Axion wind detection with the homogeneous precession domain of superfluid helium-3

Axions and axion-like particles may couple to nuclear spins like a weak oscillating effective magnetic field, the"axion wind."Existing proposals for detecting the axion wind sourced by dark matter exploit analogies to nuclear magnetic resonance (NMR) and aim to detect the small transverse field generated when the axion wind resonantly tips the precessing spins in a polarized sample of material. We describe a new proposal using the homogeneous precession domain (HPD) of superfluid helium-3 as the detection medium, where the effect of the axion wind is a small shift in the precession frequency of a large-amplitude NMR signal. We argue that this setup can provide broadband detection of multiple axion masses simultaneously, and has competitive sensitivity to other axion wind experiments such as CASPEr-Wind at masses below $10^{-7}$ eV by exploiting precision frequency metrology in the readout stage.

Axions and axion-like particles may couple to nuclear spins like a weak oscillating effective magnetic field, the "axion wind."Existing proposals for detecting the axion wind sourced by dark matter exploit analogies to nuclear magnetic resonance (NMR) and aim to detect the small transverse field generated when the axion wind resonantly tips the precessing spins in a polarized sample of material.We describe a new proposal using the homogeneous precession domain (HPD) of superfluid 3 He as the detection medium, where the effect of the axion wind is a small shift in the precession frequency of a large-amplitude NMR signal.We argue that this setup can provide broadband detection of multiple axion masses simultaneously, and has competitive sensitivity to other axion wind experiments such as CASPEr-Wind at masses below 10 −7 eV by exploiting precision frequency metrology in the readout stage.
Axions and axion-like particles (ALPs) (denoted a) are CP-odd pseudo-Goldstone bosons whose couplings to matter respect a shift symmetry, and which may also make up the cosmic dark matter (DM) density [1][2][3].The ALP coupling to nuclei N , g aN N ∂ µ aN γ µ γ 5 N , reduces in the non-relativistic limit to a Hamiltonian γ B a • σ N , where acts as an effective oscillating magnetic field which couples to nuclear spins [4].Here v a ∼ 10 −3 c is the DM velocity (in what follows we will set c = 1), ω a = m a (1 + O(v 2 a )) with m a the axion mass, ρ DM = 0.3 GeV/cm 3 is the DM density, and γ is the gyromagnetic ratio of N .Because B a is proportional to the DM velocity, this interaction is known as the "axion wind."For QCD axions which could resolve the strong CP problem [5][6][7][8], g aN N ≈ 3 × 10 −8 GeV −1 (m a /eV) [9], yielding taking γ = 0.69 GeV −1 (γ/(2π) = 32.43MHz/T) for the gyromagnetic ratio of 3 He.Several experiments are aiming to detect the axion wind, including CASPEr-Wind [10][11][12] and CASPEr-ZULF [13,14], which use a polarized sample of nuclear spins as the detection medium; comagnetometers using two species of spins to reduce noise [15,16]; and QUAX, which uses polarized electron spins [17] (see Refs. [18,19] for a review of other experimental approaches).In a close analogy to nuclear magnetic resonance (NMR), when an external B-field is tuned such that the sample Larmor frequency matches m a , the spins will tip away from the external field, yielding a transverse magnetic field proportional to B a which grows linearly with time up to the axion coherence time, τ a 2π/(m a v 2 a ) as long as this is larger than the spin relaxation time.
In this Letter, we describe a new detection mechanism which converts the small amplitude ALP signal into a small frequency shift in a large-amplitude NMR signal, allowing the use of precision frequency metrology to evade amplitude noise.The sample is an unusual phase of superfluid 3 He, the homogeneous precession domain (HPD), which may be understood as a Bose-Einstein condensate (BEC) of spin-1 magnons formed by Cooper pairs of the spin-1/2 helium nuclei in the B-phase of the superfluid [20].We will first review the properties of the HPD (based largely on the review [20]), then describe the effect of the axion wind, and finally estimate the sensitivity of the setup to various ALP masses and couplings.

HPD REVIEW
3 He is a Fermi liquid which forms a superfluid below temperatures of ∼1 K; the excitations of this superfluid are spin-1 magnons.When a static magnetic field B 0 = B 0 (z)ẑ is applied, the equilibrium magnetization is M = χ B 0 , where χ ∼ 10 −7 is the susceptibility which reaches a maximum at a temperature of 2.5 mK and pressures of 34 bar [21].A transverse magnetic field pulse will torque the spins, injecting some number of spin-1 quanta (i.e.magnons), N M .In the B-phase of 3 He, the HPD is formed by the Bose condensation of these magnons after an RF pulse: a macroscopic fraction of the sample will spontaneously begin to precess coherently about B 0 with the nuclear spins tipped at the Leggett angle, β 0 = cos −1 (−1/4) ≈ 104 • (originating from dipole-dipole interactions), while the rest of the sample remains relaxed < l a t e x i t s h a 1 _ b a s e 6 4 = " t W v b 9 2 + q y u Q 1 9 / G J E a w r T / M 8 P H / 5 8 z l O I H n C g n h l 7 K 2 v r G 5 t V 3 Y K e 7 u 7 R 8 c q k f H H e G H H J M 2 9 j 2 f 9 x w k i O c y 0 p a u 9 E g v 4 A R R x y N d Z 3 K X 5 t 1 n w o X r s 0 c 5 D Y h F 0 Z i 5 I x c j m V i 2 e j X I z u j z s W

< l a t e x i t s h a 1 _ b a s e 6 4 = " F 2 e H x O I g y 3 p L + E s M i q h w c w o m X l Y = " >
< l a t e x i t s h a 1 _ b a s e 6 4 = " 0 8 J c 0 J 8 u r 7 o r K 0 g + 7 < l a t e x i t s h a 1 _ b a s e 6 4 = " A + in equilibrium [22].This is illustrated in Fig. 1: the HPD occupies a volume V HPD = N M /n M , where is the constant magnon density.
A striking feature of the HPD is that the precession frequency is spontaneously determined.One can view the HPD as a thermodynamic system at fixed particle number N M , where the chemical potential µ is set by minimizing the free energy F .In fact, µ is the local Larmor frequency, corresponding to an angular precession frequency ω L (z) = γB 0 (z), at the location z of the domain wall (DW) separating the HPD from the relaxed domain [23].Thus, even in the presence of a spatially-varying B 0 , the entire HPD precesses coherently at the same frequency.In a lossless system, the precession would continue indefinitely at the frequency established at HPD formation, but due to longitudinal spin relaxation from surface and volume losses on a timescale T 1 , N M will decrease monotonically [24].For sufficiently large T 1 compared to the spin supercurrent propagation time, the HPD will continuously minimize F at a smaller V HPD , and thus the precession frequency ω L (t) will sweep through the local Larmor frequencies corresponding to the motion z(t) of the DW, at a speed governed by T 1 .F is minimized when the HPD occupies the region of smaller B-field, so for dB 0 /dz of definite sign, ω L (t) will decrease monotonically and approximately linearly on timescales much shorter than T 1 .While the maximum attainable T 1 for the HPD is presently unknown, in the low-temperature limit magnetic modes in 3 He have exhibited T 1 as long as 1000 seconds [25,26].The Bphase is destabilized for B 0 0.55 T [27], which sets the maximum ω L of the HPD.
The aforementioned properties of the HPD may be understood through an effective description of the sample magnetization as described by the Bloch equations.With a static external magnetic field B = B 0 (z)ẑ (not necessarily homogeneous) the magnetization M in the HPD evolves as where M 0 is the equilibrium magnetization in the HPD, in T 1 has been introduced for later convenience; note that the transverse and longitudinal relaxation times are not independent because these components of the magnetization are locked together by the spin supercurrents at the constant tip angle β 0 [28].
Since the magnetization in the relaxed domain does not evolve, we can treat the equilibrium magnetization as belonging to the HPD only via M 0 = χB 0 F, where F is the HPD fraction of the sample.We parameterize M z = χB 0 F cos β 0 and M xy = χB F sin β 0 e −iθ , which are time-dependent through the HPD fraction F(t) and because B 0 is evaluated at the position z(t) of the domain wall.Suppose first that there is no transverse magnetic field, B xy = 0.For homogeneous B , Eq. ( 4) yields so the HPD fraction decays exponentially from its initial value F 0 due to relaxation.Similarly, Eq. ( 5) yields θ = γB 0 , so that precession occurs at the local Larmor frequency ω L = γB 0 .Due to the DW motion, if the external magnetic field is inhomogeneous, B 0 in Eq. (7) will acquire an effective time dependence, and thus so will ω L .To see this, consider the evolution of the HPD over a short time interval t T 1 .For a fixed cross-sectional area A of the sample container of volume V , the position of the domain wall is z(t) = −hF(t), where h = V /A is the height of the container and we have set z = 0 at the top of the container.At t = 0 we have z 0 = −hF 0 .For short times t T 1 , the DW position is where v D ≡ hF 0 /T 1 = |z 0 |/T 1 is the instantaneous DW velocity at the height z 0 .Now, since the HPD precesses according to the magnetic field B (z) at the DW, we Taylor-expand where we have defined α ≡ ∇ z B 0 /B 0 | z0 as a tuneable parameter of the experiment.For the geometry defined in Fig. 1, α < 0, and α can be taken to be constant in the region z ≈ z 0 .We further require |α(z − z 0 )| 1 so that n M is approximately constant.Combining Eqs. ( 6) and ( 7) and expanding to first order in t/T 1 , the transverse magnetization is where ω 0 L = γB 0 (z 0 ).The linear downward drift of the precession frequency, ωL ≡ θ(t) = αv D ω 0 L < 0, is the key feature of the HPD.

EFFECT OF THE AXION WIND ON THE HPD
The effect of the axion wind on the HPD evolution can be directly computed by including B a as a source in the Bloch equations.We will first describe the effect qualitatively and estimate its parametric size, and then confirm this behavior with numerical simulations in the following section.
In CASPEr, Larmor resonance is achieved when γB 0 = m a , and B a generates a tip angle δβ(t) = γB a t that grows linearly with time up to min[τ a , T * 2 ] where T * 2 is the dephasing time.The three principal challenges in this setup are (i) detecting the extremely small transverse field generated by δβ; (ii) tuning B 0 in steps of ∼ 10 −6 to achieve resonance at each possible axion mass, given the axion bandwidth m a v 2 a ∼ 10 −6 m a ; and (iii) generating a uniform B 0 to ensure that nearby spins do not dephase, thus preserving T * 2 > τ a .The HPD setup naturally sidesteps all of these challenges, as follows.Because the HPD is a macroscopic quantum state, the inhomogeneous B-field, which generates the linear frequency drift, preserves the precession phase in a macroscopic volume while scanning Larmor frequencies.This allows the HPD to probe many axion masses with the same field profile B 0 .In the HPD, the tip angle is fixed, so the torque from a transverse B a instead resonantly changes the number of magnons in the BEC when the local Larmor frequency is equal to m a , |∆N a | χB 0 B a V HPD τ a .The axion wind can thus be seen as a small chemical potential δµ a for magnons, as shown in Fig. 1.Since the magnon density in the BEC remains constant, V HPD changes with ∆N a , and hence the DW position shifts by an amount |∆z| γB a |z 0 |τ a .This leads to a frequency shift, Note that since B a ∝ m a along the QCD line, the QCD axion frequency shift is independent of the mass: While Eq. ( 12) is a very small frequency shift, it is comparable to the best sensitivity currently achieved by microwave atomic clocks [29], and could thus be detected in principle with homodyne detection of the NMR signal referenced to a local oscillator.In contrast to CASPEr, the overall amplitude of the NMR signal from the HPD is large, of order M xy (0) ∼ χB 0 , with the smallness of the axion signal appearing in the frequency rather than the amplitude.
The resonance can persist while the DW sweeps through a frequency bandwidth up to the axion bandwidth, giving a resonance timescale Define α * such that t r = τ a , e.g.
Thus, depending on the choice of α, The scaling with √ t r is a typical feature of measurements exceeding the coherence time and arises from adding in quadrature the incoherent frequency shifts in each interval τ a [30].The signal is maximized when α = α * , which recovers Eqs. ( 11) and ( 12), but choosing larger α allows more axion masses to be scanned.We can confirm the estimate in Eq. ( 11) with a perturbative analysis of the Bloch equations.Taking a transverse monochromatic axion field, B x = B a cos(m a t + φ), and writing F = F (0) (1 + F (1) ) where F (0) is the solution of Eq. ( 6) and F (1) is proportional to B a , we have for t T 1 and to leading order in B a , On resonance when m a = ω L (t), the right-hand side contains a constant term 1 2 sin φ, and thus the solution for F (1) is which grows linearly with time.In fact, integrating Eq. ( 16) we can obtain an analytic solution valid for t T 1 , where Ω ≡ αz0ω 0 L πT1 and S(x) and C(x) are the Fresnel sine and cosine integrals, respectively.
The effect of the axion can be interpreted as a first-order contribution to the DW velocity ∆v D = − 1 2 sin φ z 0 tan β 0 γB a , which then feeds back into the precession frequency because z − z 0 = hF 0 − hF (0) (1 + F (1) ) ≈ (v D + ∆v D )t.In other words, the first-order frequency shift is ∆ω a (t) = ω 0 L α∆v D t, and integrating up to τ a we recover the parametric estimate of Eq. ( 11), up to the O(1) factor 1  2 sin φ tan β 0 .There is also an additional first-order effect on the precession phase θ, but it is always parametrically smaller than the frequency shift induced from the variation of V HPD .Finally, note that a longitudinal axion field, B z = B a cos(m a t + φ), will not yield a resonance but will rather imprint fast oscillations at frequency m a on the phase θ.

SIMULATIONS, NOISE, AND SENSITIVITY
To validate the above analysis, we numerically solved the Bloch equations with an axion source, both for a monochromatic axion field and an axion field with the expected 10 −6 bandwidth from the DM velocity dispersion.Since the axion field is a random field with a known power spectrum, for the latter case, we used the timedomain axion field constructed from sampling the speed distribution according to the prescription in Ref. [31].Specifically, we took The sum is over groups of axion particles, each group having a speed ∈ [v j , v j + ∆v].Here α j is a Rayleighdistributed random variable, φ j ∈ [0, 2π) is a random phase, and f (v) is the local DM speed distribution which we take to be Maxwellian with dispersion v 0 = 220 km/sec, boosted to the lab frame by v Earth = 232.24km/sec.A key feature of the HPD setup is the possibility of observing daily modulation, since the same axion masses may be scanned multiple times throughout the day.However, for this analysis we fix v 0 to be transverse and leave a full daily modulation analysis for future work.Indeed, we use the stochastic field simply to validate our treatment of the coherence time, and we expect that our parametric estimates of the signal strength will differ by O(1) factors from a full treatment of the correlated components of ∇a [32,33].Fig. 2 shows the evolving fractional frequency shift (| θ(t)| − m a )/ω 0 L and the perturbation to the HPD fraction F (1) (t), when a transverse axion wind is on resonance with the Larmor frequency.The DW can shift either up or down depending on the phase when the resonance occurs, and hence the frequency can shift above or below the linear drift indicated by the grey lines.As mentioned earlier, if the axion wind is longitudinal, no such resonance occurs.Both the monochromatic and the stochastic axion sources show the expected linear growth in F (1)  during the resonance (given by Eq. ( 18) in the case of the monochromatic axion).The accumulated frequency shift ∆ω a /ω 0 L appears as a persistent offset at ∆t t r and matches our parametric estimates in Eq. (11).
To obtain a projected sensitivity for the HPD setup, we need to consider possible noise sources.Since we have already argued that the expected ∆ω a is (in principle) within the sensitivity of state-of-the-art atomic clocks, we will focus on irreducible noise sources.Fortunately, the unique properties of the HPD makes thermal noise a negligible concern.Unlike the case of a BEC of atoms, thermal fluctuations in 3 He do not cause the number of magnons in the condensate to fluctuate, but rather affect the normal fluid component of the 3 He superfluid.Thus, thermal noise will not lead to a frequency shift.
In fact, the leading irreducible noise source is stochastic fluctuations in magnon number.Over a time interval ∆t T 1 , losses reduce the average number of magnons by N loss = n M ∆V HPD , where ∆V HPD = v D A∆t is the change in HPD volume over ∆t.This will lead to Poissonian "shot noise" of order σ N = √ N loss , which will cause the HPD volume and hence the frequency to fluctuate.Taking ∆t = τ a , the noise on the frequency shift is We leave the precise microscopic characterization of the stochastic noise to future work, including an analysis of surface and volume loss mechanisms, but we note that the quantized fluctuation of magnons in the HPD as parameterized by T 1 is a reasonable coarse-grained characterization of many such microphysical phenomena.Since the Larmor frequency continuously changes in the experiment, the number of axion resonances covered in a single run is given by roughly T 1 /t r ∼ 10 6 (αz 0 ), where t r is given by Eq. (13).Repeating the experiment N times suppresses the noise by √ N .However, since the axion field is incoherent between the different runs, the signal-to-noise ratio SNR ≡ ∆ωa ∆ω stoch.scales as N 1/4 for the same reasons as discussed in Eq. (15).Letting N = t int /T 1 where t int is the total integration time, we have Choosing t r = τ a for the maximum sensitivity gives for couplings on the QCD line; the axion mass m a = 7 × 10 −8 eV is the largest mass which can be probed before the HPD destabilizes at large fields.Fig. 3 shows a projected sensitivity on g aN N taking SNR = 1.Also shown are the star cooling bounds (green) from SN 1987A [34] and neutron stars [35] (assuming an FIG. 3. Projected sensitivity to gaNN (SNR = 1) from the 3 He HPD (black, red, blue), for various choices of parameters.The green shaded regions correspond to the star cooling bounds from SN 1987A [34], and neutron stars [35] (assuming an axion-neutron coupling).The dashed red lines are the CASPEr-Wind projections from Ref. [11].
axion-neutron coupling), as well as the projected sensitivity (red) from CASPEr-Wind [11].As anticipated, choosing a large gradient α results in a larger coverage in axion masses, at the cost of less time spent in each resonance and hence a smaller SNR.However, comparable sensitivity to CASPEr-Wind over an order of magnitude in m a can be achieved with reasonable parameters in 1.5 years of measurement time at 1 month per B-field tune.

CONCLUSIONS
In this Letter we have described a new proposed experiment for detection of the axion wind which exploits the macroscopic coherent properties of the B-phase of 3 He.In future work we plan to further improve the projected sensitivity by taking into account the correlations between the N measurements induced by daily modulation, as well as correlations between spatially-separated samples [36], which may increase the SNR such that QCD axion detection at masses 10 −7 eV is possible.We close by emphasizing that even without further improvements to T 1 beyond those already achieved in the laboratory (black curve in Fig. 3) the HPD detection scheme could achieve world-leading limits on the axion wind coupling which exceed the most stringent astrophysical bounds.

B 0 <
l a t e x i t s h a 1 _ b a s e 6 4 = " Y p 9 p 0 d b n 1 3 n X S u q p 6 t e r 1 Q 6 1 S b + S / K M I Z n M M l e H A D d b i H J v j A Y A S v M I N 3 w s k b m Z G P 3 9 I C y X t O Y Q n k 8 w d s P I 2 O < / l a t e x i t > dB 0 dz < l a t e x i t s h a 1 _ b a s e 6 4 = " K S N H c B P d u Z 4 h U 3 z s 2 N s Q U g A L N + M = " > A A A B 9 X i c b V D L T s J A F L 3 F F + K D q k s 3 j c T E F W k N R p c E N y 4 x s U A C T T O d T n H C z L S Z m W q w 6 Z c Y d + j W T 3 H t 3 1 i w C w F P c m 9 O z r k 3 9 x E k j C p t 2 9 9 G Z W N z a 3 u n u l v b 2 z 8 4 r J t H x z 0 V p x I T F 8 c s l o M A K c K o I K 6 m m p F B I g n i A S P 9 Y H I 7 9 / t P R C o a i w c 9 T Y j H 0 V j Q i G K k C 8 k 3 6 6 N I I p y F H d / O s / A l 9 8 2 G 3 b Q X s N a J U 5 I G l O j 6 5 t c o j H H K i d C Y I a W G j p 1 o L 0 N S U 8 x I X h u l FIG. 1. Spins in the HPD (blue) precess at the local Larmor frequency set by the location of the DW, while the remainder of the sample (red) is relaxed.A transverse axion wind Ba acts as a small chemical potential δµa for magnons.
xy ≡ M x +iM y is the transverse magnetization, M xy = M x − iM y , and likewise for B xy and B xy .The factor of 5 = cos β0−1 cos β0