DMRadio Collaboration

# Electromagnetic modeling and science reach of DMRadio-m3

A. AlShirawi  Department of Physics, Stanford University, Stanford, CA 94305
C. Bartram  SLAC National Accelerator Laboratory, Menlo Park, CA 94025  Kavli
Institute for Particle Astrophysics and Cosmology, Stanford University,
Stanford, CA 94305  J. N. Benabou  Berkeley Center for Theoretical Physics,
University of California, Berkeley, CA 94720, U.S.A.  Theoretical Physics
Group, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, U.S.A.  L.
Brouwer  Accelerator Techology 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  Kavli Institute for Particle
Astrophysics and Cosmology, Stanford University, Stanford, CA 94305  J. Corbin
Department of Physics, Stanford University, Stanford, CA 94305  Kavli
Institute for Particle Astrophysics and Cosmology, Stanford University,
Stanford, CA 94305  W. Craddock  SLAC National Accelerator Laboratory, Menlo
Park, CA 94025  A. Droster<EMAIL_ADDRESS>Department of Nuclear
Engineering, University of California, Berkeley, Berkeley, CA 94720  J. W.
Foster<EMAIL_ADDRESS>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  Department of Physics, Stanford
University, Stanford, CA 94305  SLAC National Accelerator Laboratory, Menlo
Park, CA 94025  Kavli Institute for Particle Astrophysics and Cosmology,
Stanford University, Stanford, CA 94305  F. Kadribasic  Department of Physics,
Stanford University, Stanford, CA 94305  Kavli Institute for Particle
Astrophysics and Cosmology, 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  Kavli Institute for Particle Astrophysics and
Cosmology, Stanford University, Stanford, CA 94305  N. Kurita  SLAC National
Accelerator Laboratory, Menlo Park, CA 94025  A. F. Leder  Department of
Nuclear Engineering, University of California, Berkeley, Berkeley, CA 94720
Physics Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720
D. Li  SLAC National Accelerator Laboratory, Menlo Park, CA 94025  Kavli
Institute for Particle Astrophysics and Cosmology, Stanford University,
Stanford, CA 94305  J. L. Ouellet  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<EMAIL_ADDRESS>Department of Physics, Stanford University, Stanford, CA 94305  Kavli
Institute for Particle Astrophysics and Cosmology, Stanford University,
Stanford, CA 94305  B. R. Safdi  Berkeley Center for Theoretical Physics,
University of California, Berkeley, CA 94720, U.S.A.  Theoretical Physics
Group, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, U.S.A.  C.
P. Salemi  Department of Physics, Stanford University, Stanford, CA 94305
SLAC National Accelerator Laboratory, Menlo Park, CA 94025  Kavli Institute
for Particle Astrophysics and Cosmology, Stanford University, Stanford, CA
94305  M. Simanovskaia  Department of Physics, Stanford University, Stanford,
CA 94305  Kavli Institute for Particle Astrophysics and Cosmology, Stanford
University, Stanford, CA 94305  J. Singh  Department of Physics, Stanford
University, Stanford, CA 94305  Kavli Institute for Particle Astrophysics and
Cosmology, Stanford University, Stanford, CA 94305  E. C. van Assendelft
Department of Physics, Stanford University, Stanford, CA 94305  Kavli
Institute for Particle Astrophysics and Cosmology, 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

###### Abstract

DMRadio-m3 is an experiment that is designed to be sensitive to KSVZ and DFSZ
QCD axion models in the 10–200 MHz (41 neV$/c^{2}$ – 0.83 $\mu$eV/$c^{2}$)
range. The experiment uses a solenoidal dc magnetic field to convert an axion
dark-matter signal to an ac electromagnetic response in a coaxial copper
pickup. The current induced by this axion signal is measured by dc SQUIDs. In
this work, we present the electromagnetic modeling of the response of the
experiment to an axion signal over the full frequency range of DMRadio-m3,
which extends from the low-frequency, lumped-element limit to a regime where
the axion Compton wavelength is only a factor of two larger than the detector
size. With these results, we determine the live time and sensitivity of the
experiment. The primary science goal of sensitivity to DFSZ axions across
30–200 MHz can be achieved with a $3\sigma$ live scan time of 3.7 years.

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 axion is a well-motivated dark matter (DM) candidate that solves the
strong charge-parity (CP) problem of quantum chromodynamics (QCD) [1, 2, 3, 4]
while also having a favorable cosmological production mechanism that can
populate the universe with the observed cold DM abundance [5, 6, 7]. It arises
as a pseudo-Goldstone boson of a spontaneously broken Peccei-Quinn symmetry
that has a symmetry breaking energy scale $f_{a}$. Through interactions with
QCD, the axion mass is set by the inverse of the symmetry breaking scale as
$m_{a}\approx 5.7\text{ neV}(10^{15}\text{ GeV}/f_{a})$ [8]. Recent
theoretical work suggests that the value of $f_{a}$ could be as high as the
Planck scale, which motivates searches that are sensitive to axions with mass
in the range of peV$/c^{2}$–$\mu$eV$/c^{2}$ [9, 10, 11, 12, 13]. Such axion
and axion-like particle (ALP) models are of interest for grand unification
theories (GUTs) [11, 14, 15, 16, 17, 18, 19, 20, 21], string theory models
[22, 23, 24, 25, 26, 27, 28, 29], and naturalness arguments [30, 12].

One of the most sensitive techniques to search for axions takes advantage of
their coupling to standard-model photons. An axion in a dc magnetic field can
convert to a photon whose frequency matches the axion mass
($\nu_{a}=m_{a}c^{2}/h$). Due to their low mass and low temperature, dark
matter axions have a high per-state occupation number and thus behave like a
classical field. The axion-photon interaction can then be considered as a
modification to Maxwell’s equations, resulting in a new effective current that
depends on the local magnetic field and the local axion density. Neglecting
spatial gradients in the dark matter axion field, which tend to be small, we
write the effects of the axion-photon coupling as an effective current in the
presence of a dc magnetic field [31]:

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

Here $g_{a\gamma\gamma}$ is the axion-photon coupling constant,
$\rho_{\text{DM}}=0.45\text{ GeV/cm}^{3}$ is the local axion dark matter
density [32], $m_{a}$ is the axion mass, and $\mathbf{B}$ is a background dc
magnetic field. Axion-dark-matter experiments typically scan through a
frequency band to search for axions down to a specific coupling strength
$g_{a\gamma\gamma}$ at a specified signal-to-noise ratio. The metric of
sensitivity is given by the scan rate, which denotes the frequency range
covered per unit time to achieve sensitivity to axions at a given
$g_{a\gamma\gamma}$.

To date, the most sensitive limits have been set by cavity haloscopes in mass
ranges above 1 $\mu$eV$/c^{2}$ (240 MHz) [33, 34, 35]. When a microwave cavity
is placed in a dc magnetic field, axions produce an effective current that
runs along the magnetic field lines through the cavity. If the frequency of a
cavity mode that couples to this current matches that of the axion, the signal
is resonantly enhanced. Such cavity experiments become impractical at low
axion frequencies due to the size of the cavities that is required to support
cavity modes with long wavelength.

Experiments in the sub-$\mu$eV mass range decouple the resonance conditions
from the boundary conditions imposed by the physical dimensions of a simple
cavity, making it possible to access lower frequencies within practical
experimental volumes. At very low frequencies, these detectors are often
referred to as “lumped element,” because they isolate electric and magnetic
field energy into capacitive and inductive elements [36, 37]. Experiments in
operation or proposed in the lumped-element limit include ADMX-SLIC, a
solenoidal experiment [38], as well as toroidal experiments such as the
ABRACADABRA-10 cm prototype [39, 40, 41, 42], SHAFT [43], and the DMRadio-50L
experiment [44]. DMRadio-Pathfinder performed a resonant search for dark
photons in the lumped-element regime with a solenoidal pickup [45]. A toroidal
geometry is also analyzed in the regime ranging from the lumped-element limit
to higher frequencies [46]. Other ways to probe lower frequency axions in a
subwavelength space include nuclear spin coupling [47] and heterodyne
frequency conversion [48].

In this paper, we model the sensitivity of the DMRadio-m3 experiment [49],
which operates a coaxial pickup in a $\sim 4.7$ T solenoidal magnet at
frequencies both within and above the lumped-element limit, where lumped
element approximations are not accurate [46]. Here, we analyze the sensitivity
by modeling the experiment as an equivalent tuned series resistor-inductor-
capacitor (RLC) circuit with numerical results from Finite Element Modeling
(FEM) simulations. Incorporating realistic noise models for dc SQUIDs, we use
the formalism in [50] to determine the scan times required for each frequency
interval. We present a baseline design for DMRadio-m3 to achieve sensitivity
to Dine-Fischler-Srednicki-Zhitnisky (DFSZ) [51, 52] axions in the 30–200 MHz
range as the primary science goal of the experiment, as well as sensitivity to
Kim-Shifman-Vainshtein-Zakharov (KSVZ) [53, 54] axions in the 10–30 MHz range,
as the secondary science goal of the experiment. We also present a plan to
extend the search to axions with a coupling of
$g_{a\gamma\gamma}=g_{a\gamma\gamma,\text{DFSZ}}(\nu=30\text{ MHz})=1.87\times
10^{-17}\text{ GeV}^{-1}$ across 5–30 MHz, thus extending the parameter space
covered in the secondary science goal. We denote this as the extended goal.

In $\S$II we motivate and describe the coaxial design of the DMRadio-m3 pickup
structure and present the constraints imposed by the magnet. In $\S$III, we
start with an intuitive understanding of the mode structure of the coaxial
pickups. We then discuss the different coaxial pickups used for different
frequency ranges, and summarize their computed eigenmodes. We show that
DMRadio-m3 uses a single TEM-like mode, well-separated from other parasitic
modes, over the science bandwidth of each coaxial element. This approach
enables the formalism of [50] to be rigorously applied. In $\S$IV, we
numerically extract an equivalent series RLC circuit representing the scanned
resonance, which is then used to compute the science reach of DMRadio-m3. In
$\S$V, we discuss the tuning requirements and constraints for the experiment.
$\S$VI presents relevant details on the SQUID readout amplifiers as they apply
to DMRadio-m3. Finally, in $\S$VII, we determine the required $3\sigma$ live
scan times for DMRadio-m3 to achieve its science goals.

##  II DMRadio-m3 pickup geometry

Figure 1: (a) A 2D cross section of the dc magnetic field profile of
DMRadio-m3 produced by the main magnet coil and the bucking coils, as well as
the full coaxial pickup cross section. These structures exist in a shielded
vacuum space whose boundary is not shown. In this figure, the coax height,
$h_{\text{coax}}$, is $1.68$ m. The main magnet coils produce a $B_{0}\approx
5$ T region for coupling to axion signals, and the shielded box encloses a
low-magnetic-field region at the top. The low-field region has additional
magnetic and EMI shielding, and contains superconducting components (not
shown) that include the tuning element and the SQUID readout. (b) A cross-
section of the coaxial pickup showing the effective axion current (light-green
arrows) and the resulting induced surface current due to screening effects on
the inner surface of the coaxial pickup (orange arrows). This simulation was
run at 100 MHz. The nodes in the surface current can be understood as an
effect of bulk electron shuttling (see Sections IV and Appendix D for
details). The voltage and impedance of the circuit are measured across the two
rings defined by A and B respectively, as shown in the expanded view. (c) The
circuit parameters of this pickup structure are measured as a frequency-
dependent reactance $Z_{p}(\omega)$ and axion-signal voltage amplitude
$|V(\omega)|$ measured between the circles marked A and B in subfigure (b).
This impedance is modeled as a series RLC circuit ($\S$ IV) and is coupled to
the impedance of a reactive tuning element that is located in the shielded
box. At each frequency, the impedance parameters, the voltage amplitude
induced by the axion signal, and the amplifier properties determine the scan
rate.

The pickup geometry of DMRadio-m3 is coaxial. For a coaxial transmission line,
transverse electromagnetic (TEM) propagating waves have no lower cutoff
frequency. A coaxial transmission line resonator is thus a natural geometry
for use in axion experiments at wavelengths larger than the length scale of
the experiment providing access to low frequencies through a tunable resonant
mode based on a TEM propagating mode in the coaxial section. As shown
schematically in Figure 1a, DMRadio-m3 consists of a copper coaxial pickup
structure (sometimes referred to as the coax) in the region of highest
magnetic field in the bore of a solenoidal magnet. The straight coaxial
section forms the majority of the pickup, with a fixed bottom plate shorting
the bottom. The top end of the pickup necks down to a superconducting tuning
circuit in a small low-field shielded region enclosed in a pillbox-shaped
container (henceforth referred to as the shielded box). This shielded box
holds the reactive tuning elements and SQUID readout amplifiers. Figure 1a
shows a cross-section of one coaxial pickup without any tuning or readout
elements.

The tuning and readout elements for DMRadio-m3 are connected across the rings
that are denoted as points A and B in Figure 1b and are placed in the shielded
box that encloses the low-field region shown in Figure 1a. The electromagnetic
properties of the coax can be modeled and understood independently from the
tuning and readout components, as shown in the circuit model in Figure 1c.

The dc magnetic field strength and volume for DMRadio-m3 are optimized to
maximize science reach for a given magnet cost. The chosen magnet dimensions
set a maximum coaxial outer radius of $r_{o}=0.628$ m. The coaxial inner
radius is chosen to be $r_{i}=0.336$ m. Six interchangeable copper coaxial
pickups with different values of $h_{\rm coax}$ (defined in Figure 1a) are
used, the tallest of which has a height, $h_{\text{coax}}=1.68$ m. Different
coaxial lengths are optimal for use at different frequencies. For DMRadio-m3,
the lowest axion frequency of interest (5 MHz) corresponds to a Compton
wavelength of 60 m; the highest axion frequency of interest (200 MHz)
corresponds to Compton wavelength 1.5 m. Thus, DMRadio-m3 spans frequencies
well above the lumped-element limit. In the following section ($\S$III), we
discuss the coax in terms of its cavity properties; in $\S$IV we show how to
describe the coax as an equivalent circuit at each frequency, and then how to
determine its sensitivity.

##  III Modes of the DMRadio-m3 coax

DMRadio-m3 utilizes a set of six coaxial pickups to achieve the science goals
of the experiment. The underlying physical motivation behind the use of six
coaxial pickups is three-fold:

  1. (a)

The resonator is designed to be single-moded at each frequency that it scans.
In order to achieve this, the coax is operated below the frequency at which a
second mode appears.

  2. (b)

For a given length of a coaxial pickup, the impedance of the structure and
axion-induced voltage vary with frequency; these quantities in turn influence
the scan rate. The use of different sized pickups provides an acceptable scan
rate at all frequencies of interest.

  3. (c)

The tuning elements required at each frequency must be physically realizable,
and must enable tuning over a wide frequency range.

Point (a) is discussed in this section, while points (b) and (c) are discussed
in Sections §IV and §V respectively.

We first provide a simplified intuitive model to understand the resonant
features of the DMRadio-m3 coax. Subsequently we describe the eigenmodes of
the structure and how they influence the scan strategy of the experiment.

###  III.1 A coaxial toy model

To qualitatively understand DMRadio-m3, we consider the model of an idealized
lossless coaxial structure of uniform outer radius $r_{o}$, inner radius
$r_{i}$, and height $h_{\rm coax}$, shorted at the bottom and open at the top.
The axion signal due to the solenoidal magnetic field generates a voltage
between the top of the outer and inner cylinders of the coax. The impedance of
the coaxial pickup seen at its open top end is given by the equation for the
impedance of a lossless coaxial cable shorted at one end [55]:

$Z_{p}(\omega)=iX_{p}(\omega)=iZ_{c}\tan\left(\frac{\omega h_{\rm
coax}}{c}\right),$ (2)

where $X_{p}$ is the reactance of $Z_{p}$. The characteristic impedance of the
coax is:

$Z_{c}=\frac{Z_{0}}{2\pi}\ln\left(\frac{r_{o}}{r_{i}}\right),$ (3)

where $Z_{0}\approx 377\,\Omega$ is the impedance of free space. If left open,
this coax has TEM transmission line resonances at
$\nu=(2n-1)c/(4h_{\text{coax}})$, for $n=1,2,3...$, the first being the
quarter wavelength ($\lambda/4$) mode. If shorted, this coax has TEM
transmission line resonances starting at the half wavelength mode
($\lambda/2$) where $\nu=c/(2h_{\text{coax}})$.

More generally, the resonance frequency of the coaxial transmission line
resonator of Eq. 2 can be adjusted by loading it with a reactance,
$X_{\text{tuning}}.$ At a target resonance frequency, $\omega_{0}$, the tuning
reactance is chosen to be equal and opposite to the coaxial pickup reactance,
$X_{\text{tuning}}(\omega_{0})=-X_{p}(\omega_{0})$, thus satisfying the
condition for resonance where the total reactance is zero. Depending on the
sign of $X_{p}(\omega_{0})$, the tuning element may either be inductive or
capacitive.

In the lower frequency ranges of DMRadio-m3, the coaxial impedance is well
approximated as a linear inductor with $Z_{\text{coax}}\approx i\omega
L_{\text{coax}}=i\omega Z_{c}h_{\rm coax}/c$. This limit is the lumped-element
limit, in which the voltage across the output is proportional to the time
derivative of the coupled magnetic flux from the axion signal, since signals
propagate with negligible phase lag from one end of the structure to the
other. A tuning capacitance is required to achieve the resonant condition. At
higher frequencies, the reactance is no longer linear with frequency, and when
$\pi/2<\omega_{0}h_{\rm coax}/c<\pi$, $Z_{p}$ is capacitive, and a tuning
inductance is required to achieve resonance.

###  III.2 Eigenmodes of the DMRadio-m3 coax

In this paper we use the formalism developed in [50] to compute the scan rates
required to achieve our science goals. As that formalism applies to a single-
moded resonator and since coaxial structures can support multiple cavity
modes, the eigenmodes of the DMRadio-m3 pickup must be simulated and their
characteristics understood. DMRadio-m3 is designed to avoid mode degeneracies.

As discussed in the previous section, the DMRadio-m3 coaxes can be understood
by considering a simple coaxial cavity with one end shorted and the other end
reactively loaded. In the case where both ends are shorted, resonance
frequencies of all transverse electric (TE), transverse magnetic (TM), and TEM
modes can be calculated from closed-form formulas for uniform coaxes, and with
finite-element modeling for realistic detectors, thus providing a baseline for
our understanding of the mode structure.

The TM modes of DMRadio-m3 coaxes have resonance frequencies above 200 MHz and
do not influence the scan strategy in any way. However, the lowest order TE
mode, $\text{TE}_{111}$, occurs at a lower frequency, raising a concern of
mode degeneracy. The resonance frequency of the $\text{TE}_{111}$ mode for an
idealized coax with length much larger than either radial dimension can be
approximated as:

$\nu_{\text{TE}_{111}}\approx\frac{2c}{2\pi
r_{i}\left(1+\frac{r_{o}}{r_{i}}\right)},$ (4)

where $r_{i}$ and $r_{o}$ are the inner and outer radii of the coax,
respectively [55]. In this long coax limit, the values of $r_{o}=0.628$ m and
$r_{i}=0.336$ m for DMRadio-m3 support a TE111 mode with
$\nu_{\text{TE}_{111}}\approx 100$ MHz. Shorter coaxes will have higher
$\nu_{\text{TE}_{111}}$. In the case of DMRadio-m3, where the coax is shorted
at one end and tapers at the other end, these modes are neither analytically
solvable nor do they have a trivial field profile. We determine the precise
values using the FEM software COMSOL [56], which confirms that
$\nu_{\text{TE}_{111}}$ is above 100 MHz for all DMRadio-m3 coaxes, and that
shorter coaxes have higher $\nu_{\text{TE}_{111}}$. In this section we show
that careful choice of coaxes places $\nu_{\text{TE}_{111}}$ above all
operational frequencies, and with $\nu_{\text{TE}_{111}}$ at 200 MHz for the
shortest coax of $h_{\text{coax}}=0.56$ m.

All modes in such a structure still form a complete set of eigenstates and, in
an ideal case, the TE-like modes would not mix with the TEM-like mode that is
being used for an axion scan. This idealized approximation where the two modes
do not mix is broken, in part, by the readout coupling mechanism that in
practice will not have pure azimuthal symmetry. As a result, TE-like modes
have a non-zero mixing with the TEM-like mode. We design all coaxial elements
to operate only at frequencies below the TE111 mode.

Informed by the mode structure of these coaxial pickups, as well as the
results from the next section in which we use COMSOL to extract the scan rate
of the experiment, we select a set of six coaxial pickups to scan the
DMRadio-m3 science region. These coaxial pickups, frequency ranges, and their
respective TE111 mode resonances are presented in Table 1 and Figure 2. These
simulations also show that the resonant frequencies of TE-like modes shift by
less than $1\%$ when a reactive load is placed across points A and B (Figure
1b) of the coaxial pickup relative to when it is open. This means that even as
the TEM-like scanning mode is tuned across a range of frequencies, the TE-like
mode’s frequency remains sufficiently stable to be avoided.

Coax Number | $h_{\text{coax}}$ | Frequency Range(s) (MHz)
---|---|---
1 | 1.68 m |  | 5–27 MHz
---
31–47 MHz
2 | 1.40 m |  | 27–31 MHz
---
47–75 MHz
3 | 1.12 m | 100–115 MHz
4 | 0.84 m |  | 75–100 MHz
---
115–160 MHz
5 | 0.70 m | 160–180 MHz
6 | 0.56 m | 180–200 MHz
Table 1: Table of the six coaxial pickups (coaxes) to be used in DMRadio-m3
for the primary and secondary science goals as well as the extended goal.
Together, these coaxes enable frequency scans throughout the 5–200 MHz range.
The quantity $h_{\text{coax}}$ is shown in Figure 1a. The frequency range
chosen for each coaxial pickup optimizes simultaneously the constraints of
practical tuning elements and the scan rate, while requiring that the
resonator be single-moded. Some coaxes cover two discrete frequency ranges.
These conditions are further discussed in §IV.

For Coaxes 1-5 listed in Table 1, the TE111 mode is found in the $125\text{
MHz}<f<185\text{ MHz}$ range. However, by judiciously choosing the scan range
of each coax, it can be avoided in each case, as shown in Figure 2. In Coax 6,
the TE111 mode exists at 200 MHz and, as such, the end of the single-moded
frequency range for the shortest coax coincides with the upper boundary of the
DMRadio-m3 science range.

As is discussed in more detail in Appendix A, the shielded box at the top of
the coax is small enough that it has no cavity modes within the signal band,
and is large enough that it does not significantly capacitively shunt the
coaxial pickup. Numerical models indicate that, apart from these two extremes,
the performance of the detector is not significantly affected by the
dimensions of the shielded box. Radiation to free space is not a consideration
because the experiment is fully shielded.

From the cavity analysis above, we conclude that DMRadio-m3 always operates in
a single-moded regime, where the formalism from [50] applies. In the next
section, we explain how the formalism is implemented in our sensitivity
analysis.

Figure 2: Graphical representation of the frequency range of scan regions
(Table 1) and the corresponding TE111-like modes. DMRadio-m3 is designed to
avoid the TE111-like modes due to potential mode mixing with the TEM-like mode
that is used for the axion scan. Simulations have shown that the frequency of
the TE-like mode does not significantly shift from loading by the tuning
elements, and it is possible to stay at frequencies below this mode in
operation.

##  IV The Equivalent series RLC resonator

DMRadio-m3 uses a coaxial pickup similar to the toy model discussed in §III.1,
but with additional features including a neckdown region to a shielded box
that contains the tuning reactances and dc SQUID amplifiers. This design
results in an impedance that is qualitatively similar to that described by Eq.
2, but that requires numerical computation with modeling software described in
the following section. The voltage signal induced by axions is similarly
difficult to compute in closed form and is numerically computed. It is worth
noting that in typical cavity haloscopes, the axion does not couple to TE and
TEM modes [57, 58, 59] as explained using the form factor formalism [31]. As
is shown in this paper, the DMRadio-m3 TEM-like resonance does indeed couple
both in the quasistatic limit and at frequencies beyond where this
approximation applies.

Having established in the previous section that DMRadio-m3 has a single
tunable TEM-like mode at its operational frequencies, we now compute the scan
rate by a consideration of the mode impedance and the signal amplitude. From
[50], we write the scan rate equation in the form that we use in this study:

$\begin{split}\frac{d\nu}{dt}&=\frac{\pi(6.4\times
10^{5})}{\text{SNR}^{2}}\frac{\hbar^{2}}{16c^{8}m_{a}^{4}}\times\\\
&\qquad\frac{|V(m_{a},B,g_{a\gamma\gamma})|^{4}Q(\nu_{r})\bar{\mathcal{G}}[\nu_{r},T,\eta(\nu_{r})]}{L_{\text{eff}}(\nu_{r})^{2}}.\end{split}$
(5)

The numeric pre-factor depends on the physics of a standard halo model [49,
60], $|V(m_{a},B,g_{a\gamma\gamma})|$ is the axion-mass-dependent magnitude of
the induced voltage, which depends on the magnetic field strength and axion-
photon coupling, and $Q(\nu_{r})$ is the quality factor of the resonator at
the resonator frequency $\nu_{r}$.
$\bar{\mathcal{G}}[\nu_{r},T,\eta(\nu_{r})]$, which is evaluated at $T=0.02$ K
for DMRadio-m3, parametrizes noise physics and is discussed in $\S$VI, as well
as in Appendix F4 of [50] and in Eqs. (A12) and (A13) of [49].
$L_{\text{eff}}$ is the total effective inductance of the combined equivalent
RLC circuit, and SNR is the signal to noise ratio. We set $\text{SNR}=3$,
which corresponds to a $3\sigma$ axion signal.

In our simulations presented this section, we use the finite-element modeling
package COMSOL to numerically extract the impedance $Z_{p}(\omega)$ and the
voltage induced by the axion $V(\omega)$ across the coaxial pickup. We then
model $Z_{p}(\omega)$ at each frequency as a series RLC circuit. Subsequently,
combining this equivalent circuit with an equivalent circuit of the tuning
elements, we extract $L_{\text{eff}}(\nu_{r})$, $Q(\nu_{r})$, and
$|V(m_{a},B,g_{a\gamma\gamma})|$ for the computation of scan rate in Eq. 5.

###  IV.1 Extracting the effective RLC circuit

Using COMSOL, we determine both $V(\omega)$ and $Z_{p}(\omega)$ between 5 MHz
and 200 MHz at $0.1$ MHz frequency intervals for the coaxial pickups without
tuning elements. $V(\omega)$ is calculated by simulating the axion signal
inside the coaxial pickup. Using a cubic-spline interpolation to fit the real
(resistive) and imaginary (reactive) parts of the impedance near any given
frequency of interest $\omega_{0}$, we extract a function for the impedance,
$Z_{p}(\omega)=R_{p}(\omega)+iX_{p}(\omega)$, and the derivatives of the
resistance and reactance. As an example, Figure 3 shows $X_{p}(\omega)$
extracted for the DMRadio-m3 Coax 2, with $h_{\rm coax}=1.4$ m. A detailed
discussion of the COMSOL and FEM techniques used can be found in Appendix B.

Figure 3:  The reactance, $X_{p}(\nu)=\text{Im}(Z_{p}(\nu))$. numerically
extracted across $30-$200 MHz for the $h_{\text{coax}}=1.4$ m coaxial pickup
(green) and the corresponding voltage induced by a DFSZ axion with the local
dark matter density of $\rho_{\text{DM}}=0.45\text{ GeV/cm}^{3}$ (orange)
[32]. This coax length corresponds to Coax 2 in Table 1 and is plotted here as
an example. The impedance of the tuning elements is not included; these values
are for an open, untuned structure. The precise modeled reactance $X_{p}(\nu)$
is in qualitative agreement with the reactance of a propagating TEM mode in an
idealized uniform lossless coax shorted at one end (Eq. 2). The zeros and the
poles of the impedance function represent resonant modes of the untuned
coaxial structure. These are TEM-like modes. The resistive portion
$R_{p}(\nu)=\text{Re}(Z_{p}(\nu))$, not shown in this plot, also has peaks at
the mode frequencies. At each point along the $X_{p}(\nu)$ and $R_{g}(\nu)$
curves, we Laurent expand to extract an effective series RLC, which can be
used to determine the experimental sensitivity. The tuning component is
capacitive when $X_{p}(\nu_{r})>0$ and inductive when $X_{p}(\nu_{r})<0$. The
voltage curve, $|V(\nu)|$, drops to prohibitively low values in certain
frequency regions ($\sim$120–130 MHz in this untuned structure). This
influences the scan rate which scales as $|V(\nu)|^{4}$ and, thus, such
frequency regions are to be avoided.

Any impedance function $Z_{p}(\omega)$ can be represented as a Laurent
expansion around the frequency $\omega_{0}$, for small fractional frequency
deviation ($|\omega-\omega_{0}|\ll\omega_{0}$), and away from any divergences
(such as those caused by internal resonances). When the frequency-derivative
of the reactance is positive, this expansion can be fully described as a
series RLC circuit near $\omega_{0}$ (if the derivative of the reactance is
negative, its dual — a parallel RLC circuit — can be used). The derivative of
the reactance is always positive in DMRadio-m3, and the experiment is designed
to operate away from internal resonances. Using the impedance of a series RLC
circuit: $Z_{\text{RLC}}(\omega)=R+i\omega L-\frac{i}{\omega C}$, as well as
its derivative, the parameters of the series RLC equivalent circuit can be
determined. The two real parts of the series RLC (the resistance and its
derivative) map onto the real parts of $Z_{p}(\omega)$, and the reactive parts
have a non-trivial mapping:

$\displaystyle R(\omega_{0})=$ $\displaystyle R_{p}(\omega_{0})$ (6)
$\displaystyle\frac{dR}{d\omega}(\omega_{0})=$
$\displaystyle\frac{dR_{p}}{d\omega}(\omega_{0})$ $\displaystyle
L(\omega_{0})=$
$\displaystyle\frac{\omega_{0}\frac{dX_{p}}{d\omega}(\omega_{0})+X_{p}(\omega_{0})}{2\omega_{0}}$
$\displaystyle C(\omega_{0})=$
$\displaystyle\frac{2}{\omega_{0}\left(\omega_{0}\frac{dX_{p}}{d\omega}(\omega_{0})-X_{p}(\omega_{0})\right)}.$

We find that at operational frequencies,
$\frac{dR_{p}}{d\omega}\Delta\omega\ll R_{p}$, where $\Delta\omega$ is the
sensitivity bandwidth [50, 61], so the derivative of the equivalent resistance
$\frac{dR}{d\omega}$ can be taken as zero in the equivalent circuit. We then
replace $Z_{p}(\omega)$ in Figure 1c for $\omega$ near $\omega_{0}$ with the
series RLC equivalent-circuit impedance

$Z_{p}(\omega)\approx R(\omega_{0})+i\omega L(\omega_{0})-\frac{i}{\omega
C(\omega_{0})}.$ (7)

In order to achieve a resonance at frequency $\omega_{0}$, we introduce a
tuning element $Z_{\text{tuning}}(\omega)$ such that
$X_{\text{tuning}}(\omega_{0})=-X_{p}(\omega_{0})$. The tuning element may
have additional parasitic loss. The coupling inductor for the dc SQUID imposes
additional tuning inductance, which we include as part of $L_{\text{tuning}}$.
A similar Laurent expansion can be used for the tuning elements to extract
$L_{\text{tuning}}$, $C_{\text{tuning}}$, and $R_{\text{tuning}}$. We define
$L_{\text{eff}}=L+L_{\text{tuning}}$,
$1/C_{\text{eff}}=1/C+1/C_{\text{tuning}}$,
$R_{\text{eff}}=R+R_{\text{tuning}}$ such that this total effective circuit is
our resonant series RLC, with
$\omega_{0}=1/\sqrt{L_{\text{eff}}C_{\text{eff}}}$ and where
$L_{\text{eff}}(\nu_{r})$ enters into Equation 5.

###  IV.2 Calculating the quality factor

Once the pickup circuit is modeled as a tunable series RLC and the impedance
of the tuning and readout components are considered, the quality factor can be
determined. The quality factor is given by
$Q=\omega_{0}L_{\text{eff}}/R_{\text{eff}}$, where $L_{\text{eff}}$ and
$R_{\text{eff}}$ are described in the previous paragraph.

Since the tuning elements are superconducting but the coaxial pickups are not,
$R_{\text{eff}}$ tends to be dominated by $R_{p}$. Any dielectric losses in
the system are taken to be small as compared to $R_{\text{eff}}$. In order to
compute $R_{p}$, the COMSOL model starts by modeling the conductivity of room
temperature (RT) copper, with no magnetoresistance. To compute the higher ac
conductance achieved at cryogenic temperatures, and to take into account
magnetoresistance in the dc magnetic field, we increase the conductivity
provided by the FEM model by a factor of $6$, resulting in a factor of 6
enhancement in the $Q(\nu)$. The basis of this enhancement factor is discussed
in detail in Appendix C.

###  IV.3 Calculating the axion-induced signal voltage

We now evaluate the axion-induced voltage across the pickup gap defined by
points A and B in Figure 1b. The effective axion current can be simulated as a
source current proportional to the dc magnetic field strength and following
the flow of the magnetic field lines. By using the “external current density”
functionality in the COMSOL RF Module, we model the effective axion current
that induces an electromagnetic response in the coaxial pickup (further
details can be found in Appendix B). The magnetic field used here is
numerically modeled in separate simulations. It is important to note that when
dc magnetic field lines intersect a metal, the effective axion current also
penetrates and traverses the bulk of the metal, as implied by Eq. 1. As a
consequence of Ampère’s law within the bulk of the metal, a physical electron
current density flows in the opposite direction in the bulk of the copper,
following the magnetic flux lines. It is interesting that this physical
electron current penetrates deeper than the skin depth, as a rigorous
consequence of modified Maxwell’s equations in the presence of axions. This
phenomenon, which we refer to as bulk electron shuttling (BES), is derived and
discussed in more detail in Appendix D. We find that the COMSOL external
current density function accurately models the effective axion current,
including bulk electron shuttling. Thus, COMSOL can be used to extract the
axion-induced voltage across the pickup. The result of one such voltage
simulation is shown in Figure 3. This result for
$|V(m_{a},B,g_{a\gamma\gamma})|$ can then be applied to Equation 5.

##  V Reactive Tuning Considerations

As discussed in previous sections, the tuning element needed at a given
frequency may either be capacitive or inductive, depending on the sign of the
coaxial pickup reactance, $X_{p}(\omega)$, at that frequency. Since the value
of $X_{p}(\omega)$ may vary significantly over a small frequency range, care
must be taken in selecting both the coaxial pickup and the tuning elements to
minimize the number of times these elements must be swapped out. The tuning
element design is further constrained by the requirement that the tuning
elements must fit in the shielded box of the coaxial pickup.

The tuning elements and the connections between the pickup, tuner, and
amplifier all present some amount of inductance and capacitance that do not
contribute to the sensitivity of the experiment. These parasitic reactances
are carefully controlled so that they do not significantly degrade the
experiment’s sensitivity. A detailed description of the tuning mechanisms and
their corresponding parasitics will be presented in a future work.

##  VI SQUID amplifiers in DMRadio-m3

Finally, we consider the noise behavior of the dc SQUID amplifiers, which
determine the parameter $\eta$ in $\bar{\mathcal{G}}[\nu_{r},T,\eta(\nu_{r})]$
of Equation 5. Further details on the dc SQUIDs used by DMRadio-m3 can be
found in [62].

DMRadio-m3 uses a SQUID-amplifier chain consisting of a variable transformer,
a voltage-biased first-stage single SQUID, and a second-stage SQUID array with
$50\ \Omega$ output impedance that couples to a room-temperature preamplifier
with $50\ \Omega$ input impedance. The first-stage SQUID design is the same at
all frequencies. However, three different versions of the second-stage SQUID
array are required for different frequency ranges.

The input coil of the first stage dc SQUID is wired in series between the
output of the coaxial pickup and the tuning reactance (through a transformer).
At all operational frequencies, the inductance and capacitance of the full
SQUID array (both coupled and parasitic) will be small as compared to the
inductance of the coaxial pickup [62]. The coupling to the dc SQUID is also
small enough that damping (i.e. resistance) from both active and passive loss
in the SQUID is subdominant to the loss in the electrons in the anomalous skin
depth of the coaxial pickup [62].

Following the conventions in [50], the SQUID noise is described by a parameter
$\eta$ as the ratio of the noise temperature to a half-photon of quantum
noise:

$\eta(\omega_{0})\equiv\frac{kT_{N}^{min}(\omega_{0})}{\hbar\omega_{0}/2}\geq
1.$ (8)

An amplifier capable of operating at the Standard Quantum Limit (SQL) at
optimal noise match would have $\eta=1$, corresponding to a minimum of a half
photon of added noise on resonance. Ultimately, the quantity that enters the
scan rate is $\bar{\mathcal{G}}[\nu_{r},T,\eta(\nu_{r})]$, which encodes all
the noise physics of the system (described in more detail in [50, 49]).

Details of the SQUID amplifiers are presented in Table 2 and are described in
more detail in [62]. The highest bandwidth SQUID amplifier has a value of
$\eta=24$ and lower values of $\eta$ are achieved for lower bandwidth SQUID
amplifiers.

| Low BW | Medium BW | High BW
---|---|---|---
| # of SQUIDs
---
in Second Stage
144 ($48\times 3)$ | 64 $(32\times 2)$ | 16 $(16\times 1)$
$R_{\text{dyn}}$ | 50 $\Omega$ | 50 $\Omega$ | 50 $\Omega$
Bandwidth | 52 MHz | 110 MHz | 330 MHz
Freq. range | 5–30 MHz | 30–55 MHz | 55–200 MHz
$\eta$ | 9 | 12 | 24
Table 2: Parameters of the three different two-stage SQUID amplifier chains
used in DMRadio-m3: Low, Medium, and High bandwidth (BW). The first stage
consists of a single SQUID. The second stage consists of an array of SQUIDs
using one common SQUID design. Details and experimentally measured parameters
are in [62]. The input coils of all SQUIDs in the second stage are wired in
series. The output of the second stage has $N_{\rm ser}$ SQUIDs in series, and
$N_{\rm par}$ SQUIDs in parallel, for a total number of $N_{\rm 2}$ SQUIDs,
denoted in the table as $N_{\rm 2}(N_{\rm ser}\times N_{\rm par})$. The output
dynamic resistance $R_{\text{dyn}}$ of the second-stage SQUID is listed at the
bias point, and is matched to a $50\ \Omega$ coaxial line and room-temperature
preamplifier input impedance. The bandwidth is the 3 dB rolloff of each SQUID
chain, which is limited by the coupling between the first- and second SQUID
stages. The frequency range specifies which of the three SQUID chains is used
at each frequency. The last row contains values for $\eta$ (defined in
Equation 8), the total noise of the two-stage SQUID, including the referred
room-temperature preamplifier noise. It incorporates imprecision and
backaction noise, and their correlations. Frequency range and $\eta$, the last
two rows, are the only parameters needed for this analysis.

##  VII Science Reach

Figure 4: (a) The baseline scan rate for DMRadio-m3 across the primary and
secondary science goals using the set of six coaxial pickups. The primary
science goal is to search for axions at DFSZ coupling at 30–200 MHz; the
secondary goal is to search for KSVZ axions at 10–30 MHz. Above 55 MHz, the
High BW SQUID amplifier with SQUID noise parameter $\eta=24$ is used. Both the
conservative $\eta=20$ and target $\eta=12$ cases are shown in the 30-55 MHz
range. These scan rates all assume the constant $6\times$ increase in $Q$ due
to the cryogenic increase in the conductivity of copper. (b) The sensitivity
of DMRadio-m3 for the primary and secondary science goals, as well as for the
extended goal which is defined as axions with a coupling of
$g_{a\gamma\gamma}=g_{a\gamma\gamma,\text{DFSZ}}(30\text{ MHz})=1.87\times
10^{-17}\text{ GeV}^{-1}$ over 5–30 MHz. The two orange lines show the axion-
photon coupling of KSVZ (top line) and DFSZ (bottom line) axions. The coaxes
used for each region are also shown.

The primary science goal of DMRadio-m3 is to search for DFSZ axions across the
30–200 MHz region. Using the results of the previous sections, we first
calculate the scan rate across this frequency range. As indicated in Figure 4,
the full scan uses six different coaxes with their corresponding tuning
mechanisms.

| Sensitivity & Range |  |  $3\sigma$ Live Scan Time:
---
Baseline (Scaled)
| Primary
---
Science Goal
DFSZ; 30–200 MHz | 3.7 yr (3.0 yr)
| Secondary
---
Science Goal
KSVZ; 10–30 MHz | 0.9 yr (0.6 yr)
| Extended
---
Goal
|  $1.87\times 10^{-17}\text{ GeV}^{-1}$;
---
5–30 MHz
2.6 yr (1.6 yr)
Table 3: The calculated experimental $3\sigma$ live times for the primary and
secondary science goals as well as the extended goal of DMRadio-m3. The
extended goal live time is the additional time required to achieve this goal
assuming the secondary science goal has already been achieved. For the
baseline live scan time, this table assumes SQUID amplifiers with $\eta=9$ for
5–30 MHz, $\eta=20$ for 30–55 MHz, and $\eta=24$ for 55–200 MHz, and the flat
factor of $6\times$ increase in $Q(\nu)$ as discussed in §IV.2. Using the less
conservative SQUID noise of $\eta=12$ for 30–55 MHz, and the enhancement
factor described in Appendix C, which takes into account the scaling of
$Q(\nu)\sim\nu^{-1/6}$ and magnetoresistance effects, we calculate the more
aggressive live times shown in parenthesis as “Scaled.”

For the primary science goal, we calculate a “3$\sigma$ live time,” which we
define as the live time required so that a DFSZ axion is expected to present a
3$\sigma$ signal (SNR=3 in Eq. 5). This live time does not include any time
spent rescanning. The ADMX experiment determines rescan requirements with
frequentist inference [63], and spends approximately 30% of its live time on
rescans [64]. The HAYSTAC experiment uses a Bayesian analysis framework to
plan rescans and determine its final sensitivity [65]. A detailed rescan
strategy and analysis framework for DMRadio-m3 will be the subject of a future
paper.

The SQUID amplifier is exchanged once in the primary science range at 55 MHz.
In the 30–55 MHz range, the Medium BW SQUID amplifier in Table 2 is used.
Either $\eta=12$ is assumed (as shown in the table), or that same SQUID
amplifier with a conservative assumption of $\eta=20$ is substituted (scan
times for both scenarios are shown in Figure 4a). In the 55–200 MHz range, the
High BW SQUID amplifier with $\eta=24$ is used. Furthermore, across the entire
frequency range we consider the two cases for the enhanced quality factor at
cryogenic temperatures as discussed in Appendix C: a conservative estimate of
a $6\times$ increase of the RT conductivity of copper across all frequencies
and a more ambitious increase due to the frequency dependent effects on the
resistivity.

For the primary science goal of DMRadio-m3, using the scan rates shown in
Figure 4a, the $3\sigma$ live scan time for the conservative $\eta=20$ SQUID
assumption below 55 MHz is 3.7 years. Using the more ambitious Medium
bandwidth SQUID amplifier assumption of $\eta=12$ for 30–55 MHz, followed by
$\eta=24$ for 55–200 MHz, as well as the frequency-dependent resistivity for
copper across the entire frequency range, the resulting $3\sigma$ live time is
3.0 years. All relevant $3\sigma$ live times are shown in Table 3.

The secondary science goal of DMRadio-m3 searches for KSVZ axions in the 10–30
MHz range. Using the Low BW SQUID amplifier across this range, the total
$3\sigma$ live scan time for a KSVZ signal is 0.9 years, with the
corresponding scan rate also shown in Figure 4a. The total primary and
secondary science goal is presented in Figure 4b.

We also present an extended goal that searches for axions in the 5–30 MHz
range down to a constant axion-photon coupling of
$g_{a\gamma\gamma,\text{DFSZ}}(\nu=30\text{ MHz})=1.87\times 10^{-17}\text{
GeV}^{-1}$, as shown in 4b. This represents a more ambitious scan plan which
excludes more parameter space but at a longer live scan time. Again, using the
Low BW SQUID amplifier, and taking into account the scan time already allotted
to the secondary science goal, the extended goal requires an additional
baseline $3\sigma$ live scan time of 2.6 years for a
$g_{a\gamma\gamma,\text{DFSZ}}(\nu=30\text{ MHz})=1.87\times 10^{-17}\text{
GeV}^{-1}$ axion. This $3\sigma$ live time assumes that the secondary science
goal parameter space has already been covered and that that data can be used
in the extended goal scan. In all cases, the time for rescans is not included.

##  VIII Conclusion

In this work, we have presented a model of the electromagnetic performance of
DMRadio-m3. Because the Compton wavelength scanned by this experiment
approaches twice the detector size, we have applied a full numerical model of
the modified Maxwell’s equation response of the DMRadio-m3 detector, which is
valid over the full frequency range. We have extracted the impedance of the
structure, modeled it as a different series RLC circuit at each frequency of
interest, and have determined the voltage signal $V(\nu)$ induced by the
axions. Using the noise properties of the SQUID readout amplifiers [62], we
have determined the live scan times.

DMRadio-m3 operates over a significant range of well motivated QCD axion
parameter space. We have determined (see Table 3) the $3\sigma$ live time for
DFSZ axions in the 30–200 MHz range (3.0–3.7 years) and KSVZ axions in the
10-30 MHz range (0.6–0.9 years), with an extended goal to further to search
for axions at $g_{a\gamma\gamma}=1.87\times 10^{-17}\text{ GeV}^{-1}$ over
5–30 MHz.

###### 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. Stanford University and UC Berkeley gratefully acknowledge support
from the Gordon and Betty Moore Foundation, grant number 7941, and additional
support from the Heising-Simons Foundation. C. Bartram acknowledges support
from the Panofsky Fellowship at the SLAC National Accelerator Laboratory. S.
Chaudhuri acknowledges support from the R. H. Dicke Postdoctoral Fellowship
and Dave Wilkinson Fund at Princeton University. J. W. Foster was supported by
a Pappalardo Fellowship. J. T. Fry is supported by the National Science
Foundation Graduate Research Fellowship under Grant No. 2141064. P. W. Graham
acknowledges support from the Simons Investigator Award no. 824870 and the
Gordon and Betty Moore Foundation Grant no. 7946. Y. Kahn was supported in
part by DOE grant DE-SC0015655. B. R. Safdi and J. N. Benabou were supported
in part by the DOE Early Career Grant DESC0019225. C. P. Salemi is supported
by the Kavli Institute for Particle Astrophysics and Cosmology Porat
Fellowship.

##  Appendix A Shielded box considerations

As shown in Fig. 1a, the shielded box, which is shaped like a cylindrical
pillbox and houses the tuning elements and readout electronics, is located in
the magnet’s low-field region directly above the straight coaxial section.
This box shields the sensitive SQUIDs from the large dc magnetic field and
from external interference. Although the size of the low-field region is set
by the magnet and bucking coils, the dimensions of this shielded box are free
to vary in the optimization of the detector geometry. We show here that
variations in the dimensions of this shielded box around the design value have
a negligible effect on the pickup’s response to an axion signal.

Figure 5: Reactive (teal) and resistive (red) curves for a coaxial pickup of
height 1.4 m with shielded box sizes of varying radius $r$ and height $h$. The
curves, with the exception of a few lines above 175 MHz, are nearly identical
over 300 MHz thus showing that the effects of the box are not significant for
DMRadio-m3.

Because the shielded box is a right circular cylinder, its interior supports
both TE and TM modes. At large box sizes, these modes begin to intrude into
the DMRadio-m3 science frequencies. For small box sizes, capacitance between
the walls of the box and the inner conductor of the coax shorts out signal
currents. To study the effect of box sizes on signal response, simulations
were run in COMSOL with varying box dimensions. The result of four of these
simulations is shown in Fig. 5. The effect of the box on the coax’s impedance
is negligible except at a few specific frequencies which represent TE or TM
modes in the two largest boxes studied ($r=0.64~{}\text{m},h=1.4~{}\text{m}$
and $r=1.28~{}\text{m},h=2.8~{}\text{m}$). These RF modes are found slightly
higher ($\sim 1$ MHz) than the frequencies predicted by analytical formulas
for right circular cylinders due to coupling to the impedance of the coax.
Shielded boxes smaller than $r=0.17~{}\text{m}\text{ by }h=0.35~{}\text{m}$
were not studied since they are too small to contain our electronics. The
selected dimensions of the shielded box insure that none of these modes
intrude into signal frequencies. Furthermore, the shielded box does not add
significant capacitive shunting across the coax.

##  Appendix B COMSOL simulation techniques

In this appendix, we describe the procedure used to characterize the coaxial
pickup’s response to an applied axion current. With the RF Module in COMSOL
6.0, we make a 2D axisymmetric model of the detector geometry and define a
coaxial lumped port with cable terminal across the open end of the model
(points A and B in Fig. 1). The width of the lumped port is much smaller than
the characteristic wavelength of the highest frequency being simulated
($f=200~{}\text{MHz}$, corresponding to $\lambda\approx 150~{}\text{cm}$),
allowing for voltage, $V(\omega)$ and current, $I(\omega)$, to be well-defined
across the port for all frequencies. The characteristic impedance of this
lumped port in the simulation is set to $Z_{\text{ref}}=50\times
10^{9}~{}\Omega$ to ensure that the port impedance has a minimal effect on the
measured impedance of the coaxial pickup. This value is chosen such that the
port acts as a current source. An alternative simulation using a circuit
terminal for this lumped port eliminates the need to define a characteristic
port impedance and gives identical results.

Making use of the Impedance Boundary Condition feature of COMSOL, we define
the boundaries of the coaxial pickup to have the surface impedance of metallic
copper at room temperature. This feature allows us to calculate loss (and
therefore the real part of $Z_{p}(\omega$)) without solving the
electromagnetic equations within a skin depth inside the material, therefore
reducing computation time. As described elsewhere, the modeled loss is later
adjusted to account for higher conductivity at low temperatures, and the
effect of magnetoresistance.

The impedance of the coaxial pickup, $Z_{p}(\omega)$, can then be identified
by exciting an ac voltage across the lumped port in a frequency domain study.
$Z_{p}(\omega)$ is read out as the impedance across the lumped port, referred
to as emw.Zport_1 on COMSOL.

The voltage induced by the axion signal across the open end of the coaxial
pickup, $V(\omega)$, is found by turning off the wave excitation at the lumped
port and introducing an external current density excitation throughout the
entire volume of the coaxial pickup. The external current density in COMSOL
provides a phase-coherent electric current, representing the excitation due to
an axion. We define an external current density by:

$\mathbf{J_{e}}(t,\rho,\phi,z)=1\text{
A/m}^{2}\left(\frac{\mathbf{B}(\rho,\phi,z)}{1\text{ T}}\right)e^{i\omega t},$
(9)

where $\mathbf{B}$ is the vector field of the magnetic field, which adjusts
the effective axion current density for the strength of the dc magnetic field.
We then apply an additional scaling factor that depends on the coupling
strength and dark matter model.

Due to azimuthal symmetry, this field only has components along the
$\hat{\rho}$ and $\hat{z}$ directions. $B_{\rho}$ and $B_{z}$ are found
numerically by modeling the coils of the solenoidal magnet in a separate
COMSOL simulation, using the Magnetic Fields interface within the ac/dc
Module. Importing the resulting $r$ and $z$ components of the applied magnetic
field, which are defined on a grid, and then defining an interpolation for
each of the two components, we define $\mathbf{J_{e}}$ in terms of $B_{\rho}$
and $B_{z}$ as above in Eq. 9.

To extract the axion-induced voltage we use a frequency-domain simulation with
the external current density acting as the ac source. The voltage is extracted
by integrating the radial electric field along a line across the gap (points A
and B in Figure 1):

$V_{p}(\omega)=\int_{A}^{B}E_{r}dr,$ (10)

where $r$ represents the radial component in the 2D axisymmetric situation.
The integration path is always much shorter than a wavelength, which ensures
that Eq. 10 is valid.

##  Appendix C Copper coax surface resistance

The statistical noise limit of the experiment improves with the quality
factor, so it is important to accurately calculate loss and thus resonator
$Q$.

The surface resistance of copper at room temperature is given by

$Re\left\\{Z_{\rm RT}\right\\}=\sqrt{\frac{\omega\mu_{0}}{2\sigma_{\rm RT}}},$
(11)

where $\sigma_{\rm RT}$ is the room-temperature conductance [66].

In our COMSOL models, we assume standard room-temperature, high-conductivity
copper, and then increase the modeled $Q$ by a factor $Q_{\rm rat}(\omega)$,
which is the ratio of surface resistance of copper at room temperature to its
cryogenic value at angular frequency $\omega$.

At temperatures well below 4 K, the dc resistivity decreases by a large
factor. A dc Residual Resistivity Ratio (RRR) of above 400 can be achieved for
copper. However, at higher frequencies and $T<4$ K, the resonator loss
decreases by a much smaller factor due to the anomalous skin-effect [67]. In
this extreme anomalous limit, the surface resistance of the conductor is given
by

$Re\left\\{Z_{\rm
cold}\right\\}=\frac{8}{9}\left(\frac{\sqrt{3}\lambda_{\text{mfp}}\omega^{2}\mu_{0}^{2}}{16\pi\sigma_{\rm
cold}}\right)^{1/3},$ (12)

where $\sigma_{\rm cold}$ is dc conductivity at low temperatures, and
$\lambda_{\text{mfp}}$ is the mean free path of the electrons.

The ratio of room-temperature surface resistance (Eq. 11) to extreme anomalous
low-temperature surface resistance (Eq. 12), and thus the ratio of the
achieved $Q$ between room and cryogenic temperatures, is

$Q_{\rm rat}(\omega)\equiv\frac{Re\left\\{Z_{\rm
RT}\right\\}}{Re\left\\{Z_{\rm cold}\right\\}}\propto\nu^{-1/6}.$ (13)

The scaling of the ratio of loss $Q_{\rm rat}$ with frequency is found to hold
true over a large range of frequency, with an example achieved value of
$Q_{\rm rat}=4.8$ at 2.85 GHz [68]. This study of resonator $Q$ is consistent
with extensive experimental measurements conducted in cryogenic copper
cavities [69, 34] and analysis of cryogenic copper RF conductance [70]. Using
the $\nu^{-1/6}$ scaling with the reference value of $Q_{\rm rat}=4.8$ at 2.85
GHz, the predicted value of $Q_{\rm rat}$ is 13.8 at 5 MHz, 10.3 at 30 MHz,
and 7.5 at 200 MHz. The HAYSTAC experiment has achieved a $Q_{\text{rat}}$ of
approximately 4 at 1 GHz [71] for its copper cavity and the CAPP experiment
has demonstrated a $Q_{\text{rat}}$ between 2 and 3 for frequencies of 4.7-5.0
GHz [72] for its copper cavity. Using the HAYSTAC and CAPP numbers as the
reference values for the $\nu^{-1/6}$ scaling yields $Q_{\text{rat}}$ values
between 5 and 7 at 30 MHz.

Magnetoresistance can also change the value of $Q$ achieved in DMRadio-m3. The
magnetoresistance is dominated by the longitudinal (axial) component of the dc
magnetic field in the copper coax endcaps, which is orthogonal to the
direction of current flow, and to a lesser extent by the transverse (radial)
component in the cylindrical walls. In our case, the average transverse field
contributing to magnetoresistance is $\sim 1$ T. The low-temperature dc
magnetoresistance of OFHC copper with $\text{RRR}=100$ at 1 T is
$\Delta\rho/\rho\approx 10$–30% [73, 74], consistent with Kohler’s rule [75].
For the ac magnetoresistance, in the DMRadio-m3 frequency range at $B=1$ T,
the cyclotron radius is of the same order as the skin depth and mean free
path, and the ac magnetoresistance is expected to be significantly lower than
the dc value of $\sim 10$–30% [76, 77]. It should be noted that the HAYSTAC
and CAPP experiments use stronger magnetic fields, and should have somewhat
different magnetoresistance.

In the main text, we extract the main results assuming a conservative baseline
constant value of $Q_{\text{rat}}=6$ across the frequency range. However, in
Table 3 we include the results assuming the scaling in Eq. 13 with a reference
value of 4.8 at 2.85 GHz [68] as well as a 20% decrease in $Q$ due to the
effects of magnetoresistance. These combine to adjust the $Q$ by a factor of
11.1 at 5 MHz, 9.9 at 10 MHz, 8.2 at 30 MHz, and 6.0 at 200 MHz which, in
turn, lead to a decrease in live scan time (shown as the scaled values in
Table 3). These values for $Q(\nu)$ are shown in Figure 6. They include the
inductive or capacitive contributions from the tuning elements and are shown
for three cases: (a) the room temperature conductivity of copper as modeled by
COMSOL, (b) using the constant conservative $6\times$ increase which provides
the baseline scan time in Table 3, and (c) for the $\nu^{-1/6}$-dependent
scaling which provides the scaled scan time in Table 3. The loss is assumed to
be dominated by the normal electrons in the copper.

Figure 6: The total quality factor $Q(\nu)$ for the DMRadio-m3 experiment.
This quality factor combines the contributions from the coaxial copper pickups
as well as from capacitive or inductive tuning elements. We assume that the
loss in this system is dominated by the normal electrons in the copper of the
coaxial pickup. Three curves for $Q(\nu)$ are shown. In dot-dashed green we
show the quality factor using the room temperature conductivity of copper. In
dashed purple, we show the conservative $6\times$ increase in the quality
factor at cryogenic temperatures. In orange we show the frequency dependent
$\nu^{-1/6}$ scaling.

##  Appendix D Bulk Electron Shuttling

In section IV, we mention an effect arising from the axion-modified form of
Maxwell’s equations that leads to current nodes shown schematically in the
coax in Figure 1b. We refer to this phenomenon as bulk electron shuttling
(BES). In this appendix, we derive this phenomenon in the bulk of a conductor
many skin depths thick, with electrical conductivity $\sigma$. This is a
relevant limit to many axion experiments, including DMRadio-m3, in which the
dc magnetic field, and thus the axion signal, penetrates bulk conductors.

The axion-modified form of Ampère’s law inside of an electrical conductor is
given by:

$\nabla\times\mathbf{B}=\mu_{0}\epsilon_{0}\frac{\partial\mathbf{E}}{\partial
t}+\mu_{0}\sigma\mathbf{E}+\mu_{0}\mathbf{J}_{\text{eff}},$ (14)

where $\mathbf{J}_{\text{eff}}$ represents the effective axion-induced
electrical current presented in Equation 1. We have used Ohm’s law to replace
the electron current density in the conductor:

$\mathbf{J}_{e}=\sigma\mathbf{E},$ (15)

and we assume that the electrical conductivity is uniform. By taking the curl
of Equation 14 and using the Maxwell-Faraday equation:

$\nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t},$ (16)

we derive the axion-modified wave equation for the electric field:

$\nabla^{2}\mathbf{E}=\mu_{0}\epsilon_{0}\frac{\partial^{2}\mathbf{E}}{\partial
t^{2}}+\mu_{0}\sigma\frac{\partial\mathbf{E}}{\partial
t}+\mu_{0}\frac{\partial\mathbf{J}_{\text{eff}}}{\partial t}.$ (17)

Examining this equation in the Fourier domain and converting to natural units
where $c=\mu_{0}=1$, we establish a dispersion relation:

$\left(-k^{2}+\omega^{2}-i\omega\sigma\right)\mathbf{\tilde{E}}=i\omega\tilde{\mathbf{J}}_{\text{eff}},$
(18)

where $\tilde{\mathbf{E}}$ and $\tilde{\mathbf{J}}_{\text{eff}}$ are the
Fourier transforms of $\mathbf{E}$ and $\mathbf{J}_{\text{eff}}$ respectively.
We consider which terms are dominant in the operating conditions of
DMRadio-m3. In a good electrical conductor such as copper, a conductivity of
$\mathcal{O}(10^{7}\text{ S/m})$ corresponds to a frequency scale in natural
units of $\mathcal{O}(10^{18}\text{ Hz})$ and a characteristic wavelength of
$\mathcal{O}(100\text{ pm})$. For electric field oscillation frequencies less
than $10^{18}$ Hz and spatial variations in the field larger than hundreds of
pm, the left hand side of Eq. 18 is dominated by the $i\omega\sigma$ term thus
resembling a heavily over-damped harmonic oscillator. In this limit, 18
reduces to:

$\tilde{\mathbf{E}}=-\frac{1}{\sigma}\tilde{\mathbf{J}}_{\text{eff}}.$ (19)

Using Ohm’s law (Eq. 15), we identify the BES current density, which is equal
and opposite to the axion-induced effective current density:

$\tilde{\mathbf{J}}_{e}=-\tilde{\mathbf{J}}_{\text{eff}}.$ (20)

This formula holds only in the regimes where $\sigma\gg\omega$ and also when
$\sigma$ is much larger than the inverse of the length scales associated with
the variation in the fields ($\sigma\omega\gg k^{2}$). While this derivation
has assumed that $\sigma$ is uniform across the bulk, the results derived hold
for a spatially varying $\sigma(\mathbf{x})$, as long as the variation occurs
on length scales larger than the skin depth of the metal (tens of $\mu$m for
copper at the frequencies of interest); this is evident considering that in
natural units the skin depth $\delta$ is given by
$\delta=\sqrt{2/(\sigma\omega)}$ and hence the corresponding wavenumber $k$ is
$k\sim\sqrt{\sigma\omega}$. It is also worth noting that Ohm’s Law breaks down
when the mean-free-path of the electrons becomes larger than the wavelength of
the signal; for a metal like copper this occurs around $10^{15}$ Hz. Hence,
Eq. 20 only holds for low frequencies and fields with sufficiently high
spatial uniformity (varying over length scales large as compared to the skin
depth).

The axion effective current generates an ac BES current of equal magnitude and
opposite direction in the bulk of the copper pickup of DMRadio-m3, moving
charges across the bulk of the metal, even at depths much larger than the skin
depth. This phenomenon is required to understand the current nodes in Figure
1, since some screening current is “shuttled” through the bulk of the copper,
in accordance with Eq. 20. It is also interesting to note that although these
ac currents are moved across the bulk, electromagnetic interferencei is still
shielded by the skin depth of the copper. This is because the ac BES current
passing through the bulk deeper than the skin depth, according to Eq. 20, is
dependent only on the effective axion current, and is independent of any
external electromagnetic excitation (it is a stiff current transfer). This
analysis can be extended to BES in a superconductor that is penetrated by dc
magnetic flux.

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