$\mathsection$$\mathsection$footnotetext: these authors are co-first authors
who contributed equally to this work.

# Sensitivity of ultralight axion dark matter search with optical quantum
sensors

Young Jin Kim§<EMAIL_ADDRESS>Leanne Duffy§<EMAIL_ADDRESS>Igor Savukov§
<EMAIL_ADDRESS>Ping-Han Chu  Los Alamos National Laboratory, P.O. Box
1663, Los Alamos, New Mexico 87545, USA

###### Abstract

An optical quantum sensor (OQS) based on lasers and alkali-metal atoms is a
sensitive ambient-temperature magnetometer that can be used in axion dark
matter search with an inductor-capacitor (LC) circuit at kHz and MHz
frequencies. We have previously investigated the sensitivity of an LC circuit-
OQS axion detector to ultralight axion dark matter that could be achieved
using a fT-noise OQS constructed in our lab. In this paper, we investigate the
sensitivity that could be potentially reached by an OQS performing close to
the fundamental quantum noise levels of 10 aT/$\sqrt{\text{Hz}}$. To take
advantage of the quantum-limited OQS, the LC circuit has to be made of a
superconductor and cooled to low temperature of a few K. After considering the
intrinsic noise of the advanced axion detector and characterizing possible
background noises, we estimate that such an experiment could probe benchmark
QCD axion models in an unexplored mass range near 10 neV. Reaching such a high
sensitivity is a difficult task, so we have conducted some preliminary
experiments with a large-bore magnet and a prototype axion detector consisting
of a room-temperature LC circuit and a commercial OQS unit. This paper
describes the prototype experiment and its projected sensitivity to axions in
detail.

††preprint: PRD

##  I Introduction

Several mysteries in particle and astrophysics suggest that there are new
particles yet to be discovered. One of them is an elusive cosmic substance,
six times more abundant than the ordinary matter in the Universe, known as
dark matter Bahcall (2015). Another, seemingly unrelated mystery, is the fact
that the strong nuclear interactions, described by quantum chromodynamics
(QCD), are invariant under time-reversal with $10^{-10}$ precision or better.
The QCD axion, a hypothetical particle first proposed in the 1970s Peccei and
Quinn (1977a, b); Wilczek (1978); Weinberg (1978), is an excellent candidate
for the Universe’s dark matter Duffy and van Bibber (2009): if lighter than
$\sim$meV in mass, it can be produced with the correct abundance and
temperature in the early Universe to account for dark matter. Furthermore, it
provides a dynamical mechanism to suppress time-reversal asymmetries in QCD
Peccei and Quinn (1977a, b); Wilczek (1978); Weinberg (1978).

The target of many experimental designs for axion direct detection is the
extremely small signal induced by the axion’s weak coupling to
electromagnetism Du  _et al._ (2018); Sikivie  _et al._ (2014); Salemi  _et
al._ (2021); Gramolin  _et al._ (2021). To date, the benchmark QCD axion
models of Dine-Fischler-Srednicki-Zhitnitsky (DFSZ) Dine  _et al._ (1981);
Zhitnitskii (1980) and Kim-Shifman-Vainshtein-Sakharov (KSVZ) Kim (1979);
Shifman  _et al._ (1980) have both been probed in masses above 2.66 $\mu$eV by
the Axion Dark Matter eXperiment (ADMX) Du  _et al._ (2018) using a resonant
cavity haloscope Sikivie (1983, 1987). The haloscope relies on the interaction
of the dark matter axion field with a strong static magnetic field generated
from a superconducting magnet. The smallest axion mass of $\sim\mu$eV that
ADMX can search is limited by the physical size of the resonant cavity ($\sim
1$ m), and thus the magnet bore in which it must fit: the cavity size should
be comparable to the axion Compton wavelength, for example, $\sim 1$ m and
$\sim 1$ km for axion mass of $\mu$eV and neV, respectively.

Axions with mass around 10 neV are predicted by Grand Unified Theory (GUT)
models of particle physics Langacker (2012). Due to the physical lower mass
bound, cavity haloscopes cannot be used to search for these ultralight axions.
A recent proposal to extend the use of cavity searches in ADMX is the use of
reentrant cavities to reach lower masses, down to around 0.4 $\mu$eV
Chakrabarty  _et al._ (2023). At axion masses below this, a different
experimental approach is required. As a part of that, an axion detector using
an inductor-capacitor (LC) circuit coupled to a superconducting quantum
interference device (SQUID), as a senstive magnetometer, was first proposed by
Sikivie, Sullivan, and Tanner Sikivie  _et al._ (2014). We have previously
investigated using an optical quantum sensor (OQS) as a sensitive magnetometer
Chu  _et al._ (2018) and have developed an experiment prototype based on a
commercial OQS unit. During this prototype development, we realized that the
detector sensitivity can be potentially improved with a quantum-limited OQS,
with a field noise floor of the order of 10 aT/$\sqrt{\text{Hz}}$. In this
paper, we study the sensitivity of an advanced LC circuit-OQS experiment
designed to take full advantage of the noise of a quantum-limited OQS.

This study is organized as follows. In Section II, we discuss the magnetic
signature of dark matter axions in an LC circuit-OQS axion detector. In
Section III, we discuss noise sources in axion detection experiments. We
investigate the intrinsic noise of the axion detector arising from the
intrinsic noise of the OQS and the thermal Johnson noise of the LC circuit.
For the improved axion detector sensitivity with a quantum-limited OQS, the LC
circuit must be cooled to low temperature of a few K; otherwise the thermal
noise of the circuit at room temperature dominates. In this case, other
background noises could become significant. Therefore, we also investigate
possible background noises, including the backaction noise of the OQS on the
LC circuit and the thermal noise of surrounding magnetic shield. In Section
IV, we describe development of a prototype experiment with a room temperature
LC circuit and a commercial fT-noise OQS. The sensitivity of this prototype
setup to axions is discussed in Section V. Potential improvements to take
advantage of a quantum-limited OQS and the advanced axion detector’s
sensitivity that can be achieved are discussed in Section VI. We conclude our
discussion in Section VII.

##  II Axion Signal and Detection Scheme

Dark matter axions behave like a classically field oscillating at the axion
Compton frequency, $\omega_{a}=m_{a}$ with $m_{a}$ being the axion mass, and
permeating the entire Universe Battaglieri  _et al._ (2017). Thus the axion
field can be written as $a(t)=a_{0}\sin{(m_{a}t)}$, where $a_{0}$ is the local
amplitude of the axion field. Here natural units $c=\hbar=\mu_{0}=1$ are used.
The existence of axions in the presence of a static magnetic field
$\vec{B_{0}}$ gives rise to additional terms in the Maxwell equations Sikivie
_et al._ (2014). In particular, Ampere’s Law is modified to

$\displaystyle\vec{\nabla}\times\vec{B_{0}}-\frac{\partial\vec{E}}{\partial{t}}=$
$\displaystyle g\Big{(}\vec{E}\times\vec{\nabla}a-\vec{B_{0}}\frac{\partial
a}{\partial t}\Big{)}+\vec{j}_{e},$ (1)

where $g$ is the coupling constant of the axion to two photons SRE (1985) and
$\vec{j}_{e}$ is the electrical current density associated with ordinary
matter. It follows that the homogeneous axion field ($\vec{\nabla}a\approx 0$)
can induce an electrical current density along $\vec{B}_{0}$,

$\displaystyle\vec{j}_{a}(t)=-g\vec{B}_{0}[da(t)/dt]=-g\vec{B}_{0}\sqrt{2\rho_{DM}}\cos{(m_{a}t)},$
(2)

with the relation of $a_{0}=\sqrt{2\rho_{DM}}/m_{a}$, where $\rho_{DM}\approx
0.3~{}\text{GeV/cm}^{3}$ is the standard local dark matter density Patrignani
(2016). In turn, $\vec{j}_{a}(t)$ can produce a minute perpendicular magnetic
field $\vec{B}_{a}(t)$ oscillating at an angular frequency $m_{a}$ through
$\vec{\nabla}\times\vec{B}_{a}=\vec{j}_{a}$. This led Sikivie, Sullivan, and
Tanner to propose axion detection based on the LC circuit and SQUIDs Chu  _et
al._ (2018). Here we propose an LC circuit design with a gradiometer coil and
a quantum-limited OQS for detection of the axion-induced oscillating magnetic
field $\vec{B}_{a}$.

Figure 1: Sketch of the LC circuit-OQS axion detector showing electrical
connections of the LC circuit and geometrical arrangement of the OQS and
magnetic shield (not to scale).

A sketch of the axion detection concept with an OQS is shown in Fig. 1. The
axion detector is comprised of two components. The first component is the
first-order planar gradiometer input coil and a two-loop circular output coil,
the two coils connected in series with capacitors (LC circuit) to resonantly
amplify the $\vec{B}_{a}$. The second component is an OQS to detect the
amplified magnetic field $\vec{B}_{d}$.

The OQS manipulates atomic spins for sensitive magnetic sensing based on
lasers, alkali-metal vapor cells, and optical components Budker and Romalis
(2007), and operates at ambient temperatures without the need for cryogens.
Cryogen-free operation of the OQS has many advantages for various
applications, and for the axion search the main advantage is convenience for
OQS replacement and optimization. The basic principle for a typical
implementation of OQSs is shown in Fig. 2. Typically, two laser beams are
used: one circularly polarized pump beam to optically polarize the spins of
unpaired electrons of alkali-metal atoms, such as rubidium (Rb) or potassium
(K), and one linearly polarized probe beam to read out the state of the
electron spins. The wavelengths of the laser beams are tuned to or near an
atomic transition between the ground state and an excited state, most often
the $P_{1/2}$ state. The laser beams are sent to overlap in an alkali-metal
vapor cell heated to elevate an alkali-metal atom density, e.g., $\sim
10^{14}$ cm-3 in case of K atoms at a $\sim 180~{}\degree$C cell temperature.
The action of the pump beam, referred to as optical pumping, orients nearly
all of the electron spins along its propagation direction. The interaction of
a weak external magnetic field to be detected with the polarized electron
spins leads to a change in the orientations of the spins. The degree of change
is proportional to the strength of the magnetic field. The non-zero spin
projection along the probe beam results in a rotation of the light linear
polarization plane of the probe beam caused by the Faraday effect. This
optical rotation is precisely detected with a polarizing beam splitter and two
photo-detectors as a small difference in the balanced output. The OQS
frequency of the maximum sensitivity, $\nu_{m}$, is tuned by a small bias
static magnetic field $B_{b}$ through $\nu_{m}=\gamma B_{b}$, where $\gamma=7$
GHz/T is the gyromagnetic coefficient of Rb-87 or K-39/41 electron spins.

Figure 2: OQS configuration: two laser beams overlap in a vapor cell of
alkali-metal atoms. The circularly polarized pump beam orients atomic spins
along its direction (the dashed green arrow). The external magnetic field
tilts the spins by a small angle (the solid green arrow). The tilt leads to a
rotation of the plane of linear polarization of the probe beam (dotted and
solid black arrows), which is measured by a polarizing beam splitter and two
photo-detectors.

To detect the amplified magnetic axion signal $\vec{B}_{d}$, the OQS vapor
cell is placed at the center of the output coil. The input gradiometer coil is
located inside a magnet bore at room temperature, producing a static magnetic
field $B_{0}$. The $\vec{B}_{a}$ induces a voltage in the input gradiometer by
Faraday’s law, which drives a current through the output coil producing an
oscillating magnetic field $\vec{B}_{d}$ that is our observable axion signal.
As $\vec{B}_{a}$ is azimuthally symmetric, each pickup loop of the input
gradiometer covers each half-side of a horizontal plane through the magnet and
central axis. This configuration doubles the $B_{a}$ signal, but greatly
reduces the background magnetic noise that generates voltages of opposite
signs.

As this is a resonant LC circuit axion detector, many aspects are similar to
the proposal of Sikivie, Sullivan, and Tanner Sikivie  _et al._ (2014), but an
OQS is uniquely used as a sensitive magnetometer. Thus our design is modified
for the coupling to the OQS. Our initial experimental proposal is described in
detail in a previous publication Chu  _et al._ (2018).

When the LC circuit resonates at an angular frequency equal to the axion mass,
i.e., $\omega=1/\sqrt{(L_{\text{in}}+L_{\text{out}})C}=m_{a}$ with
$L_{\text{in}}$ and $L_{\text{out}}$ being the inductances of the input and
output coils and $C$ being the capacitance of the capacitor (this includes the
two coils’ self-capacitance and the capacitance of leads, as well as other
parasitic effects), the magnitude of the current in the circuit is given by

$\displaystyle|I|=\frac{Q|\Phi_{a}|}{L_{\text{in}}+L_{\text{out}}},$ (3)

where $Q=\omega(L_{\text{in}}+L_{\text{out}})/R$ is the quality factor of the
LC circuit, $R$ is the total AC resistance of the circuit, and $\Phi_{a}$ is
the axion-induced magnetic flux through the input gradiometer coil. Using Eq.
(2), cylindrical coordinates, $(z,\rho,\phi)$, and $\vec{B_{0}}=B_{0}\hat{z}$,
the axion-induced oscillating magnetic field $\vec{B}_{a}$ in Fig. 1 is Chu
_et al._ (2018)

$\displaystyle\vec{B_{a}}=-\frac{g\sqrt{2\rho_{DM}}B_{0}\rho}{2}\hat{\phi},$
(4)

and thus the magnitude of $\Phi_{a}$ is

$\displaystyle|\Phi_{a}|=|\int
2N_{\text{in}}\vec{B_{a}}\cdot{d\vec{A}}|=2N_{\text{in}}\text{V}_{\text{in}}g\sqrt{2\rho_{DM}}B_{0},$
(5)

where $N_{\text{in}}$ is the number of turns of each pickup loop of the input
gradiometer coil and $\text{V}_{\text{in}}=l_{\text{in}}r_{\text{in}}^{2}/4$
is a geometric factor for the input gradiometer coil with $l_{\text{in}}$ and
$r_{\text{in}}$ as its length and width. The output coil is composed of two
loops in series with the center-to-center spacing, $2d$. The current $I$ in
Eq. (3) flowing through the output coil produces the magnetic field
$\vec{B}_{d}$ at the location of the OQS sensing volume and thus the magnitude
of $\vec{B}_{d}$ is

$\displaystyle|B_{d}|=$
$\displaystyle\frac{N_{\text{out}}|I|}{r_{\text{out}}[1+(d/r_{\text{out}})^{2}]^{3/2}}$
$\displaystyle=$
$\displaystyle\frac{N_{\text{out}}Q|\Phi_{a}|}{r_{\text{out}}[1+(d/r_{\text{out}})^{2}]^{3/2}(L_{\text{in}}+L_{\text{out}})},$
(6)

where $r_{\text{out}}$ and $N_{\text{out}}$ is the radius and the number of
turns of each loop of the output coil, respectively. Substituting Eq. (5) to
Eq. (6), the field magnitude is

$\displaystyle|B_{d}|=\frac{2N_{\text{in}}N_{\text{out}}Q\text{V}_{\text{in}}g\sqrt{2\rho_{DM}}B_{0}}{r_{\text{out}}[1+(d/r_{\text{out}})^{2}]^{3/2}(L_{\text{in}}+L_{\text{out}})}.$
(7)

This indicates that the large magnet bore size and strong field $B_{0}$ are
critical to improve the axion signal strength.

As the axion mass (frequency) is unknown, the axion detector must be tuned
across a range of frequencies. This will be achieved by simultaneous tuning
the OQS frequency range by changing its bias magnetic field and the LC circuit
by adjusting capacitors. When the LC circuit resonates at the axion mass, the
$\vec{B}_{d}$ is enhanced by the quality factor of the LC circuit. To find the
axion, the task is to detect an extremely small signal above noise sources. In
the following, we investigate the noise contributions to determine the
sensitivity that a quantum-limited OQS offers for axion dark matter detection.

##  III Noise Sources

The two main sources of noise in the axion detector in Fig. 1 are the
intrinsic magnetic field noise of the OQS, $\delta B_{\text{OQS}}$, and the
magnetic Johnson noise (or thermal noise) of the LC circuit, $\delta B_{J}$.
Therefore, the total magnetic noise of this detector is given by

$\displaystyle\delta B_{d}=\sqrt{\delta B_{J}^{2}+\delta B_{\text{OQS}}^{2}}.$
(8)

When the field noise of the OQS approaches the fundamental quantum limit, the
LC circuit must be cooled or its intrinsic thermal noise will dominate the
experiment. At low temperatures of the LC circuit, additional background noise
sources must also be considered, which include the backaction noise of the OQS
on the LC circuit and thermal noise from experimental magnetic shielding. This
is illustrated in Fig. 3, and discussed in detail in the following.

Figure 3: Illustration of background noise contributions using an OQS as the
magnetometer for axion detection with an LC circuit. In addition to the
intrinsic thermal (magnetic Johnson) noise of the LC circuit and the intrinsic
field noise of the OQS, additional noise contributions come from the
backaction noise of the OQS on the LC circuit and the thermal noise of
experimental magnetic shielding, here copper shield.

###  III.1 Fundamental Quantum Noise Limit of OQS

As in any quantum sensor, the intrinsic magnetic field noise of OQSs is
ultimately limited by quantum fluctuations. In this sensor, these are due to
the finite number of alkali-metal atomic spins and the probe beam photons,
both used for magnetic field sensing. Fundamentally, the three dominant
sources of quantum field noise in OQSs are: (i) the spin projection noise
caused by the finite number of alkali-metal atomic spins, (ii) the photon shot
noise resulting from the finite number of probe beam photons, and (iii) the
light-shift noise caused by fluctuations in the polarization of the probe beam
Savukov  _et al._ (2005). The quadrature sum of the three individual noise
sources determines the fundamental quantum noise limit (QNL) of so-called
radio-frequency OQS Savukov  _et al._ (2005),

$\displaystyle\delta
B_{\text{QNL}}=\frac{1}{\gamma\sqrt{nV}}\sqrt{\frac{4}{T_{2}}+\frac{R_{pr}\text{OD}}{32}+\frac{8}{R_{pr}\text{OD}T_{2}^{2}\eta}}$
(9)

operating at frequencies much above 10 kHz where spin-exchange interaction is
suppressed by optical pumping into the stretched state $|FF>$. Here $\gamma$
and $n$ are the gyromagnetic ratio and the density of alkali-metal atoms,
respectively; $V$ is the active measurement cell volume defined by the overlap
of the pump and probe beams; $T_{2}$ is the coherence time of the electron
spins of alkali-metal atoms; $R_{pr}$ is the absorption rate of photons from
the probe beam; $\eta\approx 0.8$ is the photodiode quantum efficiency in the
probe beam readout Savukov  _et al._ (2005); and OD is the optical depth of
the probe beam. If this expression inside the square root is optimized with
respect to the product of $R_{pr}$ and OD (the optimal condition in Fig. 4),
it will depend on $T_{2}$ as the first term, and the field noise of OQS will
be limited by $T_{2}$ and the number of spins. Increasing $T_{2}$ can be
obtained by pumping the electron spins into the stretched state to reduce the
dominant spin-exchange relaxation by alkali-metal atom collisions Savukov  _et
al._ (2005). The number of spins can be increased by using a large vapor cell.

Figure 4: Calculated fundamental quantum noise limit of OQS as a function of
the product of $R_{pr}$ and OD.

Figure 4 shows the fundamental quantum noise limit of OQS for operation using
K spins, calculated using Eq. (9) with $\gamma=7\times 10^{9}~{}\text{Hz/T}$,
$n=7\times 10^{13}~{}\text{cm}^{-3}$, $V=100~{}\text{cm}^{3}$, and
$T_{2}=3.5~{}\text{ms}$. It indicates that the OQS field noise of 10
aT/$\sqrt{\text{Hz}}$ can be achievable at the optimal condition. The K atoms
(natural abundance mixture of two stable isotopes K-39 and K-41 both having
the same nuclear spin 3/2) are selected owing to their lower spin-destruction
cross section between atoms Savukov  _et al._ (2005) (10 times lower than that
of Rb atoms). The optimal value of $R_{pr}\times\text{OD}$ has to be quite
large. As the $R_{pr}\times\text{OD}$ is proportional to the alkali-metal
density, the length of the vapor cell along the probe beam direction, and the
probe laser power Savukov  _et al._ (2005), the large optimal value can be
achieved by optimally increasing the three parameters. The excessive increase
of the alkali-metal density will shorten $T_{2}$; hence, the density should
increase until alkali-metal spin-destruction collisions start to dominate the
spin-destruction rate. Increasing the probe beam path length, which does not
affect $T_{2}$, can be implemented by using a vapor cell that has a long
dimension along the probe beam direction; however, this method is not
practical if a path length of greater than 20 cm is required. The probe laser
power can be increased as long as it does increase much $1/T_{2}$ Savukov  _et
al._ (2005).

###  III.2 Intrinsic Magnetic Johnson Noise of LC Circuit

The magnetic Johnson noise of the circuit $\delta B_{J}$ is the combined
thermal noise from the input gradiometer coil, the output coil, and the wire
leads between the input and output coils (Fig. 1). Due to current fluctuations
arising from the voltage Johnson noise of the LC circuit, $\delta
I_{J}=\sqrt{4k_{B}TR}/Z=Q\sqrt{4k_{B}TR}/\omega(L_{\text{in}}+L_{\text{out}})$
with $Z$ being the impedance of the resonant LC circuit, the $\delta B_{J}$ at
the location of the OQS vapor cell is given by

$\displaystyle\delta B_{J}=$
$\displaystyle\Big{[}{\omega(L_{\text{in}}+L_{\text{out}})r_{\text{out}}}[1+(d/r_{\text{out}})^{2}]^{3/2}\Big{]}^{-1}$
$\displaystyle\times$ $\displaystyle
N_{\text{out}}Q\sqrt{4k_{B}T\rho\Big{[}\frac{4(l_{\text{in}}+r_{\text{in}})N_{\text{in}}}{A_{\text{in}}}+\frac{4\pi
r_{\text{out}}N_{\text{out}}}{A_{\text{out}}}+\frac{2l_{\text{lead}}}{A_{\text{lead}}}\Big{]}},$
(10)

where $k_{B}$ is the Boltzmann’s constant; $T$ is the absolute temperature of
the LC circuit; $l_{\text{lead}}$ is the length of the wire leads; $\rho$ and
$A_{\text{in,out,lead}}$ are the the resistivity and the total cross section
area of the wires of the input gradiometer coil, output coil, and wire leads,
respectively. This indicates that the $\delta B_{J}$ reduces at lower
temperature.

###  III.3 Backaction OQS noise

We investigate backaction of the OQS, which injects the spin noise arising
from fluctuating K spins in the OQS cell into the LC circuit through the
output coil. The backaction OQS noise becomes significant when using the LC
circuit cooled to a few K due to its low thermal noise. The OQS spin noise can
be estimated in the assumption of the standard quantum limit at the magnetic
resonance:

$\delta B_{SN}=\frac{\mu_{0}\mu_{B}\sqrt{N_{s}T_{2}}}{4\pi
d^{3}}L(\nu-\nu_{0},\Delta\nu),$ (11)

where $\mu_{B}$ is the Bohr magneton, $d$ is the distance from the center of
the cell, $N_{s}$ is the number of the K spins in the cell. Here we consider
the K cell as a magnetic dipole with the fluctuating magnetic moment of
$(1/2)\mu_{B}\sqrt{N_{s}T_{2}}$ for simplification. We assume that the spin
noise scales as $\sqrt{N_{s}}$ in the standard quantum limit and it has a
Lorentzian profile $L(\nu-\nu_{0},\Delta\nu)$ near the magnetic resonance Shah
_et al._ (2010). This profile is normalized to 1 at the maximum and has the
relation of $\Delta\nu=1/(2\pi T_{2})$. For the K cell that can reach $\delta
B_{\text{QNL}}=10$ aT/$\sqrt{\text{Hz}}$ at the optimal condition, discussed
in Sec. III.1, with $V=100~{}\text{cm}^{3}$ and $n=7\times 10^{13}$ cm-3, the
spin noise is estimated to be $\delta B_{SN}=0.009$ aT/$\sqrt{\text{Hz}}$ at
$d=8$ cm, as an example.

The magnetic flux of $\delta B_{SN}$ through the two-loop circular output coil
is

$\displaystyle\delta\Phi_{SN}=$ $\displaystyle\int 2N_{\text{out}}\delta
B_{SN}dA$
$\displaystyle=\frac{\mu_{0}\mu_{B}\sqrt{N_{s}T_{2}}N_{\text{out}}}{r_{\text{out}}[1+(d/r_{\text{out}})^{2}]^{3/2}}L(\nu-\nu_{0},\delta\nu),$
(12)

which injects the OQS spin noise into the LC circuit through the relation
$\delta V_{SN}=\delta\Phi_{SN}\omega.$ The $\delta V_{SN}$ should be compared
with the thermal noise inside the LC circuit in order to check the
significance of the backaction OQS noise.

###  III.4 Possible Magnetic Background Noise

The main experimental challenge is to detect the extremely weak magnetic axion
signal above magnetic background noises. The LC circuit needs to be shielded
from external electromagnetic fields. The simplest shielding method is placing
the elements of the LC circuit into a copper shield (but not any ferromagnetic
shield, since the circuit is partially exposed to the strong magnetic field).
We estimate that the shielding factor of $10^{3}$ ($10^{6}$) can be achieved
with a copper shield thickness $x=0.5$ mm (1 mm) based on the $e^{-x/\delta}$
shielding law with the copper’s skin depth $\delta=65~{}\mu$m at 1 MHz. The
most sensitive part of the LC circuit to the electromagnetic interference is
the large input gradiometer coil. The output coil and the OQS sensor head are
not exposed to the strong field, therefore various options exist for shielding
including $\mu$-metal shield and ferrite shield that has the lowest possible
magnetic background noise due to its extremely small electrical conductivity.
Since the non-ferromagnetic conductive copper shield generates magnetic
background noise and can impose limitations on the axion detector sensitivity,
we consider the question of copper shield thermal noise in detail in this
subsection.

####  III.4.1 Copper Shield Thermal Noise: Johnson Noise and Black-Body
Radiation Noise

The thermal noise of the conductive copper shield can potentially have two
components: the Johnson noise and the black-body radiation noise. Here, we
show that actually at frequencies below $\sim$MHz, the black-body radiation
noise is identical to the Johnson noise for the copper surface, and thus its
contribution is included via the Johnson noise. While one might try to use
Plank’s law to obtain the energy density for black-body radiation

$\epsilon(\omega)\propto\frac{\omega^{3}}{e^{\hbar\omega/(k_{B}T)}-1},$ (13)

at the frequency range the assumptions which were used for the derivation are
no longer true, and also this limit is not well understood Guo (2020).
Physically thermal black-body radiation arises from thermal motion of atoms
Guo (2016), which leads to random current densities Joulain  _et al._ (2005).
By using the expression for the photon energy Bose-Einstein distribution

$\Theta(\omega,T)=\frac{\hbar\omega}{e^{\hbar\omega/(k_{B}T)}-1},$ (14)

that in the frequency limit of $\hbar\omega<<k_{B}T$ can be simplified to

$\Theta(\omega,T)=k_{B}T,$ (15)

and substituting the dielectric constant from the Drude model (see for example
Ref. Basu1  _et al._ (2009))

$\epsilon(\omega)=\epsilon_{\infty}-\frac{\sigma_{0}/\tau}{\epsilon_{0}(\omega^{2}+i\omega/\tau)},$
(16)

where $\varepsilon_{\infty}$ is the $\varepsilon(\omega)$ at very high
frequency, $\sigma_{0}$ is the electrical conductivity at zero frequency,
$\varepsilon_{0}$ is the vacuum permittivity, and $\tau$ is electron collision
time, these random current densities can be written as

$<j_{m}(r,\nu)j_{n}(r^{\prime},\nu^{\prime})>=4\sigma
k_{B}T\delta_{mn}\delta(r-r^{\prime})\delta(\nu-\nu^{\prime}).$ (17)

Here $\sigma$ is the electrical conductivity and $r$ ($r^{\prime}$) is the
$x$, $y$, $z$ ($x^{\prime}$, $y^{\prime}$, $z^{\prime}$) vector. Note that we
replaced $\delta(\omega-\omega^{\prime})$ in the original expression of Ref.
Joulain  _et al._ (2005) with $\delta(\nu-\nu^{\prime})/2\pi$ and multiplied
it by a factor of 2 for folding the negative spectrum on to the positive one.

It can be shown that the expression in Eq. (17) leads to a correct Johnson
noise voltage. The black-body radiation current in a short copper can be found
by integrating the fluctuating random current densities over the cross-section
area $A$ of the copper. This gives

$I_{BB}^{2}(z,\nu)=4\sigma
k_{B}T\delta(z-z^{\prime})\delta(\nu-\nu^{\prime}).$ (18)

The black-body radiation voltage is then $dV_{BB}=IdR=Idz/\sigma A$, and by
two-time integration of the squared voltage over the length of the conductor
$l$ along $z$,

$V_{BB}^{2}=4k_{B}TR\delta(\nu-\nu^{\prime}).$ (19)

Here $R=l/\sigma A$. The delta function means un-correlated noise in the
frequency domain, therefore double integration over some small frequency
interval $\Delta\nu$ leads to the correct expression for Johnson noise
$V_{JN}$

$V_{BB}=\sqrt{4k_{B}TR\Delta\nu}=V_{JN}.$ (20)

The magnetic noise of a small copper disk of radius $a$ at a distance $z$ much
larger than $a$ can be easily calculated analytically from Eq. (17) and the
Biot-Savart law:

$B=\int\frac{\mu_{0}}{4\pi}\frac{j\times r}{r^{3}}dxdydz.$ (21)

Since $B$ is the random function, we calculate $B^{2}$, actually $B_{z}^{2}$,
the component normal to the surface of the copper disk, which is proportional
to

$<(j_{x}y-j_{y}x)^{2}>=<j_{x},j_{x}>y^{2}+<j_{y},j_{y}>x^{2}=j^{2}_{0}(x^{2}+y^{2}),$
(22)

where $j^{2}_{0}=4\sigma k_{B}T\delta(r-r^{\prime})\delta(\nu-\nu^{\prime})$.
This can be integrated over the volume of the disk with the thickness $h$ as
well as over frequency similar to such integration in case of the copper
voltage noise to give

$B_{z}=\frac{\mu_{0}}{\sqrt{8\pi}}\sqrt{\sigma k_{B}Th}\frac{a^{2}}{z^{3}},$
(23)

which in the limit of $a\ll z$ is the exact expression for the Johnson noise
of a small disk Lee and Romalis (2008). In case of an arbitrary copper shield,
since we showed that the black-body radiation noise is identical to the
Johnson noise, it is possible just to use Johnson noise calculations in order
to estimate the thermal noise of the copper shield.

There is still a question of noise arising from dielectric materials, for
example, glass material of the OQS vapor cell. For the noise from a dielectric
material, the random current density in Eq. (17) can be obtained by replacing
$\sigma$ with $\omega\epsilon_{0}\epsilon^{\prime\prime}$, where
$\epsilon^{\prime\prime}$ is the imaginary part of the complex dielectric
permittivity. The noise ratio between copper and dielectric materials for the
same geometry will be

$\sqrt{\frac{\sigma\delta_{S}}{\omega\epsilon_{0}\epsilon^{\prime\prime}h}}.$
(24)

If we compare a copper disk with $\sigma=5.96\times 10^{7}$ S/m and the skin
depth $\delta_{S}=7\times 10^{-5}$ m with a disk made of dielectric material
with a typical $\epsilon^{\prime\prime}=10^{-3}$ and the thickness $h=1$ cm
for $\omega=2\pi\times 10^{6}$ Hz, the noise ratio is $2.7\times 10^{6}$,
meaning that the noise due to dielectric material losses is expected much
lower and can be neglected.

####  III.4.2 Magnetic Johnson Noise of Copper Shield

The magnetic field noise due to Johnson current noise in the copper shield can
be found using the method in Ref. Savukov and Kim (2020). The noise for
specific copper shield configuration can be obtained by scaling the results in
Ref. Savukov and Kim (2020). In particular, it has been shown that the noise
at considered frequency scales, where the inductance of the current path
dominates over the resistance of the path, in the following way:

$\delta B=kT^{1/2}\sigma^{-1/2}\omega^{-3/4}h^{-2},$ (25)

where $k$ is a coefficient of proportionality dependent on other parameters
such as temperature and shield geometry.

In Ref. Savukov and Kim (2020) it has been shown that at a 4.6 cm distance and
room temperature, the thermal noise of copper shield with the thickness
$h=0.2$ mm for $\omega=2\pi\times 10^{5}$ Hz, was measured to be 220
aT$\sqrt{\text{Hz}}$ and the shield Johnson noise was modeled to be 210
aT$\sqrt{\text{Hz}}$. This result confirms our claim that the thermal noise of
the copper shield at frequencies below $\sim$ MHz can be estimated by only the
Johnson noise of the copper shield. The noise value can be scaled to a 50 cm
distance (the approximate average distance from the copper shield to the coils
in our experiment) and 1 MHz frequency, giving 0.3 aT/$\sqrt{\text{Hz}}$.
Further reduction in the noise can be achieved by cooling the copper material.
If nitrogen cooling is used (77 K), the copper resistivity can be reduced by
10 times (slightly depending on purity of copper material). Taking into
consideration of the temperature reduction factor, the noise at 77 K can be
reduced by 6 times to 0.05 aT/$\sqrt{\text{Hz}}$. Cooling the copper material
to 4 K can reduce the noise another factor of 40 for high-purity copper,
reaching 0.001 aT/$\sqrt{\text{Hz}}$.

##  IV Prototype Axion Detector Development

To perform the first tests of an OQS in axion detection with an LC circuit, we
have developed and constructed an optimized room temperature design for an
existing solenoid magnet at Los Alamos National Laboratory (LANL), with a
commercial OQS. The magnet is a superconducting solenoid with a warm bore of
1-m diameter and 3-m length in the center, which can produce a static field
$B_{0}=2$ T. We employed a commercial Twinleaf OQS containing a $5\times
5\times 5~{}\text{mm}^{3}$ Rb-87 vapor cell, whose sensor head is shown in
Fig. 5(a). The measured magnetic field noise of the OQS around 300 kHz (the
target frequency range in this prototype experiment described below) was 10
fT/$\sqrt{\text{Hz}}$. As an example, the OQS field noise around 307.5 kHz is
shown in Fig. 5(b). The peak at 307.5 kHz is a known applied calibration
magnetic field to convert the OQS output voltages into magnetic fields. We
also investigated a frequency response of the OQS by applying a sinusoidally
varying magnetic field at different frequencies around 307.5 kHz, as indicated
in Fig. 5(c) (blue dot points). The data was fit to a Lorentzian function,
$f(\nu)=a_{0}+a_{1}/[4(\nu-\nu_{0})^{2}+\Delta\nu^{2}]$ with $\nu$ being the
frequency of the applied field. The fit (solid red curve) gives the bandwidth
of the OQS of $\Delta\nu_{\text{OQS}}=1.8~{}$kHz.

Figure 5: (a) Photograph of the cm-scale commercial OQS sensor head, (b) the
magnetic field noise and (c) the frequency response of the OQS. In (b), the
peak at 307.5 kHz is the known applied calibration magnetic field. The
magnetic field noise of the OQS is measured to be 10 fT/$\sqrt{\text{Hz}}$. In
(c), the data was collected by scanning applied sine field of constant
amplitude between 306 and 309 kHz; the solid curve indicates a Lorentzian fit,
giving the OQS bandwidth of 1.8 kHz. Figure 6: Optimization of the number of
turns of the input gradiometer and output coils to maximize the SNR of the
prototype axion detector. The SNR values are normalized.

To accommodate the sizes of the magnet bore and the OQS, we selected the
following experimental dimensions for the axion detector: $l_{\text{in}}=1.0$
m, $r_{\text{in}}=0.3$ m, $r_{\text{out}}=2.9$ cm, and $d=2.0$ cm. The
$l_{\text{lead}}=9.0$ m was chosen in order to locate the OQS at a position
where the residual magnetic field of the magnet is sufficiently suppressed to
the level of the Earth’s magnetic field. To reduce the AC resistance of the LC
circuit closer to the DC resistance value, the circuit was made from Litz
wires of multiple strands of thin copper wire with a 0.06 mm diameter,
recommended at our target frequency range Lit . Because the total length of
the LC circuit is dominated by the input coil and the wire leads, their wire
diameter was chosen to be $b_{\text{in}}=5.2$ mm in order to sufficiently
reduce their magnetic Johnson noise. On the other hand, the wire diameter of
the output coil was reduced to $b_{\text{out}}=1.3$ mm because of the small
output coil’s diameter.

We optimized $N_{\text{in}}$ and $N_{\text{out}}$ to maximize the signal-to-
noise ratio (SNR) of the axion detector, $|B_{d}|/\delta B_{d}$, at 300 kHz
using Eqs. (7), (8), and (10), and $\delta B_{\text{OQS}}=10$
fT/$\sqrt{\text{Hz}}$. We estimated the inductance of the input gradiometer
coil Sikivie  _et al._ (2014),

$\displaystyle
L_{\text{in}}\approx\frac{2}{\pi}N_{\text{in}}^{2}l_{\text{in}}\text{ln}(r_{\text{in}}/0.5b_{\text{in}})=N_{\text{in}}^{2}\times
3.8~{}\mu\text{H},$ (26)

and the inductance of the output coil Sikivie  _et al._ (2014),

$\displaystyle L_{\text{out}}\approx
2r_{\text{out}}N_{\text{out}}^{2}\Big{[}\text{ln}\Big{(}\frac{8r_{\text{out}}}{0.5b_{\text{out}}}\Big{)}-2\Big{]}=N_{\text{out}}^{2}\times
0.3~{}\mu\text{H},$ (27)

where we neglected the mutual inductance between the two loops for simplicity.
Figure 6 shows the calculated, normalized 2D distribution of the SNR values of
the axion detector as a function of $N_{\text{in}}$ and $N_{\text{out}}$. This
calculation implies that various optimal values exist in the red region;
however, fewer number of turns are better according to our investigations that
the noise of coils approached the theoretical value in Eq. (10) as the number
of turns was reduced Savukov  _et al._ (2009). Thus, we selected
$N_{\text{in}}=1$ and $N_{\text{out}}=4$ as optimal values, leading to the
input gradiometer consisting of two series-configured one-turn
$1.0~{}\text{m}\times 0.3~{}\text{m}$ pickup loops. Figure 7 shows the
estimated total magnetic noise of the optimized prototype axion detector as a
function of the frequency (black solid curve), indicating that the axion
detector loses sensitivity at the frequency range below $\sim$100 kHz due to
the thermal noise of the room temperature LC circuit (blue dotted curve).
Above $\sim$100 kHz, our experimental sensitivity to the axion is limited by
the OQS field noise (orange dotted curve). Based on Fig. 7, our experiment
targeted frequencies around 300 kHz.

Figure 7: Estimated total magnetic noise of the prototype axion detector with
the optimized LC circuit at 300 K and the commercial OQS unit (black solid
curve), and the advanced axion detector with the optimized superconducting LC
circuit at 2 K and the quantum-limited OQS (red solid curve). The noise below
$\sim$100 kHz and $\sim$1 MHz is dominated by the thermal noise of the LC
circuit at 300 K and 2 K, respectively, limiting the axion detector
sensitivities.

We have built a prototype axion detector with the selected experimental
dimensions and the optimized experimental parameters, as shown in Fig.
8(a)–(c). The wire leads were shielded by copper tube and the assembly of the
output coil and the OQS sensor head was shielded by a $\mu$-metal enclosure in
order to minimize interference from ambient magnetic signals. On the other
hand, the input gradiometer coil was shielded by the magnet shield composed of
an open iron rectangular enclosure located outside the magnet bore and
additional copper mesh and Faraday cage to cover the openings. When the magnet
is cooled to 4 K, the superconducting solenoid coil of the magnet could also
add additional shielding for the input gradiometer coil. The OQS output was
connected to a 24-bit data acquisition system (NI PXIe-4480) and recorded at a
sampling rate of 1 MHz using a home-built LabVIEW program.

First, we tuned the prototype axion detector around 307 kHz. This was achieved
by both tuning the OQS by applying its corresponding bias magnetic field of
$43.9~{}\mu\text{T}$ and tuning the LC circuit by using mica capacitors of 19
nF. The quality factor of the LC circuit was measured to be $Q=43$, indicating
its bandwidth of 7.1 kHz through the relation of
$\Delta\nu_{\text{LC}}=\nu/Q$. Since
$\Delta\nu_{\text{LC}}>\Delta\nu_{\text{OQS}}$, the bandwidth of the prototype
is determined by $\Delta\nu_{\text{OQS}}=1.8~{}\text{kHz}$. Before the magnet
was cooled, the sensitivity of the prototype was measured to be 50
fT/$\sqrt{\text{Hz}}$, 5 times larger than the estimated values shown in Fig.
7. We estimate that the worse sensitivity was due to the magnetic Johnson
noise from the solenoid coil of the magnet at room temperature, but this noise
decreased significantly when the solenoid coil became superconducting.

Figure 8: The Photographs of the prototype axion detector: (a) the optimized
first-order planar input gradiometer coil, (b) the combination of the
optimized circular output coil and the commercial OQS sensor head, (c) the
$\mu$-metal enclosure that houses the combination shown in (b) (the
$\mu$-metal lid is opened for illustration purposes only), and (d) its
sensitivity when the magnet is cooled to 4 K and the prototype is tuned at 307
kHz.

With the magnet cooled to 4 K where the solenoid coil becomes superconducting,
we obtained background data with the prototype (i.e., without the magnet
energized), shown in Fig. 8(d). The experimental sensitivity of the prototype
was measured to be around 10 fT/$\sqrt{\text{Hz}}$, demonstrating that with
other ambient noises sufficiently suppressed, the prototype sensitivity to the
axion dark matter is determined by the OQS noise. This means that the OQS
noise reduction is the key to success.

##  V Sensitivity Estimate of Axion detection

In principle, the SNR of the prototype axion detector can be increased with
long data integration. The total field noise of the prototype with a data
integration time, $t_{\text{int}}$, is given by

$\displaystyle\delta B_{d}^{\text{int}}=\delta
B_{d}\times(t_{c}t_{\text{int}})^{-1/4},$ (28)

where $t_{c}=(0.16~{}\text{s})\times(\text{MHz}/\nu)$ is the axion signal
coherence time for the isothermal halo model Sikivie  _et al._ (2014); Budker
_et al._ (2014). For example, the $\delta B_{d}$ can be reduced by a factor of
11 and 8 with a 7-hours integration time at 300 kHz and 1 MHz, respectively.
The axion signal coherence time limits the experimental noise reduction with
long data integration.

Figure 9: Estimated sensitivity of our prototype axion detector (blue dotted
curve) to the axion-photon coupling $g$ with $t_{\text{int}}=7~{}\text{hours}$
and 95% confidence level to the axion mass range from $10^{-11}$ to $4\times
10^{-8}$ eV. In similar range, the constraints were set by the CAST
Anastassopoulos  _et al._ (2017), ABRACADABRA Salemi  _et al._ (2021), and
SHAFT Gramolin  _et al._ (2021) experiments (gray solid curves). Estimated
sensitivities of the DM-Radio experiments (gray dotted curves) are also shown
Brouwer  _et al._ (2022). The yellow band encompasses various QCD axion
models, including the benchmark KSVZ Kim (1979); Shifman  _et al._ (1980) and
DFSZ Dine  _et al._ (1981); Zhitnitskii (1980) models, predicting the QCD
axion coupling. Furthermore, the constraints set by the ADMX Du  _et al._
(2018); Braine  _et al._ (2020); Bartram  _et al._ (2021) and ADMX-SLIC
Crisosto  _et al._ (2020) experiments are shown. The red dotted curve shows
the estimated sensitivity of our proposed axion detector optimized to an
quantum-limited OQS with 10 aT/$\sqrt{\text{Hz}}$ field noise and a
superconducting LC circuit with $t_{\text{int}}=15$ s.

Based on Eqs. (7), (8), (10), and (28), we can estimate the sensitivity of the
prototype axion detector to the axion-photon coupling $g$,

$\displaystyle g$
$\displaystyle=\text{S}\frac{r_{\text{out}}[1+(d/r_{\text{out}})^{2}]^{3/2}(L_{\text{in}}+L_{\text{out}})\delta
B_{d}^{\text{int}}}{2QN_{\text{in}}N_{\text{out}}V_{\text{in}}\sqrt{2\rho_{DM}}B_{0}}$
$\displaystyle=\text{S}\Big{(}\frac{\delta
B_{d}^{\text{int}}}{10^{-15}~{}\text{T}}\Big{)}\Big{(}\frac{\text{GeV/cm}^{3}}{\rho_{DM}}\Big{)}^{\frac{1}{2}}\Big{(}\frac{10^{3}}{Q}\Big{)}\Big{(}\frac{L}{\mu\text{H}}\Big{)}\Big{(}\frac{\text{T}}{B_{0}}\Big{)}\Big{(}\frac{1}{N_{\text{out}}}\Big{)}$
$\displaystyle\times\Big{(}\frac{1}{N_{\text{in}}}\Big{)}\Big{(}\frac{r_{\text{out}}[1+(d/r_{\text{out}})^{2}]^{\frac{3}{2}}}{\text{cm}}\Big{)}\Big{(}\frac{\text{m}^{3}}{V_{\text{in}}}\Big{)}(2\times
10^{-16}~{}\text{GeV}^{-1}),$ (29)

where S is the SNR, taken as 2 ($2\sigma$ or 95% confidence level; note that
once the axion signal is discovered, the measurement can be repeated many
times to verify the discovery with higher confidence level), and
$L=L_{\text{in}}+L_{\text{out}}$ is the total inductance of the LC circuit.
Fig. 9 shows our estimated sensitivity of the prototype (blue dotted curve)
from the background data obtained with 7-hours integration at each observation
frequency. The upper end of our search range, around 10 MHz, is limited by the
combination of inductance and stray capacitance of the chosen specific
configuration of the LC circuit. The size of the coil can be reduced in
principle to extend the search to higher frequencies, but this would require
redesigning the coil and will also reduce the axion flux and hence
sensitivity. The sensitivity loss at masses larger than 10-9 eV is due to the
noise reduction limitation from the axion signal coherence time, described
above. Our prototype sensitivity could improve the current constraints (gray
solid curves) set by the CERN Axion Solar Telescope (CAST) Anastassopoulos
_et al._ (2017), Broadband/Resonant Approach to Cosmic Axion Detection with an
Amplifying B-field Ring Apparatus (ABRACADABRA) Salemi  _et al._ (2021), and
Search for Halo Axions with Ferromagnetic Toroids (SHAFT) Gramolin  _et al._
(2021) experiments on an axion mass range between 10-10 eV and 10-7 eV,
corresponding to the frequency range between 10 kHz and 10 MHz, in particular
by 1 order of magnitude at masses of around 10-9 eV. For comparison, projected
sensitivities of the DM-Radio experiment are shown with two gray dotted lines
Brouwer  _et al._ (2022). The yellow band indicates a broad range of the
axion-photon coupling $g$ for the QCD axion predicted by various axion models.
As benchmark examples, the KSVZ Kim (1979); Shifman  _et al._ (1980) and DFSZ
Dine  _et al._ (1981); Zhitnitskii (1980) axion models are included. The
constraints set by the ADMX Du  _et al._ (2018); Braine  _et al._ (2020);
Bartram  _et al._ (2021) and ADMX-SLIC (Superconducting LC Circuit
Investigating Cold Axions) Crisosto  _et al._ (2020) experiments are also
shown. Our experiment will be able to probe the axion dark matter on the mass
range between 10-11 eV and 10-7 eV.

##  VI Potential Improvement of Axion Detector Sensitivity

Figure 7 shows the magnetic Johnson noise of optimized LC circuits at 300 K
(top blue dotted curve) and 2 K (bottom blue dotted curve), and the OQS noise
limit of the commercial unit (top orange dotted curve), and a quantum-limited
OQS (bottom range dotted curve). It is clear that to take advantage of the
quantum-limited OQS in axion detection, the detector thermal noise must be
reduced by cooling the LC circuit. In this regime, the backaction noise from
the OQS and the thermal noise from the surrounding copper shield must also be
considered, as discussed in Section III. An optimized design of an axion
detector with a quantum-limited OQS is considered in this section.

###  VI.1 Axion Detector with Quantum-limited OQS

Improvement over the prototype design can be achieved with a quantum-limited
OQS with $\delta B_{\text{OQS}}=10$ aT/$\sqrt{\text{Hz}}$. To take full
advantage of a quantum-limited OQS, we consider the LC circuit made of a pure
superconducting wire [e.g., 3-mil niobium (Nb) wire] and cooled to 4 K and
below, which can considerably reduce the magnetic Johnson noise of the LC
circuit. We investigate the sensitivity of an improved axion detector with
$l_{\text{in}}=2.9$ m and $r_{\text{in}}=0.45$ m, more closely matching the
dimension of the bore of the magnet to maximize the axion-induced magnetic
flux through the input gradiometer coil. Considering the K cell volume of
$V=100~{}\text{cm}^{3}$, we selected $r_{\text{out}}=5.0$ cm and $d=8.0$ cm to
allow space between the cooled output coil and the K cell. We also selected
$l_{\text{lead}}=4$ m where the residual magnetic field of the magnet is
reduced to the level of the Earth’s magnetic field.

The inductance of the superconducting input and output coils was estimated
using a three-dimensional inductance extraction program in superconducting
structures Kamon  _et al._ (1994): $L_{\text{in}}=N_{\text{in}}^{2}\times
26.0~{}\mu\text{H}$ and $L_{\text{out}}=N_{\text{out}}^{2}\times
0.4~{}\mu\text{H}$. The parasitic inductance of the twisted-pair leads of
3-mil superconducting Nb wires was measured to be 2.3 nH/cm Kim and Clayton
(2013), giving the negligible inductance of the wire leads of $0.9~{}\mu$H.
While a pure superconducting wire has zero electrical resistance at low
frequencies, the resistance at high frequencies is extremely small but non-
zero Sup . The high-frequency resistance $R_{Nb}$ of superconducting Nb at
around 2 K is on the order of n$\Omega$ Sup . This $R_{Nb}=1~{}\text{n}\Omega$
resistance can achieve $Q\approx 10^{11}$, however we will detune the
superconducting LC circuit to reach $Q_{\text{eff}}\approx 10^{6}$ in order to
have a reasonable axion detector bandwidth. Based on Eq. (10), the
corresponding thermal magnetic Johnson noise at the location of the OQS vapor
cell is given by

$\displaystyle\delta
B_{J}=\frac{N_{\text{out}}Q_{\text{eff}}\sqrt{4k_{B}TR_{Nb}}}{\omega(L_{\text{in}}+L_{\text{out}})r_{\text{out}}[1+(d/r_{\text{out}})^{2}]^{3/2}}.$
(30)

Figure 10: Optimization of the number of turns of the input gradiometer and
output coils made of 3-mil Nb wire at 2 K to maximize the SNR of the axion
detector with a quantum-limited OQS. The SNR values are normalized.

For the axion detector with the quantum-limited OQS, we re-optimized
$N_{\text{in}}$ and $N_{\text{out}}$ to maximize the SNR of the detector,
$|B_{d}|/\delta B_{d}$, at 10 MHz using Eqs. (7), (8), and (30) with $\delta
B_{\text{OQS}}=10$ aT/$\sqrt{\text{Hz}}$ and $Q_{\text{eff}}=2\times 10^{6}$.
Figure 10 shows the calculated, normalized 2D distribution of the SNR values
of the improved axion detector as a function of $N_{\text{in}}$ and
$N_{\text{out}}$. Among various optimal values in the red region, we selected
$N_{\text{in}}=2$ and $N_{\text{out}}=11$ as optimal values, which do not
significantly increase the coils’ width because of using 3-mil Nb wire.

Based on these optimal parameters, Fig. 7 shows the estimated total magnetic
noise of the axion detector with the quantum-limited OQS as a function of the
frequency (red solid curve), indicating that the axion detector loses
sensitivity at the frequency range below $\sim$1 MHz due to the dominant
thermal noise of the superconducting LC circuit (blue dotted curve). On the
other hand, above $\sim$1 MHz, it is limited by the OQS fundamental quantum
field noise (orange dotted curve), thus enhancing the OQS field noise is the
key to improve the axion detector sensitivity in this frequency range. The
projected sensitivity of this optimized axion detector with
$t_{\text{int}}=15$ s integration time at each observation frequency is shown
in Fig. 9 (red dotted curve). We anticipate that with sensitivity up to 7
orders of magnitude beyond the current best limit, the improved axion detector
with the quantum-limited OQS can potentially access the compelling targets
below the KSVZ QCD axion band and probe the QCD axion parameter space in a
mass range near 10-8 eV, corresponding to the frequency range of $\sim$1 MHz.

###  VI.2 Mitigation of Backaction OQS Noise

For the optimized superconducting output coil with $r_{\text{out}}=~{}5.0$ cm,
$d=8.0$ cm, and $N_{\text{out}}=11$, the magnetic flux of the backaction OQS
noise through the output coil in Eq. (12) is $\delta\Phi_{SN}=~{}1.9\times
10^{-21}~{}\text{T}\cdot\text{m}^{2}/\sqrt{\text{Hz}}$. For 1 MHz frequency,
we compare the backaction noise $\delta V_{SN}=\delta\Phi_{SN}\omega=1.2\times
10^{-14}$ V/$\sqrt{\text{Hz}}$ with the thermal Johnson noise inside the
superconducting LC circuit $\delta V_{J}=\sqrt{4k_{B}TR_{Nb}}=3.3\times
10^{-16}$ V/$\sqrt{\text{Hz}}$, assuming $R_{Nb}=1~{}\text{n}\Omega$ at 2 K.
This indicates that the backaction OQS noise can become significant. Hence,
proper detuning of the OQS magnetic resonance can be helpful to make the two
noises comparable. The OQS field noise and the backaction noise follow the
Lorentzian shape, and when the detuning exceeds the resonance width they
decrease as $[2\pi(\nu-\nu_{0})T_{2}]^{-1}$. For example, a detuning from the
resonance of the LC circuit by 4.6 kHz can reduce the backaction noise by 100
times, resulting in $\delta V_{SN}\approx\delta V_{J}$, while the OQS field
noise will increase to 1 fT/$\sqrt{\text{Hz}}$. Reaching 10 aT field noise
will give some extra sensitivity for detuning to decrease the backaction and
increase the axion detector bandwidth, accelerating scanning Brouwer  _et al._
(2022).

Spin squeezing can be a promising mitigation method to considerably suppress
the backaction noise without sacrificing the OQS field noise. Significant
reduction of the OQS spin noise by spin squeezing has been demonstrated, e.g.,
70$\%$ noise reduction in Ref. Kuzmich  _et al._ (2010), a factor of 6.4 in
Ref. Budker and Kimball (2013), and even a factor of 100 in Ref. Hosten  _et
al._ (2016). In fact, the OQS field noise and the backaction noise are
correlated, hence conducting OQS measurements during the minimum of the
oscillating spin noise can reduce both the backaction noise and the spin
projection noise in Eq. (9). In continuous non-demolition measurement Shah
_et al._ (2010), the data can be processed to weigh signals when the spins are
directed toward the output coil (i.e., parallel to its symmetry axis). This
measurement will have the minimal backaction noise if at the same time the
spin state is read out with the probe laser beam. During this measurement, the
orthogonal spin noise component reaches the maximum; however, it is
perpendicular to the output coil’s symmetry axis and thus is not contributed
to the backaction noise. As a result, it could be possible to suppress the
backaction noise without sacrificing the OQS field noise. In contrast to the
previous papers where the goal of using spin squeezing was to demonstrate the
reduction in OQS field noise, here the spin squeezing will be essential for
the backaction noise reduction. In addition, it is important to note that for
a long continuous measurement the advantage of spin squeezing is removed, but
for short measurements $<T_{2}$, the advantage can be on the order of the spin
squeezing. Because of the coherence of the axion signal over 0.25 s, if we
periodically implement the protocol of spin squeezing with the period matching
that of the axion signal, then the averaging of multiple measurements can lead
to $1/\sqrt{N}$ noise reduction, with $N$ being proportional to the
measurement time, with similar improvement of the SNR to long measurements of
coherent signals.

##  VII Conclusion

We have built a prototype axion detector operating at room temperature,
comprised of the optimized LC circuit and the commercial OQS with 10
fT/$\sqrt{\text{Hz}}$ field noise. The LC circuit contained the first-order
one-turn planar gradiometer input coil located inside the large bore of the
2-T superconducting magnet and the two-loop four-turn circular output coil
coupled to the commercial OQS sensor head. We tuned the prototype at about 300
kHz and obtained background data. We investigated the sensitivity of the
prototype based on the background data and showed that the prototype
experiment can probe the axion dark matter on the significant mass range
between 10-11 eV and 10-7 eV. The sensitivity, limited by the OQS field noise
of 10 fT/$\sqrt{\text{Hz}}$, is up to 1 order of magnitude better than the
current best limit. We also investigated the potential sensitivity of an axion
detector based on the superconducting LC circuit, the OQS reaching the
fundamental quantum noise limit of 10 aT/$\sqrt{\text{Hz}}$, and the existing
2-T magnet at LANL. The improved axion detector can potentially enhance the
experimental sensitivity up to 7 orders of magnitude beyond the current best
limit, allowing us to probe the QCD axion parameter space in a mass range near
10-8 eV. The improved experiment will be limited by the quantum noise limit of
the OQS. In addition, we characterized possible background noises in the
experiment including the backaction OQS noise and the thermal noise of the
copper shield, showing that these noises can be reduced below the thermal
noise of the superconducting LC circuit.

###### Acknowledgements.

The authors gratefully acknowledge the support by the Los Alamos National
Laboratory LDRD office through Grants No. 20190113ER, 20210254ER, and
20230633ER. The authors are grateful for helpful discussion and assistance
with operating the magnet and constructing the prototype axion detector with
Dr. Leonardo Civale, Jaren Cordova, Leonard Gonzales, Larry Burney, and Shaun
Newman. The authors are also grateful to Dr. Daniele Alves for useful
comments.

## References

  * Bahcall (2015) N. A. Bahcall, Proceedings of the National Academy of Sciences 112, 12243 (2015).
  * Peccei and Quinn (1977a) R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. 38, 1440 (1977a).
  * Peccei and Quinn (1977b) R. D. Peccei and H. R. Quinn, Phys. Rev. D 16, 1791 (1977b).
  * Wilczek (1978) F. Wilczek, Phys. Rev. Lett. 40, 279 (1978).
  * Weinberg (1978) S. Weinberg, Phys. Rev. Lett. 40, 223 (1978).
  * Duffy and van Bibber (2009) L. D. Duffy and K. van Bibber, New Journal of Physics 11, 105008 (2009).
  * Du  _et al._ (2018) N. Du _et al._ (ADMX Collaboration), Phys. Rev. Lett. 120, 151301 (2018).
  * Sikivie  _et al._ (2014) P. Sikivie, N. Sullivan, and D. B. Tanner, Phys. Rev. Lett. 112, 131301 (2014).
  * Salemi  _et al._ (2021) C. P. Salemi, J. W. Foster, J. L. Ouellet, A. Gavin, K. M. W. Pappas, S. Cheng, K. A. Richardson, R. Henning, Y. Kahn, R. Nguyen, N. L. Rodd, B. R. Safdi, and L. Winslow, Phys. Rev. Lett. 127, 081801 (2021).
  * Gramolin  _et al._ (2021) A. V. Gramolin, D. Aybas, D. Johnson, J. Adam, and A. O. Sushkov, Nature Physics 17, 79 (2021).
  * Dine  _et al._ (1981) M. Dine, W. Fischler, and M. Srednicki, Physics Letters B 104, 199 (1981).
  * Zhitnitskii (1980) A. P. Zhitnitskii, Sov. J. Nucl. Phys. 31, 260 (1980).
  * Kim (1979) J. E. Kim, Phys. Rev. Lett. 43, 103 (1979).
  * Shifman  _et al._ (1980) M. Shifman, A. Vainshtein, and V. Zakharov, Nuclear Physics B 166, 493 (1980).
  * Sikivie (1983) P. Sikivie, Phys. Rev. Lett. 51, 1415 (1983).
  * Sikivie (1987) P. Sikivie, Phys. Rev. D 36, 974 (1987).
  * Langacker (2012) P. Langacker, Scholarpedia 7, 11419 (2012), revision #137135.
  * Chakrabarty  _et al._ (2023) S. Chakrabarty, J. Gleason, Y. Han, A. Hipp, M. Solano, P. Sikivie, N. Sullivan, D. Tanner, M. Goryachev, E. Hartman, _et al._ , arXiv preprint arXiv:2303.07116 (2023).
  * Chu  _et al._ (2018) P.-H. Chu, L. D. Duffy, Y. J. Kim, and I. M. Savukov, Phys. Rev. D 97, 072011 (2018).
  * Battaglieri  _et al._ (2017) M. Battaglieri _et al._ , “Us cosmic visions: New ideas in dark matter 2017: Community report,”  (2017).
  * SRE (1985) Nuclear Physics B 260, 689 (1985).
  * Patrignani (2016) C. Patrignani, Chinese Physics C 40, 100001 (2016).
  * Budker and Romalis (2007) D. Budker and M. Romalis, Nature Physics 3, 227 (2007).
  * Savukov  _et al._ (2005) I. M. Savukov, S. J. Seltzer, M. V. Romalis, and K. L. Sauer, Phys. Rev. Lett. 95, 063004 (2005).
  * Shah  _et al._ (2010) V. Shah, G. Vasilakis, and M. V. Romalis, Phys. Rev. Lett. 104, 013601 (2010).
  * Guo (2020) W. Guo, J. Opt. Soc. Am. A 37, 1428 (2020).
  * Guo (2016) W. Guo, J. Opt. Soc. Am. A 33, 1141 (2016).
  * Joulain  _et al._ (2005) K. Joulain, J. Mulet, F. Marquier, R. Carminati, and J. J. Greffet, Surface Science Reports 57, 59–112 (2005).
  * Basu1  _et al._ (2009) S. Basu1, Z. M. Zhang, and C. J. Fu, Int. J. Energy Res. 33, 1203–1232 (2009).
  * Lee and Romalis (2008) S.-K. Lee and M. V. Romalis, J. Appl. Phys. 103, 084904 (2008).
  * Savukov and Kim (2020) I. Savukov and Y. J. Kim, Journal of Applied Physics 128, 234501 (2020).
  * (32) https://mwswire.com/specialty-wire/litz-wire/.
  * Savukov  _et al._ (2009) I. Savukov, V. Zotev, P. Volegov, M. Espy, A. Matlashov, J. Gomez, and R. Kraus, Journal of Magnetic Resonance 199, 188 (2009).
  * Budker  _et al._ (2014) D. Budker, P. W. Graham, M. Ledbetter, S. Rajendran, and A. O. Sushkov, Phys. Rev. X 4, 021030 (2014).
  * Anastassopoulos  _et al._ (2017) V. Anastassopoulos _et al._ (CAST Collaboration), Nature Physics 13, 584 (2017).
  * Brouwer  _et al._ (2022) L. Brouwer, S. Chaudhuri, H.-M. Cho, J. Corbin, C. S. Dawson, A. Droster, J. W. Foster, J. T. Fry, P. W. Graham, R. Henning, K. D. Irwin, F. Kadribasic, Y. Kahn, A. Keller, R. Kolevatov, S. Kuenstner, A. F. Leder, D. Li, J. L. Ouellet, K. M. W. Pappas, A. Phipps, N. M. Rapidis, B. R. Safdi, C. P. Salemi, M. Simanovskaia, J. Singh, E. C. van Assendelft, K. van Bibber, K. Wells, L. Winslow, W. J. Wisniewski, and B. A. Young (DMRadio Collaboration), Phys. Rev. D 106, 112003 (2022).
  * Braine  _et al._ (2020) T. Braine _et al._ (ADMX Collaboration), Phys. Rev. Lett. 124, 101303 (2020).
  * Bartram  _et al._ (2021) C. Bartram _et al._ (ADMX Collaboration), Phys. Rev. Lett. 127, 261803 (2021).
  * Crisosto  _et al._ (2020) N. Crisosto, P. Sikivie, N. S. Sullivan, D. B. Tanner, J. Yang, and G. Rybka, Phys. Rev. Lett. 124, 241101 (2020).
  * Kamon  _et al._ (1994) M. Kamon, M. Tsuk, and J. White, IEEE Transactions on Microwave Theory and Techniques 42, 1750 (1994).
  * Kim and Clayton (2013) Y. J. Kim and S. M. Clayton, IEEE Transactions on Applied Superconductivity 23, 2500104 (2013).
  * (42) https://cds.cern.ch/record/941325/files/p183.pdf.
  * Kuzmich  _et al._ (2010) A. Kuzmich, L. Mandel, and N. P. Bigelow, Phys. Rev. Lett. 85, 1594 (2010).
  * Budker and Kimball (2013) D. Budker and D. F. J. Kimball, eds., “Optical magnetometry,” (Cambridge University Press, New York, 2013) Chap. Noise, squeezing, and entanglement.
  * Hosten  _et al._ (2016) O. Hosten, N. J. Engelsen, R. Krishnakumar, and M. A. Kasevich, Nature 529, 505 (2016).