BREAD Collaboration

# Broadband solenoidal haloscope for terahertz axion detection

Jesse Liu<EMAIL_ADDRESS>Cavendish Laboratory, University of
Cambridge, Cambridge CB3 0HE, UK  Department of Physics, University of
Chicago, Chicago, IL 60637, USA  Kristin Dona  Department of Physics,
University of Chicago, Chicago, IL 60637, USA  Gabe Hoshino  Department of
Physics, University of Chicago, Chicago, IL 60637, USA  Fermi National
Accelerator Laboratory, Batavia, IL 60510, USA  Stefan Knirck<EMAIL_ADDRESS>Fermi National Accelerator Laboratory, Batavia, IL 60510, USA  Noah Kurinsky
<EMAIL_ADDRESS>SLAC National Accelerator Laboratory, Menlo Park,
CA 94025, USA  Fermi National Accelerator Laboratory, Batavia, IL 60510, USA
Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL
60637, USA  Matthew Malaker  Fermi National Accelerator Laboratory, Batavia,
IL 60510, USA  David W. Miller<EMAIL_ADDRESS>Department of
Physics, University of Chicago, Chicago, IL 60637, USA  Enrico Fermi
Institute, University of Chicago, Chicago, IL 60637, USA  Andrew Sonnenschein
<EMAIL_ADDRESS>Fermi National Accelerator Laboratory, Batavia, IL 60510,
USA  Mohamed H. Awida  Fermi National Accelerator Laboratory, Batavia, IL
60510, USA  Peter S. Barry  Argonne National Laboratory, Lemont, IL 60439, USA
Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL
60637, USA  Karl K. Berggren  Massachusetts Institute of Technology,
Cambridge, MA 02139, USA  Daniel Bowring  Fermi National Accelerator
Laboratory, Batavia, IL 60510, USA  Gianpaolo Carosi  Lawrence Livermore
National Laboratory, Livermore, CA 94551, USA  Clarence Chang  Argonne
National Laboratory, Lemont, IL 60439, USA  Kavli Institute for Cosmological
Physics, University of Chicago, Chicago, IL 60637, USA  Aaron Chou  Fermi
National Accelerator Laboratory, Batavia, IL 60510, USA  Rakshya Khatiwada
Fermi National Accelerator Laboratory, Batavia, IL 60510, USA  Department of
Physics, Illinois Institute of Technology, Chicago, IL 60616, USA  Samantha
Lewis  Fermi National Accelerator Laboratory, Batavia, IL 60510, USA  Juliang
Li  Argonne National Laboratory, Lemont, IL 60439, USA  Sae Woo Nam  National
Institute of Standards and Technology, Boulder, CO 80305, USA  Omid Noroozian
NASA Goddard Space Flight Center, Grenbelt, MD 20771, USA  Tony X. Zhou
Massachusetts Institute of Technology, Cambridge, MA 02139, USA

###### Abstract

We introduce the Broadband Reflector Experiment for Axion Detection (BREAD)
conceptual design and science program. This haloscope plans to search for
bosonic dark matter across the $[10^{-3},1]$ eV ($[0.24,240]$ THz) mass range.
BREAD proposes a cylindrical metal barrel to convert dark matter into photons,
which a novel parabolic reflector design focuses onto a photosensor. This
unique geometry enables enclosure in standard cryostats and high-field
solenoids, overcoming limitations of current dish antennas. A pilot 0.7 m2
barrel experiment planned at Fermilab is projected to surpass existing dark
photon coupling constraints by over a decade with one-day runtime. Axion
sensitivity requires $<10^{-20}$ W/$\sqrt{\rm Hz}$ sensor noise equivalent
power with a 10 T solenoid and $10$ m2 barrel. We project BREAD sensitivity
for various sensor technologies and discuss future prospects.

##  I Introduction

Astrophysical evidence for dark matter (DM) is unambiguous Rubin and Ford
(1970); Tyson  _et al._ (1998); Tegmark  _et al._ (2004); Clowe  _et al._
(2006); Aghanim  _et al._ (2020); Bertone  _et al._ (2005), but its particle
properties remain enigmatic. Recent efforts are expanding bosonic DM searches
for $m_{\text{DM}}\lesssim 1$ eV masses 111Natural units are used, where the
(reduced) Planck’s constant $\hbar$, speed of light $c$, vacuum permittivity
$\varepsilon_{0}$ and permeability $\mu_{0}$ are set to unity
$\hbar=c=\varepsilon_{0}=\mu_{0}=1$, as conventional in particle physics.
predicted by many extensions of the Standard Model (SM) Arvanitaki  _et al._
(2010); Jaeckel and Ringwald (2010); Essig  _et al._ (2013); Baker  _et al._
(2013); Battaglieri  _et al._ (2017); Ahmed  _et al._ (2018); Irastorza and
Redondo (2018), complementing higher-mass searches Akerib  _et al._ (2017);
Tan  _et al._ (2016); Aprile  _et al._ (2018); Agnese  _et al._ (2018);
Aguilar-Arevalo  _et al._ (2020); Izaguirre  _et al._ (2015); Boveia and
Doglioni (2018); ATLAS Collaboration (2020, 2019, 2021). Notably, the
unobserved neutron electric dipole moment Harris  _et al._ (1999); Baker  _et
al._ (2006); Pendlebury  _et al._ (2015); Abel  _et al._ (2020) motivates the
quantum chromodynamics (QCD) axion $a$ predicted by the Peccei-Quinn solution
of the strong charge-parity problem Peccei and Quinn (1977); Wilczek (1978);
Weinberg (1978). Dark photons $A^{\prime}$ are also sought-after candidates
arising in string theory scenarios Holdom (1986); Dienes  _et al._ (1997);
Pospelov (2009); Goodsell  _et al._ (2009). These states have compelling
early-Universe production mechanisms and their field oscillations with
frequency $\nu=m_{\text{DM}}/2\pi$ exhibit DM properties Preskill  _et al._
(1983); Abbott and Sikivie (1983); Dine and Fischler (1983); Nelson and
Scholtz (2011); Arias  _et al._ (2012); Chadha-Day and Marsh (2022). Nonzero
DM-photon couplings enable laboratory detection via electromagnetic (EM)
effects.

The most-sensitive detection strategy today is the radio-frequency resonant-
cavity haloscope Sikivie (1983); De Panfilis  _et al._ (1987); Wuensch  _et
al._ (1989); Hagmann  _et al._ (1990), where ADMX Asztalos  _et al._ (2001,
2002, 2010); Wagner  _et al._ (2010); Du  _et al._ (2018); Braine  _et al._
(2020); Bartram  _et al._ (2021), CAPP Lee  _et al._ (2020); Jeong  _et al._
(2020); Kwon  _et al._ (2021), HAYSTAC Al Kenany  _et al._ (2017); Brubaker
_et al._ (2017); Zhong  _et al._ (2018); Backes  _et al._ (2021) probe QCD
axions within $[1.8,24]\,\mu$eV masses. However, this strategy has long-
standing obstacles from (i) narrow band tuning to unknown $m_{\text{DM}}$ and
(ii) impractical high-mass scaling for $m_{\text{DM}}\gtrsim 40\,\mu$eV. Scan
rates fall precipitously with photon frequency
$R_{\text{scan}}\sim\nu^{-14/3}$ Backes  _et al._ (2021) and the number of
required resonators scales unfavorably with effective volume $\sim
m_{\text{DM}}^{3}$. Proposed dielectric haloscopes could probe
$[40,400]\,\mu$eV Caldwell  _et al._ (2017); Millar  _et al._ (2017); Brun
_et al._ (2019a) and $[0.1,10]$ eV Baryakhtar  _et al._ (2018) masses, while
topological insulators target $[0.7,3.5]$ meV Marsh  _et al._ (2019).
Significant sensitivity gaps persist across $[10^{-4},1]$ eV masses, favored
by several theoretical scenarios Graham  _et al._ (2016); Gorghetto  _et al._
(2021); Co  _et al._ (2021), motivating broadband approaches.

This Letter introduces the Broadband Reflector Experiment for Axion Detection
(BREAD) conceptual design to search multiple decades of DM mass without tuning
to $m_{\text{DM}}$. BREAD proposes a novel experimental design that optimally
realizes dish-antenna haloscopes Horns  _et al._ (2013). Its hallmark is a
cylindrical metal barrel for broadband DM-to-photon conversion with a coaxial
parabolic reflector that focuses signal photons onto a sensor. In contrast to
existing dish antennas using spherical or flat surfaces Suzuki  _et al._
(2015); Knirck  _et al._ (2018a); Tomita  _et al._ (2020); Andrianavalomahefa
_et al._ (2020); BRA , our geometry is optimized for enclosure in standard
cryostats and compact high-field solenoids. This enhances signal to noise,
ensures the emitting surface and magnetic field are parallel, and keeps costs
practical. We delineate the optical properties of this novel geometry with
detailed ray tracing and numerical simulation. While photoconversion is
broadband, photosensor performance governs final discovery reach. We assess
various state-of-the-art sensors and discuss the advances required in quantum
sensing technology for next-generation devices to fulfill BREAD science goals
with broad anticipated impact in astronomy and beyond Dhillon  _et al._
(2017).

Figure 1: (a) BREAD reflector geometry: rays (yellow lines) emitted from the
cylindrical barrel, which is parallel to an external magnetic field
$\mathbf{B}_{\rm ext}$ from a surrounding solenoid (not shown) and focused at
the vertex by a parabolic surface of revolution. (b) Radial intensity
distribution $rI(r)$ expected from DM velocity effects in the $xy$ plane at
the focal spot using ray tracing, for the BREAD geometry as in (a) with
$R=20\,{\rm cm}$ (blue) and for a conventional plane-parabolic mirror setup
used in other experiments Suzuki  _et al._ (2015); Knirck  _et al._ (2018a);
Tomita  _et al._ (2020); Andrianavalomahefa  _et al._ (2020); BRA  with the
same emitting surface area (gray). (c) Full field simulation at around
$15\,{\rm GHz}$ including a preliminary coaxial horn design. (d) Electric
(blue) and magnetic (orange) field distribution and time-averaged Poynting
flux along the $z$ direction in the $xy$ plane at the focal spot. (e)
Schematic setup in cryostat for pilot dark photon searches.

##  II Dark matter signal

Sub-eV-mass bosonic DM behave as classical fields, whose coherent oscillations
generate the local halo energy density $\rho_{\text{DM}}$, which we assume to
be $0.45$ GeV cm-3 222This value of $\rho_{\text{DM}}$ is typically adopted in
axion haloscope literature Du  _et al._ (2018), but we note its significant
uncertainties that range from $[0.2,0.6]$ GeV cm-3 from global methods to
$[0.4,1.5]$ GeV cm-3 using recent astrometry data Read (2014); Tanabashi  _et
al._ (2018); Brown  _et al._ (2018); Buch  _et al._ (2019).. We consider
scenarios where either axions or dark photons exclusively saturate the halo
DM. The DM-photon interaction augments the Ampère-Maxwell equation with an
effective source current $\mathbf{J}_{\text{DM}}$ Jaeckel and Ringwald (2010)

$\nabla\times\mathbf{B}-\partial_{t}\mathbf{E}=\mathbf{J}_{\text{DM}}.$ (1)

A nonzero $\mathbf{J}_{\text{DM}}$ induces a small EM field that causes a
discontinuity at the interface of media with different electric permittivity,
such as a conducting dish in vacuum. To satisfy the $\mathbf{E}_{\parallel}=0$
boundary condition parallel to the dish surface, a compensating EM wave with
amplitude $|\mathbf{E}_{0}|$ must be emitted perpendicular to the surface.
These waves transmit
$P_{\text{DM}}=\frac{1}{2}|\mathbf{E}_{0}|^{2}A_{\text{dish}}$ of power for
dish area $A_{\text{dish}}$. For axions with $g_{a\gamma\gamma}$ coupling to
photons, the current is
$\mathbf{J}_{a}=g_{a\gamma\gamma}\sqrt{2\rho_{\text{DM}}}\mathbf{B}_{\text{ext}}^{\parallel}\cos(m_{a}t)$
given an external magnetic field $\mathbf{B}_{\text{ext}}^{\parallel}$ with
nonzero component parallel to the plate, resulting in
$P_{a}=\frac{1}{2}\rho_{\text{DM}}(B_{\text{ext}}^{\parallel}g_{a\gamma\gamma}/m_{a})^{2}A_{\text{dish}}$
emitted power Horns  _et al._ (2013). QCD axion models Dine  _et al._ (1981);
Zhitnitsky (1980); Kim (1979); Shifman  _et al._ (1980); Grilli di Cortona
_et al._ (2016) relate $g_{a\gamma\gamma}$ to the mass by
$g_{a\gamma\gamma}\sim 10^{-13}(m_{a}/\text{meV})$ GeV-1, giving
$m_{a}$-independent power. For dark photons with $A^{\prime}$-SM kinetic
mixing $\kappa$ and polarization $\mathbf{\hat{n}}$, the current is
$\mathbf{J}_{A^{\prime}}=\kappa
m_{A^{\prime}}\sqrt{2\rho_{\text{DM}}}\mathbf{\hat{n}}\cos(m_{A^{\prime}}t)$,
yielding
$P_{A^{\prime}}=\frac{1}{2}\rho_{\text{DM}}\kappa^{2}A_{\text{dish}}\alpha_{\text{pol}}^{2}$
power. The factor $\alpha_{\text{pol}}=\sqrt{2/3}$ averages over $A^{\prime}$
polarizations Horns  _et al._ (2013). $P_{A^{\prime}}$ is
$m_{A^{\prime}}$-independent and persists even when
$\mathbf{B}_{\text{ext}}=\mathbf{0}$. Signal emission occurs independent of
frequency in principle, allowing searches across several mass decades in
single runs.

Practically, DM-detection sensitivity also depends on the signal emission-to-
detection efficiency $\epsilon_{s}$, photosensor noise equivalent power (NEP),
and runtime $\Delta t$. NEP is defined as the incident signal power required
to achieve unit signal-to-noise ratio (SNR) in a one Hertz bandwidth. We
estimate sensitivity to $g_{a\gamma\gamma}$ and $\kappa$ (squared) as the SNR
exceeding five $\text{SNR}=(\epsilon_{s}P_{\text{DM}}\sqrt{\Delta
t})/\text{NEP}>5$, where we assume sensors have sufficiently fast readout
bandwidth $\mathcal{O}(100\,{\rm kHz})$:

$\displaystyle\left\\{\\!\\!\begin{array}[]{c}\left(\frac{g_{a\gamma\gamma}}{10^{-11}}\right)^{2}\\\
\left(\frac{\kappa}{10^{-14}}\right)^{2}\end{array}\\!\\!\right\\}$
$\displaystyle=\left\\{\\!\\!\begin{array}[]{c}\frac{1.9}{\text{GeV}^{2}}\\!\left(\frac{m_{a}}{\text{meV}}\frac{10\,\text{T}}{B_{\text{ext}}}\right)^{2}\\\
7.6\frac{2/3}{\alpha_{\text{pol}}^{2}}\end{array}\\!\\!\right\\}\frac{10\,\text{m}^{2}}{A_{\text{dish}}}\left(\frac{\text{hour}}{\Delta
t}\right)^{1/2}$ (6)
$\displaystyle\\!\\!\\!\\!\\!\\!\times\\!\frac{\text{SNR}}{5}\frac{0.5}{\epsilon_{s}}\frac{\text{NEP}}{10^{-21}\,\text{W}/\sqrt{\text{Hz}}}\frac{0.45\,\text{GeV/cm}^{3}}{\rho_{\text{DM}}}.$
(7)

At high masses, shot noise is relevant due to insufficient signal photons
$N_{\text{signal}}=(\epsilon_{s}P_{\text{DM}}\Delta t)/m_{\text{DM}}<5$. For
the nominal $A_{\text{dish}}=10$ m2, $B_{\text{ext}}=10$ T configuration, QCD
axions induce a few 1 eV photons week-1 so month-long runtimes render shot
noise subdominant for $m_{\text{DM}}\lesssim 1$ eV.

In photon-counting regimes, sensors with dark count rate DCR detect photons
emitted at rate $R_{\text{DM}}=P_{\text{DM}}/m_{\text{DM}}$. We use the
counting-statistics significance
$Z=N_{\text{signal}}/\sqrt{N_{\text{noise}}}=(\epsilon_{s}R_{\text{DM}}\Delta
t)/\sqrt{\text{DCR}\Delta t}>5$ to estimate sensitivity in the background-
limited regime. In the background-free photon-counting limit, the coupling
sensitivity scales faster $g_{a\gamma\gamma}^{\rm sens}\propto(\Delta
t)^{-1/2}$. With nominal photoconversion rates down to 1 photon per day,
scaling as $R_{\gamma}\approx 10^{-5}(1\,\mathrm{eV}/m_{a})\,\mathrm{Hz}$, the
photosensors considered are background limited. We thus constrain our
projections to this scenario, where appendix .1 discusses requirements of
background-free experiments.

##  III Coaxial haloscope design

BREAD proposes a cylindrical barrel as the emitting surface and a novel
reflector geometry comprising a coaxial parabolic surface of rotation around
its tangent. This focuses the emitted radiation to a photosensor located on-
axis at the parabola’s vertex as shown in Fig. 1 (a). DM-to-photon conversion
also occurs at the parabolic surface but is not focused on the vertex. For a
barrel with radius $R$ and length $L=2\sqrt{2}R$, the effective emitting area
is $A_{\text{dish}}=2\pi R\,L$. This aspect ratio suits enclosure in
conventional high-field solenoid magnets and ensures $\mathbf{B}_{\text{ext}}$
is parallel to the emitting surface. Such magnets are widely used in basic or
applied applications with fields reaching $10\,{\rm T}$ or higher Budinger and
Bird (2018); Bird (2020).

While photoconversion occurs regardless of $m_{\text{DM}}$, sensitivity is
limited at high (low) masses by focusing (diffraction) effects. Both effects
broaden the focal spot and reduce the geometric signal efficiency due to
finite photosensor size. In the high-mass limit $\lambda_{\rm dB}\ll R$, DM-
to-photon conversion occurs incoherently as the DM de Broglie wavelength
$\lambda_{\rm dB}$ is smaller than the radius of the barrel $R\sim 1$ m. Here,
the DM-halo velocity $v\simeq 10^{-3}$ smears out the focal spot size Jaeckel
and Redondo (2013); Jaeckel and Knirck (2016, 2017) on length scales larger
than the signal-photon wavelength $\lambda_{\rm sig}$, rendering diffraction
effects negligible. The blue line in Fig. 1 (b) shows the expected intensity
distribution at the focal spot for the most conservative case where the DM
wind points along the least favorable direction. The gray line refers to a
planar conversion surface of the same area comparable to other dish-antenna
experiments Suzuki  _et al._ (2015); Knirck  _et al._ (2018a); Tomita  _et
al._ (2020); Andrianavalomahefa  _et al._ (2020); BRA  with an on-axis
parabolic mirror at $1\,{\rm m}$ distance. Since rays impinge the focal plane
from a larger solid angle, BREAD achieves improved focusing.

In the opposite low-mass limit $\lambda_{\rm dB}\gg R$, such defocusing
effects are negligible and the signal can be detected coherently. Figure 1 (c)
shows the result of a COMSOL simulation at around $15\,{\rm GHz}$. Here the
full modified Maxwell wave equation is solved to verify that there are no
spurious sources or resonances excited that may interfere destructively with
the signal. Figure 1 (d) shows the diffraction-limited electromagnetic fields
at the focal plane. The electric field polarized along the radial direction
can be picked up by coaxial horn antennas Barros  _et al._ (2013); Bykov  _et
al._ (2008). Receiver designs based on microwave and submillimeter astronomy
projects could be considered for signal collection. Proof-of-principle pilot
experiments near both these limits are in preparation. The radio-frequency
pilot targeting 10s of GHz masses, called GigaBREAD, will be detailed in
future work.

Figure 1 (e) shows the proposed experimental design for the pilot $A^{\prime}$
search at infrared (IR) frequencies, called InfraBREAD. Cooling the conducting
surfaces to 4 K suppresses thermal noise and we identified a large cryostat at
Fermilab built to test ADMX resonators, which will be available in 2022. The
barrel is constructed from aluminum with $A_{\text{dish}}=0.7\,{\rm
m}^{2}~{}(10\,{\rm m}^{2})$ for the pilot (upgrade). A 4 K blackbody with
$0.7\,{\rm m}^{2}$ area and unit emissivity emits $\sim 10^{-8}$ W of power
above 1 THz (4 meV). Simulation shows that thermal radiation is evenly
distributed across the focal plane and so suppressed by
$A_{\text{sens}}/A_{\text{dish}}\sim 10^{-6}$ for active sensor area
$A_{\text{sens}}$. For $A_{\text{dish}}=0.7\,{\rm m}^{2}$,
$A_{\text{sens}}=0.5\times 0.5$ mm2 yields 50% (25%) signal efficiency
$\epsilon_{s}$ for optimistic (pessimistic) DM-wind alignment; see appendix .3
for further discussion. For absolute alignment of the photosensor in the
reflector, we propose a piezoelectric motion stage to fine-tune the sensor
position at the focus. Off focus, the signal is not enhanced. This enables _in
situ_ noise measurements by moving the single photosensor off axis or
installing a second off-axis photosensor. A monochromatic laser or bandpass-
filtered blackbody source can inject photons via a small hole in the barrel
for absolute calibration of the reflector-photosensor setup. A room-
temperature spectrometer at UChicago is available to characterize sources and
filters Dona  _et al._ (2021).

Various upgrades and optimizations could be implemented to improve sensitivity
of the proposed experimental concept. A small secondary mirror near the focal
point could guide the signal toward a low-field region where, e.g. a chopper
and/or spectrometer could be installed. The optics may optimize the radiation
polarization and incident angle on the photosensors. A detector array or
photon imager Zhao  _et al._ (2017) could also provide spatial resolution to
correlate any observed signals with the astrophysical DM distribution.
Specifically, the focal point and signal undergo diurnal and annual
modulations due to the rotating DM velocity vector in the lab frame Knirck
_et al._ (2018b) and possibly $A^{\prime}$ polarization Caputo  _et al._
(2021). The total $A_{\rm dish}$ could be increased within the same available
volume by combining the signals from an array of smaller BREAD-like barrels,
but this increases complexity significantly. Cosmic-ray muons are a suggested
noise source for photon counters J. Allmaras and K. Berggren and I. Charaev
and C. Chang and J. Chiles and B. Korzh and A. Lita and J. Luskin, S. W. Nam
and V. Novosad and M. Shaw and V. Verma and E. Wollman (2020); _in situ_
vetoes at the sensor or barrel exterior, and/or underground operation are
mitigation strategies. Studying these options is deferred to future work.

##  IV Photosensor technologies

The expected DM signal rates and optical geometry imply stringent photosensor
requirements: broad spectral response $\Delta E/E>1$, ultralow noise NEP
$<10^{-20}\,{\rm W}\,{\rm Hz}^{-1/2}$ or DCR $<10^{-3}$ Hz, and millimeter-
size active area. Bolometers are promising because they directly measure
absorbed photon power, only the absorbing material or structures limit
spectral response, and are established technology with diverse applications
Richards (1994); Pirro and Mauskopf (2017). They comprise a thermally isolated
absorbing element with low heat capacity, sensitive thermometer and weak link
to a cold thermal bath. They can measure photon energies from $10^{-5}\,{\rm
eV}$ Kokkoniemi  _et al._ (2020) to $10^{6}\,{\rm eV}$. Bolometers are
typically insensitive to static (DC) radiation due to instrumental low-
frequency ($1/f$) noise, so input signals must be time-modulated with, e.g. a
chopper that shifts the signal to a frequency where $1/f$ noise is subdominant
and SNR increases $\propto(\Delta t)^{1/2}$.

Photon-counting devices (photocounters) are potentially more sensitive than
total-power bolometry, since simple signal-processing techniques, e.g.
thresholding and pulse fitting, can suppress noise. Small devices achieve
nearly background-free single-photon counting at thresholds $\gtrsim 1$ eV.
For lower energies, numerous devices exploit athermal breaking of Cooper
pairs, including kinetic inductance detectors (KID), superconducting nanowire
single photon detectors (SNSPD), and quantum capacitance detectors (QCDet)
333QCDet representing quantum capacitance detector avoids ambiguity with QCD
denoting quantum chromodynamics..

Table 1:  Illustrative photosensor performance: spectral energy $E$, operating temperature $T_{\text{op}}$, active area $A_{\text{sens}}$. Bolometers (photocounters) report noise equivalent power NEP (dark count rate DCR).  Photosensor | $\dfrac{E}{\text{meV}}$ | $\dfrac{T_{\text{op}}}{\text{K}}$ | $\dfrac{\text{NEP}}{\text{W}/\sqrt{\text{Hz}}}$ | $\dfrac{A_{\text{sens}}}{\text{mm}^{2}}$
---|---|---|---|---
Gentec Gentec Electro-Optics  | [0.4, 120] | 293 | $1\cdot 10^{-8}$ | $\pi 2.5^{2}$
IR Labs Infrared Laboratories  | [0.24, 248] | 1.6 | $5\cdot 10^{-14}$ | $1.5^{2}$
KID/TES Ridder  _et al._ (2016); Baselmans  _et al._ (2017) | [0.2, 125] | 0.3 | $2\cdot 10^{-19}$ | $0.2^{2}$
QCDet Echternach  _et al._ (2018, 2021) | [2, 125] | 0.015 | $\frac{{\rm DCR}}{{\rm Hz}}=4$ | $0.06^{2}$
SNSPD Hochberg  _et al._ (2019); Verma  _et al._ (2020) | [124, 830] | 0.3 | $\frac{{\rm DCR}}{{\rm Hz}}=10^{-4}$ | $0.4^{2}$

We now discuss specific technologies motivating the values in Table 1 assumed
for our projections. We display typical $A_{\text{sens}}$, but later set
$\epsilon_{s}=50\%$ for simplicity assuming sensor development will enable
scaling to required sizes. Room-temperature (Gentec pyroelectric Gentec
Electro-Optics ) and cryogenic (IR Labs semiconducting thermistor Infrared
Laboratories ) devices exemplify commercial performance.

Superconducting titanium-gold transition edge sensors (TES) Irwin (1995);
Irwin and Hilton ; Gerrits  _et al._ (2016) report down to $2\times
10^{-19}\,{\rm W}\,{\rm Hz}^{-1/2}$ NEP in arrays of $200\times 200\,\mu{\rm
m}^{2}$ pixels Ridder  _et al._ (2016). TESs have broad spectral response,
where a molybdenum-gold device reporting $4\times 10^{-19}\,{\rm W}\,{\rm
Hz}^{-1/2}$ NEP covers 1–4 $\mu$m to 160–960 $\mu$m (eV to meV) Goldie  _et
al._ (2011). Elsewhere, small 10 $\mu$m superconducting–normal-metal junction
bolometers report $2\times 10^{-20}\,{\rm W}\,{\rm Hz}^{-1/2}$ NEP Kokkoniemi
_et al._ (2019), which may be promising if active areas are scalable to
millimeters Zhang  _et al._ (2021).

Figure 2: Projected BREAD sensitivity by sensor technology (bold labels, see
Table 1) for dark photons $A^{\prime}$ (left) and axions $a$ (right). This
assumes signal-to-noise ratio $\mathrm{SNR}=5$ (significance $Z=5$ for
photocounters), signal efficiency $\epsilon_{\text{sig}}=0.5$, and dish area
$A_{\text{dish}}=10$ m2. Blue shading shows existing constraints from Ref.
Caputo  _et al._ (2021). Benchmark axion predictions include QCD axion models
Di Luzio  _et al._ (2017) (green band), cogenesis Co  _et al._ (2021) (green
dots), KSVZ Kim (1979); Shifman  _et al._ (1980) and DFSZ Dine  _et al._
(1981); Zhitnitsky (1980) (green lines). Sensitivity scaling assumes
background-limited operation where signal-power limits scale as $\sqrt{\Delta
t}$ for runtime $\Delta t$ and linearly with improved NEP.

KIDs Day  _et al._ (2003); Leduc  _et al._ (2010); Zmuidzinas (2012) are thin-
film resonators, whose surface inductance is sensitive to Cooper-pair-breaking
photons above the band gap $\Delta\simeq 0.2\,$meV. Titanium-nitride KIDs are
scalable to $50\times 50$ mm2 kilopixel arrays with $3\times 10^{-19}\,{\rm
W\,Hz}^{-1/2}$ NEP Baselmans  _et al._ (2017), which are antenna coupled and
optimized to [3.4, 12] meV Jackson  _et al._ (2012). For cosmic microwave
background (CMB) applications ($0.2\lesssim E\lesssim 2$ meV), KIDs are
limited by signal power rather than sensor noise at ${\rm NEP}\sim
10^{-17}\,{\rm W\,Hz}^{-1/2}$ Abitbol  _et al._ (2017), and therefore could
have better performance in such frequencies than current sensors targeting CMB
science. Given KID and TES devices report similar NEP in each application, we
amalgamate their presentation in our projections for simplicity. We
extrapolate the $2\times 10^{-19}\,{\rm W\,Hz}^{-1/2}$ NEP Ridder  _et al._
(2016) into the [0.2, 125] meV range where we expect KID/TES devices to
operate bolometrically, but this will require experimental demonstration.

QCDets Shaw  _et al._ (2009); Bueno  _et al._ (2010); Echternach  _et al._
(2013) recently report $3\times 10^{-21}\,{\rm W}\,{\rm Hz}^{-1/2}$ NEP at 1.5
THz (6.2 meV) Echternach  _et al._ (2018). These are scalable to 441 pixel
arrays and simulation indicates 1-4$\times 10^{-20}\,{\rm W}\,{\rm Hz}^{-1/2}$
NEP for [2, 125] meV Echternach  _et al._ (2021), driven by e.g. Origins Space
Telescope goals Leisawitz  _et al._ (2021). Such performance is promising, and
for simplicity, we assume constant $3\times 10^{-21}\,{\rm W}\,{\rm
Hz}^{-1/2}$ NEP in our projections. We convert this to ${\rm DCR}=4\,{\rm Hz}$
using $\text{NEP}=(E/\eta)\sqrt{2\cdot\text{DCR}}$ Li  _et al._ (2018) for
$E=6.2\,{\rm meV}$ and optical efficiency $\eta=0.9$ Echternach  _et al._
(2018).

SNSPDs Gol’tsman  _et al._ (2001); Natarajan  _et al._ (2012); Li  _et al._
(2018) comprise sub-micron-width wires wound across thin-film substrates that
count photons above an energy threshold. Superconductivity is momentarily lost
upon photon absorption, leading to a measurable voltage pulse. Such devices
achieve $>90\%$ efficiency Marsili  _et al._ (2013) and recently, a $400\times
400$ $\mu$m2 tungsten-silicide device reports DCR $<10^{-4}$ Hz for 0.8 eV
threshold Hochberg  _et al._ (2019). Using Fermilab refrigerators Hernandez
_et al._ (2020), we are preparing to test similar SNSPDs fabricated at MIT.
Recent advances important for BREAD include extending up to 10 $\mu$m (0.12
eV) Verma  _et al._ (2020) and developing large $3.1\times 3.1$ mm2 single
pixels Wollman  _et al._ (2021). Continued research to lower thresholds is
motivated given axions with $m_{a}\lesssim 60\,{\rm meV}$ are disfavored by
supernova constraints Chang  _et al._ (2018).

Photocounting is also possible using KIDs Gao  _et al._ (2012); de Visser  _et
al._ (2021) and TESs Miller  _et al._ (2003); Lita  _et al._ (2008); Karasik
_et al._ (2012), with the benefit of per-photon energy resolution. With e.g.
10% energy resolution determined by detector resolution, the monochromatic DM
signal occupies one energy bin but noise can be spread across 10 bins,
improving SNR after sufficient $\Delta t$ up to a trials factor. Exploring
this in BREAD requires more detailed resolution and noise models, which is
deferred to future work.

##  V Sensitivity and discussion

We project BREAD sensitivity to dark photons in Fig. 2 (left) using Eq. (7)
assuming the spectral and noise benchmarks in Table 1. Existing constraints
following Ref. Caputo  _et al._ (2021) (blue shading) include stellar
astrophysics Redondo and Raffelt (2013); Vinyoles  _et al._ (2015), cosmology
Arias  _et al._ (2012); McDermott and Witte (2020); Caputo  _et al._ (2020),
and $\gamma\to A^{\prime}$ conversion that includes laboratory probes Betz
_et al._ (2013); Kroff and Malta (2020). With just $1$ day runtime assuming
$A_{\text{dish}}=0.7$ m2, the gray thin line shows the BREAD pilot could
surpass existing $\kappa$ constraints by one decade around 1 meV using the
commercial IR Labs sensor. The reach of BREAD is substantially broader
compared with two existing dish antennas, SHUKET Brun  _et al._ (2019b) and
Tokyo Tomita  _et al._ (2020) (dark blue). Importantly, BREAD probes higher
masses than existing haloscopes ADMX Asztalos  _et al._ (2002, 2010), CAPP Lee
_et al._ (2020); Jeong  _et al._ (2020); Kwon  _et al._ (2021), HAYSTAC Zhong
_et al._ (2018); Backes  _et al._ (2021), transmon qubit Dixit  _et al._
(2021), and WISPDMX Nguyen  _et al._ (2019), whose results are recasted for
$A^{\prime}$ following Ref. Caputo  _et al._ (2021). Scaling to
$A_{\text{dish}}=10$ m2 and using KID/TES sensors could open two decades
further $\kappa$ sensitivity, while SNSPDs could achieve three decades gain
for $m_{A^{\prime}}\gtrsim 0.1$ eV. Extending runtime to $\Delta t=10^{3}$
days enables $\kappa$ sensitivity to reach six (four) decades beyond existing
constraints for $m_{A^{\prime}}\sim 0.4~{}(200)$ meV.

Axion sensitivity is illustrated in Fig. 2 (right). Existing constraints
Caputo  _et al._ (2021) additionally include the CAST helioscope Arik  _et
al._ (2014); Anastassopoulos  _et al._ (2017), telescopes Grin  _et al._
(2007); Regis  _et al._ (2021), neutron stars Foster  _et al._ (2020); Darling
(2020); Battye  _et al._ (2022), alongside ORGAN McAllister  _et al._ (2017)
QUAX Alesini  _et al._ (2019, 2021) RADES Melcón  _et al._ (2021) and URF De
Panfilis  _et al._ (1987); Wuensch  _et al._ (1989); Hagmann  _et al._ (1990)
haloscopes. While challenging with commercial devices, 10 day runtimes using
KID/TES sensors with NEP $\sim 10^{-19}\,{\rm W}\,{\rm Hz}^{-1/2}$ could
surpass CAST sensitivity for $m_{a}\lesssim 10$ meV. Longer runtimes could
test cogenesis predictions for the $c_{a\gamma\gamma}=1$ benchmark Co  _et
al._ (2021). Increasing $A_{\text{dish}}$ ($B_{\text{ext}}$) beyond 10 m2 (10
T) is financially unfavorable, requiring custom cryostats and magnets. Thus
practically probing QCD axion models Di Luzio  _et al._ (2017) requires longer
runtime and lower sensor noise. Coupling sensitivity scales slowly with
runtime $g_{a\gamma\gamma}^{\text{sens}}\sim(\Delta t)^{-1/4}$, i.e. halving
$g_{a\gamma\gamma}^{\text{sens}}$ requires $16\times$ longer runtimes. For
$\Delta t=10^{3}$ days, reaching KSVZ Kim (1979); Shifman  _et al._ (1980)
(DFSZ Dine  _et al._ (1981); Zhitnitsky (1980)) demands $1~{}(0.2)\times
10^{-22}\,{\rm W}\,{\rm Hz}^{-1/2}$ NEP. Achieving this NEP for wide spectral
ranges is challenging and a key science driver for sensor development. This
may be attainable above 0.1 meV for photocounters, e.g. SNSPDs, motivating
dedicated measurements in preparation, and next-generation bolometers at lower
masses given a recent TES-based device reports $8\times 10^{-22}\,{\rm
W}\,{\rm Hz}^{-1/2}$ electrical NEP Nagler  _et al._ (2020). Maintaining
signal efficiency when upgrading $A_{\text{dish}}=0.7\to 10$ m2 requires
quadrupling the active sensor width. Overcoming these challenges promises
significant scientific payoff given the multidecade improvements in search
coverage that has long eluded cavity haloscopes. Post discovery, the DM signal
will always persist, enabling cross checks with resonant techniques and
measurements to elucidate its particle physics and astrophysical properties
Knirck  _et al._ (2018a); Caputo  _et al._ (2021).

In summary, we proposed BREAD to improve sub-eV-mass DM reach by several
decades. We introduced the novel coaxial design optimized for embedding in
standard solenoids and cryostats, in contrast to existing dish antennas, then
detailed numerical optics simulation and examined photosensor candidates.
Realizing BREAD into a cornerstone DM experiment will catalyze synergies
across quantum technology and astroparticle physics.

### Acknowledgments

We thank Ankur Agrawal, Israel Alatorre, Kate Azar, Lydia Beresford, Pierre
Echternach, Juan Estrada, Amanda Farah, Casey Frantz, Mary Heintz, Chris Hill,
Matthew Hollister, Reina Maruyama, Jan Offermann, Mark Oreglia, Jessica
Schmidt, Sadie Seddon-Stettler, Danielle Speller, Emily Smith, Cecilia
Tosciri, Liliana Valle, Joaquin Viera, and Steven Zoltowski for interesting
and helpful discussions. This work is funded in part by the Department of
Energy through the program for Quantum Information Science Enabled Discovery
(QuantISED) for High Energy Physics and the resources of the Fermi National
Accelerator Laboratory (Fermilab), a U.S. Department of Energy, Office of
Science, HEP User Facility. Fermilab is managed by Fermi Research Alliance,
LLC (FRA), acting under Contract No. DE-AC02-07CH11359. Work at Argonne
National Laboratory is supported by the U.S. Department of Energy, Office of
High Energy Physics, under contract DE-AC02-06CH11357. We acknowledge support
by the Kavli Institute for Cosmological Physics at the University of Chicago
through grant NSF PHY-1125897 and an endowment from the Kavli Foundation and
its founder Fred Kavli. Work at Lawrence Livermore National Laboratory is
supported under contract number DE-AC52-07NA27344; Release #LLNL-JRNL-828670.
The MIT co-authors acknowledge support from the Fermi Research Alliance, LLC
(FRA) and the US Department of Energy (DOE) under contract No. DE-
AC02-07CH11359. We thank the Aspen Center for Physics, which is supported by
National Science Foundation grant PHY-1607611, for hosting the Quantum
Information Science for Fundamental Physics meeting (2020), and the University
of Washington CENPA for the Axions Beyond Gen2 workshop (2021) supported by
the Heising-Simons Foundation. JL acknowledges a University of Chicago
fellowship supported by the Grainger Foundation, where this work began, and a
Junior Research Fellowship at Trinity College, University of Cambridge.

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## Appendix

This appendix first reviews the axion and dark photon signal rate and
sensitivity in bolometric and photocounting regimes. Next, we present
additional discussion on the cryostat and magnet followed by dark matter
velocity effects on the signal spot size. Then, we expand discussion on
photosensor performance and noise sources before commenting further on
bolometers. Finally, we discuss a potential experimental staging for BREAD.

###  .1 Signal rate

The interaction Lagrangian coupling SM photons to dark photons $A^{\prime}$ (a
spin 1 vector) and axions $a$ (a spin 0 pseudoscalar) is given respectively by
Jaeckel and Ringwald (2010); Horns  _et al._ (2013)

$\mathcal{L}_{A^{\prime}}=-\frac{1}{4}\kappa
F_{\mu\nu}^{\prime}F^{\mu\nu},\quad\mathcal{L}_{a}=-\frac{1}{4}g_{a\gamma\gamma}aF_{\mu\nu}\tilde{F}^{\mu\nu},$
(8)

where $\tilde{F}^{\mu\nu}=\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$ with
$\epsilon^{\mu\nu\rho\sigma}$ being the totally antisymmetric tensor,
$g_{a\gamma\gamma}$ is the axion–photon coupling, $\kappa$ is the kinetic
mixing parameter, and
$F_{\mu\nu}^{(^{\prime})}=\partial_{[\mu}A^{(^{\prime})}_{\nu]}$ is the SM
(dark) photon field strength. Upon solving the Lagrangian equations of motion,
the consequential modification to SM electrodynamics is an additional
effective source current $\mathbf{J}_{\text{DM}}$ in the Ampère–Maxwell
equation Sikivie (1983)

$\nabla\times\mathbf{B}-\partial_{t}\mathbf{E}=\mathbf{J}_{\text{DM}}.$ (9)

The nonzero current $\mathbf{J}_{\text{DM}}$ from the DM–photon interaction
Lagrangian of Eq. (8) is given by

$\mathbf{J}_{A^{\prime}}=\kappa
m_{A^{\prime}}^{2}\mathbf{A}^{\prime}(t),\quad\mathbf{J}_{a}=g_{a\gamma\gamma}\mathbf{B}_{\text{ext}}m_{a}a(t),$
(10)

where $\mathbf{A}^{\prime}(t)$ and $a(t)$ are the dynamical dark photon and
axion fields, respectively, and $\mathbf{B}_{\text{ext}}$ is the external
magnetic field applied in the laboratory. In general, one considers wave
solutions of the form $\exp[\mathrm{i}(\mathbf{k}\cdot\mathbf{x}-\omega t)]$,
where $\omega=m_{\text{DM}}$. As the local DM halo has nonrelativistic
velocity $v\simeq 10^{-3}$, the momentum wavevector approximately vanishes
$\mathbf{k}\to 0$ and thus $\nabla\times\mathbf{B}$ in Eq. (9) induced by DM
is negligible. From the cosmology of bosonic condensate DM, the local halo
energy density $\rho_{\text{DM}}$ is related to the DM mass and fields by

$\rho^{(A^{\prime})}_{\text{DM}}=\frac{1}{2}m_{A^{\prime}}^{2}|\mathbf{A}^{\prime}(t)|^{2},\quad\rho^{(a)}_{\text{DM}}=\frac{1}{2}m_{a}^{2}a^{2}(t).$
(11)

Using this in Eq. (10) gives the resulting source current for axions has the
form
$\mathbf{J}_{a}=g_{a\gamma\gamma}\sqrt{2\rho_{\text{DM}}}\mathbf{B}_{\text{ext}}\cos(m_{a}t)$.
For dark photons, the source current is $\mathbf{J}_{A^{\prime}}=\kappa
m_{A^{\prime}}\sqrt{2\rho_{\text{DM}}}\mathbf{\hat{n}}\cos(m_{A^{\prime}}t)$,
where $\mathbf{\hat{n}}$ is the $A^{\prime}$ polarization. The electric field
induced by $\mathbf{J}_{\text{DM}}$ is then deduced from Eq. (9) and has
magnitude

$|\mathbf{E}_{A^{\prime}}|=\kappa\sqrt{2\rho_{\text{DM}}},\quad|\mathbf{E}_{a}|=\left(g_{a\gamma\gamma}|\mathbf{B}_{\text{ext}}^{||}|/m_{a}\right)\sqrt{2\rho_{\text{DM}}},$
(12)

where $\mathbf{B}_{\text{ext}}^{||}$ is the magnetic field component parallel
to the emitting barrel surface. This nonzero electric field causes a
discontinuity at the interface of a conducting dish in vacuum. To satisfy the
$\mathbf{E}_{\parallel}=0$ boundary condition parallel to the dish surface
(from Faraday’s law $\nabla\times\mathbf{E}+\partial_{t}\mathbf{B}=0$, which
remains unchanged), a compensating electromagnetic wave with amplitude
$\mathbf{E}_{0}$ must be emitted perpendicular to the surface. The power per
unit area $P/A_{\text{dish}}$ of the emitted waves is given by
$P/A_{\text{dish}}=\frac{1}{2}|\mathbf{E}_{0}|^{2}$ and applying this to the
axion-induced electric field of Eq. (12) gives
$P_{a}=\frac{1}{2}\rho_{\text{DM}}(B_{\text{ext}}^{\parallel}g_{a\gamma\gamma}/m_{a})^{2}A_{\text{dish}}$
for dish area $A_{\text{dish}}$. Normalizing the power to experimental
parameters, the power emitted for axion and dark photon DM is, respectively

$\displaystyle\left\\{\\!\begin{array}[]{c}\dfrac{P_{a}}{8.8\cdot
10^{-23}\,\text{W}}\\\\[8.5359pt] \dfrac{P_{A^{\prime}}}{2.2\cdot
10^{-23}\,\text{W}}\end{array}\\!\right\\}$
$\displaystyle=\left\\{\\!\\!\begin{array}[]{c}\left(\frac{g_{a\gamma\gamma}}{10^{-11}\,\text{GeV}^{-1}}\frac{\text{meV}}{m_{a}}\right)^{2}\left(\dfrac{B_{\text{ext}}}{10\,\text{T}}\right)^{2}\\\\[8.5359pt]
\dfrac{\alpha_{\text{pol}}^{2}}{2/3}\left(\dfrac{\kappa}{10^{-14}}\right)^{2}\end{array}\\!\\!\right\\}$
(17)
$\displaystyle\quad\times\frac{\rho_{\text{DM}}}{0.45\,\text{GeV/cm}^{3}}\frac{A_{\text{dish}}}{10\,\text{m}^{2}}.$
(18)

Figure 3: Photon emission rate of QCD axion benchmarks using Eq. (37). Thick
blue (thin orange) solid lines shows the KSVZ (DFSZ) models. Dotted (dashed)
variations shows the dish area (magnetic field) reduced by a factor of 10.

Figure 4: Sensitivity (${\rm SNR=5}$) to axion–photon coupling normalized to
the DFSZ prediction
$\left|g_{a\gamma\gamma}/g_{a\gamma\gamma}^{\text{DSFZ}}\right|$ as a function
of various experimental parameters following Eq. (7): (left) product of dish
area and magnetic field $A_{\text{dish}}\cdot B_{\text{ext}}^{2}$, (center)
signal emission-to-detection efficiency, (right) noise equivalent power (NEP).
For the $A_{\text{dish}}\cdot B_{\text{ext}}^{2}$ and NEP plots, two
integration times of 10 and 1000 days are shown. For the signal efficiency
plot, two NEP values of $10^{-21}$ and $10^{-23}$ W Hz-1/2 are considered. The
KSVZ (DFSZ) predictions are shown as dot-dashed (dashed) gray horizontal
lines.

In photon counting regimes, it is more convenient to consider the DM-induced
rate $R_{\text{DM}}$ of emitted photons given by
$P_{\text{DM}}/m_{\text{DM}}$:

$\displaystyle\left\\{\\!\begin{array}[]{c}\dfrac{R_{a}}{0.55\,\text{Hz}}\\\\[8.5359pt]
\dfrac{R_{A^{\prime}}}{0.14\,\text{Hz}}\end{array}\\!\right\\}$
$\displaystyle=\left\\{\\!\\!\begin{array}[]{c}\left(\dfrac{g_{a\gamma\gamma}}{10^{-11}\,\text{GeV}^{-1}}\right)^{2}\left(\dfrac{\text{meV}}{m_{a}}\right)^{3}\left(\dfrac{B_{\text{ext}}}{10\,\text{T}}\right)^{2}\\\\[8.5359pt]
\dfrac{\alpha_{\text{pol}}^{2}}{2/3}\left(\dfrac{\kappa}{10^{-14}}\right)^{2}\dfrac{\text{meV}}{m_{A^{\prime}}}\end{array}\\!\\!\right\\}$
(23)
$\displaystyle\quad\times\frac{\rho_{\text{DM}}}{0.45\,\text{GeV/cm}^{3}}\frac{A_{\text{dish}}}{10\,\text{m}^{2}}.$
(24)

For counting statistics, the relevant figure-of-merit is the significance $Z$
given a runtime $\Delta t$, which for a photosensor with dark count rate DCR,
is estimated as

$Z=\frac{N_{\text{signal}}}{\sqrt{N_{\text{noise}}}}=\frac{\epsilon_{s}R_{\text{DM}}\Delta
t}{\sqrt{\text{DCR}\Delta t}}.$ (25)

Formally, $N_{\text{signal}}/\sqrt{N_{\text{noise}}}$ holds in the Gaussian
regime where $N_{\text{noise}}\gtrsim 10$, below which Poissonian statistics
applies. For zero noise counts $N_{\text{noise}}=0$, one has 95% CL exclusion
sensitivity for three or more signal events.

Requiring $Z=5$ for DM reach implies the coupling sensitivity is related to
the DCR by

$\displaystyle\left\\{\\!\\!\begin{array}[]{c}\left(\dfrac{g_{a\gamma\gamma}}{10^{-12}}\right)^{2}\\\
\left(\dfrac{\kappa}{10^{-15}}\right)^{2}\end{array}\\!\\!\right\\}$
$\displaystyle=\left\\{\\!\\!\begin{array}[]{c}\dfrac{3.0}{\text{GeV}^{2}}\\!\left(\dfrac{m_{a}}{\text{meV}}\right)^{3}\left(\dfrac{10\,\text{T}}{B_{\text{ext}}}\right)^{2}\\\
11.9\dfrac{2/3}{\alpha_{\text{pol}}^{2}}\dfrac{m_{A^{\prime}}}{\text{meV}}\end{array}\\!\\!\right\\}\\!\left(\frac{\text{hour}}{\Delta
t}\right)^{1/2}$ (30)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\times\\!\frac{10\,\text{m}^{2}}{A_{\text{dish}}}\frac{Z}{5}\frac{0.5}{\epsilon_{s}}\left(\frac{\text{DCR}}{10^{-2}\,\text{Hz}}\right)^{1/2}\frac{0.45\,\text{GeV/cm}^{3}}{\rho_{\text{DM}}}.$
(31)

While measuring DCR is more relevant for photosensors operating in photon
counting regimes, device physics literature often converts to NEP using
$\text{NEP}=(E/\epsilon)\sqrt{2\cdot\text{DCR}}$ to facilitate comparisons Li
_et al._ (2018), where $\epsilon$ is the detection efficiency.

Canonical QCD axion scenarios Grilli di Cortona  _et al._ (2016) fix the
relation between the axion coupling and mass to
$g_{a\gamma\gamma}^{\text{KSVZ}}=-3.9\times
10^{-13}(m_{a}/\text{meV})~{}\text{GeV}^{-1}$ and
$g_{a\gamma\gamma}^{\text{DFSZ}}=1.5\times
10^{-13}(m_{a}/\text{meV})~{}\text{GeV}^{-1}$. The prefactor depends on the
theoretical details of the ultraviolet completion assumed in the KSVZ and DFSZ
models Dine  _et al._ (1981); Zhitnitsky (1980); Kim (1979); Shifman  _et al._
(1980). Using Eq. (24), we find the photon emission rate is

$\displaystyle\left\\{\begin{array}[]{c}R_{a}^{\text{KSVZ}}\\\\[8.5359pt]
R_{a}^{\text{DFSZ}}\end{array}\right\\}$
$\displaystyle=\left\\{\begin{array}[]{c}\dfrac{72.2}{{\rm day}}\\\\[8.5359pt]
\dfrac{10.6}{{\rm
day}}\end{array}\right\\}\dfrac{\text{meV}}{m_{a}}\left(\dfrac{B_{\text{ext}}}{10\,\text{T}}\right)^{2}$
(36)
$\displaystyle\quad\quad\times\frac{\rho_{\text{DM}}}{0.45\,\text{GeV\,cm}^{-3}}\frac{A_{\text{dish}}}{10\,\text{m}^{2}}.$
(37)

Figure 3 shows the photon emission rate
$R_{\gamma}=P_{\text{DM}}/m_{\text{DM}}$ for the QCD axion scenarios assuming
$A_{\text{dish}}=10$ m2 and $B_{\text{ext}}=10$ T, where the impact of
reducing these parameters by a factor of ten is shown. In this configuration,
we estimate on the order of 100 photons per day for 1 THz (4 meV) photons
considering the KSVZ scenario. Higher masses $m_{a}\gtrsim 1$ eV requires
longer runtimes $\Delta t\gtrsim{\rm months}$ to overcome shot noise.

###  .2 Cryostat and magnet considerations

The BREAD geometry is advantageous over recent dish antenna designs that use
spherical or flat emitting surfaces Suzuki  _et al._ (2015); Knirck  _et al._
(2018a); Tomita  _et al._ (2020); Andrianavalomahefa  _et al._ (2020); BRA
because the latter are not suited for enclosure in a solenoid and typically
require custom magnet designs. Solenoids are the leading magnet design that
allows access to large bore sizes and high magnetic fields. An extant large-
bore of radius $R=3$ m solenoid in particle physics is the CMS experiment at
CERN but with modest fields $B=3.8$ T, while the ITER project features an
$R=1.3$ m, $B=13$ T magnet Budinger and Bird (2018); Bird (2020). However,
these meter-scale magnets cost around three orders of magnitude more than
magnets used for state-of-the-art DM experiments e.g. ADMX.

The dish area and magnetic field are two haloscope parameters that bound the
instrument dimensions and govern experimental sensitivity of BREAD. The
coupling sensitivity improves as $\sim A_{\text{dish}}^{1/2}B_{\text{ext}}$.
One may consider advances in both existing cryostat and magnets size possible
with 10 years of research-and-development. Figure 4 (left) shows how the
sensitivity to the axion–photon coupling varies with the product
$A_{\text{dish}}\cdot B_{\text{ext}}^{2}$ for 1 and 1000 days runtime. For the
proposed aspect ratio where the barrel length $L$ is related to its radius $R$
by $L=2\sqrt{2}R$, the radius is fully determined from the dish area
$A_{\text{dish}}$ by $R=\sqrt{A_{\text{dish}}/(4\pi\sqrt{2})}$. So as $R\sim
A^{1/2}$ for BREAD, coupling sensitivity scales linearly with both bore radius
and field strength $B_{\text{ext}}R$.

The radius for the pilot $A_{\text{dish}}=0.7$ m2 is $R=0.2$ m and the nominal
experiment with $A_{\text{dish}}=10$ m2 has $R=0.75$ m. While a
$A_{\text{dish}}=100$ m2 barrel that has $R=2.4$ m is within engineering
feasibility, the financial cost of the large cryostat and magnet required is
challenging. The energy stored in the magnetic field in the solenoid is
$(B^{2}V)/(2\mu_{0})=(\pi 2\sqrt{2}R^{3}B^{2})/(2\mu_{0})$, where $\mu_{0}$ is
the vacuum magnetic permeability, which corresponds to 150 MJ for $R=0.75\,$m,
$B=10\,$T. For comparison, standard medical magnetic-resonance-imaging (MRI)
magnets typically have a bore radius of around 0.35 m. Constructing a
dedicated solenoid for BREAD is financially impractical. We have identified an
available 9.4 T solenoid originally constructed for MRI research at the
University of Illinois at Chicago, which we plan to repurpose for next-
generation axion DM experiments at Fermilab including BREAD. Given coupling
sensitivity scales quickest with the magnetic field, this reinforces the
motivation for continued research-and-development into high-field large-bore
solenoids. Once the haloscope design parameters
$A_{\text{dish}},B_{\text{ext}}$ are maximized within engineering and
financial feasibility, photosensor performance then governs DM discovery
reach.

Figure 5:  Projected sensitivity for dark photons $A^{\prime}$ (left) and
axions $a$ (right) assuming generic bolometers for different noise equivalent
power (NEP) and photocounters for various dark count rate (DCR) with $\Delta
t=1000$ days runtime. Analogous to Fig. 2, blue shading shows existing
constraints from Ref. Caputo  _et al._ (2021) while benchmark axion
predictions include QCD axion models Di Luzio  _et al._ (2017) (green band),
KSVZ Kim (1979); Shifman  _et al._ (1980) (dark green line), DFSZ Dine  _et
al._ (1981); Zhitnitsky (1980) (lighter green line), and cogenesis Co  _et
al._ (2021) (green dots).

###  .3 Focusing and velocity effects

In principle, the parabolic reflector focuses EM radiation emitted
perpendicularly to the barrel onto a point, but in practice, the resulting
spot size is smeared. This is due to a nonzero DM velocity causing the
direction of photon emission to deviate from the surface normal by a small
angle. Therefore, if the photosensor area $A_{\text{sens}}$ is smaller than
the signal spot area $A_{\text{spot}}$, $A_{\text{sens}}<A_{\text{spot}}$, the
signal detection efficiency is reduced.

Figure 6 shows the impact of DM velocity on the geometric signal efficiency
$A_{\text{sens}}/A_{\text{spot}}$ assuming the sensor has saturated efficiency
i.e. each photon that arrives within $A_{\text{sens}}$ is absorbed and a
detection reported. The upper (lower) axis shows the detector radius assuming
the barrel radius is $R=0.75~{}(0.2)$ m. The dashed (solid) line shows the
most optimistic (pessimistic) scenario for the DM wind velocity aligned in the
$z$ ($x/y$) direction. For the pilot experiment with $R=0.2$ m with
$A_{\text{dish}}=0.7$ m2, achieving 50% signal efficiency assuming $z$ ($x/y$)
DM-wind alignment requires a 0.25 mm (0.5 mm) sensor radius. The larger
nominal radius of $R=0.75$ m requires a 1 mm (2 mm) radius sensor. As
discussed in the main text, such sensor dimensions are readily fabricated for
commercial bolometers but state-of-the-art superconducting photosensors are
typically on the order of $\sim 0.1$ mm in size or smaller. Figure 4 (center)
shows how the sensitivity varies with detection efficiency, showing the modest
$\sqrt{\epsilon_{s}}$ dependence in Eq. (7). Indeed, the main text sets an
efficiency of $\epsilon_{s}=50\%$ for simplicity. Because the coupling
sensitivity from Eq. (31) only scales as $\sqrt{\epsilon_{s}}$, if instead
$\epsilon_{s}=5\%$, the sensitivity is only reduced by a factor of around
three. Nonetheless, this sets an important BREAD physics-driven
instrumentation goal of scaling the candidate sensors to millimeter dimensions
while maintaining low noise.

Figure 6: Detection efficiency in the short-wavelength limit as a function of
detector radius $r$ due to DM velocity effects smearing out the focal spot.
The solid (dashed) blue line corresponds to the efficiency assuming the DM
wind velocity is aligned with the $z$ ($x/y$) axes.

###  .4 Photosensor performance and noise

The main text identifies various candidate technologies that could meet the
stringent requirements of BREAD. The values reported in the literature are
often optimized to certain spectral bands or optical configurations, and while
the sensor technologies typically have broadband response, this requires
experimental demonstration. Photosensors are also typically fabricated as a
flat plane for perpendicular illumination, whereas the photons in BREAD can
arrive at steep incidence angles. As the photons are emitted perpendicularly
to the surface (and magnetic field), they also have specific polarization.
Specifically for BREAD, it is important to characterize how both signal
efficiency and noise (dark count) performance scales with not only frequency
but also angle-of-incidence and polarization. Many of these photosensor
properties are challenging to simulate, therefore dedicated in-situ
measurements of candidate photosensor technologies are required and under
preparation. Ensuring signal efficiency and noise performance are demonstrated
while scaling sensor active area to millimeter-sized pixels and/or arrays are
key research goals for BREAD.

BREAD | Pilot | Stage 1 | Stage 2a | Stage 2b
---|---|---|---|---
Axion $a$ | — | $\checkmark$ | $\checkmark$ | $\checkmark$
Dark photon $A^{\prime}$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$
Experimental parameters |
$A_{\text{dish}}$ [m2] | 0.7 | 10 | 10 | 10
$B_{\text{ext}}$ [T] | — | 10 | 10 | 10
$\epsilon_{s}$ | 0.5 | 0.5 | 0.5 | 0.5
$\Delta t$ [days] | 10 | 10 | 1000 | 1000
NEP [W Hz-1/2] | $10^{-14}$ | $10^{-18}$ | $10^{-20}$ | $10^{-22}$
Coupling sensitivity (SNR = 5) |
$\left|g_{a\gamma\gamma}/g_{a\gamma\gamma}^{\text{KSVZ}}\right|$ | — | 280 | 9.0 | 0.90
$\left|g_{a\gamma\gamma}/g_{a\gamma\gamma}^{\text{DFSZ}}\right|$ | — | 740 | 23 | 2.3
$\kappa/10^{-14}$ | 8400 | 22 | 0.7 | 0.07
Table 2: Benchmark experimental parameters for staging the BREAD science
program. Shown are the dish area $A_{\text{dish}}$, external magnetic field
$B_{\text{ext}}$, signal emission-to-detector efficiency $\epsilon_{s}$,
runtime $\Delta t$, and photosensor noise equivalent power (NEP). The lower
rows show the sensitivity assuming signal-to-noise ratio SNR $=5$ to
axion–photon couplings $g_{a\gamma\gamma}$ normalized to the KSVZ/DFSZ
predictions and dark photon kinetic mixing $\kappa$ using Eq. (7).

The noise budget and sources can be categorized by: _intrinsic_ to sensor such
as readout, internal thermal fluctuations; _extrinsic_ to sensor such as
environmental thermal emission and cosmic rays. Devices that act like
bolometers are limited by intrinsic sources such as thermal fluctuations in
TESs, and generation-recombination noise of pairs in KIDs. Single photon
counters such as SNSPDs are reaching very low noise that they are limited by
extrinsic sources. Carefully understanding these noise sources experimentally
for BREAD could help design strategies for their mitigation.

Environmental conditions that would affect long-term low-background
measurements are challenges not unique to BREAD but also other DM experiments.
Cosmic-ray backgrounds are stochastic, are not expected to have long-term
variations, and can be actively rejected with in situ veto systems. The
alignment of the barrel-reflector-sensor setup could be subject to time-
dependent environmental variations such as temperature and humidity or
mechanical vibrations. These can be rejected by correlation with active
monitoring systems, and further accommodated with regular dedicated
calibration and alignment runs. Detailed studies of this requires in situ data
of the experimental site that is the subject of future work.

To provide concrete targets driven by BREAD for photosensor noise, we can
estimate the required NEP for sensitivity to the KSVZ and DFSZ axion assuming
different runtimes in Figure 4 (right) following Eq. (7). The state-of-the-art
NEP today is around $3\times 10^{-21}{\rm\,W\,Hz}^{-1/2}$ for quantum
capacitance detectors Echternach  _et al._ (2018). There are also recent
claims TES-based device could achieve $8\times 10^{-22}$ W Hz-1/2 electrical
NEP Nagler  _et al._ (2020). With this noise performance, it remains
challenging to probe the KSVZ scenario assuming 1000 days runtime, which
requires $1\times 10^{-22}{\rm\,W\,Hz}^{-1/2}$. Probing DFSZ requires an order
of magnitude improvement in NEP to $2\times 10^{-23}{\rm\,W\,Hz}^{-1/2}$,
which is very challenging but motivates a target for sensor research-and-
development.

Figure 5 shows the projected reach for dark photons and axions in the coupling
vs. mass plane for generic bolometric photosensors or photocounters, assuming
various NEP and DCR for $\Delta t=1000$ days runtime. Experimentally
demonstrating that the required NEP is achievable across several decades of
frequency with high efficiency is a challenging requirement for BREAD to probe
QCD axion models. The scientific potential for realizing this is significant,
given the multidecade mass sensitivity without needing to tune the experiment
to an unknown DM mass. Photocounters such as SNSPDs have promising dark count
rates, but their low-mass (longer wavelength) reach are currently limited to
$E>125$ meV photon energies ($\lambda<10$ $\mu$m wavelengths) Verma  _et al._
(2020). Lowering these photon energy thresholds is a key sensor research-and-
development target that would have a significant payoff in BREAD.

###  .5 Further bolometry discussion

We expand on details related to bolometers. For the SNR calculation with
bolometers, the bandwidth of the sensor does not enter directly. A chopper or
moving mirror modulates the signal, allowing noise mitigation by rejecting DC
power components. Thus the bandwidth appearing in the signal-to-noise
calculation is the bandwidth of the modulated signal, which becomes narrower
with integration time, giving an SNR that improves with $\sqrt{\Delta t}$. The
sensor bandwidth of a bolometer is measured with pulse rise and decay time.
This is only relevant in that it restricts the speed that the signal can be
modulated with a chopper. For example, the chopper acquired with the IR Labs
bolometer operates at 100 Hz implying a 10 millisecond or faster bolometer
response is required.

The noise equivalent power NEP is related to the thermal relaxation time of a
bolometer. Therefore, the design goals of NEP and large bandwidth are in
tension and the lowest noise bolometers will be slow. Bolometers need to have
sufficiently fast response such that a chopper can modulate the signal at a
frequency that makes $1/f$ components of the system noise subdominant. We
anticipate this does depend on the specific device technology, and
characterizing this for candidate sensors with experimental measurements is an
important part of the BREAD research program.

###  .6 Experimental staging

Table 2 summarizes benchmark experimental parameters for a staged approach to
BREAD. The corresponding sensitivity to the axion $g_{a\gamma\gamma}$ and dark
photon $\kappa$ couplings are normalized to benchmark targets. We consider the
prototypical pilot dark photon search with 10 days runtime. This demonstrator
forms a nearer-term science goal that will give valuable experimental
experience, already provide meaningful physics results, and serve as a proof-
of-principle for BREAD. Longer term, stage 1 shows the nominal experiment with
the full-scale experiment inside a 10 T magnet assuming only a $10^{-18}$ W
Hz-1/2 NEP photosensor can be installed and demonstrated, which can start to
probe currently unconstrained axion parameter space. The planned stage 2a (2b)
considers 1000 days runtime and experimental work to successfully develop and
couple a $10^{-20}~{}(10^{-22})$ W Hz-1/2 NEP photosensor. This could probe
QCD axion scenarios assuming ongoing research-and-development can demonstrate
multiple order-of-magnitude improvements in sensor NEP.