Constraints on bosonic dark matter from ... - Science Advances · bands around theJ-coupling doublet as shown in Fig. 1C. The sidebands are separated from the carrier peaks by ±w

SC I ENCE ADVANCES | R E S EARCH ART I C L E

PHYS I CS

1Johannes Gutenberg-Universität, Mainz 55099, Germany. 2Helmholtz-InstitutMainz, 55099 Mainz, Germany. 3Department of Physics, Stanford Institute for The-oretical Physics, Stanford University, Stanford, CA 94305, USA. 4Department ofPhysics, California State University East Bay, Hayward, CA 94542-3084, USA. 5De-partment of Physics, University of California, Berkeley, Berkeley, CA 94720-7300,USA 6Department of Physics, Boston University, Boston, MA 02215, USA.*Corresponding author. Email: [email protected]

Garcon et al., Sci. Adv. 2019;5 : eaax4539 25 October 2019

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Constraints on bosonic dark matter from ultralow-fieldnuclear magnetic resonanceAntoine Garcon1,2, John W. Blanchard2*, Gary P. Centers1,2, Nataniel L. Figueroa1,2,Peter W. Graham3, Derek F. Jackson Kimball4, Surjeet Rajendran5, Alexander O. Sushkov6,Yevgeny V. Stadnik1,2, Arne Wickenbrock1,2, Teng Wu1,2, Dmitry Budker1,2,5

The nature of dark matter, the invisible substance making up over 80% of the matter in the universe, is one ofthe most fundamental mysteries of modern physics. Ultralight bosons such as axions, axion-like particles, ordark photons could make up most of the dark matter. Couplings between such bosons and nuclear spinsmay enable their direct detection via nuclear magnetic resonance (NMR) spectroscopy: As nuclear spins movethrough the galactic dark-matter halo, they couple to dark matter and behave as if they were in an oscillatingmagnetic field, generating a dark-matter–driven NMR signal. As part of the cosmic axion spin precession experiment(CASPEr), an NMR-based dark-matter search, we use ultralow-field NMR to probe the axion-fermion “wind” couplingand dark-photon couplings to nuclear spins. No dark matter signal was detected above background, establishingnew experimental bounds for dark matter bosons with masses ranging from 1.8 × 10−16 to 7.8 × 10−14 eV.

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INTRODUCTIONUltralight bosonic dark matterThe nature of dark matter, the invisible substance that makes up over80% of the matter in the universe (1), is one of the most intriguingmysteries of modern physics (2). Elucidating the nature of dark matterwill profoundly affect our understanding of cosmology, astrophysics,and particle physics, providing insights into the evolution of the uni-verse and potentially uncovering new physical laws and fundamentalforces beyond the standard model. While the observational evidencefor dark matter is derived from its gravitational effects at the galacticscale and larger, the key to solving the mystery of its nature lies indirectly measuring nongravitational interactions of dark matter withStandard-Model particles and fields.

To date, experimental efforts to directly detect dark matter havelargely focused on weakly interacting massive particles (WIMPs), withmasses between 10 and 1000 GeV (3, 4). Despite considerable efforts,there have been no conclusive signs of WIMP interactions with ordi-nary matter. The absence of evidence for WIMPs has reinvigoratedefforts to search for ultralight bosonic fields, another class of theore-tically well-motivated darkmatter candidates (5), composed of bosonswith masses smaller than a few electron volts. A wide variety of theoriespredict new spin-0 bosons such as axions and axion-like particles(ALPs) as well as spin-1 bosons such as dark photons (1, 6). Their ex-istencemayhelp to answer other open questions in particle physics suchas why the strong force respects the combined charge-conjugation andparity-inversion (CP) symmetry to such a high degree (7), the relativeweakness of the gravitational interaction (8), and how to unify the the-ories of quantum mechanics and general relativity (9).

Bosonic fields can be detected through their interactions withStandard-Model particles. Most experiments searching for bosonicfields seek to detect photons created by the conversion of these bosons instrong electromagnetic fields via the Primakoff effect (10–16). Another

method to search for dark-matter bosonic fields was recently proposed:dark-matter–driven spin precession (17–19), detected via nuclearmagnetic resonance (NMR) techniques (18–22). These concepts wererecently applied to data measuring the permanent electric dipole mo-ment (EDM) of the neutron and successfully constrained ALP darkmatter with masses ≲ 10−17 eV (22).

The Cosmic Axion Spin Precession Experiment (CASPEr) is amultifaceted research program using NMR techniques to search fordark-matter–driven spin precession (18). Efforts within this programusing zero- to ultralow-field (ZULF) NMR spectroscopy (23) are col-lectively referred to as CASPEr-ZULF. We report here experimentalresults from a CASPEr-ZULF search for ultralight bosonic dark-matterfields, probing bosons with masses ranging from 1.8 × 10−16 to 7.8 ×10−14 eV (corresponding to Compton frequencies ranging from∼45 mHz to 19 Hz).

In the following section, we explain how dark-matter bosonic fields,in particular axion, ALP, and dark-photon fields, can be detected byexamining ZULF NMR spectra. The subsequent sections describe themeasurement scheme and the data processing techniques used duringthis search. Last, we present new laboratory results for bosonic darkmatter, complementing astrophysical constraints obtained from su-pernova 1987A [we note that the constraints based on SN1987A datacontinue to be reexamined; see, e.g., (24, 25)] (26).

RESULTSDark-matter field propertiesIf darkmatter predominantly consists of particleswithmassesmDMc

2≲10 eV, making up the totality of the average local dark-matter density,then they must be bosons with a large mode occupation number. Itwould be impossible for fermions with such low masses to accountfor the observed galactic darkmatter density because the Pauli exclusionprinciple prevents them from having the required mode occupation.

In this scenario, axion and ALP bosonic dark matter is well de-scribed by a classical field a(t), oscillating at the Compton frequency(wDM ≈ mDMc

2/ℏ) (27–29)

aðtÞ≈ a0cos ðwDMtÞ ð1Þ

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where c is the velocity of light, ℏ is the reduced Planck constant, anda0 is the amplitude of the bosonic field.

The temporal coherence of the bosonic field is limited by the relativemotion of the Earth through random spatial fluctuations of the field.The characteristic coherence time tDM, during which the bosonic darkmatter field remains phase coherent, corresponds to ∼106 periods ofoscillation of the field (18).

The amplitude a0 can be estimated by assuming that the fieldenergy density constitutes the totality of the average local dark matterenergy density (rDM≈ 0.4GeV/cm3) (30). Then, a0 is related to the darkmatter density through

rDM ¼ 12c2

ℏ2 mDM2a20 ð2Þ

Wenote that a0 is expected to fluctuate and behave stochastically due toself-interference of the field (31); however, for simplicity, we assumea0 to be constant.

Dark-matter couplings to nuclear spinsCASPEr-Wind is sensitive to any field such that its interaction withnuclear spins can be written in the form

HDM ¼ �ℏgNIN⋅D ð3Þ

where gN is a coupling constant that parametrizes the coupling of theeffective field, D, to nuclear spins represented by the operator IN. Inanalogy with the Zeeman interaction,HZeeman = − ℏgNIN ·B, such aneffective fieldDmay be thought of as a pseudo-magnetic field inter-acting with nuclear spins, where the nuclear gyromagnetic ratio, gN,is replaced by the coupling constant, gN.

For clarity, we focus this discussion on the so-called axion windinteraction with effective fieldDwind = −∇ a(r, t) and coupling constantgaNN. A number of other possible couplings between nuclear spins andbosonic dark matter fields take a form similar to Eq. 3. These includecouplings to the “dark” electric (with coupling constant gdEDM andeffective field DdEDM) and magnetic (with coupling constant gdMDM

and effective field DdMDM) fields mediated by spin-1 bosons such asdark photons (5, 32) or a quadratic “wind” coupling to an ALP field(with coupling constant gquad and effective field Dquad) (33). Theseare discussed further in Materials and Methods.

As the Earth orbits around the Sun (itselfmoving toward theCygnusconstellation at velocity, v, comparable to the local galactic virial velocity∼10−3c), it moves through the galactic dark matter halo and an interac-tion between axions and ALPs with a given nucleon, N, can arise.Assuming that ALPs make up all of the dark matter energy density,rDM, the axion dark matter halo to be static with respect to the centerof the galaxy [a further Monte Carlo–based analysis was performed byassuming a virialized ALP dark matter halo for which the bosonic fieldhas anonzero velocity and a stochastic amplitude due to self-interferencesof the field (31)] (see section S12), and that the dominant interactionwith nucleon spins is linear in∇a(r, t), Eqs. 1 and 2 can be used to writethe effective field as

DwindðtÞ ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ℏcrDM

psin ðwDMtÞv ð4Þ

Then, given the local galactic virial velocity, v, only two free param-eters remain in the Hamiltonian in Eq. 3: the coupling constant, gaNN,


and the field’s oscillation frequency, wDM, fixed by the boson mass. ACASPEr search consists of probing this parameter space over a bosonicmass range defined by the bandwidth of the experiment. To calibratethe experiment, we apply known magnetic fields and the experiment’ssensitivity to magnetic fields directly translates to sensitivity to the cou-pling constant. If no ALP field is detected, upper bounds on the cou-pling constant can be determined on the basis of the overall sensitivity ofthe experiment.

Dark matter signatures in ZULF NMRThe results presented in thisworkwere obtained by applying techniquesof ZULFNMR [see the “Experimental parameters” section inMaterialsand Methods; a review of ZULF NMR can be found in (23)]. Thesample—13C-formic acid, effectively a two-spin 1H–13C system—ispre-polarized in a 1.8-T permanent magnet and pneumaticallyshuttled to a magnetically shielded environment for magnetizationevolution and detection.

The spin Hamiltonian describing the system is

H ¼ ℏJCHI˙S� ℏðgHIþ gCSÞ˙B

þ ℏ gappI� 13gannS

� �˙DwindðtÞ ð5Þ

where the electron-mediated spin-spin coupling JCH/2p ≈ 221 Hz forformic acid and I and S are the nuclear-spin operators for 1H and13C, respectively. In addition, gH and gC are the gyromagnetic ratiosof the 1H and 13C spins, B is an applied magnetic field, gapp is theALP-proton coupling strength, and gann is the ALP-neutron couplingstrength. We assume gapp = gann = gaNN (22).

In the absence of external fields (B =Dwind(t) = 0), the nuclear spinenergy eigenstates are a singlet with total angular momentum F = 0 andthree degenerate triplet states with F= 1, separated byℏJCH. The observ-able in our experiment is the ymagnetization, leading to selection rulesDF = 0, ± 1 and DmF = ± 1, as in (34). The zero-field spectrum thusconsists of a single Lorentzian located at JCH/2p ≈ 221 Hz, as shownin Fig. 1A.

In the presence of a static magnetic field,B ¼ Bzez, applied along z,the mF = 0 states are unaffected, while the mF = ± 1 triplet states’degeneracy is lifted. The corresponding spectrum exhibits two peaksat JCH/2p ± Bz(gH + gC)/4p, as shown in Fig. 1B.

So long as∣JCH∣≫∣ gHB∣≫∣ gaNNDwind∣, themF= 0 states areunaffected and the mF = ± 1 states are shifted by

DEðmF ¼ ±1ÞðtÞ ¼ ∓ℏ

2BzðgH þ gCÞ±

ℏ

223gaNNDzðtÞ ð6Þ

where Dz(t) is the projection ofDwind(t) along the axis of the appliedmagnetic field. The time dependence of Dwind(t) leads to an oscil-latory modulation of themF = ± 1 energy levels, giving rise to side-bands around the J-coupling doublet as shown in Fig. 1C. Thesidebands are separated from the carrier peaks by ±wDM/2p andhave an amplitude proportional to the modulation index (gaNN/wDM). Dark matter fields with sufficiently strong coupling to nuclearspins can then be detected by searching for frequency modulation–induced sidebands in the well-defined ZULF NMR spectrum of for-mic acid.

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Coherent averagingThe expected dark matter coherence time (~14 hours for a particlewith 19-Hz Compton frequency) is much longer than the nuclear spincoherence time in 13C-formic acid (~10 s). Taking advantage of thismismatch, we introduce the post-processing phase-cycling techniqueshown in Fig. 2 (see the “Signal processing” section in Materials andMethods), which consists of incrementally phase shifting the transientspectra and subsequently averaging them together. If the phase incre-ment matches the phase accumulated by the oscillating field betweeneach transient acquisition, the sidebands add constructively. This allowscoherent averaging of the complex spectra such that the signal-to-noiseratio (SNR) scales asN1/2, whereN is the number of transients. Becausethe dark matter Compton frequency is unknown, it is necessary torepeat this operation for a large number of different phase increments(at least many as the number of transient acquisitions).

CalibrationThe energy shifts produced by Dz(t) (Eq. 6) are equivalent to thoseproduced by a real magnetic field with amplitude

BwindðtÞ ¼ 23

gaNNgH þ gC

˙DzðtÞº gaNNgH þ gC

ffiffiffiffiffiffiffiffiffirDM

psin ðwDMtÞv˙ez ð7Þ

Similar relationships for dark photon and quadratic wind cou-plings are provided in Materials and Methods. Based on Eq. 7, thesensitivity of the experiment to darkmatter was calibrated by applying areal oscillating magnetic field of known amplitude and frequency andmeasuring the amplitude of the resulting sidebands in the coherentlyaveraged spectrum. Further details are provided in the SupplementaryMaterials.

Search and exclusion methodThe dark matter search data were acquired and processed as describedabove but without a calibration AC magnetic field applied. For eachCompton frequency, the appropriate phase increment is computed,which identifies the corresponding coherently averaged spectrum tobe analyzed. The noise in the spectrum defines a detection threshold


at the 90% confidence level (further details in section S7). When thesignal amplitude at the given frequency is below the threshold, we setlimits on the darkmatter couplings to nuclear spins at levels determinedby the calibration and effective field conversion factors (see the “Bosonicdark matter effective fields” section in Materials and Methods). If thesignal is above the threshold, a more stringent analysis is performedby fitting the coherently averaged spectrum to a four-sideband model(further details in sections S8 and S9). When the fit rules out detection,the threshold level is again used to set limits. In case of an apparentdetection, further repeat measurements would need to be performedto check for false alarm and confirm that the signal is persistent andexhibits expected sidereal and annual variations.

CASPEr-ZULF search results: Constraints on bosonicdark matterThe results of the CASPEr-ZULF search for ALPs are given in Fig. 3.The frequencies presenting sharp losses in sensitivities at 0.21, 1.69, and2.16 Hz were the ones for which the nearest optimal phase incrementwas close to zero, thus presenting maximal amplitude J-coupling peaks,raising the detection threshold (see discussion in section S3). The redshaded area labeled “CASPEr-ZULF” corresponds to upper boundson nuclear-spin couplings to dark matter consisting of ALPs at the90% confidence level. This represents our current sensitivity limitationafter 850 30-s transient acquisitions using samples thermally polarizedat ~1.8 T. The “CASPEr-ZULF Phase II” line corresponds to theprojected sensitivity of a future iteration of thiswork thatwill use amoresensitive magnetometry scheme to measure a larger sample withenhanced (nonequilibrium) nuclear spin polarization.

Figures 4 and 5 show the search results for the ALP quadratic inter-action and dark photon interactions, respectively. No signal consistentwith axion, ALP, or dark-photon fields has been observed in the redshaded areas. The two different limits given in Fig. 5 were obtainedusing the same dataset but analyzed by assuming two orthogonal initialpolarizations of the dark photon field (see section S12).

In all cases, the search bandwidth was limited from below by thefinite linewidth of the J-resonance peaks, preventing resolution of side-bands at frequencies lower than 45mHz. Because of the finite coherencetime of the dark matter fields (corresponding to ~106 oscillations), the

1JCH

F=1

F=0

J-coupling J-coupling + DC field J-coupling + DC field + AC field

2ωac

2ωac

F=1

F=0

F=1

F=0

Frequency (Hz) Frequency (Hz) Frequency (Hz)215 220 225 230 215 220 225 230215 220 225 230

A B C

131

(γH+γC)B

Real partMagnitude

Fig. 1. Nuclear spin energy levels and NMR spectra of 13C-formic acid measured in three different field conditions. (A) At zero magnetic field, the F = 1 levels aredegenerate, resulting in a spectrum exhibiting a single peak at the J-coupling frequency. (B) In the presence of a DC magnetic field Bz ≈ 50 nT, the mF = ±1 degeneracyis lifted. The spectrum exhibits two split J-resonances. The splitting is equal to ℏBz(gC + gH). The asymmetry of the resonances is due to the influence of the applied fieldon the response characteristics of the atomic magnetometer. (C) Addition of an oscillating magnetic field along Bz modulates the mF = ±1 energy levels, resulting insidebands located at J/2p ± Bz(gC + gH)/2p ± wAC with amplitude proportional to the modulation index: As º BAC(gC + gH)/(2wAC).

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bandwidth’s upper limit (19 Hz) is the highest frequency that can becoherently averaged after 14 hours of integration time. The sensitivityfall off is due to the sidebands’ amplitude scaling as the modulationindex. Further details are given in sections S6 and S11.

DISCUSSIONThis work constitutes demonstration of a dark matter search usingNMR techniques with a coupled heteronuclear spin system. The resultsprovidenew laboratory-basedupper bounds for bosonic darkmatterwithmasses ranging from 1.86 × 10−16 to 7.85 × 10−14 eV, complementingastrophysical bounds obtained from supernova SN1987A (26, 35).

Our data analysis provides a method to perform coherent averagingof the bosonic-field–induced transient signals. This method shouldprove useful for other experiments seeking to measure external fields ofunknown frequency using a detector with a comparatively short


coherence time. Conveniently, this phase-cycling approach also sup-presses the carrier-frequency signals, which would otherwise increasethe detection threshold via spectral leakage. As this method is appliedduring post-processing, it does not require modification of theexperiments provided that the data to be analyzed have been timestamped.

Phase II sensitivity improvementsTo search a greater region of the bosonic darkmatter parameter spaces,several sensitivity-enhancing improvements are planned for the nextphase of the experiment. In this work, nuclear spin polarization wasachieved by allowing the sample to equilibrate in a 1.8-T permanentmagnet, which yields a 1H polarization ≲10−5. For the next phase ofthe experiment, a substantial sensitivity improvement can be obtainedby using so-called hyperpolarization methods to achieve much higher,nonequilibrium nuclear-spin polarization. Current efforts are focusedon the implementation of nonhydrogenative parahydrogen-induced

Signal

FFT

Phaseshift

Average

Average

218 220 222 224−3

−2

−1

0

1

2

3A B

C

218220

222224

226 −3−2

−10

1

32

Frequency (Hz)226224222220218

Am

plitu

de (f

T)

120

100

80

60

40

20

0Frequency (Hz)

2.93 rad0 rad

(sidebands coherent)(J peaks coherent)

Fig. 2. Signal acquisition and processing schemes. Top: Signal acquisition scheme with simulated spectra. After polarization, each transient acquisition starts following amagnetic p-pulse (corresponding to a 180° flip of the 13C spin along any direction). The external AC magnetic field’s phase varies between transient acquisitions (orange). As aresult, the sidebands generally have different phases in each transient spectrum. Averaging the transients yields a spectrum in which the sidebands are destructively averagedout (purple). Shifting each transient by a phase equal to the external field’s accumulated phase restores the sidebands’ phase coherence, yielding a spectrum with high SNRsidebands (orange). For clarity, only one of the two Zeeman-split J-coupling peaks and its two sidebands are shown. Bottom: Result of the phase-shifting procedure for actualdata. (A) Transients are averaged using 2001 phase increments and stacked into a two-dimensional plot. (B) Side view of (A); sidebands are rescaled by a factor 10 for clarity.(C) Averaging with f = 0 rad corresponds to averaging the transients without phase shift; sidebands are averaged out and carrier peaks appear with maximum amplitude.When the optimal phase (for w/2p = 0.73 Hz, f = 2.93 rad) is approached, sidebands appear. These spectra were acquired in an experiment during which the AC fieldfrequency and amplitude were set to 0.73 Hz and 0.24 nT. Transient acquisitions of 30 s were repeated 850 times with a time interval between each transient of t = 61 s.

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polarization (NH-PHIP) [NH-PHIP methods are often referred to assignal amplification by reversible exchange of parahydrogen (SABRE)](36). Signal enhancement via NH-PHIP has been demonstrated at zerofield (37) and after optimization is expected to increase nuclear spin po-larization levels to at least 1%. Because parahydrogen can be flowed con-tinuously into the sample, a steady-state polarization enhancement canbe achieved (38), improving the experimental duty cycle.Combining con-tinuous NH-PHIP with a feedback system to produce a self-oscillatingnuclear spin resonator (39) could conceivably offer further improvementand simplify data analysis.

Additional sensitivity enhancement will be provided by magne-tometer improvements and use of a larger sample. In the experimentsreported here, only about 50 ml of the sample contributed to the signal,which was detected from below with an atomic magnetometer with anoise floor around10 fT=

ffiffiffiffiffiffiHz

p. With a larger (≳1ml) sample hyperpo-

larized via NH-PHIP detected via a gradiometric magnetometer arraywith optimized geometry and sensitivity below 1 fT=

ffiffiffiffiffiffiHz

p, we antici-

pate an improvement by ≳105 relative to the results presented here.Future experiments will be carried out with increased integration

time. We recall that the bosonic dark matter fields are coherent for atime tDM on the order of 106 periods of oscillation. The phase-cyclingprocedure depicted in Fig. 2 is valid for datasets with total time lessthan tDM. For integration times longer than the coherence time of thebosonic field Ttot > tDM, the datasets have to be coherently averaged insets of duration tDM using the phase-cycling procedure. This yieldsTtot/tDM sets of coherently averaged data. To profit from longer integra-tion time and further increase the SNR, these sets can be incoherentlyaveraged by averaging their power spectra, yielding a further SNRscaling as (TtottDM)

1/4 [see supplementary materials in (18)].

Fig. 3. ALP wind nucleon linear coupling parameter space. The CASPEr-ZULF region is excluded by this work (90% confidence level) using a thermally polarizedsample (data averaged over 850 transient acquisitions of 30 s each). The “New Force” region is excluded by searches for new spin-dependent forces (50). The SN1987Aregion represents existing limits from supernova SN1987A cooling (26, 35). The nn/nHg region is excluded (at 95% confidence level) by measurements of the ratio ofneutron and 199Hg Larmor precession frequencies (22). The dashed line corresponds to the sensitivity of a planned second phase of CASPEr-ZULF, with a projected ~105

factor increase in sensitivity, and the bandwidth extended toward lower frequencies by using a comagnetometer technique (43) and longer integration times. Thedotted lines show limits assuming a virialized ALP dark matter halo for which the field velocity and amplitude fluctuate (see section S12).

Fig. 4. ALP wind nucleon quadratic coupling parameter space. The CASPEr-ZULF region is excluded by this work (90% confidence level) using a thermallypolarized sample (data averaged over 850 transient acquisitions of 30 s each).Other regions of this figure are defined in the caption of Fig. 3.

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Complementary searchesTo increase the bandwidth of the experiment (see section S11), we pro-pose complementarymeasurement procedures. As the amplitude of thesidebands scales as 1/wDM, the sensitivity of the experiment decreasesfor higher frequencies. To probe frequencies-ranging from~20 to 500Hz(corresponding to bosonic masses of∼8 × 10−14 to 2 × 10−12 eV), it wasshown in (40) that a resonant detection method will be more sensitivethan the current frequency-modulation–induced sideband measure-ment scheme.ResonantAC fields can induce phase shifts in the J-couplingpeaks (41); cosmic fields can induce the same effect. By gradually vary-ing themagnitude of a leadingmagnetic field, one can tune the splittingof the J-coupling multiplets to match the dark-matter field frequency.Such a resonance would manifest itself by shifting the phase of theJ-coupling peak.

For frequencies below ~45mHz (corresponding to bosonic masses2 × 10−16 eV), the sidebands are located inside of the J-coupling peaksand the experimental sensitivity drops rapidly. This represents thelower limit of the bandwidth accessible by the frequency modulation–induced sideband measurement scheme presented in this work. Toprobe down to arbitrarily low frequencies, another measurement schemehas been implemented on the basis of a single-component liquid-statenuclear-spin comagnetometer (42). Further details and results of thisscheme are presented elsewhere (43).

MATERIALS AND METHODSExperimental parametersThe experimental setup used in this experiment was the one describedin (44). Additional descriptions of similar ZULF NMR setups can befound in (45).


The NMR sample consisted of ~100 ml of liquid 13C-formic acid(13CHOOH) obtained from ISOTEC Stable Isotopes (MilliporeSigma),degassed by several freeze-pump-thaw cycles under vacuum, andflame-sealed in a standard 5-mm glass NMR tube.

The sample was thermally polarized for ~30 s in a 1.8-T permanentmagnet, after which the NMR tube was pneumatically shuttled into thezero-field region. After the guiding magnetic field was turned off, amagnetic pulse (corresponding to a p rotation of the 13C spin) was ap-plied to initiate magnetization evolution.

Following each transient acquisition, the sample was shuttledback into the polarizing magnet and the experiment was repeated.To increase the SNR, the transient signals were averaged using thephase-cycling technique described in the “Signal processing” section.

Bosonic dark matter effective fieldsIn the case of the ALP-wind linear coupling, the field acting on the1H-13C spins induces an energy shift equal to the one produced by amagnetic field with amplitude

BeffALP;zðtÞ ¼ � 2

3gaNN

gH þ gC

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ℏcrDM

p�sin ðwDMt � k˙rþ φÞv˙ez ð8Þ

wherewDM≈mDMc2/ℏ is the ALP Compton frequency, k≈mDMv/ℏ

is the wave vector (v is the relative velocity),mDM is the rest mass ofthe ALP, φ is an unknown phase, and ez is the axis along which theleading DC magnetic field is applied.

It is theoretically possible that interaction of nuclear spins with ∇acan be suppressed (33, 46), in which case the dominant axion wind

Fig. 5. Dark photon-nucleon couplings. Left: Dark photon–nucleon dEDM coupling parameter space. The SN1987A region represents existing limits for ALPs fromsupernova SN1987A cooling (26, 35) adjusted to constrain dark photons as discussed in (51). Right: Dark photon–nucleon dMDM coupling parameter space. The CASPEr-ZULF regions are excluded by this work (90% confidence level) using a thermally polarized sample (data averaged over 850 transient acquisitions of 30 s each). The red andpurple lines correspond to the case where the dark photon field polarization is along the e1 and e3 axes of the nonrotating Celestial frame, respectively (see section S12). Thedashed lines correspond to the sensitivity of a planned second phase of CASPEr-ZULF, with a projected ~105 factor increase in sensitivity.

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interaction, referred to as the quadratic wind coupling, is related to∇a2. In the case of the ALP-wind quadratic coupling, the equivalentmagnetic field amplitude is

Beffquad;zðtÞ ¼ � 4

3ℏc2

gquad2

gH þ gC

rDMwDM

�sin ð2wDMt � 2k˙rþ φÞv˙ez ð9Þ

where gquad, having dimensions of inverse energy, parameterizes theALP quadratic coupling strength to nuclear spins.

There are two possible interactions of dark photons with nuclearspins that can be detected with CASPEr-ZULF: the coupling of the darkelectric field to the dark EDM (dEDM) and the coupling of the darkmagnetic field to the dark magnetic dipole moment (dMDM). Theequivalent magnetic field amplitudes are

BeffdEDMðtÞ ¼

23

gdEDMgC þ gH

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ℏc3rDM

p� cos ðwDMt þ φÞe⋅ez ð10Þ

and

BeffdMDMðtÞ ¼

23

gdMDM

gC þ gH

vc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ℏc3rDM

p� cos ðwDMt þ φÞe⋅ez ð11Þ

with coupling constants gdEDM and gdMDM (having dimensions ofinverse energy) and dark photon field polarization e.

The experimental sensitivity to real magnetic fields then directlytranslates to sensitivity to the coupling constants gaNN, gquad, gdEDM,and gdMDM. Inverting Eqs. 8 to 11 yields the corresponding conversionfactors from magnetic field to the dark matter coupling constants

dgaNNðwÞ≈ 1:3� 108GeV�1

T

� �dBðwÞ ð12Þ

dgquadðwÞ≈ 190GeV�1ffiffiffiT

p ffiffiffiffiffiffiffiffiffiffiffirad=s

p" # ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

w⋅dBðwÞp

ð13Þ

dgdEDMðwÞ≈ 1:3� 105GeV�1

T


dgdMDMðwÞ≈ 1:3� 108GeV�1

T


Here,wehaveused gC/2p =10.70MHz.T−1 andgH/2p =42.57MHz.T−1,v≈ 10−3c, and rDM≈ 0.4 GeV/cm3. The full derivation of these expres-sions is given in section S2.3.

Signal processingFor each transient acquisition, the sample was prepared in the same ini-tial state, which determined the phase of the J-coupling peaks. Whenaveraging the transients together, the J-coupling peaks’ amplitude and


phase remain constant, while the uncorrelated noise is averaged away,thus increasing the SNR as the square root of the total integration timeTtot, i.e., SNR ðTtotÞºT1=2

tot .However, the dark-matter–related information resides not in

the J-coupling peaks but in their sidebands. The external bosonic fieldoscillates at an unknown frequency, and its phase at the beginning ofeach transient acquisition is unknown. This phase directly translatesinto the phase of the sidebands in the transient spectra: While thephase of the J-coupling peaks is identical from one transient acqui-sition to another, the phase of the sidebands varies. As a result, na-ively averaging the transient spectra averages the sidebands away,thus removing the dark-matter–related information from the result-ing spectrum.

Here, we used a post-processing phase-cycling technique thatenables coherent averaging of the spectra in the frequency domain,even for transient signals for which no obvious experimental phase-locking can be achieved due to the unknown frequency of the signal.The method is similar to acquisition techniques in which an externalclock is used to register the times of the transient acquisitions andpost-processing phase shifting of the transient signals is used to recoverthe external field’s phase (47–49).

The method relies on the fact that the bosonic field’s phase at thebeginning of each transient acquisition is unknown but not random.We recall that the bosonic fields remain phase coherent for ~106 os-cillations, which for frequencies below 19 Hz is longer than the totalintegration time (14 hours). Thus, precise knowledge of the transientsignal acquisition starting times enables recovery of the phase of the bo-sonic field.

A full description of this averaging method is given in section S4.Each transient spectrum is incrementally phase shifted before aver-aging. If the phase shift is equal to the phase accumulated by the bosonicfield between two transient acquisitions, then the phase stability of theJ-coupling peaks is shifted to their sidebands, which can thus be co-herently averaged.

Considering that the frequency of the bosonic field is unknown, thecorrect phase shift is also unknown. Thus, we repeated the operation for2001 different phase increments between [−p, p], yielding 2001 aver-aged spectra, one of which being averaged with the phase incrementsuch that the sidebands are coherent.

To demonstrate the viability of this method, a small magnetic fieldwas applied with amplitude 0.24 nT oscillating at 0.73 Hz to simulate adark matter field. Using this processing technique, the SNR of the side-bands scales as SNRðTtotÞºT1=2

tot (see fig. S3), as expected during a co-herent averaging procedure. This is a marked improvement over thealternative power spectrum averaging (typically implemented for setsof incoherent spectra), which would yield a T1=4

tot scaling.

Sensitivity analysisThe experimental sensitivity is defined by the ability to observe dark-matter–induced sidebands above the magnetometer noise floor. Toshow that the sideband amplitude scales with the modulation index,B/w, and does not present unusual scalings due to experimental errors,two calibration experimentswere performed.A first calibrationwas per-formed by varying the amplitude of the AC field from 95 to 310 pTwhileholding the frequency constant at w = 2p × 0.73 Hz. Then the amplitudewas held at 160 pT while varying the frequency from 0.45 to 1.7 Hz.

The results of this experiment are shown in fig. S1. Similar ex-periments were performed to determine the minimum detectable fre-quency (see section S11). On the basis of this calibration, we can

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extrapolate the expected sideband amplitude, As(B, w), for any field ofamplitude B and frequency w:

AsðB;wÞ ¼ 5:53� 10�6 rad:s�1 � B½T�w½rad:s�1� ð16Þ

The magnetometer noise level then determines the smallest detectabledriving field, which is converted to dark matter coupling bounds viaEqs. 12 to 15.

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SUPPLEMENTARY MATERIALSSupplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/10/eaax4539/DC1Section S1. Experimental setupSection S2. Measurement schemeSection S3. Dark matter effective fieldsSection S4. Signal processing: Post-processing phase cyclingSection S5. Sideband amplitude determinationSection S6. Calibration: Signal scaling versus bosonic field amplitude and frequencySection S7. Detection threshold determinationSection S8. Search and exclusion methodSection S9. False alarms and false negativesSection S10. Coherent averaging: Signal scaling with integration timeSection S11. Bandwidth: Accessible bosonic mass rangeSection S12. Dark matter field modelsSection S13. Search data time stampFig. S1. Calibration data.Fig. S2. Detection threshold determination.Fig. S3. Sideband amplitude, detection threshold, and SNR versus integration time.Fig. S4. Sideband amplitude scaling with calibration field frequency.References (52–56)

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Acknowledgments: We thank D. Antypas, D. Aybas, A. Derevianko, M. Engler, P. Fadeev,M. Lawson, H. Masia Roig, and J. Smiga for useful discussions and comments. Funding: Thisproject has received funding from the European Research Council under the EuropeanUnions Horizon 2020 research and innovation program (grant agreement no. 695405). Weacknowledge the support of the Simons and Heising-Simons Foundations and the DFGReinhart Koselleck project. P.W.G. acknowledges support from DOE Grant DE-SC0012012, NSFGrant PHY-1720397, DOE HEP QuantISED award no. 100495, and the Gordon and Betty MooreFoundation Grant GBMF794. D.F.J.K. acknowledges the support of the U.S. National ScienceFoundation under grant number PHY-1707875. Y.V.S. was supported by the HumboldtResearch Fellowship. Author contributions: J.W.B., D.B., D.F.J.K., and A.O.S. conceived theproject. J.W.B. oversaw and managed the progress of the project. J.W.B. and T.W. designed,constructed, and calibrated the experimental apparatus and software controls. J.W.B., D.B., andA.G. conceived the data acquisition scheme. A.G. acquired and analyzed the data withcontribution of J.W.B., G.P.C., N.L.F., and T.W. A.G. wrote the manuscript with the contributionof J.W.B., D.B., G.P.C., N.L.F., D.F.J.K., A.W., and T.W. P.W.G., D.F.J.K., S.R., and Y.V.S. conceived thetheory related to axion and hidden photon coupling to nuclear spins. All authors have readand contributed to the final form of the manuscript. Competing interests: The authors declarethat they have no competing interests. Data and materials availability: All data needed toevaluate the conclusions in the paper are present in the paper and/or the SupplementaryMaterials. Additional data related to this paper may be requested from the authors.

Submitted 25 March 2019Accepted 16 September 2019Published 25 October 201910.1126/sciadv.aax4539

Citation: A. Garcon, J. W. Blanchard, G. P. Centers, N. L. Figueroa, P. W. Graham, D. F. JacksonKimball, S. Rajendran, A. O. Sushkov, Y. V. Stadnik, A. Wickenbrock, T. Wu, D. Budker,Constraints on bosonic dark matter from ultralow-field nuclear magnetic resonance. Sci.Adv. 5, eaax4539 (2019).

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Constraints on bosonic dark matter from ultralow-field nuclear magnetic resonance

Surjeet Rajendran, Alexander O. Sushkov, Yevgeny V. Stadnik, Arne Wickenbrock, Teng Wu and Dmitry BudkerAntoine Garcon, John W. Blanchard, Gary P. Centers, Nataniel L. Figueroa, Peter W. Graham, Derek F. Jackson Kimball,

DOI: 10.1126/sciadv.aax4539 (10), eaax4539.5Sci Adv

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MATERIALSSUPPLEMENTARY http://advances.sciencemag.org/content/suppl/2019/10/21/5.10.eaax4539.DC1

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Constraints on bosonic dark matter from ... - Science Advances · bands around theJ-coupling doublet as shown in Fig. 1C. The sidebands are separated from the carrier peaks by ±w