9*0.75             3*0.25               4         0.0265    $\pm$ 0.0009        1.3146 $\pm$ 0.0605     0.4388 $\pm$ 0.0731

6         0.0268    $\pm$ 0.0009        1.3439 $\pm$ 0.0778     0.6006 $\pm$ 0.2169

8         0.0235    $\pm$ 0.0005        1.2071 $\pm$ 0.048      0.5044 $\pm$ 0.1404

3*0.5              4         0.0270     $\pm$ 0.0007        1.3345 $\pm$ 0.0477     0.4345 $\pm$ 0.0665

6         0.0271    $\pm$ 0.0007        1.3469 $\pm$ 0.0479     0.6300   $\pm$ 0.1526

8         0.0241    $\pm$ 0.0012        1.234  $\pm$ 0.0702     0.4843 $\pm$ 0.2099

3*0.75               4         0.0273    $\pm$ 0.0006        1.3624 $\pm$ 0.0485     0.4885 $\pm$ 0.1032

6         0.0267    $\pm$ 0.0009        1.3373 $\pm$ 0.0566     0.5509 $\pm$ 0.1314

8         0.0237    $\pm$ 0.0005        1.2369 $\pm$ 0.065      0.5501 $\pm$ 0.1776

XGBoost models with different data subsample ratios, feature subsample ratios and max depths between  2014-07-04 (Era 601) and 2018-04-27 (Era 800)

Max Depth     L2-reg     L1-reg     Mean Corr     Sharpe     Calmar

9*4             3*0.0
0.0        0.0276    $\pm$ 0.0007        1.3829 $\pm$ 0.0515     0.5305 $\pm$ 0.1077

0.001      0.0274    $\pm$ 0.0006        1.3764 $\pm$ 0.0485     0.5161 $\pm$ 0.1213

0.1        0.0268    $\pm$ 0.0008        1.3283 $\pm$ 0.0627     0.4224 $\pm$ 0.0937

3*0.001      0.0        0.0276    $\pm$ 0.0007        1.3829 $\pm$ 0.0515     0.5305 $\pm$ 0.1077

0.001      0.0274    $\pm$ 0.0006        1.3764 $\pm$ 0.0485     0.5161 $\pm$ 0.1213

0.1        0.0268    $\pm$ 0.0008        1.3283 $\pm$ 0.0627     0.4224 $\pm$ 0.0938

3*0.1       0.0        0.0271    $\pm$ 0.0002        1.3489 $\pm$ 0.0253     0.4771 $\pm$ 0.0824

0.001      0.0274    $\pm$ 0.0004        1.368  $\pm$ 0.0358     0.4977 $\pm$ 0.1186

0.1        0.0274    $\pm$ 0.0004        1.3694 $\pm$ 0.0362     0.4833 $\pm$ 0.0918

9*6             3*0.0
0.0        0.0266    $\pm$ 0.0008        1.3354 $\pm$ 0.0453     0.5574 $\pm$ 0.1282

0.001      0.0267    $\pm$ 0.0011        1.3353 $\pm$ 0.061      0.5422 $\pm$ 0.1697

0.1        0.0267    $\pm$ 0.0011        1.3201 $\pm$ 0.0638     0.5665 $\pm$ 0.1578

3*0.001      0.0        0.0265    $\pm$ 0.0007        1.3347 $\pm$ 0.0441     0.5711 $\pm$ 0.1244

0.001      0.0268    $\pm$ 0.0011        1.3403 $\pm$ 0.0657     0.5184 $\pm$ 0.1226

0.1        0.0267    $\pm$ 0.0011        1.3201 $\pm$ 0.0638     0.5665 $\pm$ 0.1578

3*0.1        0.0        0.0269  $\pm$ 0.0009        1.3551 $\pm$ 0.0559     0.6187 $\pm$ 0.1823

0.001      0.0267    $\pm$ 0.0013        1.3307 $\pm$ 0.0818     0.4871 $\pm$ 0.1164

0.1        0.0269    $\pm$ 0.0009        1.3638 $\pm$ 0.0563     0.5304 $\pm$ 0.0598

9*8             3*0.0
0.0        0.0237    $\pm$ 0.0008        1.2226 $\pm$ 0.0592     0.5127 $\pm$ 0.0875

0.001      0.0238    $\pm$ 0.0003        1.2445 $\pm$ 0.0623     0.5269 $\pm$ 0.1949

0.1        0.0237    $\pm$ 0.0004        1.2408 $\pm$ 0.0633     0.5298 $\pm$ 0.0905

3*0.001      0.0        0.0238    $\pm$ 0.0007        1.243  $\pm$ 0.0942     0.5657 $\pm$ 0.2615

0.001      0.0238    $\pm$ 0.0004        1.2383 $\pm$ 0.0254     0.4905 $\pm$ 0.1505

0.1        0.0238    $\pm$ 0.0003        1.2429 $\pm$ 0.0748     0.6487 $\pm$ 0.1551

3*0.1        0.0        0.0235    $\pm$ 0.0006        1.2102 $\pm$ 0.0655     0.6036 $\pm$ 0.2693

0.001      0.024     $\pm$ 0.0003        1.2461 $\pm$ 0.0552     0.53   $\pm$ 0.1934

0.1        0.0235    $\pm$ 0.0008        1.2438 $\pm$ 0.1056     0.5426 $\pm$ 0.2102

XGBoost models with different L1 and L2 regularisation with fixed data and feature sub-sampling ratios of $75\%$ between  2014-07-04 (Era 601) and 2018-04-27 (Era 800)

§.§ Additional Results

[Mean Corr]

Performances, (a) Mean Corr, (b) Sharpe ratio, and (c) Calmar ratio of the deep IL XGBoost models with Jackknife feature sampling under different market regimes.

[Mean Corr]

Performances, (a) Mean Corr, (b) Sharpe ratio, and (c) Calmar ratio of the deep IL XGBoost models with random feature sampling under different market regimes.

[Mean Corr]

Performances, (a) Mean Corr, (b) Sharpe ratio, and (c) Calmar ratio of the deep IL XGBoost models with different training sizes under different market regimes.

[Mean Corr]

Performances, (a) Mean Corr, (b) Sharpe ratio, and (c) Calmar ratio of the deep IL XGBoost models with different learning rates under different market regimes.

[Mean Corr]

Performances, (a) Mean Corr, (b) Sharpe ratio, and (c) Calmar ratio of the deep IL XGBoost models with different targets and learning rates under different market regimes.

[Mean Corr]

Comparing the performances of two data sampling schemes $S_1$ and $S_2$ with different number of boosting rounds $B=500,1000,2500,5000$ for risk metrics (a) Mean Corr, (b) Sharpe ratio, (c) Calmar ratio, under different market regimes, over 10 different hyperparameter settings

[Mean Corr]

Comparing the performances of two data sampling schemes with different number of boosting rounds $B=500,1000,2500,5000$ for risk metrics (a) Mean Corr, (b) Sharpe ratio, (c) Calmar ratio, under different market regimes, using the Ansatz hyperparameters Tree Depth = 4 and Ratio of feature sampling per tree = 0.75.

Learning curves of XGBoost models with different learning rates for different number of boosting rounds $B=1000,2500,5000,50000$ for risk metric Mean Corr under different market regimes

Learning curves of XGBoost models with different learning rates for different number of boosting rounds $B=1000,2500,5000,50000$ for risk metric Sharpe ratio under different market regimes

Learning curves of XGBoost models with different learning rates for different number of boosting rounds $B=1000,2500,5000,50000$ for risk metric Calmar ratio under different market regimes

Learning curves of benchmark XGBoost models with different number of boosting rounds $B=1000,5000,10000,25000,50000$ for risk metric Mean Corr under different market regimes

Learning curves of benchmark XGBoost models with different number of boosting rounds $B=1000,5000,10000,25000,50000$ for risk metric Sharpe under different market regimes

Learning curves of benchmark XGBoost models with different number of boosting rounds $B=1000,5000,10000,25000,50000$ for risk metric Calmar under different market regimes

[EMA models]

[Feature Transform factor-timing models]

Sharpe ratio of EMA and Feature Transform factor-timing models under different market regimes

[EMA models]

[Feature Transform factor-timing models]

Calmar ratio of EMA and Feature Transform factor-timing models under different market regimes

Regime     Strategy     Mean Corr     Sharpe     Max Drawdown
5*Test      Example Model               0.0264      0.9626      0.2608
Baseline Model              0.0257      1.0983      0.1839
Tail Risk Model             0.0022      0.1607      0.2161
Static Hedged Model         0.0176      1.0507      0.0586
Dynamic Hedged Model        0.0224      1.2378      0.0415
5*Bull      Example Model               0.0307      1.2512      0.0693
Baseline Model              0.0294      1.3962      0.0493
Tail Risk Model             0.0011      0.0794      0.2161
Static Hedged Model         0.0195      1.1858      0.0504
Dynamic Hedged Model        0.0246      1.3731      0.0415
5*Bear      Example Model               -0.0060     -0.2306     0.2608
Baseline Model              -0.0018     -0.0831     0.1839
Tail Risk Model             0.0106      1.0225      0.0027
Static Hedged Model         0.0033      0.2970      0.0586
Dynamic Hedged Model        0.0057      0.7733      0.0184

Performances of Dynamic Hedged deep IL XGBoost ensemble model based on random feature sampling and V4.2 Example Model from Era 901 to Era 1070 under different market regimes.

Regime     Strategy     Mean Corr     Sharpe     Max Drawdown
5*Test      Example Model               0.0264      0.9626      0.2608
Baseline Model              0.0266      1.1559      0.1646
Tail Risk Model             0.0016      0.1068      0.3728
Static Hedged Model         0.0207      1.1337      0.0742
Dynamic Hedged Model        0.0225      1.2646      0.0330
5*Bull      Example Model               0.0307      1.2512      0.0693
Baseline Model              0.0301      1.4305      0.0351
Tail Risk Model             0.0007      0.0481      0.3728
Static Hedged Model         0.0227      1.2941      0.0380
Dynamic Hedged Model        0.0246      1.4305      0.0330
5*Bear      Example Model               -0.0060     -0.2306     0.2608
Baseline Model              0.0003      0.0151      0.1646
Tail Risk Model             0.0081      0.5826      0.0219
Static Hedged Model         0.0054      0.3409      0.0742
Dynamic Hedged Model        0.0069      0.4871      0.0312

Performances of Dynamic Hedged deep IL XGBoost ensemble model based on different training set sizes and V4.2 Example Model from Era 901 to Era 1070 under different market regimes.

Regime     Strategy     Mean Corr     Sharpe     Max Drawdown
5*Test      Example Model               0.0264      0.9626      0.2608
Baseline Model              0.0265      1.1943      0.1562
Tail Risk Model             0.0015      0.1044      0.1754
Static Hedged Model         0.0199      0.9978      0.1460
Dynamic Hedged Model        0.0207      1.0760      0.0871
5*Bull      Example Model               0.0307      1.2512      0.0693
Baseline Model              0.0300      1.5073      0.0343
Tail Risk Model             0.0011      0.0743      0.1754
Static Hedged Model         0.0227      1.2135      0.0377
Dynamic Hedged Model        0.0233      1.2755      0.0434
5*Bear      Example Model               -0.0060     -0.2306     0.2608
Baseline Model              -0.0001     -0.0053     0.1562
Tail Risk Model             0.0051      0.2925      0.0743
Static Hedged Model         -0.0008     -0.0483     0.1460
Dynamic Hedged Model        0.0015      0.0998      0.0871

Performances of Dynamic Hedged deep IL XGBoost ensemble model based on different learning rates and V4.2 Example Model from Era 901 to Era 1070 under different market regimes.

Regime     Strategy     Mean Corr     Sharpe     Max Drawdown
5*Test      Example Model               0.0264      0.9626      0.2608
Baseline Model              0.0247      1.1915      0.1026
Tail Risk Model             -0.0001      -0.0064      0.5444
Static Hedged Model         0.0189      0.9937      0.0401
Dynamic Hedged Model        0.0220      1.2399      0.0457
5*Bull      Example Model               0.0307      1.2512      0.0693
Baseline Model              0.0277      1.4344      0.0411
Tail Risk Model             -0.0017      -0.0990      0.5444
Static Hedged Model         0.0204      1.0636      0.0401
Dynamic Hedged Model        0.0234      1.3173      0.0457
5*Bear      Example Model               -0.0060     -0.2306     0.2608
Baseline Model              0.0020     0.1219     0.1026
Tail Risk Model             0.0118      0.8287      0.0148
Static Hedged Model         0.0078     0.5788     0.0383
Dynamic Hedged Model        0.0115      0.8475      0.0244

Performances of Dynamic Hedged deep IL XGBoost ensemble model based on different targets and V4.2 Example Model from Era 901 to Era 1070 under different market regimes.


# NASDUCK SERF: New constraints on axion-like dark matter
from a SERF comagnetometer

Itay M. Bloch  Berkeley Center for Theoretical Physics, University of
California, Berkeley, CA 94720, U.S.A.  Theory Group, Lawrence Berkeley
National Laboratory, Berkeley, CA 94720, U.S.A.  Roy Shaham  Rafael Ltd.,
31021 Haifa, Israel  Department of Physics of Complex Systems, Weizmann
Institute of Science, Rehovot 76100, Israel  Yonit Hochberg  Racah Institute
of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel  Eric
Kuflik  Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem
91904, Israel  Tomer Volansky  Department of Physics, Tel Aviv University, Tel
Aviv, Israel  Or Katz<EMAIL_ADDRESS>Duke Quantum Center, Duke University,
Durham, NC 27701  Department of Electrical and Computer Engineering, Duke
University, Durham, NC 27708

###### Abstract

Ultralight axion-like particles are well-motivated relics that might compose
the cosmological dark matter and source anomalous time-dependent magnetic
fields. We report on new terrestrial bounds on the coupling of axion-like
particles to neutrons and protons. The detector uses nuclei of noble-gas and
alkali-metal atoms and operates in the Spin-Exchange Relaxation-Free (SERF)
regime, achieving high sensitivity to axion-like dark matter fields.
Conducting a month-long search, we cover the mass range of $1.4\times
10^{-12}$ eV/$c^{2}$ to $2\times 10^{-10}$ eV/$c^{2}$ and provide world-
leading limits which supersede robust astrophysical bounds, and improve upon
previous terrestrial constraints by up to two orders of magnitudes for many
masses within this range. These are the first reliable terrestrial bounds
reported on the coupling of protons with axion-like dark matter, covering a
new and unexplored terrain in its parameter space.

##  I Introduction

It has been known for nearly a century that most of the matter in our universe
is non-luminous. The presence of this so called Dark Matter (DM) has been
confirmed by a wide range of observations on all scales. However, to date, all
observations rely on the DM’s gravitational interactions, precluding the
determination of its particle identity, including its mass and interactions. A
highly motivated and well-studied class of DM candidates are Axion-Like
Particles (ALPs) in the ultralight mass regime [1, 2]. Originally introduced
to solve the strong CP problem of the Standard Model of particle physics [3,
4, 5], the axion and its generalizations can explain the observed DM relic
abundance, and at the same time, predict feeble non-gravitational interactions
with visible matter which can be probed experimentally [1].

ALPs may interact with gluons or photons, or with fermions such as protons,
neutrons and electrons. Various experiments explore constraints on these
possible couplings of the ALP DM [6, 7, 8]. Most classical direct detection
experiments search for DM from masses at the ${\rm eV}/c^{2}$ scale and above
via absorption by or scattering from target materials [9]. While ultralight
ALPs cannot be detected using such techniques, significant progress has been
made recently in the search for them using experiments on earth [10, 11, 12,
13, 14, 15, 16, 17, 18, 19, 20, 21]. For light enough ALPs, the predicted
density is high enough so that it can be treated as a classical background
field which may be detectable by other means. In particular, when interacting
with fermions, the ALP exhibits itself as a coherent narrow-bandwidth time-
dependent, anomalous field which interacts with the spins of target fermions
[22].

Figure 1: Search for ALP dark matter using a SERF comagnetometer. (a) As the
Earth moves through the dark matter halo, matter on Earth can potentially
interact with the Axion-Like Particles (ALPs) dark matter in a non-
gravitational manner. Such interactions can manifest through the gradient
field ${\bf\mathcal{A}}={\bf\nabla}a$ of the ALP coupling to other particles.
(b) ALPs coupling to atoms. The coupling to alkali-metal and noble-gas nuclear
spins is manifested as an anomalous-magnetic field which drives the precession
of the spins. The coupling to noble-gas spins is predominantly through their
valence neutron whereas the coupling to alkali-metal nuclei is predominantly
through their proton hole in their nucleus [23]. (c) Detector configuration.
Spin-polarized 39K (red) and 3He (blue) gases are contained in a spherical
glass cell, and their precession at a constant magnetic field $B$ is optically
monitored. The magnetic field determines the resonance precession frequency of
the atoms and thus also the ALP masses for which the detector is sensitive.
(d) Operation in SERF regime. Operation with high potassium density renders
random spin-exchange collisions between pairs of 39K atoms very frequent, but
also suppresses their relaxation (Spin-Exchange Relaxation Free). Collisions
between 3He and 39K coherently couple the spin gases and enables to superpose
the signal of the 3He on the optically measured 39K.

Atomic magnetometers, Nuclear Magnetic Resonance (NMR) sensors and co-
magnetometers are terrestrial detectors which are utilized for the search of
ALPs-sourced anomalous fields and other signatures of physics beyond the
Standard Model [17, 12, 21, 24, 18, 25, 26, 19, 27, 28, 29, 11, 30, 31]. These
sensors use spin-polarized atoms in a gaseous, liquid, or solid phase which
collectively respond to the anomalous-magnetic field of the ALP and can thus
detect or constrain the couplings of ALP to fermions. Typically, the mass
range in these searches is set by the relaxation rate of the spins and their
resonance frequencies, which determine the range in which the spins are most
sensitive to oscillating fields.

Previous searches have utilized dual atomic species to search for ALP DM at
small masses, usually using nuclear spins as at least one of the species [12,
21, 32, 19, 33, 24]. They relied on simultaneous overlapping of the resonance
frequencies of the two species at the oscillating frequency of the ALP, which
limited the measurement bandwidth to $f\lesssim 1$ Hz (about $4\times
10^{-15}$ eV$/h$). Owing to the limited search duration (typically up to
several months), previous bounds were cast on $m_{\rm DM}\lesssim 4\times
10^{-12}$ eV$/c^{2}$ [34, 11, 32]. To constrain higher ALP masses one may
focus on the response of ALPs at a single broadband resonance and operate in
the SERF regime [35, 36, 37] to suppress non-magnetic noise sources. While
several techniques have been proposed previously [38], to this day none has
been directly implemented above $m_{\rm DM}>4\times 10^{-14}$ eV$/c^{2}$ [32].

Furthermore, the most stringent terrestrial constraints of the coupling
between ALPs and fermions in the ultra-light mass regime are based on atomic
detectors that use nuclear spins of noble gases. These spins are composed
predominantly by their valence neutrons, which enables measurable ALP-neutron
interactions. However, cases where the ALP-proton interaction dominates are
also theoretically motivated [39]. For instance, in models of hadronic axions
(e.g. the KSVZ model [40, 41]), ALPs do not couple to quarks and leptons at
high energies, and as a consequence their coupling to neutrons is small or
could possibly vanish, while their coupling to protons remains sizeable [39].
Constraining the coupling between ALPs and protons requires nuclear spins
whose proton contribution is large and known to a sufficient accuracy.
Therefore to this day, no reliable bounds on ALP DM interaction with protons
were reported in any terrestrial technique.

Here we report on new experimental constraints on ALP DM interactions with
protons and neutrons. The results rely on a month long search using a dual-
specie spin ensembles of polarized 3He and 39K (potassium) atoms. The former
is sensitive to ALP-neutron coupling and the latter to ALP-proton coupling,
utilizing the nonzero neutron or proton contributions to the spin in their
nuclei. Operating the detector in the Spin-Exchange Relaxation-Free (SERF)
regime with a high number-density for the potassium, we suppress the effect of
photon shot noise and improve on the current terrestrial limits of the
coupling to neutrons and protons by as much as two orders of magnitude in the
mass range $1.4\times 10^{-12}-2\times 10^{-10}$ eV/$c^{2}$. Moreover, barring
the uncertain supernova constraints, the ALP-proton bound improves on all
existing terrestrial and astrophysical limits, partially closing the region
for couplings in the range $5\times 10^{-6}{{\rm\ GeV^{-1}}}$ to $2\times
10^{-5}{{\rm\ GeV^{-1}}}$.

##  II ALP Interactions with Fermions

We use a gaseous mixture of alkali-metal and noble-gas spins whose nuclear
spins can couple to the ALPs as shown in Fig. 1. The interaction Hamiltonian
of ALP fields with noble-gas spins is given by

$H_{\textrm{ALP-He}}=g_{\textrm{N}}\bm{\mathcal{A}}\cdot\textbf{N}$ (1)

where $\textbf{N}=\sum_{n}\textbf{N}_{n}$ is the collective nuclear spin
operators of the noble-gas ensemble, summing over the operators of all noble-
gas spins in the measurement volume. The vector
$\bm{\mathcal{A}}=\bm{\nabla}a$ denotes the gradient of the ALP field $a$,
which is a stochastic variable whose spectral content depends on the energy
density and the velocity distribution of DM ${\bf v}_{\rm DM}$ [42], and is
concentrated in a narrow band of frequencies near $\omega_{\rm DM}=(m_{\rm
DM}c^{2}+\langle{\bf v}_{\rm DM}^{2}\rangle/2)/\hbar$. The factor
$g_{\textrm{N}}$ represents the coupling between ALPs and the particles that
compose the nuclear spin. For 3He (spin $1/2$), the dominant contribution
comes from its single valence neutron, such that
$g_{\textrm{N}}=\epsilon_{\textrm{N}}g_{a\textrm{NN}}$. We take
$\epsilon_{\textrm{N}}\approx 0.85$ for the fractional contribution of neutron
spin to the nuclear spin of 3He atoms [43]. $g_{a\textrm{NN}}$ is the ALP-
neutron coupling coefficient we aim to bound.

We use spins of alkali-metal atoms to detect the Helium response as an optical
SERF magnetometer, and concurrently, to constrain the coupling between ALPs
and protons as illustrated in Fig. 1b. ${}^{39}{\rm K}$ atoms have a single
valence electron (spin-$1/2$) as well as a nonzero nuclear spin. The
interaction Hamiltonian of ALP fields with alkali-metal nuclei is given by

$H_{\textrm{ALP-K}}=g_{\textrm{I}}\bm{\mathcal{A}}\cdot\textbf{I}.$ (2)

Here $\textbf{I}=\sum_{i}\textbf{I}_{i}$ denotes the collective nuclear spin
operator of the alkali-metal atoms, summing over all alkali-metal atoms in the
measurement volume. We use natural abundant potassium atoms (predominantly 39K
with spin-$3/2$) which have a proton hole that dominates the nuclear spin
[23]. We then cast $g_{\textrm{I}}=\epsilon_{\textrm{P}}g_{a\textrm{PP}}$
where $\epsilon_{\textrm{P}}\approx 0.16\pm 0.04$ is the fractional
contribution of proton spin to the nuclear spin for 39K where the uncertainty
denotes the spread of prediction by the models reviewed in Ref. [23]. These
models also report on uncertain and much smaller fractional contribution of
neutron spin, which is consistent with or is exactly zero; in this search we
neglect this contribution. $g_{a\textrm{PP}}$ is the ALP-proton coupling
coefficient we aim to bound.

Figure 2: Constraints on ALP-neutron and ALP-proton couplings. In this work we
use a NASDUCK collaboration SERF comagnetometer to constrain ALP couplings
over a broad range of ALP masses. We present 95% C.L. limits. The light
transparent red region shows the exclusion region for the ALP-neutron
couplings (left) and ALP-proton couplings (right). Due to the finite
resolution of the figure and given the dense set of measurements, the limits
appear as a bright red band. The width of this band denotes the strongest and
weakest values around each mass point. The precise tabulated bounds can be
found in [44]. The black solid line shows a binned average of the bound while
its 1$\sigma$ variation is shown as the red band (both calculated in log-log
space at a binning resolution of $1\%$ of the mass). Other terrestrial
constraints from searches for long-range forces (which do not search for dark
matter) [25, 45] (olive-green region) and from NASDUCK-Floquet [11] (blue-
line) are presented. In beige we depict the excluded regions from solar ALPs
unobserved at the Solar Neutrino Observatory (SNO) [46]; for the neutron
couplings in the left panel, we also show the complementary stellar
constraints from neutron stars cooling [47, 48, 49, 50] (exceeding the scale
of the figure). We do not show the model-dependent limits from supernova (SN)
cooling considerations and neutrino flux measurements [1, 51, 52], which would
constrain the entire range presented in the figure, since they notably rely on
the unknown SN collapse mechanism [53].

##  III Searching for New Physics with SERF Magnetometers

###  III.1 Apparatus

The detector is comprised of a naturally abundant potassium vapor and $1500$
Torr of 3He gas that are enclosed in a spherical glass cell111The cell also
contains $40$ Torr of $\textrm{N}_{2}$ gas to mitigate radiation trapping and
assist the optical pumping process. Further details on the apparatus are given
in Ref. [54]. as shown in Fig. 1c. The two spin ensembles are initially
unpolarized; we orient the spins by continuous optical-pumping of the
potassium spins, reaching a polarization degree of $P_{\textrm{K}}\approx
85\%$. Subsequently, the helium spins are polarized by collisions with the
optically-pumped potassium atoms, in a process known as spin-exchange optical-
pumping (SEOP) [55], reaching a polarization degree of $P_{\textrm{He}}\approx
20\%$. The spins are aligned along an externally applied magnetic field
$B_{z}\hat{z}$ and are magnetically shielded from the ambient field. The
magnetic field together with applied light-shifts and the spin-exchange shifts
generated by the two polarized spin gases determine the electron paramagnetic
resonance (EPR) frequency $\omega_{\textrm{K}}$ of the potassium spins and the
nuclear magnetic resonance (NMR) frequency $\omega_{\textrm{He}}$ of the
helium spins. We use an off-resonant optical probe to measure the collective
total spin of the alkali-metal atoms along the $\hat{x}$ direction via
polarimetry technique, using a pair of photo-diodes in a homodyne
configuration [56].

We operate the detector in the SERF regime, for which relaxation by spin-
exchange collisions between alkali-metal pairs and also by spin-destruction
collisions is highly suppressed [57, 58, 37, 59, 60]. This regime is realized
by using an elevated potassium density for which the EPR frequencies we
consider are much slower than the rapid rate of spin-exchange collisions
$R_{\mathrm{SE}}\approx 5\times 10^{5}\,\textrm{s}^{-1}$. It allows us to use
a large number of potassium atoms $N_{\textrm{K}}\approx 5\times 10^{14}$ to
increase the apparatus sensitivity, yet maintain a low decoherence rate; e.g.
at $\omega_{\textrm{K}}=10$ kHz, the measured magnetic decoherence rate222This
is the total decoherence rate including power broadening by continuous
optical-pumping, residual spin-exchange relaxation and other relaxation
processes. is $\Gamma_{\textrm{K}}=770$ Hz, about three orders of magnitudes
smaller than $R_{\mathrm{SE}}$. The sensitivity of the apparatus to magnetic
or anomalous fields oscillating near the EPR resonance is better than $\sim
2\,\text{ fT/}\sqrt{\text{Hz}}$, but at most search frequencies, the
sensitivity to anomalous fields is dominated by a magnetic-field noise floor
of about $10\,\text{ fT/}\sqrt{\text{Hz}}$ (see Methods).

###  III.2 Response to ALP field

We can describe the response of the collective spins of the alkali-metal and
noble-gas spins to the ALP field with the Bloch-equations. A weak oscillatory
field in the $xy$ plane
${\mathcal{A}}_{+}={\mathcal{A}}_{x}+i{\mathcal{A}}_{y}$ exerts a torque that
rotates the orientation of the collective spins of the two gases off the
$\hat{z}$ axis, and generates a steady precession. The response of the spins
depends on the amplitude of the ALP field and on the frequency difference of
$\omega_{\rm DM}$ from their magnetic resonance. Anomalous fields that couple
to neutrons tilt the helium spins and produce an oscillating spin component
with amplitude

$\langle\textrm{N}_{+}\rangle=\frac{P_{\textrm{He}}N_{\textrm{He}}}{2}\frac{g_{\textrm{N}}{\mathcal{A}}_{+}}{\omega_{\textrm{He}}-\omega_{\rm
DM}-i\Gamma_{\textrm{He}}}.$ (3)

Here
$\langle\textrm{N}_{+}\rangle=\langle\textrm{N}_{x}\rangle+i\langle\textrm{N}_{y}\rangle$
denotes the mean collective spin of the helium ensemble in the transverse
direction and $\Gamma_{\textrm{He}}<0.1$ Hz is the spin decoherence rate. In
our search range
$\omega_{\textrm{DM}}\gg\omega_{\textrm{He}}\gg\Gamma_{\textrm{He}}$.

Owing to the great separation between the EPR and NMR resonances considered in
this search
($\omega_{\textrm{{DM}}},\omega_{\textrm{K}}\gg\omega_{\textrm{He}}$) the
dynamics of the two spin gases is weakly-coupled [61, 54, 62, 63].
Consequently, the total collective spin of the potassium
$\langle\textbf{F}\rangle=\sum_{i}\left(\langle\textbf{I}_{i}\rangle+\langle{\textbf{S}}_{i}\rangle\right)$,
comprised of summation over the potassiums’ nuclear and electron spins,
simultaneously responds to the torque exerted by the precessing helium and by
its direct coupling to the ALP. Representing the collective spin of the
alkali-metal vapor in a complex form
$\langle\textrm{F}_{+}\rangle=\langle\textrm{F}_{x}\rangle+i\langle\textrm{F}_{y}\rangle$,
we derive the total spin response to ALP fields

$\langle\textrm{F}_{+}\rangle=\frac{P_{\textrm{K}}N_{\textrm{K}}}{2}\frac{(\zeta
g_{a\rm PP}-\xi g_{a\rm NN}){\mathcal{A}}_{+}}{\omega_{\textrm{K}}-\omega_{\rm
DM}-i\Gamma_{\textrm{K}}},$ (4)

where the unitless parameter $\xi$ depends inversely on $\omega_{\rm DM}$ and
varies in the search between $46$ at low frequencies to $0.29$ at high
frequencies and $\zeta=0.69$ (see Methods). The potassium spin component
$\langle F_{x}\rangle$ is directly detected by the optical probe, allowing to
simultaneously measure $g_{a\rm PP}$ and $g_{a\rm NN}$.

##  IV The Search

###  IV.1 Data acquisition

We searched for ALP fields with $f_{\rm DM}$ in the range of
$0.33-50\leavevmode\nobreak\ {\rm kHz}$ (corresponding to a mass range of
$1.4$ to $200\leavevmode\nobreak\ {\rm peV}/c^{2}$) by recording the
precession of the potassium in the absence of any applied oscillating fields.
We varied the magnetic field at few tens of discrete points, and at each point
recorded the detector’s response for a duration that is longer than the
coherence time of the ALP and is about $10^{7}-10^{8}$ oscillations of the EPR
frequency at that field. Over a one-month period, we completed 5 different
scans with different samplings of the magnetic field in the range
$B_{z}=10-70$ mG to gain sensitivity for ALP fields that oscillate in the
reported search range. Each measurement was preceded by initial pumping of the
helium spins at $B_{z}=70$ mG, and was preceded and appended by a calibration
of the magnetic response and the helium spin polarization. The first and
fourth scans were designated to optimize the analysis procedure and were not
used to cast bounds, as was decided before unblinding.

###  IV.2 Search results

We use the log-likelihood ratio test to constrain the presence of ALP DM with
95% confidence level (C.L.) bounds, presented in Fig 2 for the ALP-neutron
coupling (left) and ALP-proton coupling (right). The stochastic nature of the
ALPs was treated using the method of Ref. [11] (see also Refs. [10, 26, 64]
for similar procedures). The search only aims to exclude ALPs and does not
attempt at a possible discovery. These constraints cover the mass range
between $1.4\leavevmode\nobreak\ {\rm peV}/c^{2}$ and $200\leavevmode\nobreak\
{\rm peV}/c^{2}$, measured with a resolution slightly higher than the line-
width of the ALP wavepacket (about $3\times 10^{-7}\leavevmode\nobreak\ m_{a}$
taking $m_{a}$ as the ALP mass). While the entire bounds as a function of mass
is tabulated in Ref. [44], we illustrate the limits by their geometric mean in
log-spaced bins of size $1\%$ of the mass (black lines), surrounded by the
standard deviation (computed in log-space) spread of bounds in the same bins
(red), further surrounded by the minimal and maximal bounds found in each bin
(bright red). All couplings above the bright red bands are excluded (light
transparent red region).

The olive green regions show constraints from other long-range searches on
ALP-neutron [25] and ALP-proton [45] couplings (which do not search for dark
matter). The blue line is the mean exclusion line of the NASDUCK-Floquet
experiment [11]. Our results provide complementary probes to stellar
constraints. The beige regions come from searches for ALP-emission from the
sun by the Solar Neutrino Observatory [46] and from neutron star (NS) cooling
considerations [47, 48, 49, 50]. The latter extends throughout the entire
plotted region of the ALP-neutron coupling. For both neutron and proton
couplings, the shown regions are constrained by model-dependent supernova (SN)
cooling considerations or neutrino flux measurements from SN1987A [1, 51, 52].
Since the theoretical reliability of such limits is arguable due to the uknown
SN collapse mechanism [53], we do not plot them here. The newly derived limits
of this search, dubbed NASDUCK-SERF, on the ALP couplings to neutrons, improve
the existing terrestrial limits over nearly the entire mass range by up to two
orders of magnitude, providing a strong complementary probe to stellar
constraints from NS and SN. Furthermore, our bound on ALP-protons interactions
in a large part of the mass range from ${2\leavevmode\nobreak\ {\rm
peV}/c^{2}}$ to $100\leavevmode\nobreak\ {\rm peV}/c^{2}$ lies in a region
that is otherwise only excluded by model-dependent arguable SN constraints.

##  V Discussion and Outlook

It is interesting to compare the operation and sensitivity of our detector
with self-compensating co-magnetometers that hold the strongest terrestrial
bounds on neutron coupling at low frequencies [12, 65, 66]. Self-compensating
co-magnetometers use similar mixtures of alkali-metal and noble-gas spins to
detect anomalous fields but set the EPR frequency $\omega_{\textrm{K}}$ near
resonance with the NMR frequency $\omega_{\textrm{He}}$ to realize damped and
near-critically coupled dynamics. This operation enhances the signal to noise
ratio for detecting oscillating anomalous fields that act on the noble-gas
spins at very low frequencies below $\omega_{\rm DM}\lesssim 10$ Hz, by
suppressing magnetic noises [67]. However, for the range of ALP frequencies
that we consider in this work, $\omega_{\rm DM}\gg\omega_{\textrm{He}}$ such
that the low-frequency enhancement and suppression mechanisms are rendered
ineffective, and neither the alkali-metal nor the noble-gas spins are resonant
with the driving field. Our detector instead, sets the EPR resonance near the
frequency of the searched ALP, and by that fully exploits the high sensitivity
of the SERF magnetometer. This operation therefore improves the sensitivity to
anomalous fields by about a factor of $\omega_{\rm DM}/\Gamma_{\textrm{K}}$
compared to self-compensating co-magnetometers, which is more than 100-fold at
the high-frequency end of our search range.

The constraints set by our search are predominantly limited by the magnetic
field noise that is produced by our innermost magnetic shield layer.
Replacement of this layer by a low-noise material (e.g. MnZn feritte [68]) is
expected to reduce the magnetic noise level by a factor of ${\cal O}(5)$.
Furthermore, it is possible to improve the bounds on neutrons by using a
denser noble-gas and hybrid SEOP, which are expected to further improve the
noble-gas spin magnetization.

###### Acknowledgements.

Acknowledgments. The work of YH is supported by the Israel Science Foundation
(grants No. 1112/17 and 1818/22), by the Binational Science Foundation (grant
No. 2016155), by the Azrieli Foundation and by an ERC STG grant (grant No.
101040019). EK is supported by the US-Israeli Binational Science Foundation
(grant No. 2020220) and by the Israel Science Foundation (grant No. 1111/17).
TV is supported by the Israel Science Foundation (grant No. 1862/21), by the
Binational Science Foundation (grant No. 2020220) and by the European Research
Council (ERC) under the EU Horizon 2020 Programme (ERC-CoG-2015 - Proposal n.
682676 LDMThExp). This project has received funding from the European Research
Council (ERC) under the European Union’s Horizon Europe research and
innovation programme (grant agreement No. 101040019). Views and opinions
expressed are however those of the author(s) only and do not necessarily
reflect those of the European Union. The European Union cannot be held
responsible for them.

## Methods

###  V.1 Spin dynamics in the SERF regime

The electron and nuclear spins of the potassium are coupled via the strong
hyperfine interaction
$H_{\textrm{hpf}}=A_{\textrm{hpf}}\textbf{I}_{i}\cdot\textbf{S}_{i}$, which
sets the total spin in the electronic ground-state
$\textbf{F}_{i}=\textbf{I}_{i}+\textbf{S}_{i}$ of the $i$th atom as an
operator with good quantum numbers. We can describe the dynamics of the
collective spin of the potassium ensemble $\textbf{F}=\sum_{i}\textbf{F}_{i}$
and the helium spin ensemble $\textbf{N}=\sum_{i}\textbf{N}_{i}$ in the $xy$
plane using the coupled Bloch equations [69]

$\displaystyle\frac{d}{dt}\langle\textbf{F}\rangle$
$\displaystyle=-\left(\gamma_{e}\textbf{B}+\frac{2qA_{\textrm{He}}}{N_{\textrm{He}}}\langle\textbf{N}\rangle\right)\times\langle\textbf{S}\rangle-
g_{\textrm{I}}\bm{\mathcal{A}}\times\langle\textbf{I}\rangle-\Gamma_{\textrm{K}}\langle\textbf{F}\rangle,$
(5) $\displaystyle\frac{d}{dt}\langle\textbf{N}\rangle$
$\displaystyle=-\left(\gamma_{\textrm{N}}\textbf{B}+\frac{2A_{\textrm{K}}}{N_{\textrm{K}}}\langle\textbf{S}\rangle\right)\times\langle\textbf{N}\rangle-
g_{\textrm{N}}\bm{\mathcal{A}}\times\langle\textbf{N}\rangle-\Gamma_{\textrm{He}}\langle\textbf{N}\rangle.$
(6)

Here $\gamma_{\textrm{e}}$, $\gamma_{\textrm{N}}$ are the gyro-magnetic ratios
of the bare electron and helium spins respectively, and
$\textbf{B}=B_{z}\hat{\textbf{z}}$ is the magnetic field.
${q}(P_{\textrm{K}})$ is the slowing-down-factor [70], monotonically
decreasing from $q(P_{\textrm{K}}=0)=6$ to $q(P_{\textrm{K}}=1)=4$ as a
function of $P_{\textrm{K}}$, the potassium degree of polarization. It is
experimentally determined by measurement of the gyro-magnetic ratio of the
potassium in our setup, giving $q=4.3$. $A_{\textrm{He}}$ ($A_{\textrm{K}}$)
are the total spin-exchange shifts of the magnetic levels had all helium
(potassium) spins were completely polarized, i.e.
$|\langle\textbf{N}\rangle|=N_{\textrm{He}}/2$
($|\langle\textbf{S}\rangle|=N_{\textrm{He}}/2$), and in our experiment
$A_{\textrm{He}}\approx 18$ kHz and $A_{\textrm{K}}\approx 0.8$ Hz. We denote
the collective spins of the electrons and nuclei of alkali-metal atoms
$\langle\textbf{S}\rangle=\sum_{i}\langle\textbf{S}_{i}\rangle$ and
$\langle\textbf{I}\rangle=\sum_{i}\langle\textbf{I}_{i}\rangle$. We denote by
$\Gamma_{\mathrm{He}}$ and $\Gamma_{\mathrm{K}}$ the transverse relaxation
rates of the helium and potassium spins respectively.

The ALP fields interacting with the alkali-metal nuclei exert a torque on the
collective nuclear spin
$\langle\textbf{I}\rangle=\sum_{i}\langle\textbf{I}_{i}\rangle$. Here we
consider a dense alkali ensemble in the SERF regime, for which rapid spin-
exchange collisions between pairs of alkali-metal atoms drive the spin state
to follow a spin temperature distribution, and correlate the mean collective
spins by
$\langle\textbf{F}\rangle=q\langle\textbf{S}\rangle=\tfrac{q}{q-1}\langle\textbf{I}\rangle$
[70]. The number of polarized atoms are
$P_{\textrm{K}}N_{\textrm{K}}=2|\langle\textbf{S}\rangle|$ for the potassium
and $P_{\textrm{He}}N_{\textrm{He}}=2\langle\textbf{N}\rangle$ for the helium
[69, 71]. In our experimental setup, the spins are nearly aligned with the $z$
direction, such that
$|\langle\textbf{S}\rangle|\approx|\langle\textrm{S}_{z}\rangle|$ and
$|\langle\textbf{N}\rangle|\approx|\langle\textrm{N}_{z}\rangle|$.

In the regime we operate the apparatus, the EPR frequency associated with the
precession of the total alkali-metal spin
$\omega_{\textrm{K}}=|\tfrac{1}{q}B\gamma_{\textrm{e}}+P_{\textrm{He}}A_{\textrm{He}}|$
is about two orders of magnitude larger than the NMR frequency of the helium
spins
$\omega_{\textrm{He}}=|\gamma_{\textrm{N}}B+P_{\textrm{K}}A_{\textrm{K}}|$.
This different scaling originates from the difference in the gyromagnetic
ratios, and the application of magnetic field that is aligned with spin-
exchange field produced by the helium ($A_{\textrm{He}}$), in contrast with
the operation of self compensating co-magnetometers which use an inverted
magnetic field [63, 65]. Considering the spectral response of the helium spins
to oscillating dark matter fields that are near the resonance of the potassium
spins, we obtain Eq. (3) as the fourier transform of Eq. (6) after neglecting
the small effect of $\langle\textrm{S}_{+}\rangle$ on the noble-gas dynamics.

The response of the transverse collective spin $\langle\textrm{F}_{+}\rangle$
can similarly be represented in the frequency domain by

$\langle\text{F}_{+}(\omega)\rangle=\frac{P_{\textrm{K}}N_{\textrm{K}}}{2}\frac{\frac{2qA_{\textrm{He}}}{N_{\textrm{He}}}\langle\text{N}_{+}(\omega)\rangle+g_{\textrm{I}}\left(q-1\right)\mathcal{A}_{+}(\omega)}{\omega_{\textrm{K}}-\omega-i\Gamma_{\textrm{K}}}.$
(7)

Substitution of Eq. (3) in Eq. (7) directly yields Eq. (4), with the effective
coupling coefficient reading

$g_{\rm eff}\equiv(q-1)\epsilon_{\rm P}g_{a\rm PP}-\xi g_{a\rm NN}.$ (8)

We can therefore identify the first unitless coefficient
$\zeta=(q-1)\epsilon_{\rm P}\approx 0.69$ and the second unitless coefficient
as

$\xi(\omega)=\frac{q\epsilon_{\rm
N}P_{\textrm{He}}A_{\textrm{He}}}{\omega-\omega_{\textrm{He}}+i\Gamma_{\textrm{He}}}\approx\frac{q\epsilon_{\rm
N}P_{\textrm{He}}A_{\textrm{He}}}{\omega}.$ (9)

The last approximation pertains to the spectral content far from the helium
resonance, because in our search range
$\omega\gg\omega_{\textrm{He}},\Gamma_{\textrm{He}}$ as $\omega$ is near
$\omega_{\textrm{K}}$.

###  V.2 Background and signal sensitivity

The detector is sensitive to both real and anomalous magnetic fields. In this
section, we present and characterize the noise model for the detector and
analyze the sources of noise that limit its detection sensitivity in most
frequencies. The dominant sources of noise are the magnetic field noise and
noise that is associated with our optical detection, primarily polarization
noise of the probe beam that is associated with its shot noise.

The magnetometer signal is proportional to the collective spin component
$\langle F_{x}\rangle=\textrm{Re}\left(\langle F_{+}\rangle\right)$ whose
response to ALPs is given in Eq. (4). In the presence of noise, the collective
spin measured by the detector reads as $\langle F_{+}\rangle\rightarrow\langle
F_{+}\rangle+\delta F_{+}$ where the noise-driven contribution at frequency
$\omega\gg\omega_{\textrm{He}},\Gamma_{\textrm{He}}$ is given by

$\delta
F_{+}(\omega)=\left(\gamma_{e}-\frac{\xi(\omega)\gamma_{\textrm{He}}}{\epsilon_{\rm
N}}\right)\frac{P_{\textrm{K}}N_{\textrm{K}}}{2}\frac{\delta
B_{+}(\omega)}{\omega_{\textrm{K}}-\omega-i\Gamma_{\textrm{K}}}+W(\omega).$
(10)

The first term describes the effect of magnetic field noise $\delta
B_{+}=\delta B_{x}+i\delta B_{y}$ at frequency $\omega$ when included in Eqs.
(5-6) by $\textbf{B}=\delta B_{x}\hat{\textbf{x}}+\delta
B_{y}\hat{\textbf{y}}+B_{z}\hat{\textbf{z}}$. It describes tilting of the
total potassium spin that is driven by coupling of the electron spin to the
magnetic field (with gyromagnetic ratio $\gamma_{\textrm{e}}=2.8$ MHz/G, first
term in Eq. 10), or to the transverse spin-exchange shift that is exerted by
the tilted collective spin of the helium (with gyromagnetic-ratio
$\gamma_{\textrm{He}}=3.2$ kHz/G, second term in Eq. 10). The last term
$W(\omega)$ denotes the technical noise originating primarily from measurement
of the probe beam.

In Fig. 3 we exemplify the noise characteristics of the detector by presenting
the square root of the power spectral density (PSD) of two recordings using
Welch’s method. The recordings were taken at two different magnetic fields
$B_{z}=19\leavevmode\nobreak\ {\rm mG}$ (pink circles) and
$B_{z}=50\leavevmode\nobreak\ {\rm mG}$ (brown circles) for $5$ seconds each.
The two noise spectra are presented in the raw units of the measuring device.
The detector response to transverse oscillating magnetic-fields at these
conditions are independently measured, and shown with dashed, and dotted-
dashed lines respectively. We find an approximately white magnetic noise floor
of about $\sim 10\,\text{ fT/}\sqrt{\text{Hz}}$ accompanied with several
narrow spikes, and also technical noise floor of about $\sim 2\,\text{
fT/}\sqrt{\text{Hz}}$. The magnetic noise is consistent with the estimated
noise produced by our innermost magnetic shield in the cell location [72], and
is the limiting noise of our near-resonance measurement at most frequencies.
The technical noise variance determines the realized sensitivity of our
detector; It is proportional to the power of the probe beam and is considered
as spin-independent because it is almost unchanged by turn off of the pump
beam.

Figure 3: Noise spectral density and magnetic response. Measured noise
spectral density $\rm PSD$ of the detector (left axis) at two different
magnetic fields in the search ($B_{z}=19$ mG, pink circles and $B_{z}=50$ mG,
brown circles). Apart for some peaks, the spectrum generally follows an offset
Lorentzian profile, owing to magnetic field noise and limited sensitivity of
the detector. Dashed and dotted-dashed curves show the measured detector
response to weakly oscillating transverse magnetic fields along the $y$
direction for $B_{z}=19,\,50$ mG respectively (right axis).

###  V.3 Calibrations

We have routinely calibrated the slowing down factor $q$ by measurement of the
EPR frequency as a function of $B_{z}$. The slope of the linear fit yields the
gyromagnetic ratio of the polarized potassium which corresponds to
$\gamma_{\textrm{e}}/q$.

We estimated the spin-exchange field produced by the helium spins using the
measured data as detailed in the data processing section. This estimation was
validated by a measurement of the EPR the frequency during the spin-exchange
optical-pumping stage (several hours process, preceded by application of
strong magnetic fields that zerored the initial helium polarization). We
fitted it to the function

$\omega_{\mathrm{K}}(t)=\omega_{\mathrm{K}}(0)+A_{\textrm{He}}P_{\textrm{He}}(1-e^{-t/\tau}).$
(11)

We calibrated the sensitivity of the magnetometer to weakly oscillating
magnetic fields at the beginning and at the end of every measurement round, in
between changes of the magnetic field value. We applied an additional
calibration field $B=B_{0}\cos(\omega_{y}t)\hat{y}$ with amplitude
$B_{0}=2.7{\mu{\rm G}}$, and measured the response for several $\omega_{y}$
sampled around the EPR resonance. This measurement enabled the construction of
the spectral form of the response function. The width and calibration factor
remained very stable ($<10\%$ drift) during a single measurement. The EPR
resonance however was decreased during the measurement because of temporal
change in $P_{\textrm{He}}(t)$; While the helium spins were polarized at
$B=70$ mG at the onset of each measurement, at lower magnetic fields the
decoherence of the helium spins rate was moderately increased, primarily due
to spatial inhomogeneity in the alkali-metal spins polarization [54]. The
temporal variation in the spin-exchange field which shifted the EPR frequency
was included in the analysis.

###  V.4 Data processing

We analyzed the data using the log-likelihood ratio test to exclude the
presence of ALPs at frequency $\omega_{\rm DM}$ with a width determined by the
signal coherence time and the effects of earth’s rotation on the sensitive
axes of the detector. To accurately account for the velocity distribution of
the Dark Matter, we carried the analysis procedure that is similar to Ref.
[11] except for a few changes listed below. In short, we take the total ALP
field as a superposition of the plane-waves representing the individual
particles, whose momentum distribution follows the Standard Halo Model [42]
and phases are uncorrelated and randomly distributed. To calculate the signal
generated by the total ALP gradient, we compute the response of the detector
in the frequency domain (which is represented by the $\alpha$ matrices as
defined in Ref. [11]). Once the spectral content of an ALP has been
determined, we use frequencies outside the predicted spectral content of the
ALP to estimate the level of noise for the white noise hypothesis used to
generate the bounds.

All analysis procedures and cuts were designed in a blind fashion, and decided
in advance before looking at the data, to eliminate bias. We emphasize that
this is an exclusion search, and no discovery was attempted. We carried a
similar procedure for the quality cuts as described in Ref. [11], primarily to
avoid a change in the analysis procedure, which presumably mitigated low-
frequency noise; Indeed the noise spectrum of the unblinded data was nearly
unchanged. We also note that owing to the long measurement time, spectral
representation of the ALP required fewer points that in Ref. [11], rendering
its computation more efficient.

The magnetization of the helium has decayed during the measurement, owing to
its magnetic-field dependent lifetime. This rendered both $\xi$ and
$\omega_{\textrm{K}}$ time-dependent, slowly-varying throughout the
measurement at the scale of many minutes. Because this time scale is much
longer than $1/\omega_{\rm DM}$, we assumed that $P_{\rm He}$, and
$\omega_{\rm K}$ change adiabatically, such that Eqs. (3-7) remain valid at
short time scales. We first classified if a decay is negligible based on the
following criterion: we compared the EPR frequencies of the magnetic-response
function measured at the calibration preceding and appending the recording and
checked for variation that is greater than $200$ Hz. This criterion classified
most runs with $\omega_{\rm K}(B_{z})\gtrsim 20\leavevmode\nobreak\ {\rm kHz}$
as cases with negligible decay, where we took the average of the two EPR
frequencies and the average of helium polarizations as constant values (note
that in this regime $\Gamma_{\mathrm{K}}\gtrsim 800$ Hz). For the other cases,
we have estimated the spin-exchange field of the helium spins as a function of
time; we computed the spectrum of the data in a window of about one second
every few seconds, and fitted it to an amalgam of the response functions
(taken during the calibration stages) whose central frequency follows the
central peak of the noise spectrum. Because the fitting function is several
orders of magnitude wider in frequency than ALP signals, this procedure
introduces no bias to the search. The time-dependence of the response was then
used in computation of $\alpha$ (see [11]).

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