Probing Ghz Gravitational Waves with Graviton–Magnon Resonance

Eur. Phys. J. C (2020) 80:179 https://doi.org/10.1140/epjc/s10052-020-7735-y Letter Probing GHz gravitational waves with graviton–magnon resonance Asuka Itoa, Tomonori Ikedab, Kentaro Miuchic, Jiro Sodad Department of Physics, Kobe University, Kobe 657-8501, Japan Received: 30 November 2019 / Accepted: 9 February 2020 / Published online: 27 February 2020 © The Author(s) 2020 Abstract A novel method for extending the frequency length of gravitational waves exceeds the current Hubble frontier in gravitational wave observations is proposed. It horizon. Measuring the temperature anisotropy and the B- is shown that gravitational waves can excite a magnon. mode polarization of the cosmic microwave background Thus, gravitational waves can be probed by a graviton– [3,4], we can probe gravitational waves with frequencies magnon detector which measures resonance fluorescence of between 10−18 and 10−16 Hz. Astrometry of extragalactic magnons. Searching for gravitational waves with a wave radio sources is sensitive to gravitational waves with fre- length λ by using a ferromagnetic sample with a dimension quencies between 10−16 and 10−9 Hz [5,6]. The pulsar tim- l, the sensitivity of the graviton–magnon detector reaches ing arrays, like EPTA [7,8], IPTA [9] and NANOGrav [10], −22 l −2 −1/2 spectral densities, around 5.4 × 10 × ( λ/ π ) [Hz ] observe the gravitational waves in the frequency band from 2 − − . × −21 × ( l )−2 [ −1/2] 10 9 Hz to 10 7 Hz. Doppler tracking of a space craft, which at 14 GHz and 8 6 10 λ/2π Hz at 8.2 GHz, respectively. uses a measurement similar to the pulsar timing arrays, can search for gravitational waves in the frequency band from 10−7 to 10−3 Hz [11]. The space interferometers LISA [12] −3 1 Introduction and DECIGO [13] can cover the range between 10 and 10 Hz. The interferometer detectors LIGO [14], Virgo [15], and In 2015, the gravitational wave interferometer detector KAGRA [16] with km size arm lengths can search for gravi- LIGO [1] opened up full-blown multi-messenger astronomy tational waves with frequencies from 10 Hz to 1 kHz. In this and cosmology, where electromagnetic waves, gravitational frequency band, resonant bar experiments [17] are comple- waves, neutrinos, and cosmic rays are utilized to explore the mentary to the interferometers [18]. Furthermore, interfer- universe. In future, as the history of electromagnetic wave ometers can be used to measure gravitational waves with the astronomy tells us, multi-frequency gravitational wave obser- frequencies between 1 kHz and 100 MHz. In fact, recently, vations will be required to boost the multi-messenger astron- the limit on gravitational waves at MHz was reported [19]. omy and cosmology. To the best of our knowledge, the measurement of 100 MHz The purpose of this letter is to present a novel idea for gravitational waves with a 0.75 m arm length interferometer extending the frequency frontier in gravitational wave obser- [20] is the highest frequency gravitational wave experiment vations and to report the first limit on GHz gravitational to date. Thus, the frequency range higher than 100 MHz is waves. As we will see below, there are experimental and remaining to be explored. Given this experimental situation, theoretical motivations to probe GHz gravitational waves. GHz experiments are desired to extend the frequency frontier. It is useful to review the current status of gravitational Theoretically, GHz gravitational waves are interesting wave observations [2]. It should be stressed that there exists from various points of view. As is well known, inflation a lowest measurable frequency. Indeed, the lowest frequency can produce primordial gravitational waves. Among the fea- we can measure is around 10−18 Hz below which the wave tures of primordial gravitational waves, the most clear sig- nature is the break of the spectrum determined by the energy scale of the inflation, which may locate at around GHz a e-mail: [email protected] (corresponding author) [17]. Moreover, at the end of inflation or just after infla- b e-mail: [email protected] tion, there may be a high frequency peak of gravitational c e-mail: [email protected] waves [21,22]. Remarkably, there is a chance to observe d e-mail: [email protected] 123 179 Page 2 of 5 Eur. Phys. J. C (2020) 80 :179 the non-classical nature of primordial gravitational waves Heisenberg model: with frequency between MHz and GHz [23]. On the other H =−μ ˆz + ˆ j − ˆ · ˆ , hand, there are many astrophysical sources producing high g B Bz 2S(i) hzjS(i) JijS(i) S( j) frequency gravitational waves [24]. Among them, primor- i i, j dial black holes are the most interesting ones because they (3) give a hint of the information loss problem. Exotic signals where an external magnetic field Bz is applied along the z- from extra dimensions may exist in the GHz band [25,26]. direction and i specifies each of the sites of the spins. The Hence, GHz gravitational waves could be a window to the second term represents the interactions between spins with extra dimensions [27]. Thus, it is worth investigating GHz coupling constants Jij. gravitational waves to understand the astrophysical process, Let us consider planar gravitational waves propagating the early universe, and quantum gravity. in the z–x plane, namely, the wave number vector of the In this letter, we propose a novel method for detecting GHz gravitational waves k has a direction kˆ = (sin θ,0, cos θ). gravitational waves with a magnon detector. First, we show Moreover, we assume that the wave length of the gravitational that gravitational waves excite magnons in a ferromagnetic waves is much larger than the dimension of the sample. This insulator. Furthermore, using experimental results of mea- is the case of cavity experiments which we utilize in the next surement of resonance fluorescence of magnons [28,29], we section. We can expand the metric perturbations in terms of demonstrate that the sensitivity to the spectral density of grav- (σ) (σ ) linear polarization tensors satisfying e e = δσσ as . × −22 × ( l )−2 [ −1/2] ij ij itational waves are around 5 4 10 λ/2π Hz . × −21 × ( l )−2 [ −1/2] ( ) = (+)( ) (+) + (×)( ) (×), at 14 GHz and 8 6 10 λ/2π Hz at 8.2 GHz, hij t h t eij h t eij (4) respectively, where l is the dimension of the ferromagnetic where we used the fact that the amplitude is approximately insulator and λ is the wave length of the gravitational waves. uniform over the sample. More explicitly, we took the repre- sentation (+) (+) h − w w 2 Graviton–magnon resonance h (t) = e i h t + ei h t , (5) 2 (×) The Dirac equation in curved spacetime with a metric gμν is (×) h − (w +α) (w +α) h (t) = e i h t + ei h t , (6) given by 2 μ γ αˆ ∂ + + ψ = ψ, where ωh is an angular frequency of the gravitational waves i eαˆ μ μ ieAμ m (1) and α represents a difference of the phases of polarizations. αˆ where γ , e, Aμ are the gamma matrices, the electronic Note that the polarization tensors can be explicitly con- μ charge, and a vector potential, respectively. A tetrad eαˆ sat- structed as ˆ ⎛ ⎞ αˆ β θ 2 − θ θ isfies eμeν ηαˆ βˆ = gμν. The spin connection is defined by cos 0 cos sin (+) 1 ⎝ ⎠ 1 αˆ νβˆ ν λβˆ 1 e = √ 0 −10 , (7) μ = e σ ˆ ∂μe + e , where σ ˆ = [γαˆ ,γˆ] ij 2 ν αˆ β λμ αˆ β 4 β 2 − θ θ θ 2 μ cos sin 0sin is a generator of the Lorentz group and νλ is the Christoffel ⎛ ⎞ 0 cos θ 0 symbol. (×) 1 e = √ ⎝ cos θ 0 − sin θ ⎠ . (8) In the non-relativistic limit, one can obtain interaction ij 2 − θ terms between a magnetic field and a spin: 0 sin 0 In Eqs. (7) and (8), we defined the + mode as a deformation H −μ 2δ + h Sî B j , (2) spin B ij ij in the y-direction. ˆ where μB =|e|/2m, S, and B are the Bohr magneton, the It is well known that the spin system (3) can be rewritten spin of the electron, and an external magnetic field, respec- by using the Holstein–Primakoff transformation [32]: 1 ⎧ tively. hij( 1) describes “effective” gravitational waves. ˆz 1 ˆ † ˆ ⎪S( ) = − C Ci , The second term shows that gravitational waves can interact ⎨ i 2 i ˆ+ ˆ † ˆ ˆ with the spin in the presence of external magnetic fields [31]. S( ) = 1 − C Ci Ci , (9) ⎪ i i We now consider a ferromagnetic sample which has N ⎩ ˆ− ˆ † ˆ † ˆ S( ) = C 1 − C Ci , electronic spins. Such a system is well described by the i i i ˆ ˆ † where the bosonic operators Ci and Ci satisfy the commuta- 1 The discussion should be developed in the proper detector frame. [ ˆ , ˆ †]=δ ± = x ± y k l k tion relations Ci C j ij and S( j) S( j) iS( j) are the Then hij is given by the Riemann tensor like Rikjlx x ,wherex is the spatial coordinate of a Fermi normal coordinate [30]. Consequently, a ladder operators. The bosonic operators describe spin waves ( l )2 with dispersion relations determined by B and J . Further- suppression factor λ/2π appears when we read off the “true” gravi- z ij tational wave from hij at the final result. more, provided that contributions from the surface of the 123 Eur.

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