Magnetic Resonance Searches

Abstract

Ultralight bosonic dark matter (UBDM), such as axions and axionlike particles (ALPs), can interact with Standard Model particles via a variety of portals. One type of portal induces electric dipole moments (EDMs) of nuclei and electrons and another type generates torques on nuclear and electronic spins. Several experiments search for interactions of spins with the galactic dark matter background via these portals, comprising a new class of dark matter haloscopes based on magnetic resonance.

6.1 Searching for Axionlike Dark Matter via Nuclear Magnetic Resonance

Searches for ultralight bosonic dark matter (UBDM) based on magnetic resonance rely on the interaction between the UBDM field and spin. The possible forms of these interactions are detailed in Chap. 2 (see Table 2.1 and surrounding discussion). Here we focus on axions and axionlike particles (ALPs) and write these interactions in a format that aids the description of relevant experiments. We also assume that the axion or ALP field is the dominant component of the dark matter energy density. Chapter 8 covers searches that do not make this assumption.

6.1.1 Interactions with Nuclear Spins

The electric dipole moment (EDM) interaction of the axion field a with nuclear spin I is described by the non-relativistic Hamiltonian:

$$\displaystyle \begin{aligned} \mathcal{H}_{\mathrm{EDM}} = g_da\boldsymbol{E}^*\cdot\boldsymbol{I}/I, {} \end{aligned} $$

(6.1)

where g_d is the coupling constant and E^∗ is the effective electric field, see below. The gradient interaction is described by the non-relativistic Hamiltonian (see discussion in Sect. 2.4.3):

$$\displaystyle \begin{aligned} \mathcal{H}_{\mathrm{gr}} = \hbar cg_{\mathrm{aNN}}\boldsymbol{\nabla}a\cdot\boldsymbol{I}, {} \end{aligned} $$

(6.2)

where g_aNN is the coupling constant. Both $\mathcal {H}_{\mathrm {EDM}}$ and $\mathcal {H}_{\mathrm {gr}}$ have the same form as the Zeeman Hamiltonian, $\mathcal {H}_Z = -\hbar \gamma \boldsymbol {B}^*\cdot \boldsymbol {I}$, where γ is the nuclear spin gyromagnetic ratio, and B^∗ is an effective magnetic field, proportional either to aE^∗ or to ∇a. To a first approximation, the axionlike dark matter field $a(t)=a_0\cos {(\omega _at)}$ oscillates at the Compton angular frequency ω_a = m_ac²∕ħ, where c is the speed of light in vacuum, ħ is the reduced Planck constant, and m_a is the unknown ALP mass. Therefore the effective magnetic field B^∗ also oscillates at this angular frequency.

We consider here magnetic resonance experiments that search for spin energy shifts or for spin precession induced by these oscillating interactions. The energy shift is quantified by the expectation value of the relevant Hamiltonian, ${\langle \mathcal {H}\rangle }$, where $\mathcal {H}$ could be either $\mathcal {H}{{ }_{\mbox{ EDM}}}$ or $\mathcal {H}{{ }_{\mbox{ gr}}}$. To quantify the rate of spin precession, it is convenient to define the Rabi frequency Ω_a. Assuming that the effective magnetic field B^∗ is linearly polarized, in the rotating wave approximation this Rabi frequency is given by

$$\displaystyle \begin{aligned} \Omega_a = \frac{{\langle \mathcal{H}\rangle}}{2\hbar}~. {} \end{aligned} $$

(6.3)

6.1.1.1 The EDM Interaction with P,T-odd Moments of Nucleons and Nuclei

As discussed in Chap. 2, the axion concept was invented as a solution of the strong CP problem of quantum chromodynamics (QCD). It is the QCD axion interaction with the gluon field that achieves this goal, by relating the θ parameter of the QCD Lagrangian to the axion field: θ = a∕f_a. In the presence of a dynamical background axionlike field a(t), the oscillating θ(t) can lead to experimentally observable effects. For an isolated nucleon, θ induces an electric dipole moment (EDM):

$$\displaystyle \begin{aligned} d_n = g_d a = 2.4\times 10^{-16}\,\theta {\, \mathrm{e}\cdot \mathrm{cm}} = 2.4\times 10^{-3}\,\theta {\, \mathrm{e}\cdot \mathrm{fm}}, {} \end{aligned} $$

(6.4)

calculated with 40% accuracy [1, 2]. We note the unit conversion: 1 e ⋅cm = 1.5 × 10¹³ GeV⁻¹. The neutron EDM experiment discussed in Ref. [3] and the proton storage ring experiment described in Ref. [4] search for these oscillating nucleon EDMs.

When the nucleon is bound inside an atomic nucleus, this EDM is screened. According to the Schiff theorem, the EDM of a point-like nucleus is completely screened by atomic electrons in the low-frequency limit [5]. Taking into account finite nuclear size, the nuclear Schiff moment is given by

$$\displaystyle \begin{aligned} \boldsymbol{S} = \frac{e}{10}\left( \langle r^2\boldsymbol{r}\rangle - \frac{5}{3Z}\langle r^2\rangle\langle\boldsymbol{r}\rangle \right), {} \end{aligned} $$

(6.5)

where e is the elementary electric charge, Z is the atomic number, and $\langle r^k \rangle = \int r^k\rho (\boldsymbol {r})d^3r$ are the moments of nuclear charge density ρ(r). The Schiff moment sources the (P)arity- and (T)ime-odd electrostatic potential

$$\displaystyle \begin{aligned} \varphi(\boldsymbol{r}) = 4\pi(\boldsymbol{S}\cdot\nabla)\delta(\boldsymbol{r}). {} \end{aligned} $$

(6.6)

Let us consider two contributions to the nuclear Schiff moment: (1) the permanent nucleon EDM d_n, and (2) P,T-odd nuclear forces.

• 1. The contribution due to d_n arises because of non-coincident densities of nuclear charge and dipole moment. It can be estimated using Eq. (8.76) from Ref. [6]:

  $$\displaystyle \begin{aligned} 4\pi S_{\mathrm{EDM}} \approx d_n\times \frac{4\pi}{25}\frac{(K+1)I}{I(I+1)}r_0^2, {} \end{aligned} $$

  (6.7)

  where K = (ℓ − 1)(2I + 1) = 1 and r₀ = 1.25A^1∕3, ℓ being the orbital angular momentum of the valence nucleon. Note that the definition of the Schiff moment in Ref. [6] differs from ours by a factor of 4π, which appears on the left-hand side.

• 2. The P,T-odd nuclear interaction of a non-relativistic nucleon with nuclear core is parametrized by strength η [7]:

  $$\displaystyle \begin{aligned} W = \frac{G_F}{\sqrt{2}}\frac{\eta}{2m_n}\boldsymbol{\sigma}\cdot\nabla \rho(\boldsymbol{r}), {} \end{aligned} $$

  (6.8)

  where G_F ≈ 10⁻⁵ GeV⁻² is the Fermi constant, m_n is the nucleon mass, σ is its spin, and ρ(r) is the density of core nucleons. A vacuum θ angle gives rise to this interaction via the P,T-odd pion-nucleon coupling constant [6, 8]:

  $$\displaystyle \begin{aligned} \eta = 1.8\times 10^6\,\theta. {} \end{aligned} $$

  (6.9)

  Nuclear physics calculations express the nuclear Schiff moment of a particular nucleus in terms of parameter η, see, for example, Ref. [9].

Once the nuclear Schiff moment is expressed in terms of θ, atomic calculations are used to connect physical observables to the value of the Schiff moment. For example, the observable can be the value of the energy shift of a nuclear spin state in an applied electric field. The connection can then be made to the effective electric field E^∗ defined in Eq. (6.1). This calculation is performed for ²⁰⁷Pb nuclear spins in ferroelectric crystals in Refs. [10, 11]. For ferroelectrically poled PMN-PT (lead magnesium niobate-lead titanate) it is found that the effective electric field is E^∗ = 340 kV∕cm with estimated uncertainty ≈50% [12]. The effect on nuclear spins is equivalent to that of an effective magnetic field

$$\displaystyle \begin{aligned} \boldsymbol{B}^*_{\text{EDM}} =-\frac{g_d a\boldsymbol{E}^*}{\hbar\gamma I}, {} \end{aligned} $$

(6.10)

where γ is the spin gyromagnetic ratio.

6.1.1.2 The Gradient Interaction

In order to calculate the gradient of the axionlike field a that appears in Eq. (6.2), it is necessary to consider the integral over the velocity distribution of the axionlike galactic dark matter field. Importantly, there are contributions from both the lab velocity with respect to the galactic rest reference frame, and from the spread of the dark matter virial velocity distribution. The effect on nuclear spins is equivalent to that of an effective magnetic field

$$\displaystyle \begin{aligned} \boldsymbol{B}^*_{\text{aNN}} =-(cg_{\text{aNN}}/\gamma)\boldsymbol{\nabla}a, {} \end{aligned} $$

(6.11)

where γ is the spin gyromagnetic ratio.

6.1.2 Interactions with Electron Spins

Electron spins can also couple to the ALP field via the derivative fermion coupling. This gradient interaction generates the electron spin Hamiltonian with the same form as Eq. (6.11). For an electron spin the coupling constant in the Lagrangian is often written as g_aee∕(2m_e), where m_e is the electron mass, and g_aee is unitless. However, the physics is exactly the same—an electron spin experiences a torque due to the gradient ∇a, which acts as an effective magnetic field, whose magnitude is proportional to g_aee. There are stringent astrophysical limits on the coupling constant g_aee, and the QUest for AXions experiment (QUAX) is a laboratory search for this interaction at frequencies near 10 GHz [13].

6.2 Basics of NMR

Nuclear magnetic resonance (NMR) encompasses a broad and versatile set of techniques that have found application in a wide range of disciplines. A typical NMR experiment involves measurement of nuclear spin dynamics in an applied bias magnetic field (Fig. 6.1).

Fig. 6.1

A schematic of a typical NMR experiment. M is the nuclear spin magnetization of the sample. B₀ is the bias magnetic field, and $B_1\sin {\omega _1t}$ is the “pseudo-magnetic” field due to interaction with ultralight dark matter. The spin-1/2 level diagram indicates spin polarization as larger population in the ground spin sublevel, and spin coherence induced by the excitation field B₁, if it is resonant with the spin Larmor frequency

In pulsed magnetic resonance experiments, the spins are perturbed by resonant radiofrequency (RF) magnetic field pulses, and the subsequent spin evolution is detected. Since the introduction of digital fast Fourier transform algorithms, most modern applications of NMR utilize the pulsed scheme [14, 15]. Searches for permanent electric dipole moments are an example of pulsed NMR experiments in the fundamental-physics context [16]. In continuous wave (CW) magnetic resonance experiments, the excitation field is present continuously and the bias field is varied. Spin-based dark matter haloscope experiments usually employ the CW scheme [17]. Here we provide a basic introduction to NMR, in order to help readers make sense of nuclear-spin-based dark matter searches. For a more thorough treatment, the reader is referred to Refs. [18,19,20].

6.2.1 Nuclear Magnetism

In virtually all cases,¹ detection of an NMR signal involves measurement of the magnetic field produced by nuclear spins in a sample. The magnetic moment of a nucleus with non-zero spin is given by

$$\displaystyle \begin{aligned} \boldsymbol{\mu}=\frac{g_I\mu_N \boldsymbol{I}}{\hbar}, {} \end{aligned} $$

(6.12)

where g_I is the g-factor of the nuclear spin I. In NMR it is typically convenient to rewrite the nuclear magnetic moment in terms of a nucleus’ gyromagnetic ratio,

$$\displaystyle \begin{aligned} \gamma=\frac{g_I \mu_N}{\hbar}, \end{aligned} $$

(6.13)

where g_I is the nuclear g-factor and μ_N is the nuclear magneton, such that the magnetic moment may be written in condensed form as

$$\displaystyle \begin{aligned} \boldsymbol{\mu}=\gamma \boldsymbol{I}. \end{aligned} $$

(6.14)

The total magnetization of an ensemble of spins can then be written as

$$\displaystyle \begin{aligned} \boldsymbol{M}=N \hbar \gamma P_0 \boldsymbol{I}, \end{aligned} $$

(6.15)

where N is the number density of nuclear spins, and P₀ is the ratio of the spin state population difference to the total population, generally referred to as the spin polarization. Explicitly, for spin-1∕2 nuclei,

(6.16)

where represents spins with m_I = +1∕2, and represents spins with m_I = −1∕2. For nuclear spin polarization at thermal equilibrium, the Boltzmann distribution gives

(6.17)

(6.18)

where E_± is the energy of the state with m_I = ±1∕2, k_B is the Boltzmann constant, and T is the temperature of the system. In a large magnetic field, the dominant energy term is the Zeeman interaction,

$$\displaystyle \begin{aligned} \mathcal{H}_Z=-\hbar \gamma \boldsymbol{B}\cdot \boldsymbol{I}, {} \end{aligned} $$

(6.19)

where B is the applied magnetic field. For a magnetic field B₀ in the $\hat {\boldsymbol {z}}$ direction, the energy is

$$\displaystyle \begin{aligned} E=-\hbar \gamma B_0 m_I, \end{aligned} $$

(6.20)

so Eq. (6.16) may be written as

$$\displaystyle \begin{aligned} P_0=\frac{e^{\hbar \gamma B_0/(2 k_B T)}-e^{-\hbar \gamma B_0/(2 k_B T)}}{e^{\hbar \gamma B_0/(2 k_B T)}+e^{-\hbar \gamma B_0/(2 k_B T)}} = \tanh\left( \frac{\hbar \gamma B_0}{2 k_B T} \right)~. {} \end{aligned} $$

(6.21)

Under practically achievable conditions, the so-called high-temperature approximation is valid, so we keep only the leading term of the Taylor expansion of the hyperbolic tangent,

$$\displaystyle \begin{aligned} P_0 \approx \frac{\hbar \gamma B_0}{2 k_B T}~, \end{aligned} $$

(6.22)

such that

$$\displaystyle \begin{aligned} \boldsymbol{M} \approx \frac{N \hbar^2 \gamma^2 B_0}{2 k_B T} \boldsymbol{I}~. {} \end{aligned} $$

(6.23)

From this, we can see that the magnitude of the observable NMR signal depends on the spin density, gyromagnetic ratio, and (assuming thermal spin polarization) the magnetic field strength.

Because the energy difference between nuclear spin states is generally small compared to thermal energy, ħγB₀ ≪ k_BT, spin polarization at thermal equilibrium is many orders of magnitude smaller than unity. Considering that the NMR signal is proportional to polarization, sensitivity (either in dark matter searches or in chemical analysis) can be greatly enhanced through the use of so-called hyperpolarization techniques, which generate non-equilibrium spin states with polarization approaching 100%. Examples of such techniques include dynamic nuclear polarization (DNP) [26, 27], spin-exchange optical pumping (SEOP) [24], metastability exchange optical pumping (MEOP) [28], and parahydrogen-induced polarization (PHIP) [29,30,31].

6.2.2 Nuclear Spin Dynamics

A phenomenological description of the evolution an ensemble of nuclear spins in a magnetic field is given by the Bloch equation:

$$\displaystyle \begin{aligned} \frac{d\boldsymbol{M}}{dt}= \gamma \boldsymbol{M} \times \boldsymbol{B} - \frac{M_x \hat{\boldsymbol{x}} + M_y \hat{\boldsymbol{y}}}{T_2} - \frac{M_z - M_0}{T_1}\hat{\boldsymbol{z}}~, {} \end{aligned} $$

(6.24)

where γ is the nuclear gyromagnetic ratio, M is the nuclear magnetization vector, M₀ is the magnitude of the equilibrium magnetization, as derived in Eq. (6.23), and B is the magnetic field, oriented along the z-axis. We have also introduced two characteristic relaxation times: the longitudinal relaxation time T₁ and the transverse relaxation time T₂. The magnetization dynamics are a combination of relaxation and precession about the magnetic field at the Larmor frequency

$$\displaystyle \begin{aligned} \omega_L=\gamma B~. \end{aligned} $$

(6.25)

The longitudinal relaxation time may be interpreted as the characteristic time it takes for the spin system to reach equilibrium. For example, if an unpolarized sample is placed into a magnetic field, the magnetization will build up as

$$\displaystyle \begin{aligned} M(t)=M_0 \left(1-e^{-t/T_1}\right). \end{aligned} $$

(6.26)

For hyperpolarized spin systems, the magnetization generally decays to its equilibrium value with characteristic time T₁ as well.

The transverse relaxation time may be thought of as the nuclear spin coherence lifetime, corresponding to the exponential decay of precessing magnetization in the xy plane. As an example, we consider the case where we have prepared a spin state where our sample is magnetized along $\hat {\boldsymbol {x}}$ (by applying, for example, a − π∕2 pulse along $\hat {\boldsymbol {y}}$ to rotate spins initially oriented along $\hat {\boldsymbol {z}}$ to along $\hat {\boldsymbol {x}}$). Then the magnetization along $\hat {\boldsymbol {x}}$ will be

$$\displaystyle \begin{aligned} M_x(t)=M_0 \cos{\left(\gamma B t\right)} e^{-t/T_2}. \end{aligned} $$

(6.27)

The Fourier transform of this signal yields a Lorentzian peak at the Larmor frequency with full-width at half maximum (FWHM)

$$\displaystyle \begin{aligned} w_{1/2}=\frac{1}{\pi T_2}. \end{aligned} $$

(6.28)

More specifically, T₂ refers to the “intrinsic” dephasing time that would occur in a perfectly homogeneous magnetic field. The transverse relaxation time in a real magnetic field (i.e., possessing some inhomogeneity) is $T_2^*$.

To understand how nuclear spins respond to oscillating magnetic (or axion) fields, it is useful to transform into a rotating reference frame. Consider a driving field along the x-axis: $B_x(t) = 2B_1\cos {\omega t}$, which can be decomposed into two counter-rotating components. In the reference frame rotating around the z-axis at the frequency ω the magnetization components are $\tilde {M}_x$, $\tilde {M}_y$, and the M_z component is unchanged. The connection between the laboratory-frame and the rotating-frame magnetization components is:

$$\displaystyle \begin{aligned} \begin{array}{rcl} M_x & =&\displaystyle \tilde{M}_x\cos{\omega t} - \tilde{M}_y\sin{\omega t}{} \end{array} \end{aligned} $$

(6.29)

$$\displaystyle \begin{aligned} \begin{array}{rcl} M_y & =&\displaystyle \tilde{M}_x\sin{\omega t} + \tilde{M}_y\cos{\omega t}. \end{array} \end{aligned} $$

(6.30)

The steady-state solution of the Bloch equations in the rotating frame is:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \tilde{M}_x & =&\displaystyle \frac{\Delta\omega \gamma B_1 T_2^2}{1+ (T_2 \Delta \omega)^2+\gamma^2 B_1^2 T_1 T_2} M_0{} \end{array} \end{aligned} $$

(6.31)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \tilde{M}_y & =&\displaystyle \frac{\gamma B_1 T_2}{1+ (T_2 \Delta \omega)^2+\gamma^2 B_1^2 T_1 T_2} M_0 \end{array} \end{aligned} $$

(6.32)

$$\displaystyle \begin{aligned} \begin{array}{rcl} M_z & =&\displaystyle \frac{1+ (\Delta\omega T_2)^2}{1+ (T_2 \Delta \omega)^2+\gamma^2 B_1^2 T_1 T_2} M_0, \end{array} \end{aligned} $$

(6.33)

where Δω = ω − ω_L is the drive detuning.

For a resonant (Δω = 0) driving field that is far from saturation ($\gamma B_1 \ll 1/\sqrt {T_1 T_2}$), these equations become substantially simpler. We see that $\tilde {M}_x=0$, $\tilde {M}_z \approx M_0$, and

$$\displaystyle \begin{aligned} M_y \approx \gamma B_1 T_2 M_0\cos{\omega t}. {} \end{aligned} $$

(6.34)

Together with Eqs. (6.10) and (6.11), this result allows us to convert the strength of the axionlike dark matter EDM or gradient interaction to an experimental observable: the transverse nuclear magnetization.

6.2.3 Nuclear Spin Interactions

We will now consider some basic features of NMR spectra that arise due to various spin interactions. An understanding of NMR spectra is needed for design and calibration of NMR-based dark matter searches.

6.2.3.1 Chemical Shielding

Because most experiments are not conducted with bare nuclei, it is necessary to generalize the Zeeman interaction of Eq. (6.19) to include local susceptibility effects:

$$\displaystyle \begin{aligned} \mathcal{H}_{CS} = \hbar \gamma \boldsymbol{I} \cdot (1 - \boldsymbol{\sigma}{{}_{\mbox{ cs}}}) \cdot \boldsymbol{B}_0, {} \end{aligned} $$

(6.35)

where the chemical shielding tensor σ_cs describes the effect of electrons producing counteracting magnetic fields that “shield” the nucleus from the external field. This chemical shielding interaction is particularly useful for chemists: different chemical environments within a given molecule, labeled by subscript j, give rise to different shielding σ_cs,j, and therefore different peak shifts, which can be interpreted in terms of electron density. For practical reasons, it is often convenient to refer to the chemical shift relative to some reference, defined as δ = σ_ref −σ_cs. The shift is typically measured in units of “parts per million” (ppm). For example, δ = 10⁻⁶ corresponds to a 1 ppm shift to higher frequency.² From the perspective of dark matter searches, these shifts can be problematic, as maximum sensitivity is achieved if all spins in a sample have the same Larmor frequency. For example, consider the case of ethanol, CH₃CH₂OH, which contains 102.8 moles of ¹H per liter: 1∕2 of the hydrogens are in the CH₃ environment, 1∕3 are in the CH₂ environment, and 1∕6 of all hydrogens are in the OH environment. For comparison, methanol, CH₃OH, contains 98.9 moles of ¹H per liter, and 3∕4 of them are in the CH₃ environment. So on a per-peak basis, methanol gives roughly 50% more signal for the peaks associated with the CH₃ environment, as illustrated in Fig. 6.2b.

Fig. 6.2

¹H NMR spectra of ethanol (blue) and methanol (red), acquired at 1.4 T (60 MHz ¹H Larmor frequency), with the chemical-shift scale referenced to tetramethylsilane (TMS). For simplicity, the hydroxyl (-OH) signals at 4.8 ppm are not shown in parts (b) and (c). (a) In the low-resolution case—corresponding here to a line width of 10 ppm—ethanol yields slightly more signal than methanol due the larger concentration of ¹H. (b) For line widths on the order of 1 ppm (the line width shown is 15 Hz, or 0.25 ppm), chemical shifts can be resolved. Because the (non-hydroxyl) hydrogens in ethanol are in two different chemical environments, the signal is separated into two peaks—the signal at 1.2 ppm corresponds to the CH₃ hydrogens and the signal at 3.7 ppm corresponds to the CH₂ hydrogens—each of which is smaller than the methanol signal. (c) At higher resolution (the line width shown here is approximately 1.8 Hz, or 30 ppb), further structure due to J-couplings can be seen in ethanol: the CH₃ signal is split into a triplet by the CH₂ hydrogens, and the CH₂ signal is split into a quartet by the CH₃ hydrogens. Note that the hydroxyl protons do not induce splittings due to rapid chemical exchange

The tensor nature of the chemical shielding is also of note, as its principal axis system is defined in the molecular frame. In liquid samples, rapid molecular tumbling averages out all components except for the isotropic part of the tensor, σ_iso. In the case of solid samples, however, the anisotropy of the chemical shielding tensor persists, and different molecular orientations yield different NMR frequencies. In single-crystal samples, NMR spectra consist of a countable number of peaks, which shift depending on crystal orientation relative to the magnetic field. Polycrystalline or powder samples, however, are composed of a large number of randomly oriented crystallites, which leads to a distribution of NMR resonances. Such distributions are often hundreds of ppm wide, which reduces sensitivity. It is worth pointing out that the so-called powder broadening is an example of inhomogeneous broadening—the signal is broad not because of T₂ relaxation, but because of the distribution of orientations.

6.2.3.2 Direct Dipole-Dipole Coupling

The Hamiltonian describing the direct dipole-dipole coupling of two nuclear spins I₁ and I₂ is given by

$$\displaystyle \begin{aligned} \mathcal{H}_{12} = \frac{\mu_0}{4\pi} \frac{\hbar^2\gamma_1 \gamma_2 }{r_{12}^3} \left[ \boldsymbol{I}_1 \cdot \boldsymbol{I}_2 - 3 \left( \boldsymbol{I}_1 \cdot \hat{\boldsymbol{r}}_{12} \right) \left( \boldsymbol{I}_2 \cdot \hat{\boldsymbol{r}}_{12} \right) \right], {} \end{aligned} $$

(6.36)

where γ₁, γ₂ are the gyromagnetic ratios of the two spins and r₁₂ is the vector between them. In magnetic fields much larger than the characteristic dipolar field, the part of this Hamiltonian that commutes with the Zeeman interaction is:

$$\displaystyle \begin{aligned} \mathcal{H}_{12} \approx \frac{\mu_0}{4\pi} \frac{\hbar^2\gamma_1 \gamma_2 }{r_{12}^3} \left(1-3\cos^2{\alpha}\right)\left[ \hat{I}_{1z} \hat{I}_{2z} - \frac{1}{4}\left(\hat{I}_{1+} \hat{I}_{2-} + \hat{I}_{1-} \hat{I}_{2+} \right) \right], {} \end{aligned} $$

(6.37)

where α is the angle between r₁₂ and the magnetic field vector, $\hat {I}_{1\pm }$ and $\hat {I}_{2\pm }$ are the raising and lowering operators for the respective nuclear spins 1 and 2, where

$$\displaystyle \begin{aligned} \hat{I}_+ & = \hat{I}_x + i \hat{I}_y~, \end{aligned} $$

(6.38)

$$\displaystyle \begin{aligned} \hat{I}_- & = \hat{I}_x - i \hat{I}_y~. \end{aligned} $$

(6.39)

In NMR literature the dipolar Hamiltonian acting on the spin ensemble is often written as

$$\displaystyle \begin{aligned} \mathcal{H}_D = \hbar \sum_{N;N^\prime>N} \boldsymbol{I}_N \cdot \boldsymbol{D}_{NN^\prime} \cdot \boldsymbol{I}_{N^\prime}, {} \end{aligned} $$

(6.40)

where $\boldsymbol {D}_{NN^\prime }$ is a rank-2 symmetric tensor, and the sum is over all spin pairs in the ensemble. Because the dipole-dipole coupling tensor is traceless, it is fully averaged out by molecular tumbling in isotropic liquids. In “dilute” powdered solids composed of isolated spin pairs, it is possible to resolve a double-peaked feature called “Pake doublet” that arises due to the distribution of angles α for different spin pairs. Solids with high nuclear spin density, which are of greater interest for dark matter searches, cannot be considered in terms of isolated spin pairs, so one must also consider the distribution of distances r_IS. This generally gives rise to broad “blobby” NMR resonances (a more detailed analysis of solid-state NMR lineshapes is given in Ref. [20]).

In some experiments dipolar broadening can be greatly reduced using a technique called magic angle spinning (MAS) [32]. By rotating a sample rapidly³ about an axis, tensor components transverse to the spinning axis are averaged to zero. By setting the angle between the spinning axis and the magnetic field such that $1-3 \cos ^2\alpha = 0$, the effective time-averaged dipole-dipole coupling is zero. This technique can also be used to average out powder broadening due to chemical-shift anisotropy. While the added complexity of MAS may seem daunting for dark matter experiments, the improved signal intensity may prove useful.

6.2.3.3 Indirect Spin-Spin Coupling

The indirect dipole-dipole coupling, known as J-coupling, may be thought of as a second-order hyperfine effect where one nucleus N affects the electronic state of a molecule through hyperfine couplings to the molecular electron density, and this perturbation is then transmitted from the molecular electronic state to a second nucleus N^′ through its hyperfine interaction with the molecular electron density. The Hamiltonian may be written in the form

$$\displaystyle \begin{aligned} \mathcal{H}_J = \hbar \sum_{N;N^\prime \neq N} \boldsymbol{I}_N \cdot \boldsymbol{J}_{NN^\prime} \cdot \boldsymbol{I}_{N^\prime}~, {} \end{aligned} $$

(6.41)

where I_N and $\boldsymbol {I}_{N^\prime }$ are the spins of nuclei N and N^′, and $\boldsymbol {J}_{NN^\prime }$ is the second-rank J-coupling tensor. $\boldsymbol {J}_{NN^\prime }$ may in general be represented as a sum of irreducible spherical tensors

$$\displaystyle \begin{aligned} \boldsymbol{J}_{NN^\prime}=\boldsymbol{J}_{NN^\prime}^{(0)}+\boldsymbol{J}_{NN^\prime}^{(1)}+\boldsymbol{J}_{NN^\prime}^{(2)}~, \end{aligned} $$

(6.42)

where the isotropic component $\boldsymbol {J}_{NN^\prime }^{(0)}$ transforms as a scalar, the antisymmetric component $\boldsymbol {J}_{NN^\prime }^{(1)}$ transforms as a pseudovector, and the symmetric component $\boldsymbol {J}_{NN^\prime }^{(2)}$ transforms as a symmetric rank-2 spherical tensor. The J-coupling is often referred to as the “scalar” coupling because typical high-resolution NMR experiments utilize isotropic liquid samples where rapid molecular tumbling averages higher-order tensor components to zero. In solid-state experiments, $\boldsymbol {J}_{NN^\prime }^{(2)}$ adds to the dipole-dipole coupling and transforms the same way. The rank-1 component does not commute with the high-field Zeeman Hamiltonian and has never been conclusively measured.

In terms of dark matter searches, the J-coupling is most important for experiments with liquid samples [33, 34], where it causes peak splitting. Consider again the cases of ethanol (CH₃CH₂OH) and methanol (CH₃OH). Assuming the presence of even a small amount of water, the OH hydrogens undergo rapid exchange and thus their couplings to the other spins are averaged to zero. The three CH₃ hydrogens in methanol are magnetically equivalent, so there are no observable J-couplings and the methanol spectrum consists of a single resonance. In ethanol, there are two sets of equivalent spins, the three CH₃ hydrogens and the two CH₂ hydrogens. The two CH₂ hydrogens split the CH₃ resonance into a triplet with relative amplitudes 1:2:1, and the three CH₃ hydrogens split the CH₂ resonance into a quartet with relative amplitudes 1:3:3:1. As a result, the largest peak in ethanol is about three times smaller than the methanol peak, as shown in Fig. 6.2c.

6.2.3.4 Quadrupolar Coupling

The quadrupolar coupling Hamiltonian has the form

$$\displaystyle \begin{aligned} \mathcal{H}_Q (\Theta)= \frac{eQ}{2I(2I-1)}\boldsymbol{I}\cdot\boldsymbol{V}(\Theta)\cdot\boldsymbol{I}, {} \end{aligned} $$

(6.43)

where e is the electric charge, Q is the nuclear quadrupole moment, I is the nuclear spin quantum number, and V ( Θ) is the electric field gradient tensor for an arbitrary molecular orientation Θ [19]. As a rank-2 interaction, the quadrupolar coupling is non-zero only for nuclei with spin I ≥ 1. This is one of the notable advantages of studying spin-1∕2 nuclei, because the coupling of quadrupolar nuclei to the electric field gradient is a major source of relaxation, leading to short coherence times and thus to broad resonance lines.

6.2.4 Zero-to-Ultralow-Field NMR

While the vast majority of NMR experiments are performed in a large applied magnetic field, an alternative method, zero-to-ultralow-field (ZULF) NMR [35], also exists and has found use in dark matter searches [33, 34]. In ZULF NMR, the magnetic field is small enough that the Zeeman interaction may be treated as a perturbation on other spin couplings. As of the time of this writing, all ZULF NMR dark matter searches have relied on the electron-mediated J-coupling as the primary interaction, perturbed by a small bias magnetic field [33, 34].

6.3 Detecting Spin Evolution due to Axionlike Dark Matter

The most sensitive scheme for NMR detection depends on the frequency range being explored. In general, the goal is to search for oscillating magnetic fields originating from the evolution of spins due to interaction with ALP dark matter. Spins act as transducers for the cosmic signal, converting the oscillating ALP field to an oscillating magnetic field at the same frequency. The optimal detection modality is the most sensitive magnetometer for the frequency corresponding to the ALP mass range under investigation. Additionally, the working environment of the device has to be considered. For example, the most sensitive vapor cell magnetometers require that the ambient background magnetic field is below 100 nT in order to operate. This requirement might be at odds with the conditions needed for the spin sample in an ALP search: for instance, in the case of ALP Compton frequencies ω_a in the ∼100 MHz range, applied magnetic fields of several tesla are required in order for the NMR resonant frequency to match ω_a. Superconducting quantum interference devices (SQUIDs) require cryogenic temperatures which demands either a stand-off distance for thermal insulation or a cold nuclear sample. The latter might be problematic due the inherently broader magnetic resonance linewidth of solids.

This section gives an overview of the most sensitive detection modalities used in NMR spectroscopy at various frequencies. The frequency range to be considered is as open as the mass range of the potential dark matter candidates. However, NMR measurements at the highest possible frequencies require the largest possible magnetic field and the field range of commercially available magnets is limited. Commercially available high-field magnet systems feature proton frequencies up to 1.2 GHz corresponding to a field of 28.2 T in a 54 mm diameter bore. The maximum demonstrated dc magnetic field in a research facility at the time of writing was 45.5 T [36] corresponding to a proton Larmor frequency of 1.93 GHz and therefore an ALP mass of 8 μeV. An overview of even higher magnetic fields that can be produced for short times can be found in Ref. [37]. Here we begin with general considerations of the axion-induced NMR signals and then review the working principles of a selection of sensitive devices used to search for NMR signals, namely optical atomic magnetometers in the spin-exchange-relaxation free (SERF) regime for the lowest bandwidth and highest magnetic field sensitivity, RF vapor cell magnetometers in an intermediate regime, followed by SQUIDs, and finally inductive pick-up coils. The section closes with a brief discussion of magnetic noise suppression techniques. Table 6.1 summarizes the different detection modalities mentioned in this chapter and their basic properties.

Table 6.1 Example of different NMR detection modalities with their frequency range and sensitivities to oscillating magnetic fields

6.3.1 Axion-Induced NMR Signals

Figure 6.3 illustrates how to detect oscillating magnetic signals of unknown origin and frequency using nuclear magnetic resonance. As discussed in Sect. 6.1.1, the interaction of nuclear spins with UBDM appears as an effective magnetic interaction Hamiltonian [Eqs. (6.1) and (6.10) for the EDM coupling and Eqs. (6.2) and (6.11) for the gradient coupling] modulated at the ALP field’s Compton frequency. This interaction can be detected as an oscillating torque on the magnetization in a frame co-rotating at the Larmor frequency of the nuclear sample, Fig. 6.3a. Depending on the experimental setup an ALP field can also be detected as a periodic modulation of the magnetic resonance frequency itself, see Fig. 6.3b. This can be searched for by observing the magnetic resonance frequency, for example, by probing it with another oscillating magnetic field at frequency ω_RF and observing the out-of-phase quadrature (Y) component with lock-in detection demodulated at ω_RF. The direction of the ALP pseudo-magnetic field depends on the coupling that is being investigated. In case of the gradient coupling, the oscillating pseudo-magnetic field is a property of the axion field itself. EDM coupling searches require an additional applied electric field which determines the direction of the pseudo-magnetic field.

Fig. 6.3

Two ways to detect ALPs with nuclear magnetic resonance using a spherical sample of polarized nuclei and a sensor. (a–c) Continuous wave nuclear magnetic resonance (CW-NMR) with the cosmic ALP field driving spin population. The energy levels are scanned with the applied magnetic field and a resonance occurs if the Larmor frequency of the nuclear sample corresponds to the Compton frequency of the ALP. This detection mechanism is relevant for dipole moment and gradient coupling. In gradient coupling searches this detection is sensitive to two directions of the signal (x and y) for leading B₀ along z. In oscillating electric dipole moment searches an additional electric field needs to be applied along the magnetic background field. (d–f) Modulation of the magnetic field. Note that for gradient coupling searches the properties of the ALP pseudo-magnetic field are given by the field itself, while for EDM coupling searches an additional electric field needs to be applied. And here different directions with respect to the leading magnetic field can be chosen, so that it is possible to choose between situation (a) and (d)

Generally speaking, in order to obtain the biggest possible NMR signal from the ALP interaction one aims to have the largest number of nuclear spins subjected to the same magnetic field, such that they precess at the same Larmor frequency ω_L. The signal for the measurement configuration displayed in Fig. 6.3a is proportional to the magnitude of the oscillating component of the transverse magnetization M_x. The overall magnetization magnitude M, Eq. (6.23), results from N particles with magnetic moment μ, Eq. (6.12). The fundamental limitations on NMR sensitivity were already pointed out by Felix Bloch in his 1946 paper [42]: assuming uncorrelated, randomly oriented spin-1/2 particles in a volume V : there will be a statistically incomplete cancellation of the magnetization in any direction of order

$$\displaystyle \begin{aligned} M_{\text{SPN}} \approx \mu\frac{\sqrt{N}}{V}~. \end{aligned} $$

(6.44)

So even without any ALP signal, there will be a fluctuating signal at the detector proportional to this magnetization: this is the spin-projection noise (SPN). The maximum signal-to-noise therefore scales as

$$\displaystyle \begin{aligned} \frac{M}{M_{\text{SPN}}} \approx \sqrt{N}. \end{aligned} $$

(6.45)

This means the larger the volume and the density of nuclear spins in a homogeneous magnetic field the more sensitive is the ALP search. The average power spectrum of the spin-noise signal will feature a Lorentzian lineshape with a width given by the transverse relaxation time $T_2^*$ of the spin ensemble and a center frequency given by the Larmor frequency,

$$\displaystyle \begin{aligned} \omega_L=\gamma B_0~, \end{aligned} $$

(6.46)

which depends on the applied background field B₀ and the gyromagnetic ratio γ of the nuclear species. Protons feature the highest known nuclear gyromagnetic ratio (of all stable nuclei) with γ_p∕(2π) = 42.6 MHz/T. In a magnetic resonance search, the accessible Larmor frequencies ω_L determine the sensitive mass range of the ALP-search experiment. Searches following Fig. 6.3b schematic, i.e., measurements of an oscillating center frequency of a magnetic resonance, result in a slightly different limit that depends on the linewidth of the magnetic resonance as well.

6.3.2 Inductive Coil Detection

The most basic approach to detect an NMR signal is to use a pick-up coil coupled to the spin ensemble. The transverse magnetization precesses around the leading field at the Larmor frequency, creating an oscillating magnetic flux, which induces a Faraday voltage across the coil. A resonant circuit is often used to couple this voltage to a sensitive amplifier. There are many configurations analyzed in the NMR literature, see, for example, Ref. [43]. One example of such circuit is shown in Fig. 6.4. In this series capacitor-matched circuit the values of capacitors C₁ and C₂ are chosen so that the circuit resonance ω₀ is near the spin Larmor frequency, and probe impedance is matched to the transmission line and the amplifier input impedance. The resistance R includes dissipation due to the circuit elements, as well as the spin ensemble itself. The circuit dissipation can be quantified by the quality factor Q = ω₀L∕R.

Fig. 6.4

A schematic of an NMR series capacitor-matched detection circuit. The blue dashed box denotes a transmission line (such as a coaxial cable) that couples the resonant detection probe on the left to the amplifier on the right

Inductive pick-up coils have been engineered since the invention of nuclear magnetic resonance and are highly sophisticated devices, commercially available in a broad range of frequencies (up to several GHz), Fig. 6.5. The voltage induced in the pick-up coil by the precessing spin magnetization is shown as a voltage source V_s in Fig. 6.4. This voltage is proportional to the Larmor frequency, due to Faraday’s law. On resonance, and in the limit of small circuit dissipation (Q ≫ 1), we can write a simple expression for the resulting voltage that appears at the input of the amplifier:

$$\displaystyle \begin{aligned} V^{\prime}_s = \frac{V_s}{2}\sqrt{\frac{QR_a}{\omega_0L}}, {} \end{aligned} $$

(6.47)

where R_a is the amplifier input impedance, usually matched to the 50  Ω transmission line impedance.

Fig. 6.5

Different coil geometries for inductive detection. Image and caption taken with permission from Ref. [44]. Common NMR coil geometries: (a) solenoid, (b) saddle coil, (c) inductively coupled high-temperature superconducting coils (drawings by Jason Kitchen of the National High Magnetic Field Laboratory)

Consider the signal-to-noise ratio that can be achieved with this detection method. One of the noise sources is the thermal Nyquist noise due to the dissipation in the circuit, which appears as a noise voltage source in series with the signal source V_s, and with power spectral density $\tilde {V}^2_{th}(\omega ) = 2Rk_BT/\pi$, where k_B is the Boltzmann constant and T is the circuit temperature. Another noise source is the amplifier noise, which usually referred to the amplifier input. The relative importance of these, and other noise sources, depends on the details of the NMR sample and of the probe circuit. Many of the modern NMR machines are limited by the thermal Nyquist noise, and therefore probes are sometimes cooled to reduce this noise.

6.3.3 Superconducting Quantum Interference Devices

Superconducting quantum interference devices (SQUIDs) can be used as current sensors to detect magnetic resonance signals. Since SQUIDs are usually low-frequency devices, optimization of the coupling circuit is, in general, different from the high-frequency inductive detection shown in Fig. 6.5. For example, in the case of non-resonant coupling the pick-up coil inductance should be roughly matched to the SQUID input coil self-inductance [45].

The basic building block of a dc SQUID is a loop of superconducting material interrupted via one or two Josephson junctions, i.e., non-conducting barriers in the superconducting loop. This loop is inductively coupled to a sensing coil, which is often superconducting itself, as seen in Fig. 6.6. The SQUID is a complicated nonlinear superconducting device, whose operation is treated in Refs. [47, 48], which also present a comprehensive introduction to SQUID technology and applications. However, the SQUID characteristics can be linearized by operating it in feedback mode, where a feedback loop supplies a signal that cancels the flux from the sensing coil.

Fig. 6.6

Superconducting quantum interference device (SQUID) magnetometer. Image and caption taken with permission from Ref. [46]. (a) Photograph of a thin-film SQUID fabricated at Berkeley (right) and a close up of the Josephson junction area (left). (b) Configuration of a flux transformer coupled to a SQUID to form a magnetometer

SQUIDs have been used to measure NMR since the 1990s [49,50,51] in a wide variety of fields. SQUID sensitivity can be characterized in terms of magnetic field at the location of the pick-up coil. A wideband ultra-sensitive magnetometer [41] from 2017 demonstrates a noise floor of 150 aT/$\sqrt {\text{Hz}}$ in a frequency range between 20 kHz and 2.5 MHz (Fig. 6.7). Similar performance has been achieved in the Search for Halo Axions with Ferromagnetic Toroids (SHAFT) experiment that searches for electromagnetic coupling of ultralight axionlike dark matter [45]. We note that the superconducting nature of SQUIDs limit their use to low-temperature environments, usually near 4 K (although there are high-T_c SQUIDs, whose performance is not quite as good). The SQUID sensors should also be carefully shielded from external magnetic fields. There are also other superconducting devices in development whose performance may offer significant improvements compared to the SQUID characteristics [52].

Fig. 6.7

Amplitude noise spectrum of a SQUID magnetometer and a SQUID gradiometer. Image and caption taken with permission from Ref. [41]. Measured magnetic flux density noise $S^{1/2}_{B,m}$ for the two setups with 45 mm diameter pick-up coils: magnetometer (solid green curve) and gradiometer (solid blue curve). The calculated intrinsic SQUID noise levels $S^{1/2}_{B,i}$ are given by the dotted curves. For the gradiometer, the noise is referred to the bottom pick-up loop, and the gradient noise is shown on the right

6.3.4 Atomic Vapor Sensors

Optical atomic magnetometry [54] is based on the manipulation of atomic spins with laser light and the subsequent observation of their evolution under the influence of a magnetic field. Overall the principles of optical atomic magnetometry are very similar to those of NMR discussed in earlier sections, however, in atoms the nuclear spin and electron angular momentum (orbital and spin) are coupled and so the dynamics involve the total atomic angular momentum. For alkali atoms, the ground state magnetic moment is dominated by the electron spin of the valence electron. Atomic magnetometers have been around since the 1970s [55]. Excellent review articles and books have been written on this topic [54, 56, 57]. Admitting numerous variations, Fig. 6.8 illustrates the most common ingredients of an optical atomic magnetometer. Figure 6.8a shows a vapor cell filled with a dilute vapor of alkali atoms. Figure 6.8b shows an experimental configuration using crossed probe and pump beams orthogonal to an applied background magnetic field B₀, and an oscillating field close to the Larmor frequency ω_L to excite the magnetic resonance. A (truncated) atomic level scheme and optical transitions for ⁸⁷Rb can be seen in Fig. 6.8c. Applying an on-resonant circular polarized light beam along the direction of the magnetic field optically pumps the atoms, through consecutive absorption and spontaneous emission cycles, into the highest magnetic sublevel of the ground state. The atomic spins in the vapor are thus oriented and can be measured as a macroscopic, collective (electron spin) magnetization. The evolution of these spins due to the magnetic field occurs at the driving frequency ω_RF and a magnetic resonance centered around the Larmor frequency ω_L is observable using demodulation with a lock-in amplifier. The width and amplitude of this complex Lorentzian are the key quantities to optimize for sensitive magnetometry. The center frequency of the Lorentzian, i.e., the Larmor frequency, is a measure of the magnetic field. For small magnetic fields (in the regime of the linear Zeeman effect) it can be written as:

$$\displaystyle \begin{aligned} \hbar\omega_L=\mu_B g_F \left|\boldsymbol{B}\right|, \end{aligned} $$

(6.48)

where μ_B∕ħ = 2π × 14 GHz/T is the Bohr magneton, g_F the Landé factor, and $\left |\boldsymbol {B}\right |$ the magnitude of the background magnetic field. This oscillation is then read out, for example, with an off-resonant laser beam via modulation of its polarization due to the Faraday effect [58]. The cells can be evacuated (i.e., contain a low density of single species alkali atoms), include various wall coatings or buffer gases to reduce relaxation rates and contain different species or combination of species of alkali atoms. The fundamental atomic shot-noise limited sensitivity δB_SNL of such a magnetometer is dominated by two quantities: the spin-relaxation rate Γ_rel and the number of spins N that are measured simultaneously. For measurement times τ ≫ 1∕ Γ_rel [59] it is given by:

$$\displaystyle \begin{aligned} \delta B{{}_{\mbox{ SNL}}}\approx\frac{1}{\gamma}\sqrt{\frac{\Gamma{{}_{\mbox{ rel}}}}{N\tau}}~. \end{aligned} $$

(6.49)

Fig. 6.8

Optical magnetometry with alkali vapor cells (a) An example of a glass vapor cell used for magnetometry. It is two centimeters in diameter, has a reservoir for Rb and a stem to separate the sensing volume and the reservoir. In this particular case, the inner walls of the cell are coated with an alkene film [53] enabling coherence times of up to 77 s. (b) In the cell atoms fly ballistically with a large velocity and are interrogated and manipulated with laser beams. (c) The atomic energy level of ⁸⁷Rb atoms with interactions. (d) The resulting magnetic resonance is often demodulated with a lock-in and is well described by a complex Lorentzian centered around the Larmor frequency ω_L

Similar to NMR-based ALP searches, the fundamental sensitivity improves with the number of atoms and with a reduction in the relaxation rate. The longer the more spins can be observed precessing the better the magnetic field resolution. Unfortunately, N and Γ_rel are often correlated. For example, increasing the vapor pressure of the alkali atoms by heating the cell (and therefore the number of atoms to be interrogated) also increases spin-exchange- and spin-destruction-collision rates, which in turn increase the relaxation rate.

Radiofrequency vapor cell magnetometry has been used to measure NMR at 60 kHz [40]. A complication is that, due to different gyromagnetic ratios, the nuclear spins and the alkali atoms cannot be subjected to the same magnetic field. This problem can be solved by placing the nuclear sample in a magnetic solenoid coil penetrating the magnetic shield of the vapor cell magnetometer [60].

6.3.4.1 Spin-Exchange-Collision-Free (SERF) Magnetometry

Increasing the density of the alkali vapor and therefore the spin-exchange collision rate leads to an interesting new regime. First, magnetic resonances broaden as a function of spin-exchange rate, and then begin to get narrower. This is the so-called spin-exchange relaxation free regime (SERF). If the spin-exchange collision rate is much higher than the Larmor precession frequency, this decoherence mechanism effectively averages out. This behavior was first reported in Ref. [61], explained by the same group [62], and has been used for magnetometry since 2002 [63]. In 2010, the Romalis group at Princeton demonstrated a record-breaking sensitivity of 160 aT$/\sqrt {\text{Hz}}$ in a gradiometric configuration [38]. SERF magnetometers can be used for ZULF NMR [35, 64, 65], see Sect. 6.2.4. Due to the fact that SERF magnetometers employ a magnetic resonance centered around a near-zero magnetic field, the accessible frequency range depends on the linewidth of the resonance, normally below 1 kHz. Searches for UBDM using SERF magnetometers include the Cosmic Axion Spin Precession Experiment (CASPEr) ZULF sidebands and CASPEr comagnetometer experiments, as shown in Fig. 6.9.

Fig. 6.9

Existing bounds and sensitivity projections for the: (a) EDM and (b) gradient coupling of axionlike dark matter taken from [12] with permission. The purple line shows the QCD axion coupling band in (a) and (b). The darker purple color shows the mass range motivated by theory [2]. The blue regions mark the mass ranges where the ADMX and HAYSTAC experiments (see Chap. 4) have probed the QCD axion-photon coupling [75, 76]. The green region is excluded by analysis of cooling in supernova SN1987A (see Chap. 3), with color gradient indicating theoretical uncertainty [2].The region shaded in red is the exclusion at 95% confidence level placed by CASPEr-E in [12]. The dashed green line marks the projected 5σ sensitivity of the CASPEr-E search with a 4.6 mm sample, as used in [12]. The dashed blue line marks the projected 5σ sensitivity of the CASPEr-E search with an 80 cm sample, operating at 100 mK temperature. The black dashed line marks the sensitivity limited by quantum spin-projection noise [77]. This would be sufficient to detect the EDM coupling of the QCD axion across a 6-decade mass range from ≈0.3 peV to ≈500 neV. The other bounds are as follows. (a) The pink region is excluded by the neutron EDM (nEDM) experiment [78]. The blue region is excluded by the HfF⁺ EDM experiment [79]. The yellow region is excluded by analysis of Big Bang nucleosynthesis (BBN) [80]. (b) The pink region is excluded by the neutron EDM (nEDM) experiment [78]. The blue region is excluded by the zero-to-ultralow field comagnetometer (ZULF CM) experiment [33]. The gray region is excluded by the zero-to-ultralow field sideband (ZULF SB) experiment [34]. The yellow region is excluded by the new-force search with K-³He comagnetometer [81]. The bounds are shown as published, although corrections should be made to some of the low-mass limits, due to stochastic fluctuations of the axionlike dark matter field [82]

6.3.5 Magnetic Noise Suppression

As discussed in Sects. 6.1.1.1 and 6.1.1.2, a UBDM field that couples to atomic spins mimics an oscillating magnetic field, therefore real oscillating magnetic fields are one of the most important sources of systematic errors. And, of course, magnetic fields are everywhere: the Earth’s magnetic field itself is ∼10¹⁰ times larger than the smallest field that can be detected by commercially available atomic magnetometers (averaged over 1 s), radio waves over a wide range of frequencies travel through space, and electronic currents generate associated magnetic fields. Thus a conventional laboratory environment is teeming with complicated patterns of oscillating magnetic fields and magnetic field gradients of many orders at many frequencies, especially at the power line frequency and its harmonics. Most sensitive magnetometers require, therefore, a sophisticated shielded environment to avoid being saturated by magnetic noise. The effort to invest in magnetic shielding depends on the UBDM candidate mass range that is being searched. While static and slowly varying magnetic fields require complicated, multilayered magnetic shields constructed from materials with a high magnetic permeability to guide the magnetic flux around the sample or possibly superconducting shields, higher frequency magnetic noise can be effectively shielded by conductive enclosures. For sub-kHz magnetic noise, most vapor cell magnetometer are still limited by intrinsic magnetic field noise of the shield. Note that, in many cases (such as if the coupling of the UBDM is primarily to nuclei), signals from UBDM fields are unaffected by the magnetic shielding [66], while in some cases the magnetic shielding can in fact have significant effects (e.g., for hidden photons as discussed in Chap. 7). Detailed discussion of magnetic shielding can be found in Chapter 12 of Ref. [54].

To further improve the sensitivity of UBDM searches relying on magnetic resonance, other mechanisms have to be deployed to extend the discovery reach of the apparatus. One method of reducing sensitivity to local magnetic field changes is comagnetometry [67], i.e., where the magnetic field (and the UBDM signal) is simultaneously measured in two (or more) different ways in the same volume such that the magnetic responses and UBDM responses of the two methods are distinct. As elucidated in Sect. 6.1, the overall coupling strength of an ALP field to an atomic nucleus depends on the nuclear composition and the electronic state of the sample [68]. Consider the case where the sample consists of two ALP-sensitive species with opposing signs for the ALP interaction and identical signs for the magnetic interaction. When signals from both species are measured simultaneously and independently, subtracting the resulting signals reduces the effects of magnetic fields while enhancing the measurable effects of an ALP-spin interaction. This method has been successfully deployed in the experiments described in Refs. [3, 69, 70].

As a last thought for this section, if comagnetometer measurements are able to achieve sufficient insensitivity to magnetic fields, what will be the next most important systematic? The answer could be given by a common application of nuclear comagnetometry: gyroscopy [71, 72]. In fact, systematic errors due to rotations are major impediments to recent comagnetometer experiments searching for Lorentz invariance [73, 74].

6.4 Experimental Searches

Finally, we present a selection of magnetic-resonance-based experiments constraining ALP parameter space and place them in context with respect to astrophysical limits (see Chap. 3). The Cosmic Axion Spin Precession experiment (CASPEr) is a multi-pronged approach proposed in 2014 [77] to search for ALP dark matter over a wide range of ALP masses using NMR and undertaken in Boston, USA, and Mainz, Germany. The nEDM experiment searches for a permanent electric dipole moment of the neutron (nEDM) at the Paul Scherrer Institut in Switzerland. Reanalyzing their accumulated years of nEDM data (including a data set collected at the Institut Laue-Langevin in Grenoble, France between 1998 and 2002) for oscillating signals allowed the team to place tight constraints on both the EDM and the gradient coupling for low mass ALPs. The HfF⁺ EDM experiment at the Joint Institute for Laboratory Astrophysics (JILA) in Boulder, USA, by the group of Eric Cornell searches for a permanent electron EDM. It is a precision experiment measuring electron spin precession with trapped molecular ions. Data collected in 2016 and 2017 were reanalyzed for oscillating signals and used to place constraints on the EDM coupling. Figure 6.9 summarizes the constraints on EDM and gradient ALP couplings. Other closely related magnetic resonance experiments include the QUAX experiment [13, 83, 84] that searches for electron-ALP interactions as well as a proposal to use antiferromagnetically doped topological insulators [85] to search for high mass axions in the 0.7–3.5 meV range.