A Formalism for Magnon Gravitational Wave Detectors

Eur. Phys. J. C (2020) 80:545 https://doi.org/10.1140/epjc/s10052-020-8092-6

Regular Article - Theoretical Physics

A formalism for magnon gravitational wave detectors

Asuka Itoa, Jiro Sodab Department of Physics, Kobe University, Kobe 657-8501, Japan

Received: 18 April 2020 / Accepted: 27 May 2020 / Published online: 16 June 2020 © The Author(s) 2020

Abstract In order to detect high frequency gravitational verse. In the future, as the history of electromagnetic wave waves, we need a new detection method. In this paper, we astronomy tells us, multi-frequency gravitational wave obser- develop a formalism for a gravitational wave detector using vations will be required to boost the multi-messenger astron- magnons in a cavity. Using Fermi normal coordinates and omy. taking the non-relativistic limit, we obtain a Hamiltonian It is useful to review the current status of gravitational for magnons in gravitational wave backgrounds. Given the wave observations [1]. Note that the lowest frequency we can Hamiltonian, we show how to use the magnons for detect- measure is around 10−18 Hz, below which the wavelength ing high frequency gravitational waves. Furthermore, as a of gravitational waves exceeds the current Hubble horizon. demonstration of the magnon gravitational wave detector, Measuring the temperature anisotropy and the B-mode polar- we give upper limits on GHz gravitational waves by utiliz- ization of the cosmic microwave background [2,3], we can ing known results of magnon experiments for an axion dark probe gravitational waves with frequencies between 10−18 matter search. and 10−16 Hz. Astrometry of extragalactic radio sources is sensitive to gravitational waves with frequencies between 10−16 and 10−9 Hz [4,5]. The pulsar timing arrays, like Contents EPTA [6,7], IPTA [8] and NANOGrav [9], observe gravitational waves in the frequency band from 10−9 to 10−7 1 Introduction ...... 1 Hz. Doppler tracking of a space craft, which uses a mea- 2 Dirac field in Fermi normal coordinates ...... 2 surement method similar to the pulsar timing arrays, can 3 Non-relativistic limit of Dirac equation ...... 3 search for gravitational waves in the frequency band from 4 Magnon gravitational wave detectors ...... 6 10−7 to 10−3 Hz [10]. The space interferometers LISA [11] 5 Conclusion ...... 9 and DECIGO [12] can cover the range between 10−3 and Acknowledgements ...... 9 10 Hz. The interferometer detectors LIGO [13], Virgo [14], Appendix A: Fermi normal coordinates ...... 10 and KAGRA [15] with km size arm lengths can search for Appendix B: Proper detector frame ...... 11 gravitational waves with frequencies from 10 to 1 kHz. In iS −iS Appendix C: Expansion formula for e He and this frequency band, resonant bar experiments [16] are com- ∂ iS −iS ∂t e e ...... 14 plementary to the interferometers [17]. Furthermore, inter- References ...... 14 ferometers can be used to measure gravitational waves with the frequencies between 1 kHz and 100 MHz. Recently, a limit on gravitational waves at MHz was reported [18] and 1 Introduction a0.75 m arm length interferometer gave an upper limit on 100 MHz gravitational waves [19]. At 100 MHz, there is a The discovery of gravitational waves by the interferome- waveguide experiment using an interaction between gravita- ter detector LIGO in 2015 [1] opened up multi-messenger tional waves and electromagnetic fields [20]. The interaction astronomy, where electromagnetic waves, gravitational waves, of gravitational waves with electromagnetic fields is useful neutrinos, and cosmic rays are utilized to explore the uni- to explore high frequency gravitational waves and has been studied extensively [21,22]. Indeed, the interaction is utilized a e-mail: [email protected] (corresponding author) to constrain very high frequency gravitational waves higher 14 b e-mail: [email protected] than 10 Hz [23]. Although gravitational waves in the GHz 123 545 Page 2 of 15 Eur. Phys. J. C (2020) 80 :545 range are theoretically interesting [24], no detector for GHz defined by gravitational waves has been constructed. i αˆ νβˆ ν λβˆ In order to explore the GHz range, it would be useful to μ = eν σ ˆ ∂μe + λμe , (3) 2 αˆ β consider condensed matter systems. In our previous work, σ = i [γ ,γ ] we pointed out that magnons in a cavity can be utilized to where αˆ βˆ 4 αˆ βˆ is a generator of the Lorentz group μ detect GHz gravitational waves [24]. There, we gave obser- and νλ is the Christoffel symbol. vational constraints on GHz gravitational waves for the first Since there is the equivalence principle for gravity, the time. In this paper, we present the method in detail. To treat choice of coordinates is quite important. We should consider the general coordinate invariance appropriately, we need to a proper reference frame, which coincides with the coordi- use Fermi normal coordinates, or more precisely detector nates used in an experiment. Actually, the proper reference coordinates. Furthermore, we study non-relativistic fermions frame can be approximated by Fermi normal coordinates (see to reveal the interaction between magnons and gravitational Appendix A) because the effects of the earth are negligible waves. As a result, we obtain a formalism for non-relativistic for our purposes, as discussed in Appendix B. fermions in curved spacetime, including a gravitational wave In Appendix A, we have derived an explicit expression of background as a special case. Finally, as a demonstration, we the metric in Fermi normal coordinates: will give upper limits on the spectral density of continuous =− − i j , gravitational waves (95% CL): 7.5 × 10−19 [Hz−1/2] at 14 g00 1 R0i0 j x x (4) − −1/2 GHz and 8.7 × 10 18 [Hz ] at 8.2 GHz, respectively, by 2 j k g0i =− R0 jikx x , (5) utilizing results of magnon experiments conducted recently 3 [25,26]. 1 k l gij = δij − Rikjlx x , (6) The organization of the paper is as follows. In Sect. 2, 3 we study the Dirac equation in Fermi normal coordinates. In where the Riemann tensor is evaluated at x = 0 and thus it Sect. 3, we take the non-relativistic limit to obtain a Hamil- only depends on time x0. Moreover, the inverse of the metric tonian of the fermions. In Sect. 4, we explain how to use is approximately given by magnons for detecting high frequency gravitational waves. 00 =− + 0i0 j , Furthermore, we give upper limits on continuous gravita- g 1 R xi x j (7) tional waves in the GHz range. The final section is devoted 0i 2 0 jik g =+ R x j xk , (8) to the conclusion. In Appendices A and B, we review how to 3 derive Fermi normal coordinates and proper detector coordi- ij 1 ikjl g = δij + R xk xl , (9) nates, respectively. In particular in Appendix B, the reason 3 why one can neglect gravity of the earth and use the Fermi where we neglected higher order terms with respect to normal coordinates as the proper detector frames approxi- the curvature. From the metric (4)Ð(9), one can obtain the mately will be clarified. We also give a simple mathematical Christoffel symbols: formula for calculations in Appendix C. 0 0 j 0 1 k = 0 ,= R i j x ,= R ijk + R jik x , 00 0i 0 0 ij 3 0 0 i = j ,i = k , 00 R0i0 j x 0 j R0kjix 2 Dirac field in Fermi normal coordinates i 1 l = Rkijl + R jikl x . (10) jk 3 In order to study the effects of gravity on a fermion, we consider the Dirac equation in curved spacetime described The tetrad is constructed using Eq. (2): by a metric gμν.Itisgivenby αˆ αˆ 1 αˆ α k l e = δ − δα R x x , (11) 0 0 2 k0l γ αˆ μ ∂ − − ψ = ψ, i eαˆ μ μ ieAμ m (1) αˆ αˆ 1 αˆ α k l e = δ − δα R x x . (12) i i 6 kil αˆ where γ , e, m, Aμ are the gamma matrices, the elementary 0 0 1 0 0 1 j k l e = δ + δ R − δ R jk0l x x , (13) charge, the mass of the fermion, and the vector potential of αˆ αˆ 2 αˆ k0l 6 αˆ μ U( ) e 1 1 j 1 gauge theory, respectively. The tetrad αˆ satisfies ei = δi − δ0 R0 i xk xl + δ Ri xk xl . (14) αˆ αˆ 2 αˆ kl 6 αˆ kjl αˆ βˆ eμeν ηαˆ βˆ = gμν . (2) Substituting Eqs. (10)Ð(14) into Eq. (3), we can evaluate the spin connection as

Note that ηαˆ βˆ is the Minkowski metric of a local inertial 1 0ˆ iˆ j 1 iˆ jˆ k = γ γ R i j x + γ γ Rij k x , (15) frame and a hat is used for the frame. The spin connection is 0 2 0 0 4 0 123 Eur. Phys. J. C (2020) 80 :545 Page 3 of 15 545 ˆ ˆ ˆ ˆ 1 0 j k 1 j k l 0ˆ jˆ 1 k l 1 0ˆ aˆ k l i = γ γ R jikx + γ γ R jkilx . (16) + γ γ + R x x + γ γ R x x 4 0 8 0kjl jkal 2 6 μ αˆ μˆ Here we have rewritten δ γ as γ and we will do so 1 0ˆ jˆ k l 1 aˆ jˆ k l αˆ + γ γ R0k0l x x − γ γ Rak0l x x throughout. 2 6 On the other hand, the Dirac equation (1) can be rewritten −i∂ − eA j j as ˆ 1 ˆ 1 ˆ + γ 0 + γ 0 R xk xl − γ a R xk xl m . (23) 0 0 0k0l ak0l iγ ∂0ψ = iγ (0 + ieA0) 2 6 Furthermore, substituting Eqs. (15) and (16) into the above −iγ j ∂ − − ieA + m ψ j j j Hamiltonian and rearranging terms, we have = γ 0 Hψ, (17) i 0ˆ iˆ j i iˆ jˆ k H = γ γ R0i0 j x + γ γ R0ikjx where we deﬁned a Hamiltonian H and the gamma matrices 2 4 μ μ αˆ in curved spacetime γ = e γ satisfying the relation i 0ˆ iˆ jˆ kˆ l αˆ + γ γ γ γ R jkilx − eA0 μ ν μν 8 {γ ,γ }=−2g . (18) + γ 0ˆ γ iˆ δ j + θ j i i Let us express the Hamiltonian in terms of the gamma matrices of the local inertial frame instead of those of curved space- 1 k l 1 iˆ jˆ k l + R kjlx x − γ γ Rik l x x −i∂ j − eAj time. Because of γ 0γ 0 =−g00, we obtain 2 0 6 0 00 −1 00 0ˆ 1 k l 1 0ˆ iˆ k l H = (g ) ig (0 + ieA0) +γ 1 + R0k0l x x − γ γ Rik0l x x m , (24) 2 6 + γ 0γ j ∂ − − − γ 0 . i j j ieAj m (19) where we deﬁned

Using Eqs. (13) and (14), we calculate j 1 j k l 1 k l θ = δ R k l x x + R jkilx x . (25) i 2 i 0 0 6 γ 0γ j = 0 γ αˆ j γ βˆ eαˆ e ˆ β The result is consistent with the earlier work [27] where the ˆ ˆ j ˆ j ˆ Hamiltonian (24) was obtained to examine the effects of grav- = e0γ 0 + e0γ a e γ 0 + e γ b 0ˆ aˆ 0ˆ bˆ ity on an atom. ˆ ˆ 1 ˆ ˆ × = γ 0γ j − γ 0γ 0 R0 j xk xl The Hamiltonian we have obtained is the 4 4matrix 2 kl including both the particle and the anti-particle. What we 1 ˆ ˆ + γ 0γ b R j xk xl will consider is the non-relativistic fermion. To take the non- 6 kbl relativistic limit of the Hamiltonian of the fermion, we have +1γ 0ˆ γ jˆ 0 k l − 1γ aˆ γ jˆ k l . to separate the particle and the anti-particle while expanding R k l x x Rak0l x x (20) 2 0 6 the Hamiltonian in powers of 1/m. We will explicitly see Together with Eq. (7), we have how to perform this in the next section.

00 −1 0 j 0ˆ jˆ 1 k l (g ) γ γ −γ γ − R kjlx x 2 0 3 Non-relativistic limit of Dirac equation 1 0ˆ aˆ k l − γ γ R jkalx x 6 In the previous section, we derived the Hamiltonian of a Dirac 1 0ˆ jˆ k l − γ γ R k l x x 2 0 0 field in general curved spacetime with Fermi normal coordinates. Assuming that a fermion has a velocity well below 1 aˆ jˆ k l + γ γ Rak l x x . (21) 6 0 the speed of light, which is the situation we will consider in the Sect. 4, we take the non-relativistic limit of the Hamilto- Similarly, one can obtain nian. The procedure in flat spacetime is known as the FoldyÐ 00 −1 0 0ˆ 1 0ˆ k l Wouthuysen transformation [28,29]. We generalize it to the (g ) γ −γ − γ R k l x x 2 0 0 case of curved spacetime. 1 aˆ k l We first separate the Hamiltonian (24) into the even part, + γ Rak0l x x . (22) 6 the odd part and the terms multiplied by m as Therefore, from Eqs. (19), (21) and (22), the Hamiltonian = i αi j − i αi α j αk l expressed in the local inertial coordinates becomes H R0i0 j x R jkilx 2 8 0ˆ jˆ +αi δ j + θ j H = i0 + iγ γ j − eA0 i i j 123 545 Page 4 of 15 Eur. Phys. J. C (2020) 80 :545

i i j k obtain −eA0 − α α R0ikjx 4 = + , − 1 , , 1 k l 1 i j k l H H i S H S S H + R0kjlx x + α α Rik0l x x j 2 2 6 i − S, S, S, H +··· 1 k l 1 i k l 6 + β 1 + R0k0l x x − βα Rik0l x x m 2 6 i −S˙ − S, S˙ +··· . (30) 1 k l 2 = O + E + β 1 + R0k0l x x 2 First, let us eliminate the off-diagonal part of the Hamilto- 1 i k l nian (26) at the order of m by a unitary transformation. Then − βα Rik l x x m , (26) 6 0 we will drop the higher order terms with respect to the Rie- mann tensor, which only depends on time, and derivatives of 0ˆ i 0ˆ iˆ where we have defined β = γ , α = γ γ and j = the Riemann tensor with respect to the time by assuming that 1 −i∂ j − eAj for brevity. The even part, E, means that the they are small enough. matrix has only block diagonal elements and the odd part, To cancel the last term in the square bracket of (26), we O, means that the matrix has only block off-diagonal ele- take ments. Any product of two even (odd) matrices is even and i 1 i k l S =− β − βα Rik l x x m . (31) a product of even (odd) and odd (even) matrices becomes 2m 6 0 odd. To take the non-relativistic limit of the Hamiltonian, we have to diagonalize the Hamiltonian (26) and expand the We then obtain / upper block diagonal part in powers of 1 m. More precisely, 1 i k l 1 i j k l i S, H βα Rik l x x m − α ,α Rik l x x j 1/m expansion is recognized as an expansion with respect 6 0 12 0 ( )−1 v/ to two parameters, mx and c. Here, x represents a i i j k i i j k + α α R ikjx + α α R jikx . (32) typical length scale of the system which can be specified by 6 0 12 0 ∼ i v Fermi normal coordinates, i.e., x x xi , is the velocity Therefore, from Eqs. (30) and (32), we have the transformed of the fermion and c denotes the speed of light. Assuming Hamiltonian as 1/mx 1 and v/c 1, which hold in the situation of the Sect. 4, we will perform the 1/m expansion. It is known H H + i S, H that this can be done in flat spacetime by repeating unitary i i j i i j k l i j j α R i j x − α α α R jkilx + α δ + θ j transformations order by order in powers of 1/m [28,29]. 2 0 0 8 i i Let us generalize the method to the case of curved spacetime i k 2 k l −eA0 − R0ikix + R0kilx x i in Fermi normal coordinates. 6 3 We consider a unitary transformation, 1 k l +β 1 + R0k0l x x m 2 ψ = eiSψ, (27) 1 k l = O + E + β 1 + R0k0l x x m , (33) where S is a time-dependent Hermitian 4 × 4 matrix. Observ- 2 ing that where we have used the relation αi ,αj = 2δij. One can ∂ψ ∂ see that only even terms remain at the order of m, as expected. = iSψ i i e Next, we focus on the order of m0 and eliminate the odd ∂t ∂t ∂ψ ∂ terms by a unitary transformation. In order to do so, we = eiS i + i eiS ψ ∂ ∂ choose the Hermitian operator to be t t ∂ i 1 = eiSHe−iS + i eiS e−iS ψ , S =− β O − α j R xk xl ∂ (28) 0k0l j t 2m 2 i j k we find that the Hamiltonian after the unitary transformation + α R0k0 j x . (34) is given by 2 It is straightfoward to obtain − ∂ − H = eiSHe iS + i eiS e iS . (29) ∂ 1 1 t i S , H −O + βO2 + β O, E m 2m We now assume that S is proportional to powers of 1/m and expand the transformed Hamiltonian (29)inpowersofS up 1 Then the Hermiticity of the non-relativistic Hamiltonian is guaranteed to the order of 1/m. Using Eqs. (C4) and (C7)inEq.(29), we [30]. 123 Eur. Phys. J. C (2020) 80 :545 Page 5 of 15 545

1 i j k l i k Therefore, we have the transformed Hamiltonian as − βα α R0k0l x x i j + β R0k0i x i 2m m + , i i j k 1 H H i S H + β α ,α R0k0i x j + β R0i0i 4m 4m 1 k l E + β 1 + R0k0l x x m , (41) i j k l 2 − βα R k l x x ∂ j eA , (35) 4m 0 0 0 where E is given by Furthermore, up to the order of 1/m, we can deduce i 2 1 1 1 E =− − k + k l + βO2 − S , S , H − S , iO eA0 R0ikix R0kilx x i 2 2 6 3 2m 1 1 i j k l − βO2 − βα α R0k0l x x i j 2m 4m i i 1 i j k l + β R xk + β αi ,αj R xk + βα α R k l x x i j 0k0i i 0k0i j 4m 0 0 2m 8m 1 i k + β R . (42) − β R k i x i 0i0i 2m 0 0 8m i i j k Moreover, the fourth term in the first line of Eq. (42) can be − β α ,α R0k0i x j 8m evaluated as − 1 β , R0i0i (36) 1 2 i i j m 8m βO β α ,α ilmeB δlj + 2θlj 2m 8m and − i β αi ,αj 1 + δl m Rlmji j R0i0m x l ˙ i ˙ i j k l ˙ 4m 4 − S βO + βα R0k0l x x eA j . (37) 2m 4m 1 + β δij + 2θij i j Therefore, the Hamiltonian after the unitary transformation 2m is given by i j 1 + β Rkikj x i + β R0i0i 12m 4m + , − 1 , , − ˙ H H i S H S S H S 1 i j k l 2 + βα α α α Rijkl 16m − i βαj k l + 1 β O, E + O˙ R0k0l x x eEj i i i j k l m 4m 2m − β α ,α α α Rkljmx i , (43) 16m 1 2 1 i j k l +E + βO − βα α R k l x x i j 2m 4m 0 0 i ≡ 1 ijk(∂ − ∂ ) where B 2 j Ak k A j is a magnetic field. Using i i i j k k i j k Eqs. (42), (43) and the relation α ,α = 2iijkσ in the + β R0k0i x i + β α ,α R0k0i x j 2m 8m transformed Hamiltonian (41), we finally arrive at the Hamil- 1 tonian for a non-relativistic fermion up to the order of 1/m + β R0i0i 8m as

+β + 1 k l 1 R0k0l x x m 1 k l i k 2 H = 1 + R k l x x m − eA − R ikix 2 0 0 0 6 0 = O + E + β + 1 k l , 1 R0k0l x x m (38) 2 k l 2 + R0kilx x i 3 where E ≡ ∂ A − A˙ is an electric ﬁeld. We see that O 1 1 j j 0 j + δ 1 + R xk xl /m ij 0k0l has only terms of order of 1 , so that odd terms at the order 2m 2 of m0 have been eliminated. 1 k l O + R jkilx x i j Finally, we will eliminate the odd term and then the 3 Hamiltonian will consist of only even terms up to the order e i j 1 k l of 1/m, which we want to get. To this end, we now choose − σ B δij 1 + R k l x x 2m 2 0 0 the Hermitian operator of a unitary transformation as 1 k l 1 k l + Rmkmlx x − Rikjlx x i i i k l 6 6 S =− β O − βα eEi R k l x x . (39) 2m 4m 0 0 1 k l + ijkσ Rijlm + 2δ jm R0i0l x m Then, up to the order of 1/m,wehave 8m 1 , −O . + 3R i i − Rijij i S H (40) 8m 0 0 123 545 Page 6 of 15 Eur. Phys. J. C (2020) 80 :545

i 1 i where ημν stands for a flat spacetime metric and hμν repre- + R0i0 j − Rkikj x j . (44) 2m 3 sents a deviation from the flat spacetime. Because the Rie- The first term is the rest mass and its correction from the mann tensor (47) is invariant under gauge transformations, gravity at a point xi . The third term represents gravitational we can use any coordinate to evaluate the Riemann tensor redshift, namely energy shift due to gravity. The first term included in Eq. (44). We then take the transverse traceless = = = in the last line gives the same effect at the order of 1/m. gauge, i.e., h0μ hii hij, j 0. As a result, one can We find that the fourth and the fifth terms describe gravita- obtain tional effects on the motion of a particle. However, we notice 1 ¨ R i j =− hij , that the former contains the time derivative of the curvature, 0 0 2 which has been assumed to be small. The second term in the 1 ˙ ˙ R ijk = hij,k − hik, j , last line is also small. The third line represents interactions 0 2 between gravity and a spin in the presence of an external 1 Rijkl = hil, jk + h jk,il − h jl,ik − hik, jl . (48) magnetic field. This is what causes the spin resonance and/or 2 the excitation of magnons as we will see in the next section. Note that they are evaluated at the origin, xi = 0, so that they The fourth line is a spinÐorbit coupling mediated by gravity. do not depend on spatial coordinates. Substituting (48)into In vacuum, the Riemann tensor coincides with the Weyl (44), we finally obtain tensor. Then it may be useful to rewrite the Riemann tensor of the Hamiltonian (44) in terms of the electric Eij and magnetic 1 ¨ i j ρσ H = 1 − hijx x m − eA0 Hij components of the Weyl tensor Cμν , defined by 4 1 ρσ ρσ [ρ |0| [ρ σ] + − k l Rμν = Cμν = 4 γ[μ − δ[μ δ| | Eν] hki,l hkl,i x x i 0 3 [ρ σ]α ρσα0 0 +2μνα δ H + 2 δ Hν]α , (45) 1 1 ¨ k l 0 0 [μ + δij 1 − hkl x x 2m 4 where γμν is the induced three dimensional metric, i.e., γ00 = 1 k l γ = 0,γ δ . Substituting the above relation into the + h jl,ki + hki, jl − hkl,ij − hij,kl x x i j 0i ij ij 6 Hamiltonian (44), we obtain e i j 1 ¨ k l − σ B δij 1 − hkl x x 1 k l i k 2m 3 H = 1 + Ekl x x m − eA0 − Hilkilx 2 6 1 k l − hil,kj + hkj,il − hkl,ij − hij,kl x x 2 k l + Hkmilmx x i 12 3 1 k l + σ , − , − δ ¨ 1 1 k l ijk him jl hil jm jmhil x m + δij 1 + Ekl x x 8m 2m 2 i ¨ i − hijx j , (49) 4 k l 6m + δ[ j|[i El]|k]x x i j 3 where we have used the equation of motion for gravitational e i j 1 k l waves, i.e., h = 0. − σ B δij 1 + Ekl x x ij 2m 2 In the next section, we will see that gravitational waves 2 k l 2 k l excite magnons, which are collective excitation of spins + δ[m|[m El]|k]x x − δ[i|[ j El]|k]x x 3 3 through the interaction in the third line in Eq. (49). 1 k l + ijkσ 2δ[i|[l Em]| j] + δ jmEil x m 4m 1 3 4 Magnon gravitational wave detectors + Eii − δ[i|[i E j]| j] 2m 4 In Sect. 3, we revealed gravitational effects on a non- i 4 i + Eij − δ[k|[k E j]|i] x j . (46) relativistic Dirac fermion in Fermi normal coordinates. As 2m 3 you can see in Eq. (49), if one consider a freely falling point Although Eq. (44) is applicable to a general curved space- particle and set Fermi normal coordinates, the particle does time, let us focus on gravitational waves as gravitational not feel perturbative gravity hij at the origin because of the effects from now on. The Riemann tensor for a perturbed equivalence principle. However, gravitational effects are can- metric gμν = ημν + hμν at the linear order is given by celed, of course, only at one point and thus an object with finite dimension feels gravitation. In the case of magnons, α 1 α ,α α ,α R μβν = (h ν,μβ − h − h β,μν + h ), (47) 2 μν β μβ ,ν we prepare, for example, a ferromagnetic sample in an exter- 123 Eur. Phys. J. C (2020) 80 :545 Page 7 of 15 545

ˆ † ˆ nal magnetic field and then the sample feels gravity since the transformation (54). Wenote that Ci Ci represents the par- it has finite size. Thus, magnons can be excited by gravita- ticle numbers of the boson created by the creation operator ˆ † tional waves. To examine the effect of gravitational waves Ci . The bosonic operators describe spin waves with disper- on magnons, it is appropriate to set a Fermi normal coordi- sion relations determined by Bz and Jij. Furthermore, pro- nate with the origin placed at the center of the ferromagnetic vided that contributions from the surface of the sample are sample. Then we can use the discussion of Sect. 3. negligible, one can expand the bosonic operators by plane We consider a ferromagnetic sample in an external mag- waves as netic field. Such a system is described by the Heisenberg −i k·ri model [31]: ˆ e Ci = √ cˆk , (55) N =− μ ˆz − ˆ · ˆ , k Hspin 2 B Bz S(i) JijS(i) S( j) (50) i i, j where ri is the position vector of the i spin. The excitation ˆ† where the Bohr magneton μB = e/2me is defined by the ele- of the spin waves created by ck is called a magnon. mentary charge e and the mass of electrons me. We applied We now rewrite the spin system (53) by magnons with the an external magnetic field along the z-direction, Bz, with- HolsteinÐPrimakoff transformation (54) and then we only out loss of generality because of isotropy. Here, i specifies focus on the homogeneous mode of magnons, i.e., k = 0 each site of spins. The first term is the conventional Pauli mode. Then the second term in the total Hamiltonian (53)is term, which turns the spin direction to be along the exter- irrelevant because it does not contribute to the homogeneous nal magnetic field. The second term represents the exchange mode. Furthermore, because Qzz does not contribute to the interactions between spins with the strength Jij. resonance of the spins, namely excitation of magnons, we Next, we take into account the effect of gravitational waves will drop it. Thus we have on the system. From Eq. (49), the interaction Hamiltonian ˆ + ˆ † between gravitational waves and a spin in the ferromagnetic ˆ † ˆ Ci Ci Htot = μB Bz 2C Ci + Qzx sample is i 2 i =−μ ˆb , ˆ − ˆ † HGW B Ba S(i) Qab (51) Ci Ci + Qzy . (56) 2i where we have defined

2 ¨ k l Now let us consider a planar gravitational wave propa- Qij =− δijhkl|x=0 x x 3 gating in the zÐx plane, namely, the wave number vector of 1 k l the gravitational wave k has a direction kˆ = (sin θ,0, cos θ). − hil,kj + hkj,il − hkl,ij − hij,kl |x=0 x x . (52) 6 Moreover, we postulate that the wavelength of the gravita- It represents the effect of gravitational waves on a spin located tional wave is much longer than the dimension of the sam- at xi in Fermi normal coordinates. Indeed, at the origin, xi = ple and it is necessary for the validity of the Fermi normal 0, we see that Qij = 0. From Eqs. (50) and (51), the total coordinates. This situation is actually satisfied in the case of Hamiltonian of the system is usual cavity experiments for magnons. We can expand the gravitational wave hij in terms of linear polarization tensors Htot = Hspin + HGW (σ) (σ ) satisfying e e = δσσ as =−μ ( δ + ) â − ˆ · ˆ . ij ij B 2 za Qza Bz S(i) JijS(i) S( j) i i, j ( , ) = (+)( , ) (+) + (×)( , ) (×) . hij x t h x t eij h x t eij (57) (53) More explicitly, we took the representation The spin system (53) can be rewritten by using the HolsteinÐPrimakoff transformation [32]: ⎧ (+) ⎧ ⎪ (+) h − (w − · ) (w − · ) ⎪ ˆz = 1 − ˆ † ˆ , ⎪ h (x, t) = e i h t k x + ei h t k x , (58) ⎨⎪S(i) 2 Ci Ci ⎨ 2 ˆ+ ˆ † ˆ ˆ (×) S( ) = 1 − C Ci Ci , (54) ⎪ (×) h −i(w t−k·x+α) i(w t−k·x+α) ⎪ i i ⎪ h (x, t) = e h + e h , ⎩ − † † ⎪ 2 Sˆ = Cˆ 1 − Cˆ Cˆ , ⎩ (i) i i i (59) ˆ ˆ † where bosonic operators Ci and Ci satisfy commutation [ ˆ , ˆ †]=δ ˆ± = ˆ x ± ˆ y ω relations Ci C j ij and S( j) S( j) i S( j) are the where h is an angular frequency of the gravitational ladder operators. It is easy to check that the SU(2) algebra, wave and α represents a difference of the phases of polar- î ˆ j ˆk [S , S ]=iijkS (i, j, k = x, y, z), is satisfied even after izations. Note that the polarization tensors can be explicitly 123 545 Page 8 of 15 Eur. Phys. J. C (2020) 80 :545 constructed as stokes parameters Q and U transform as ⎛ ⎞ cos θ 2 0 − cos θ sin θ Q ψ ψ Q (+) 1 = cos 4 sin 4 = √ ⎝ − ⎠ , (67) eij 0 10 (60) U − sin 4ψ cos 4ψ U 2 − cos θ sin θ 0sinθ 2 ⎛ ⎞ where ψ is the rotation angle around k. θ 0 cos 0 The second term in Eq. (62) shows that planar gravita- (×) = √1 ⎝ θ − θ ⎠ . eij cos 0 sin (61) tional waves induce the resonant spin precessions and/or the 2 − θ 0 sin 0 excitation of magnons if the angular frequency of the gravita- In Eqs. (60) and (61), we defined the + mode as a deformation tional waves is near the Lamor frequency, 2μB Bz.Itisworth in the y-direction. noting that the situation is similar to the resonant bar experi- Then substituting Eqs. (57)Ð(61) into the total Hamilto- ments [16] where planar gravitational waves excite phonons nian (56), moving on to the Fourier space and using the rotat- in a bar detector. ing wave approximation, one can deduce Let us show the ability of magnon gravitational detectors by giving constraints on high frequency gravitational † † −iωh t iωh t Htot 2μB Bzcˆ cˆ + gef f cˆ e +ˆce , (62) waves. Recently, measurements of resonance fluorescence of magnons induced by the axion dark matter was con- ˆ =ˆ where c ck=0 and ducted and upper bounds on an axionÐelectron coupling con- √ π 2 2 √ stant have been obtained [25,26]. Such an axionÐmagnon 2 l 2 (+) 2 gef f = μB Bz sin θ N cos θ(h ) resonance [33] has a similar mechanism to our gravitonÐ 60 λ / magnon resonance. Therefore, we can utilize these experi- (×) (+) (×) 1 2 +(h )2 + 2 cos θ sin α h h , (63) mental results to give the upper bounds on the amplitude of GHz gravitational waves [24]. is an effective coupling constant between the gravitational The interaction hamiltonian which describe the axionÐ λ = π/ω waves and the magnons. The parameters l and 2 h magnon resonance is given by are the radius of the (spherical) ferromagnetic sample and † −ima t ima t the wavelength of the gravitational wave. We note that the Ha =˜geff cˆ e +ˆce , (68) sum over the spin sites i was evaluated as where g˜eff is an effective coupling constant between an 1 l xx = yy = zz r 2 axion and a magnon. Notice that the axion oscillates with L3 i i i 0 a frequency determined by the axion mass ma. One can see that this form is the same as the interaction term in 4π l5 sin ζ (r cos ζ )2 drdζ dφ = , (64) Eq. (62). Through the hamiltonian (68), g˜ is related to an 15 L3 eff axionÐelectron coupling constant in [25,26]. Then the axionÐ where L is a lattice constant, which is related to the number electron coupling constant can be converted to g˜eff by using = 4π 3 / 3 of spins as N 3 l L . parameters, such as the energy density of the axion dark From Eq. (63), we see that√ the effective coupling constant matter, which are explicitly given in [25,26]. Therefore con- N has gotten a huge factor . Moreover, in order to obtain straints on g˜ef f (95% CL) can be read from the constraints a coordinate-independent expression of gef f ,itisusefulto on the axionÐelectron coupling constant given in [25,26], use the Stokes parameters: respectively, as follows: √ 2 2 √ 2π l . × −12 , g = μ B sin θ N 3 5 10 eV eff λ B z g˜eff < (69) 60 3.1 × 10−11 eV . 1/2 1 + cos2 θ sin2 θ × I − Q + cos θ V , (65) It is easy to convert the above constraints to those on the 2 2 amplitude of gravitational waves appearing in the effective where the Stokes parameters are defined by coupling constant (65). Indeed, we can read off the exter- ⎧ nal magnetic field B and the number of electrons N as (+) 2 (×) 2 z ⎪I = (h ) + (h ) , 19 ⎪ (Bz, N) = (0.5T, 5.6 × 10 ) from [25] and (Bz, N) = ⎨ (+) (×) Q = (h )2 − (h )2 , (0.3T, 9.2 × 1019) from [26], respectively. The external ⎪ (+) (×) (66) ⎪U = 2 cos α h h , magnetic field Bz determines the frequency of gravitational ⎩⎪ V = 2sinα h(+)h(×) . waves we can detect. Therefore, using Eqs. (65), (69) and the above parameters, one can put upper limits on gravita- 2 2 2 2 They satisfy I = U + Q + V . We see that the effective tional waves at frequencies determined by Bz. Since [25] and coupling constant depends on the polarizations. Note that the [26] focused on the direction of Cygnus and set the external 123 Eur. Phys. J. C (2020) 80 :545 Page 9 of 15 545

5 Conclusion

In order to detect high frequency gravitational waves, we developed a new detection method. Using Fermi normal coordinates and taking the non-relativistic limit, we obtained the Hamiltonian for non-relativistic fermions in Fermi normal coordinates for general curved spacetime. This Hamiltonian is applicable for any curved spacetime background as long as one can treat a curvature perturbatively. Therefore, our formalism is useful to consider gravitational effects on non- relativistic fermions, which is usual in condensed matter systems. In Sect. 4, we focused on the interaction between a spin of a fermion and gravitational waves, expressed by the third line in Eq. (49). It turned out that gravitational waves can excite Fig. 1 Several experimental sensitivities and constraints on high fre- magnons. Moreover, we explicitly demonstrated how to use quency gravitational waves are depicted. The blue color represents an upper limit on stochastic gravitational waves by waveguide experiment magnons for detecting high frequency gravitational waves using an interaction between electromagnetic fields and gravitational and gave upper limits on the spectral density of continu- waves [20]. The green one is the upper limit on stochastic gravitational ous gravitational waves (95 % CL): 7.5 × 10−19 [Hz−1/2] waves, obtained by the 0.75 m interferometer [19]. Our new constraints at 14 GHz and 8.7 × 10−18 [Hz−1/2] at 8.2 GHz, respec- on continuous gravitational waves are plotted with a red color, which also represent the sensitivity of the magnon gravitational wave detector tively, by utilizing results of magnon experiments. Interest- for stochastic gravitational waves ingly, there are several theoretical models predicting high frequency gravitational waves which are within the scope of our method [1]. magnetic fields to be perpendicular to it, we probe continu- The gravitonÐmagnon resonance is also useful for prob- θ = π ing stochastic gravitational waves with almost the same sen- ous gravitational waves coming from Cygnus with 2 (more precisely, sin θ = 0.9in[26]).Wealsoassumeno sitivity illustrated in Fig.1. Although the current sensitivity linear and circular polarizations, i.e., Q = U = V = 0. is still not sufficient for putting a meaningful constraint on Consequently, experimental data [25] and [26] enable us to stochastic gravitational waves, it is important to pursue the constrain the characteristic amplitude of gravitational waves high frequency stochastic gravitational wave search for future (+) (×) defined by hc = h = h as gravitational wave physics. Moreover, we can probe a burst of gravitational waves of any wave form if the duration time is smaller than the relaxation time of a system. The situa- . × −13 , ∼ 1 3 10 at 14 GHz tion is the same as for resonant bar detectors [34,35]. For hc − (70) 1.1 × 10 12 at 8.2 GHz , instance, in the measurements [25,26], the relaxation time is about 0.1 µs which is determined by the line width of the at 95% CL, respectively. In terms of the spectral density ferromagnetic sample and the cavity. If the duration of the = 2/ burst of gravitational waves is smaller than 0.1 µs, we can defined by Sh hc 2 f and the energy density parame- = π 2 2 2/ 2 detect it. Furthermore, improving the line width of the sample ter defined by GW 2 f hc 3H0 (H0 is the Hubble parameter), the upper limits at 95% CL are and the cavity not only leads to detecting a burst of gravitational waves but also to increasing the sensitivity. As another way to improve the sensitivity of the magnon gravitational . × −19 [ −1/2] , ∼ 7 5 10 Hz at 14 GHz wave detector, quantum nondemolition measurement may be Sh − − / (71) 8.7 × 10 18 [Hz 1 2] at 8.2 GHz , promising [36Ð38]. and Acknowledgements A.I. was supported by Grant-in-Aid for JSPS Research Fellow and JSPS KAKENHI Grant No.JP17J00216. J.S. was in part supported by JSPS KAKENHI Grant Numbers JP17H02894, 2.1 × 1029 at 14 GHz , JP17K18778. This research was supported by the Munich Institute for h2 ∼ (72) Astro Ð and Particle Physics (MIAPP) which is funded by the Deutsche 0 GW 30 5.5 × 10 at 8.2 GHz . Forschungsgemeinschaft (DFG,German Research Foundation) under Germany’s Excellence Strategy-EXC-2094-390783311. We depict the limits on the spectral density with several other gravitational wave experiments in Fig.1. 123 545 Page 10 of 15 Eur. Phys. J. C (2020) 80 :545

Data Availability Statement This manuscript has no associated data or the data will not be deposited. [Authors’ comment: There are no experimental data associated with the article.] ∂ Open Access This article is licensed under a Creative Commons Attri- ∂τ bution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, pro- γs vide a link to the Creative Commons licence, and indicate if changes P (τ) were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indi- cated otherwise in a credit line to the material. If material is not ∂ included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permit- ∂s ted use, you will need to obtain permission directly from the copy- right holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/. Funded by SCOAP3. γτ

Appendix A: Fermi normal coordinates Fig. 2 A timelike geodesic γτ is parametrized by a proper time τ and aspacelikegeodesicγs is parametrized by a proper distance s,which One can construct local inertial coordinates along a geodesic is orthogonal to γτ at the crossing point of a particle, the so-called Fermi normal coordinates [39]. An observer on the earth is freely falling when gravity of Fermi normal coordinates are parallelly transformed along the earth, which will be taken into account in Appendix B, γτ ,wehave is negligible. Thus, the Fermi normal coordinates describe μ α the frame used in an experiment. In this appendix, we briefly ∂ ∂ 0 = review how to construct the Fermi normal coordinates [39]. ∂xν ∂τ ;α We consider a timelike geodesic γτ parametrized by a μ α = δν δ proper time τ and specify a point on the geodesic by P(τ). ;α 0 = μ ( 0 = τ, i = ) We also consider a spacelike geodesic γs orthogonal to γτ ν0 x x 0 (τ) 2 = μ | , at P , which is parametrized by a proper distance s. We ν0 γτ (A3) set the crossing point as s = 0. The situation is illustrated in Fig. 2. where we used the fact that the vector components of the ∂ μ = δμ Then the Fermi normal coordinates which are locally inertial bases in Fermi normal coordinates are ∂xν ν .On γ frames along γτ are defined as follows: the other hand, on the spacelike geodesic s, the geodesic equation x0 = τ, xi = αi s . (A1) 2 μ α β d x μ dx dx ∂ + = The bases of the Fermi normal coordinates, μ , are paral- αβ 0(A4) ∂x ds2 ds ds lelly transported along the geodesic γτ and the αi are the ∂ is satisfied. Using (A1)inEq.(A4), we obtain components of the tangent vector ∂s , namely, μ ∂ ∂ (x0 = τ,xi = αi s)αi α j = 0 . (A5) = αi . (A2) ij ∂s ∂xi In particular on γτ , namely at s = 0, we conclude that ∂ Also, the bases, ∂xμ , are taken to be orthonormal by utilizing μ μ i ( 0 = τ, i = ) = | = . the degree of rescaling α . Thus, the metric in Fermi normal ij x x 0 ij γτ 0 (A6) coordinates is given by ημν on the geodesic γτ .3 Therefore, from Eqs. (A3) and (A6), we see that the Christof- Let us show that the Fermi normal coordinates (A1)are fel symbols vanish along the timelike geodesic γτ . indeed local inertial frames, namely, the Christoffel symbols Now one can calculate the metric components in the vicin- are zero along the geodesic γτ . First, because the bases of the ity of the geodesic γτ in Fermi normal coordinates. In a situation that a curvature scale is much larger than that of a 2 One can use an affine parameter instead of s, which does not change the following discussion. system we treat, we can expand the metric in terms of the coordinates xμ. The first order term vanishes by definition. 3 Note that orthonormality holds at any point on the geodesic γτ if it is satisfied at one point on γτ , because the parallel transport keeps For our purpose, it is enough to calculate the metric up to the orthonormality. second order. 123 Eur. Phys. J. C (2020) 80 :545 Page 11 of 15 545

Note that the Christoffel symbols vanish along the From the definition of the Christoffel symbol, we have geodesic γτ α α gμν,λ = gμανλ + gναμλ . (A16) μ |γ = 0 . (A7) σ νλ,0 τ Differentiating it with respect to x leads to Then, from the definition of the Riemann tensor, we find α α gμν,λσ |γτ = ημανλ,σ |γτ + ηναμλ,σ |γτ . (A17) μ | = μ | . ν0,λ γτ R νλ0 γτ (A8) Using Eqs. (A7), (A8) and (A15)inEq.(A17), one can deduce gμν, λ = 0 and the following: To go further, we use the geodesic deviation equation: 0 =− | , 2 μ α g00,ij 2R0i0 j γτ d ξ dξ μ β + 2 u 2 λ2 λ αβ =− | + | , d d g0i, jk R0 jik γτ R0kij γτ μ μ μ δ μ δ 3 + R αγβ + αγ,β + βδαγ − γδαβ 1 g , =− R |γ + R |γ . (A18) α β γ ij kl ikjl τ iljk τ u u ξ = 0 , (A9) 3 Thus, up to the quadratic order of the coordinates, the metric λ τ γ where can be either or s. We notice that a point on s is components in Fermi normal coordinates are given by specified by the parameters (τ, s,αi ). Then, as to the space- i j γ g00 =−1 − R0i0 j |γτ x x , (A19) like geodesic s, one can consider two deviation vectors; one ∂ ∂ ∂ is and the other is . The vector rep- 2 j k ∂τ s,αi ∂αi ∂τ s,αi g i =− R jik|γτ x x , (A20) τ,s 0 3 0 resents a deviation between two spacelike geodesics which ∂ 1 k l γ gij = δij − Rikjl|γτ x x . (A21) stem from different points on τ and ∂αi represents a τ,s 3 deviation between two spacelike geodesics which stem from μ ∂ μ μ We note that the Riemann tensor is evaluated on the time- (τ) γτ ξ = = δ 0 the same point P on . Substituting ∂τ s,αi 0 like geodesic γτ , hence it only depends on x . It should be into Eq. (A9) yields mentioned that the Riemann tensor in Eqs. (A19)Ð(A21)is μ μ j k calculated in Fermi normal coordinates. However, the Rie- |γ − R |γ α α = 0 , (A10) i0, j τ ij0 τ mann tensor for the linear perturbations around the flat spacetime background is invariant under gauge transformations. which is automatically satisfied because of Eq. (A8). On the ξ μ = ∂ = δμ In this case, the Riemann tensor constructed in Fermi nor- other hand, substituting ∂αi s into Eq. (A9), τ,s i mal coordinates is the same as that in the transverse trace- we obtain less gauge. Therefore, one can use (47)inEqs.(A19)Ð(A21) μ j μ j k μ j k 2 when we consider gravitational waves on the flat spacetime 2 α + sR |γ α α + s |γ α α + O(s ) = 0 . ij jik τ ij,k τ background. (A11)

TheﬁrstterminEq.(A11) can be expanded in powers of s as Appendix B: Proper detector frame

∂ μ j μ j μ j α = |γ α + s α In Appendix A, we constructed local inertial coordinates 2 ij 2 ij τ 2 ∂ ij s at γτ along a geodesic for a freely falling observer, namely Fermi = μ | α j αk . normal coordinates. However, an observer bound on the earth 2s ij,k γτ (A12) is not freely falling. This is because the observer accelerates From Eqs. (A11) and (A12), we find the relation against the center of the earth with g = 9.8m/s2 and has the rotational motion since the earth is rotating. In this appendix, μ 1 μ j k |γτ + R jik|γτ α α = 0 . (A13) we take into account these effects of the earth [40,41]. We ij,k 3 will see that these effects are negligible in the discussion in It implies that the symmetric part of the indices j and k in the text. the parenthesis should be zero, i.e., The procedure is almost the same as the case of the Fermi normal coordinates. We first consider a timelike curve γτ μ μ 1 μ μ |γτ + |γτ =− R jik|γτ + R kij|γτ . (A14) parametrized by τ and construct a spacelike geodesic γ ij,k ik, j 3 s parametrized by a proper distance s, which is orthogonal to After a little algebra, this can be solved as the geodesic γτ at s = 0. The situation is illustrated in Fig.2. The difference from the Fermi normal coordinates appears in μ | =−1 μ | + μ | . , γτ R ijk γτ R jik γτ (A15) eμ ij k 3 the transportation of the orthonormal bases which cover 123 545 Page 12 of 15 Eur. Phys. J. C (2020) 80 :545 small region around a point on the curve γτ . The bases eμ Using the above and the explicit relation γ d = are parallelly transported along τ , i.e. τ eμ 0, in the con- μ 0 0 1 1 2 d a aμ =−a a + a a = g , (B7) struction of the Fermi normal coordinates,4 while the eμ in the present case are transported as follows [40]: we can obtain the following equations: D 0 = u0 = 1 , eμ =− · eα a dτ gu τ 1 (B8) d 1 u 0 μν a = τ = gu . = eμ ∧ eν · eα d α =−eα , (B1) A solution of Eq. (B8) is given by μ −1 μν t = g sinh (gτ) , where is an infinitesimal Lorentz transformation defined (B9) x1 = g−1 cosh (gτ) , by 2 2 −2 μν μ ν ν μ αβμν which is a hyperbolic world curve, indeed, x − t = g . = a u − a u + uαωβ The hyperbolic curve is invariant under a Lorentz boost from = μν + μν . (F) (R) (B2) the inertial coordinate (t, x1) to another one. Since τ depen- dence appears in Eq. (B9), one can construct the rest frame Here, we defined the four velocity for the accelerating observer at instant τ by doing a Lorentz μ μ dx boost transformation depending on τ. Such a Lorentz boost, u = , (B3) dτ which is a four dimensional rotation of a plane spanned by μ μ μν the four acceleration u and a , would be expressed by (F) . Indeed, for an 1 μ observer accelerating along the x -direction, we have μ du a = , (B4) 0x1 dτ (F) =−g , (B10)

ωμ μν and represents an angular velocity of rotation of spa- and the other components of (F) vanish. Thus, the four μ tial bases ei . Note that the orthonormality of the bases vector x = (τ, 0, 0, 0), after the infinitesimal Lorentz trans- μν holds under the evolution (B1) as a consequence of the anti- formation conducted by (F) , is given by symmetricity of μν. μν 0 0 μ One can see that (R) represents just a three dimen- d x − x = x (F)μ0 dτ sional rotation in a four dimensional covariant form. In fact, = 0 . (B11) in the rest frame, i.e. uμ = (1, 0, 0, 0), we obtain α γβα Hence, we get − eα (R)μ =−eαuγ ωβ μ 0 0ij dx = ωi e j μ = 0 . (B12) dτ = ω × e , (B5) j μ=k This is consistent with the first equation in (B9) when gτ where we identified the label of the bases eμ as the component 1. Similarly, we obtain of them due to orthonormality to obtain the last equality and 1 1 μ d x − x = x ( )μ dτ μ = k represents the fact that μ takes a spatial index. For the F 1 observer on the earth, ω is the angular velocity of the earth. = τgdτ, (B13) μν The transformation ( ) is called FermiÐWalker trans- F which leads to port. Consider an accelerating observer with magnitude of μ 1 the gravity of the earth, a aμ = g2, along x1-coordinate dx = gτ. (B14) in an inertial frame.5 Then, because an acceleration vector dτ defined by (B4) is orthogonal to the four velocity, we have This is consistent with the second equation in (B9) when μν μ 0 0 i i gτ 1. Therefore, we find that (F) correctly represents a uμ =−a u + a u = 0 . (B6) an infinitesimal Lorentz transformation which connects a rest frame to an accelerating frame relative to the rest frame. Now, 4 Here, the coordinate bases are not restricted to those given by we can understand the meaning of the FermiÐWalker trans- Eq.(A1). γ 2 i port in Eq. (B1). At one point on τ , one can construct a rest 5 In the rest frame for the observer, the Newton equation d x −g = 0 (dx0)2 frame for an accelerating observer, but after a certain dura- holds. Here, we used the fact that the 0-component of aμ is zero because μ μ μ μ tion the frame is not a rest frame for the observer anymore. a is orthogonal to u and u = δ in the rest frame. Therefore, the 0 μ τ relation of the relativistically invariant quantity, a aμ = g, is satisfied In order to keep a frame as a rest frame at any time ,the as in Newtonian gravity. bases of the frame should be developed by the FermiÐWalker 123 Eur. Phys. J. C (2020) 80 :545 Page 13 of 15 545 transport. Thus, we obtain a coordinate system moving with where a dot represents a derivative with respect to τ.Onthe an accelerating observer. geodesic γτ , from the definition of the Riemann tensor, we From now on, we use coordinate bases specified by find Eq. (A1): μ = μ + μ − μ α + μ α . ν0,λ Rνλ0 νλ,0 λα ν0 0α νλ (B25) x0 = τ, xi = αi s , (B15) Substituting Eqs. (B24) into Eq. (B25), we can deduce and get an explicit expression for the metric in the proper 0 | =˙i + j ωk0ijk , detector coordinate which is moving with an accelerating 00,i γτ a a observer due to the earth. The procedure is similar to the 0 | = 0 | − i j , i0, j γτ Rij0 γτ a a case of the Fermi normal coordinates in Appendix A. i i 0ijk i j i j k |γ = R |γ −˙ω + a a + ω ω − δ ω ω , From Eq. (B1), we obtain the relation 00, j τ 0 j0 τ k ij k i i j 0ikl |γ = R |γ + a ω . (B26) α = α . j0,k τ jk0 τ l μ0 μ (B16) μ In order to obtain an expression for |γ , one can utilize Using uμ = (1, 0, 0, 0) and aμ = (0, ai ) in the definition ij,k τ a geodesic deviation equation for γ and the procedure is (B2), we have s completely the same as that in the construction of the Fermi 0 = ,i =−0ijkω . i ai j k (B17) normal coordinates. Thus, the result is given by Eq. (A15): Thus, together with Eqs. (B16) and (B17), we obtain μ | =−1 μ | + μ | . ij,k γτ Rijk γτ R jik γτ (B27) 0 0 i i i 0ijk 3 = 0 ,|γτ = |γτ = a ,|γτ =−ωk . 00 i0 00 j0 Finally, we express the second order derivative of the met- (B18) ric by the Christoffel symbols and their first derivatives, and We see that the proper reference frame is not a local iner- then relations between the second derivatives of the met- tial frame anymore. Furthermore, considering a spacelike ric and the Riemann tensor are obtained. Differentiating σ geodesic equation along γs, Eq. (B22) with respect to x , we obtain the relation μ α β α α 2 gμν,λσ |γ = ημα |γ + ηνα |γ d x + μ dx dx = , τ νλ,σ τ μλ,σ τ 2 αβ 0 (B19) α α ds ds ds +gμα,σ |γτ νλ|γτ + gνα,σ |γτ μλ|γτ . (B28) we can deduce Using Eqs. (B26) and (B27)inEq.(B28), we can deduce the μ ( 0 = τ, i = αi )αi α j = . ij x x s 0 (B20) following equations:

Especially, at s = 0, we obtain gμν,00 = 0 , μ i | = . g00,0i =−2a˙ , ij γτ 0 (B21) 0 i j i j k g00,ij =−2R − 2a a − 2ω ω + 2δijω ωk , From Eqs. (B18), (B21) and the relation between the metric ij0 =˙ω 0ijk , and the Christoffel symbol g0i,0 j k α α =−2 | + | , gμν,λ = gμα + gνα , (B22) g0i, jk R0 jik γτ R0kij γτ νλ μλ 3 = , the ﬁrst order derivative of the metric reads gij,0k 0 =−1 | + | . gμν, = 0 , gij,kl Rikjl γτ Riljk γτ (B29) 0 3 i g , =−2a , 00 i Therefore, in a proper reference coordinate system, up to the 0ijk g0i, j =−ωk , quadratic order of the coordinates, the metric is given by = , gij,k 0 (B23) 2 2 i i i i i g00 =−1 − 2ai x − a x − ω x along the timelike curve γτ . i i j j i j Next, we evaluate the second order derivatives of the met- +ω ω x x − R0i0 j |γτ x x , (B30) τ ric. Differentiating Eqs. (B18) and (B21) with respect to , 0ijk j 2 j k g i =−ωk x − R jik|γτ x x , (B31) we get 0 3 0 0 | = μ | = , 1 k l , γτ , γτ 0 gij = δij − Rikjl|γτ x x . (B32) 00 0 ij 0 3 0 | = i | =˙i , i0,0 γτ 00,0 γτ a We see that the effects of the Earth enter even at the linear i | =−˙ω 0ijk , i ωi j0,0 γτ k (B24) order of a and . However, we can neglect these effects. 123 545 Page 14 of 15 Eur. Phys. J. C (2020) 80 :545

2 For example, assuming the scale of the experimental appa- ∂ g(λ, S) λ − λ = ei Si2 S, S˙ e i S , ratus to be xi ∼ 1 m and using the values ai ∼ 9.8m/s2, ∂λ2 ωi ∼ . × −7 / i i ∼ . × −16 . 2 0 10 rad s, we can estimate a x 1 1 10 . and ωi xi ∼ 6.7 × 10−16. These corrections are negligi- ∂n+1 f (λ, S) ble in experiments because they are small and their effects = eiλSin+1 S, S,..., S, S˙ ··· e−iλS . ∂λn+1 are static. Indeed, the effects of the earth are negligible in (C6) magnon experiments because we utilize the phenomenon of resonance between gravitational waves and magnons to Note that the right-hand side of the last equation has n pieces detect gravitational waves. Therefore, we use the Fermi nor- of S. Hence, we ﬁnally arrive at mal coordinates approximately for an observer on the earth. ∂ − eiS e iS = g(1, S) ∂t ∞ + iS −iS in 1 Appendix C: Expansion formula for e He and = S, S,..., S, S˙ ··· . ∂ iS −iS (n + 1)! ∂t e e n=0 (C7) Let us introduce a parameter λ by (λ, ) ≡ iλS −iλS . f S e He (C1) References We set λ = 1 after the calculations. Expanding it with respect to λ, we obtain 1. K. Kuroda, W.-T. Ni, W.-P. Pan, Int. J. Mod. Phys. D 24, 1530031 (2015). arXiv:1511.00231 [gr-qc] ∞ 2. Y.Akrami et al. (Planck), (2018). arXiv:1807.06211 [astro-ph.CO] λn ∂n f (λ, S) (λ, ) = . 3. P.A.R. Ade et al., (BICEP2, Planck). Phys. Rev. Lett. 114, 101301 f S n (C2) n! ∂λ λ= (2015). arXiv:1502.00612 [astro-ph.CO] n=0 0 4. C.R. Gwinn, T.M. Eubanks, T. Pyne, M. Birkinshaw, D.N. Mat- We ﬁnd that sakis, Astrophys. J. 485, 87 (1997). arXiv:astro-ph/9610086 [astro- ph] ∂ f (λ, S) = eiλSi S, H e−iλS , 5. J. Darling, A.E. Truebenbach, J. Paine, Astrophys. J. 861, 113 ∂λ (2018). arXiv:1804.06986 [astro-ph.IM] 2 6. L. Lentati et al., Mon. Not. R. Astron. Soc. 453, 2576 (2015). ∂ f (λ, S) λ − λ = ei Si2 S, S, H e i S , arXiv:1504.03692 [astro-ph.CO] 2 ∂λ 7. S. Babak et al., Mon. Not. R. Astron. Soc. 455, 1665 (2016). . . arXiv:1509.02165 [astro-ph.CO] 8. B.B.P. Perera et al., Mon. Not. R. Astron. Soc. 490, 4666 (2019). n ∂ f (λ, S) λ − λ = ei Sin S, S,..., S, H ··· e i S . (C3) https://doi.org/10.1093/mnras/stz2857. arXiv:1909.04534 [astro- ∂λn ph.HE] 9. Z. Arzoumanian et al., (NANOGRAV). Astrophys. J. 859,47 Therefore, one can deduce (2018). arXiv:1801.02617 [astro-ph.HE] iS −iS 10. J.W. Armstrong, L. Iess, P. Tortora, B. Bertotti, Astrophys. J. 599, e He = f (1, S) 806 (2003). https://doi.org/10.1086/379505 ∞ in 11. P. Amaro-Seoane et al., GW Notes 6, 4 (2013). arXiv:1201.3621 = S, S,..., S, H ]··· . (C4) [astro-ph.CO] n! n=0 12. N. Seto, S. Kawamura, T. Nakamura, Phys. Rev. Lett. 87, 221103 (2001). https://doi.org/10.1103/PhysRevLett.87.221103. Equation (C4) is called the CampbellÐBakerÐHausdorff for- arXiv:astro-ph/0108011 [astro-ph] mula. 13. Ligo, https://www.ligo.caltech.edu/page/study-work ∂ Next, let us consider the expansion of eiS e−iS in pow- 14. Virgo, http://www.virgo-gw.eu/ ∂t 15. K. Somiya, (KAGRA), Class. Quantum Gravity 29, 124007 ers of S. Again we introduce a parameter λ and expand g with (2012). https://doi.org/10.1088/0264-9381/29/12/124007. respect to λ as arXiv:1111.7185 [gr-qc] 16. M. Maggiore, Phys. Rep. 331, 283 (2000). arXiv:gr-qc/9909001 ∂ λ − λ [gr-qc] g(λ, S) = ei S e i S ∂t 17. F. Acernese et al., (VIRGO AURIGA-EXPLORER-NAUTILUS). Class. Quantum Gravity 25, 205007 (2008). arXiv:0710.3752 [gr- ∞ λn ∂n (λ, ) = g S . qc] n (C5) 18. A.S. Chou et al., (Holometer). Phys. Rev. D 95, 063002 (2017). n! ∂λ λ= n=0 0 arXiv:1611.05560 [astro-ph.IM] We see that 19. T. Akutsu et al., Phys. Rev. Lett. 101, 101101 (2008). arXiv:0803.4094 [gr-qc] ∂g(λ, S) 20. A.M. Cruise, R.M.J. Ingley, Class. Quantum Gravity 23, 6185 = eiλS i Se˙ −iλS , ∂λ (2006) 123 Eur. Phys. J. C (2020) 80 :545 Page 15 of 15 545

21. F. Li, R.M.L. Baker Jr., Z. Fang, G.V. Stephenson, Z. Chen, 30. X. Huang, L. Parker, Phys. Rev. D 79, 024020 (2009). https://doi. Eur. Phys. J. C 56, 407 (2008). https://doi.org/10.1140/epjc/ org/10.1103/PhysRevD.79.024020. arXiv:0811.2296 [hep-th] s10052-008-0656-9. arXiv:0806.1989 [gr-qc] 31. W. Heisenberg, Zeitschrift für Physik 38, 411 (1926) 22. F. Li, N. Yang, Z. Fang, R.M.L. Baker Jr., G.V. Stephenson, H. 32. T. Holstein, H. Primakoff, Phys. Rev. 58, 1098 (1940) Wen, Phys. Rev. D 80, 064013 (2009). https://doi.org/10.1103/ 33. R. Barbieri, M. Cerdonio, G. Fiorentini, S. Vitale, Phys. Lett. B PhysRevD.80.064013. arXiv:0909.4118 [gr-qc] 226, 357 (1989) 23. A. Ejlli, D. Ejlli, A.M. Cruise, G. Pisano, H. Grote, Eur. 34. M. Maggiore, Gravitational Waves, vol 1: Theory and Experiments. Phys. J. C 79, 1032 (2019). https://doi.org/10.1140/epjc/ Oxford Master Series in Physics (Oxford University Press, Oxford, s10052-019-7542-5. arXiv:1908.00232 [gr-qc] 2007) 24. A. Ito, T. Ikeda, K. Miuchi, J. Soda, Eur. Phys. J. C 35. P. Astone et al., Phys. Rev. D 82, 022003 (2010). https://doi.org/ 80, 179 (2020). https://doi.org/10.1140/epjc/s10052-020-7735-y. 10.1103/PhysRevD.82.022003. arXiv:1002.3515 [gr-qc] arXiv:1903.04843 [gr-qc] 36. Y. Tabuchi et al., Science 349, 405 (2015) 25. N. Crescini et al., Eur. Phys. J. C 78, 703 (2018). arXiv:1806.00310 37. Y. Tabuchi et al., Comptes Rendus Physique 17, 729 (2016). [hep-ex] (Erratum: Eur. Phys. J. C 78, no.9, 813 (2018)) (quantum microwaves/Micro-ondes quantiques) 26. G. Flower, J. Bourhill, M. Goryachev, M.E. Tobar, (2018), 38. D. Lachance-Quirion et al., Sci. Adv. 3, (2017) arXiv:1811.09348 [physics.ins-det] 39. F.K. Manasse, C.W. Misner, J. Math. Phys. 4, 735 (1963). https:// 27. L. Parker, Phys. Rev. D 22, 1922 (1980). https://doi.org/10.1103/ doi.org/10.1063/1.1724316 PhysRevD.22.1922 40. C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation (W.H.Free- 28. L.L. Foldy, S.A. Wouthuysen, Phys. Rev. 78, 29 (1950). https:// man, San Francisco, 1973) doi.org/10.1103/PhysRev.78.29 41. W.-T. Ni, M. Zimmermann, Phys. Rev. D 17, 1473 (1978). https:// 29. J. D. Bjorken, S. D. Drell (1965) doi.org/10.1103/PhysRevD.17.1473

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