Dark matter detection using nuclear magnetization in magnet with hyperfine interaction

We consider the possibility to detect cosmic light dark matter (DM), i.e., axions and dark photons, of mass $\sim 10^{-6}$ eV and $\sim 10^{-4}$ eV, by magnetic excitation in a magnet with strong hyperfine interaction. In particular, we consider a canted anti-ferromagnet, MnCO$_3$, as a concrete candidate material. With spin transfer between nuclear and electron spins allowed by the hyperfine interaction, nuclear spins become naturally highly polarized due to an effective (electron-spin-induced) magnetic field, and have long-range interactions with each other. The collective precession of nuclear spins, i.e., a nuclear magnon, can be generated by the DM field through the nucleon-DM interaction, while they are also sensitive to the electron-DM interaction through the electron-nuclear spin mixing. Compared with conventional nuclear-spin precession experiments, this system as a DM sensor is sensitive to higher frequency needing only a small static magnetic field applied. The system also has collective precession of electron spins, mixed with nuclear spins, as the additional channels that can be used for DM probes. We estimate the sensitivity under appropriate readout setups such as an inductive pick-up loop associated with an LC resonant circuit, or a photon cavity with a photon counting device. We show that this method covers an unexplored parameter region of light bosonic DM.


Introduction
Dark matter (DM) has been the unanswered mystery of particle physics for decades. Its existence, covering 80% of matter in the universe, has been vividly suggested and supported by both astrophysical and cosmological evidences (for a review, see, e.g., Ref. [1]). However, DM cannot be reasonably accounted for by particles already known in the standard model (SM), and its particle-physics properties are still unknown. Undoubtedly, DM would provide us priceless hints for understanding physics beyond the standard model (BSM), even if we discover just a few secrets of its anatomy directly.
Light bosonic particles with mass 10 −22 eV-keV are possible candidates of DM. These include axions, pseudo Nambu-Goldstone bosons emerging from the Peccei-Quinn (PQ) symmetry breaking [2][3][4][5], including axion-like-particles (ALPs) motivated from string theory [6][7][8], and dark photons [9][10][11][12], spin-1 vector bosons kinetically mixing with ordinary photons. Generally, they have small mass with large number density which causes them to effectively behave like a coherent classical field. This type of DM is out of reach from the nuclear (electron) recoil experiments, e.g., CDMS [13][14][15], XENON [16,17], and PANDA [18,19]; these experiments have excellent sensitivity for ∼ GeV (MeV) DM, however, for the DM of mass below GeV (MeV) the sensitivities become worse rapidly because the recoil energy is severely suppressed due to the smallness of the momentum of DM. So far, constraints on the sub-MeV and light DM are weak and require dedicated methods for their detection.
In this paper, we consider the nuclear magnetization system with a strong hyperfine interaction [56][57][58][59][60], which allows the mixing between electron spin and nuclear spin. At low temperature, nuclear spins are highly polarized due to the electron-spin-induced effective magnetic field of O(10) T. Besides, nuclear spins have a long range interaction with each other via the so-called Suhl-Nakamura exchange interaction [61,62], which ensures the existence of nuclear spin waves (or nuclear magnons in the quantized version [60]). The presence of DM can excite the mixed state between electron and nuclear spin precessions, resulting in macroscopically observable magnetization, which is enhanced by a factor of O(10) − O(10 3 ) compared to the typical nuclear magnetization signal. According to the eigenfrequency profile of material with a strong hyperfine interaction such as MnCO 3 [58,59,63,64], CsMnF 3 [65][66][67], CoCO 3 [57] and FeBO 3 [57], they may probe ∼ µeV -meV DM with good sensitivity to various interaction parameters through either electron-DM or nucleon-DM coupling. #1 This kind of system also has electron-spin precession modes, mixed with nuclear spins, as an additional channels that can be used for DM probes. In order to show the possibility of using magnetically-ordered material with strong hyperfine interaction for DM detection, we focus on the canted antiferromagnetic material MnCO 3 , which has strong hyperfine interaction with 100% magnetic isotope, as a concrete example. We explore in detail its resonance profile and response to the DM field composed of axions or dark photons under the sensible readout setup such as an inductive pick-up loop associated with LC resonant circuit, or a photon cavity with a photon counting device. We show that a sizable signal from DM axion or dark photons can be expected.
We summarize the advantage of a magnetic system with hyperfine interaction as: 1. nuclear spins are naturally highly polarized, which leads to large resonance signal 2. sensitivity at high DM mass is achievable with a small applied static field, compared to a nuclear magnetic system without hyperfine interaction 3. there are sensitivities to both electron-DM or nucleon-DM couplings 4. readout is enhanced or becomes possible by the presence of mixing of electron spins to nuclear spins The construction of this paper is as follows. We start with a review of DM focusing on the two promising candidates, axions and dark photons in Sec. 2. We elaborate the properties of the material in Sec. 3: the magnetic material with strong hyperfine interaction; and excitation of hybridized precession state of electron and nuclear spins induced by DM. In Sec. 4, we show the sensitivity of light DM detection with several concrete proposals of experimental setup. We conclude in Sec. 5.

Dark matter target
We consider two candidates of light bosonic DM, axions and dark photons, whose masses correspond to the energy which can cause the magnetic resonance in a magnet with hyperfine interaction. Because their masses are small, their number densities are large such that they behave coherently as a classical field and may act as an oscillating magnetic field in the view of spins of SM particles. If the coupling is strong enough, spins inside magnetic material can be perturbed from the ground state and macroscopically produce an observable signal as oscillating magnetization. We review models of axions and dark photons, and describe the oscillating effective magnetic field induced by them. We illustrate in Sec. 3 the response of magnet excited by DM, based on MnCO 3 as a concrete example material with strong hyperfine interaction.
In this section, natural unit is adopted. #1 To achieve sensitivity in such a range of frequency, technically, a nuclear magnetic system without a strong hyperfine interaction needs an applied magnetic field of magnitude larger than 30 T for the experimental setup.

DM axion
QCD axions and ALPs are candidates for the DM particle. QCD axions are pseudo Nambu-Goldstone bosons arising from the PQ symmetry breaking [2][3][4][5]. They are proposed to solve the strong CP problem. QCD axions interact with gluons, photons, and SM fermions with interaction strength determined by the axion mass. On the other hand, ALPs are the generalization of QCD axions; they are particles that interact with SM particles with a similar form of interactions but do not have roles to solve the strong CP problem and do not have a specific relation between the mass and the coupling with SM particles. We collectively call QCD axions and ALPs as simply "axions".

Axion model and effective Lagrangian
Typically, the QCD axion is introduced as the Nambu-Goldstone boson from spontaneous symmetry breaking of a new U(1) symmetry, the so-called U(1) PQ symmetry, with chiral anomaly associated with the color SU(3) symmetry and also electromagnetic symmetry in some models. With this mechanism called the PQ mechanism, axion effective Lagrangian reads where a, N (≡ p, n), and e represent axion, nucleon (proton, neutron), and electron fields, respectively, m a is the axion mass, and F µν is the field strength tensor of photons. Coupling constants g aγγ , g aN N , and g aee are model dependent and can be written in the form g app = m p c ap f a , g ann = m n c an f a , g aee = m e c ae f a , (2.2) showing the dependence on the axion decay constant f a and model-dependent constants c af with f = p, n, e. The QCD axion mass m a is related to scale f a by the relation [68]: m a ≃ 5.7 × 10 12 GeV f a µeV, (2.3) which implies a model-dependent relation between the coupling constant g af f and the axion mass m a .
As explicit examples of the QCD axion model, there are several famous ones adopted as building blocks of others. The main difference lies in how the anomaly of the U(1) PQ symmetry is introduced. The model of Kim-Shifman-Vainshtein-Zakharov (KSVZ) type introduces heavy quarks Qs which transform under PQ charge chirally [69,70]. The other one is the Dine-Fischler-Srednicki-Zhitnitsky (DFSZ) type model which contains two Higgs doublets responsible for the electroweak symmetry breaking and assigns the PQ charges to SM particles and to Higgs bosons [71,72]. For the KSVZ model of axions, the modeldependent parameters are where β is the ratio of the vacuum expectation values of two Higgs doublets. It should be noted that the value of parameter β is constrained by the perturbativity of the Yukawa coupling as 0.28 < tan β < 140 [73]. Recently flavorful axion models are also considered [74,75], which also predict sizable axion-fermion couplings.
On the other hand, ALPs in general are expected to possess the same Lagrangian as shown in Eq. (2.1) while there is no relationship between its mass m a and the coupling g af f .

Axion-induced magnetic field
We now estimate the magnitude of the axion-induced effective magnetic field acting on spins of SM particles assuming that all of the DM is composed of axions. Recall that the axionnucleon interaction is given by the fourth term of Eq. (2.1). In the non-relativistic limit, the interaction reduces to the form with ⃗ S N representing the nucleon spin. This shows us that the axion field acts as an effective magnetic field interacting with nuclear spins. Below, we estimate the amplitude of the effective magnetic field from the axion properties.
We consider the axion DM in the mass range 10 −6 eV ≲ m a ≲ 10 −4 eV. We adopt the standard cold DM velocity profiles that the DM velocity v DM is expected to be ∼ 10 −3 with spreading ∆v DM of the same order. The de Broglie wavelength of axion DM, is much longer than the size of the magnetic material used for the experiment. The occupation number of the axion DM is also large due to the small mass, so we treat the axion DM as a classical field that interacts coherently with nuclear spins inside the magnetic material within coherence time (2.10) We parameterize the axion DM classical field as where a 0 is the oscillation amplitude with which the energy density is given by ρ a = 1 2 m a a 2 0 , while δ is a random phase. We assume ρ a = ρ DM , where ρ DM is the local DM density around the earth. In our numerical calculation, we take ρ DM = 0.4 GeV cm 3 . (2.12) We assume that the oscillation persists in its coherent phase for time interval τ DM , each interval connected with a discrete jump of ⃗ v DM and δ. Due to the velocity distribution, the bandwidth of the field is given by which translates to quality factor (2.14) To consider the interaction between nuclear spin ⃗ I and the axion, we need to take into account the nucleon spin contribution to the nuclear spin. This is shown in detail in Appendix A. Matching the nucleon spin-axion interaction to the interaction between the nuclear magnetic moment ⃗ µ I and an arbitrary magnetic field ⃗ h of the form L int = ⃗ µ I ⋅ ⃗ h, we can define the effective magnetic field induced by the axion, which is felt by the nuclear spin as whereg aI ≡ g app σ p + g ann σ n with σ p,n denoting the spin contribution of proton and neutron to nuclear spin, and γ n ≡ g I µ N is the nuclear gyromagnetic ratio defined by the nuclear g-factor g I and the nuclear magneton µ N ≡ e 2m p . Note that the value of γ n depends on magnetic isotope, while the typical order of magnitude is O(10 7 ) rad ⋅ s −1 ⋅ T −1 .
On the other hand, the axion also interacts with an electron through the last term of Eq. (2.1). Similarly to the case of the nuclear spin, we introduce the axion-induced effective magnetic field for the electron spin as where γ e = 1.760 × 10 11 rad ⋅ s −1 ⋅ T −1 is the electron gyromagnetic ratio. As a result, we obtain the amplitude of the effective magnetic field as: for g app = g ann = g aN N .

DM dark photon
The dark photon is introduced as a new gauge boson of a new dark U(1) gauge symmetry in addition to the SM symmetry. Even when SM particles do not have dark U(1) charges, kinetic mixing between the dark photon and the SM photon becomes a portal for the SM particles to interact with dark photons. The massive dark photon that is light enough and weakly interact with SM particles is a viable candidate of DM from the viewpoint of the up-to-present observable constraints. See Refs. [76][77][78][79][80][81][82][83][84][85][86] for production mechanisms of dark photon DM. The general review of the dark photon is given in Ref. [12], while the careful treatment of dark photon polarization and its effect on actual experiments are discussed in Ref. [87].

Dark photon model
We introduce the dark photon, a massive vector field (denoted as A ′ µ ) which couples to the SM fields only through the kinetic mixing with the ordinary photon. The Lagrangian includes the following terms: where A µ , A ′ µ are the gauge fields associated with the electromagnetic U(1) symmetry and dark U(1) symmetry, respectively, µ are the corresponding field strength tensors, while J EM µ is the electromagnetic current of the SM particles. In addition, is the kinetic mixing parameter that should be much smaller than unity. In this basis, there is no direct coupling between the dark photon and the SM fermions.
We can also work in the basis in which the kinetic mixing of dark and ordinary photons vanishes. There is a mass eigenstate with zero mass, which corresponds to the ordinary photon as The Lagrangian can be expressed as Here and hereafter, we neglect terms of O( 2 ) because ≪ 1. We can see that, in this basis, the dark photon A ′ µ obeys the Proca Lagrangian with eJ EM µ A ′ µ as a source term. By taking the Lorenz gauge for the SM photon, the equations of motion are given by

Dark-photon-induced magnetic field
Let us focus on the interaction between the dark photon and the SM particles in the basis of mass eigenstates: We assume the following form of the vector potential of dark photon: The spread of amplitude in frequency space is given also by Eq. (2.13) with mass replaced by m γ ′ . Then, assuming that the whole DM abundance is explained by the dark photon, we Similarly to the ordinary relation between the vector potential and the magnetic field, the effective magnetic field induced by the dark photon is expressed as where the hat symbol represents the unit vector pointing in the direction of the original vector. Numerically, we obtain the amplitude of the effective magnetic field as where sin ϕ ≡v DM ×Â ′ is the angle between the DM velocity and the vector potential ⃗ A ′ . Note that the effective magnetic/electric field induced by the dark photon is significantly affected by the conductor shield around the experimental apparatus [88]. As a rough estimate, for the typical length scale of the shield L shield , the effective magnetic field (2.26) should be multiplied by an additional factor of (m γ ′ L shield v DM ) −1 for 1 < m γ ′ L shield < v −1 DM and by m γ ′ L shield v DM for m γ ′ L shield < 1, which can either enhance or suppress the signal depending on m γ ′ L shield . Note that v DM ∼ 10 −3 . Numerically, the signal is suppressed when which could be avoided by using a reasonably large magnetic shield. On the other hand, thanks to the absence of a suppression factor v DM in the shielding effect, the effective magnetic field (2.26) can be enhanced up to a factor 10 3 . In this paper, we do not discuss a real experimental apparatus including the shield and simply use Eq. (2.26) as a conservative estimate.

Magnetization dynamics in MnCO 3
The DM-induced magnetic field discussed in the previous section can cause magnetic resonance in the material and give an observable oscillating magnetization signal. We propose to use materials with strong hyperfine interaction that allows spin transfer between nuclear and electron spins for DM detection. The hyperfine interaction between electron and nuclear spins originates from the magnetic dipole interaction between them. The following properties are realized due to this interaction. Nuclear spins are highly polarized by the large effective magnetic field due to electron spin even in the low external static magnetic field. It provides large magnetic signals in the high-frequency region not easily accessible by other approaches using ordinary nuclear magnetization. Through the exchange interaction with electron spins, nuclear spins effectively obtain an exchange interaction among themselves called the Suhl-Nakamura interaction [61,62], to realize the nuclear spin wave modes (or the nuclear magnons in the quantum picture). In this mode dominated by the precession of nuclear spins, the electron spins with a larger gyromagnetic ratio also contribute to the total observable magnetization enhancing the overall magnetization signal compared to nuclear magnetization alone. On the other hand, there is also an electron-spin-dominated mode mixed with nuclear spins. They are sensitive to both the DM-electron and the DM-nucleon interactions. As a concrete example, we focus on a canted anti-ferromagnetic material MnCO 3 .
We introduce the property of MnCO 3 in Sec. 3.1. Then, we illustrate the magnetic dynamics of MnCO 3 in Sec. 3.2 and its response to the DM coupled only to either the nuclear spin or the electron spin in Sec. 3.3.
The calculation in the following sections is done in the SI unit. However, we simply omit the vacuum permeability factor µ 0 in the formulae for convenience. #2 The factor µ 0 is restored in Sec. 4.

Introduction to MnCO 3
MnCO 3 is a canted anti-ferromagnetic material. It has large nuclear spins (with I = 5 2) associated with 100% magnetic isotope of 55 Mn, which couple to localized electron spins (with S = 5 2) through strong hyperfine interactions. The electron ground state configuration of 55 Mn is [Ar] 4s 2 3d 5 with multiplet 6 S 5 2 .
The MnCO 3 lattice structure can be represented by the rhombohedral unit cell. The parameters for this representation are interval length a rh = 5.84 A with angle α = 47°. In the following, lattice directions such as [111] or [101] are to be understood as of the rhombohedral representation. For the lattice structure, see, e.g., Refs. [58,89,90].
The hard-axis anisotropy presents in the [111] lattice direction, forcing electron spins, which is localized at each Mn ion, to lie in the basal plane ((111) plane). Importantly, due to an exchange interaction between electron spins, they align anti-parallel to the one in the nearest site, forming an anti-ferromagnetic order below the Nèel temperature (T * ≈ 35 K [58]). There are also weak anisotropic fields present in the basal plane. Besides, the Dzyaloshinskii-Moriya interaction produces the weak-ferromagnetic properties of MnCO 3 in the basal plane. #2 The formulae are then the same as that in the CGS unit except for the definition of susceptibility.
In general, the interaction between nuclear spin ⃗ I and electron spin ⃗ s originates from the dipolar interaction between them. It is given as follows [91]: where γ e and γ n are gyromagnetic ratios of electron and nucleus, respectively; ⃗ l is the orbital angular momentum of electron and ⃗ r is the position vector of electron with the origin ⃗ r = 0 being the nucleus position. In the MnCO 3 case, the unpaired 3d electrons have Coulomb interaction with the 2s electrons of the opposite spin more efficiently than the 2s electrons of the same spin. Thus the spin-polarization occurs at the core and is proportional to the magnitude of 3d electron spins. The last term of Eq. (3.1) shows the interaction between electron-spin polarized core and nuclear spins, which indicates an effective but strong interaction between 3d electron spins and nuclear spins of the form H hy ∝ ⃗ I ⋅ ⃗ S where ⃗ S is the total electron spin associated with each Mn atom.
In MnCO 3 , the nuclear spin is sensitive to a hyperfine field and becomes highly polarized pointing to the direction correlated to the electron spin. Besides, nuclear magnetic resonance occurs at a very high frequency ∼ 500 MHz compared to that of typical nuclear spin precession. Under the presence of the strong hyperfine interaction and the exchange interaction of electron spins, there exists an effective exchange interaction between nuclear spins ensuring the existence of nuclear spin wave. On the other hand, the system can also be viewed as a hybrid system of nuclear and electron spins. Detailed dynamics of such a system and its response to the DM will be discussed in the following subsections.
Materials with 55 Mn ions are frequently used in nuclear-spin-wave experiments. The reasons are the following: • 55 Mn magnetic isotopes have large localized nuclear and electron spins (I = 5 2, S = 5 2).
• The electron spin wave modes have low eigenfrequency close to that of nuclear modes, leading to a large mixing between nuclear and electron spins. #3 Other example materials with 55 Mn ions besides the canted antiferromagnet MnCO 3 #4 [58,59,64,92,93], include the hexagonal antiferromagnet CsMnF 3 [65][66][67] with biaxial anisotropy, the antiferromagnet RbMnF 3 [94,95] and KMnF 3 [96] with cubic crystalline anisotropy, the antiferromagnet MnF 2 [97] with uniaxial anisotropy, and the ferrimagnet MnFe 2 O 4 [98]. Other materials are also studied in the context of nuclear spin waves: CoCO 3 [57] since 59 Co #3 Note that the 3d electrons of 55 Mn are all unpaired and fill each 3d shell, and hence the total orbital angular momentum of the electron spin associated with each atom is zero. This only leads to a weak magnetocrystalline anisotropy, which causes a small gap of the electron-spin system.
#4 MnCO 3 has a weak-ferromagnetic property due to the Dzyaloshinkii-Moriya interaction, which forces the ground state spins to be canted even in the absence of an applied static magnetic field. Therefore, there is no need to worry about the phase transition from the antiferromagnetic phase to the spin-flop phase as in usual antiferromagnetic materials.
Recently, Shiomi et al. [58] and Kikkawa et al. [59] reported experiments combining nuclear spin waves with spintronics in MnCO 3 , showing nuclear spin pumping effect and nuclear spin Seebeck effect of the system and establishing a new area of spin technology, nuclear spintronics.
The spin transfer between electron and nuclear spins allows us to deal with nuclear spins easier through more-accessible electron spins. Here, for DM detection, it provides a unique probe for nucleon-DM and electron-DM interactions of the favorable frequency range, with a "nature" tool (electron spins and their magnetization) supporting signal readout. At the same time, the precession of electron spins, mixed with nuclear spins, is also sensitive to DM (specifically to electron-DM coupling) and we then include it in the discussion. Next, we move to the details of the magnetic system of MnCO 3 .

Magnetic system of MnCO 3
We discuss the (macroscopic) magnetization dynamics of MnCO 3 within a classical theory, with some details shown in Appendix B. We also show that the results are consistent with those derived in the quantum magnon picture in Appendix C.
We define ⃗ M 1 and ⃗ M 2 as electron magnetizations vectors and ⃗ m 1 and ⃗ m 2 as nuclear magnetization vectors, where the subscripts indicate sub-lattices they belong to. Magnetization is defined as the magnetic dipole moment per unit volume, and thus #5 where ⃗ S i and ⃗ I i are the electron-spin operator and nuclear-spin operator at the spin site i, respectively, V is the volume of the sample. The summations of spins run over the sub-lattice 1 and 2 for the corresponding magnetization vector. At the ground state, the magnitude of magnetization of each lattice is assumed to be M 0 and m 0 for electron magnetization and nuclear magnetization, respectively; they can be expressed by where N total is the total number of Mn ions (which is equal to the number of spin sites), S = 5 2 is the total value of electron spins localized at each Mn ion, and ⟨I⟩ is thermal average of nuclear spins localized at each Mn ion, which generally takes a smaller value than I = 5 2 as we will argue later. #5 The gyromagnetic ratio of electron is negative. However, we define γ e and γ n to be positive and hence the additional minus sign appears for electron magnetization.
In the anti-ferromagnetic phase, the potential per unit volume for magnetizations in MnCO 3 is given by The subscripts x, y, z attached to the magnetization parameters refer to the component of the vector which points in those directions. The potential contains the (1) anti-ferromagnetic exchange interaction, (2) Dzyaloshinskii-Moriya interaction, (3) hard-axis anisotropy, (4) inplane uniaxial anisotropy, (5) Zeeman effect, and (6) hyperfine interaction. The coefficients H E , H D , H K , H K ′ and A hy correspond to constants associated with the exchange interaction, Dzyaloshinskii-Moriya interaction, hard-axis anisotropic effect, in-plane anisotropic effect, and hyperfine interaction, respectively. The coordinate setup is assumed such that the ydirection is the hard-axis corresponding to the [111] lattice direction. External magnetic fields are assumed to include an applied static field ⃗ H pointing in the [101] lattice-direction (which we call x): with ⃗ e i being the unit vector pointing in the i-direction, an oscillating field ⃗ h e coupling to electron spins, and an oscillating field ⃗ h n coupling to nuclear spins. The latter two account for the exotic fields originating from DM. For convenience of later discussion, we define effective fields which come from hyperfine interactions and are felt by electron and nuclear spins in their ground state, respectively. In the ground state of this system, there are two sub-lattices of spins pointing in almost anti-parallel directions along the z-axis and in the basal plane (xz-plane) due to the in-plane and hard-axis anisotropic effects, respectively. The Dyaloshinskii-Moriya interaction and an applied magnetic field in the x-direction tilt the spins of two sub-lattices; the tilt angle is denoted as ψ. Then, the magnetic system shows effective ferromagnetism in the basal plane. The schematic picture of the ground state and effective fields are shown in Fig. 1. For convenience, we use magnetization parameters based on the coordinate systems tilted by angles ψ, π − ψ. For electron magnetization, we apply and similarly, for nuclear magnetization, we apply In these tilted frames with the ground state expectation values are given by M x 1 ,y 1 The coordinate system adopted in the transformation is shown in Fig. 2. The polarization factor ⟨I⟩ I is expressed by the thermal average in the presence of magnetic field H n exerted on the nuclear spins: We plot it in Fig. 3, which clearly shows a paramagnetic property of nuclear spins. However, due to the large magnetic field H n ∼ 60 T through the hyperfine interaction, the polarization is naturally large without any other external field. This is one benefit of the strong hyperfine interaction. For concrete estimation, we adopt the temperature of the sample and of nuclear spins to be T = 0.1 K. With this coordinate, the hyperfine interaction (the last line of (3.5)) can be written in the form where The term U ∥ represents the hyperfine interaction in the direction of spin alignment in the ground state, which makes the Larmor frequency of both nuclear and electron spin precessions higher. On the other hand, the term U mix causes the mixing of the nuclear and electron spin precessions. In the presence of an effective oscillating magnetic field induced by DM, the magnetic resonance of this system may occur. We derive equation of motion for magnetization similarly as done in Refs. [93,99]. Under the potential U ( ⃗ M 1,2 , ⃗ m 1,2 ), the magnetizations ⃗ M 1,2 and ⃗ m 1,2 feel the effective magnetic fields determined by from which they receive torque and precess according to the equations of motion given by We can linearize these equations by focusing on the small perturbations around the ground state. We consider the precession of magnetization with a small deflect angle such that Similarly, we consider a situation where the precession angle of the nuclear magnetization vector from the ground state is small such that m to be m 0 . Defining "plus" and "minus" modes as with α = x, y, z, it turns out that the equation of motions for magnetization vector are decoupled for + and − combinations, which are usually called the in-phase and out-phase modes, respectively. For electron magnetization, we obtain the equation of motion for the in-phase mode as and for the out-phase mode as where the magnetic field h x,y,z e is an exotic field interacting only with electron spins and oscillates in the x-, y-, or z-direction, respectively. Note that the ⃗ M + modes can only be excited by oscillating fields in y-and z-directions. On the other hand, the ⃗ M − can be excited only via the x-direction.
For nuclear magnetization, the equations of motion for the in-phase mode are and those for the out-phase mode are which include an exotic oscillating field h x,y,z n that interacts only with nuclear spins and oscillates in the x-, y-or z-direction, respectively.
The eigensystem of the in-phase mode can be solved with the ansatz: in the absence of exotic fields (h e,n = 0). The problem is reduced to an ordinary eigenproblem. The detailed calculation is shown in Appendix B. The same can be done for the out-phase mode. For convenience, we give first the eigenfrequency of the in-phase/out-phase precession mode of the electron and nuclear magnetization without mixing between them (i.e., neglecting U mix ). The eigenfrequncies of the electron-magnetization system are 26) and the eigenfrequency of the nuclear-magnetization system is Note that nuclear magnetization precessions have two degenerate modes (with angular frequency ω n ) in the absence of the mixing term U mix . Once we take into account of the mixing U mix , we obtain the following. For the out-phase mode, the eigenfrequencies are

Parameters
Magnitude which correspond to the eigenmodes dominated by electron and nuclear spins, respectively. For the in-phase mode, we can obtain a similar expression: corresponding to the electron-and nuclear-dominated modes, respectively. Note that the nuclear-dominated modes receive the so-called pulling effect through the hyperfine mixing between electron and nuclear spins, such that its eigenfrequency is bent by some amount depending on the mixing angle and strength.
The value of the gyromagnetic ratio γ n of 55 Mn is taken from Refs. [100,101], where values of the other parameters in the Hamiltonian is taken from Refs. [64,102,103]. We show them in Table 1. There is a relation among magnitudes of effective fields The eigenfrequencies of the magnetic precession in MnCO 3 are plotted in Fig. 4. Because of the large difference between the eigenfrequencies of the out-phase modes of nuclear and electron magnetizations, the mixing angle between their precessions due to the hyperfine interaction is expected to be small. On the other hand, the in-phase modes of nuclear and electron magnetizations, which have similar eigenfrequencies, significantly mix with each  other. Therefore, from now we focus on and discuss only the in-phase modes and their response to the external oscillating field, while neglecting the out-phase modes.
Note that the magnetic resonance of the system has a relatively high frequency compared to nuclear magnetic system without hyperfine interaction. For instance, 3 He needs a magnetic field of 30 T to realize a magnetic resonance of the same frequency.

Response to DM field
As discussed in Sec. 2.1.2, when the entire DM is composed of axions, a magnetized sample is affected by oscillating magnetic fields ⃗ h n = ⃗ h axion n and ⃗ h e = ⃗ h axion e defined in Eqs. (2.15) and (2.16), respectively. It should be noted that, based on the nuclear shell model of nuclei with odd atomic number, the nuclear spin of MnCO 3 mainly comes from the proton spin [104,105] and hence is sensitive to the axion-proton coupling g app , not to the axion-neutron coupling g ann . For simplicity, we take σ p = 0.1, σ n = 0.0 as a spin contribution from protons and neutrons for the numerical estimation of axion-induced magnetic field ⃗ h axion n . See Appendix A for more detail. For the case of dark photon DM, we have instead Here we show the response of the system by evaluating the magnetization and power absorbed by the system, in Secs. 3.3.1 and 3.3.2, respectively, when the DM mass is equal to the excitation energy of the magnetic material.

Magnetization signal
Now suppose the existence of a non-zero oscillating field from axion or dark-photon DM, ⃗ h n,e ∝ sin ω DM t with ̵ hω DM = m a , m γ ′ . We can write the response in terms of susceptibility χ defined by the relation of total magnetization ⃗ M total and external oscillating field ⃗ h n and ⃗ h e as M α Table 2: The signal susceptibility for the exotic perturbation ⃗ h n that interacts only with nuclear spins. The stars indicate sensitive channels for probing the axion-proton coupling g app . The magnetic system is excited when ω DM − ωñ ,+ ≲ 1 T 2n or ω DM − ωẽ ,+ ≲ 1 T 2e corresponding to the first or second column, respectively. indicate sensitive channels for probing the axion-electron coupling g aee and kinetic mixing parameter . The magnetic system is excited when ω DM −ωñ ,+ ≲ 1 T 2n or ω DM − ωẽ ,+ ≲ 1 T 2e corresponding to the first or second column, respectively.
where α, β = x, y, z denoting the direction of vectors ⃗ M total , ⃗ h n , ⃗ h e or the corresponding component of the tensor χ. The magnetization susceptibility χ can be derived by solving the special solution of the equation of motion for magnetization with the initial condition representing the ground state of the magnetic system. The relaxation time of magnetization should also be taken into account. In the present assumption that MnCO 3 sample is magnetized along the x-direction and that the direction of the oscillating field induced by DM is the y-or z-direction, the precession of total magnetization occurs in the yz-plane for the in-phase mode. We define a magnetization signal M signal as the z-component of the total magnetization vector: Note that because the gyromagnetic ratio of the electron spin is much larger than that of the nuclear spin, a large part of magnetization of the sample is induced by electrons. One can then estimate M z total from M z total ≃ M z 1 + M z 2 = M x + sin ψ. The detailed calculation for signal susceptibility is shown in Appendix B. When the DM mass is close to the precession frequency, in particular ω DM − ωñ ,+ ≲ 1 T 2n or ω DM − ωẽ ,+ ≲ 1 T 2e , the collective spins would be excited, where T 2n and T 2e are the relaxation times for the nuclear-dominated mode and electron-dominated mode, respectively. We assume that these relaxation times are much smaller than the coherence time τ DM of the DM field (T 2e,2n ≪ τ DM ), which is actually the case for MnCO 3 . We assume the values of relaxation time at T = 0.1 K as T 2n = 1 µs and T 2e = 1 ns [57,58,63,92,106]. Within DM coherence time t ≲ τ DM , the magnetic field from DM is a coherent driving field, so the magnetization signal rotates coherently with frequency ω DM (whereas its amplitude is determined by the relaxation time T 2e,2n ). The spread ∆ω signal in frequency space of the signal is determined by that of DM spectrum: The susceptibility on resonance is shown in Tables 2 and 3. The star symbols represent the sensitive channels for probing the listed coupling parameters of DM. We introduce a parameter η defined by which is an enhancement factor of signal magnetization compared to nuclear magnetization, in addition to the effect that nuclear spins are highly polarized by the hyperfine interaction with electron spins. Note that η → 0 for sin ψ → 0 (see Eq. (3.10)). At T = 0.1 K, we obtain η ≈ 50. For example, at H 0 = 1 T corresponding to ωñ ,+ 2π ≈ 500 MHz and ωẽ ,+ 2π ≈ 40 GHz, we obtain numerically for sensitive channels: The order of magnitude does not change around the frequency range of interest (corresponding to H 0 ∼ 0.5-2 T) where the mixing between precessions of electron spins and nuclear spins remains to be large. (See, e.g., Fig. 4 for the relation between H 0 and the eigenfrequency of the system.) The y-direction is the most sensitive to the DM-nuclear interaction, while only the zor both y-and z-directions are to the DM-electron interaction depending on which mode is excited. We define the angle between the sensitive direction and the polarization direction of the DM-induced magnetic field by a parameter θ. To account for the unknown polarization (equivalently the unknown direction of the velocity of DM), we perform a substitution cos 2 θ → 1 3 if the system is sensitive to a single polarization direction, or by cos 2 θ → 2 3 if it is sensitive to any polarization direction in a plane [87].

Power absorbed to the magnetized sample
The absorbed power into the material can be estimated by the relation where V is the sample volume. In Tables 4 and 5, we show the time-averaged power absorbed under the resonance condition for each channel. For example, at H 0 = 1 T corresponding to ωñ ,+ 2π ≈ 500 MHz and ωẽ ,+ 2π ≈ 40 GHz, numerically, we obtain for the most sensitive y-direction of the DM-nucleon interaction (h y n ≠ 0): where W MnCO3 is the total mass of the MnCO 3 sample. Also, at H 0 = 1 T, we obtain for the y-and z−directions of the DM-electron interaction (h y,z e ≠ 0): Within the frequency range of interest (corresponding to H 0 ∼ 0.5-2 T), magnitude of absorption power only slightly changes. In Sec. 4, we suggest experimental ways to detect the signal through a magnetic system with the hyperfine interaction. We estimate the sensitivity of the method based on the magnetization signal and power absorbed into the system, which we compare with power of relevant noises.

Dark matter detection with nuclear magnetic excitation
In this section, we present the sensitivity of DM detection by excitation of hybridized spin precession modes under a low-temperature environment T = 0.1 K. Our basic strategy is as follows. We can set an eigenfrequency of the magnetic system to a desired value by tuning the applied magnetic field H 0 . When the DM mass is close to the eigenfrequency, collective motion of spins can be excited, since the field of DM axions or DM dark photons oscillates with the frequency ω DM ≃ m a,γ ′ ̵ h. Then, a wide DM mass range can be searched for by sweeping H 0 . The probe range is divided into two much different scales corresponding to the two bands of the nuclear-and electron-dominated modes in MnCO 3 . We discuss them in Secs. 4.1 and 4.2, respectively.
n ω n m 0 T 2e γ n 2 Table 4: Power absorbed in the magnetized sample on resonance in the presence of the DMinduced magnetic field interacting only with nuclear spins. The stars indicate sensitive channels for probing the axion-proton coupling g app . The magnetic system is excited when ω DM − ωñ ,+ ≲ 1 T 2n or ω DM − ωẽ ,+ ≲ 1 T 2e corresponding to the first or second column, respectively. Table 5: Power absorbed in the magnetized sample on resonance in the presence of the DMinduced magnetic field interacting only with electron spins. The stars indicate sensitive channels for probing the axion-electron coupling g aee and the kinetic mixing parameter . The magnetic system is excited when ω DM −ωñ ,+ ≲ 1 T 2n or ω DM −ωẽ ,+ ≲ 1 T 2e corresponding to the first or second column, respectively.

Sensitivity at the nuclear resonance frequency
We consider the LC resonant circuit with a pick-up loop similar to Ref [107], combined with the microstrip SQUID amplifier #6 [112][113][114] to detect magnetization induced by DM. It has flexibility to tune the resonance frequency in the range of interest and the high quality factor can be achieved. Before going into the details of the measurement method and setup, let us consider the frequency range of interest and the quality factor of the resonator.
At T = 0.1 K, the spectrum of the nuclear-dominant magnetization precession of MnCO 3 covers a high frequency range from O(10) MHz to ∼ 600 MHz (see Fig. 4). For a frequency range ω 2π ≳ 500 MHz, the observed intrinsic relaxation time T 2n from spin echoes is of order O(10) µs at a low temperature O(1) K, while below 500 MHz the relaxation time T 2n #6 In the frequency range of interest, the dc-SQUID is not favorable as a magnetometer because there is a severe parasitic capacitance between dc-SQUID washer and the input coil. Detection with a reactive (dissipationless) ac-SQUID and the microwave resonator [108,109] is proposed to detect dark photons in the frequency range from 10 MHz to 1 GHz in [110]. Putting the input coil inside the hole in SQUID washer might cure the parasitic capacitance problem for the dc-SQUID as well [111]. Using a microstrip-coupled dc-SQUID as the amplifier is also a way to deal with this issue; we pursue this possibility in this work. drops rapidly and the signal from DM might be suppressed (see, e.g., Refs. [57,58,92,106]). Here, for simplicity we focus on a frequencty range 500-600 MHz, which can be covered by sweeping the static field H 0 from 0.7 T to 2 T (see Fig. 4), while assuming constant T 2n = 1 µs.
An important quantity for quantifying the signal and noise in the resonant approach is the quality factor Q r of the circuit, which is related to the circuit bandwidth ∆ω r and resonant frequency ω as ∆ω r = ω Q r . (4.1) We desire the value of Q r as high as possible up to Q DM = 10 6 so that the signal is highly resonant while the circuit bandwidth ∆ω r still covers bandwidth ∆ω DM of DM signal. Here, in numerical calculation we assume Q r = 10 5 , while a higher value such as Q r ∼ 10 6 might also be achieved by the LC circuit, which is discussed in Ref. [88].
The following steps can be used to scan DM masses within the frequency range ω 2π = 500-600 MHz: 1. Match eigenfrequencies of the magnetic system and the resonant circuit to the frequency ω of interest at the same time. The former can be tuned by changing the external static field H 0 , while the latter by changing the values of resistance and capacitance of the circuit.
2. Measure the magnetization signal from the magnetic material MnCO 3 for an interrogation time ∆t.
3. Shift the frequency ω by an interval ∆ω determined by Eq. (4.1) after each step of the signal measurement until the frequency range of interest is fully covered.
The interrogation time ∆t is determined by the total observation time T total and the number of scan steps determined as a function of Q r . Now we move to the discussion of experimental setup. We consider picking up the signal inductively by a pick-up loop L that is in parallel with the resistance R and capacitor C, while the resonant circuit is capacitively coupled to the amplifier through the capacitor C c (Fig. 5). When DM induces a magnetic excitation in the magnetized sample, the oscillation of magnetization produces an oscillating flux through the pick-up loop, which generates a voltage V p that can be detected through the load resistance R load associated with the amplifier.
Let us estimate the voltage over the load resistance R load (which is the input voltage of the amplifier) so that we can compare the power due to the magnetization signal with that of the thermal noise to derive the signal-to-noise ratio (SNR). We apply the Thevenin theorem to simplify the task. For convenience, the Thevenin equivalent circuit is illustrated in Fig. 5 (c). The Thevenin impedance Z(ω) is the impedance of the circuit when we look from the terminal of R load with V p neglected. It is given by    The parameters are to be chosen such that the impedance is matched to the amplifier impedance R load = 50 Ω at resonance. Requiring Z = Z 0 = 50 Ω at an arbitrary resonance frequency ω 0 , we can choose [107,115] where Q 0 is the unloaded quality factor (or, equivalently, the quality factor of the RLC circuit without C c and R load ). Since the impedance is matched, the loaded quality factor Q r is determined to be Q r = Q 0 2 [115]. On the other hand, according to the Thevenin theorem, the equivalent voltage V s is equal to the voltage between terminals of R load when it is replaced by open terminals (the unloaded probe). Equivalently, this is the case when we take the limit of R load → ∞. Therefore, we obtain the Thevenin equivalent voltage V s in the frequency space as where Q 0 ω 0 L ≫ Z 0 is assumed, and Φ p is the flux signal at the pick-up coil induced by the transverse magnetization M signal of the sample. The tilde symbol represents the value in the frequency space. One can derive the flux signal Φ p from the Faraday induction law [116] and the reciprocity theorem: where β is the rate of flux induction from one unit current. With the geometry of a one-turn pick-up loop with the sample at the center, we obtain where a is the loop radius, d the distance of the sample from the loop and V the sample volume. Note that the sample volume V is limited by the size of the loop. Assuming a sample of cylindrical shape shrunk by ratio ν from a cylinder of radius a and height 2a, the sample volume is given by V = νπa 2 (2a). (4.9) We arrive at the Thevenin equivalent circuit with V s given by Eq. (4.6) and impedance Z(ω) given by Eq. (4.2), whose value on resonance is set to R load . Since the impedance is matched, half of the voltage V s is applied to the input amplifier. (See Fig. 5 (c) for the Thevenin equivalent circuit.) We consider the Johnson-Nyquist thermal noise from the resonator and amplifier with voltageṼ noise = k B (T + T a )Z 0 at the input amplifier, where T a is the noise temperature of the amplifier, T the resonator temperature, and k B the Boltzmann constant. Combining contributions from the DM signal and noise, we obtain the power density at the input amplifier asP (ω) =P s (ω) +P noise (ω), (4.10) At the temperature T = 0.1 K, with the microstrip SQUID amplifier tuned for a frequency ∼ O(100) MHz, combined with the hetero-structure field-effect transistors amplifier [117], the noise temperature of the amplifier is less than T a = 0.1 K.
The SNR is estimated from the Dicke radiometer equation [118] with Eqs. (4.8) and (4.11). We obtain where ∆f is the bandwidth, ∆t is the observation time at each frequency or the interrogation time mentioned above, and the overall coefficient 1 2 is added to account for the time average. By requiring SNR ≥ 1, we obtain the expected sensitivity for the coupling constant between DM and nucleons or electrons.
Here we assume that the material is placed in the proper direction to read out M signal ≡ M z total , and is appropriately shielded from unwanted perturbations. The sensitivities based on sensitive channels of MnCO 3 discussed in Sec. 3.3, along with the present constraints of the relevant DM parameters, are shown in Figs. 6, 7, and 8. We have taken T = T a = 0.1 K, the pick-up coil inductance L ≈ 0.45×(a 5 cm) µH, ν = 0.5, a = 5 cm and the total observation time T total = 1 year. Note that ν = 0.5 and a = 5 cm correspond to the total MnCO 3 mass of 1.5 kg. The quality factor Q r is assumed to be 10 5 . Although the quality factor of circuit might be affected by the relaxation time of the magnet through the coupling between the circuit and magnet, we neglect this effect just for simplicity. The interrogation time at each scan step is ∆t = 1.7 × 10 3 s. The relaxation time T 2n is assumed to be 1 µs for a realistic setup and, in addition, 10 µs for an optimistic one. The latter choice corresponds to a situation where the (effective) magnetic relaxation time is longer thanks to the reduced inhomogeneity in material and applied field.
For the dark photon DM detection, the observed flux at the pick-up loop is due to both the dark photon field itself h γ ′ and the field generated by the magnetization M signal . Since typically χ e ≳ 1, the pick-up magnetic flux caused by M signal is larger than that of h γ ′ ; thus, the sensitivity to is derived taking into account only the contribution of M signal , neglecting that of h γ ′ .

Sensitivity at the electron resonance frequency
The material MnCO 3 is also sensitive to another frequency range of ω 2π ≳ 10 GHz. It is due to excitation of the electron-dominated mode (see Fig. 4). However, because of the hyperfine interaction, it also contains a nuclear-spin component, and hence we also expect a sensitivity to g app . Here we consider the frequency range ω DM 2π ≈ ωẽ ,+ 2π = 45-55 GHz. The relaxation time of this mode is mainly determined by that of the electron spin which is found to be T 2e ≈ O(0.1) ns at T = 4.2 K [63] and T 2e ≈ O(1) ns at T = 1.5 K [58]. We use T 2e = 1 ns for estimating the sensitivity at T = 0.1 K in the following.
Our strategy is to use a microwave cavity strongly coupled to the magnetization of the material [21,48]. When the DM mass is the same as the excitation energy of the homogeneous magnetic precession, the collective motion of spins is resonantly excited. With the setup of the microwave cavity strongly coupled to the magnetization of the material, half of the energy of the magnetic resonance is transferred to the excitation of cavity photons which can be f (Hz) Figure 6: The sensitivity to the axion-proton coupling g app of an experimental setup using a 1.5 kg canted antiferromagnet MnCO 3 . Two hybridized oscillation modes of nuclear and electron spins are considered with each total observation time T total = 1 year. Green (pale green) projection (derived in Sec. 4.1): the nuclear-dominated mode χ zy n (ωñ ,+ ) is used with ωñ ,+ 2π = 500-600 MHz, T 2n = 1(10) µs. The macroscopic magnetization signal is assumed to be detected by the inductive pick-up loop associated with the LC resonant circuit, of which the quality factor Q r = 10 5 is expected. Each scan step at a fixed frequency takes ∆t ≈ 1.7 × 10 3 s. Red (orange) projection (derived in Sec. 4.2.2): electron-dominated mode resonance χ zy n (ωẽ ,+ ) is used with ωẽ ,+ 2π = 45-55 GHz, T 2e = 1(10) ns and ∆t = 5 × 10 5 (5 × 10 4 ) s. In this case, the cavity photons excited in cavity by spin systems are the expected signal. For both projections, the angular average cos 2 θ → 1 3 is assumed to account for the unknown polarization of DM-induced magnetic field. The gray area represents constraints from astrophysical consideration: SN1987A [119][120][121], neutron star cooling [122]. The blue band shows the theoretical prediction from the DFSZ axion model, with 0.28 < tan β < 140 [73].
detected as the signal. The readout can be done by a linear amplifier or photon counter, which are discussed in subsections 4.2.1 and 4.2.2, respectively. However, our aim is to utilize the latter, a photon counter, to avoid the quantum limit that significantly suppresses the sensitivity at the frequency of interest when we use a linear amplifier [129].
The microwave cavity frequency should be tuned to the electron precession frequency, realizing a strong coupling between the magnet and the cavity. The overall relaxation time (ωẽ ,+ ) is used with ωẽ ,+ 2π = 45-55 GHz, T 2e = 1(10) ns and ∆t = 5 × 10 5 (5 × 10 4 ) s. In this case, the cavity photons excited by spin systems are the expected signal. The angular average cos 2 θ → 1 3 and 2 3 are assumed for the former and latter projections, respectively. The gray area represents constraints from astrophysical consideration: The tip of red giants [123], luminosity function of white dwarfs [124,125]; and the constraints from underground detectors searching for solar axions [126][127][128]. The blue band shows the theoretical prediction from the DFSZ axion model, with 0.28 < tan β < 140 [73]. of a single mode of this coupled system is given by (4.13) where τ cav is the cavity lifetime, which we assume to be O(1) µs as in Ref. [21]. The strongcoupling between the magnet and the cavity affects the signal calculation discussed in Sec Green (pale green) projection (derived in Sec. 4.1): the nuclear-dominated mode χ zz e (ωñ ,+ ) is used with ωñ ,+ 2π = 500-600 MHz, T 2n = 1(10) µs. The macroscopic magnetization signal is assumed to be detected by the inductive pick-up loop associated with the LC resonant circuit, of which the quality factor Q r = 10 5 is expected. Each scan step at a fixed frequency takes time ∆t ≈ 1.7 × 10 3 s. Red (orange) projection (derived in Sec. 4.2.2): electron-dominated mode resonance χ zz,zy e (ωẽ ,+ ) is used with ωẽ ,+ 2π = 45-55 GHz, T 2e = 1(10) ns and ∆t = 5 × 10 5 (5 × 10 4 ) s. In this case, the cavity photons excited by spin systems are the expected signal. The angular average sin 2 ϕ → 1 2 is assumed for both projections, while cos 2 θ → 1 3 and cos 2 θ → 2 3 are assumed for the former and latter, respectively. The gray area covered by the bold line represents constraints from cosmological considerations [11], i.e., distortion of CMB and effective neutrino species. The other gray areas are constraints from haloscope searches [87].
is reduced by a factor √ 2 due to the maximal mixing, which results in 50% reduction of power absorbed to the material when only a single mode is excited #7 . Finally, the coupled relaxation time T coupled also determines the bandwidth of the magnon-cavity coupled mode: (4.14) Similarly to the case of the nuclear-frequency range, the following steps can be used to #7 However, with T coupled ≈ 2T 2e , the two mentioned contributions cancel with each other, giving the same absorption power P absorb as originally derived in Sec. 3.3. scan the DM mass over the frequency range ω 2π = 45-55 GHz: 1. Match eigenfrequencies of the magnetic system and the cavity photon to the frequency ω of interest at the same time, realizing the strong coupling between them. The former can be tuned by changing the external static field H 0 , while the latter by changing the size of the cavity.
2. Measure the power of cavity photons induced by the magnetic material MnCO 3 for an interrogation time ∆t.
3. Shift the frequency ω by an interval ∆ω = 2 T coupled after each step of the signal measurement until the frequency range of interest is fully covered.
The interrogation time ∆t is determined by the total observation time T total and the number of scan steps, which is fixed from ∆ω and the whole frequency range of interest.

Microwave cavity with linear amplifier
Using a linear amplifier for probing ω DM 2π ∼ 50 GHz, the effective temperature of the quantum noise is T Q = ̵ hω DM k B ≃ 2.3 K. Therefore, under an experimental setup with the temperature T = 0.1 K, we assume the quantum noise dominates over others.
Let us assume that the resonance condition is satisfied: ω DM ≈ ωẽ ,+ . Input power is discussed in Sec. 3.3 with some subtle care from the magnet-cavity mixing mentioned below Eq. (4.13). In the case of the strong coupling where half of power is transferred to the cavity [21,48], we have P out = P absorb 2.
The following is output power calculated at ω DM 2π = 48 GHz. For detecting the axioninduced field through the axion-proton coupling g app and axion-electron coupling g aee , χ zy n (ωẽ ,+ ) and χ zy,zz e (ωẽ ,+ ) are the most important, respectively; the corresponding output powers are On the other hand, power of the quantum noise is with ∆f ≈ 0.16 GHz and ∆t = 5 × 10 5 s, corresponding to the case that T 2e = 1 ns and T total = 1 year. We can see that this method is not sensitive enough for detecting the DM signal, suffering from the quantum noise from the linear amplifier.

Microwave cavity with photon counter
Instead of using a linear amplifier, we may utilize a single photon counter avoiding the standard quantum limit [129]. In this case, the only relevant noise is the shot noise of the cavity photon of frequency ω = ω DM , whose rate is determined by the effective relaxation time of the coupled system T coupled and the cavity temperature T c as [21] In this setup, we define the SNR as The constraints on DM couplings are given by requiring SNR ≳ 1. The output power P out transferring to the cavity photon system is related to the excitation rate R signal (of spin-flipinduced photons) by the relation The sensitivities obtained for ω 2π = 45-55 GHz, based on the sensitive channels of MnCO 3 discussed in Sec. 3.3 are shown in Fig. 6, 7 and 8, together with the relevant constraints. We assume the relaxation time T 2e = 1 ns for a realistic setup. In addition, we add the case with T 2e = 10 ns, illustrating the optimistic situation when inhomogeneity in the material and the applied field could be reduced and hence the effective magnetic relaxation time is increased. The interrogation time at each scan step is ∆t = 5 × 10 5 s and ∆t = 5 × 10 4 s for the two cases, respectively. Again, the volume of MnCO 3 is assumed to be given by Eq. (4.9) with ν = 0.5 and a = 5 cm, corresponding to the total MnCO 3 mass of 1.5 kg.

Conclusion and discussion
We have discussed a possibility of detecting axion or dark photon DM using nuclear magnetization in a magnet, which has a strong hyperfine coupling between electron and nuclear spins. We focus on the canted antiferromagnet MnCO 3 as a concrete example of such a magnet. Both axion and dark photon DMs may interact with nuclear and electron spins and exert (effective) oscillating magnetic fields on them. This can induce magnetic resonance when the DM mass is equal to the excitation energy of the magnet. Due to strong magnetic fields generated by electron spins through the hyperfine interaction, nuclear spins are highly polarized and can give a sizable resonance signal under relatively low external magnetic field.
There is also an electron magnetic resonance, which is also sensitive to DM. Due to the hyperfine interaction, the nuclear and electron resonance modes are mixed with each other. Specifically in MnCO 3 , both modes are sensitive to frequencies much higher than typical values achieved by ordinary nuclear spin precession experiments. Other materials such as CsMnF 3 [65][66][67], CoCO 3 [57], FeBO 3 [57], and Nd 2 CuO 4 [130,131] are expected to share the same properties.
We investigated the observable magnetization and energy stored in the magnetic material due to the DM-induced oscillating magnetic field. The details of measurement setup and strategy are discussed in Sec. 4 for two frequency ranges: the nuclear frequency ω DM 2π ∼ 500-600 MHz corresponding to m DM ∼ 10 −6 eV (Sec. 4.1) and the electron frequency ω DM 2π ∼ 45-55 GHz corresponding to m DM ∼ 10 −4 eV (Sec. 4.2). As detection schemes, a pick-up loop with LC resonant circuit and cavity photons coupled with the spin systems are considered for the former and latter channels, respectively. The sensitivity for the 1-year measurement with 1.5 kg of MnCO 3 under temperature 0.1 K is shown in Figs. 6, 7, and 8. We summarize the results in the following.
• This system is sensitive to the axion-proton interaction. The axion of mass ∼ 10 −6 eV with g app ≳ 10 −11 could be probed by our method through the nuclear-dominated mode. So far it is hardly reached by other experiments. For example, a nuclear-precession experiment like CASPEr is sensitive to lower mass regions due to the limitation of the magnitude of the external magnetic field. We have also examined the electron magnetic resonance excited through the mixing with the nuclear-spins, which indeed has a sensitivity to the nucleon-DM interaction at the mass scale ∼ 10 −4 eV or O(10) GHz. However, it turns out that this electron-dominated mode is not sensitive enough to constrain the axion parameter space beyond astrophysical constraints, because the signal is suppressed by the short relaxation scale T 2e of the material. Other materials with longer relaxation times should be able to achieve better sensitivity, which is left as a future work.
• The system is also sensitive to the axion-electron interaction assuming an optimistic magnetic relaxation time achieved by small inhomogeneity of the material and the applied magnetic field. The nuclear-dominated mode can be used to explore the axion parameter region with g aee ≈ 10 −14 and m a ∼ 10 −6 eV. This region still survives the astrophysical constraints from stellar evolution such as white dwarf luminosity function and the tip of red giant branch. In addition, by using the electron magnetic resonance mixed with nuclear spins, we can search for the axion mass of 10 −4 eV with parameter g aee ∼ 10 −13 .
• For the kinetic mixing parameter of the dark photon, there are already haloscope experiments that are searching for the mass range m γ ′ ∼ 10 −6 eV corresponding to the frequency regions to which the nuclear-dominated mode is sensitive. Even though we can only probe the already scanned regions with our method, a complementary check of such regions is plausible for ∼ 10 −13 . In addition, with the electron magnetic resonance mixed with nuclear spins, we might be able to probe the parameter region down to ∼ 10 −13 at m γ ′ ∼ 10 −4 eV, which is beyond the current experimental limits and cosmological constraints.
As shown above, DM detection with a magnet with a strong hyperfine interaction provides possibilities to detect both axion DM and dark photon DM. The unique feature of DM detection via the proposed system is the allowed spin transfer between electron and nuclear spins, which provides several available channels and supports the readout of the nuclear signal relying on more accessible electron spins. Nevertheless, further study is still needed to investigate its real potential and improvability. Further study of parameters of magnetic materials is encouraged, such as the magnetic relaxation time T 2n,2e and the susceptibility of materials with a strong hyperfine coupling other than MnCO 3 . Taking into account the statistical behavior of the DM field would also help one to estimate the result more precisely. Further study to cure the weak points associated with the limited band of nucleardominated modes and the short relaxation time of electron-spin systems is also definitely useful to maximize the competence of this method.

A Exotic interaction between nuclear spins of Mn and dark matter
In this Appendix, we consider the interaction between nuclear spin ⃗ I and axions based on the interaction between the nucleon spin and axion given by Eq. (2.8). We mainly follow Ref. [132] in the discussion below. Natural unit is adopted in this Appendix.
Both the spin and orbital angular momentum of protons and neutrons contribute to the nuclear spin ⃗ I and nuclear magnetic moment ⃗ µ I : where ⃗ S p,n and ⃗ L p,n are the total spin and orbital angular momentum operators for protons or neutrons, respectively. Spin and orbital g-factors are given by g p = 5.586, g n = −3.826, g lp = 1, and g ln = 0. For convenience, using contribution variable σ ξ ≡ ⟨ ⃗ I ⋅ ⃗ ξ⟩ ⟨ ⃗ I ⋅ ⃗ I⟩ with the expectation value taken with respect to a nuclear state, and defining the nuclear g-factor g I from ⟨µ zI ⟩ = g I µ N ⟨I z ⟩, we obtain the relation 1 = σ p + σ n + σ lp + σ ln , (A.3) g I = g p σ p + g n σ n + g lp σ lp + g ln σ ln .
(A.4) Now let us discuss the axion interaction with the nuclear spin. One should note that the axion only couples to the nucleon spin, not to its orbital motion. Starting from the axion interaction of the form (g app ⃗ S p + g ann ⃗ S n ) ⋅ ⃗ ∇a, we obtain the interaction between the nuclear spin ⃗ I and the axion of the form whereg aI ≡ g app σ p +g ann σ n . Matching this to the magnetic interaction of the form L int = ⃗ µ I ⋅ ⃗ h, with ⃗ h being the magnetic field and ⃗ µ I the nuclear magnetic moment, we can define the effective magnetic field caused by the axion and felt by the nucleus as where ρ DM and ⃗ v DM are density and velocity vector of DM, respectively. We have used ⃗ ∇a ≈ m⃗ v DM a(t) and assumed that DM is composed solely of axions, hence ρ DM = m 2 a a 2 0 2 where a 0 is the amplitude of axion coherent oscillation.
The remaining task is to find a spin contribution for the case of magnetic isotope of interest; here we focus on 55 Mn. For example, we can adopt the semi-empirical approach following Engel and Vogel [133], by assuming that the nucleon species with even number (in this case, neutrons) contributes little to the angular momentum of the system, and taking g I as the observed value. With g I = 1.38 as observed nuclear g-factor for 55 Mn and σ n = 0, σ ln = 0, we obtain the spin contribution as σ p ≈ 0.1, σ lp ≈ 0.9.
However, there are some uncertainties in the above semi-empirical estimation. One of them comes from the deformed shell structure of the 55 Mn nucleus [105,134], which implies the collective rotation of the nucleus in addition to the intrinsic state of nucleons. This can affect the ratio of the spin and orbital contribution to nuclear spin. In particular, concerning the z-direction contribution to the magnetic moment, the nuclear magnetic moment is given by [135,136] The first term is the magnetic moment from the angular momentum of proton in the state Ω⟩, while the second term accounts for the collective rotation of nucleons with the rotational g-factor g R . Due to the deformed configuration of 55 Mn nucleus, particles experience axiallysymmetric potential (with z being the axial direction), and degenerate spherical shells split. Almost all nucleons couple in pair-wise contributing little to the angular momentum, and the last odd proton occupies the Nilsson state with the asymptotic quantum number [312]5 2 − . (For the Nilsson state [N n z Λ]Ω π of a nucleon [135], N is the total quantum number, n z the number of oscillator quanta in the z-direction, Λ the orbital angular momentum in the z-direction, Ω the total angular momentum in the z-direction, and π the parity.) This last odd proton and collective rotation of nucleons contribute to the nuclear spin I = 5 2. For the ground state of 55 Mn, the state of the last proton can be approximated by Ω⟩ = l z = 2, S z = 1 2⟩. Since ⟨Ω µ z Ω⟩ = 1 2 g p + 2g lp , (A.8) and the rotation g-factor g R of nucleus should not have a contribution from the spin of the nucleons, we know that σ p = 0.5 (I + 1) ≈ 0.14 is the proton-spin contribution realizing the magnetic moment µ. The value of σ p from this deformed shell model is slightly higher than the one obtained from the simple semi-empirical model. Another uncertainty is the neutron-spin contribution which has been neglected so far. For the axion coupling to the nuclear spin, the above two approximations imply that the 55 Mn nucleus is sensitive to the axion-proton coupling constant g app , but not to the axionneutron coupling g ann . However, in reality there might also be a significant spin contribution from the core neutrons through its spin polarization via the spin-spin interaction [137]. It may make the 55 Mn nucleus also sensitive to the axion-neutron coupling. In this paper, we neglect this effect for simplicity.
In this work, we use the value to numerically estimate the magnitude of the axion-induced magnetic field acting on the 55 Mn nuclear spin.

B Detailed calculation of magnetic dynamics in MnCO 3
In this Appendix we calculate the magnetic dynamics of MnCO 3 under the DM background.
In this Appendix and the following, SI unit is adopted with vacuum permeability factor µ 0 is omitted in the formulae for convenience.
To estimate the eigensystem of the magnetic system in MnCO 3 , we first note that there is the orders-of-magnitude relation of the effective fields and other constants: where δ ∼ 10 −2 roughly accounts for the hierarchy among these parameters. Below we will consider only the leading order result in δ, while δ appearance is omitted for notational simplicity.
Recall that potential of the system and equation of motions are given by Eqs. (3.5) and (3.17). The dynamics of magnetic system concerning only in-phase modes following Eqs. (3.19) and (3.21) reads where ⃗ u ≡ (M x + , M y + , m x + , m y + ) T represents a collect of the perturbation of magnetization parameters defined by Eq. (3.18), and at leading order Term ⃗ R represents relaxation of the magnetizations. ⃗ D is source term due to the oscillating magnetic field induced by DM and is given by In the case that ⃗ R = 0 and ⃗ D = 0, with the ansatz the equation of motion reads iω⃗ u = Ω⃗ u of which the solution give eigenvalues of the precession mode of magnetization as ω 1,3 = ±ωñ ,+ , ω 2,4 = ±ωẽ ,+ , (B.8) where ωñ ,+ and ωẽ ,+ are given by Eqs. (3.31) and (3.30). The corresponding eigenmodes are up to an arbitrary normalization factor. Now, we take into account the damping effect ⃗ R and source term ⃗ D. We adopt here for simplicity relaxation term of the form where T 1,3 = T 2n and T 2,4 = T 2e denoting the relaxation time scale of nuclear-dominated mode and electron-dominated mode, respectively. With ansatz the equation of motion reduces to the problem finding c i 's satisfying The signal magnetization (M signal ≡ M z total ) is then given by When ω DM ∼ ω 1 , the i = 1 term is dominant with corresponding steady-state solution of the form It is approximately the solution at time t ≳ T 2n after begin excited. On the other hand, when ω DM ∼ ω 2 the i = 2 term is dominant with steady-state solution of the form which is a good approximation for the solution at the time t ≳ T 2e after being excited. That is the resonance occurs when ω DM − ωñ ,+ ≲ 1 T 2n or ω DM − ωẽ ,+ ≲ 1 T 2e . Note that when t ≲ τ DM , the magnetization rotates with a coherent phase with frequency ω DM , where τ DM is the coherence time of DM assumed to be larger than T 2n,2e . Due to the velocity distribution of DM, the signal spread with bandwidth ∆ω DM in the frequency space. For power estimation, we take into account the relevant component of magnetization and use Eq. (3.39). The signal susceptibility and power are listed in table 2, 3, and 4, 5, respectively. If one use relaxation term R = (−1 τ 2n , − τ 2n , − τ 2e , −τ 2e ) T of Bloch's kind, the effective relaxation time T 2n , T 2e can be expressed as T 2n = τ 2n τ 2e (H a γ e ω n (2H E γ e ω n − ω 2 e,+ ) + ω 4 e,+ ) τ 2e ω 4 e,+ + H a τ 2n γ e ω n (2H E γ e ω n − ω 2 e,+ ) , (B.16) T 2e = τ 2n τ 2e (H a γ e ω n (2H E γ e ω n − ω 2 e,+ ) + ω 4 e,+ ) τ 2n ω 4 e,+ + H a τ 2e γ e ω n (2H E γ e ω n − ω 2 e,+ ) . (B.17)

C Nuclear magnon picture
In this Appendix, we discuss the dynamics of magnetic system of MnCO 3 using the quantum magnon picture. Based on the following derivation, we check that the response derived in the magnon picture is consistent with that derived in the classical picture illustrated in the main text (Sec. 3).

C.1 Hamiltonian in the magnon picture
The following is the Hamiltonian for the spin of MnCO 3 in the antiferromagnetic phase [63,93,99,102]: where i and j represent lattice cites in two sublattices; J, K, K ′ andÃ hy are positive constants, and γ n , γ e are gyromagnetic ratios for nuclear and electron spin, respectively. The Hamiltonian includes (1) isotropic exchange interaction between electron spins of nearest sites between ⃗ S i and ⃗ S j ; (2) Dzyaloshinskii-Moriya interactions; (3) easy-plane anisotropy; (4) in-plane uniaxial anisotropy; (5) Zeeman effects for electron spins ⃗ S and nuclear spins ⃗ I; and (6) hyperfine interaction between electron spins ⃗ S and nuclear spins ⃗ I. Here, we discuss the situation where there are static field applied in x direction and exotic oscillating field induced by DM of which ⃗ h e (t) interacting with electron spin and ⃗ h n (t) interacting with nuclear spin. The coordinate is set to coincide with situation discussed in the main text using classical theory. The relations between spins and magnetizations are given by Eqs. (3.2) and (3.3). It reduces to the classical Hamiltonian (3.5) when all the spins in each sublattice are aligned, which gives the relation Note that the direction between magnetization and spin of electron is opposite while we take gyromagnetic ratio parameter γ e , γ n to be positive.
In the ground state, electron spins align in the xz plane pointing (almost in) to +z and −z direction for each sublattice due to easy-plane and in-plane anisotropy. They slightly tilt toward −x direction due to the applied static field H 0 and the Dzyaloshinskii-Moriya interactions. For nuclear spins, they align opposite to electron spin of the same Mn due to hyperfine interaction. The configuration (of magnetizations) is shown in Fig. 1.
For convenience to consider fluctuation around ground state, we rotate our spin parameters for each sub-lattice: where the ground state expectation value gives ⟨S z 1 i ⟩ = ⟨S z 2 j ⟩ = −S = −5 2 and ⟨S x 1,2 ,y i,j ⟩ = 0. The coordinate transformation is shown in Fig. 2. Due to hyperfine interaction, at ground state nuclear spins lie in the opposite direction to the electron spins of the same sites and similarly form two sub-lattices. We perform the same coordinate transformation for nuclear spins ⃗ I (Eq. (C.4) but with S ↔ I). In this case, the thermal expectation value for nuclear spin temperature T is ⟨I z 1 i ⟩ = ⟨I z 2 j ⟩ = ⟨I⟩ (C.5) given by Eq. (3.11), and ⟨I x 1,2 ,y i,j ⟩ = 0. The function ⟨I⟩ I is shown in Fig. 3. Now the hyperfine interaction can be written in the form The term H ∥ represents the hyperfine interaction in the direction of spin alignment in the ground state, which make both the eigenfrequency of nuclear and electron spin precession higher. On the other hand, the term H mix causes the mixing between the nuclear and electron spin precession modes. Then we perform the Holstein-Primakoff transformation for expressing perturbations in terms of magnons: for the first sublattice, where a, a † and c, c † are magnon annihilation, creation operators satisfying bosonic commutation relation. Here, we consider only small perturbation and hence the higher order of creation/annihilation in the Holstein-Primakoff are neglected. We perform the same transformation for the other sublattice by using b j and d j representing the electron and nuclear magnon operators respectively. We also perform Fourier transformation: with N denoting number of spin sites in each sublattice. However, we focus only on the ⃗ k = 0 mode since it is dominantly excited by DM which is assumed to be almost spatially uniform. As long as only the k = 0 mode is considered, we simply omit the subscript k for the annihilation and creation operators.

C.2 Diagonalization
The quadratic terms in the Hamiltonian can be divided into several terms as H = H e + H n + H mix + H DM , (C.11) where H e is the electron-magnon term including H ∥ part of the hyperfine interaction, H n is the nuclear-magnon term, H mix is the mixing term given by Eq. (C. 8), and H DM is the interaction term between the spin and DM. where ω n =Ã hy S ̵ h. (C.14) The interaction Hamiltonian between the DM and spin is with the following elements: The diagonalized Hamiltonian is written as H e ̵ h = ω e,− α † α + ω e,+ β † β, (C.20) where ω e,− and ω e,+ are equal to those given by Eqs. (3.25) and (3.26), respectively. Magnons α and β correspond to the out-phase and in-phase mode of electron-spin precession, respectively (see Sec. 3.2). Because the eigenenergy of out-phase mode is far from that of the nuclear magnon, its mixing with the nuclear magnon is expected to be small. We then focus only on the in-phase mode β.

Nuclear magnons and electron magnons mix with each other by the precession component of hyperfine interaction:
H mix = −Ã hy S⟨I⟩(ac + a † c † + bd + b † d † ). (C.21) When we focus only on the in-phase mode β of the electron magnon, we obtain H mix = −Ã hy S⟨I⟩ (Q 12 β + Q 14 β † )η + (Q 12 β † + Q 14 β)η † , (C. 22) where η ≡ (c + d) Let us focus on the in-phase magnons. First neglecting the DM part, the Hamiltonian of interest is given by where Y ≡ (β η β † η † ) T and W ≡ βηβ †η † T . We find that matrix R that preserves canonical quantization of magnon operators is in the form (C. 28) with the elements given by , (C.31) and Note that ω 1 = ωẽ ,+ and ω 2 = ωñ ,+ , which are equal to those given by Eqs. For convenience of later discussion, we define the following mixing angle matrix φ ± : which can be obtained from matrix R defined in Eq. (C.28).

(C.38)
One can then apply the linear excitation theory to find expectation value of operators in Eq. (C.37), and hence magnetization signal. Focusing on one particular mode ζ =β,η, the Hamiltonian is (C.39) where g ζ is coupling constant depending on the amplitude of DM-induced field ⃗ h e , ⃗ h n (see Eqs. (2.15),(2.16) and (2.26)), while ω DM is equal to axion mass m a or dark photon mass m γ ′ . The Heisenberg equation of motion reads with T 2ζ representing the relaxation time of precession mode ζ which is twice a value of the magnon lifetime. The stationary steady-state solution for this system is given by which is a good approximation for the solution at t ≳ T 2ζ . Substituting the expectation value to the expression of magnetization signal (Eq. (C.37)), one can derive the response of the system to DM. Due to velocity distribution of DM, the magnetization signal spreads with width ∆ω DM determined by that of DM field. We find that the response of the system derived from quantum magnon picture is consistent to that of classical theory given in table 2 and 3. On the other hand, the power of the system can be derived from the relation P = ̵ hω ζ ⟨n ζ ⟩2 T 2ζ where n ζ = ζ † ζ is the number operator.