Search for topological defect dark matter using the global network of optical magnetometers for exotic physics searches (GNOME)
Samer Afach, Ben C. Buchler, Dmitry Budker, Conner Dailey, Andrei Derevianko, Vincent Dumont, Nataniel L. Figueroa, Ilja Gerhardt, Zoran D. Gruji?, Hong Guo, Chuanpeng Hao, Paul S. Hamilton, Morgan Hedges, Derek F. Jackson Kimball, Dongok Kim, Sami Khamis, Thomas Kornack, Victor Lebedev, Zheng-Tian Lu, Hector Masia-Roig, Madeline Monroy, Mikhail Padniuk, Christopher A. Palm, Sun Yool Park, Karun V. Paul, Alexander Penaflor, Xiang Peng, Maxim Pospelov, Rayshaun Preston, Szymon Pustelny, Theo Scholtes, Perrin C. Segura, Yannis K. Semertzidis, Dong Sheng, Yun Chang Shin, Joseph A. Smiga, Jason E. Stalnaker, Ibrahim Sulai, Dhruv Tandon, Tao Wang, Antoine Weis, Arne Wickenbrock, Tatum Wilson, Teng Wu, David Wurm, Wei Xiao, Yucheng Yang, Dongrui Yu, Jianwei Zhang
SSearch for topological defect dark matter using the global network of opticalmagnetometers for exotic physics searches (GNOME)
Samer Afach,
1, 2
Ben C. Buchler, Dmitry Budker,
1, 2, 4
Conner Dailey, ∗ Andrei Derevianko, Vincent Dumont, Nataniel L. Figueroa,
1, 2
Ilja Gerhardt, Zoran D. Gruji´c,
8, 9
Hong Guo, Chuanpeng Hao, PaulS. Hamilton, Morgan Hedges, Derek F. Jackson Kimball, Dongok Kim,
14, 15
Sami Khamis, ThomasKornack, Victor Lebedev, Zheng-Tian Lu, Hector Masia-Roig,
1, 2, † Madeline Monroy,
4, 13
MikhailPadniuk, Christopher A. Palm, Sun Yool Park, ‡ Karun V. Paul, Alexander Penaflor, XiangPeng, Maxim Pospelov,
20, 21
Rayshaun Preston, Szymon Pustelny, Theo Scholtes,
9, 22
Perrin C.Segura, § Yannis K. Semertzidis,
14, 15
Dong Sheng, Yun Chang Shin, Joseph A. Smiga,
1, 2, ¶ Jason E.Stalnaker, Ibrahim Sulai, Dhruv Tandon, Tao Wang, Antoine Weis, Arne Wickenbrock,
1, 2
TatumWilson, Teng Wu, David Wurm, Wei Xiao, Yucheng Yang, Dongrui Yu, and Jianwei Zhang Johannes Gutenberg-Universit¨at Mainz, 55128 Mainz, Germany Helmholtz-Institut Mainz, GSI Helmholtzzentrum f¨ur Schwerionenforschung, 55128 Mainz, Germany Centre for Quantum Computation and Communication Technology,Research School of Physics, The Australian National University, Acton 2601, Australia Department of Physics, University of California at Berkeley, Berkeley, California 94720-7300, USA Department of Physics, University of Nevada, Reno, Nevada 89557, USA Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA Institute for Quantum Science and Technology (IQST), 3rd Institute of Physics,and Max Planck Institute for Solid State Research, D-70569 Stuttgart, Germany Institute of Physic Belgrade, University of Belgrade, 11080 Belgrade, Serbia Physics Department, University of Fribourg, Chemin du Mus´ee 3, CH-1700 Fribourg, Switzerland State Key Laboratory of Advanced Optical Communication Systems and Networks, Department of Electronics,and Center for Quantum Information Technology, Peking University, Beijing 100871, China Department of Precision Machinery and Precision Instrumentation,University of Science and Technology of China, Hefei 230026, P. R. China Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA Department of Physics, California State University – East Bay, Hayward, California 94542-3084, USA Center for Axion and Precision Physics Research, IBS, Daejeon 34051, Republic of Korea Department of Physics, KAIST, Daejeon 34141, Republic of Korea Twinleaf LLC, 300 Deer Creek Drive, Plainsboro, NJ 08536, USA Hefei National Laboratory for Physical Sciences at the Microscale,University of Science and Technology of China, Hefei 230026, P. R. China Institute of Physics, Jagiellonian University in Krakow,prof. Stanis(cid:32)lawa (cid:32)Lojasiewicza 11, 30-348, Krak´ow, Poland Department of Physics and Astronomy, Oberlin College, Oberlin, OH 44074, USA School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA William I. Fine Theoretical Physics Institute, School of Physics and Astronomy,University of Minnesota, Minneapolis, MN 55455, USA Leibniz Institute of Photonic Technology, Albert-Einstein-Straße 9, D-07745 Jena, Germany Department of Physics and Astronomy, One Dent Drive,Bucknell University, Lewisburg, Pennsylvania 17837, USA Department of Physics, Princeton University, Princeton, New Jersey, 08544, USA Technische Universit¨at M¨unchen, 85748 Garching, Germany (Dated: March 2, 2021)Results are reported from the first full-scale search for transient signals from exotic fields ofastrophysical origin using data from a newly constructed Earth-scale detector: the Global Networkof Optical Magnetometers for Exotic physics searches (GNOME). Data collected by the GNOMEconsist of correlated measurements from optical atomic magnetometers located in laboratories allover the world. GNOME data are searched for patterns of signals propagating through the networkconsistent with exotic fields composed of ultralight bosons such as axion-like particles (ALPs).Analysis of data from a continuous month-long operation of the GNOME finds no statisticallysignificant signals consistent with those expected due to encounters with topological defects (axiondomain walls), placing new experimental constraints on such dark matter scenarios. a r X i v : . [ a s t r o - ph . C O ] M a r I. INTRODUCTION
The nature of dark matter, an invisible substance com-prising over 80% of the mass of the universe [1–3], is oneof the most profound mysteries of modern physics. Al-though evidence for the existence of dark matter comesfrom its gravitational interactions, unraveling its naturelikely requires observing non-gravitational interactionsbetween dark matter and ordinary matter [4]. One ofthe leading hypotheses is that dark matter consists ofultralight bosons such as axions [5–7] or axion-like par-ticles (ALPs) [8–10]. Axions and ALPs arise from spon-taneous symmetry breaking at an unknown energy scale f SB , which, along with their mass m a , determines manyof their physical properties.ALPs can form stable, macroscopic field configurationsin the form of topological defects [11–14] or compositeobjects bound together by self-interactions such as bo-son stars [15–17]. Such ALP field configurations couldconcentrate the dark matter density into many distinct,compact spatial regions that are small compared to thegalaxy but much larger than the Earth. In such scenar-ios, Earth-bound detectors would only be able to measuresignals associated with dark matter interactions on occa-sions when the Earth passes through such a dark-matterobject. It turns out that there is a wide range of param-eter space, consistent with observations, for which suchdark-matter objects can have the required size and abun-dance such that the characteristic time between encoun-ters could be on the order of one year or less [13, 17, 18].This opens up the possibility of searches with terrestrialdetectors. Here we present the results of such a search forALP domain walls, a class of topological defects whichcan form between regions of space with different vacuaof an ALP field [11–14]. Since ALPs can interact with atomic spins [4], the pas-sage of Earth through an ALP domain wall affects atomicspins similarly to a transient magnetic field pulse [13, 17].Considering a linear coupling between the ALP field gra-dient ∇ a ( r , t ) and atomic spin S , the interaction Hamil- ∗ Currently: University of Waterloo, Department of Physics andAstronomy, N2L 3G1,Ontario, CA † Electronic address: [email protected] ‡ Currently: JILA, NIST and University of Colorado, and Depart-ment of Physics, University of Colorado, Boulder Colorado 80309-0440, USA § Currently: Department of Physics, Harvard University, Cam-bridge, MA 02138 ¶ Electronic address: [email protected] A different type of bosonic field configuration is the Q-ball [19,20] which is spherically symmetric having nearly constant fieldmagnitude in the bulk with a rapid fall off at the boundaries.It is another possible dark matter candidate. We note that thesignal pattern produced in GNOME for sufficiently large Q-ballswith couplings to atomic spins can be similar to that of ALPdomain walls [17]. tonian can be written as H lin = − ( (cid:126) c ) / ξf SB S (cid:107) S (cid:107) · ∇ a ( r , t ) , (1)where (cid:126) is the reduced Planck’s constant, c is the speedof light, r is the position of the spin, t is the time, and f SB /ξ ≡ f int is the coupling constant in units of energydescribed with respect to the symmetry-breaking scale f SB [21], where ξ is unitless. In most theories, the cou-pling constants f int describing the interaction betweenStandard Model fermions and the ALP field are propor-tional to f SB ; however, f int can differ between electrons,neutrons, and protons by model-dependent factors thatcan be significant in some cases [4, 8].In analogy with Eq. (1), the Zeeman Hamiltonian de-scribing the interaction of a magnetic field B with anatomic spin S can be written as H Z = − γ S · B , (2)where γ is the gyromagnetic ratio. Since Eqs. (1) and (2)have the same structure, the gradient of the ALP fieldcan be treated as a pseudo-magnetic field. An importantdistinction between the ALP-spin interaction [Eq. (1)]and the Zeeman interaction [Eq. (2)] is that while γ tendsto scale inversely with the fermion mass, no such scalingof the ALP-spin interaction is expected [4].The amplitude, direction, and duration of the pseudo-magnetic field pulse associated with the transit of theEarth through an ALP domain wall depends on manyunknown parameters such as the energy density stored inthe ALP field, the coupling constant f int , the thicknessof the domain wall, and the relative velocity v betweenEarth and the domain wall. The dynamical parameters,such as the velocities of the dark matter objects, are ex-pected to randomly vary from encounter-to-encounter.We assume that they are described by the Standard HaloModel for virialized dark matter [22]. Furthermore, theabundance of domain walls in the galaxy is limited byphysical constants, m a and f SB , as these determine theenergy contained in the wall and the total energy of alldomain walls is constrained by estimates of the local darkmatter density [23]. The expected temporal form of thepseudo-magnetic field pulse can depend both on the the-oretical model describing the ALP domain wall as well asparticular details of the terrestrial encounter such as theorientation of the Earth. The relationships between theseparameters and characteristics of the pseudo-magneticfield pulses searched for in our analysis are discussed inAppendix A and Refs. [13, 17, 21].The Global Network of Optical Magnetometers for Ex-otic physics searches (GNOME) is a worldwide networksearching for correlated signals heralding beyond-the-Standard-Model physics which currently consists of morethan a dozen optical atomic magnetometers, with sta-tions in Europe, North America, Asia, the Middle East,and Australia. A schematic of a domain-wall encounterwith the GNOME with stations used in this study is b)a) South
FIG. 1: (a) Visualization of an ALP domain wall crossing theEarth. The red arrows indicate the position and sensitive axesof the GNOME magnetometers during Science Run 2 (Sec. I).(b) Simulation of the signals expected to be observed from adomain-wall crossing. shown in Fig. 1. The measurements from the magne-tometers are recorded with data-acquisition systems, syn-chronized to the Global Positioning System (GPS) time,and uploaded to servers located in Mainz, Germany, andDaejeon, South Korea. Descriptions of the operationalprinciples and characteristics of GNOME magnetome-ters are presented in Sec. IV and Ref. [24], details onthe GNOME data acquisition system are discussed inRef. [25] (see also Table I).The active field sensor at the heart of every GNOMEmagnetometer is an optically pumped and probed gasof alkali atoms. Magnetic fields are measured through variations in the Larmor spin precession of the opticallypolarized atoms. The vapor cells containing the alkaliatoms are placed inside multi-layer magnetic shieldingsystems which reduce background magnetic noise by or-ders of magnitude [26] while retaining sensitivity to ex-otic spin-couplings between ALP dark matter and atomicnuclei. Since all GNOME magnetometers presently useatoms whose nuclei have a valence proton, the signal am-plitudes measured by the GNOME due to an ALP-spininteraction are proportional to the relative contributionof the proton spin to the nuclear spin (as discussed inAppendix B and Ref. [28]). This pattern of signal ampli-tudes [Eq. (1)] can be characterized by a pseudo-magneticfield B j measured with sensor j : B j = σ j η j g F,j B p , (3)where B p ( r , t ) = ( (cid:126) c ) / ξµ B f SB ∇ a ( r , t ) (4)is the normalized pseudo-magnetic field describing theeffect of the ALP domain wall on proton spins and µ B is the Bohr magneton. The ratio between the Land´e g-factor and the effective proton spin ( g F,j /σ j ) accounts forthe specific proton-spin coupling in the respective sensor.This ratio depends on the atomic and nuclear structureas well as details of the magnetometry scheme, see Ap-pendix B. Since each GNOME magnetometer measuresthe projection of the field along a particular sensitiveaxis, the factor η j is introduced to account for the direc-tional sensitivity. This factor, given by the cosine of theangle between B p and the sensitive axes, takes on valuesbetween ± Signals due to ALP interactions with electron spins can be coun-teracted by effects from interaction between the ALP field andelectron spins in the magnetic shielding material [27].
Here we report the first results of a dark matter searchwith the GNOME: a search for transient couplings ofatomic spins to macroscopic dark-matter objects, andtherefore demonstrate the ability of the GNOME to ex-plore parameter space previously unconstrained by directlaboratory experiments. Searches for macroscopic dark-matter objects based on similar ideas were carried outusing atomic clock networks [18, 31–33], and there are anumber of experimental proposals utilizing other sensornetworks [34–38]. All of these networks are sensitive tobosonic dark matter with a scalar coupling to StandardModel particles [4]. The GNOME is sensitive to a differ-ent class of dark matter: bosons with pseudoscalar cou-plings to Standard Model particles. Pseudoscalar bosonicdark matter generally produces no observable effects inclock networks [4] but does couple to atomic spins via theinteraction described by Eq. (1). Thus the GNOME issensitive to a distinct, so far mostly unconstrained, classof interactions as compared to other sensor networks.
II. RESULTS
There have been four GNOME Science Runs between2017 and 2020 as discussed in Sec. IV. Here we analyzethe data from Science Run 2, which had comparativelygood overall noise characteristics and consistent networkoperation (as seen in Fig. 5). Nine magnetometers tookpart in Science Run 2 that spanned from 29 November2017 to 22 December 2017. The characteristics of themagnetometers are summarized in Table I.Before the data are searched for evidence of domain-wall signatures, they are preprocessed by applying arolling average, high-pass filters, and notch filters to theraw data. The averaging enhances the signal-to-noise ra-tio for certain pulse durations, avoids complications aris-ing from different magnetometers having different band-widths, and reduces the amount of data to be analyzed.The high-pass and notch filters reduce the effects of long-term drifts and noisy frequency bands. We refer to the fil-tered and rolling-averaged data set as the “search data.”The search data are examined for evidence of collectivesignal patterns corresponding to planes with uniform,non-zero thickness, crossing Earth at constant veloci-ties. The imprinted pattern of amplitudes depends on thedomain-wall crossing velocity [29]. As noted in Sec. I, weassume that the domain-wall-velocity probability densityfunction follows the Standard Halo Model for virializeddark matter. The signature of a domain-wall crossingthe magnetometer network depends on the perpendicu-lar component to the domain-wall plane of the relativevelocity between the Earth and the domain wall, v ⊥ . Alattice of points in velocity space is constructed such thatthe search algorithm covers 97.5% of the velocity proba-bility density function. The algorithm scans over the ve-locity lattice and, for every velocity, the data from eachmagnetometer are appropriately time-shifted so that thesignals in different magnetometers from a hypothetical domain-wall crossing with the given velocity occur at thesame time. For each velocity and at each measurementtime, the amplitudes measured by each magnetometerare fit to the ALP domain-wall crossing model describedin Ref. [29]. As a result, estimations for signal magnitudeand domain-wall direction, along with associated uncer-tainties, are obtained for each measurement time and alllattice velocities. The magnitude-to-uncertainty ratio ofan event is given by the ratio between the signal magni-tude and its associated uncertainty.The search algorithm uses two different tests to eval-uate if a given event is likely to have been producedby an ALP domain-wall-crossing: a domain-wall modeltest and a directional-consistency test [29]. The domain-wall model test evaluates whether the event amplitudesmeasured by the GNOME magnetometers match the sig-nal amplitudes predicted by the ALP domain-wall cross-ing model, and is quantified by the p -value as discussedin Sec. IV and Ref. [29]. The directional-consistencytest checks the agreement between the direction of thescanned velocity and the estimated domain wall direc-tion, and is quantified by the angle between the two di-rections normalized by the angle between adjacent latticevelocities.In order to evaluate the detection probability of thesearch algorithm, a well-characterized data set that in-cludes domain-wall-crossing signals with known proper-ties is required. For this purpose, we generate a back-ground data set by randomly time-shuffling the searchdata so that the relative timing of measurements fromdifferent GNOME stations is shifted by amounts so largethat no true-positive events could occur. By repeatingthe process of time shuffling, the length of the back-ground data can be made to far exceed the search data.This method is used to generate background data withnoise characteristics closely reproducing those of thesearch data [39]. A set of pseudo-magnetic field pulsesmatching the expected amplitude and timing patternproduced by the passages of Earth through ALP domainwalls are inserted into the background data to create thetest data. The test data are used to evaluate the true-positive probability for the search algorithm for differentthresholds on the p -value and the normalized angular dif-ference. The detection probability as a function of thethresholds is studied in order to find a combination ofthresholds that ensures the detection of 97.5% of the in-serted domain walls (see Sec. IV and Fig. 6). This resultsin an overall detection efficiency (cid:15) ≥
95% for the searchalgorithm, considering both the incomplete velocity lat-tice coverage and the detection probability.The search data are analyzed for domain-wall encoun-ters using the algorithm presented in Ref. [29]. Thecumulative distribution of candidate events as a func-tion of their magnitude-to-uncertainty ratio is shown asa solid green line in Fig. 2. The background data areanalyzed in the same way as the search data to estimatethe expected number of background events during Sci-ence Run 2. Time-shuffling is used to create 10 . C u m u l a t i v e nu m be r o f e v en t s Significance ( ) E v en t r a t e ( y ea r ) BackgroundSearch
Magnitude-to-uncertainty ratio
FIG. 2: The blue dashed line represents the number eventsexpected from the background in the twenty-three days ofdata from Science Run 2. 10.7 years of time-shuffled data areused to evaluate the background. The solid green line repre-sents the number of events measured in Science Run 2. Thered crosses indicate at which magnitude-to-uncertainty rationew events are found. The upper axis indicates the statis-tical significance in units of Gaussian standard deviations offinding one event in the search data. The event with greatestmagnitude-to-uncertainty ratio is found at 12.6. of background data. The number of candidate eventsmeasured in the background data re-scaled to the dura-tion of Science Run 2 is shown as a dashed blue line inFig. 2. The significance of a candidate event in the searchdata is given by the probability of detecting one or morebackground events at a magnitude-to-uncertainty ratioabove that of the candidate event [see Eq. (10)]. Theprobability of finding a number of candidate events inthe background data follows Poisson statistics.The candidate event in the search data with the largestmagnitude-to-uncertainty ratio (= 12.6) had a signifi-cance of less than one sigma. Therefore, we find no ev-idence of an ALP domain-wall crossing during ScienceRun 2.In order to evaluate the domain-wall characteristics ex-cluded by this result, the observable domain-wall cross-ing parameters above 12.6 magnitude-to-uncertainty ra-tio during Science Run 2 are determined. The GNOMEhas nonuniform directional sensitivity [29]; we conserva-tively estimate the network sensitivity assuming the do-main wall comes from the least-sensitive direction. Fig-ure 3 shows the active time ˜ T (∆ t, B (cid:48) p ), i.e., how longthe network was sensitive to domain-walls as a functionof pseudo-magnetic field-pulse-magnitude sensitivity, B (cid:48) p ,and pulse duration, ∆ t . A signal with pseudo-magneticfield magnitude B p produces a magnitude-to-uncertaintyratio of ζ = B p / B (cid:48) p . The characteristic shape of the sen-sitive region is a result of the filtering and averaging ofthe raw data as described in Sec. IV. Averaging reducesthe sensitivity of the search data to short pulse durationsand high-pass filtering suppresses sensitivity for long ∆ t .The GNOME sensitivity varies in time as the number of Signal duration t (s) S en s i t i v i t y p ( p T ) D a ys s en s i t i v e ALP mass m a (eV/ c ), for v = 10 c FIG. 3: Amount of time ˜ T that the GNOME is sensitive to do-main walls with a given duration ∆ t and normalized pseudo-magnetic field magnitude sensitivity B (cid:48) p throughout ScienceRun 2, the magnitude of an event needed to induce a signalwith a magnitude-to-uncertainty ratio of one [see Eq. (11)].Only the worst-case direction is considered. The plot assumesthe parameters of the analysis: 20 s averaging time, 1.67 mHzfirst-order zero-phase Butterworth filter, and 50 Hz and 60 Hzzero-phase notch filters with a Q -factor of 60. active GNOME magnetometers recording data and theirbackground noise change. The active time, ˜ T (∆ t, B (cid:48) p ),can be used to constrain ALP domain-wall parameterspace as discussed in Appendix A.If one assumes a probability distribution for the num-ber of domain-wall encounters, an upper bound on therate R C of such encounters can be calculated with a con-fidence level C . We assume a Poisson probability dis-tribution for the domain-wall crossings. Since the ex-cess number of events in the search data as comparedto the background data was not statistically significant,the upper bound on the observable rate is given by theprobability of measuring no events during the effectivetime [40]. Note that since ˜ T depends on the parametersof the domain-wall crossing, our constraint on the ob-served rate depends on the ALP properties. We choosethe confidence level to be C = 90%. III. DISCUSSION
The analysis of the GNOME data did not find any sta-tistically significant excess of events above backgroundduring Science Run 2 that could point to the existenceof ALP domain walls, as seen in Fig. 2. The expectedrate of domain-wall encounters, r , depends on the ALPmass, m a , the domain-wall energy density in the MilkyWay, ρ DW , the typical relative domain-wall speed ¯ v givenby the Standard Halo Model, and the symmetry break-ing scale, f SB . The region of parameter space to whichGNOME is sensitive is defined by the ALP parametersexpected to produce signals above 12.6 magnitude-to-uncertainty ratio with rates r ≥ R during ScienceRun 2 (see Fig. 3). Based on the null result of our search,the sensitive region is interpreted as excluded ALP pa-rameter space.The ALP parameters and the phenomenological pa-rameters describing the ALP domain walls in our galaxy,namely the thickness ∆ x , the surface tension or energyper unit area σ DW , and the average separation ¯ L can berelated through the ALP domain wall model describedin Refs. [13, 21]. A full derivation of how observableparameters are related to ALP parameters is given inAppendix A and summarized below.The domain-wall thickness is determined by the ALPmass, and is on the order of the ALP reduced Comptonwavelength λ a [21],∆ x ≈ √ λ a = 2 √ (cid:126) m a c . (5)The constant prefactor of 2 √ v ⊥ , the signal durationis ∆ t = ∆ xv ⊥ ∝ m − a . (6)We assume that domain walls comprise the dominantcomponent of dark matter. Thus with the energy density ρ DW ≈ . / cm in the Milky Way [23], the energyper unit area (surface tension) in a domain wall, σ DW ,determines the average separation between domain walls,¯ L . The surface tension σ DW is related to the symmetrybreaking scale [13], σ DW = 8 (cid:126) m a f . (7)The average domain-wall separation is then approxi-mated by ¯ L ≈ σ DW ρ DW = 8 (cid:126) m a f ρ DW , (8)which results in the average domain-wall encounter rate, r = ¯ v/ ¯ L ∝ (cid:0) m a f (cid:1) − . (9)The colored region in Fig. 4a describes the symmetrybreaking scales up to which GNOME was sensitive with90% confidence (as detailed in Appendix A). The param-eter space is spanned by ALP mass, maximum symme-try breaking scale, and ratio between symmetry breakingscale and coupling constant. The shape of the sensitivearea shown in Fig. 4a is determined by the event with thelargest magnitude-to-uncertainty and the characteristicsof the preprocessing applied to the raw data. Figure 4b shows various cross-sections of Fig. 4a fordifferent ratios between the symmetry breaking scale andthe coupling constant indicated by the dashed lines. Theupper bound of f SB that can be observed by the net-work is shown in Fig. 4b for different ratios ξ ≡ f SB /f int .Because B p ∝ m a [see Eq. (A.10)], there is a sharpcut-off for low ALP mass where the corresponding fieldmagnitude falls below the network sensitivity. Eventhough B p increases for large m a , the mean rate ofdomain-wall encounters decreases with increasing mass[see Eqs. (8) and (9)]. Correspondingly, the upper limitfor the symmetry-breaking scale f SB is ∝ / √ m a . Giventhat no events were found, the sensitive region of ALP-domain-wall parameter space during Science Run 2 canbe excluded.Our experiment explores ALP parameter space up to f int ∼ × GeV (see Fig. 4). This goes beyond that ex-cluded by previous direct laboratory experiments search-ing for ALP-mediated exotic pseudoscalar interactionsbetween protons which have shown that f int (cid:38)
300 GeVover the ALP mass range probed by the GNOME [41].Although astrophysical observations suggest that f int (cid:38) × GeV, there are a variety of scenarios in whichsuch astrophysical constraints can be evaded [42, 43].Future work of the GNOME collaboration will focusboth on upgrades to our experimental apparatus andnew data-analysis strategies. One of our key goals isto improve overall reliability and the duration of con-tinuous operation of GNOME magnetometers . Addi-tionally, magnetometers varied in their bandwidths andreliability, and stability of their calibration. These chal-lenges were addressed in Science Run 4 through a va-riety of magnetometer upgrades and instituting dailyworldwide test and calibration pulse sequences. How-ever GNOME suffered disruptions due to the worldwideCOVID-19 pandemic. We plan to carry out ScienceRun 5 in 2021 to take full advantage of the improve-ments. Furthermore, by upgrading to noble-gas-basedcomagnetometers [44, 45] for future science runs (Ad-vanced GNOME), we expect to significantly improve thesensitivity to ALP domain walls. Additionally, GNOMEdata can be searched for other signatures of physics be-yond the Standard Model such as boson stars [17], relax-ion halos [46], and bursts of exotic low-mass fields fromblack-hole mergers [30]. In terms of the data-analysis al-gorithm used to search for ALP domain walls, in futurework we aim to improve the efficiency of the scan over thevelocity lattice. The number of points in the velocity lat-tice to reliably cover a fixed fraction (e.g., 97.5%) of theALP velocity probability distribution grows as (∆ t ) − [where ∆ t is given by Eq. (6)]. This makes the algo-rithm computationally intensive. We are investigating The intermittent operation of some magnetometers due to techni-cal difficulties during Science Runs 1-3 made it difficult to searchfor signals persisting (cid:38) Mass (eV/c ) f i n t ( G e V ) b) f SB / f int Mass (eV/c ) f S B / f i n t a) M a x f i n t ( G e V ) FIG. 4: Parameter space of the ALP constrained by the presented analysis of Science Run 2 with 90% confidence level (seeSec. II). The relationship between ALP theory parameters and measured quantities are discussed in Sec. III and Appendix A.(a) In color, upper bound on the the interaction scale, f int , to which the GNOME was sensitive as a function of m a and the ratio ξ ≡ f SB /f int . (b) Cross-sections of the excluded parameter volume in (a) for different ratios ξ represented by the correspondingdashed lines of the matching color in (a). a variety of new analysis approaches, such as machine-learning-based algorithms, to address these issues. IV. METHODS
The GNOME consists of over a dozen optical atomicmagnetometers, each enclosed within a multi-layer mag-netic shield, distributed around the world [26]. GNOMEmagnetometers are based on a variety of different atomicspecies, optical transitions, and measurement techniques:some are frequency- or amplitude-modulated nonlinearmagneto-optical rotation magnetometers (NMOR) [47,48], some are rf-driven optical magnetometers [24], whileothers are spin-exchange-relaxation-free magnetometers(SERF) [49]. A detailed description and characterizationof six GNOME magnetometers is given in Ref. [24]. Asummary of the properties of the GNOME magnetome-ters active during Science Run 2 is presented in Table I.Each GNOME station is equipped with auxiliarysensors, including accelerometers, gyroscopes, and un-shielded magnetometers, to measure local perturbationsthat could mimic a dark matter signal. Suspicious dataare flagged [24] and discarded during the analysis.The number of active GNOME magnetometers duringthe four Science Runs and the combined network noise asdefined in Ref. [29] is shown as a function of time in Fig. 5.The data in Science Run 4, although they extended for alonger period of time, featured poorer noise characteris-tics and consistency of operation to Science Run 2. Sincemany GNOME stations underwent upgrades in 2018 and2019, further characterization of Science Run 4 data isneeded, and results will be presented in future work.Here we provide more details on the analysis procedure discussed in Sec. II. It is composed of three stages to iden-tify events likely to be produced by ALP domain-wallcrossings: preprocessing, velocity scanning, and post-selection [29]. First, in the preprocessing stage, a rollingaverage and filters are applied to the GNOME magne-tometer raw data which are originally recorded by theGPS-synchronized data-acquisition system at a rate of512 samples/s [25]. The rolling average is characterizedby a 20 s time constant. Noisy frequency bands are sup-pressed using a first-order Butterworth high-pass filter at1.67 mHz together with the notch filters correspondingto 50 Hz and 60 Hz power line frequencies with a qualityfactor of 60. These filters are applied forward and back-ward to remove phase effects. This limits the observablepulse properties to a frequency region to which all mag-netometers are sensitive. Additionally, it guarantees thatthe duration of the signal is the same in all sensors. Wenote that these filter settings may be changed in futureanalysis.The local standard deviation around each point inthe magnetometer’s data is determined using an itera-tive process. Outliers are discarded until the standarddeviation of the data in the segment converges. Thelocal standard deviation is calculated taking 100 down-sampled points around each data point.Additionally, auxiliary measurements have shown thatthe calibration factors used by each magnetometer toconvert raw data into magnetic field units experiencechange over time due to, for example, changes in the en-vironmental conditions. Upper limits on the calibrationfactor errors due to such drifts over the course of ScienceRun 2 have been evaluated and are listed in Table I. Cali-bration factor errors result in magnetic field measurementerrors proportional to the magnetic field B j . The uncer- J un J u l N o i s e p ( p T ) Run 1 D e c Run 2 J un J u l A ug S ep O c t N o v D e c J an F eb M a r A p r M a y DateRun 3 F eb M a r A p r M a y Run 4 b) Active a c t i v e a) FIG. 5: Summary of the GNOME performance during the four Science Runs from 2017 to 2020. The raw magnetometer dataare averaged for 20 s and their standard deviation is calculated over a minimum of one and a maximum of two hours segmentsdepending on the availability of continuous data segments. For each binned point, the combined network noise consideringthe worse case domain-wall crossing direction is evaluated as defined in Ref. [29]. (a) One-day rolling average of the numberof active sensors. (b) Multi-colored solid line represents the one-day rolling average of the combined network noise and themulti-colored dashes show the noise of the individual sampled segments. The data are preprocessed with the same filters usedfor the analysis. The number of magnetometers active is indicated by the color of the line and dashes.TABLE I: Characteristics of the magnetometers active during Science Run 2 (see, e.g., Fig. 5). The station name, location inlongitude and latitude, orientation of the sensitive axis, type of magnetometer (NMOR [47, 48], rf-driven [24], or SERF [49]),and probed transition are listed. The bandwidth indicates the measured -3 dB point of the magnetometers’ frequency responseto oscillating magnetic fields. The calibration error takes into account potential temporal variation of the magnetometers’calibration over the course of Science Run 2, and is estimated based on auxiliary measurements. The rightmost column liststhe estimated ratio between the effective proton spin polarization and the Land´e g -factor for the magnetometer, σ p /g , whichdepends on the atomic species and the magnetometry scheme as described in Appendix B. The σ p /g value is used to relatethe measured magnetic field to the signal expected from the interaction of an ALP field with proton spins. The indicateduncertainty describes the range of values from different theoretical calculations [28].Location OrientationStation Longitude Latitude Az Alt Type Probed transition Bandwidth Cal. Error σ p /g Beijing 116 . ° E 40 . ° N +251 ° ° NMOR
Cs D2 F=4 115 Hz 20% − . +0 . − . Berkeley 122 . ° W 37 . ° N 0 ° +90 ° NMOR
Cs D2 F=4 7 Hz 40% − . +0 . − . Daejeon 127 . ° E 36 . ° N 0 ° +90 ° NMOR
Cs D2 F=4 10 Hz 20% − . +0 . − . Fribourg 7 . ° E 46 . ° N +190 ° ° rf-driven Cs D1 F=4 94 Hz 5% − . +0 . − . Hayward 122 . ° W 37 . ° N 0 ° − ° NMOR Rb D2 F=3 37 Hz 5% − . +0 . − . Hefei 117 . ° E 31 . ° N +90 ° ° SERF Rb & Rb D1 127 Hz 5% − . +0 . − . Krakow 19 . ° E 50 . ° N +45 ° ° NMOR Rb D1 F=2 3 Hz 20% 0 . +0 . − . Lewisburg 76 . ° W 40 . ° N 0 ° +90 ° SERF Rb D2 200 Hz 10% 0 . +0 . − . Mainz 8 . ° E 49 . ° N 0 ° − ° NMOR Rb D2 F=2 99 Hz 2% 0 . +0 . − . tainty resulting from the calibration error is later used todetermine agreement with the domain-wall model, butnot in the magnitude-to-uncertainty ratio estimate re-sulting from the model, since the calibration error affectssignal and noise in the same way.Second, in the velocity-scanning stage, the data from the individual magnetometers are time-shifted accordingto different relative velocities between Earth and the ALPdomain walls. In order to sample 97.5% of the velocityprobability distribution, a scan of the speeds from 53.7to 770 km/s with directions covering the full 4 π solidangle is chosen. Note that this distribution considers justthe observable perpendicular component of the relativedomain-wall velocity and neglects the orbital motion ofthe Earth around the Sun. For low relative velocities,both the time between signals at different magnetometersas well as the signal duration diverge. So the velocityrange is determined by the chosen 97.5% coverage andthe maximum relative speed of domain walls travellingat the galactic escape speed.The corresponding time-shifted data along with theirlocal standard deviation estimate are fetched from eachmagnetometer’s rolling averaged full-rate data at a rateof 0.1 samples/s. This reduces the amount of data toprocess, while keeping the full timing resolution.The step size used in the speed scan is chosen so thata single step in speed corresponds to time-shift differ-ences of less than the down-sampled sampling period.For each speed, a lattice of directions covering the full4 π solid angle is constructed. The angular difference be-tween adjacent directions is informed by sampling rateand speed [29] such that, as for the speed scan, a singlestep in direction results in time-shift differences of lessthan the down-sampled sampling period. With the set-tings used, the velocity-scanning lattice consists of 1661points. This number scales with the down-sampled sam-pling rate cubed.After time-shifting, the pulses produced by a domain-wall crossing appear simultaneously as if all the magne-tometers were placed at the Earth’s center. This processresults in a time-shifted data set for each lattice veloc-ity on which for each time point a χ -minimization isperformed to estimate the domain-wall parameters. AnALP domain-wall-crossing direction and magnitude, B p ,with the corresponding p -value quantifying the agree-ment is obtained. The p -value is evaluated as the prob-ability of obtaining the given χ value or higher fromthe χ -minimization. The p -value is calculated using thequadrature sum of the standard deviation of the data andthe uncertainty due to drifts of the calibration factors.All data points in every time-shifted data set are consid-ered potential events, characterized by time, p -value aswell as direction and magnitude B p with their associateduncertainties. The magnitude-to-uncertainty ratio of anevent ζ is the ratio between this B p and its associateduncertainty.Third, in the post-selection stage, two tests are carriedout to check if a potential event is consistent with anALP domain-wall crossing. The domain-wall model testevaluates if the observed signal amplitudes are consistentwith the expected pattern of a domain-wall crossing fromany possible direction. It is quantified by the aforemen-tioned p -value. The directional-consistency test is basedon the angular difference between the estimated domain-wall crossing direction and the direction of the velocitycorresponding to the particular time-shifted data set be-ing analyzed. In a real domain-wall crossing event, thesetwo directions should be aligned.To evaluate the consistency of a potential event witha domain-wall crossing, we impose thresholds on the p - value and the angular difference normalized with respectto the angular spacing of the lattice of velocity points forthat speed. The thresholds are chosen to guarantee a de-tection probability of 97.5% with the minimum possiblefalse-positive probability. The false-positive analysis isperformed on the background data. The true-positiveanalysis is performed on test data consisting of back-ground data with randomly inserted domain wall signalsas we describe below.A single signal pattern may appear as multiple po-tential events in the analysis, whereas we are seekingto characterize a single underlying domain-wall crossingevent. For example, a signal consistent with a domain-wall crossing lasting for multiple sampling periods wouldappear as multiple potential events in a single time-shifted data set. Furthermore, even if such a signal lastsfor only a single sampling period, corresponding potentialevents appear in different time-shifted data sets. Sinceit is assumed that domain-wall crossings occur rarely,such clusters of potential events are classified as a sin-gle “event.” In order to reduce double-counting of theseevents, conditions are imposed. If potential events pass-ing the thresholds occur at the same time in differenttime-shifted data sets or are contiguous in time, the po-tential event with the greatest magnitude-to-uncertaintyratio is classified as the corresponding single event.The true-positive analysis studies the detection prob-ability as a function of the thresholds. Multiple testdata sets are created featuring domain-wall-signal pat-terns with random parameters by inserting Lorentzian-shaped pulses into the background data of the differentGNOME magnetometers. The domain-crossing eventshave magnitudes of B p randomly selected between 0 . pT and durations randomly selected between0 .
01 s and 10 s. The distributions of the these ran-domized parameters are chosen to be flat on a logarith-mic scale. Additionally, the signals are inserted at ran-dom times with random directions. In order to simulatecalibration error effects, the pulse amplitudes insertedin each magnetometer are weighted by a random factorwhose range is given in Table I. The crossing velocity isalso randomized within the range covered by the veloc-ity lattice. For each inserted domain-wall-crossing event,the p -value, the normalized angular difference, and themagnitude-to-uncertainty ratio are computed.Figure 6a shows the detection probability as a func-tion of the threshold on the lower-limit of the p -valueand the threshold on the upper-limit of the normalizedangular difference. We restrict the analysis in Fig. 6ato events inserted with a magnitude-to-uncertainty ra-tio between 5 and 10. This enables reliable determina-tion of the true-positive detection probability withoutsignificant contamination by false positive events sincethe background event probability above ζ = 5 is below0.01% in a 10 s sampling interval. Since the detectionprobability increases with the signal magnitude, we fo-cus on the events below ζ = 10. The detection proba-bility is then the number of detected events divided by0 a)b)
10 100 10 Magnitude-to-uncertainty ratio D e t e c t i on p r obab ili t y ( % ) Directional consistency threshold p - v a l u e t h r e s h o l d D e t e c t i o n p r o b a b ili t y FIG. 6: Summary of the true-positive analysis results. (a)shows the probability of detecting a domain-wall-crossingevent with randomized parameters (as discussed in the text)as a function of p -value and directional-consistency thresh-olds. The inserted events have a magnitude-to-uncertaintyratio between 5 and 10. The black line indicates the com-bination of parameters corresponding to a 97.5% detectionprobability. The white dot indicates the particular thresholdschosen for the analysis. (b) Shows the mean detection prob-ability reached for different magnitude-to-uncertainty ratiosfor the chosen thresholds. the number of inserted events. The black line marks thenumerically evaluated boundary of the area guaranteeingat least 97.5% detection. All points along this black linewill yield the desired detection probability, so the partic-ular choice is made to minimize the number of candidateevents when applying the search algorithm to backgrounddata. These values are 0 .
001 for the p -value thresholdand 3 . (cid:15) ≥ p -value and directionalconsistency tests, we perform a false-positive study onbackground data. The analysis algorithm is applied toT b =10.7 years of time-shuffled data in order to establishthe rate of events expected solely from background. Be- cause of the larger amount of background data analyzed,lower rates and larger magnitude-to-uncertainty ratiosare accessible as compared to the search data. Based onthe false positive study, the probability of finding one ormore events in the search data above ζ , is [50], P ( ≥ ζ ) = 1 − exp (cid:18) − TT b [1 + n b ( ζ )] (cid:19) , (10)where T = 23 days is the duration of Science Run 2and n b ( ζ ) is the number of candidate events found in thebackground data above ζ . The significance is then de-fined as, S = −√ − [1 − − P )], where erf − is theinverse error function. The significance is given in unitsof the Gaussian standard deviation which corresponds toa one-sided probability of P .After characterizing the background for Science Run 2,the search data are analyzed. The results are representedas a solid green line in Fig. 2. For ζ > ζ max was measured at 12.6 followedby additional events at 6 . .
6. From Eq. (10), thesignificance associated with finding one or more eventsproduced by the background featuring at least ζ max islower than one sigma. This null result defines the sensi-tivity of the search and is used to set constraints on theparameter space describing ALP domain walls.The observable rate of domain-wall crossings dependson how long GNOME was sensitive to different signal du-rations and magnitudes. For the evaluation of this effec-tive time, the raw data of each magnetometer are dividedinto continuous segments between one to two hours dura-tion depending on the availability of the data. The pre-processing steps are applied to each segment. Then thedata are binned by taking the average in 20 s intervals.To estimate the noise in each magnetometer, the stan-dard deviation in each binned segment is calculated todefine the covariance matrix Σ s . The domain-wall mag-nitude, crossing with the worse case direction ˆ m , neededto produce ζ = 1 is calculated as in Ref. [29], B (cid:48) p (∆ t ) = (cid:113) ˆ m (cid:0) D T ∆ t Σ − s D ∆ t (cid:1) − ˆ m , (11)for each bin. The matrix D ∆ t contains the sensitivityaxes of the magnetometers, the factor σ p /g , and the ef-fects of the preprocessing as a function of the signal dura-tion as described in Ref. [29]. Such prepocessing effectsrely on a Lorentzian-shaped signal and give rise to thecharacteristic shape of Fig. 3. The effective time, ˜ T , isdefined as the amount of time the network can measurea domain wall with duration ∆ t and magnitude B (cid:48) p pro-ducing ζ ≥
1. Monte Carlo simulations analyzing seg-ments with inserted domain-wall encounters on raw datashow a good agreement with the sensitivity estimation inEq. (11).Assuming that the domain-wall encounters follow Pois-son statistics, a bound on the observable rate of events1above ζ max with 90% confidence is set as [40], R = − log (0 . (cid:15) ˜ T (∆ t, B (cid:48) p ) . (12)The ALP parameter space is constrained by imposingthat r ≥ R . The experimental constraint on thecoupling constant is written as (see Eq. (A.13) in Ap-pendix A), f int ≤ (cid:126) ξ (cid:114) ¯ vρ DW (cid:15) m a log (0 .
1) ˜ T (∆ t, B (cid:48) p ) (13)The signal duration can be written in terms of the massof the hypothetical ALP particle and the specific domain-wall crossing speed, ∆ t = √ (cid:126) ¯ vm a c . When calculating theconstraints on f int , we fix the domain-wall crossing speedto the typical relative speed from the Standard HaloModel, ¯ v = 300 km s − [31]. In contrast to the sig-nal duration, the pseudo-magnetic field signal dependson all parameters of the ALPs, the mass and the ra-tio between the coupling and symmetry breaking con-stants, B (cid:48) p = m a c ξµ B ζ . Figure 4 is given by Eq. (13) taking ζ = 12 .
6. The shape of the constrained space is given bythe fact that ˜ T varies depending on the target m a and ξ .A detailed derivation of Eq. (13) is given in Appendix A. Acknowledgments
The authors are grateful to Chris Pankow, JoshuaR. Smith, Jocelyn Read, Menachem Givon, Ron Fol-man, Wojciech Gawlik, Kathryn Grimm, Grzegorz(cid:32)Lukasiewicz, Peter Fierlinger, Volkmar Schultze, TilmanSander-Th¨omes, and Holger M¨uller for insightful discus-sions.This work was supported by the U.S. National Sci-ence Foundation under grants PHY-1707875 and PHY-1707803, the Swiss National Science Foundation undergrant No. 200021 172686, the German research founda-tion under grant no. 439720477, the European ResearchCouncil under the European Union’s Horizon 2020 Re-search and Innovative Program under Grant agreementNo. 695405, the Cluster of Excellence PRISMA+, DFGReinhart Koselleck Project, Simons Foundation, a Fun-damental Physics Innovation Award from the Gordonand Betty Moore Foundation, Heising-Simons Founda-tion, the National Science Centre of Poland within theOPUS program (Project No. 2015/19/B/ST2/02129),and IBS-R017-D1-2021-a00 of the Republic of Korea. [1] G. Bertone, D. 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Domain walls form when a field can monotonically varyacross vacuum states; two degenerate vacua or possiblythe same state if, for example, the field takes values onthe 1-sphere. This is the case for ALPs which arise fromthe angular part of a complex scalar field [51].The following Lagrangian terms are considered in nat-ural units ( (cid:126) = c = 1 here and throughout this appendix)for a new complex scalar field φ , L ⊃ | ∂ µ φ | − λS N − | N/ φ N + S N | , (A.1)where λ is a unitless constant and S is a constant withunits of energy [13].The axion field is obtained by reparameterizing thecomplex field φ in Eq. (A.1) in terms of the real field S (Higgs) and a (axion), φ = S √ ia/S ) . The second term in Eq. (A.1) will break the U (1) sym-metry of the complex field into a discrete Z N symmetry, φ → exp(2 πik/N ) φ for integer k . This corresponds tothe axion shift-symmetry, a → a + πS N k . The Higgsmode obtains a vacuum expectation value S → S , andthe axion field has degenerate vacua or ground energy Other references may use a minus sign in front of the S N term,which results in a similar potential, up to a phase. The end resultof Eq. (A.1) is that the axion potential will have a maximum atzero, while the minus-convention will have a vacuum at zero. a = πS N (2 k + 1) for integer k . One can definethe symmetry-breaking scale as f SB = S /N . Reparame-terizing the complex scalar field in Eq. (A.1) and settingthe Higgs mode to the vacuum expectation value, theaxion Lagrangian is L a = 12 ( ∂ µ a ) − m a f cos (cid:18) a f SB (cid:19) , (A.2)where the axion mass is m a = N S √ λ . This can be seenby matching the second-derivative of the cosine term atthe minima to a scalar mass term.For simplicity, a static domain wall in the yz -planeseparating domains of − πf SB and + πf SB is considered.Solving the classical field equations, one finds a ( x ) = 2 f SB arcsin [tanh( m a x )] . (A.3)The gradient of the field is then dadx ( x ) = 2 f SB m a cosh( m a x ) . (A.4)This has the full width at half maximum,∆ x = 2 cosh − (2) m a ≈ √ m a . (A.5)Using the domain-wall solution [Eq. (A.3)] and integrat-ing the energy density of the domain wall over x yieldsthe surface tension (energy per unit area) [13], σ DW = 8 m a f . (A.6)Interactions observable by magnetometers involve cou-pling between the axion field gradient and the axial-vector current of a fermionic field. For a fermion field ψ , the interaction is L int = i ( φ∂ µ φ ∗ − ∂ µ φφ ∗ ) S f int ¯ ψγ µ γ ψ S → S −−−−→ ∂ µ af int ¯ ψγ µ γ ψ . (A.7)The axial-vector current is related to the spin S , so thatthe interaction Hamiltonian becomes H int = 1 f int ∇ a · S (cid:107) S (cid:107) , (A.8)i.e., for spin- / particles, 1 / (cid:107) S (cid:107) = 2.Optical magnetometers operate by measuring thechange in atomic energy levels between two energy stateswith magnetic quantum numbers differing by ∆ m F . Aleading magnetic field is applied to the atoms, and vari-ations in the magnetic field along the leading field aremeasured. The spin coupling from Eq. (A.8) can in-duce a similar shift in energy levels to a magnetic field.The maximum energy shift is determined by plugging thelargest gradient from Eq. (A.4) into Eq. (A.8),∆ E = (cid:88) i ∈ e,p,n,. . . ησ ( i ) ∆ m F (cid:107) S ( i ) (cid:107) f SB f ( i )int m a , (A.9) where i labels the species of fermion, σ ( i ) = (cid:104) S ( i ) · F ( i ) (cid:105) F i ) is the projected spin coupling, η = cos θ for the an-gle between the axion gradient and sensitive axis θ , and f ( i )int is the interaction coupling to particle i . In general,we will combine the σ ( i ) (cid:107) S ( i ) (cid:107) f ( i )int terms into an effective ra-tio σ j f int , where j now labels the magnetometer. Com-paring this to the energy shift due to a magnetic field,∆ E B = g F,j µ B ∆ m F B j , one obtains a relationship for anormalized pseudo-magnetic field, g F,j B j σ j η j = 4 µ B m a ξ ≡ B p , (A.10)for ξ ≡ f SB f int , g F,j being the g -factor for the magnetometer j , and (cid:107) S ( i ) (cid:107) = 1 / / particles. Here, the normalization is such that B p is the same for all magnetometers, though each individualsensor may observe a different pseudo-magnetic field, B j .There are two factors that must be considered whendetermining if axion domain walls are observable by thenetwork: the magnitude of the signal B p and the rate ofsignals. Domain walls are assumed to exist in a static(or virialized) network across the galaxy through whichEarth traverses. For a domain-wall velocity v , the du-ration of a signal is ∆ t = ∆ x/v . Filters and bandwidthlimitations generally reduce the magnitude by a factordependent on the signal duration, which affects the sen-sitivity of the network (see Appendix in Ref. [29]).Meanwhile, if the domain walls induce a strong enoughsignal to be observed, but are so infrequent that one isunlikely to be found over the course of a measurement,then the network is effectively insensitive. If the energydensity of domain walls across the galaxy is ρ DW , thenthe average rate of domain walls passing through Earthis given by r = ¯ vρ DW σ DW = ¯ vρ DW m a f , (A.11)where ¯ v is the typical relative speed.The physical parameters describing the ALP domainwalls ( m a , f SB , and f int ) must be related to the parame-ters observable by the network ( B p and ∆ t ). The energydensity ρ DW and the typical relative speed ¯ v are assumedaccording to the observed dark matter energy density andthe galactic rotation velocity, respectively.In order to determine if a set of physical parameters isobservable, the likelihood that no events are found mustbe constrained. This constraint defines the confidencelevel of the detection. The probability of observing k events given that one expects to observe µ events is givenby the Poisson probability mass function, P ( k ; µ ) = µ k k ! e − µ . However, the network also has some detection efficiency (cid:15) <
1, so there could be multiple domain-wall-crossing4events, but no detection. In particular, the chance ofmissing no events given that there were k events is (1 − (cid:15) ) k . For an event rate r and measurement time T , theprobability that no events are detected is then ∞ (cid:88) k =0 (1 − (cid:15) ) k ( rT ) k k ! e − rT = e − (cid:15)rT . A bound on the event rate R C at confidence level C isthen given by demanding the probability of observing atleast one event 1 − e − (cid:15)R C T ≥ C , likewise, one wouldexpect to observe event rates r ≥ R C ≡ − log (1 − C ) (cid:15)T . (A.12)The physical parameter space of the ALPs is constrainedby demanding that r ≥ R C . Similar arguments for defin-ing constraints can be found, e.g., Ref. [40]. The totaltime that the network is sensitive to the measurable pa-rameters, ˜ T (∆ t, B (cid:48) p ), may be less than the total measure-ment time. These parameters are related to the physicalparameters via Eq. (A.5) and Eq. (A.10). One finds asensitivity bound for f int in terms of m a and ξ , f int ≤ ξ (cid:114) − ¯ vρ DW (cid:15) m a log (1 − C ) ˜ T (cid:0) ∆ t, B (cid:48) p (cid:1) (A.13)for ∆ t = 2 √ vm a and B (cid:48) p = 4 m a ξµ B ζ , where B (cid:48) p is the sensitivity of the network and ζ is themagnitude-to-noise ratio induced by a signal with mag-nitude B p . The main calculation from the network dataneeded for this sensitivity bound is ˜ T . This is calculatedby measuring the sensitivity of the network over timefor different signal durations ∆ t and integrating over thetime during which the network is sensitive to B p = ζ B (cid:48) p .Finally, if, after analyzing the data, no domain walls arefound, Eq. (A.13) defines an exclusion region. Appendix B: Conversion between magnetic fieldunits and proton spin coupling
The amplitude of a signal appearing in the magneticfield data from a GNOME magnetometer due to interac-tion of atomic spins with an ALP field a via the linearcoupling described by Eq. (1) varies based on the atomicspecies. In every GNOME magnetometer, the atomic va-por cell is located within a multi-layer magnetic shieldmade of mu-metal and, in some cases a ferrite inner-most layer. Interactions of the ALP field with electronspins in the magnetic shielding material can generate acompensating magnetic field that could partially cancelthe energy shift due to the interaction of the ALP fieldwith atomic electrons in the vapor cell, as discussed inRef. [27]. For this reason, GNOME magnetometers aremost sensitive to interactions of the ALP field with nu-clear spins.
1. Deriving spin-projection
All GNOME magnetometers active during ScienceRun 2 measure spin-dependent interactions of alkaliatoms whose nuclei have valence protons. Thus theGNOME is primarily sensitive to spin-dependent interac-tions of ALP fields with proton spins. Consequently, theexpected signal amplitude measured by a GNOME mag-netometer due to the pseudo-magnetic field pulse frompassage of Earth through an ALP domain wall mustbe rescaled by the ratio of the proton spin content ofthe probed ground-state hyperfine level(s) to their gyro-magnetic ratio. Some GNOME magnetometers opticallypump and probe a single ground-state hyperfine level,while others rely on the technique of spin-exchange re-laxation free (SERF) magnetometry in which the spin-exchange collision rate is much faster than the Larmorprecession frequency [49, 52, 53]. For SERF magnetome-ters a weighted average of the ground-state Zeeman sub-levels over both ground-state hyperfine levels is opticallypumped and probed.Table II shows the relevant factors needed to convertthe magnetic field signal recorded by GNOME magne-tometers into the expected pseudo-magnetic field due tointeraction of an ALP field with the proton spin. Detailedcalculations are carried out in Ref. [28]. The relationshipof the expectation value for total atomic angular momen-tum (cid:104) F (cid:105) to the nuclear spin (cid:104) I (cid:105) can be estimated basedon the Russell-Saunders LS -coupling scheme: (cid:104) F (cid:105) = (cid:104) S e (cid:105) + (cid:104) L (cid:105) + (cid:104) I (cid:105) , = (cid:104) S e · F (cid:105) F ( F + 1) (cid:104) F (cid:105) + (cid:104) L · F (cid:105) F ( F + 1) (cid:104) F (cid:105) + (cid:104) I · F (cid:105) F ( F + 1) (cid:104) F (cid:105) , (B.1)where S e is the electronic spin and L is the orbital an-gular momentum. GNOME magnetometers pump andprobe atomic states with L = 0, which simplifies theabove equation to: (cid:104) F (cid:105) = (cid:104) S e · F (cid:105) F ( F + 1) (cid:104) F (cid:105) + (cid:104) I · F (cid:105) F ( F + 1) (cid:104) F (cid:105) . (B.2)For L = 0 the projection of I on F is given by (cid:104) I · F (cid:105) = 12 [ F ( F + 1) + I ( I + 1) − S e ( S e + 1)] . (B.3)The above relations define the fractional spin polariza-tion of the nucleus relative to the spin polarization of theatom: σ N,F ≡ (cid:104) I · F (cid:105) F ( F + 1) . (B.4)The next step is to relate σ N,F to the spin polarizationof the valence proton σ p,F for a particular F . As dis-cussed in Ref. [28], a reasonable estimate for K, Rb, andCs nuclei can be obtained from the nuclear shell model5 TABLE II: Fractional proton spin polarization σ p,F , Land´e g -factors g F , and their ratios for the ground state hyperfine levelsused in GNOME, and the weighted average of these values across both hyperfine levels ( (cid:104) σ p (cid:105) hf / (cid:104) g (cid:105) hf ) applicable to SERFmagnetometers in the low-spin-polarization limit. The estimates are based on the single-particle Schmidt model for nuclearspin [54] and the Russell-Saunders scheme for the atomic states. Uncertainties in the values for σ p,F describe the range ofdifferent results from calculations based on the Schmidt model, semi-empirical models [55, 56], and large-scale nuclear shellmodel calculations where available [57–59]. The uncertainties in σ p,F and (cid:104) σ p (cid:105) hf are one-sided because alternative methods tothe Schmidt model generally predict smaller absolute values of the proton spin polarization. See Ref. [28] for further details.Atom (state) σ p,F g F σ p,F /g F (cid:104) σ p (cid:105) hf / (cid:104) g (cid:105) hf K ( F = 2) − . +0 . − . . − . +0 . − . -0.5 +0 . − . K ( F = 1) − . +0 . − . − .
50 0 . +0 . − . Rb ( F = 3) − . +0 . − . . − . +0 . − . -0.8 +0 . − . Rb ( F = 2) − . +0 . − . − .
33 0 . +0 . − . Rb ( F = 2) 0 . +0 . − . .
50 0 . +0 . − . +0 . − . Rb ( F = 1) 0 . +0 . − . − . − . +0 . − . Cs ( F = 4) − . +0 . − . . − . +0 . − . -1.2 +0 . − . Cs ( F = 3) − . +0 . − . − .
25 0 . +0 . − . by assuming that the nuclear spin I is due to the or-bital motion and intrinsic spin of only the valence nu-cleon and that the spin and orbital angular momenta ofall other nucleons sum to zero. This is the assumption ofthe Schmidt or single-particle model [54]. In the Schmidtmodel, the nuclear spin I is generated by a combinationof the valence nucleon spin ( S p ) and the valence nucleonorbital angular momentum (cid:96) , so that we have σ p,F = (cid:104) S p · I (cid:105) I ( I + 1) σ N,F , = S p ( S p + 1) + I ( I + 1) − (cid:96) ( (cid:96) + 1)2 I ( I + 1) σ N,F , = σ p σ N,F , (B.5)where it is assumed that the valence nucleon is in a well-defined state of (cid:96) and S p , and σ p is defined to be the frac-tional proton spin polarization for a given nucleus [28].For comparison between GNOME magnetometers us-ing different atomic species, it is essential to evaluate theuncertainty in the estimate of σ p,F based on the Schmidtmodel. To estimate this uncertainty, we compare calcula-tions of σ p,F based on the Schmidt model to the results ofthe semi-empirical calculations described in Refs. [55, 56]and to the results of detailed nuclear shell-model calcula-tions where available [57–59]. Conservatively, we assignthe uncertainty in σ p,F to be given by the full range (max-imum to minimum) of the values of σ p,F calculated bythese various methods. It turns out that in each consid-ered case, the estimate based on the Schmidt model givesthe largest value of | σ p,F | , causing the theoretical uncer-tainties in estimates of σ p,F to be one-sided as shown inTable II. Further details are discussed in Ref. [28].
2. SERF magnetometers
SERF magnetometers operate in a regime where theLarmor frequency is small compared to the spin-exchangerate, so that the rapid spin-exchange locks together theexpectation values of the angular momentum projection (cid:104) M F (cid:105) in both ground-state hyperfine levels of the alkaliatom. Because the Land´e g -factors g F in the two ground-state hyperfine levels have nearly equal magnitudes butopposite signs, the magnitude of the effective Land´e g -factor in a SERF magnetometer, (cid:104) g (cid:105) hf , is reduced com-pared to that in optical atomic magnetometers where asingle ground-state hyperfine level is probed.To calculate the effective Land´e g -factor averaged overhyperfine levels, (cid:104) g (cid:105) hf , for a SERF magnetometer, it isinstructive to consider the equation describing the mag-netic torque on an alkali atom, g s µ B B × (cid:104) S e (cid:105) ≈ d (cid:104) F (cid:105) dt , (B.6)where g s ≈ g -factor and we have ig-nored the contribution of the nuclear magnetic moment.In the SERF regime, where the alkali vapor is in spin-exchange equilibrium, the populations of the Zeemansublevels correspond to the spin-temperature distribu-tion [60] described by a density matrix in the Zeemanbasis given by [61, 62] ρ = Ce β · F , (B.7)where C is a normalization constant and β is the spin-temperature vector defined to point in the direction ofthe spin polarization P with magnitude β = ln (cid:18) P − P (cid:19) . (B.8)6In the low-spin-polarization limit, ρ ≈ C (1 + β · F ) . (B.9)If follows that (cid:104) S e (cid:105) = Tr( ρ S e ) = 13 S e ( S e + 1) β = 14 β , (B.10)and (cid:104) I (cid:105) = Tr( ρ I ) = 13 I ( I + 1) β . (B.11)Substituting the above expressions into Eq. (B.6) yields (cid:104) g (cid:105) hf µ B B × β ≈ d β dt , (B.12)where (cid:104) g (cid:105) hf = 3 g s I ( I + 1) . (B.13)Equation (B.13) can be compared to the g -factor for aparticular alkali ground-state hyperfine level [63], g F = ± g s I + 1 . (B.14)The effective proton spin polarization (cid:104) σ p (cid:105) hf for SERFmagnetometers can also be derived by considering therelevant torque equation (cid:18) f int ∇ a (cid:19) × (cid:104) S p (cid:105)(cid:107) S p (cid:107) = d (cid:104) F (cid:105) dt , (cid:18) f int ∇ a (cid:19) × σ p (cid:107) S p (cid:107) (cid:104) I (cid:105) = d (cid:104) F (cid:105) dt , (B.15)where we have used the fact that GNOME as configuredfor Science Run 2 is sensitive to the coupling of the ALPfield to proton spins. In the low-spin-polarization limit,based on Eqs. (B.10) and (B.11), (cid:18) f int ∇ a (cid:19) × (cid:104) σ p (cid:105) hf (cid:107) S p (cid:107) β = d β dt , (B.16)where (cid:104) σ p (cid:105) hf = σ p I ( I + 1)3 + 4 I ( I + 1) . (B.17)Table II shows the ratio between the effective protonspin polarization averaged over both ground-state hyper-fine levels, (cid:104) σ p (cid:105) hf , to the effective Land´e g -factor, (cid:104) g (cid:105) hf , in the low-spin-polarization regime. Note that the mag-nitudes of (cid:104) σ p (cid:105) hf / (cid:104) g (cid:105) hf are in general similar or slightlylarger than the magnitudes of σ p,F /g F for a single hy-perfine level.To determine the actual values of (cid:104) σ p (cid:105) hf / (cid:104) g (cid:105) hf for theSERF magnetometers used in GNOME’s Science Run 2,more detailed considerations are required.The values for the ratio of the effective proton spinpolarization to the effective Land´e g -factors for eachGNOME magnetometer active during Science Run 2 aregiven in Table I. a. Hefei magnetometer The Hefei GNOME station employs a SERF mag-netometer in a closed-loop, single-beam configuration,where the laser light is resonant with the Rb D1 line(pressure-broadened by 600 torr of nitrogen gas to alinewidth of ∼
10 GHz). The Hefei SERF magnetome-ter operates in the low-spin-polarization mode. The va-por cell contains K, Rb, and Rb atoms in naturalabundance, so spin-exchange collisions average over bothground-state hyperfine levels of all three species. Tak-ing into account the relative abundances of the differ-ent atomic species at the cell temperature of ≈ ◦ C( ≈ K, ≈ . Rb, ≈ . Rb), we findthat (cid:104) g (cid:105) hf ≈ . (cid:104) σ p (cid:105) hf = − . +0 . − . , and thus (cid:104) σ p (cid:105) hf / (cid:104) g (cid:105) hf = − . +0 . − . for the Hefei magnetometer. b. Lewisburg magnetometer The Lewisburg GNOME station employs a SERF mag-netometer in a closed-loop, two-beam configuration. Thevapor cell contains only Rb atoms. The LewisburgSERF magnetometer operates with a spin-polarization P ≈ .
5, outside the low-spin-polarization regime. Dis-cussions of the high-polarization regime are given inRefs. [62, 64]. For a nucleus with I = 3 /
2, the effectiveLand´e g -factor is given by (cid:104) g (cid:105) hf = g s P P , (B.18)and the effective proton spin polarization is given by (cid:104) σ p (cid:105) hf = σ p P P . (B.19)Based on Eqs. (B.18) and (B.19), we find that for theLewisburg magnetometer (cid:104) σ p (cid:105) hf / (cid:104) g (cid:105) hf = 0 . +0 . − .15