DMRadio Collaboration

# Projected Sensitivity of DMRadio-m3: A Search for the QCD Axion Below
$1\,\mu$eV

L. Brouwer  Accelerator Technology and Applied Physics Division, Lawrence
Berkeley National Laboratory, Berkeley, CA 94720  S. Chaudhuri  Department of
Physics, Princeton University, Princeton, NJ 08544  H.-M. Cho  SLAC National
Accelerator Laboratory, Menlo Park, CA 94025  J. Corbin  Department of
Physics, Stanford University, Stanford, CA 94305  W. Craddock  SLAC National
Accelerator Laboratory, Menlo Park, CA 94025  C. S. Dawson  Department of
Physics, Stanford University, Stanford, CA 94305  A. Droster  Department of
Nuclear Engineering, University of California, Berkeley, Berkeley, CA 94720
J. W. Foster  Center for Theoretical Physics, Massachusetts Institute of
Technology, Cambridge, MA 02139  J. T. Fry  Laboratory of Nuclear Science,
Massachusetts Institute of Technology, Cambridge, MA 02139  P. W. Graham
Department of Physics, Stanford University, Stanford, CA 94305  R. Henning
Department of Physics and Astronomy, University of North Carolina, Chapel
Hill, Chapel Hill, North Carolina, 27599  Triangle Universities Nuclear
Laboratory, Durham, NC 27710  K. D. Irwin<EMAIL_ADDRESS>Department of
Physics, Stanford University, Stanford, CA 94305  SLAC National Accelerator
Laboratory, Menlo Park, CA 94025  F. Kadribasic  Department of Physics,
Stanford University, Stanford, CA 94305  Y. Kahn  Department of Physics,
University of Illinois at Urbana-Champaign, Urbana, IL 61801  A. Keller
Department of Nuclear Engineering, University of California, Berkeley,
Berkeley, CA 94720  R. Kolevatov  Department of Physics, Princeton University,
Princeton, NJ 08544  S. Kuenstner  Department of Physics, Stanford University,
Stanford, CA 94305  A. F. Leder  Department of Nuclear Engineering, University
of California, Berkeley, Berkeley, CA 94720  Accelerator Technology and
Applied Physics Division, Lawrence Berkeley National Laboratory, Berkeley, CA
94720  D. Li  SLAC National Accelerator Laboratory, Menlo Park, CA 94025  J.
L. Ouellet<EMAIL_ADDRESS>Laboratory of Nuclear Science, Massachusetts
Institute of Technology, Cambridge, MA 02139  K. M. W. Pappas  Laboratory of
Nuclear Science, Massachusetts Institute of Technology, Cambridge, MA 02139
A. Phipps  Department of Physics, California State University, East Bay,
Hayward, CA 94542  N. M. Rapidis  Department of Physics, Stanford University,
Stanford, CA 94305  B. R. Safdi  Department of Physics, University of
California, Berkeley, Berkeley, CA 94720  C. P. Salemi  Laboratory of Nuclear
Science, Massachusetts Institute of Technology, Cambridge, MA 02139  M.
Simanovskaia  Department of Physics, Stanford University, Stanford, CA 94305
J. Singh  Department of Physics, Stanford University, Stanford, CA 94305  E.
C. van Assendelft  Department of Physics, Stanford University, Stanford, CA
94305  K. van Bibber  Department of Nuclear Engineering, University of
California, Berkeley, Berkeley, CA 94720  K. Wells  Department of Physics,
Stanford University, Stanford, CA 94305  L. Winslow  Laboratory of Nuclear
Science, Massachusetts Institute of Technology, Cambridge, MA 02139  W. J.
Wisniewski  SLAC National Accelerator Laboratory, Menlo Park, CA 94025  B. A.
Young  Department of Physics, Santa Clara University, Santa Clara, CA 95053

(April 28, 2022)

###### Abstract

The QCD axion is one of the most compelling candidates to explain the dark
matter abundance of the universe. With its extremely small mass ($\ll
1\,\mathrm{eV}/c^{2}$), axion dark matter interacts as a classical field
rather than a particle. Its coupling to photons leads to a modification of
Maxwell’s equations that can be measured with extremely sensitive readout
circuits. DMRadio-m3 is a next-generation search for axion dark matter below
$1\,\mu$eV using a $>4$ T static magnetic field, a coaxial inductive pickup, a
tunable LC resonator, and a DC-SQUID readout. It is designed to search for QCD
axion dark matter over the range $20\,\mathrm{neV}\lesssim m_{a}c^{2}\lesssim
800$ neV ($5\,\mathrm{MHz}<\nu<200$ MHz). The primary science goal aims to
achieve DFSZ sensitivity above $m_{a}c^{2}\approx 120$ neV (30 MHz), with a
secondary science goal of probing KSVZ axions down to $m_{a}c^{2}\approx 40$
neV (10 MHz).

SM

    Standard Model
QED

    quantum electrodynamics
QCD

    quantum chromodynamics
BSM

    beyond the standard model
DM

    dark matter
CDM

    cold dark matter
GUT

    grand unification theory
WIMP

    weakly interacting massive particle
SHM

    Standard Halo Model
ppm

    part-per-million
ppb

    part-per-billion
ADM

    axion dark matter
ALP

    axion-like particle
PQ

    Peccei-Quinn
PQWW

    Peccei-Quinn-Wilczek-Weinberg
KSVZ

    Kim-Shifman–Vainshtein–Zakharov
DFSZ

    Dine–Fischler–Srednicki–Zhitnitsky
LSW

    light shining through wall
DP

    dark photon
DR

    dilution refrigerator
PT

    pulse tube
OFHC

    oxygen-free, high-conductivity
TE

    transverse electric
TM

    transverse magnetic
TEM

    transverse electromagnetic
MQS

    magneto-quasistatic
SQL

    standard quantum limit
QND

    quantum non-demolition
DFT

    discrete Fourier transform
FFT

    fast Fourier transform
SNR

    signal-to-noise ratio
PSD

    power spectral density

##  I Introduction

The Strong CP problem describes an unnaturally fine-tuned symmetry of nature
that suggests an explanation beyond the Standard Model (SM) of particle
physics. The leading solution to this problem is the introduction of a new
Peccei-Quinn symmetry, which is spontaneously broken at some high energy scale
$f_{a}$ producing a pseudo-Goldstone boson, the axion $a$ [1, 2, 3, 4].
Interactions with quantum chromodynamics (QCD) give the axion a potential,
which solves the Strong CP problem, gives the axion mass $m_{a}c^{2}\approx
5.7\,\mathrm{neV}(10^{15}\,\mathrm{GeV}/f_{a})$ [5], and produces a relic
axion abundance in the early universe that satisfies the conditions to be dark
matter (DM) [6, 7, 8]. Over the last few years, the axion has emerged as a
leading DM candidate.

Recent theoretical work has opened a wide range of interesting parameter space
for both QCD axion and axion-like particle (ALP) models [9, 10, 11, 12, 13].
In particular, the mass range $1\,\mathrm{neV}\lesssim m_{a}c^{2}\lesssim
1\,\mu\mathrm{eV}$ is interesting for grand unification theories [14, 15, 16,
17, 18, 19, 20, 21, 22, 11], String Theory models [23, 24, 25, 26, 27, 28, 29,
30], and naturalness arguments [31, 12]. Other ALP models in this mass range
simultaneously explain both the DM abundance and matter-antimatter asymmetry
[32].

Its small mass and cold temperature give axion dark matter (ADM) a very high
per-state occupation making it interact as a wave-like classical field.
Depending on the precise model, the axion may couple to any SM particle, but
one of the least model-dependent couplings is the axion-photon coupling
$g_{a\gamma\gamma}$ [14]. For the QCD axion, $g_{a\gamma\gamma}$ is directly
proportional to $m_{a}$, with an uncertainty on the proportionality constant
spanning an order of magnitude in uncertainty and represented by the Kim-
Shifman–Vainshtein–Zakharov (KSVZ) [33, 34] and
Dine–Fischler–Srednicki–Zhitnitsky (DFSZ) [35, 36] models. A more generic
class of ALPs break this proportionality and can have a wide range of masses
and $g_{a\gamma\gamma}$ couplings.

The axion-photon coupling produces a modified Ampère’s law, behaving as an
effective current that can be written approximately as

$\mathbf{J}_{\rm eff}\approx g_{a\gamma\gamma}\frac{\sqrt{\hbar
c}}{\mu_{0}}\sqrt{2\rho_{\rm
DM}}\cos\left(\frac{m_{a}c^{2}t}{\hbar}\right)\mathbf{B},$ (1)

with local DM density $\rho_{\rm DM}\approx 0.45\,\mathrm{GeV/cm^{3}}$ [37]
and axion frequency $\nu_{a}=m_{a}c^{2}/(2\pi\hbar)$. A powerful approach to
searching for ADM is to deploy a large, static $\mathbf{B}$-field to drive
$\mathbf{J}_{\rm eff}$ through a pickup structure and search for excess power
in a narrow axion signal bandwidth $\Delta\nu_{\rm sig}/\nu_{a}\approx
10^{-6}$ set by the Standard Halo Model (SHM) [38].

(a)

(b)

(c)

Figure 1: Left: Cartoon of the DMRadio-m3 coaxial pickup in a uniform magnetic
field. The magnet (not shown) drives an oscillating $\mathbf{J}_{\rm eff}$
(blue arrows) along the axis of the coax, which drives an azimuthal magnetic
field (purple arrows), and a current along the coax (red arrows). Voltage
accumulates across the slit on the top. Middle: Cross section of DMRadio-m3
with major components labeled. Total height is $\sim 2.5$ m. Right: Conceptual
magnetic field design that produces high field across the volume of the coax
(gold box) and a low field region for sensitive electronics (gray dashed
region). Color shading represents the vertical component of the primary field
$B_{z}$, while the red lines indicate the magnetic field flux lines. The field
is generated by the solenoid coils in the black-hashed region, and bucking
coils with counter flowing current in the black-dotted region.

Historically, ADM experiments have focused on the mass range
$1\,\mu\mathrm{eV}\lesssim m_{a}c^{2}\lesssim 40\,\mu\mathrm{eV}$, where the
axion is able to resonantly excite a microwave cavity [39, 40, 41, 42, 43, 44,
45, 46]. However, the cavity resonance requires a detector size $L$ comparable
to the axion Compton wavelength, $m_{a}cL/\hbar\sim 1$ and makes probing
masses below $1\,\mu\mathrm{eV}$ impractical. At lower axion masses, the
magneto-quasistatic (MQS) limit applies and $\mathbf{J}_{\rm eff}$ can be
treated as inducing magnetic fields that are detectable with an inductive
pickup and enhanced with a lumped-element resonator [47, 48, 49, 50, 51]. This
has been experimentally realized with a toroidal magnet with the
ABRACADABRA-10 cm prototype [52, 53, 54] and SHAFT [55] and in a solenoidal
geometry with ADMX-SLIC [56] and the BASE Penning trap [57]. DMRadio-
Pathfinder also performed a resonant search for dark photons in the MQS regime
with a solenoidal pickup [58]. In this letter, we present an optimized lumped-
element experimental design called DMRadio-m3, capable of probing ADM over the
mass range $20\,\mathrm{neV}\lesssim m_{a}c^{2}\lesssim 800\,\mathrm{neV}$
($5\,\mathrm{MHz}<\nu<200$ MHz) with a 5 yr scan time, achieving DFSZ
sensitivity for masses above $\sim\\!120\,\mathrm{neV}/c^{2}$ (30 MHz).

##  II Lumped-Element Detection

The challenge facing any ADM experiment is one of signal-to-noise ratio (SNR).
The ADM field contains enough power per square meter to illuminate an LED, but
the self-impedance of photon-electron coupling implies that any practical
receiver will only extract a tiny fraction of this power [59]. Resonant
circuits – either microwave cavities or LC lumped-element circuits – have much
lower impedance on resonance and can extract more power from the axion field.

For a readout circuit inductively coupled to $\mathbf{J}_{\rm eff}$, the
induced voltage can be expressed as

$|\tilde{V}_{pp}|^{2}=4g_{a\gamma\gamma}^{2}\rho_{\rm
DM}\left(\frac{c^{5}}{\hbar\mu_{0}}\right)m_{a}^{2}c_{\rm PU}^{2}L_{\rm
eff}B_{0}^{2}V_{\rm PU}^{5/3}$ (2)

(see Appendix). Here $B_{0}$ and $V_{\rm PU}$ are the characteristic magnetic
field and pickup volume, $L_{\rm eff}$ is the effective inductance of the
pickup and readout circuit at the axion frequency $2\pi\nu_{a}=m_{a}/\hbar$
and should ideally be dominated by the effective inductance of the pickup,
$L_{\rm PU}$. $c_{\rm PU}$ is a dimensionless proportionality constant that
depends on the geometry. When coupled to a capacitor, the impedance of the
resulting circuit $Z_{L}(\nu)$ decreases significantly at the resonance
frequency, $\nu_{r}$. If $\nu_{r}$ is tuned close to the axion driving
frequency, the current driven through the circuit from axion conversion,
$\tilde{I}_{\rm sig}$, can be enhanced by multiple orders of magnitude –
expressed in terms of the resonator quality $Q$:

$|\tilde{I}_{\rm sig}|^{2}=4g_{a\gamma\gamma}^{2}\rho_{\rm
DM}\left(\frac{\hbar c}{\mu_{0}}\right)\frac{c_{\rm PU}^{2}B_{0}^{2}V_{\rm
PU}^{5/3}}{L_{\rm eff}}\left(\frac{Q^{2}}{1+(2Q\Delta)^{2}}\right)$ (3)

which holds for small fractional detunings ($\Delta\ll 1$), and assumes that
the pickup dominates the inductance of the circuit. The signal follows the
Lorentzian response of the resonator circuit and depends on the fractional
detuning of resonance from the axion frequency,
$\Delta=(2\pi\hbar\nu_{r}-m_{a}c^{2})/m_{a}c^{2}$. This gives the resonator
characteristic width $\Delta\nu_{r}=\nu_{r}/Q$.

The sensitivity for a resonant search is determined by its scan rate: the rate
at which one can scan through frequencies searching for the axion induced
signal current. The scan rate is set by the desired $g_{a\gamma\gamma}$
sensitivity and SNR at each resonator tuning, and the total equivalent current
noise $|\tilde{I}_{N}|^{2}$ at that frequency. At a single resonator tuning,
the integration time, $\tau$, required to achieve this SNR at
$g_{a\gamma\gamma}$ is given by a modified Dicke radiometer equation

$\mathrm{SNR}(\nu)=\frac{|\tilde{I}_{\rm
sig}|^{2}}{|\tilde{I}_{N}(\nu)|^{2}}\sqrt{\tau\Delta\nu_{\rm sig}}.$ (4)

The current noise $|\tilde{I}_{N}(\nu)|^{2}$ is dominated by thermal Johnson-
Nyquist noise and the noise of the first stage amplifier, $\eta_{A}(\nu)$. The
amplifier noise consists of both the imprecision and backaction noise (see
Appendix).

For a search with many discrete resonator tunings, the scan rate is
approximated by the continuous function

$\begin{split}\frac{d\log\nu_{r}}{dt}=\pi&(6.4\times
10^{5})\left(\frac{c^{2}}{\mu_{0}^{2}}\right)\frac{g_{a\gamma\gamma}^{4}\rho_{\rm
DM}^{2}}{\mathrm{SNR}^{2}\nu_{r}}\times\\\ &c_{\rm PU}^{4}QB_{0}^{4}V_{\rm
PU}^{10/3}\bar{\mathcal{G}}(\nu_{r},k_{B}T,\eta_{A}(\nu_{r}))\end{split}$ (5)

$\bar{\mathcal{G}}(\nu_{r},k_{B}T,\eta_{A}(\nu_{r}))$ is a dimensionless
factor that defines the effect of the amplifier matching on the scan rate (see
Appendix). Explicitly, it is a trade-off between the Lorentzian signal gain
and the frequency dependent system noise. Together, these two set the
sensitivity bandwidth, $\Delta\nu_{\rm sens}$, which determines the spacing
between adjacent resonator tunings. Since the thermal noise and amplifier
backaction components follow the same Lorentzian lineshape as the signal gain,
were they the only noise sources, we would have constant SNR over all
frequencies and $\Delta\nu_{\rm sens}$ could grow arbitrarily large. However,
at frequencies sufficiently far from $\nu_{r}$, the amplifier imprecision
noise dominates, decreasing the SNR. For a scattering-mode circuit tuned for
optimal single-frequency power transfer, the sensitivity bandwidth nearly
matches the resonator line width, $\Delta\nu_{\rm sens}\approx\nu_{r}/Q$.
Alternatively, for a matching circuit tuned for optimal scan speed, modestly
reducing on-resonance SNR for a larger $\Delta\nu_{\rm sens}$ can
significantly improve the scan rate. In the thermal noise limit, the optimal
coupling yields $\bar{\mathcal{G}}(\nu_{r},k_{B}T,\eta)\approx
2\pi\hbar\nu_{r}(6\sqrt{3}k_{B}T\eta)^{-1}$.

##  III Detector Design

Figure 2: Effective DMRadio-m3 circuit model. $\mathbf{J}_{\rm eff}$ is
inductively coupled to a coaxial pickup, $L_{\rm PU}$, which, along with a
tunable capacitor, $C$, form the resonant circuit. An inductive transformer
$\kappa$ couples the signal to the SQUID amplifier. Loss in the readout is
represented by resistor, $R_{p}$. On resonance, $\nu_{r}\approx
1/(2\pi\sqrt{L_{\rm PU}C})$, the circuit impedance drops and axion power is
more efficiently transferred to the receiver. Figure 3: Projected DMRadio-m3
ADM sensitivity with a 5 year scan, with projected sensitivity for
DMRadio-50L, and existing bounds below $1\,\mu$eV from ABRACADABRA-10 cm,
SHAFT, ADMX-SLIC (spike around 200 neV), BASE (spike around 3 neV), and bounds
from microwave cavities above $1\,\mu$eV. DMRadio-m3 targets DFSZ sensitivity
above $\approx 120\,$neV and secondarily KSVZ above $\approx 40\,$neV. Plot
generated using code from [60].

DMRadio-m3 uses a static $>$4 T solenoid magnet to drive $\mathbf{J}_{\rm
eff}$ along the axis of a coaxial inductive pickup (see Fig. 1b.) The coaxial
pickup couples to a tunable capacitor to form an LC resonator circuit,
targeting a quality factor of $Q>10^{5}$. The circuit is read out by
conventional DC-SQUIDs with a target noise level of $20\times$ the standard
quantum limit (SQL). The coaxial pickup and capacitor are cooled to an
operating temperature of $T\approx 20$ mK, while the magnet sits at 4 K. We
discuss each of these in detail below.

DMRadio-m3 builds on the experience of ABRACADABRA-10 cm, DMRadio-Pathfinder,
and DMRadio-50L (currently under construction.) All three detectors use a
superconducting inductive pickup to search for ADM or DPs below 20 neV.
DMRadio-Pathfinder and DMRadio-50L [61] utilize a resonant readout approach,
and ABRACADABRA-10 cm and DMRadio-50L utilize a toroidal magnet. The toroidal
geometry works well at low frequencies (i.e. low axion masses) because it
completely encloses the magnetic field, allowing the use of lossless
superconducting materials. Loss is only introduced through coupling to lossy
materials near the detector, which can be efficiently screened. However, at
frequencies above $\sim$50 MHz, irreducible capacitances associated with the
readout begin to short the signal and rapidly reduce the signal sensitivity at
higher frequencies [62]. A change of geometry is required to probe higher
axion masses.

DMRadio-m3 uses a solenoid magnet, with a bore volume of $V\approx
2\,\mathrm{m}^{3}$, and a characteristic field of $B_{0}>4\,\mathrm{T}$. This
geometry creates a large-volume, high-field region and drives $\mathbf{J}_{\rm
eff}$ along the axis of the solenoid (see Fig. 1a.) Unlike the toroid, the
solenoid exposes the magnetic field, requiring that the pickup sit in the
large primary field. Because SQUIDs cannot operate in such a field, DMRadio-m3
uses a set of bucking coils to bend the field lines outward, producing a low-
field region directly above the high-field region. The resulting magnetic
field has significant variation within the bore of the magnet, reaching a peak
value of $\approx 7\,\mathrm{T}$ off-axis; but with a characteristic value of
$B_{0}>4\,\mathrm{T}$ along the central axis in the high-field region. The
low-field region sits approximately 20 cm above the high-field region and has
a maximum field of $|\mathbf{B}|\lesssim 400\,\mathrm{mT}$ (see Fig. 1c).
These remaining fields are further reduced with additional bucking coils and
superconducting shielding.

DMRadio-m3 detects the axion-induced $\mathbf{J}_{\rm eff}$ via a coaxial
inductive pickup in the high-field region of the magnet (see Fig. 1a). The
geometry maximizes the coupling to $\mathbf{J}_{\rm eff}$, while minimizing
the pickup inductance $L_{\rm PU}$. Because it sits inside the magnetic field,
it is challenging to make the pickup from lossless superconducting material.
Instead, the pickup is made from oxygen-free, high-conductivity (OFHC) copper.
The finite conductivity of copper sets an upper limit on the achievable
quality factor of the LC resonator. At DMRadio-m3 frequencies and at
temperatures $<100\,\mathrm{mK}$, the anomalous surface resistance of copper
is lower than at $\sim$GHz frequencies and, combined with the large volume-to-
surface ratio, the resonator goal of $Q>10^{5}$ is achievable.

While the inductive pickup can sit in the primary field without unacceptably
degrading $Q$, the same does not extend to the tunable capacitor, which must
be superconducting, or to the SQUIDs, which require an ultra-low field to
operate. These are instead placed in the low-field region above the coaxial
pickup. To bridge the distance between the pickup and the low-field region
without introducing excessive parasitic inductance, the coax funnels to a
narrow neck coupled to the matching circuit above. To probe the mass range
from $20\,\mathrm{neV}\lesssim m_{a}c^{2}\lesssim 800\,\mathrm{neV}$,
DMRadio-m3 uses a set of interchangeable coaxial pickups that are exchanged
during the run of the experiment. The coaxial pickups are currently being
designed, and aim to keep $c_{\rm PU}$ & $V_{\rm PU}$ maximized over the
entire search range, while avoiding resonant modes that may short our signal.

The matching circuit must optimally couple to the pickup and convey the signal
to the SQUID readout (see Fig. 2) and consists of the coupling transformer, a
tunable capacitor and the inductive coupling to the SQUID amplifier. Any
additional inductance in the circuit that is not directly coupled to the
signal flux reduces the coupled power. In the series transformer configuration
of [48, 49], the coupled axion power is maximized by minimizing the
transformer inductance $L_{0}\ll L_{\rm PU}$, which is technically challenging
given the nominal $L_{\rm PU}\lesssim 200\,\mathrm{nH}$. However, in the
parallel transformer configuration of Fig. 2, axion signal power is maximized
by making the transformer inductance $L_{0}\gg L_{\rm PU}$, which can be
achieved more readily, with a transformer inductance of $L_{0}\gtrsim
1\,\mu\mathrm{H}$.

The baseline tunable capacitor consists of an array of parallel
superconducting plates and sapphire dielectrics. The tuning must be able to
scan the range $10\,\mathrm{pF}\lesssim C\lesssim 5\,\mathrm{nF}$ with part-
per-million (ppm) stepping precision. This can be achieved using a multiple
capacitor design with a coarse tuning to give large capacitance swing to cover
the full frequency range, and fine tuning to give ppm frequency precision. If
the loss in the insertable sapphire dielectric proves to be too large,
degrading the resonator $Q$ to an unacceptable level, a fallback option
includes using movable superconducting capacitor plates alone to adjust
conductor overlap and resultant capacitance. The design of the tunable
capacitor will build off the one currently under construction for DMRadio-50L
where many of these questions are being addressed.

Finally, the signal is amplified and read out through a phase-insensitive DC
SQUID readout. To maximize the frequency-integrated axion sensitivity, the
coupling transformer coefficient $\kappa$ must maintain an optimal tradeoff
between imprecision and backaction noise. This optimization is worked out for
a simple series circuit in detail in [63]. For the sensitivity projections
presented here, we target an amplifier noise level of $\eta_{A}=20$, i.e.
$20\times$ the SQL, which has been demonstrated in [64].

The optimal coupling $\kappa$ is in general frequency dependent, reflecting
the fact that at $T\approx 20\,\mathrm{mK}$ the number of thermal noise
photons $n_{T}=(\mathrm{exp}(2\pi\hbar\nu/k_{B}T)-1)^{-1}$ depends strongly on
frequency, and varies from $n_{T}\approx 90$ at 5 MHz to $n_{T}\approx 2$ at
200 MHz. Therefore, DMRadio-m3 utilizes a transformer capable of in situ
variation of the coupling to maintain optimal matching to the readout
amplifier.

The entire pickup structure is cooled to $T\approx 20\,\mathrm{mK}$ using a
dilution refrigerator (DR), while the magnet is held at 4 K. DRs cooling
payloads of similar heat capacity and volume to lower temperatures have been
implemented previously [65, 66]. DMRadio-m3 will be located in Building 750 at
SLAC National Accelerator Laboratory, the former site of the SLD detector.

##  IV Sensitivity

DMRadio-m3 will scan the full range $20\,\mathrm{neV}\lesssim
m_{a}c^{2}\lesssim 800\,\mathrm{neV}$ in stages, by sequentially stepping
through a series of detector configurations, each with a different coaxial
pickup. For the lowest frequency coax $c_{\rm PU}\approx 0.16$, with the exact
value depending on details of the final design.

The projected sensitivity for DMRadio-m3 is shown in Fig. 3 alongside the
sensitivity of the upcoming DMRadio-50L, which will the subject of future
publications. The shape of the sensitivity reach is set by the scan strategy,
which aims to cover the full mass range with as much DFSZ sensitivity as
achievable in 5 yr of scan time, leading to the corner at $\nu\approx 30$ MHz
or $m_{a}c^{2}\approx 120\,\mathrm{neV}$.

An intriguing feature of the sensitivity is that much of the scan time is
spent at lower masses above the DFSZ line. Above the 120 neV corner, the
target $g_{a\gamma\gamma}$ sensitivity increases $\propto\nu$, and the scan
speed increases $\propto\nu^{3}$ (see Eq. 5). The scan rate will be further
enhanced because of the growth of $\bar{\mathcal{G}}$, due to the decreasing
impact of thermal noise at higher frequencies – though this growth is slightly
slower than $\propto\nu_{r}$ and will be offset by a gradual decrease in SQUID
sensitivity at higher frequency.

##  V Conclusion

The axion is among the best motivated candidates to explain the DM abundance
of the universe. At present, the parameter space of ADM models is relatively
unexplored. Recent advances in quantum sensors, magnet technology, and
cryogenics have made searching the full ADM parameter space more feasible than
ever. The region of ADM parameter space with $m_{a}\lesssim 1\,\mu$eV is well
motivated and particularly attractive due to its connection with GUT models.

In this letter, we have presented the baseline design and sensitivity reach of
the DMRadio-m3 experiment, which uses a lumped-element approach to search for
QCD ADM over mass range $20\,\mathrm{neV}\lesssim m_{a}c^{2}\lesssim
800\,\mathrm{neV}$. This design is capable of probing at or below the KSVZ
sensitivity over the full range and DFSZ sensitivity above 120 neV with a 5 yr
scan time. This will make DMRadio-m3 a key component of the next generation of
experimental searches for ADM.

Future upgrades to DMRadio-m3 could moderately extend the DFSZ sensitivity
below the 120 neV corner through the use of backaction-evading quantum sensors
[67]. Future proposed experiments like DMRadio-GUT could probe DFSZ models to
even lower masses through the use of beyond-SQL sensors and high-field, high-
volume magnets [68].

###### Acknowledgements.

The authors acknowledge support for DMRadio-m3 as part of the DOE Dark Matter
New Initiatives program under SLAC FWP 100559. Members of the DMRadio
Collaboration acknowledge support from the NSF under awards 2110720 and
2014215. S. Chaudhuri acknowledges support from the R. H. Dicke Postdoctoral
Fellowship and Dave Wilkinson Fund at Princeton University. C. P. Salemi is
supported in part by the National Science Foundation Graduate Research
Fellowship under Grant No. 1122374. Y. Kahn was supported in part by DOE grant
DE-SC0015655. B. R. Safdi was supported in part by the DOE Early Career Grant
DESC0019225. P. W. Graham acknowledges support from the Simons Investigator
Award no. 824870 and the Gordon and Betty Moore Foundation Grant no. 7946. J.
W. Foster was supported by a Pappalardo Fellowship.

##  Appendix A Appendix: Scan Rate Calculation

The general derivation of Eq. 5 and $\bar{\mathcal{G}}$ can be found in [63].
Here we summarize the result for convenience.

In general, the voltage across an inductive pickup, $L_{\rm eff}$, coupled to
$\mathbf{J}_{\rm eff}$ will be proportional to the coupled energy $U_{\rm DM}$
[63]

$|V_{pp}^{2}|=4\left(\frac{c^{4}}{\hbar^{2}}\right)m_{a}^{2}L_{\rm eff}U_{\rm
DM}$ (6)

The coupled energy, defined as

$U_{\rm DM}=k^{2}\frac{g_{a\gamma\gamma}^{2}\rho_{\rm
DM}}{m_{a}^{2}}\frac{\hbar^{3}}{c\mu_{0}}\int|\mathbf{B}(\mathbf{x})|^{2}dV,$
(7)

gives a measure of the amount of energy coupled from the axion field into the
pickup. The volume integral is performed over all space and is proportional to
the total stored energy of the magnet. The dimensionless proportionality
constant, $k$, is a geometric factor containing all the information about
mutual coupling between the inductor and $\mathbf{J}_{\rm eff}$. This constant
generalizes the $C_{nml}$ form factor in a cavity haloscope. Energy
conservation dictates that $k^{2}\leq\frac{1}{2}$ [59]. However, it is also
clear that the voltage driven across an inductive element should go to zero as
the driving frequency goes to zero ($m_{a}\rightarrow 0$), and that therefore
$k$ should have a dependence on $m_{a}$. Given this, it is convenient to
define a new dimensionless constant, $c_{\rm PU}$, that separates this mass
dependence

$k^{2}=\left(\frac{c}{\hbar}\right)^{2}m_{a}^{2}V_{\rm
PU}^{2/3}\left(\frac{B_{0}^{2}V_{\rm
PU}}{\int|\mathbf{B}(\mathbf{x})|^{2}\,dV}\right)c_{\rm PU}^{2}$ (8)

and is both frequency and scale invariant. This constant can be calculated
explicitly in the MQS limit, when the Compton wavelength is large compared to
the size of the detector. In this limit, we can use the relationship
$V=\partial\Phi/\partial t$ and write
$\tilde{V}_{pp}=2\pi\nu_{a}i\tilde{\Phi}_{a}$ in the frequency domain, where
$\Phi_{a}$ is the axion induced flux through the pickup. This flux can be
extracted from the Biot-Savart law as

$\begin{split}\tilde{\Phi}_{a}={}&\frac{g_{a\gamma\gamma}\sqrt{\hbar
c}\sqrt{2\rho_{\rm DM}}}{4\pi}\\\ &\times\int_{A}dA\int
dV^{\prime}\mathbf{\hat{n}}\cdot\frac{\mathbf{B}_{0}(\mathbf{r}^{\prime})\times(\mathbf{r}-\mathbf{r}^{\prime})}{|\mathbf{r}-\mathbf{r}^{\prime}|^{3}}.\end{split}$
(9)

where the integral $dA$ is taken over the area of the pickup $A$, and the
integral $dV^{\prime}$ is performed over the shielding volume containing the
coax. Combining Eqns. 6, 7, 8, and 9, we can write $c_{\rm PU}$ in the MQS
limit as

$c_{\rm
PU}^{2}=\mu_{0}\frac{\left|\int\,dA\int\,dV^{\prime}\mathbf{\hat{n}}\cdot\frac{\mathbf{B}_{0}(\mathbf{r}^{\prime})\times\mathbf{(\mathbf{r}-\mathbf{r}^{\prime})}}{|\mathbf{r}-\mathbf{r}^{\prime}|^{3}}\right|^{2}}{2(4\pi)^{2}B_{0}^{2}V_{\rm
PU}^{5/3}L_{\rm eff}}.$ (10)

The scale invariance can be seen explicitly by noting the scaling $L_{\rm
eff}\approx L_{\rm PU}\propto V^{1/3}$ for a constant geometry. Typical values
for the DMRadio-m3 geometry are $c_{\rm PU}\approx 0.1-0.2$.

This formulation also generalizes beyond the validity of the lumped element
approximation. At frequencies where the Compton wavelength becomes comparable
to the size of the detector, $2\pi\hbar/m_{a}c\approx V_{\rm PU}^{1/3}$, the
coaxial pickup no longer behaves purely inductively and we must incorporate
its full complex impedance into the calculation: $2\pi i\nu L_{\rm
PU}\rightarrow Z_{\rm PU}(\nu)$. We can Taylor expand any resonant circuit’s
impedance around the resonant frequency to find an effective inductance,

$Z(\nu)\approx R_{\rm eff}+4\pi iL_{\rm
eff}(\nu-\nu_{r})+O\left((\nu-\nu_{r})^{2}\right)$ (11)

This expansion is generally valid over frequency bandwidths small compared to
the resonance frequency, i.e. for small fractional detunings $\Delta\ll 1$111A
more complete discussion of lumped elements detectors beyond the validity of
the MQS approximation is in preparation [69]..

The resonant enhancement of the circuit comes directly from its impedance
$Z(\nu)$

$|\tilde{I}_{\rm sig}|^{2}=\frac{|\tilde{V}_{pp}|^{2}}{|Z(\nu)|^{2}}.$ (12)

For a generic RLC circuit, $Z(\nu)=R+i(2\pi\nu L+(2\pi\nu C)^{-1})$, we can
evaluate

$\begin{split}\frac{1}{|Z(\nu)|^{2}}&{}=\frac{Q^{2}}{4\pi^{2}\nu_{r}^{2}L^{2}\left(1+Q^{2}\frac{\nu_{r}^{2}}{\nu^{2}}\left(\frac{\nu^{2}}{\nu_{r}^{2}}-1\right)^{2}\right)}\\\
&\approx\frac{Q^{2}}{4\pi^{2}\nu_{r}^{2}L^{2}\left(1+4Q^{2}\Delta^{2}\right)}\\\
\end{split}$ (13)

where $\nu_{r}=(2\pi\sqrt{LC})^{-1}$ and $Q=\sqrt{L/CR^{2}}$, and the
approximation holds for $\nu\approx\nu_{r}$. We see the Lorentzian behavior of
the resonator stems directly from the impedance of the readout circuit.

Figure 4: $\bar{\mathcal{G}}(\nu,k_{B}T,\eta_{A})$ as a function of frequency
for several choices of physical temperature, $T$, and amplifier noise
$\eta_{A}$. The solid lines show the full evaluation, while the dotted lines
show the linear behavior, which holds for the region where
$k_{B}T/\hbar\nu\gg\frac{1}{2}\eta_{A},1$.

The noise power $|\tilde{I}_{N}(\nu)|^{2}$ can be expressed in terms of a
current noise power spectral density at the input coil of the readout SQUID,

$S_{II}^{\rm tot}(\nu)=\frac{|\tilde{I}_{N}(\nu)|^{2}}{\Delta\nu}.$ (14)

This noise power can be decomposed into thermal, amplifier, and vacuum
components

$\begin{split}S_{II}^{\rm
tot}(\nu)={}&8\pi\hbar\nu\left(n_{T}+\frac{1}{2}\right)\frac{\mathrm{Re}[Z(\nu)]}{|Z(\nu)|^{2}}\\\
&+S_{II}^{\rm Imp}+\frac{S_{VV}^{\rm BA}}{|Z(\nu)|^{2}}\end{split}$ (15)

Where the first term in the parenthesis corresponds to the number of thermal
noise photons
$n_{T}(\nu,k_{B}T)=\left(\exp(2\pi\hbar\nu/k_{B}T)-1\right)^{-1}$ at physical
temperature $T$. DMRadio-m3 targets a physical temperature of $T=20$ mK,
corresponding to $n_{T}\approx 90$ noise photons at 5 MHz and $n_{T}\approx 2$
noise photons at 200 MHz. The $\frac{1}{2}$ corresponds to vacuum noise,
intrinsic to any phase-insensitive measurement. $S_{II}^{\rm Imp}$ is the
amplifier imprecision noise, expressed in terms of current noise at the
amplifier input, and $S_{VV}^{\rm BA}$ is the amplifier backaction noise,
expressed as a voltage source at the amplifier input and are both independent
of frequency in this form. Equation 15 demonstrates the different spectral
behavior of the various noise sources. The thermal, vacuum, and backaction
noise are all shaped by the resonator Lorentzian, while the imprecision noise
has a spectrally flat distribution. In practice, a DC-SQUID will have an
additional type of noise term, $S_{IV}^{\rm corr}$, that defines correlations
between the imprecision and backaction noise [70, 71]. These correlations
complicate the optimization and are included in our sensitivity calculation,
but do not qualitatively change the approach and so are omitted here for
simplicity.

The amplifier noise parameter, $\eta_{A}$ can be written in terms of these
noise powers as

$\eta_{A}=\frac{\sqrt{S_{II}^{\rm Imp}S_{VV}^{\rm BA}}}{2\pi\hbar\nu}.$ (16)

At the SQL, $\eta_{A}=1$. For DMRadio-m3, the amplifier noise target is
$\eta_{A}=20$, including the correlated noise that we have omitted here. This
corresponds to $n_{A}=\eta_{A}/2=10$ added noise photons.

The amplifier noise parameter $\eta_{A}$ depends on the amplifier imprecision
and backaction noise as well as $\kappa$ coupling in Fig. 2, which regulates
the tradeoff between the two. In particular, $S_{II}^{\rm
Imp}\propto\eta_{A}/\kappa^{2}$, while $S_{VV}^{BA}\propto\eta_{A}\kappa^{2}$.
A stronger coupling increases the backaction noise and decreases the input
referred imprecision noise. Because of the different spectral responses of
these two noise terms, the coupling $\kappa$ can be optimized to give the
fastest possible scan rate. For a careful and extensive analysis of this
optimization, including noise correlations, see [63]. But the resulting effect
on the scan rate can be written in terms of a single parameter

$\bar{\mathcal{G}}\left(\nu_{r},\bar{\alpha},k_{B}T,\eta_{A}\right)=\frac{\bar{\alpha}}{\left[\bar{\alpha}^{2}+2(2n_{T}+1)\bar{\alpha}+\eta_{A}^{2}\right]^{3/2}}$
(17)

where

$\bar{\alpha}=\frac{2\eta_{A}^{2}}{2n_{T}+1+\sqrt{(2n_{T}+1)^{2}+8\eta_{A}^{2}}}.$
(18)

In the limit where $n_{T}\gg\frac{1}{2}\eta_{A},1$, or equivalently
$k_{B}T/\hbar\nu\gg\frac{1}{2}\eta_{A},1$, $\bar{\mathcal{G}}$ has an
approximately linear dependence on $\nu$:

$\bar{\mathcal{G}}\approx\frac{2\pi\hbar\nu}{6\sqrt{3}k_{B}T\eta_{A}},\hskip
19.91692pt\frac{k_{B}T}{\hbar\nu}\gg\frac{1}{2}\eta_{A},1.$ (19)

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  *[SM]: Standard Model
  *[QCD]: quantum chromodynamics
  *[DM]: dark matter
  *[ALP]: axion-like particle
  *[ADM]: axion dark matter
  *[KSVZ]: Kim-Shifman–Vainshtein–Zakharov
  *[DFSZ]: Dine–Fischler–Srednicki–Zhitnitsky
  *[ALPs]: axion-like particle
  *[SHM]: Standard Halo Model
  *[MQS]: magneto-quasistatic
  *[SNR]: signal-to-noise ratio
  *[SQL]: standard quantum limit
  *[DPs]: dark photon
  *[OFHC]: oxygen-free, high-conductivity
  *[ppm]: part-per-million
  *[DR]: dilution refrigerator
  *[DRs]: dilution refrigerator
  *[GUT]: grand unification theory