Tunable Entanglement in Cavity-Magnon Optomechanics
Ming-Yue Liu Xian-Xian Huang Wei Xiong





Department of Physics, Wenzhou University, Zhejiang 325035, China


Department of Physics, Wenzhou University, Zhejiang 325035, China






xiongweiphys@wzu.edu.cn
Department of Physics, Wenzhou University, Zhejiang 325035, China

































Cavity optomechanics, providing an inherently nonlinear interaction between photons and phonons, have shown enomerous potential in generating macroscopic quantum entanglement. Here we propose to realize diverse bipartite and tripartite entanglement in cavity-magnon optomechanics. By introducing magnons to standard cavity optomechanics, not only tunable optomechanical entanglement and magnon-magnon entanglement can be achieved, but also flexible tripartite entanglement including magnon-photon-phonon entanglement, magnon-magnon-photon and -phonon entanglement can be generated. Moreover, optimal bipartite and tripartite entanglement can be achieved by tuning parameters. We further show that all entanglement can be enhanced via engineering the magnon-photon coupling, and is proven to be robust against the bath temperature within the survival temperature. Besides, we find that the optomechanical entanglement can be protected or restored by bad magnons with large decay rate, while other entanglement is severely reduced. The results indicate that our proposal provides a novel avenue to explore and control tunable macroscopic quantum effects in hybrid cavity-magnon optomechanics.




Tunable Entanglement in Cavity-Magnon Optomechanics
    Wei Xiong
    April 29, 2024
===================================================





§ INTRODUCTION


Magnons <cit.>, quanta of collective spin-wave excitations in magnetically ordered materials like yttrium iron garnet (YIG), have garnered considerable interest in quantum physics and condensed matter physics <cit.>. With unique properties such as high spin density and low energy loss <cit.>, diverse novel phenomena including strong <cit.> and ultrastrong coupling <cit.>, qubit-magnon coupling <cit.>, magnon dark modes and gradient memory <cit.>, magnon Kerr effect <cit.>, spin current manipulation <cit.>, exceptional points <cit.>, axion detection <cit.>, superadiant phase transition <cit.>, dissipative coupling <cit.>, and strong spin coupling <cit.>. Recently, special attention has been paid to macroscopic quantum entanglement associated with magnons <cit.>. This is because it is crucial for understanding the classical-quantum boundary <cit.> and can be regarded as an important resource for quantum information science <cit.>. However, to produce such macroscopic entanglement, nonlinear effects are always prerequired <cit.>. The extensively explored cavity optomechanics <cit.>, hybridized by photons in a cavity mode weakly coupled to phonons in a mechanical resonator through radiation pressure <cit.>, can provide the natural nonlinearity for investigating optomechanical entanglement <cit.> and other intriguing effects such as sensing <cit.>, ground-state cooling <cit.>, squeezed light generation <cit.>, nonreciprocity <cit.>, optomechanically induced transparency <cit.>, coupling enhancement <cit.>, multistability <cit.>, and high-order exceptional points <cit.>. This indicates that combination of magnonics and optomechanics is a potential path to investigate macroscopic magnon entanglement.

Owing to the strong photon-magnon coupling in planar <cit.> and 3D cavities <cit.>, hybridizing magnons with optomechanics to form cavity-magnon optomechanics becomes feasible <cit.> for utilizing their individual advantages. In this work, we propose to generate diverse bipartite and tripartite entanglement in hybrid cavity-magnon optomechanics. Compared to the standard cavity optomechanics, the optomechanical entanglement can be adjusted flexibly by the introduced magnons in the Kittle mode of a single sphere. Specifically, the optimal optomechanical entanglement can be predicted by tuning the detuning of the Kittle mode from the driving field to be resonant with the mechanical mode (i.e., Δ_m=ω_b). But when we tune this detuning to be anti-resonant with the mechanical mode (i.e., Δ_m=-ω_b), the optomechanical entanglement is weakest, while the magnon-photon entanglement, the magnon-phonon, as well as the magnon-photon-phonon entanglement are approximately optimal. These entanglement can be further enhanced by engineering the coupling between magnons and photons. In addition, the tunable magnon-magnon entanglement and the tripartite entanglement including the magnon-magnon-photon entanglement and the magnon-magnon-phonon entanglement can also be generated in the proposed cavity-magnon optomechanics with two spheres. Tuning one Kittle mode to be resonant with the mechanical mode (i.e., Δ_1=ω_b), while the other to be anti-resonant with the mechanical mode (i.e., Δ_2=-ω_b), or vice versa, the optimal magnon-magnon entanglement is predicted, and its value approximately equals to half the optomechanical entanglement. But to have optimal magnon-magnon-photon and -phonon entanglement, two Kittle modes are needed to be resonant (i.e., Δ_1=Δ_2). We further show that the optomechanical entanglement can be protected or restored by using bad magnons with large decay rate, but the magnon-magnon entanglement as well as the magnon-magnon-photon and -phonon entanglement are severly reduced. Besides, the bath temperature effect on these entanglement are also investigated. We find that for the weaker magnon-photon coupling (i.e., g_m<G), the stronger optomechanical entanglement than that of the magnon-magnon entanglement can be obtained within the survival temperature, but the situation is reversed when the magnon-photon coupling becomes comparable with the optomechanical coupling (i.e., g_m≈ G). Moreover, the survival temperature of the magnon-magnon entanglement is more robust against the magnon-photon coupling than that of the optomechanical entanglement. For the tripartite entanglement, we show that the magnon-magnon-phonon entanglement is stronger than the magnon-magno-photon entanglement within their survival temperature, owing to the mechanical cooling. All achieved entanglement exhibit robust against the bath temperature. Our proposal opens an alternative path to study tunable macroscopic quantum effect in hybrid cavity-magnon optomechanics.

This rest paper is organized as follows: In Sec. <ref>, the model
is described and the entanglement quantification is given.
Then in Sec. <ref>, we study optomechanical entanglement mediated by magnons in the proposed cavity-magnon optomechanics, where only a single sphere is considered. In Sec. <ref>, tunable magnon-magnon entanglement induced by the optomechanical interface is investigated via examing the effect of system parameters. In Sec. <ref>, we further study tripartite entanglement including magnon-magnon-photon and magnon-magnon-phonon entanglement.  Finally, a conclusion is given in Sec. <ref>.



§ MODEL AND ENTANGLEMENT QUANTIFICATION



 §.§ Model



We consider a hybrid cavity-magnon optomechanics consisting of a mechanical resonator weakly coupled to a cavity containing two separated YIG spheres, as depicted in Fig. <ref>(a). The magnons in the Kittle modes of two spheres are strongly coupled to the photons in the cavity mode with the uniform coupling strength g_1=g_2=g_m. By imposing an external field, with the frequency ω_0 and the amplitude Ω, to the cavity, the Hamiltonian of the proposed hybrid system with respect to ω_0, can be given by (setting ħ=1)

    H_ sys=   H_ OM+∑_j=1,2H_ CM^(j)+i Ω(a^†-a),

where H_ OM=Δ_a a^† a+ω_b/2(q^2+p^2)+g_0 a^† a q denotes the Hamiltonia of the standard cavity optomechanics, describing the interaction between the cavity mode and the mechanical mode via the radiation pressure, with the single-photon optomechanical coupling strength g_0. Δ_a=ω_a-ω_0, with ω_a being the frequency of the cavity mode, is the frequency detuning of the cavity mode from the driving field, and ω_b is the frequency of the mechanical mode. The Hamiltonian H_ CM^(j)=Δ_j m^†_j m_j+g_j(m_j a^†+m_j^† a) charaterizes the interaction between the cavity mode and the Kittle mode of the jth sphere. Δ_j=ω_j-ω_0 is the frequency detuning of the Kittle mode of the jth sphere from the driving field. The creation (annihilation) operators the cavity mode and the Kittle mode of the jth sphere are denoted by a^† (a) and m_j^† (m_j), respectively. q and p are dimensionless position and momentum quadratures of the mechanical mode.



 §.§ Quantum Langevin equation


By taking input noises and dissipations into account, the dynamics of the proposed system can be governed by the quantum Langevin equations, i.e.,

    q̇=   ω_b p,
    ṗ=   -ω_b q-g_0 a^† a-γ_b p+ξ,
    ṁ_̇j̇=   -(κ_j+i Δ_j) m_j-i g_j a+√(2 κ_j)ζ_j,
    ȧ=   -(κ_a+iΔ_a) a-ig_0 aq-ig_j m_j+Ω+√(2 κ_a)μ,

where κ_a, κ_j, and γ_b are the decay rates of the cavity mode, the Kittle mode of the jth sphere, and the mechanical mode, respectively. For simplicity, we assume that the Kittle modes of two spheres have the identical decay rate κ_1=κ_2=κ_m below. μ, ζ_j and ξ are the vaccum input noise operators for the cavity mode, the mechanical mode, and the Kittle mode of the jth sphere, having the zero mean values ⟨μ⟩=⟨ζ_j⟩=⟨ξ⟩=0. Under the Markovian approximation, two-time correlation functions of these input noise operators are

    ⟨μ(t) μ^†(t^')⟩=   (n̅_a+1) δ(t-t^'),
    ⟨μ^†(t) μ(t^')⟩=   n̅_a δ(t-t^'),
    ⟨ζ_j(t) ζ_j^†(t^')⟩=   (n̅_j+1) δ(t-t^'),
    ⟨ζ_j^†(t) ζ_j(t^')⟩=   n̅_j δ(t-t^'),
    ⟨ξ(t) ξ(t^')+ξ(t^') ξ(t)⟩≃    2γ_b(2 n̅_b+1) δ(t-t^'),

where n̅_σ=[exp(ħω_σ / k_B T)-1]^-1 (σ=a,b,j) is the mean thermal excitation number, with k_B being the Boltzmann constant and T the bath temperature.

In fact, Eq. (<ref>) can be linearized when the external driving field is strong. For this, we first rewrite the operator O=⟨ O⟩+δ O (O=q,p,a,m_j), where ⟨ O⟩ is the steady-state value and δ O is the fluctuation operator. Then we substitue the redefined operator O into Eq. (<ref>) and neglect the high-order fluctuation terms,  resulting in
	
    δ̇ ̇q̇=   ω_b δ p,
    δ̇ ̇ṗ=   -ω_b δ q-(G^*δ a+G δ a^†)/√(2)-γ_b δ p+ξ,
    δ̇ ̇ṁ_̇j̇=   -(κ_j+i Δ_j) δ m_j-i g_j δ a+√(2 κ_j)ζ_j,
    δ̇ ̇ȧ=   -(κ_a+iΔ̃_a) δ a-iG q/√(2)-ig_jδ m_j+√(2 κ_a)μ,

where Δ̃_a=Δ_a+g_0⟨ q⟩ is the effective frequency detuning, induced by the displacement of the mechanical mode, and G=√(2)g_0⟨ a⟩ is the effectively enhanced optomechanical coupling strength between the cavity mode and the mechanical mode by the factor ⟨ a⟩. The steady-state values ⟨ O⟩ in Eq. (<ref>) are given by ⟨̇ ̇Ȯ⟩̇=0. Specifically, ⟨ p⟩=0, ⟨ q⟩=-g_0|⟨ a⟩|^2/ω_b, ⟨ m_j⟩≈- g_j⟨ a⟩/Δ_j, and ⟨ a⟩≈i Ω/(g_j^2/ Δ_j-Δ̃_a), where |Δ_j|,|Δ̃_a|≫κ_j,κ_a are taken. Experimentally, these conditions can be easily achieved by tuning the frequency of the driving field. Also, the value of ⟨ a⟩ can be real via tuning the initial phase of the strong driving field. Hereafter, we assume ⟨ a⟩ to be real for simplicity. Thus, the effective Hamiltonian of the linearized hybrid system can be written as

    H_ eff=   Δ̃_a a^† a+ω_b/2(q^2+p^2)+G/√(2)(a+a^†)q
       +∑_j=1,2Δ_j m_j^† m_j+g_j(m_j a^†+a m_j^†),

which effectively describes the interaction among the cavity mode, the mechanical mode, and the Kittle modes, as illustrated in Fig. <ref>(b). Notably, all parameters in the Hamiltonian (<ref>) are adjustable except for the mechanical frequency ω_b. The fluctuation symbol δ is omitted for convenience.


 §.§ Covariance matrix

By further defining fluctuation quadratures of the Kittle and the cavity modes as

    x_j=   (m_j+m_j^†) / √(2),  y_j=i(m_j^†-m_j) / √(2),
    
    	 x_a=   (a+a^†) / √(2),  y_a=i(a^†-a) / √(2),

and the associated input noise operators as

    X_j=   (ζ_j+ζ_j^†) / √(2),  Y_j=i(ζ_j^†-ζ_j) /√(2),
    
    	 X_μ=   (μ+μ^†) / √(2),  Y_μ=i(μ^†-μ) / √(2),

the dynamics in Eq. (<ref>) can be equivalently written as the matrix form,

    u̇(t)=A u(t)+n(t),

where u(t)=[x_1,y_1, x_2,y_2, x_a,y_a,q,p]^T is the vector operator of the system, n(t)=[√(2κ_1)X_1,√(2κ_1)Y_1,√(2κ_2)X_2,
√(2κ_2)Y_2,√(2κ_a)X_μ,√(2κ_a)Y_μ,0,ξ]^T is the vector operator of the input noise, and
 

    A=([  -κ_1   Δ_1     0     0     0   g_1     0     0;  -Δ_1  -κ_1     0     0  -g_1     0     0     0;     0     0  -κ_2   Δ_2     0   g_2     0     0;     0     0  -Δ_2  -κ_2  -g_2     0     0     0;     0   g_1     0   g_2  -κ_a  Δ̃_a     0     0;  -g_1     0  -g_2     0 -Δ̃_a  -κ_a    -G     0;     0     0     0     0     0     0     0   ω_b;     0     0     0     0    -G     0  -ω_b   γ_b ])

 










is the drift matrix. According to the Routh-Hurwitz criteria<cit.>, the system is stable only when all the eigenvalues of A have negative real parts. The stability is ensured numerically in the following.

Since the input quantum noises are zero-mean quantum Gaussian noises, the quantum steady state for the
fluctuations is a zero-mean continuous variable Gaussian state, fully characterized by an 8×8 covariance matrix 𝒱_lk(t)=1/2⟨ u_l(t) u_k(t^')+u_k(t^') u_l(t)⟩(l,k=1,2, …, 8). The matrix 𝒱 can be obtained 
by directly solving the Lyapunov equation<cit.> 

    A 𝒱+𝒱 A^T+𝒟 =0,

where 𝒟=diag[κ_1(2 n̅_1+1), κ_1(2 n̅_1+1),κ_2(2 n̅_2+1), κ_2(2 n̅_2+1),κ_a(2 n̅_a+1),κ_a(2 n̅_a+1), 0, γ_b(2 n̅_b+1)]
is defined by ⟨n̅_l(t) n̅_k(t^') +n̅_k(t^') n̅_l(t)⟩=2𝒟_lkδ(t-t^').
 



 §.§ Bipartite and tripartite entanglement


Once the matrix 𝒱 is obtained by solving Eq. (<ref>), arbitrary bipartite entanglement can be quantified by the logarithmic negativity (LN) <cit.>,

    E_N≡max [0,-ln 2η^-]

with

    η^-=2^-1/2[Σ-(Σ^2-4 det V_4)^1/2]^1/2,

where Σ=det𝒜+detℬ-2det𝒞 and 
𝒱_4=[   𝒜   𝒞; 𝒞^𝒯   ℬ;     ]
is the 4 ×4 block form of the correlation matrix, associated with two modes of interest <cit.>. 𝒜, ℬ, and 𝒞 are the 2×2 blocks of 𝒱_4. E_N>0 means that the interested two modes are entangled. 

Additionally, tripartite entanglement among arbitrary three interested modes can also be studied by the function of the minimal residual contangle (MRC) <cit.>:

    ℛ_τ≡min[ℛ_τ^r | s t, ℛ_τ^s | r t, ℛ_τ^t | r s],

where

    ℛ_τ^r | s t≡ C_r | s t-C_r | s-C_r | t,

with C_u | v being the contangle of a subsystem of u and v (v contains one
or two modes) and {r,s,t}∈{a,b,m_1,m_2}, is a proper entanglement monotone defined as the squared LN <cit.>. ℛ_τ> 0 indicates that the interested three modes are entangled.





§ OPTOMECHANICAL ENTANGLEMENT


In this part, we study the magnon-mediated optomechanical entanglement. To be more clear, we first re-examine the optomechanical entanglement without magnons, i.e., g_1=0 and g_2=0. Then we investigate the magnon-mediated optomechanical entanglement by taking a single sphere for example. Without loss of generaity, we assume g_1=g_m≠0 and g_2=0.



 §.§ Without magnons



When no spheres are placed in the cavity (i.e., g_1=g_2=0), the proposed cavity-magnon optomechanics reduces to a standard optomechanical configuration comprising solely a cavity and a mechanical resonator. In this setup, optomechanical entanglement can be analyzed by plotting LN (E_N^ab) vs the normalized cavity detuning Δ̃_a/ω_b with different linearized coupling strengths G in Fig. <ref>(a). Here, we employ experimentally accessible parameters<cit.>: ω_a/2π = 10 GHz, ω_b/2π = 10 MHz, κ_a/2π=1 MHz, γ_b/2π=10^2 Hz, and T=20 mK. Evidently, the optomechanical entanglement increases first and then decreases with the cavity detuning Δ̃_a/ω_b. Around Δ̃_a≈ω_b, the optimal optomechanical entanglement is observed. By gradually increasing the optomechanical coupling, the optomechanical entanglement can be significantly enhanced (as depicted by the three curves). The generated optomechanical entanglement arises from both the two-mode squeezing effect (a^† b^†+ab, where q∝ b+b^†) and the beam-splitter effect (a^† b+ab^†). While the former generates entanglement, and the latter cools the mechanical resonator to sustain the entanglement. In Fig. <ref>(b), the effect of the bath temperature on the optomechanical entanglement is further elucidated by fixing Δ̃_a=0.9ω_b and G=4κ_a. It is shown that the optomechanical entanglement monotonically decreases as the bath temperature increases. The survival temperature for the optomechanical entanglement can be up to ∼ 190 mK.




 §.§ With magnons


To investigate the magnon-mediated optomechanical entanglement, we take the hybrid cavity-magnon optomechanics with a single sphere as an example, that is, we set g_1=g_m≠ 0 and g_2=0 in Fig. <ref>. Correspondingly, Δ_1=Δ_m is specified. Within the framework of cavity magnonics, the parameters such as the magnon frequency, the magnon-photon coupling, and hte magnon decay rate are all tuned, leading to tunable optomechanical entanglement generated, as numerically investigated in Fig. <ref>. From Fig. <ref>(a), it is observed that the optomechanical entanglement decreases with increasing the magnon-photon coupling while keeping other system parameters fixed. Around Δ_m≈ω_b, the optomechanical entanglement is optimal when g_m is comparable to G (g_m∼ G). This can also be evident in Fig. <ref>(b). Notably, a larger decay rate of the Kittle mode can significantly preserve the optomechanical entanglement, which can be given by comparing Figs. <ref>(a) and <ref>(b). Furthermore, Figure <ref>(c) elucidates that the optomechanical entanglement will be reduced when magnons are introduced. The reduction of the optomechanical entanglement is initially transformed into the magnon-photon entanglement (blue curve) and subsequently into the magnon-phonon entanglement (red curve). To simultaneously achieve these three entanglements, tuning the detuning of the cavity mode to be an optimal value is demanded. Additionally, we demonstrate that the magnon-photon-phonon tripartite entanglement (ℛ_τ^abm) around Δ_m≈-ω_b can be anticipated, as depicted in Fig. <ref>(d).By increasing the magnon-photon coupling, this tripartite entanglement noticeably improves, indicating that the optomechanical entanglement is also partially transfered to the tripartite entanglement among the Kittle mode, the cavity mode, and the mechanical mode.





§ MAGNON-MAGNON ENTANGLEMENT WITH THE OPTOMECHANICAL INTERFACE


Based on the aforementioned study of the optomechanical entanglement, it is evident that the optomechanical interface can serve as an effective entanglement generator. Therefore, we proceed to demonstrate the generation of magnon-magnon entanglement within the framework of two spheres included cavity-magnon optomechanics (see Fig. <ref>). 



 §.§ Magnon detuning effect


Figure <ref> plots the magnon-magnon entanglement (E_N^m_1m_2) vs the normalized frequency detunings of two Kittle modes of the spheres with κ_m=κ_a and κ_m=2κ_a, where g_m=G is fixed. The bat-shaped pattern in Fig. <ref>(a) demonstrates that the magnon-magnon entanglement can be symmetrically adjusted by varying the detunings of the Kittle modes. By varying one detuning while keeping the other unchanged, we observe that the magnon-magnon entanglement initially increases to its optimal value, and then decreases to zero. The optimal magnon-magnon entanglement can be predicted around Δ_1=-Δ_2≈±ω_b, which are symmetrically distributed on both sides of the white dashed line Δ_1=Δ_2. The similar results are also evident from the bird-shaped pattern in Fig. <ref>(b). By comparing Figs. <ref>(a) and <ref>(b), we show that the larger magnon decay rate, the weaker magnon-magnon entanglement. Moreover, with increasing the magnon decay rate, the symmetric points about the diagonal line, where the optimal magnon-magnon entanglement is predicted, will converge in the case that magnons are resonant with the driving field, i.e., Δ_1=Δ_2=0.




 §.§ Magnon-photon coupling effect


In this work, we assume that the magnons in the Kittle modes of two spheres have the identical coupling strength with the cavity mode. It is necessary to investigate the effect of two different coupling strengths g_1 and g_2 on the magnon-magnon entanglement generation. The numerical result is plotted in Fig. <ref>(a), where Δ_1=-ω_b and Δ_2=ω_b are fixed. Apparently, the optimal magnon-magnon entanglement can only be attained when two magnon-photon coupling strengths are comparable (see the white dashed line). With increasing the magnon-photon coupling strength, the magnon-magnon entanglement increases first to the optimal value, and then decreases to zero. To further show where the  magnon-magnon entanglement comes, we plot the magnon-magnon entanglemnt (E_N^m_1m_2) and the optomechanical entanglement (E_N^ab) vs the normalized coupling strenth g_m/G in Fig <ref>(b). We find that, with increasing the magnon-photon coupling to g_m ∼ G, the optomechanical entanglement decreases to the optimal value of the magnon-magnon entanglement. By further increasing the magnon-photon coupling, both the magnon-magnon entanglement and the optomechanical entanglement reduce to zero. Obviously, the former decreases much slower than the latter. The behavior in Fig. <ref>(b) reveals that the magnon-magnon entanglement is partially transfered from the optomechanical entanglement.



 §.§ Noise effect



In Fig. <ref>(a), the effects of the magnon decay rate on both the magnon-magnon and the optomechanical entanglement are investigated, where g_m=G and Δ_1=-Δ_2=-ω_b are fixed. It is observed that larger decay rates of the magnons do not promote the generation of magnon-magnon entanglement. Instead, they facilitate the production of optomechanical entanglement, consistent with the findings from Figs. <ref>(a) and <ref>(b). This not only reveals that the entanglement exchange between these two entanglement occurs, but also shows that the generated magnon-magnon entanglement can be be leveraged to restore the optomechanical entanglement by increasing the decay rate of the Kittle mode.
Since the bath temperature is crucial for entanglement survival, we respectively plot E_N^m_1m_2 and E_N^ab vs the bath temperature T in Figs. <ref>(b) and <ref>(c) with different magnon-photon coupling strengths, where κ_m=κ_a is fixed. When the magnon-photon coupling strength is weaker than the optomechanical coupling strength [see g_m=0.5G in Fig. <ref>(b)], the optomechanical entanglement (the black curve) is stronger than the magnon-magnon entanglement (the red curve) within the survival temperature. But when the magnon-photon coupling strength is comparable with the optomechanical coupling strength [see g_m=G Fig. <ref>(c)], the situation is reversed, that is, the magnon-magnon entanglement becomes much stronger than the optomechanical entanglement. Moreover, the survival temperature of the optomechanical entanglement is significantly shifted with varying the magnon-photon coupling. This indicates that the survival temperature of the magnon-magnon entanglement is more robust against the change of the magnon-photon coupling than that of the optomechanical entanglement.




§ MAGNON-BASED TRIPARTITE ENTANGLEMENT


Apart from the magnon-magnon entanglement, the optomechanical interface can also be used to produce tripatite entanglement, including the magnon-magnon-photon entanglement and the magnon-magnon-phonon entanglement, in hybrid cavity-magnon optomechanics.




 §.§ Magnon-magnon-photon entanglement


In Fig. <ref>(a), we plot the magnon-magnon-photon entanglement (ℛ_τ^am_1m_2) vs the normalized detunings Δ_1/ω_b and Δ_2/ω_b, where g_m=G=4κ_a, κ_m=κ_a, and T=20 mK. The eagle-pattern clearly indicates that the optimal magnon-magnon-photon entanglement is expected when magnons in two Kittle modes of the spheres are resonant and works on the blue-sideband, i.e., Δ_1=Δ_2≈-ω_b. When detunings deviate from this optimal point along the white diagonal line, where the resonant condition is maintained, we observe a rapid reduction in magnon-magnon-photon entanglement to zero. 
However, when nonidentical detunings are considered, this tripartite entanglement decreases slowly as detunings deviate from the optimal values, akin to a scenario where detunings stray from one of the eagle's wings. This indicates that nonidentical detunings can be used to protect this tripartite entanglement. 
Figure <ref>(b) demonstrates that the magnon-magnon-photon entanglement can be significantly enhanced by appropriately increasing the magnon-photon coupling strengths while varying the detuning Δ_m=Δ_1=Δ_2. Additionally, we observe that adjusting the magnon decay rate can also generate magnon-magnon-photon entanglement, as depicted in Fig. <ref>(c). The inset reveals the robustness of magnon-magnon-photon entanglement against the bath temperature when T<100 mK. However, at a higher temperature, the entanglement diminishes and disappears around T≈180 mK.







 §.§ Magnon-magnon-phonon entanglement


Next, we go on studying the magnon-magnon-phonon entanglement in the proposed hybrid cavity-magnon optomechanics. Note that there are no direct magnon-magnon and magnon-phonon couplings. We observe that optimal entanglement occurs around Δ_1=Δ_2≈-0.5ω_b. This point coincides with the nose of the fox-pattern depicted in Fig. <ref>(a). A comparison with Fig. <ref>(a) reveals that the magnon-magnon-phonon entanglement is stronger than that of magnon-magnon-photon entanglement, which can be regarded as that the mechanical mode can be cooled by the optomechanical beam-splitter interaction, or alternatively, the noise effect on the magnon-magnon-phonon entanglement is weaker than the magnon-magnon-photon entanglement because of the indirect interaction among magnons and phonons. We also find that such a tripartite entanglement can be improved by enhancing the magnon-photon coupling [see Fig. <ref>(b)]. Upon examining the entanglement with respect to the decay rate of the Kittle mode, we plot the magnon-magnon-phonon entanglement (ℛ_τ^bm_1m_2) vs the normalized decay rate κ_m/κ_a. It is shown that a weaker decay rate does not benefit the generation of magnon-magnon-phonon entanglement. By gradually increasing the decay rate, the magnon-magnon-phonon entanglement increases to its maximum and then decreases to zero. Comparing Figs. <ref>(c) and <ref>(c), we find that both the magnon-magnon-phonon entanglement and the magnon-magnon-photon entanglement have the similar resilience against the magnon decay rate. However, the magnon-magnon-phonon entanglement exhibits more robust againt the bath temperature thant that of the magnon-magnon-photon entanglement, as shown in the insets of Figs <ref>(c) and <ref>(c). In fact, the magnon-magnon-phonon entanglement is about two times than the magnon-magnon-photon entanglement within the survival temperature∼180 mK.



§ CONCLUSION


In summary, we propose to generate diverse bipartite and tripartite entanglement in hybrid cavity-magnon optomechanics. Unlike standard cavity optomechanics, our proposal not only enables magnon-mediated optomechanical entanglement but also allows for flexible magnon-magnon entanglement. It can even generate tripartite entanglement, including magnon-photon-phonon, magnon-magnon-photon, and magnon-magnon-phonon configurations. Moreover, optimal bipartite and tripartite entanglement can be achieved by tuning parameters. We further show that all entanglement can be enhanced via engineering the magnon-photon coupling, and are robust against the bath temperature within the survival temperature. In addition, we find that the optomechanical entanglement can be protected or restored by bad magnons with large decay rate, while other entanglement are significantly diminished. These findings suggest that our proposal offers a novel avenue to explore and control tunable macroscopic quantum effects in hybrid cavity-magnon optomechanics.




This work is supported by Zhejiang Provincial Natural Science Foundation of China under Grant
No. LY24A040004.





