Controllable Fano-type optical response and four-wave mixing via magnetoelastic coupling in a opto-magnomechanical system

Cavity optomechanics is a rapidly growing area of theoretical and experimental research that explores the coherent coupling between mechanical and optical modes through the radiation pressure force of photons being trapped inside optomechanical cavities Mey . It has been responsible for significant advances in quantum technology, particularly when non-linearities are available. We mention quantum information processing Fio , quantum walk implementation Jalil , ultrahigh-precision measurementTeu ; AliM , higher-order sidebands generation Qian , optical nonlinearity Lud , single photon blockade SS1 ; SS2 , quantum entanglement AM0 ; MYP ; SKS ; BTT , gravitation-wave detectionArva ; AliM2 , optomechanically induced transparency (OMIT) MYPOM1 ; OM1 ; OM2 , and optomechanically induced absorption (OMIA) MYPOM2 including optomechanically induced stochastic resonance Faraz . In OMIT, analogously to electromagnetic induced transparency (EIT) Imam , the radiation pressure-induced mechanical oscillations lead to destructive interference and hence significantly alter the optical response as well as its group delay Saif ; Saiff .

Another important and fascinating non-linear phenomenon is the interference and quantum coherence, called Fano resonance Yuri , where the prominent feature is a steep and sharp asymmetric line profile. Fano resonance were theoretically discovered by Ugo Fano Fano , who found that the interaction of a discrete excited state of an atom with a continuum of scattering states, is entirely different from the resonance phenomena delineated by the Lorentzian formula Fano . Fano interference has important applications such as in lasing without population inversion Gav , as a probe to study decoherence Kuh , and as a way to improve the efficiency of heat engines Chap . Moreover, the radiating two coupled-resonators model, in which the two resonators have quite different lifetime can be used to explain the Fano line shapes Tass ; Alim2 . So the lifetimes of these modes are very significant in observing the Fano line shapes Gal and can be easily obtained in plasmonic systems Gall and metamaterial structures Ali ; AliM5 . Fano resonance has been recently studied in cavity magnomechanics Kamran and magnon-qubit system Sabur .

Four-wave mixing (FWM), as the most engrossing nonlinear optical effect, is widely studied in optomechanical systems FWM1 . FWM is based on interference and quantum coherence and is extensively used to enlarge the frequency span of coherent light sources to ultraviolet and infrared FWM2 ; AliM3 . Multi-wave or FWM-mixing processes allow many vital physical processes, such as realizing optical bistability FWM3 , normal mode splitting FWM4A or generating two coexisting pairs of narrowband biphotons FWM5 . Consequently, the FWM effect has gained much attention from researchers. Jiang et al. examined the controllability and tunability of the FWM process by driving the two cavities in a two-mode cavity optomechanical system at their respective red sidebands FWM6 . Wang and Chen studied the enhancement of the FWM response FWM7 . It must be pointed out that FWM is a weak phenomenon, which makes it hard to be realized in experiments. Therefore new directions are being proposed to improve the strength of those nonlinear phenomena.

The hybridization of different modes in cavity magnomechancis (CMM) opens up a promising platform to explore numerous non-linear quantum effects Tang . Magnomechanical systems consider a geometry such that an optical mode directly (indirectly) couples with a magnon mode (mechanical mode). They are based totally on magnons-quanta of spin waves in Ferrimagnetic materials Heyroth , such as yttrium iron garnet (YIG), which provide a versatile and new platform to investigate strong light-matter interactions Gros . Strong interactions are due to two main reasons: (1) The Spin density of the YIG material is very high; (2) The damping rate of the YIG material is very low. Therefore, in the last decade, CMM has been explored in theoretical as well as an experimental domain because of its potential applications for observing macroscopic quantum states Jiee , quantum sensing Usami and magnomechanically induced transparency (MMIT) Kamran ; AliM4 ; XLi . In CMM systems, the microwave (MW) cavity field is coupled to a magnon mode in a ferromagnetic YIG sphere MYP as well as the magnon mode also couples to a (vibrational) mechanical mode of the sphere through the magnetostrictive force Deng . Such magnomechanical interaction is dispersive Tang ; Rome in a similar fashion to radiation pressure interaction, leading to several theoretical and experimental investigations to study various quantum phenomena in cavity magnomechanical systems analogous to cavity optomechanics, such as magnomechanically induced transparency (MMIT) Tang ; Kamran , magnetically tunable slow light Cui ; XLi , exceptional points Den , tripartite magnon-photon-phonon entanglement and Einstein-Podolsky-Rosen steerable nonlocality Tann , cooling of mechanical motion Asjad , phonon lasing Cong and parity-time-related phenomena Annl .

In this paper, we propose an unconventional and advantageous opto-magnomechanical system (OMMS) setup which is a composite architecture based on the indirect coupling of optical photons and magnon modes via the phononic mode (vibrational mode of the magnon) as shown in Fig. 1(a). The magnetoelastic coupling is responsible for the geometric deformation of the magnon in a YIG bridge which forms the vibrational motion. We consider that the phonon frequency is much smaller than the frequency of the magnon and in this situation, the dispersive interaction between the phonon and magnon modes becomes dominant (See Appendix A for details). This present scheme can be well fabricated by placing a high reflectivity small size mirror on the surface (edge) of the YIG bridge and therefore, the vibrational motion of the magnon can be easily coupled to an optical cavity through the radiation pressure, assembling an optomechanical cavity. However, the tiny size of the mirror does not affect the vibrations of the YIG bridge. In addition, such kind of indirect coupling between optical photons and magnons can easily be realized by assuming the low mechanical damping rate $\gamma_{b}$ both in the optomechanical and magnomechanical cooperativities $C_{om}=\frac{G_{c}^{2}}{\kappa_{c}\gamma_{b}}>1$ ($\kappa_{c}$ being the cavity decay rate), and $C_{mm}=\frac{G_{m}^{2}}{\kappa_{m}\gamma_{b}}>1$ ($\kappa_{m}$ being the magnon decay rate) respectively, under present-day technology Mey ; Tang ; CAPE . Moreover, experimentally feasible parameters have been taken for the current scheme. In this paper, we have presented several interesting results for generating/controlling Fano profiles and the FWM by varying many system parameters.

The current article is arranged as follows. In Sect. 2, we present the theoretical model and the corresponding Hamiltonian of the OMMS. Section 3 designates the dynamics of the OMMS by exploiting the standard quantum Langevin approach, to get the expression for the optical response and the FWM process. In Sect. 4, we analytically discuss the optical response and the FWM process in detail. Finally, the conclusion is given in the last Sect. 5.

The opto-magnomechanical system under consideration consists of an optical mode $c$ and a slab of YIG nanoresonator in which collective excitation of a number of spins are coupled to mechanical vibrations Heyroth . It is vital to mention here that a micrometer size YIG bridge can well support a gigahertz (megahertz) magnon mode (vibrational mode) Heyroth . Therefore, such a magnomechanical system has a strong magnon-phonon dispersive coupling disp2 ; mag ; Kittel . The present scheme can be accomplished by placing a small mirror on the edge (surface) of the YIG bridge as shown in Fig. 1 (a), and in this way, an optical field can be coupled to the magnetostriction-induced mechanical displacement via radiation pressure force. The fabricated mirror should be highly refractive, light, and small such that it will not affect the mechanical properties of the ferrimagnet. Therefore, in this way the magnomechanical displacement can be probed by light exploiting the optomechanical interaction, which is further used to determine the magnon population ZRCS . Moreover, the deformation mode (displacement of the mechanical resonator) is along the direction of the sticky surface area of the slab, such that the tightly coupled YIG bridge and the mirror resonate with the same frequency. In addition, the dispersive magnetostrictive interaction between magnon and phonon modes depends on the resonance frequencies of these coupled modes Zha .

It is important to mention here that Fig.1 is only a diagrammatic sketch of one of the possible configurations to couple the magnomechanically induced displacement to an optical cavity. In addition, one doesn’t have to attach a mirror (on the surface of the microbridge) to form the optical cavity. Alternatively, one can also adopt the analogy of taking a ”membrane-in-the-middle” configuration by positioning the YIG sample inside the optical cavity. In this setup, the magnomechanically induced displacement can also be dispersively coupled to the optical cavity. In this case, no additional damping of the mechanical mode is needed. In the current study, we consider a large size crystal such that the magnon frequency is considered to be much greater than the vibrational frequency of the phononic mode such that the dispersive interaction between magnon and phonon modes, becomes dominant Tang ; Ballestero . The Hamiltonian of the present OMMS reads

$$H/\hbar=H_{\circ}+H_{in}+H_{dr},$$

(1)

where

	$\displaystyle H_{\circ}$	$\displaystyle=$	$\displaystyle\omega_{c}c^{\dagger}c+\omega_{m}m^{\dagger}m+\frac{\omega_{b}}{2}(q^{2}+p^{2}),$		(2)
	$\displaystyle H_{in}$	$\displaystyle=$	$\displaystyle g_{mb}m^{\dagger}mq-g_{cb}c^{\dagger}cq,$		(3)
	$\displaystyle H_{dr}$	$\displaystyle=$	$\displaystyle i\Omega(m^{\dagger}e^{-i\omega_{\circ}t}-me^{i\omega_{\circ}t})$		(4)
			$\displaystyle+i\sum_{f=L,p}\varepsilon_{f}(c^{\dagger}e^{-i\omega_{f}t}-ce^{i\omega_{f}t}).$

where $c(m)$ and $c^{{\dagger}}\left(m^{{\dagger}}\right)$ are the respective annihilation and creation operators for the cavity (magnon) mode. $p$ and $q$ are the dimensionless momentum and position of the phononic mode (deformation vibrational mode), considered as a mechanical resonator, which simultaneously couples to the cavity (magnon) mode via radiation pressure interaction $g_{cb}$ (dispersive magnetostrictive interaction $g_{mb}$). The details of the optomechanical coupling $g_{cb}$ can be seen in CGEN while the detailed discussion about the magnomechanical coupling $g_{mb}$ is present in Appendix A. In addition, $\omega_{c}$($\omega_{m}$)[$\omega_{b}$] is the resonance frequencies of the cavity mode (magnon mode) [phononic mode]. $\omega_{m}$ can be flexibly tuned via $\omega_{m}=\gamma_{0}H_{m}$, where $H_{m}$ is the bias magnetic field and $\gamma_{0}$ is the gyromagnetic ratio. The Rabi frequency $\Omega=\left(\sqrt{5}/4\right)\gamma_{0}\sqrt{N_{s}}B_{0}$ Jiee represents the coupling strength of the input laser drive field with frequency $\omega_{0}$, where $\gamma_{0}=28$GHz/T, $B_{0}=3.9\times 10^{-9}$T and the total number of spins $N_{s}=\rho V$, where $\rho=4.22\times 10^{27}m^{-3}$ denotes the spin density and $V$ being the volume of the YIG bridge. In addition, we apply the Holstein-Primakoff transformation to the bosonic operators for the magnon ($m$ and $m^{\dagger}$), which basically form from the collective motion of the spins. In fact, $\Omega$ can be derived by the basic low-lying excitations assumption i.e., $\langle m^{\dagger}m\rangle\ll 2Ns$, where $N$ ($s=\frac{5}{2}$) is the total number of spin states (the spin number of the ground state Fe³⁺ ion) in YIG Holstein . Furthermore, we also drive the optical cavity through external laser field with amplitude $\varepsilon_{L}=\sqrt{\kappa_{c}\wp_{L}/\hbar\omega_{L}}$, where $\kappa_{c}$ is the decay rate of the optical cavity mode, $\omega_{L}$ ($\wp_{L}$) is the frequency (power) of the external input laser field. At the same time, the cavity mode is also driven by a weak probe field with amplitude $\varepsilon_{L}$ and frequency $\wp_{p}$. The corresponding Hamiltonian of the OMMS, after the rotating wave approximation at the coupling frequency $\omega_{L}$, is given by

	$\displaystyle H/\hbar$	$\displaystyle=$	$\displaystyle\Delta_{c}^{0}c^{{\dagger}}c+\Delta_{m}^{0}m^{{\dagger}}m+\frac{\omega_{b}}{2}(q^{2}+p^{2})+g_{mb}m^{{\dagger}}mq$		(5)
			$\displaystyle-g_{cb}c^{\dagger}cq+i\Omega(m^{{\dagger}}-m)+i\varepsilon_{L}(c^{\dagger}-c)$
			$\displaystyle+i\varepsilon_{p}(c^{\dagger}e^{-i\delta t}-ce^{i\delta t}),$

where $\Delta_{m}^{0}=\omega_{m}-\omega_{L}$, $\Delta_{c}^{0}=\omega_{c}-\omega_{L}$ and $\delta=\omega_{p}-\omega_{L}$.

Now, we discuss the dynamics of this OMMS by writing the corresponding QLEs, which are given by

	$\displaystyle\dot{q}$	$\displaystyle=$	$\displaystyle\omega_{b}p,$
	$\displaystyle\dot{p}$	$\displaystyle=$	$\displaystyle\omega_{b}q-\gamma_{b}p-g_{mb}m^{\dagger}m+g_{cb}c^{\dagger}c+\xi,$
	$\displaystyle\dot{c}$	$\displaystyle=$	$\displaystyle-(i\Delta_{c}^{0}+\kappa_{c})c+ig_{cb}cq+\varepsilon_{L}+\varepsilon_{p}e^{-i\delta t}+\sqrt{\kappa_{c}}c^{in},$
	$\displaystyle\dot{m}$	$\displaystyle=$	$\displaystyle-(i\Delta_{m}^{0}+\kappa_{m})m-ig_{mb}mq+\Omega+\sqrt{\kappa_{m}}m^{in},$		(6)

where $\kappa_{c}$($\kappa_{m}$) is the decay rate of the cavity mode (magnon mode), and $\gamma_{b}$ is the damping rate of the vibrational mode. Furthermore, $\xi$, $c^{in}$ and $m^{in}$ are, respectively, the input noise operators for the vibrational mode, cavity modes and magnon mode. The above QLEs can be linearized by writing $z=z_{s}+\delta z$, $(z=p,q,c,m)$, where $z_{s}$ ($\delta z$) is the steady state value (fluctuation) of any operator. For a sufficiently large amplitude of magnon and cavity modes, average value should be much greater than the fluctuation. Substituting the above ansatz into Eq.(6) and ignoring the higher order perturbations, the average values of the dynamical operators are given by

	$\displaystyle p_{s}$	$\displaystyle=$	$\displaystyle 0,$
	$\displaystyle q_{s}$	$\displaystyle=$	$\displaystyle g_{cb}\frac{\left|c_{s}\right|^{2}}{\omega_{b}}-g_{cb}\frac{\left|m_{s}\right|^{2}}{\omega_{b}},$
	$\displaystyle c_{s}$	$\displaystyle=$	$\displaystyle\frac{\varepsilon_{L}}{(i\Delta_{c}+\kappa_{c})},$
	$\displaystyle m_{s}$	$\displaystyle=$	$\displaystyle\frac{\Omega}{(i\Delta_{m}+\kappa_{m})},$		(7)

where $\Delta_{c}=\Delta_{c}^{0}-g_{cb}q_{s}$ and $\Delta_{m}=\Delta_{m}^{0}+g_{mb}q_{s}$ are the effective cavity and magnon mode detunings, respectively, which include the slight shift of frequency. The corresponding linearized QLEs, after ignoring quantum and thermal noise terms, are:

	$\displaystyle\delta\dot{q}$	$\displaystyle=$	$\displaystyle\omega_{b}\delta p,$
	$\displaystyle\delta\dot{p}$	$\displaystyle=$	$\displaystyle\omega_{b}\delta q-\gamma_{b}\delta p-G_{m}\delta m^{\dagger}-G_{m}^{\ast}\delta m+G_{c}\delta c^{\dagger}+G_{c}^{\ast}\delta c,$
	$\displaystyle\delta\dot{c}$	$\displaystyle=$	$\displaystyle-(i\Delta_{c}+\kappa_{c})\delta c+iG_{c}\delta q+\varepsilon_{p}e^{-i\delta t},$
	$\displaystyle\delta\dot{m}$	$\displaystyle=$	$\displaystyle-(i\Delta_{m}+\kappa_{m})\delta m-iG_{m}\delta q.$		(8)

Here, $G_{mb}=\sqrt{2}g_{mb}m_{s}$ and $G_{cb}=\sqrt{2}g_{cb}c_{s}$ are the effective magnomechanical and optomechanical coupling, respectively. The obtained fluctuation operators can then be solved by setting: $\delta z=z_{-}e^{-i\delta t}+z_{+}e^{i\delta t}$ where $z_{-}$ and $z_{+}$ (with $z=q,p,c,m$) are much smaller than $z_{s}$. It is noteworthy to mention that the fluctuation operators contain many Fourier components after expansion. Moreover, the high-order terms can safely be neglected in the limit of weak signal field. Substituting the above ansatz into Eq. (8) and taking the lowest order in $\varepsilon_{p}$, we establish the final solution for the coefficients of optical response and FWM process as

$$c_{-}=\frac{[\omega_{b}^{2}-\delta^{2}-i\Theta_{n}]\varepsilon_{p}}{[\omega_{b}^{2}-\delta^{2}-i\Theta_{n}]\Omega_{2}^{c}-i\omega_{b}G_{cb}^{2}},$$

(9)

and

$$c_{+}=\frac{iG_{cb}^{2}\omega_{b}\Theta_{p}\varepsilon_{p}}{[\omega_{b}^{2}-\delta^{2}+i\Theta_{p}]\left(\Omega_{1}^{c}\right)^{\ast}-i\omega_{b}G_{cb}^{2}},$$

(10)

where

	$\displaystyle\Theta_{n}$	$\displaystyle=$	$\displaystyle\delta\gamma_{b}-\omega_{b}\chi_{12},\ \ \ \ \ \ \ \ \ \Theta_{p}=\delta\gamma_{b}+\omega_{b}\alpha_{12},$
	$\displaystyle\chi_{12}$	$\displaystyle=$	$\displaystyle G_{mb}^{2}\chi_{1}+G_{cb}^{2}\chi_{2},\ \ \ \chi_{1}=[\Omega_{2}^{m}-\Omega_{1}^{m}][\Omega_{1}^{m}\Omega_{2}^{m}]^{-1},$
	$\displaystyle\chi_{2}$	$\displaystyle=$	$\displaystyle[\Omega_{1}^{c}]^{-1},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \alpha_{2}=[\left(\Omega_{2}^{c}\right)^{\ast}]^{-1},$
	$\displaystyle\alpha_{12}$	$\displaystyle=$	$\displaystyle G_{mb}^{2}\alpha_{1}+G_{ab}^{2}\alpha_{2},$
	$\displaystyle\alpha_{1}$	$\displaystyle=$	$\displaystyle[\left(\Omega_{1}^{m}\right)^{\ast}-\left(\Omega_{2}^{m}\right)^{\ast}][\left(\Omega_{1}^{m}\right)^{\ast}\left(\Omega_{2}^{m}\right)^{\ast}]^{-1},$
	$\displaystyle\Omega_{1}^{m}$	$\displaystyle=$	$\displaystyle\kappa_{m}-i(\Delta_{m}+\delta),\ \ \ \ \ \ \ \ \ \Omega_{2}^{m}=\kappa_{m}+i(\Delta_{m}-\delta),$
	$\displaystyle\Omega_{1}^{c}$	$\displaystyle=$	$\displaystyle\kappa_{c}-i(\Delta_{c}+\delta),\ \ \ \ \ \ \ \ \ \ \ \ \ \Omega_{2}^{c}=\kappa_{c}+i(\Delta_{c}-\delta).$

To investigate the optical properties of the output field and the FWM process in OMMS, we start with the following standard input-output relation DFW

$$c_{out}(t)={c}_{in}(t)-\sqrt{2\kappa_{c}}c(t).$$

(11)

In the above formula, ${c}_{in}$ and $c_{out}$ are, respectively the input field and output field operators. Furthermore, the expectation value of the output field is given by

	$\displaystyle c_{out}(t)$	$\displaystyle=$	$\displaystyle\left(\sqrt{2\kappa_{c}}c_{0}-\frac{\varepsilon_{L}}{\sqrt{2\kappa_{c}}}\right)e^{-i\omega_{L}t}$		(12)
			$\displaystyle+\left(\sqrt{2\kappa_{c}}c_{-}-\frac{\varepsilon_{p}}{\sqrt{2\kappa_{c}}}\right)e^{-i(\delta+\omega_{L})t}$
			$\displaystyle+\sqrt{2\kappa_{c}}c_{+}e^{i(\delta-\omega_{L})t},$
		$\displaystyle=$	$\displaystyle\left(\sqrt{2\kappa_{c}}c_{0}-\frac{\varepsilon_{L}}{\sqrt{2\kappa_{c}}}\right)e^{-i\omega_{L}t}$
			$\displaystyle+\left(\sqrt{2\kappa_{c}}c_{-}-\frac{\varepsilon_{p}}{\sqrt{2\kappa_{c}}}\right)e^{-i\omega_{p}t}$
			$\displaystyle+\sqrt{2\kappa_{c}}c_{+}e^{i(\omega_{p}-2\omega_{L})t},$

where the second (third) term corresponds to the components at the frequencies $\omega_{p}$ ($2\omega_{L}-\omega_{p}$) representing the optical (FWM) response of the system induced by the anti-Stokes (Stokes) scattering process. In addition, $c_{out}(t)$ can also be expanded as

	$\displaystyle\left\langle c_{out}(t)\right\rangle$	$\displaystyle=$	$\displaystyle c_{out,0}e^{-i\omega_{L}t}+c_{out,-}e^{-i(\delta+\omega_{L})t}$		(13)
			$\displaystyle+c_{out,+}e^{i(\delta-\omega_{L})t},$

Comparing Eq. (12) and Eq. (13), we easily arrive at

	$\displaystyle c_{out,-}$	$\displaystyle=$	$\displaystyle\sqrt{2\kappa_{c}}c_{-}-\frac{\varepsilon_{p}}{\sqrt{2\kappa_{c}}},$
	$\displaystyle c_{out,+}$	$\displaystyle=$	$\displaystyle\sqrt{2\kappa_{c}}c_{+}.$		(14)

Finally, we obtain the following total output field relation at the probing frequency $\omega_{p}$

$\displaystyle\varepsilon_{T}$

$\displaystyle=$

$\displaystyle\frac{\sqrt{2\kappa_{c}}c_{out,-}+\varepsilon_{p}}{\varepsilon_{p}}=\frac{2\kappa_{c}c_{-}}{\varepsilon_{p}}=\Lambda+i\tilde{\Lambda},$

(15)

where $\Lambda$=Re[$\varepsilon_{T}$] and $\tilde{\Lambda}$=Im[$\varepsilon_{T}$] are, respectively, represent the in-phase and out-of-phase quadratures of the output probe field. In addition, one can obtained the absorptive (dispersive) behavior of the output probe field from $\Lambda$($\tilde{\Lambda}$) AMEPJD .

Similarly, at Stokes scattering, the FWM intensity can be obtained from output stokes field as

$$\varepsilon_{FWM}=\left|\frac{\sqrt{2\kappa_{c}}c_{out,+}}{\varepsilon_{p}}\right|^{2}=\left|\frac{2\kappa_{c}c_{+}}{\varepsilon_{p}}\right|^{2}.$$

(16)

In conclusion, we have introduced a model consisting of an optomechanical system directly coupled with a ferromagnetic material. The deformation of the mechanical mode is taken simultaneously due to an optomechanical coupling and the magnetoelasticity of the ferromagnet. We report that the magnetostrictively induced displacement evolves to transfigure the standard OMIT window to Fano profiles in the output field at the probe frequency. The asymmetry of the Fano profiles can be well tuned/controlled by appropriate adjusting the magnon (cavity) detuning and the effective magnomechanical coupling. Remarkably, it is found that the magnetoelastic interaction also gives rise to the FWM phenomenon. FWM intensity can be resonantly tuned by the system’s parameters. For example, modulating the effective magnomechanical coupling can realize conversion among a double and triple peak signals. Similarly, in the case of magnon detuning, FWM exhibits the characteristics of symmetric double, triple and even quadruple peak signals. In addition, one can also control the strength of the FWM signals by employing the magnon (cavity) detuning and effective magnomechanical coupling. Moreover, the FWM spectrum exhibits suppressive behavior upon increasing (decreasing) the magnon (cavity) decay. The present OMMS opens new perspectives in highly sensitive detection and quantum information processing, which shall be addressed elsewhere.

The magnetoelastic coupling is fully explained by the interaction between the elastic strain and magnetization of the magnetic material RBWD ; Kittel . The magnetoelastic energy density for a cubic crystal is given by KIT

	$\displaystyle F_{me}$	$\displaystyle=$	$\displaystyle\frac{2B_{1}}{M_{s}^{2}}\left(M_{x}M_{y}\epsilon_{xy}+M_{x}M_{z}\epsilon_{xz}+M_{y}M_{z}\epsilon_{yz}\right)$		(17)
			$\displaystyle+\frac{B_{2}}{M_{s}^{2}}\left(M_{x}^{2}\epsilon_{xx}+M_{y}^{2}\epsilon_{yy}+M_{z}^{2}\epsilon_{zz}\right),$

where the magnetoelastic coupling coefficients are represented by $B_{1}$ and $B_{2}$. $M_{s}$ being the saturation magnetization. In addition, magnetization components are represented by $M_{x}$, $M_{y}$ and $M_{z}$. Furthermore, $\epsilon_{nm}$ ($n,m\in{x,y,z}$) defines the strain tensor of the cubic magnetic crystal.

Now, the magnetization can be quantized using the Holstein-Primakoff transformation

$$m=\sqrt{\frac{V}{2\hbar\gamma M_{s}}}\left(M_{x}-iM_{y}\right),$$

(18)

where $V$ is the volume of the cubic crystal and $m$ defines the magnon mode operator. After rearranging, we obtain

	$\displaystyle M_{x}$	$\displaystyle=$	$\displaystyle\sqrt{\frac{\hbar\gamma M_{s}}{2V}}\left(m+m^{\dagger}\right),$
	$\displaystyle M_{y}$	$\displaystyle=$	$\displaystyle i\sqrt{\frac{\hbar\gamma M_{s}}{2V}}\left(m-m^{\dagger}\right),$		(19)

and

$$M_{z}=\left[M_{s}^{2}-M_{x}^{2}-M_{y}^{2}\right]^{\frac{1}{2}}\simeq M_{s}-\frac{\hbar\gamma}{V}m^{\dagger}m.$$

(20)

Putting the value of $M_{x,y,z}$ in Eq. (17) and then integrating over the volume of the cubic magnetic crystal, the first term in Eq. (17) yields the semiclassical magnetoelastic Hamiltonian,

	$\displaystyle H_{a}$	$\displaystyle\simeq$	$\displaystyle\frac{B_{1}}{M_{s}}\frac{\hbar\gamma}{V}m^{\dagger}m\int dl^{3}\left(\epsilon_{xx}+\epsilon_{yy}-2\epsilon_{zz}\right)$		(21)
			$\displaystyle+\frac{B_{1}}{M_{s}}\frac{\hbar\gamma}{2V}\left(m^{2}+m^{\dagger 2}\right)\int dl^{3}\left(\epsilon_{xx}-\epsilon_{yy}\right)$
			$\displaystyle+\frac{B_{1}}{M_{s}^{2}}\frac{\hbar^{2}\gamma^{2}}{V^{2}}m^{\dagger}mm^{\dagger}m\int dl^{3}\epsilon_{zz},$

and the second term of magnetoelastic energy density from Eq. (17) yields the Hamiltonian

	$\displaystyle H_{b}$	$\displaystyle\simeq$	$\displaystyle\frac{B_{2}}{M_{s}}\frac{\hbar\gamma}{V}\left(m^{2}-m^{\dagger 2}\right)\int dl^{3}\epsilon_{xy}$		(22)
			$\displaystyle+\frac{2B_{2}}{M_{s}}\sqrt{\frac{\hbar\gamma M_{s}}{2V}}\left[M_{s}-\frac{\hbar\gamma}{V}m^{\dagger}m\right]$
			$\displaystyle\times\left[m\int dl^{3}\left(\epsilon_{xz}+i\epsilon_{yz}\right)+H.c\right].$

Hence, the total magnetoelastic Hamiltonian, which is sum of $H_{a}$ and $H_{b}$, implies total magnon-phonon interactions

The magnetoelastic displacement can be defined as:

$$\vec{u}\left(x,y,z\right)=\sum\limits_{l,m,n}d^{(l,m,n)}\vec{\chi}^{(l,m,n)}\left(x,y,z\right),$$

(23)

where $\vec{\chi}^{(l,m,n)}\left(x,y,z\right)$ ($d^{(l,m,n)}$) denotes the normalized displacement eigenmode (the corresponding amplitude).

$$d^{(l,m,n)}=d_{zpm}^{(l,m,n)}\left(b_{l,m,n}+b_{l,m,n}^{\dagger}\right),$$

(24)

where $d_{zpm}^{(l,m,n)}$ represents the amplitude of the zero-point motion. In addition, $b_{l,m,n}$ and $b_{l,m,n}^{\dagger}$ are, respectively the corresponding phononic annihilation and creation operators. Substituting Eqs. (23) and (24) in Eq. (21), the dispersive kind of interaction can be obtained,

$$H_{a}=\hbar\sum\limits_{l,m,n}g_{mb}^{(l,m,n)}m^{\dagger}m\left(b_{l,m,n}+b_{l,m,n}^{\dagger}\right),$$

(25)

where $g_{mb}^{(l,m,n)}$ represents the dispersive coupling strength between magnon and phonon, is given by

	$\displaystyle g_{mb}^{(l,m,n)}$	$\displaystyle=$	$\displaystyle\frac{B_{1}}{M_{s}}\frac{\gamma}{V}\int dl^{3}d_{zmp}^{(l,m,n)}$		(26)
			$\displaystyle\times\left[\frac{\partial\chi_{x}^{(l,m,n)}}{\partial x}+\frac{\partial\chi_{y}^{(l,m,n)}}{\partial y}-2\frac{\partial\chi_{z}^{(l,m,n)}}{\partial z}\right].$

In case of a one dimensional mechanical mode oscillation, we obtain the simplified hamiltonian as:

$$H_{a}=\hbar g_{mb}m^{\dagger}m\left(b+b^{\dagger}\right),$$

(27)

which is the same as the first part of $H_{i}$ (see Eq. (3)), since $q=\frac{1}{\sqrt{2}}\left(b+b^{\dagger}\right)$. Note that we use the condition of low-lying magnon excitations to derive the above and therefore, the magnon excitation $m^{\dagger}m$ second order term (third line of Eq. (21) ) can be safely eliminated.

All data used during this study are available within the article.