Axion search with quantum nondemolition detection of magnons
Tomonori Ikeda, Asuka Ito, Kentaro Miuchi, Jiro Soda, Hisaya Kurashige, Yutaka Shikano
NNCTS-QIQC/2101, KOBE-COSMO-21-03, APS/123-QED
Axion search with quantum nondemolition detection of magnons
Tomonori Ikeda ∗ Department of Physics, Kyoto University, Kita-Shirakawa, Sakyo-ku, Kyoto 606-8502, Japan
Asuka Ito † Physics Division, National Center for Theoretical Sciences,National Tsing-Hua University, Hsinchu, 30013, Taiwan
Kentaro Miuchi, ‡ Jiro Soda, § and Hisaya Kurashige Department of Physics, Kobe University, Rokkodaicho, Nada-ku, Hyogo 657-8501, Japan
Yutaka Shikano ¶ Quantum Computing Center, Keio University, Hiyoshi,Kohoku-ku, Yokohama, Kanagawa 223-8522, JapanInstitute for Quantum Studies, Chapman University, CA 92866, USA andResearch Center for Advanced Science and Technology,The University of Tokyo, Meguro-ku, Tokyo 153-8904, Japan (Dated: February 26, 2021)The axion provides a solution for the strong CP problem and is one of the leading candidatesfor dark matter. This paper proposes an axion detection scheme based on quantum nondemolitiondetection of magnon, i.e., quanta of collective spin excitations in solid, which is expected to beexcited by the axion–electron interaction predicted by the Dine-Fischer-Srednicki-Zhitnitsky (DFSZ)model. The prototype detector is composed of a ferrimagnetic sphere as an electronic spin targetand a superconducting qubit. Both of these are embedded inside a microwave cavity, which leads toa coherent effective interaction between the uniform magnetostatic mode in the ferrimagnetic crystaland the qubit. An upper limit for the coupling constant between an axion and an electron is obtainedas g aee < . × − at the 95% confidence level for the axion mass of 33 . µ eV < m a < . µ eV. PACS numbers: 95.35.+d, 42.50.Dv
I. INTRODUCTION
The Standard Model successfully predicted the ex-istence of Higgs bosons [1–3]. However, several long-standing problems in particle physics remain to be ex-plained beyond the Standard Model or its alternative.For example, the theory of quantum chromodynamicsrequires an extremely fine tuning of parameters to ex-plain the experimentally-observed electric dipole momentof the neutron [4, 5]. This issue is considered to be thestrong CP problem. To solve this, Peccei and Quinnproposed a global U (1) symmetry that is broken at ahigh energy scale F a allowing for the restoration of theCP symmetry, which consequently gives rise to a newpseudoscalar boson called axion [6–8]. Axions can pro-vide a significant fraction of the dark matter (DM) [9].Therefore, axion DM research is an important field forastrophysics and physics beyond the Standard Model.The axion model is classified into the Kim-Shifman-Vainshtein-Zakharov (KSVZ) model [10, 11], where ax- ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] ions couple with photons and hadrons, and the Dine-Fischler-Srednicki-Zhitnitsky (DFSZ) model [12, 13],where axions also couple with electrons [14]. Most ex-periments for the axion DM research conducted so farwere based on the axion–photon coupling through thePrimakoff effect [15]. Using this approach, the ADMXexperiment excluded the axion mass range of 1 . µ eV In this section, the theory of axion–electron interactionis first introduced. The axion-induced effective magneticfield generated by the movement of the Earth throughaxion DM is discussed. After introducing collective spinexcitations in a ferrimagnetic crystal, the QND detectionscheme of magnons for the axion DM search is described.This is achieved by measuring the absorption spectrumof a superconducting qubit dispersively coupled to the Kittel mode in the ferrimagnetic crystal. A. Axion-electron interaction The axion emerges as a Nambu-Goldstone boson ofthe broken Peccei-Quinn symmetry [7, 8]. In the DFSZmodel [14], the axion field a ( x ) can interact with an elec-tron field ψ ( x ) as L int = − ig aee a ( x ) ¯ ψ ( x ) γ ψ ( x ) , (2)where g aee is a dimensionless coupling constant, inverselyproportional to the energy scale of the Peccei-Quinn sym-metry breaking. Note that we can rewrite the interactionof Eq. (2) as ˜ g aee ( ∂ µ a ) ¯ ψγ µ γ ψ ( x ) using the backgroundDirac equation, where ˜ g aee = g aee / m e . This clearlyshows the shift symmetry of the axion field. In the non-relativistic limit, the interaction term reads H int (cid:39) − g aee (cid:126) m e ˆ σ · ∇ a = − µ B ˆ S · (cid:16) g aee e ∇ a (cid:17) , (3)where m e is the electron mass, e is the elementary elec-tric charge, µ B = e (cid:126) / m e is the Bohr magneton. Theelectron spin operator ˆ S is related to the Pauli matri-ces ˆ σ with ˆ S = ˆ σ / 2. The term in the parentheses canbe considered as an effective magnetic field B a = g aee e ∇ a. (4)If the DM is composed of axions, this effective magneticfield is ubiquitous around us. Importantly, the axion DMis oscillating in time at the frequency f a that is relatedto its mass according to Eq. (1). B. Axion-induced effective magnetic field It is assumed that the axion DM forms solitonic ob-jects, oscillating coherently with the frequency (1) insideof them [29, 30]. The radius of such axion clumps, namelythe Jeans length r ob , can be estimated by applying thevirial theorem to the object. This is based on the as-sumption that the Jeans length is roughly equal to thede Broglie wavelength of the axion. This leads to r ob ∼ . × (cid:18) . µ eV m a (cid:19) / (cid:18) . 45 GeV / cm ρ ob (cid:19) / m , (5)where ρ ob is the energy density of the object. Here, ρ ob = 0 . 45 GeV / cm is assumed to be the local DM density,although it could be several orders of magnitude higher. As the objects are moving with the virial velocity v in theGalaxy (see Appendix. A for detail), where the coherence time is estimated as t ob ∼ r ob v (cid:39) . × × (cid:18) . µ eV m a (cid:19) / (cid:18) . 45 GeV / cm ρ ob (cid:19) / (cid:18) 300 km / s v (cid:19) s . (6)Notably, the coherence time, during which the effec-tive magnetic field is coherent, is considered to be muchlonger than the observation time for the present experi-ment.For the coherently oscillating object, it is expected thatthe distribution of the axion field is almost homogeneousdue to the uncertainty principle. In particular, the axionfield cannot vary in the length scale smaller than its de Broglie wavelength. Even then, the spatial gradient ofthe axion field is nonzero, as the object is moving withthe velocity v relative to the laboratory frame [31]. Wethen have a relation ∂ i a (cid:39) m a va because the proper timeof the moving object depends on the coordinates of ourframe through the Lorentz transformation. The ampli-tude of the effective magnetic field B a can be estimatedfrom Eq. (4) and the relation ρ ob ∼ m a a / 2, resulting in B a (cid:39) . × − × g aee (cid:18) ρ ob . 45 GeV / cm (cid:19) / (cid:18) v 300 km / s (cid:19) T . (7)The amplitude is negligible if we consider the astro-physics constraint of g aee (cid:46) − from the bounds onthe cooling of white dwarfs [32]. However, as demon-strated later, the coupling constant can effectively be en-hanced for collective spin excitation modes in ferrimag-netic crystals. C. Collective spin excitations Let us consider a ferrimagnetic crystal containing N electron spins. This system is described by the Heisen-berg model [33]ˆ H m − a = gµ B (cid:88) i ˆ S i · ( B + B a ) − (cid:88) i,j J ij ˆ S i · ˆ S j , (8)where, B is the external magnetic field, g is the g-factor for electron spins in the ferrimagnetic crystal, and i labels each spins. The second term represents the ex-change interaction between the neighboring spins withthe strength J ij . Considering an external magnetic field B along the z -axis and assuming, without loss of gen-erality, that the direction of the effective magnetic field B a ( B a (cid:28) B ) lies in the z - x plane, we write B = (0 , , B ) , B a (cid:39) ( | B a | sin θ, , . (9)Here, θ is the angle between the external and effectivemagnetic fields.The effective magnetic field is considered to be uni-form throughout the sample during the typical obser-vation time. Thus, the effective magnetic field can bewritten as B a ( t ) = B a sin θ (cid:0) e − iω a t + e iω a t (cid:1) (1 , , . (10)Substituting Eqs. (9) and (10) into Eq. (8) yieldsˆ H m − a = gµ B (cid:88) i (cid:20) ˆ S zi B + B a sin θ (cid:16) ˆ S − i e − iω a t + ˆ S + i e iω a t (cid:17)(cid:21) − (cid:88) i,j J ij ˆ S i · ˆ S j , (11) where S ± j = S xj ± iS yj are the spin ladder operators. Thesecond term in the right-hand side of Eq. (11) shows thatthe axion DM excites the spins if the frequency of the ef-fective magnetic field ω a is equal to the Larmor frequency ω m ≡ gµ B B / (cid:126) .The spin system of Eq. (11), including the axion-induced effective magnetic field, can be rewritten in termsof the bosonic operators ˆ C i and ˆ C † i , which satisfies thecommutation relation [ ˆ C i , ˆ C † j ] = δ ij , using the Holstein-Primakoff transformation [34]:ˆ S zi = 12 − ˆ C † i ˆ C i , ˆ S + i = (cid:113) − ˆ C † i ˆ C i ˆ C i , (12)ˆ S − i = ˆ C † i (cid:113) − ˆ C † i ˆ C i . This introduces spin waves with a dispersion relationdetermined by the amplitude of the external magneticfield B and the amplitudes J ij of the ferrimagnetic ex-change interaction. Magnons are quanta of the spin-wavemodes. Furthermore, provided that the contributionsfrom the surface of the sample are negligible, one canexpand the bosonic operators in terms of plane waves asfollows: ˆ C i = 1 √ N (cid:88) k e − i k · r i ˆ c k . (13)Here, r i is the position vector of spin i , and ˆ c k anni-hilates a magnon from the mode with a wave vector k .As the effective magnetic field is induced by axions, itis supposed to be homogeneous over the ferrimagneticcrystal for a typical experiment. Then, only the uniformmagnetostatic mode, Kittel mode, can be excited as longas the contributions from the surface are negligible [35].Substituting Eqs. (12) and (13) into the Hamiltonian ofEq. (11) yieldsˆ H m − a ≡ ˆ H m + ˆ H a , (14)ˆ H m = (cid:126) ω m ˆ c † ˆ c, (15)ˆ H a = gµ B B a sin θ √ N (cid:0) ˆ c † e − iω a t + ˆ ce iω a t (cid:1) , (16)where ˆ c ≡ ˆ c k =0 and n m ≡ (cid:104) ˆ c † ˆ c (cid:105) (cid:28) N are assumed. Onecan see that the coupling strength is enhanced by a factorof √ N . This effect enables one to potentially detect thesmall effective magnetic field B a oscillating at ω a . D. Quantum nondemolition detection of magnons To detect the excitation of a magnon in the Kittelmode, a hybrid system schematically shown in Fig. 1is used. This system consists of a spherical ferrimag-netic crystal and a superconducting qubit [36]. Theyare individually coupled to the modes of a microwavecavity through magnetic and electric dipole interactions,respectively. This leads to an effective coherent cou-pling between the Kittel mode and the qubit [28, 37–40].The transmon-type superconducting qubit [41] can be de-scribed as an anharmonic oscillator through the Hamil-tonian ˆ H q / (cid:126) = (cid:16) ω q − α (cid:17) ˆ q † ˆ q + α (cid:0) ˆ q † ˆ q (cid:1) , (17)where ω q is the frequency of the transition between theground and first-excited states of the qubit, | g (cid:105) and | e (cid:105) ,respectively. The creation and annihilation operators forthe qubit are respectively ˆ q † and ˆ q . Furthermore, the an-harmonicity α < ω q + α [42]. After adiabaticallyeliminating the microwave cavity modes from the totalHamiltonian of the hybrid system, the effective interac-tion Hamiltonian between the Kittel mode and the qubitis given by ˆ H q − m / (cid:126) = g q − m (ˆ q † ˆ c + ˆ q ˆ c † ) , (18)where g q − m is the coupling strength between the Kittelmode and the qubit [28, 37–39].Combining Eqs. (14), (17) and (18), the Hamiltonianof the hybrid quantum system, including the effectiveaxion-induced effective magnetic field, is given by:ˆ H tot / (cid:126) = ω m ˆ c † ˆ c + (cid:16) ω q − α (cid:17) ˆ q † ˆ q + α (cid:0) ˆ q † ˆ q (cid:1) + g q − m (ˆ q † ˆ c + ˆ q ˆ c † ) + g eff (cid:0) ˆ c † e − iω a t + ˆ ce iω a t (cid:1) . (19)Here, (cid:126) g eff = gµ B B a sin θ √ N , (20) is the effective coupling constant between axions andmagnons, which corresponds to the strength of the co-herent magnon drive.Let us consider the dispersive regime correspondingto a detuning ∆ q − m ≡ ω q − ω m between the qubit fre-quency ω q and the frequency of the Kittel mode ω m .That is much larger than the coupling strength g q − m suchthat the exchange of energy between the two systems ishighly suppressed [28]. For this limit, the total Hamilto-nian of Eq. (19) can be rewritten asˆ H (cid:48) tot / (cid:126) (cid:39) ω m ˆ c † ˆ c + 12 ˜ ω q ˆ σ z + χ q − m ˆ c † ˆ c ˆ σ z + g eff (cid:0) ˆ c † e − iω a t + ˆ ce iω a t (cid:1) , (21)where ˜ ω q = ω q + χ q − m is the qubit frequency shiftedby the qubit–magnon dispersive shift χ q − m , which is de-scribed by [42]. χ q − m (cid:39) αg − m ∆ q − m (∆ q − m + α ) . (22)The qubit–magnon dispersive shift can also be estimatednumerically by diagonalizing the Hamiltonian of the hy-brid system [28]. The Hamiltonian of Eq. (21) consid-ers only the first two states of the qubit through thePauli matrices: ˆ σ z = | e (cid:105) (cid:104) e | − | g (cid:105) (cid:104) g | , ˆ σ + = | e (cid:105) (cid:104) g | ,and ˆ σ − = | g (cid:105) (cid:104) e | . Furthermore, higher order terms in( g q − m / ∆ q − m ) are neglected. The third term on the right-hand side of Eq. (21) shows that the qubit frequency de-pends on the magnon occupancy through an interactionterm, which commutes with the Hamiltonian of the Kittelmode. More specifically, the qubit frequency ˜ ω q shifts by2 χ q − m for every magnon in the Kittel mode. Therefore,measuring the qubit frequency enables one to perform aQND detection for the magnon number. E. Qubit spectrum The qubit frequency can be determined, for example,by measuring its absorption spectrum S ( ω s ) = Re (cid:20) √ π (cid:90) ∞ d t (cid:104) ˆ σ − ( t )ˆ σ + (0) (cid:105) e iω s t (cid:21) , (23)where ω s is the spectroscopy frequency [42]. In this sub-section, an analytical model for the qubit spectrum inthe presence of a dispersive interaction with the Kittelmode of a ferrimagnetic crystal is provided [28, 42]. Forthis purpose, the Hamiltonian of Eq. (21) is transformedsuch that the qubit is in a reference frame that rotatesat the spectroscopy frequency ω s . Meanwhile, the Kittelmode is in a reference frame that rotates at the axionfrequency ω a . The Hamiltonian of Eq. (21) becomesˆ H / (cid:126) = (∆ a + χ q − m ) ˆ c † ˆ c + 12 ∆ s ˆ σ z + χ q − m ˆ c † ˆ c ˆ σ z + g eff (cid:0) ˆ c † + ˆ c (cid:1) + Ω s (ˆ σ + + ˆ σ − ) , (24) Ferrimagneticcrystal Superconducting qubitMicrowave cavity 𝑩 𝒂 𝑩 𝝌 𝒈 SN q-m eff FIG. 1. Schematic illustration of the detector. A spheri-cal ferrimagnetic crystal and a transmon-type superconduct-ing qubit are coherently coupled through a microwave cav-ity. The effective magnetic field of the axion DM coher-ently drives the uniform spin-precession mode (Kittel mode)in the ferrimagnetic crystal with an effective coupling con-stant g eff . In the strong dispersive regime, each magnon ex-cited in the Kittel mode shifts the resonance frequency of thequbit by 2 χ q − m + ∆ a , where χ q − m is the dispersive shift and∆ a = ω g m − ω a is the detuning between the frequency ω a ofthe axion-induced effective magnetic field and the frequency ω g m of the Kittel mode with the qubit in the ground state | g (cid:105) . where ∆ a = ω g m − ω a is the detuning between the fre-quency ω g m of the Kittel mode with the qubit in theground state | g (cid:105) and the axion frequency ω a . In ad-dition, ∆ s = ˜ ω q − ω s is the detuning between the fre-quency ˜ ω q of the qubit and the spectroscopy frequency ω s . The amplitudes of the driving terms are given by theeffective coupling constant g eff and the Rabi frequencyΩ s , respectively. When considering the decoherences forthe Kittel mode and the qubit that are characterized bylinewidths γ m and γ q , respectively, the qubit spectrum isgiven by [42] S ( ω s ) = 1 π ∞ (cid:88) n m =0 n m ! Re ( − A ) n m e A γ ( n m )q / − i (cid:16) ω s − ˜ ω ( n m )q (cid:17) , (25)with ω ( n m )q = ˜ ω q + B + 2 χ q − m n m , (26)˜ ω ( n m )q = ω ( n m )q + n m ∆ a , (27) γ ( n m )q = γ q + γ m ( n m + D ss ) , (28) A = D ss (cid:18) γ m / − i (∆ a + 2 χ q − m ) γ m / i (∆ a + 2 χ q − m ) (cid:19) , (29) B = χ q − m ( n g m + n e m − D ss ) , (30) D ss = 2 ( n g m + n e m ) χ − m ( γ m / + χ − m + ( χ q − m + ∆ a ) , (31) n g m = g ( γ m / + ∆ a , (32) n e m = g ( γ m / + (∆ a + 2 χ q − m ) . (33) Here, ˜ ω ( n m )q and γ ( n m )q are respectively the frequency andthe linewidth of the qubit with the Kittel mode in thenumber state | n m (cid:105) for a given effective coupling con-stant g eff . Equations (31) [(32)] expresses the steady-state magnon occupancy when the qubit is in the ground(excited) state, which is given by n g ( e )m .In the strong dispersive regime, which corresponds to2 χ q − m (cid:29) γ m , the qubit spectrum is given by a sum ofLorentzian functions centered at the shifted qubit fre-quency ˜ ω ( n m )q . The spectrum explicitly depends on thepopulation of the magnons potentially excited by theaxion-induced effective magnetic field B a . Therefore, theaxion DM can be probed by measuring the spectrum ofthe qubit that is coupled to the Kittel mode in the ferri-magnetic crystal.Figure 2 shows the expected qubit spectrum for thedifferent values of the coupling constant g aee consid-ering a spherical ferrimagnetic crystal of yttrium irongarnet (Y Fe O , YIG). This crystal is considered tohave a diameter of 0 . ∼ . × cm − and the following realistic parame-ters: ω g m / π = 8 . ω a / π = 8 . m a ≈ µ eV), ˜ ω q / π = 8 . γ m / π = 1 . γ q / π = 0 . χ q − m / π =10 MHz. For the calculation, the magnon number is trun-cated at n m = 10, which is justified for n g ( e )m (cid:28) 1. With-out the axion-induced magnetic field ( g aee = 0), onlyone peak appears at the qubit frequency ˜ ω (0)q = ˜ ω q . Onthe other hand, for g aee = 5 . × − , a second peak at˜ ω (1)q ≈ ˜ ω (0)q + 2 χ q − m appears. This second peak corre-sponds to the excitation of a single magnon in the Kittelmode due to the axion DM. Therefore, the observationof a peak around the frequency ˜ ω (1)q demonstrates theexistence of the axion DM. III. EXPERIMENT In Ref. [28], owing to the strong dispersive regime be-tween the uniform magnetostatic mode of the spheri-cal ferrimagnetic crystal and a superconducting qubit,quanta of the collective spin excitations were observedunder an additional resonant drive of the Kittel mode.Here, the data without an intentional drive is analyzedfor the search of axion DM. More details about the ex-periment can be found in Ref. [28]. A. Device and parameters As depicted in Fig. 1, the hybrid quantum systemconsists of a superconducting qubit and a single crys-talline sphere of YIG, both inside a three-dimensionalmicrowave cavity. The diameter of the YIG sphere is0 . B ≈ . 29 T p /2 s w Spectroscopy frequency 00.51 ) s w N o r m a li z ed qub i t s pe c t r u m S ( p /2 w~ p /2 w~ p /2 w~p )/2 a D + q-m c (2 = 0 aee g -7 · = 5.0 aee g FIG. 2. Normalized qubit spectra S ( ω s ) in the absence andpresence of a coupling with axions, shown as the blue ( g aee =0) and red ( g aee = 5 . × − ) lines, respectively. The pa-rameters used here are ω g m / π = 8 . ω a / π = 8 . ω q / π = 8 . γ m / π = 1 . γ q / π = 0 . χ q − m / π = 10 MHz. for the YIG sphere. The transmon-type superconductingqubit has a frequency ˜ ω q / π ≈ . 99 GHz.The superconducting qubit and the Kittel mode ofthe ferrimagnetic crystal are coherently coupled bytheir individual interactions with the modes of the mi-crowave cavity [28, 37, 38]. The effective couplingstrength g q − m / π = 7 . 79 MHz is experimentally deter-mined from the magnon-vacuum Rabi splitting of thequbit. This coupling strength is much larger than thepower-broadened qubit linewidth γ q and the magnonlinewidth γ m [28].The frequency of the Kittel mode is set based on thecurrent in the coil to reach the dispersive regime of theinteraction between the uniform mode and the qubit.The amplitude of the detuning | ∆ q − m | = | ˜ ω q − ω g m | ismuch larger than the coupling strength g q − m . The qubit-magnon dispersive shift χ q − m / π = 1 . ± . ω g m / π = 7 . B. Measurements and results Here we analyze the qubit spectrum obtained in theexperiment of Ref. [28] without an intentional excita-tion of the Kittel mode. The qubit absorption spectrumshown in Fig. 3 was measured in the frequency range7 . < ω s / π < . S ( ω s ) = A (cid:88) n m =0 S n m ( ω s ) + S off , (34)where A is a scaling factor, S off is an offset, and S n m ( ω s ) is the contribution of the qubit spectrum with n m magnons excited in the uniform magnetostatic mode.The small asymmetry of the qubit spectrum correspond-ing to n m = 0 is due to the finite photon occupancy of themicrowave cavity mode that is used to probe the qubit.The effect of the photon occupation in the probe modecan be considered as follows: S n m ( ω s ) ≈ S n m ,n p =0 ( ω s ) + B × S n m ,n p =1 ( ω s ) , (35)where B = 0 . 03 is the relative spectral weight betweenthe one-photon and the zero-photon peaks. To considerthe ac Stark shift of the qubit frequency by the photonsin the probe mode used to measure the qubit spectrum,the following is substituted as˜ ω q → ω ( n p =0)q = ˜ ω q + B p , (36)where ω ( n p =0)q is the ac-Stark-shifted qubit frequencywith the Kittel mode in the vacuum state. The qubitlinewidth with the Kittel mode in the vacuum state issubstituted to γ q → γ ( n p =0)q = γ q + κ p D ssp , (37)where γ ( n p =0)q is the linewidth increased by themeasurement-induced dephasing from photons in theprobe mode. The parameters fixed in the fit of thequbit spectrum are the qubit frequency ˜ ω q , the qubitlinewidth γ q , the probe mode occupancy n g p , the qubit–probe-mode dispersive shift χ q − p , the probe cavity-mode linewidth κ p , the qubit–magnon-mode dispersiveshift χ q − m , the magnon linewidth γ m , and the drive de-tuning ∆ a .To consider the uncertainties of the experimentallymeasured parameters, the chi-square function χ was de-fined with nuisance parameters α j as χ ≡ n (cid:88) i =0 ˜ S i − ˜ S (cid:16) ω ( i )s , γ (cid:48) q , n g p (cid:48) , χ (cid:48) q − p , κ p (cid:48) , χ (cid:48) q − m , γ (cid:48) m (cid:17) σ ˜ S i + (cid:88) j =0 α j . (38) TABLE I. Parameters determined in the experiment. The error ranges indicate the 95% confidence interval.Parameter Symbol ValueDressed magnon frequency ω g m / π ω ( n p =0)q / π γ ( n p =0)q / π ± κ p / π ± γ m / π ± χ q − p / π − . ± . χ q − m / π ± n g p ± with γ (cid:48) q = γ q − α σ γ q ,n g p (cid:48) = n g p − α σ n g p ,χ (cid:48) q − p = χ q − p − α σ χ q − p ,κ p (cid:48) = κ p − α σ κ p ,χ (cid:48) q − m = χ q − m − α σ χ q − m ,γ (cid:48) m = γ m − α σ γ m , where ˜ S i and σ ˜ S i are the average and standard deviationof the experimentally measured spectrum for bin i fora spectroscopy frequency ω ( i )s , respectively. Systematicerrors of γ q , n g p , χ q − p , κ p , χ q − m , and γ m are given by σ γ q , σ n g p , σ χ q − p , σ κ p , σ χ q − m , and σ γ m , respectively.The results of the fitted data are presented in Fig. 3,where the average number of magnons is fixed to n g m = 0,which is equivalent to g eff = 0. From this, χ was reducedto 167.1/193. This is consistent with the null hypothesisand there was no significant excess in the residuals foundfor the frequency ˜ ω (1)q = ˜ ω (0)q + 2 χ q − m . As a result, a95% confidence level is set for the upper limit of g aee .The upper limit of the average number of magnons n limit was calculated as follows: (cid:82) n limit L d n g m (cid:82) ∞ L d n g m = 0 . , (39)where L is defined as follows: L ≡ exp (cid:18) − χ ( n g m ) − χ (cid:19) . (40)The chi-square function χ ( n g m ) is calculated by varying n g m , while χ is the minimum χ . From this, n limit =1 . × − is obtained. The expected residuals calculatedwith n limit is shown in Fig. 3.The amplitude of the effective magnetic field is givenby B a sin θ , where θ is the angle between the direction ofthe external magnetic field B and the direction of theaxion-induced effective magnetic field. From Eqs. (20)and (32), the 95%-confidence-level upper limit on the am-plitude of the effective magnetic field at m a = 33 . µ eVcan be determined as follows: B a sin θ < . × − T . (41) ) ( a r b . ) s w ( S ~ Q ub i t s pe c t r u m p /2 w~ p /2 w~ p /2 w~ p /2 w~ DataFit - D a t a - F i t p /2 s w Spectroscopy frequency FIG. 3. (a) Measured qubit spectrum ˜ S ( ω s ) (black dots) andfit to Eq. (34) (red line). (b) Residuals between the measuredqubit spectrum and the fit. The blue dotted curve showsthe expected residual at the 95%-confidence-level upper limit(¯ n g m = 0 . Using the conventional galactic density of DM, ρ DM =0 . 45 GeV/cm [43], and the expectation value of the ve-locity of an axion clump (cid:104) v tot sin θ (cid:105) = 338 km/s (see Ap-pendix A), the upper limit of the axion–electron couplingconstant can be obtained: g aee < . × − . (42)The axion mass is swept in the range of 33 . µ eV Red giantWhite dwarf coolingCAST DFSZ axion QUAXThis work16 - - - - - - - FIG. 4. Constraints on the coupling constant g aee betweenaxions and electrons. The region excluded by this workwith 95% confidence is shown in red. The mass range is33 . µ eV < m a < . µ eV. Indirect astrophysicalbounds from the solar axion search (CAST experiment [44]),white dwarf cooling [32], and red giants [45] are illustratedwith dashed lines. The expected coupling constant for theDFSZ model is represented by a solid line. is plotted in Fig. 4 and is compared with other previouslyestablished bounds on the axion–electron coupling con-stant. IV. DISCUSSION Although this work set the best upper limits for theaxion mass 33 . µ eV < m a < . µ eV with the di-rect axion–electron interaction search, several orders ofmagnitude improvements are required to reach the the-oretical predictions. We discuss some possiblilityies toimprove the sensitivity here. The most straightforwardimprovement is to increase the statistics with a longermeasurement time and optimize the measurement condi-tion. The data used for this axion search was originallytaken for a different purpose [28]. The data acquisitiontime was roughly 4 hours for the spectroscopy window[7 . − . . − . g aee ∼ − .The sensitivity can also be improved by increasing thenumber of electron spin targets. Coupling N pieces ofYIG spheres in the uniform mode to the superconductingqubit increases the effective coupling constant by a fac-tor of √ N . The QUAX experiment deployed five piecesof Gallium doped YIG spheres of 1 mm diameter andsucceeded in increasing the number of electron spin tar-gets [24]. This technique can also be used for our caseand would be expected to improve the sensitivity for thecoupling strength.As aforementioned, Refs. [24–26] showed the upperbound of the axion-electron coupling using the magnonin the spherical ferrimagnetic crystals. Even though thismethod directly measures the magnon number, the emit-ted electromagnetic radiation is measured from a mi-crowave cavity where one cavity mode is hybridized withone or multiple uniform magnetostatic modes of ferri-magnetic spheres. Therefore, the background of magnondetection is different. There are still open questions onhow to fairly compare this method with the work pre-sented here. V. CONCLUSION Magnons can be utilized for exploring the axionDM [24–26] and gravitational waves [46, 47]. In partic-ular, the QND detection of magnons was achieved usinga hybrid quantum system consisting of a superconduct-ing qubit and a spherical ferrimagnetic crystal [28]. Weapplied to the direct axion search based on the axion–electron coupling and analyzed the background data. Nosignificant signal was detected, and an upper limit of the95% confidence level was set to be g aee < . × − for the axion–electron coupling coefficient for the axionmass 33 . µ eV < m a < . µ eV. The sensitivity ispresently limited by statistics. Increasing the dispersiveshift or reducing the power-broadened qubit linewidthand the magnon linewidth will lower the upper bound onthe axion–electron coupling. ACKNOWLEDGEMENT We would like to thank Yasunobu Nakamura and DanyLachance-Quirion for providing the data used in this pa-per. We are also grateful to them for valuable comments.We acknowledge N. Crescini and C. C. Speake for theiruseful discussions. A. I. was supported by National Cen-ter for Theoretical Sciences. J.S. was supported by JSPSKAKENHI Grant Numbers JP17H02894, JP17K18778,and JP20H01902. K. M. was supported by JSPS KAK-ENHI Grant Numbers 26104005, 16H02189, 19H05806 Appendix A: Velocity of axion clump The effective magnetic field induced by the axion DMgiven by Eq. (7) is determined by the velocity of the axionclumps. 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