Constraints on axion-like dark matter from a SERF comagnetometer

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 terrestrial bounds from the Noble And Alkali Spin Detectors for Ultralight Coherent darK matter (NASDUCK) collaboration 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 × 10−12 eV/c2 to 2 × 10−10 eV/c2 and provide limits which supersede robust astrophysical bounds, and improve upon previous terrestrial constraints by over two orders of magnitude for many masses within this range for protons, and up to two orders of magnitude for neutrons. These are the sole reliable terrestrial bounds reported on the coupling of protons with axion-like dark matter, covering an 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 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 * or.katz@duke.edu coherent narrow-bandwidth time-dependent, anomalous field which interacts with the spins of target fermions [22].
Atomic magnetometers, Nuclear Magnetic Resonance (NMR) sensors and co-magnetometers are terrestrial detectors which are utilized for the search of ALPssourced anomalous fields and other signatures of physics beyond the Standard Model [11, 12, 17-19, 21, 24-31]. 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 [12,19,21,24,32,33]. 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 1 Hz (about 4 × 10 −15 eV/h). Owing to the limited search duration (typically up to several months), previous bounds were cast on m DM 4×10 −12 eV/c 2 [11,32,34]. 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 nonmagnetic noise sources. While several techniques have been proposed previously [38], to this day none has been directly implemented above m DM > 4×10 −14 eV/c 2 [32].
Furthermore, the most stringent terrestrial constraints of the coupling between ALPs and fermions in the ultralight 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 Such interactions can manifest through the gradient field A = ∇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 39 K (red) and 3 He (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 39 K atoms very frequent, but also suppresses their relaxation (Spin-Exchange Relaxation Free). Collisions between 3 He and 39 K coherently couple the spin gases and enables to superpose the signal of the 3 He on the optically measured 39 K.
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 3 He and 39 K (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 × 10 −12 − 2 × 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 × 10 −6 GeV −1 to 2 × 10 −5 GeV −1 .

II. ALP INTERACTIONS WITH FERMIONS
We use a gaseous mixture of alkali-metal and noblegas 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 where N = n 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 A = ∇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 v DM [42], and is concentrated in a narrow band of frequencies near ω DM = (m DM c 2 + v 2 DM /2)/ . The factor g N represents the coupling between ALPs and the particles that compose the nuclear spin. For 3 He (spin 1/2), the dominant contribution comes from its single valence neutron, such that g N = N g aNN . We take N ≈ 0.85 for the fractional contribution of neutron spin to the nuclear spin of 3 He atoms [43]. g aNN 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 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 Here I = i 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 39 K with spin-3/2) which have a proton hole that dominates the nuclear spin [23]. We then cast g I = P g aPP where P ≈ 0.16 ± 0.04 is the fractional contribution of proton spin to the nuclear spin for 39 K 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 aPP is the ALP-proton coupling coefficient we aim to bound.

A. Apparatus
The detector is comprised of a naturally abundant potassium vapor and 1500 Torr of 3 He gas that are enclosed in a spherical glass cell 1 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 K ≈ 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 He ≈ 20%. The spins are aligned along an externally applied magnetic field B zẑ and are magnetically shielded from the ambient field. The magnetic field together with applied lightshifts and the spin-exchange shifts generated by the two polarized spin gases determine the electron paramagnetic resonance (EPR) frequency ω K of the potassium spins and the nuclear magnetic resonance (NMR) frequency ω He of the helium spins.
We use an off-resonant optical probe to measure the collective total spin of the alkali-metal atoms along thex 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 alkalimetal pairs and also by spin-destruction collisions is highly suppressed [37,[57][58][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 SE ≈ 5 × 10 5 s −1 . It allows us to use a large number of potassium atoms N K ≈ 5 × 10 14 to increase the apparatus sensitivity, yet maintain a low decoherence rate; e.g. at ω K = 10 kHz, the measured magnetic decoherence rate 2 is Γ K = 770 Hz, about three orders of magnitudes smaller than R SE . The sensitivity of the apparatus to magnetic or anomalous fields oscillating near the EPR resonance is better than ∼ 2 fT/ √ Hz, but at most search frequencies, the sensitivity to anomalous fields is dominated by a magnetic-field noise floor of about 10 fT/ √ Hz (see Methods).

B. 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 A + = A x + iA y exerts a torque that rotates the orientation of the collective spins of the two gases off theẑ 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 ω DM from their magnetic resonance. Anomalous fields that couple to neutrons tilt the helium spins and produce an oscillating spin component with amplitude Here N + = N x + i N y denotes the mean collective spin of the helium ensemble in the transverse direction and Γ He < 0.1 Hz is the spin decoherence rate. In our search range ω DM ω He Γ He . Owing to the great separation between the EPR and NMR resonances considered in this search (ω DM , ω K ω He ) the dynamics of the two spin gases is weaklycoupled [54,[61][62][63]. Consequently, the total collective spin of the potassium F = i ( I i + S i ), comprised of summation over the potassiums' nuclear and electron spins, simultaneously responds to the torque exerted by 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 1σ 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].
the precessing helium and by its direct coupling to the ALP. Representing the collective spin of the alkali-metal vapor in a complex form F + = F x + i F y , we derive the total spin response to ALP fields where the unitless parameter ξ depends inversely on ω DM and varies in the search between 46 at low frequencies to 0.29 at high frequencies and ζ = 0.69 (see Methods). The potassium spin component F x is directly detected by the optical probe, allowing to simultaneously measure g aPP and g aNN .

A. Data acquisition
We searched for ALP fields with f DM in the range of 0.33 − 50 kHz (corresponding to a mass range of 1.4 to 200 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.

B. 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 peV/c 2 and 200 peV/c 2 , measured with a resolution slightly higher than the line-width of the ALP wavepacket (about 3 × 10 −7 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 logspace) 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 ALPproton [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 peV/c 2 to 100 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 comagnetometers that hold the strongest terrestrial bounds on neutron coupling at low frequencies [12,65,66]. Selfcompensating co-magnetometers use similar mixtures of alkali-metal and noble-gas spins to detect anomalous fields but set the EPR frequency ω K near resonance with the NMR frequency ω He to realize damped and nearcritically 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 ω DM 10 Hz, by suppressing magnetic noises [67]. However, for the range of ALP frequencies that we consider in this work, ω DM ω 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 ω DM /Γ K compared to selfcompensating 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 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 noblegas spin magnetization.

A. Spin dynamics in the SERF regime
The electron and nuclear spins of the potassium are coupled via the strong hyperfine interaction H hpf = A hpf I i · S i , which sets the total spin in the electronic ground-state F i = I i + S i of the ith atom as an operator with good quantum numbers. We can describe the dynamics of the collective spin of the potassium ensemble F = i F i and the helium spin ensemble N = i N i in the xy plane using the coupled Bloch equations [69] Here γ e , γ N are the gyro-magnetic ratios of the bare electron and helium spins respectively, and B = B zẑ is the magnetic field. q(P K ) is the slowing-down-factor [70], monotonically decreasing from q(P K = 0) = 6 to q(P K = 1) = 4 as a function of P 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 He (A K ) are the total spin-exchange shifts of the magnetic levels had all helium (potassium) spins were completely polarized, i.e. | N | = N He /2 (| S | = N He /2), and in our experiment A He ≈ 18 kHz and A K ≈ 0.8 Hz. We denote the collective spins of the electrons and nuclei of alkali-metal atoms S = i S i and I = i I i . We denote by Γ He and Γ 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 I = i I i . 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 F = q S = q q−1 I [70]. The number of polarized atoms are P K N K = 2| S | for the potassium and P He N He = 2 N for the helium [69,71]. In our experimental setup, the spins are nearly aligned with the z direction, such that | S | ≈ | S z | and | N | ≈ | N z |.
In the regime we operate the apparatus, the EPR frequency associated with the precession of the total alkalimetal spin ω K = | 1 q Bγ e + P He A He | is about two orders of magnitude larger than the NMR frequency of the helium spins ω He = |γ N B + P K A 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 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 S + on the noble-gas dynamics.
The response of the transverse collective spin F + can similarly be represented in the frequency domain by Substitution of Eq. (3) in Eq. (7) directly yields Eq. (4), with the effective coupling coefficient reading We can therefore identify the first unitless coefficient ζ = (q − 1) P ≈ 0.69 and the second unitless coefficient as The last approximation pertains to the spectral content far from the helium resonance, because in our search range ω ω He , Γ He as ω is near ω K .

B. 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 F x = Re ( F + ) whose response to ALPs is given in Eq. (4). In the presence of noise, the collective spin measured by the detector reads as F + → F + + δF + where the noise-driven contribution at frequency ω ω He , Γ He is given by The first term describes the effect of magnetic field noise δB + = δB x +iδB y at frequency ω when included in Eqs. (5)(6) by B = δB xx + δB yŷ + B zẑ . It describes tilting of the total potassium spin that is driven by coupling of the electron spin to the magnetic field (with gyromagnetic ratio γ e = 2.8 MHz/G, first term in Eq. 10), or to the transverse spinexchange shift that is exerted by the tilted collective spin of the helium (with gyromagnetic-ratio γ He = 3.2 kHz/G, second term in Eq. 10). The last term W (ω) 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 mG (pink circles) and B z = 50 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 ∼ 10 fT/ √ Hz accompanied with several narrow spikes, and also technical noise floor of about ∼ 2 fT/ √ 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.

C. 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 γ 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 spinexchange 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 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(ω y t)ŷ with amplitude B 0 = 2.7µG, and measured the response for several ω 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 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.

D. Data processing
We analyzed the data using the log-likelihood ratio test to exclude the presence of ALPs at frequency ω 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 α 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 ξ and ω K time-dependent, slowly-varying throughout the measurement at the scale of many minutes. Because this time scale is much longer than 1/ω DM , we assumed that P He , and ω K change adiabatically, such that Eqs. (3)(4)(5)(6)(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 ω K (B z ) 20 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 Γ K 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 α (see [11]).