Searching for Earth/Solar Axion Halos

We discuss the sensitivity of the present and near-future axion dark matter experiments to a halo of axions or axion-like particles gravitationally bound to the Earth or the Sun. The existence of such halos, assuming they are formed, renders a significant gain in the sensitivity of axion searches while satisfying all the present experimental bounds. The structure and coherence properties of these halos also imply novel signals, which can depend on the latitude or orientation of the detector. We demonstrate this by analysing the sensitivity of several distinct types of axion dark matter experiments.


Introduction
Axion-like particles (ALPs) 1 can form coherently oscillating classical field configurations. Such an oscillating field behaves as non relativistic matter from the early universe, and can account for dark matter (DM) in the present universe [1][2][3].
A variety of experiments have been proposed to search for such DM candidates. Those experiments include conventional axion halo-scopes using microwave cavities [4][5][6], experiments exploiting magnetic resonance [7][8][9], interferometry [10][11][12][13], precision magnetometers and LC circuits [14][15][16][17], atomic transitions [18], searches for an oscillating neutron electric dipole moment [19], and many others (for a more complete list of axion dark matter detection experiments, see Refs. [20,21] and references therein). These experiments cover a wide range of ALP DM particle mass, from 10 −22 eV to 10 −2 eV. Since all of them are based on the assumption that the dark matter is a coherently oscillating ALP field, the experimental reach depends on the local dark matter density.
On the other hand, it is well known that throughout the cosmological history, axions may form gravitationally-bound objects, whose density can be orders of magnitude larger than the local dark matter density (see also [22][23][24] for recent discussions.). Typical examples include axion miniclusters [25,26] and boson stars [27][28][29]. Being much denser than local dark matter, these small-scale objects could boost discovery potential of aforementioned axion dark matter experiments, should they exist. To exploit a density enhancement, the encounter rate between Earth and these small scale objects should be large enough such that one may expect several transient signals within a experimental timescale. However, the encounter rate inversely depends on the density of such objects, so even if bound objects make up the entirety of the DM in the galaxy, one can only expect a mild density enhancement within a reasonable experimental timescale (see e.g. [30] for detailed discussions about the encounter rate and expected density enhancement). It was recently pointed out that microwave atomic clocks, being sensitive to the mass range around 10 −6 eV, may still benefit from the density enhancement of transient boson stars [31].
In this paper, we point out how the reach of experimental searches can be greatly increased if small-scale ALP objects become gravitationally bound to other astrophysical objects such as the Earth or the Sun. The resulting ALP density may be many orders of magnitude above the local galactic DM density and, at the same time, one need not consider encounter rates anymore as the signal is no longer transient. In addition to the density enhancement, the improvement in experimental sensitivity stems from an increased coherence time and (in a few specific cases) a modification of spatial gradients in the DM field. We illustrate the possible improvements in the experimental sensitivity by analyzing several proposed or ongoing DM experiments.
The paper is organized in the following way. In Sec. 2 , we summarize the main properties of the axion halos, based on the results previously derived for a dilaton-like (scalar) field halo in [30]. We also provide general estimates for the possible signal enhancement and describe unique signatures of this scenario. In Sec. 3 , we analyse the sensitivity of the experiments testing the couplings of the axion field to the Standard Model. In Sec. 4 , we reinterpret sensitivity to QCD axion parameters as a probe of the mass of the axion halos in the solar system, bound by the Earth or the Sun. We conclude in Sec. 5 . We work in natural units, where = c = 1, which in this case can be done without loss of generality [32].

Axion Halo Properties
We consider compact astrophysical objects composed of ALPs. In particular, we focus on scenarios where such objects are gravitationally bounded to the Earth or the Sun; we call such an object a Solar or Earth axion halo. The axion halo we are considering here shares similar properties with usual boson stars (see e.g. [27-29, 33, 34]), except that it is bounded by a gravitational potential of an external body rather than its own self-gravity. Throughout this work, we will assume that the center of axion halo coincides with the center of the external host body. In this section, we review some properties of axion halo, following and expanding on the previous work on scalar field halos [30].
We consider either the Earth or the Sun as a host of an axion halo. As the axion halo is maintained by the gravitational potential of the host, its mass M is assumed to be smaller than M ext (R ), defined as the host mass that is enclosed within the radius R of the axion halo. For clarity we will take the axion halo mass as where M max is the maximally allowed mass for axion halo in the solar system that is consistent with local measurements of gravity and solar system ephemerides [30,35,36]. A factor of two is introduced to ensure that the object is maintained due to the gravitational potential of external body rather than self-gravity.
To approximate the shape of the density profile, we used the procedure outlined in [30]: the axion halo profile is approximately exponential at R R ext ; it is Gaussian at R R ext ; and in the intermediate region we interpolate between the two. For example, when R R ext , the axion field has an exponentially decaying profile with the distance from the center of the host body r, and oscillating in time with a frequency approximately equal to the axion mass m φ in the halo rest frame: where θ(r, t) is a phase factor slowly varying with respect to the position and the time. The time scale that the phase factor changes by an order one value is the coherence time scale that we discuss below. The corresponding energy density is given by ρ m 2 φ φ 2 /2. Since the time dependence is given by the same factor cos (m φ t) throughout, for the remainder of this work we will no longer write it explicitly and refer to φ as the space-dependent part only.
The size of the halo can be determined as a balance between the repulsive gradient en- , and the attractive gravitational potential energy, U grav ∼ GM ext M /R . The resulting radius is which is independent of M . That is, the axion halo mass is a free parameter; later, in the estimations of experimental sensitivities, we will set it by Eq. (1). For R > R ext , the radius scales as R ∝ m −2 φ , while for R < R ext , it scales as R ∝ m −1/2 φ since the enclosed external mass scales as M ext (R ) ∝ R 3 . If the axion halo profile extends to sufficiently large radii, experiments on Earth's surface will benefit from a large axion halo density. For an Earth-based halo, the relevant requirement is R R ⊕ , implying m φ 10 −9 eV, while, for a Sun-based of Figure 1, we consider axion halo either hosted by Earth (Earth halo) or Sun (solar halo), and compare the field amplitude of these halos to that of the standard axion DM scenario. The outer envelopes of the red and blue regions represent the maximal axion halo mass allowed, given in Eq. (1); the contours represent smaller values of M /M ext . Note that the standard axion DM scenario is characterised by a local field value φ DM , coherence time τ DM , and virial velocity v vir given by where ρ local = 0.4 GeV/cm 3 is the local density of the background axion dark matter (in our numerical calculations we took v vir = 230 km/sec = 7.7 × 10 −4 ).
The coherence time of the axion halo requires a more careful discussion. If the axion halo is an exact ground-state solution of the equation of motion (a condensate), as is often assumed about self-gravitating axion stars [28], its oscillations remain coherent on infinite timescales. If, on the other hand, the axion halo is not the true ground state solution, but is a collection of axions with some distribution of velocities centred around v , then we can describe the axion halo as a sum of incoherent subcomponents with slightly shifted oscillation frequencies and phases.
This picture is potentially motivated by by simulations of axion star formation which observe lingering fluctuations on scales larger than λ dB [37] (see also [38] for the phenomenological implication of the picture for the large-scale bosonic DM halo). In this case, the coherence timescale of the axion halo can naively be estimated as  The radial component comes from the variation of the radial field profile, Eq. (2), which is given by This is a static gradient, oriented along the radial direction; for convenience it can be written in terms of v = (m φ R ) −1 , though this should be interpreted as a velocity dispersion rather than a net flow of axions (see relevant discussions in [39,40]). This makes sense at the level of the classical equation of motion, where in the rest frame of the halo the kinetic energy in the φ The tangential component of the gradient appears as a consequence of the relative motion between the halo and an experimental device. The corresponding field gradient is where v rel is approximately the speed of a device in the rest frame of halo. Its value depends on the halo angular momentum, which is a free parameter at the level of this analysis. If the halo is non-rotating, we can take v rel equal to the Earth orbital speed around the Sun v rel ∼ 10 −4 (for the solar halo) or to the Earth surface speed, v ⊕ rel ∼ 10 −6 at the equator (for the Earth halo). For simplicity we assume zero angular momentum for the axion halo, so that the maximum tangential gradient for an experimental apparatus is of order the optimistic estimate given here.
In Figure 2, we present the relative enhancement of the field gradient as a function of ALP mass. In the left panel, we illustrate the radial component for different choices of M (the contours correspond to M /M ext ), while, in the right panel, we compare the radial (red and blue) and tangential (gray) gradients. The tangential gradient becomes most important at small values of m φ , where the radial gradient 1/R is suppressed. This implies that the position and orientation of the device can be important to determine the signal strength.

Detector Orientation, Latitude, and Modulation of Signals
Depending on the spatial orientation of an experimental device, it may be sensitive to either radial or tangental (or both) components of the gradient, and the amplitude of the signal may modulate over the course of a day or year. For the Earth halo, both gradient components are constant; the radial gradient can be maximized by choice of orientation of the experimental apparatus with respect to the surface of the Earth, whereas the tangential gradient depends in addition on the latitude of the detector, as the relative velocity v ⊕ rel is faster at the equator and goes to zero at Earth's poles. For the solar halo, both gradient components modulate strongly with a period of one solar day, with a weaker additional annual modulation, because as the Earth rotates and revolves around the sun, the orientation of the detector rotates as well.
Suppose for concreteness that an experiment is searching for a signal proportional to r det ·∇φ, where r det (denoted as the "detector orientation" in what follows) is some vector quantity in the experiment (e.g. it is related to the orientation of nuclear spin in a nuclear magnetic resonance experiment). Though we focus on the example of a dot product, it is simple to extrapolate to the case where the signal is given by the cross product r det × ∇φ instead.
In the case of an Earth-based halo, both components of the gradient are constant with time.
The radial gradient will be maximized when r det is oriented perpendicular to the surface of the Earth, whereas the tangential gradient is maximized when r det is instead parallel to the surface of Earth, oriented along the direction of Earth's rotation about its axis 2 . Further, the latter depends on the latitude of the detector, because the linear speed of Earth's rotation is maximized at the Equator and minimized at the poles. If r det is pointed along the lines of latitude, then we can parametrize the dependence on the latitude of the detector simply aŝ where the hat denotes a unit vector. Note that in our notation, a latitude of x • N is denoted by = +x and y • S is denoted by = −y.
The case of the solar halo is more complex. As before, the effect depends on the latitude of the detector, but as the Earth rotates and revolves around the sun, a detector oriented alonĝ r det will also see modulation of the signal along both the radial and tangential directions. To The existence of separate radial and tangential gradients is a unique prediction of the axion halo model. This implies a method to distinguish a bound axion halo from background DM by using directional information, as the detector orientation modifies the signal. We see further

Experimental Probes
We demonstrate possible improvements in experimental sensitivity due to an axion halo by investigating a set of proposed axion DM experiments. Among many other axion DM searches, we choose experiments sensitive to the axion mass range 10 −16 eV < m φ < 10 −6 eV, a range in which the axion halo could be sufficiently large such that these searches could exploit a large density enhancement. We re-evaluate the reach on coupling constants for these searches, and illustrate the main differences between the axion halo case and the axion DM scenario.

ALP Search using nuclear magnetic resonance
We begin with ALP DM searches using nuclear magnetic resonance (NMR) techniques. From the UV perspective all the relevant interactions stem, to the leading order, from a single operator Since the zero mode of axion introduces CPV interactions in the chiral Lagrangian, it gives rise to nucleon electric dipole moment, for instance, through a pion loop diagram with an insertion of CPV nucleon-pion vertex. In addition, the CPV nucleon-pion vertex also induces where N is the nucleon field, F µν is the electromagnetic field strength tensor, g φN N and g d are the coupling constants, and the dots correspond to other terms, for instance the nucleon-nucleon contact term. The first term represents, for example, the coupling between axion and nuclear spin, while the second term is the axion-dependent nucleon electric dipole moment (EDM).
These operators of the effective nuclear action produces CP-violating nuclear Schiff moment which actually induces atomic EDM d a which is measured in experiments [39]. According to the Schiff theorem, nuclear EDM is screened by atomic electrons [42]. Atomic EDM is actually produced by a higher order electric multipole called the nuclear Schiff moment [43]. In atoms with zero electron angular momentum, the atomic EDM d a and magnetic moment µ a are directed along the nuclear spin I. Therefore, the effective interaction of the nuclear spin with external electric and magnetic fields may be presented in the following form: where µ n is the nuclear magnetic dipole moment. The axion-dependent atomic EDM and the effective magnetic field are d a = g ad φ I , where γ n is the gyromagnetic ratio of a given nucleus, and d a is the ALP-induced atomic electric dipole moment. We simply express g ad = s g d following [7]. Note that the gyromagnetic ratio cancels in Eq. (11) (since µ n ∝ γ n ), so that the ALP-induced force to the nuclear spin is in fact independent of γ n .
The Cosmic Axion Spin Precession Experiment (CASPEr) has been proposed to probe the above interactions using NMR techniques [7,8]; we briefly review the concept of this experiment here. We discuss how CASPEr can probe the effective axion-induced atomic EDM (CASPEr-Electric). A polarized material is placed in the external magnetic field background, B ext , while its direction is aligned along the direction of the polarization. An external electric field E is also applied orthogonal to the direction of external magnetic field. Basically the atom has an axion-induced EDM, thus, the electric field exerts a torque to nuclear spins, which develop a non-zero angle with respect to the external magnetic field, and hence, a non-zero transverse magnetization is obtained.
If the spin precession frequency, i.e. the Larmor frequency Ω = 2µ n B ext , matches to the frequency of the oscillating EDM ω m φ , the amplitude of the transverse magnetization is and static EDM searches (gray). The input parameters were those used in [7]. resonantly enhanced as where n is the nuclear spin density, p is the material polarization. In Figure 5, we show a comparison between the reach of CASPEr, both for the axion halo and for the standard axion DM scenario. We use the parameters of Ref. [7], which defines both Phase 1 and Phase 2 of the experiment. While the Phase 2 upgrade is necessary to probe a portion of the QCD axion band for the background DM model, Phase 1 would be sufficient to probe a wide range of QCD axion masses below 10 −7 eV in the presence of an Earth-based or Sun-based halo. The main reason for this is a higher energy density of a halo as well as the longer coherence time. CASPEr is also able to probe the axial-vector coupling of axion to nucleon in a similar fashion, but without an external electric field (this version is known as CASPEr-Wind). A spin-polarized material is similarly prepared, and an external magnetic field B ext is applied along the direction of the nuclear magnetic moment µ n . In the absence of the background axion field, there is no spin precession in the material because µ n × B ext = 0. In the presence of the background axion, the material experiences additional torques, developing nonvanishing transverse magnetization. If the resonance condition is met, 2µ n B ext = m φ , the amplitude of the induced transverse magnetization is resonantly enhanced as We remind the reader that there are two contributions to the axion-induced magnetic field, For the solar halo, the relevant relative velocity is given by Earth's orbital velocity around the Sun, roughly v rel 10 −4 . The induced signal will be proportional to the torque | µ n × B φ |, which depends on Of course, the orientation of the nuclear spin with respect to the motion of the Earth depends on the time of day (as the Earth rotates) and the time of year (as the Earth moves through its not-quite-circular orbit around the Sun). For simplicity, in estimating sensitivity we will take a fixed value | v tot | ∼ v + v rel (see Section 2.1 for further details).
For the Earth-based halo, there is an analogous relative velocity originating in the rotation of the Earth through the axion halo. In the current instantiation of the CASPEr-Wind experiment, the nuclear magnetic moment µ is oriented perpendicular to the surface of Earth, implying that µ n × ∇φ µ n × ∇ ⊥ φ, and so the signal will come from the tangential gradient only. Therefore in our sensitivity estimates below, we use |∇ ⊥ φ| m φ φ v ⊕ rel with v ⊕ rel 1.5 × 10 −6 . Note however that the ideal setup to optimize sensitivity to the axion halo would be to orient µ n parallel to the surface of Earth (maximizing the radial gradient) and perpendicular to Earth's lines of latitude (maximizing the tangential gradient also).
In order to isolate the change in sensitivity in the presence of an axion halo, we analyze only the sensitivity enhancement for the axion-spin coupling g φN N , rather than the absolute sensitivity. In Figure 5, we present the ratio (SNR) halo /(SNR) local , which is the enhancement in the signal-to-noise ratio in presence of an axion halo; see Appendix A for more details. For a solar halo, the enhancement can be 1 − 2 orders of magnitude, whereas for the Earth halo it could exceed a factor of 10 5 in increased sensitivity. For the projected sensitivities in the presence of local DM, we refer readers to [8,44].   [48]. 3 If the Earth encounters the axion stars with a relative velocity v, all GNOME sensors would register a transient signal within the time T ∼ 2R ⊕ /v. GNOME is sensitive to the axion coupling to nucleon spin [48], which is given in Eq. (10).

ALP Search using a global network of magnetometers
In the non-relativistic limit, the Hamiltonian can be written as, H − µ n · B φ cos(m φ t), and thus a nuclear spin experiences an oscillating energy shift, ∆E = µ n B φ in the presence of an axion background. The signal-to-noise ratio for an optical magnetometer is where S 1/2 is the magnetometer sensitivity. The function F (t, τ φ ) describes the behavior of signal-to-noise as a function of total measurement time t and a coherence time τ φ , and it is given as  [46], as interpreted in the presence of a solar axion halo. The green shaded region is the current exclusion limit from CAST [47].
where the behavior for t > τ φ is discussed in [7]. Given the current instantiation of GNOME, the magnetometer sensitivity is S We only consider the solar axion halo since the GNOME experiment is sensitive to axion masses lighter than roughly m φ 10 −13 eV. Below m φ 10 −14 eV, the coherence time of the axion halo could be longer than a year (see Figure 1), but we use the total measurement time to be a year for the estimate. Requiring SNR 1, we show the projected sensitivity of GNOME to g φN N in the presence of a solar halo in Figure 7. We observe that over a large range of m φ , GNOME and especially Advanced GNOME can probe the axion-nucleon spin coupling with a sensitivity that is several orders of magnitude better than existing limits from CAST [47].
We comment on three unique signatures that we may expect from GNOME, compared to other experiments we have considered. First, we have taken the sensitivity estimate of a single GNOME (co)magnetometer, but the experiment consists in N > 10 such detectors (with more coming online in the near future). Second, the axion halo that we are considering has macroscopic spatial coherence length, extending far beyond the radius of Earth's orbit, and thus, the signals in all GNOME stations would be coherent. As a result, we expect that in the presence of a signal, the correlation between different GNOME stations will give an enhancement, likely improving the sensitivity proportionally to √ N . Third, the multi-station design of GNOME lends itself perfectly to test a particular prediction of this scenario, namely, the existence of nearly-independent radial and tangential components of the gradient ∇φ. As of this writing, GNOME has stations around the world with different orientations relative to the surface of the Earth, allowing access to different proportions of the two gradient components. We emphasize that a dedicated search for a axion solar halo, including both the effect of detector latitude as well as orientation, would surely improve sensitivity to the signal as well.
In addition to GNOME, we also consider decade-old data from 3 He-K noble-alkali comagnetometers [49][50][51], which was recently used to constrain the ALP coupling to nucleons g φN N [46].
In the presence of an axion halo, the gradient measured by the "old" comagnetometers is enhanced by the factor (∇φ/∇φ DM ) (φ/φ DM ) (v vir /v tot ), which are essentially fixed by the results of Figures 1 and 2. In Figure 7, we show the constraint on g φN N for the case of the solar halo (red lines). In the presence of a solar halo, the current GNOME sensitivity is weaker than these old data, except in a narrow region around m φ ∼ 2 × 10 −16 eV; again, this may be improved in a dedicated analysis using the full power of all GNOME stations. With a planned upgrade, Advance GNOME will be sensitive over a wider range of masses m φ , and more sensitive to g φN N than the old comagnetometers by an order of magnitude or more.

ALP Search using an axion-induced current
The third class of experiments we consider is based on the axion-induced effective current, which originates from the following anomalous coupling of axion to photon: where F µν = µνρσ F ρσ /2 is a dual electromagnetic field strength. We treat g φγγ as an independent parameter to analyse possible constraints. In the presence of the above anomalous coupling, one can write the Euler-Lagrange equation ∂ µ (F µν + g φγγ φ F µν ) = 0 in terms of electromagnetic field as From the above modified Maxwell's equation, it is clear that a background magnetic field B 0 would induce an effective current A Broadband/Resonant Approach to Cosmic Axion Detection with an Amplifying B-field Ring Apparatus (ABRACADABRA) has been proposed to utilize this axion-induced current to probe the axion DM [16], and has recently set its first experimental limit on g φγγ [52]. The experimental setup consists of a toroidal geometry generating a background magnetic field along the azimuthal direction, and a pickup coil coupled to a SQUID magnetometer, located at the center of a toroidal magnet. The effective current j eff is generated along B 0 according to Eq. (20), and due For ABRACADABRA, we take the same parameters used in [16] for a broadband search. In all cases we have also restricted the integration time to 1 yr.
to Maxwell's equation, an oscillating component of magnetic field is developed orthogonal to B 0 , whose flux is measured by a pickup circuit (for detailed experimental setup, see [16]). The signal-to-noise is given as where the function F is defined in Eq. (17), Φ is are the magnetic flux through SQUID, and S 1/2 = 10 −6 Φ 0 / √ Hz is the sensitivity of commercial SQUID with Φ 0 = h/(2e). The amplitude of the magnetic flux is approximately given as [16,52] where V is the volume of the toroid, B max is the maximum magnetic field inside the toroid, and G is a geometric factor.
Using the parameters given in [16,52] and requiring SNR 1, we find current and projected sensitivities in Figure 8. The solid, dashed, dotted lines correspond to the projected sensitivity on g φγγ in the current (solid lines) and optimistic eventual (dashed lines) iterations of the experiment; the latter implies a significant upgrade in toroid size, magnetic field strength, and integration time. The black lines assume the standard background DM density, whereas the blue (red) lines are for an Earth-based (Sun-based) axion halo. The black (blue) shaded region corresponds to the current constraint from ABRA10cm assuming local DM (maximal axion halo) parameters [52]. It is intriguing that with the current level of sensitivity, ABRA10cm is already probing maximal axion Earth halos composed of QCD axions of mass m φ ∼ neV.
The current and projected sensitivities shown in Figure 8 are obtained for broadband searches. Although ABRACADABRA may also conduct resonant searches, the current and near-future versions do not appear to be optimized for this mode of running, and thus, we have only shown the result for broadband search strategy. It is worth noting that there exists another proposal, DM-Radio, searching for the same coupling constant g φγγ based on the same principle. DM-Radio focuses on resonant search, claiming that this is the superior strategy on general theoretical grounds [53]. In any case, in the scenario of a bound axion halo the logic determining the relative benefits of broadband vs. resonant searches must be reconsidered carefully.
At present, there are no detailed estimates in publication regarding the DM-Radio sensitivity to light scalars, as the current focus of the collaboration is on the DM-Radio Pathfinder, a prototype experiment optimized to search for dark photons in a similar mass range [17]. For this reason, we do not show the projected sensitivity of DM-Radio in this paper, though the experiment can in principle enjoy the same density enhancement from axion halo that we have described for ABRACADABRA.

Constraining the Halo Mass
In a predictive model, the ALP couplings to matter can be directly computed from the particle

Outlook
We have analyzed the effect of an axion halo, gravitationally bound to our solar system, on In a previous work [30], we performed an analogous analysis for fields in a similar mass range but with scalar couplings, those that give rise to effective oscillation of fundamental constants like the electron mass or fine-structure constant. Such oscillations are typically probed by atomic physics experiments, which are currently achieving impressive levels of sensitivity to low-amplitude oscillations. It is interesting to point out that some models of scalar field DM, notably the relaxion [54], that is an axion-like particle that mixes with the Higgs due to the presence of CP violation [55], predicts simultaneous presence of scalar and pseudoscalar couplings.
It is possible that a combined search across different and diverse experimental programs may represent a unique probe of these models. A significant task to the scenario we have assumed in this work will be to outline a detailed formation history of scalar fields bound to objects in the solar system; thus far, we have only analysed the phenomenological consequences, should such a formation occur. On this point we merely mention that simulations have, only very recently, begun to shed light on the formation of the self-gravitating counterpart of our axion halos, known as boson stars. We speculate that, in the presence of a large external gravitational potential (e.g. that of the Sun), the very same relaxation processes that leads to boson star formation might instead lead to the formation of an axion halo (around the Sun), the latter being the ground state of the composite system. The details of this and other possible formation histories for bound axion halos will be pursued in the near future.

A CASPEr
CASPEr is a resonant search. At each frequency, we assume a measurement time as t s ≈ ∆ω m φ t int , where ∆ω is determined as (∆ω) −1 = Min [τ φ , T 2 ]. And, t int is the total running time of the experiment, which is taken to be 3 years. For CASPEr-Electric, the maximal size of transverse magnetization is limited to The resonantly enhanced magnetization is measured by pickup look coupled to SQUID magnetometer. The magnetic flux through SQUID is estimated as Φ = (4πA eff )M φ , where A eff 0.3 cm 2 is effective geometrical area for the given experimental setup [7]. On the other hand, the sensitivity (noise power spectral density) of commercial SQUID magnetometer is S 1/2 10 −6 Φ 0 / √ Hz with Φ 0 = h/(2e). Therefore, the signal-to-noise ratio is where F is defined as (17). Note that we have assumed that the sensitivity will not depend on the spin relaxation time T 2 , because if T 2 < τ φ it is possible to recover coherent enhancement up to t = τ φ through the phase-cycling technique applied in the signal post-processing; see [56] for details. The projected sensitivities in the main text is obtained by requiring SNR 1.
The sensitivity of CASPEr-Wind can be obtained by the same procedure. Since the noise sources are not explicitly discussed in [44], we instead compute the relative enhancement of sensitivity for the halo compared to local dark matter scenario for CASPEr-Wind. The enhancement in sensitivities due to density contrast is estimated as We use the parameters of [8], including T 2 = 100 sec, and the shot time (t s ) is chosen as discussed above for CASPEr-Electric.
We emphasize that the axion mass being probed in CASPEr is proportional to the external magnetic field applied, so at low frequencies, it is necessary to use weak magnetic field to achieve the resonance effect discussed above. In this regime standard NMR techniques are not ideal, and so the sensitivity estimates we have used may not be appropriate. However, the apparatus can be modified through the use of hyper-polarization techniques as well as modified encoding and detection methods not based on Faraday induction [44]. Such techniques are being employed in CASPEr-ZULF (Zero to Ultra-Low Field) [56,57] as an extension of CASPEr-Wind, and will allow the experiment to probe axion masses in the range 10 −22 eV m φ 10 −13 eV. In this case the phase-cycling technique mentioned above is crucial, since typically τ φ T 2 for the polarized materials used in the experiment.