_{0}$, where $h$ is the characteristic strain of
the GW. Such signals were recently investigated in Ref. Berlin  _et al._
(2021b).

GWs are typically classified into two different categories: stochastic GW
backgrounds, and GWs with a preferred direction of propagation and well-
defined frequency.444Such sources are often transient, and thus the frequency
typically evolves in time, with the particular timescale depending on the
source of emission. We concentrate on the latter case, since the detection
prospects are typically better. A detailed theoretical analysis of the
sensitivity to stochastic signals is ongoing.

Figure 17: The projected sensitivity of axion cavity haloscope experiments to
high-frequency coherent gravitational waves.

The setup we focus on in this section is a cavity that is permeated by a
static external $B$-field. The incoming GWs can interact with the static
$B$-field of the cavity and excite a specific cavity mode, as shown in Fig.
16. Such a setup is most sensitive near resonance, so that the resonant
frequency matches the GW frequency, $\omega_{\rm
sig}\simeq\omega_{g}\sim\text{GHz}$. Possible sources of GWs in this frequency
range are binary mergers of ultralight compact objects with mass $M\lesssim
10^{-11}M_{\odot}$ and GWs emitted by the superradiant axion cloud around
spinning primordial black holes, corresponding to masses of $M\sim
10^{-4}\,M_{\odot}$. For a review on high frequency GW sources and other
recent proposals, see, e.g. Ref. Aggarwal  _et al._ (2021); Domcke  _et al._
(2022).

Following Ref. Berlin  _et al._ (2021b), the sensitivity of the described
setup to the GW strain $h$ is

$\displaystyle h$ $\displaystyle\gtrsim$ $\displaystyle 3\times
10^{-22}\times\bigg{(}\frac{1\
\text{GHz}}{\omega_{g}/2\pi}\bigg{)}^{\frac{3}{2}}\bigg{(}\frac{0.1}{\eta_{n}}\bigg{)}\bigg{(}\frac{8\
\text{T}}{B_{0}}\bigg{)}\bigg{(}\frac{0.1\
\text{m}^{3}}{V}\bigg{)}^{\frac{5}{6}}$ (19)
$\displaystyle\bigg{(}\frac{10^{5}}{Q}\bigg{)}^{\frac{1}{2}}\bigg{(}\frac{T_{\text{sys}}}{1\
\text{K}}\bigg{)}^{\frac{1}{2}}\bigg{(}\frac{\Delta\nu}{10\
\text{kHz}}\bigg{)}^{\frac{1}{4}}\bigg{(}\frac{1\
\text{min}}{t_{\text{int}}}\bigg{)}^{\frac{1}{4}}~{},$

where $\omega_{g}$ is the GW frequency, $\eta_{n}$ the coupling coefficient
defined below, $B_{0}$ the magnitude of the external $B$-field, $V$ the cavity
volume, $Q$ the cavity quality factor, $T_{\text{sys}}$ the system
temperature, $\Delta\nu$ the signal bandwidth, and $t_{\text{int}}$ the
integration time. In Fig. 17 we show the sensitivity of cavity experiments
originally designed to detect axion DM. All experiments shown (except that
labelled “SQMS” which corresponds to a cavity under development) only need to
reanalyze data already recorded for axion DM searches in order to set limits
on GWs in the GHz regime.

The coupling coefficient $\eta_{n}$ in the sensitivity estimate of Eq. (19) is
defined as

$\displaystyle\eta_{n}\equiv\frac{\Big{|}\int_{V}d^{3}\mathbf{x}~{}\bm{E}_{n}^{*}\cdot\hat{\boldsymbol{j}}_{+,\times}\Big{|}}{V^{1/2}\,\big{(}\int_{V}d^{3}\mathbf{x}~{}|\bm{E}_{n}|^{2}\big{)}^{1/2}}~{},$
(20)

where the index $n$ stands for a generic mode of the cavity,
$\bm{E}_{n}(\bm{x})$ the spatial cavity mode function, and
$\hat{\boldsymbol{j}}_{+,\times}$ the effective current induced by the GW. In
particular, $\hat{\boldsymbol{j}}_{+,\times}$ depends on the GWs polarization,
which is indicated with a $+$ or $\times$ sign. The full definition can be
found in Ref. Berlin  _et al._ (2021b). The coupling coefficient is best
evaluated in the local inertial frame of the cavity. This frame is often
referred to as the proper detector frame or Fermi normal coordinate system
Fermi (1922a, b); Manasse and Misner (1963); Marzlin (1994); Rakhmanov (2014).
Furthermore, resonant cavities operate in the regime where the GW frequency is
comparable to the inverse geometric size of the cavity, i.e.,
$\omega_{g}L_{\rm det}\sim 1$. In this case, evaluating the signal strength
requires the resummation of an infinite series of derivatives acting on the
Riemann tensor Berlin  _et al._ (2021b).

In Fig. 18 we show the coupling coefficient as a function of the incidence
angle $\beta$ of the GW. The external $B$-field is taken to be coaxial with
the cavity bore ($\alpha=0$). We consider two modes of a cylindrical cavity:
TM01p (left) and TE21p (right). The coupling coefficient is non-zero in the
TM010 mode used by most axion haloscopes, such that already-recorded data can
be reanalysed to search for GWs. From Fig. 18 it is clear that the detailed
spatial structure and mode-dependence of the signal enables sensitivity to the
direction and polarization of the GW.

Figure 18: Polar plot for the coupling coefficient $\eta_{n}$ for TM01p and
TE21p modes. The polar angle is $\beta$, the orientation of the GW relative to
the bore of the cavity, as depicted in Fig. 16.

High-frequency GWs can also be detected by the interaction of GWs with a
pumped cavity mode, analogous to the upconversion axion search in Sec. III.4.
In this case, the GWs induce the transition of photons from the pumped cavity
mode into an empty signal mode Braginskii  _et al._ (1973); Grishchuk and
Sazhin (1975); Pegoraro  _et al._ (1978a, b); Caves (1979); Bernard  _et al._
(2001, 2002); Ballantini  _et al._ (2003, 2005); Mensky and Rudenko (2009). A
detailed analysis taking into account the proper detector frame, and
interactions of the GW with mechanical modes of the cavity is currently
ongoing. Since this setup shares many features with multi-mode axion DM
searches, such as in Sec. III.4, the R&D for these two goals is synergistic.

##  V Capabilities and Infrastructure

SRF cavities, as we show above, provide a rich new arena to explore new
physics. In this section we give examples of new capabilities and new
supporting infrastructure. In particular we present ongoing and future
research on optical-RF transduction, which would have an impact on single
photon sensing and senor networks. We also comment on opportunities to scale-
up cavity based sensing experiments, including a planned cryogenic platform at
SQMS in Fermilab which would be the world’s biggest of its kind.

###  V.1 Enhanced sensitivity by High-Efficient Microwave-Optical transduction

The strong effort to utilize SRF cavities for QIS at SQMS has inspired a
research on high efficiency microwave-optical transduction based on high-$Q$
SRF resonators. A high RF-to-optical transduction efficiency may enhance new
physics searches. For example, up/down photon conversion at high efficiency
can extend new physics particle searches over a wider energy spectrum, with
the possibility to detect single photons in both microwave and optical
regimes, as well as performing measurements below the SQL. Up/down photon
conversion may also enable highly sensitive axion and dark photon haloscope
searches in the THz regime. One possibility is to let the axion or dark photon
convert to a SM photon in a THz cavity. Transduction would then allow this
signal to accumulate coherently in a much larger GHz range microwave cavity,
greatly enhancing the SNR of a possible DM signal.

Fermilab is supporting a preliminary study on quantum transduction through the
Laboratory Directed R&D (LDRD) program. We are exploring hybrid coherent
resonance systems and bi-directional quantum transduction schemes to up/down-
convert the microwave information to/from the optical regime and enhance the
conversion efficiency at the quamtum threshold.

Non-centrosymmetric crystals are used to create interactions between microwave
and optical fields using photonic RF three-wave-mixing processes or electro-
optic modulation. An electric field, applied on these crystals, modulates the
refractive index and the incident optical field linearly with the RF voltage
($\chi^{(2)}$), which is known as the Pockels effect. In microwave-optical
transduction, the self-heterodyne techniques have the advantage of eliminating
the local oscillator (LO) used in conventional super-heterodyne mixers,
reducing therefore complexity, power consumption, added noise, and thermal
quasiparticle poisoning. In nonlinear electro-optic materials, the RF (i.e.,
the excitation mode of the resonant cavity) and the sidebands are mixed
through the second order nonlinearity in the transmitted optical power:
$P_{\text{opt}}\propto V^{2}_{\text{RF}}$. We are interested in detecting the
sidebands of the optical signal. The fraction of transmitted optical power
related to the sideband of the RF signal is proportional to the optical field
$|E_{\text{opt}}|^{2}$. If the sideband of the microwave signal
$V_{\text{RF}}$ is modulated in phase $\varphi_{b}(t)$, this phase modulation
is also detected as
$P_{\text{opt}}(t)\propto(1+\cos(\varphi_{b}(t))+\cos(2\varphi_{b}(t)))$.

The integration between different technologies in a Hybrid Quantum System
(HQS) is a major challenge which goes beyond the design, implementation, and
the control of single components. The Hamiltonian to model this system
electro-optic interaction in a triple-resonance scheme can be reduced to the
following form:

$\hat{H}=\hslash g_{eo}(pa^{\dagger}+p^{\dagger}a)(b+b^{\dagger}),$ (21)

the pump optical mode ($p$) is driven to coherently couple the optical signal
mode ($a$) with the microwave mode ($b$), with the electro-optic (EO) coupling
strength $g_{eo}$ Kurizki  _et al._ (2015).

(a)

(b)

Figure 19: (a) Bulk Niobium cavity, superconducting at cryogenic temperature
coupled to $\text{L}\text{i}\text{N}\text{b}\text{O}_{\text{3}}$ whispering-
gallery mode crystal, with a prism for the optical coupling with fiber-optics,
operating in a dilution refrigerator at 15 mK. (b) Cross-section of a hybrid
microwave-optical cavity.

Limiting factors in the attainment of high-efficiency in the microwave-optical
transduction are:

  * •

Quality factor of the microwave cavity – While the optical $Q_{a}$ can reach
$10^{7}$, microwave $Q_{b}$ is limited in the current schemes.

  * •

Single-photon microwave-optic coupling coefficient ($g_{eo}$) – This paramter
is determined by the overlap between microwave and optical fields.

  * •

Pump power – High pump power leads to overheating of the device, and thermal
quasiparticles poisoning.

(a)

(b)

Figure 20: (a) Quantum transduction efficiency ($\eta_{i}$) and (b) Spectral
density (S) with reference to the SQL, with back-action noise cancellation.
Both plots evaluate the parameters with increasing RF quality factors (Q).

The bi-directional efficiency depends on the microwave-optical cooperativty
and losses:

$\eta=\frac{\kappa_{a,ex}}{\kappa_{a}}\frac{\kappa_{b,ex}}{\kappa_{b}}\times\frac{4C}{(1+C)^{2}}~{}~{},~{}~{}C=\frac{4n_{p}{g}_{eo}^{2}}{\kappa_{a}\kappa_{b}}$
(22)

where, for a generic $m$ mode ($m=a,b$), $\kappa_{m}$ is the total loss rate,
consisting of both the intrinsic and external losses:
$\kappa_{m}=\kappa_{m,ex}+\kappa_{m,i}$. $C$ is the cooperativity between the
optical and the microwave modes, which depends on the photon number in the
pump mode ($n_{p}$). The second term of Eq. 22 is the internal efficiency:
$\eta_{i}=4C/(1+C)^{2}$.

We evaluate the single photon coupling as

$g_{eo}=\frac{1}{8}n^{2}\omega_{p}r_{33}E_{\text{max}}\sqrt{\frac{\hbar\omega_{m}}{W}}$
(23)

where $n$ is the reflective index of the non-centrosymmetric crystal,
$\omega_{p}$ is the optical pump power, $r_{33}$ is the electro-optic
coefficient, $E_{\text{max}}$ is the maximum electric field that overlaps with
the optical mode at the edge of the crystal, $\omega_{m}$ is the microwave
frequency, and $W$ is the stored energy in the microwave cavity. Lithium
niobate ($\text{L}\text{i}\text{N}\text{b}\text{O}_{\text{3}}$) is among the
best materials exhibiting the Pockels effect, with a large electro-optic
coefficient ($r_{33}=31\ \text{pm}/\text{V}$ at 9 GHz) and low optical losses.
In Fig. 19, we show the diagram of a microwave-optical device based on bulk
resonators, along with the 3D model of a possible SRF hybrid cavity. The
design focuses on the optimal overlap between the microwave and the optical
fields, maximizing the quality factor and $E_{\text{max}}$, to achieve high
efficiency and cooperativity, with low added noise. Analyzing microwave
simulations we observe that hybrid devices with higher $Q$ achieve maximum
conversion efficiency at lower pump powers, as shown in Fig. 20(a). Therefore
longer coherence in the microwave cavity will ensure low-noise for few-photon
quantum transduction, which is different from the conventional trade-off
between efficiency and noise Rueda  _et al._ (2016); Hease  _et al._ (2020);
Holzgrafe  _et al._ (2020).

A combination of high-efficiency and quantum noise reduction schemes can be
proposed to increase the sensitivity, such as squeezed light and back-action
noise cancellation techniques Nazmiev  _et al._ (2021); Tsang (2010). Fig.
20(b) shows spectral densities with reference to the SQL. A back-action noise
evading scheme is applied to measure a combination of in-phase and quadrature
RF signals, achieving a sensitivity better than the SQL. We observe that by
increasing the $Q$ in the microwave cavity, the SQL is reached at lower pump
powers.

###  V.2 Cryogenic Platform for Scaled-up Sensing Experiments

The SQMS center at Fermilab is developing a cryogenic platform capable of
reaching millikelvin temperatures in an experimental volume of 2 meters
diameter by 1.5 meters in height Hollister  _et al._ (2021). The platform is
designed to host a three-dimensional qubit architecture based on SRF
technology, as well as sensing experiments.

The system, which is sketched in Figure 21, will consist of several cooling
stages of which we quote here specifications for two. The 20 mK space will
consist of a cooled volume of 4.7 m3 and with a cooling power of 300 $\mu$W,
while the 2K stage (superfluid Helium) will simultaneously have a cooling
power of order 30 Watt. The system can be operated as a 2-K platform only, in
which configuration the cold volume will be of order 10 m3. The specifications
for other stages of the platform are listed in Hollister  _et al._ (2021).

Figure 21: A CAD rendering of the planned SQMS cryogenic platform for quantum
computing and sensing. Specifications of the system are presented in Hollister
_et al._ (2021).

##  VI Outlook and future R&D

The development of high quality superconducting cavities for accelerators
presents some low hanging fruit to explore new fundamental physics. Off-the-
shelf systems and existing test facilities are already probing new ground, for
example, with Dark SRF’s dark photon search.

Taking full advantage of high quality systems and reaping the full set of
physics opportunities presented in this document requires dedicated R&D. An
incomplete list of research directions that are planned or needed include:

  * •

Material science for improved $Q$ – The new physics searches outlined here
benefit from high quality factors. Ongoing research aimed at enhancing the
coherence time of SRF cavities for use as quantum devices (e.g., Ref.
Romanenko  _et al._ (2020)) is synergistic with the sensing goals (including
achieving high $Q$ when a cavity is in a multi-tesla magnetic field).

  * •

Quantum engineering of cavity states – Sensing goals can be aided by preparing
nontrivial quantum states in electromagnetic cavities (e.g., squeezed states
Zheng  _et al._ (2016); Malnou  _et al._ (2019); Zhong  _et al._ (2018)). As
we discussed, further enhancement can be achieved with a network of entangled
cavities Brady  _et al._ (2022).

  * •

Cavities and Qubits – The integration of a qubit into an ultra high cavity may
enable new schemes for quantum computing et al (2022) and synergetically allow
for employing a photon counting non-demolition measurement for DM searches, as
exemplified in Ref. Dixit  _et al._ (2021b). For certain DM search schemes, it
would also be beneficial to have qubits that can operate successfully even in
high magnetic fields.

  * •

Nonlinearities in superconducting cavities – Searches with multiple cavity
modes, for axions or gravitational waves, depend on observing a faint signal
in the presence of an excited mode (or modes) in the background. Nonlinear
effects within the cavity walls can mimic such a signal, particularly if the
signal mode is near a harmonic. Nonlinear effects have recently been studied
and shown to depend on both the disorder parameter and the temperature of the
superconductor Sauls (2022). Experimental studies to measure and mitigate such
nonlinearities are well-motivated.

  * •

Nonlinearites due to TLS and other impurities – Nonlinear effects can also
come from impurites on the cavity walls. Two-level systems (TLS) are a known
source of loss in superconducting devices and are likely to induce nonlinear
current responses at some level. Measurements and estimates of TLS-induced
nonlinearity is required. Mitigating TLS altogether, or minimizing their
overlap with cavity modes is well-motivated for our sensing goals, as well as
to enhance the coherence of superconducting devices for quantum applications.

  * •

Frequency stability – the cavity frequency can change during measurements, for
example due to microphonics (i.e., mechanical vibrations, helium pressure
fluctuations, etc.) or Lorentz force detuning. Developing methods for
mitigating the effects of these phenomena will be especially important when
operating with a narrow bandwidth, which will help to increase SNR.

  * •

Cavity and antenna design – cavity search schemes often involve different
modes, and in many cases, it is beneficial to use different antennas to couple
to different modes, so that each couples strongly to the mode of interest, but
weakly to the others. This can help avoid nonlinear excitation and other
phenomena that can cause a spurious signal.

  * •

Frequency tuning – to scan over a mass range of interest, tuners are used to
vary the resonant frequency of cavities. There are a variety of tuning systems
that are implemented, with different mechanisms, tuning ranges, tuning speeds,
and other qualities such as hysteresis. In millikelvin systems, a critical
parameter to consider is heat generation. A tuner that uses an electrical
motor may overwhelm the cooling of such a system. Even a piezo tuner may still
generate enough heat to raise the temperature of the cavity after a tuning
step, requiring cooldown time. Innovation in low dissipation, wide range
frequency tuning systems would be beneficial.

  * •

Vibration noise – New tools for improving isolation of systems under test from
sources of vibrations will be helpful for future experiments.

  * •

RF sources – RF noise sources can include not just thermal photons, but also
crosstalk, amplifier noise, and nonlinearities. New techniques and tools to
reduce these sources of noise will be important, in addition to diligence when
it comes to established practices to limit RF noise. Many such techniques have
already been developed in the context of precision metrology, and their
incorporation into SRF cavity searches represents an important opportunity for
collaboration between these fields.

  * •

Quantum transduction – Microwave cavities with long coherence will increase
the microwave-optical transduction cooperativity and efficiency, ensuring low-
noise conversion at the quantum threshold. Sensitivity below the SQL is also
enhanced by the higher microwave $Q$. The up/down conversion may enable highly
sensitive axion and dark photon haloscope searches in the THz regime.

We conclude that advances in SRF and quantum technology present exciting
opportunities to probe fundamental physics. We have presented examples,
including searches for new particles, searches for DM, and new approaches to
detection of gravitational waves. The first searches employing SRF cavities
are underway, both in the classical and quantum regime. It is notable that
many of the ideas presented here have been proposed in the past few years, and
that dedicated R&D, such as work that is planned in the SQMS center at
Fermilab, has potential to make significant advances. In this context, these
R&D directions are synergistic with the quest to build SRF cavity-based
quantum computers.

Many of the opportunities presented here are small-scale experiments. In the
long term, the most successful schemes may have room for growth and
multiplexing. Modern SRF-based accelerators, such as PIP-II, LCLS-II, and the
European XFEL are large integrated sytems with a large total cavity volume in
excess of 10 m3, and a total cooling power of order several kW. The well-
motivated physics goals presented here may be significantly advanced by an
efforts on this scale. A promising path towards ambitious science goals is to
test search schemes with single-cavity setups in existing test stands. The
most successful of these can then be scaled up.

##  VII Acknowledgements

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 the contract
No. DE-AC02-07CH11359. Fermilab is operated by the Fermi Research Alliance,
LLC under contract No. DE-AC02-07CH11359 with the United States Department of
Energy.

DB is supported by a ‘Ayuda Beatriz Galindo Senior’ from the Spanish
‘Ministerio de Universidades’, grant BG20/00228. IFAE is partially funded by
the CERCA program of the Generalitat de Catalunya. The research leading to
these results has received funding from the Spanish Ministry of Science and
Innovation (PID2020-115845GB-I00/AEI/10.13039/501100011033).

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