In this work, we present a hybrid cavity opto-magnomechanical system to generate distant bipartite entanglement between different quantum carriers. Accordingly, the system consists of two cavity photons, a phonon of yttrium iron garnet (YIG), a magnon, and a phonon of membrane. Specifically, the two cavities are coupled through an optical fiber, and one of the optical cavities consists of a membrane coupled with the cavity photon through radiation pressure force. While the other cavity contains a YIG, the magnon mode connects to the cavity photon via magnetic dipole interaction and, simultaneously, couples to the mechanical resonator of the YIG through magnetostrictive interaction. We show that entanglement generation can be realized under indirectly coupled bipartitions for parameters and detunings within appropriate regimes. Furthermore, for various bipartitions, we also obtain appropriate cavity and magnon detuning values for a considerable remote entanglement. Moreover, the generation of distant bipartite entanglements and entanglement transfer between subsystems is predominantly influenced by the coupling strength. Remarkably, the distant bipartite entanglement is strongly contrary to the environmental temperature. Thus, optimizing the system’s parameters allows for the maximum possible entanglement between various quantum carriers. We believe our results could provide more stable bipartite entanglements and serve as a potential quantum interface to realize particularly long-range entanglement transfers.

## I. INTRODUCTION

In recent years, quantum entanglement has been recognized as the most interesting feature of quantum formalism for its vast applications in quantum networking, quantum information processing, quantum-dense coding, and quantum-enhanced metrology.^{1,2} It can be realized in a variety of macroscopic mechanical systems, such as atomic ensembles, optomechanical systems, and magnomechanical systems.^{3–8} In particular, cavity optomechanical systems effectively couple the interaction between optical and mechanical degrees of freedom mediated by radiation pressure force.^{9} As a result, several new quantum phenomena are proposed, including control of the quantum state of a macroscopic object, gravitational wave detection, tiny displacement measurements, and accuracy in the measurement of weak forces and fields.^{10–14} In addition, the interaction of the mechanical resonator with the cavity field leads to the generation of entanglement between a cavity field and a macroscopic mechanical resonator.^{15–20}

Recently, magnomechanical systems, which are developed from cavity quantum electrodynamics systems comprising a microwave cavity and a material called yttrium iron garnet (YIG, Y_{3}Fe_{5}O_{12}) sphere, give a new approach to couple various quantum information carriers.^{21–24} In such systems, the size of the sphere is significantly less than the microwave wavelength; as a result, the radiation pressure effect is minimal, and the cavity field and mechanical resonator modes do not directly interact. However, the magnon mode (collective excitation of the spins) interacts with the cavity field through the magnetic dipole interaction. In addition, it interacts with the phonon mode of a mechanical oscillator via a magnetostrictive interaction. Due to its high spin density and minimal losses, the yttrium iron garnet (YIG) sphere is a great ferrimagnetic material for quantum information processing. Thus, the cavity magnomechanical system provides an excellent framework for studying the strong interaction between matter and light.^{25–30}

Most recently, quantum entanglement in magnomechanical systems has drawn much interest. For instance, Li *et al.*^{31} showed how to generate bipartite and tripartite entanglements in a cavity magnomechanical system. In addition, Amazioug *et al.*^{32} proposed a scheme to enhance magnon-photon-phonon entanglement in cavity magnomechanics using a coherent feedback loop. Furthermore, an opto-magnomechanical hybrid system was proposed by Wang *et al.*^{33} to generate entanglements between various bipartitions in optical and microwave domains. Similarly, Li *et al.*^{34} studied a microwave cavity with a silicon-nitride membrane and the YIG sphere in a hybrid magnomechanical system to generate bipartite and tripartite entanglements. In addition, Qamar *et al.*,^{35} provide a method to generate distant bipartite and tripartite entanglements between the YIG sphere and an atomic ensemble in coupled microwave cavities. Moreover, a coupled cavity magnomechanical system with a single photon hooping connecting two cavities was proposed by Sohail *et al.*^{23} This system demonstrates the transfer of entanglement and quantum steering within subsystems that are presently the focus of very few theoretical works. Consequently, such magnomechanical systems would have potential applications in quantum information processing, quantum information science, and quantum networks. As a result, the study of the generation of distant bipartite entanglements under indirectly coupled bipartitions to transfer information across the subsystems is an active research field, and relatively few studies have been conducted.

In this work, we investigated theoretically the generation of distant bipartite entanglements between different quantum carriers of a hybrid cavity opto-magnomechanical system. Thus, the hybrid system consists of two cavity photons, a magnon, a phonon of YIG, and a phonon of a membrane, where the two cavities are coupled via an optical fiber. In particular, one of the optical cavities consists of a membrane that interacts linearly with the cavity photons through the radiation pressure force. However, the interaction between the other cavity photons and the magnon is mediated by the magnetic dipole interaction, and the magnetostrictive interaction couples the magnon with the phonon mode of the YIG. We employ logarithmic negativity to quantify the degree of entanglement between different bipartitions. Our results show significant distant bipartite entanglements between different subsystems that are remotely coupled for experimentally feasible parameters. In addition, we also obtain suitable detuning parameter regimes for both cavities and magnons. Furthermore, we notice the strong impact of cavity coupling strength on the degree of entanglement and entanglement transfer between different bipartitions. Notably, the distant bipartite entanglements are robust to certain environmental temperatures. As a result, we believe that the proposed scheme provides a promising platform for generating and transferring entanglement across distant bipartitions for the future advancement of technology.

## II. THEORETICAL MODEL

The hybrid opto-magnomechanical system shown in Fig. 1 comprises two coupled microwave cavities with resonance frequencies *ω*_{j}, where *j* = 1, 2 represents cavity-1 and cavity-2. The left side of the microwave cavity consists of a membrane, and the YIG sphere is embedded in the right side of the microwave cavity. The membrane interacts linearly with the cavity field through radiation pressure force with coupling rates *g*_{ab}. In addition, the phonon that arises from the YIG sphere’s geometry deformation is coupled with the magnon through the magnetostrictive force, with a coupling strength of *g*_{md}. On the other hand, the magnon interacts with the magnetic field of the cavity by using magnetic dipole interaction with the coupling strength *g*_{am}.^{24,36} The decay rates of the two cavity modes, magnon mode, and phonon modes of the mechanical resonator and YIG are denoted by *κ*_{a}, *κ*_{m}, *γ*_{b}, and *γ*_{d}, respectively.

*ω*

_{j}and

*ω*

_{m}, respectively; $(q\u0302,x\u0302)$ and $(p\u0302,y\u0302)$ are the dimensionless position operators and momentum operators of the membrane and YIG sphere with frequencies

*ω*

_{b}and

*ω*

_{d}, respectively; and

*J*is the coupling strength of two cavities. The Rabi frequency $\Omega l=2P\kappa a/\u210f\omega l$ represents the strength of the cavity driving field, which is reliant on the input power

*P*and the frequency

*ω*

_{l}of the drive field, as well as the decay rate (

*κ*

_{a}) of the cavity. Moreover, the Rabi frequency $\Omega m=54\Gamma NsB0$ indicates the strength of the drive field of YIG sphere with frequency

*ω*

_{l},

^{37}where

*B*

_{0}= 3.9 × 10

^{−9}T is the amplitude of the drive field,

*N*

_{s}=

*ρV*

_{s}is the total number of spins with the spin density of the YIG sphere, i.e.,

*ρ*= 4.22 × 10

^{27}m

^{−3},

*V*

_{s}represents the volume of the YIG, and Γ = 2π × 28 GHz/T is the gyromagnetic ratio. Moreover, the Rabi frequency Ω

_{m}is obtained under the fundamental assumption of low-laying excitations, i.e., $m\u0302\u2020m\u0302\u226a2Ns\u03c2$, where

*ς*= 5/2 denotes the spin number of the ground state Fe

^{3+}ion in the YIG sphere.

^{38}The Hamiltonian that describes the system under a frame rotating with the drive field frequency

*ω*

_{l}is given by

_{j}=

*ω*

_{j}−

*ω*

_{l}and Δ

_{m}=

*ω*

_{m}−

*ω*

_{l}are the detuning frequencies of the cavities and magnon mode, respectively. Due to the interaction of the system and its surroundings, the system will be subjected to cavity decay, magnon decay, and mechanical damping. These dissipative elements allow the dynamics of the system to be represented by a set of quantum Langevin equations, which are as follows:

*ξ*

_{i}(

*i*=

*b*,

*d*) are the input noise operators for cavities, magnon, and mechanical modes for the YIG sphere and membrane, respectively, with their mean values being zero, whereas their non-zero correlation functions satisfy

^{12,13,39}$a\u0302jin(t)a\u0302jin\u2020(t\u2032)=nj(\omega j)+1\delta (t\u2212t\u2032)$ and $m\u0302in(t)m\u0302in\u2020(t\u2032)=nm(\omega m)+1\delta (t\u2212t\u2032)$, and $\xi i(t)\xi i(t\u2032)+\xi i(t\u2032)\xi i(t)/2\u2248\gamma i[2Ni(\omega i)+1]\delta (t\u2212t\u2032)$, where a Markovian approximation has been made that is valid for a large mechanical quality factor

*Q*=

*ω*

_{i}/

*γ*

_{i}≫ 1.

^{40}Moreover, the equilibrium mean number of thermal photon, magnon, and phonon modes of the membrane and YIG sphere is given by $nw=exp(\u210f\omega w/kBT)\u22121\u22121(w=a1,a2,m,b,d)$, with

*k*

_{B}being the Boltzmann constant and

*T*being the environmental temperature. Considering that there is strong excitation in the magnon mode, it suggests that |

*m*| ≫ 1. Furthermore, both microwave cavity fields exhibit large amplitudes because cavity-1 is driven strongly and cavity-2 interacts with the magnon modes via the beam splitter interaction. Thus, the quantum Langevin equations of the system can be written as the sum of their mean value and small fluctuation operator with a zero mean value as $O\u0302=Os+\delta O\u0302(O\u0302=a\u03021,a\u03022,m\u0302,q\u0302,p\u0302,x\u0302,y\u0302)$.

^{41,42}The steady-state solutions to Eq. (3) are obtained as

^{24}

*u*(

*t*) is the column vector of the fluctuations and

*n*(

*t*) is the column vector of the noise source. Their transposes are given by

$u(t)T=\delta x\u03021,\delta y\u03021,\delta x\u03022,\delta y\u03022,\delta x\u03023,\delta y\u03023,\delta x\u0302,\delta y\u0302,\delta q\u0302,\delta p\u0302$, and $n(t)T=[2\kappa a\delta x\u03021in,2\kappa a\delta y\u03021in,2\kappa a\delta x\u03022in,2\kappa a\delta y\u03022in,2\kappa m\delta x\u03023in,2\kappa m\delta y\u03023in,0,\xi d,0,\xi b]$.

*M*can be given by

*M*of Eq. (7). It should be noted that when all of the real components of the eigenvalues of the drift matrix

*M*are negative, the system can achieve a stable steady-state condition. The stability condition can be obtained by using the Routh–Hurwitz criterion.

^{24}Therefore, the steady state of the quantum fluctuations is a continuous-variable Gaussian state.

^{43,44}This state is fully characterized by a 10 × 10 covariance matrix with corresponding components defined as $Vij=ui(t)uj(t\u2032)+uj(t)ui(t\u2032)/2$, which can be obtained by solving the following Lyapunov equation,

^{40,45}

*E*

_{N}), which can be defined as

^{32,46,47}

*σ*= det

*A*+ det

*B*− 2 det

*C*, and we have considered the bipartite correlation matrix in a 2 × 2 block form as

*A*,

*B*, and

*C*are 2 × 2 sub-block matrices. If

*E*

_{N}> 0, then the two subsystems are entangled.

## III. RESULTS AND DISCUSSION

In this section, employing the linearized quantum Langevin equations to describe steady states and the Lyapunov equation to quantum fluctuation, we aim to analyze the bipartite entanglements of five different modes in coupled microwave cavities, which consist of a membrane and YIG sphere. Therefore, we may get bipartite entanglement in any of the two modes; however, our main objective is to generate bipartite entanglement that takes place in spatially distant subsystems. We adopt practically feasible parameters reported in Refs. 31 and 48–^{49}: $\omega m/2\pi =\omega a1/2\pi =\omega a2/2\pi =10GHz$, *ω*_{b}/2*π* = *ω*_{d}/2*π* = 10 MHz, *κ*_{m}/2*π* = *κ*_{a}/2π = 1 MHz, $gam/2\pi =gmd/2\pi =0.4Hz=Gmd/2\pi =3.2MHz$, *γ*_{b}/2*π* = *γ*_{d}/2*π* = 100 Hz, *T* = 10 mK, *P* = 8.9 mW, *m* = 5 × 10^{−12} kg (mass of membrane), and *l* = 1 mm (length of cavity).

In Fig. 2, we generate four different distant bipartite entanglements as a function of $\Delta 1\u2032/\omega d$ and Δ_{2}/*ω*_{d}. By employing the coupling strength *J* = 0.8*ω*_{d} and effective magnon detuning $\Delta m\u2032=0.9\omega d$ (resonant with a blue sideband), we obtain the maximum entanglement between cavity-1 photon and magnon $(ENa1m)$ at two different regions. While the entanglement between cavity-1 photon and phonon of the YIG sphere $(ENa1d)$ shows the optimum value in a certain region of normalized cavity detunings, as shown in Figs. 2(a) and 2(b). Specifically, Fig. 2(a) illustrates that the entanglement $ENa1m$ reaches the optimum when either both cavity detunings resonate near the resonance or both cavity detunings resonate in the red sideband regime around $\Delta 1\u2032=\u22121.8\omega d$ and Δ_{2} = −1.57*ω*_{d}. In addition, the entanglement $ENa1d$ reaches optimum when both cavity detunings resonate near the drive field, as shown in Fig. 2(b). Furthermore, in Figs. 2(c) and 2(d), we generate the distant bipartite entanglements between the magnon and phonon of a membrane $(ENmb)$ and the phonon of a membrane and phonon of the YIG sphere $(ENdb)$. In this case, when the effective magnon detuning is in the red sideband regime, we observed that the entanglement $ENmb$ attains maximum values around $\Delta 1\u2032=\omega d$ and Δ_{2} is kept either in the red sideband regime or blue sideband regime. On the other hand, the entanglement $ENdb$ reaches maximum near $\Delta 1\u2032=0.5\omega d$ and Δ_{2} = 0. From those results, we realized that by eventually varying both cavity detunings, we could transfer entanglement from one subsystem to another.

Next, Fig. 3 shows three different distant bipartite entanglements between cavity-1 photon and magnon $(ENa1m)$, cavity-1 photon and phonon of YIG sphere $(ENa1d)$, and cavity-1 and cavity-2 photons $(ENa1a2)$ as a function of $\Delta 1\u2032/\omega d$ and Δ_{2}/*ω*_{d} for different coupling strength *J*. We have used the effective magnon detuning in resonance with the blue sideband regime. Specifically, for *J* = 0.5*ω*_{d}, the entanglement $ENa1m$ reaches the maximum value near the resonance around $\Delta 1\u2032=\u22120.2\omega d$ and Δ_{2} = −0.36*ω*_{d}, as shown in Fig. 3(a). In addition, the bipartite entanglement $ENa1d$ attains the maximum value near the resonance almost for Δ_{1} ≈ 0 and Δ_{2} = −0.34*ω*_{d}, as observed in Fig. 3(b). It can be seen that the bipartite entanglement $ENa1a2$ attains the optimal value near $\Delta 1\u2032=0.5\omega d$ and Δ_{2} = −0.5*ω*_{d}, as observed in Fig. 3(c). For *J* = 0.8*ω*_{d}, the bipartite entanglement $ENa1m$ becomes maximum for two distinct instances, i.e., either for both cavity detunings to be in the red sideband regime for $\Delta 1\u2032=\u22121.8\omega d$ and Δ_{2} = −1.57*ω*_{d} or near the resonance $\Delta 1\u2032=\u22120.04\omega d$ and Δ_{2} = −0.28*ω*_{d} as given in Fig. 3(d). However, the bipartite entanglement $ENa1d$ attains the optimal value, near the resonance, as shown in Fig. 3(e). In addition, the entanglement $ENa1a2$ reaches its optimum value when both cavity detunings are in the red sideband regime around $\Delta 1\u2032=\u22120.3\omega d$ and Δ_{2} = −1.4*ω*_{d}, as shown in Fig. 3(f). Further increasing *J* = *ω*_{d}, the region where the bipartite entanglement $ENa1m$ reaches the maximum value, corresponding to the red sideband regime, starts to decrease, whereas the region corresponding to resonance cavities starts to shift to the blue sideband, as shown in Fig. 3(g). Furthermore, the region where the bipartite entanglement $ENa1d$ reaches the maximum value near the resonance slightly changes to the blue sideband regime, as shown in Fig. 3(h). Moreover, the entanglement $ENa1a2$ reaches the optimum in the red sideband regimes near the resonance, as seen in Fig. 3(i). In general, Fig. 3 shows that the concentration of bipartite entanglements in density plots decreases significantly as the cavity coupling *J* increases. This is due to a specific range that exists in which there is a positive correlation between the coupling strength and the bipartite entanglement. Nevertheless, if it keeps rising, the system will deteriorate and bipartite entanglement will decline.

In Fig. 4, we discuss the impact of the cavity coupling strength *J*/*ω*_{d} and normalized cavities detunings Δ_{a}/*ω*_{d} on the distant bipartite entanglements $ENa1m$, $ENa1d$, and $ENa1a2$. Notably, we have studied two methods in this instance: the non-symmetric case $\Delta a/\omega d=\u2212\Delta 1\u2032/\omega d=\Delta 2/\omega d$, which represents (*a*, *b*, *c*), and the symmetric case $\Delta a/\omega d=\Delta 1\u2032/\omega d=\Delta 2/\omega d$, referring to (*d*, *e*, *f*). For the non-symmetric case, the bipartite entanglement $ENa1m$ attains the maximum value when the cavity detunings are in the red sideband regime around the resonance Δ_{a} = −0.5*ω*_{d} and the cavity coupling strength *J* varies in between 0.5*ω*_{d} and 1.4*ω*_{d}; after this range, the entanglement vanishes as shown in Fig. 4(a). In addition, the entanglement $ENa1d$ reaches a maximum value around Δ_{a} = −0.3*ω*_{d} and for *J* varying in the range of 0.5*ω*_{d} to 1.4*ω*_{d} as shown in Fig. 4(b). Furthermore, the bipartite entanglement $ENa1a2$ reaches its maximum around Δ_{a} = −1.2*ω*_{d} and *J* is in between 0.5*ω*_{d} and 1.6*ω*_{d}; while, after this, it drops very quickly on slowly varying Δ_{a}/*ω*_{d} and *J*/*ω*_{d}, as observed in Fig. 4(c). For the symmetric case, the bipartite entanglement $ENa1m$ reaches the maximum value when cavity detunings Δ_{a} are in the red sideband regime and *J* = 0.5*ω*_{d}, while in the blue sideband, the regime reaches finite value on varying Δ_{a}/*ω*_{d} and *J*/*ω*_{d} as shown in Fig. 4(d). Moreover, the bipartite entanglement $ENa1d$ attains the maximum value near the resonance, i.e., Δ_{a} = 0 and the value of the cavity coupling strength becomes *J* = *ω*_{d} as shown in Fig. 4(e). Furthermore, the entanglement $ENa1a2$ attains the maximum value for Δ_{a} varying in the negative region, whereas the coupling strength ranges from 0.5*ω*_{d} to 1.5*ω*_{d}, as observed in Fig. 4(f). As we observed, the density plots of distinct subsystems display entirely different outcomes based on whether the phonon of the YIG sphere deformation causes the cavity detunings to be resonant with the blue or red sideband regime.

In Fig. 5, we study the effect of coupling strength on different bipartite entanglements between cavity-1 and cavity-2 $(ENa1a2)$, magnon and phonon of YIG sphere $(ENmd)$, cavity-2 and magnon $(ENa2m)$, cavity-2 and phonon of YIG sphere $(ENa2d)$, cavity-1 and magnon $(ENa1m)$, and cavity-1 and phonon of YIG sphere $(ENa1d)$. Here, we plot bipartite entanglements vs normalized cavity detunings Δ_{a}/*ω*_{d} for $\Delta a=\Delta 2=\u2212\Delta 1\u2032$ considering different coupling strengths *J*. For *J* = 0.2*ω*_{d}, the entanglement $ENa1a2$ reaches the optimum value of 0.04 for slowly changing Δ_{a} around 0.2*ω*_{d}; apart from this point, the entanglement decreases and becomes zero, as seen in Fig. 5(a). In addition, the entanglement $ENmd$ attains the maximum value of 0.27 for Δ_{a} near the resonance and reaches a certain value for positive Δ_{a}/*ω*_{d}. Furthermore, the entanglement $ENa2d$ attains its maximum value of 0.23 for negative Δ_{a}/*ω*_{d}, whereas almost zero with positive Δ_{a}/*ω*_{d}. Similarly, the entanglement $ENa2m$ attain its optimum value of 0.13 for negative Δ_{a}/*ω*_{d}, while zero with positive Δ_{a}/*ω*_{d}. Here, we have realized that the result for entanglements $ENmd$, $ENa2d$, and $ENa2m$ in line with the result reported by Li *et al.*^{31} for magnon-photon-phonon entanglement in a single-cavity magnomechanical system. Moreover, the entanglement $ENa1d$ attains the maximum value of 0.03 for varying Δ_{a} in between −1.3*ω*_{d} and 1.07*ω*_{d}; after this range, it almost becomes zero, whereas the bipartite entanglement $ENa1m$ becomes zero for both positive and negative Δ_{a}/*ω*_{d} for the mentioned coupling strength. It is realized that for *J* = 0.2*ω*_{d}, there is a considerable amount of entanglement transfer from $ENmd$ to $ENa2m$ and $ENa2d$ around Δ_{a} = −1.07*ω*_{d}, as shown in Fig. 5(a). For *J* = 0.5*ω*_{d}, the bipartite entanglement $ENa1a2$ decreases for both positive and negative Δ_{a}/*ω*_{d}, as seen in Fig. 5(b). It is observed that the bipartite entanglement $ENmd$ attains the maximum value of 0.22 for negative Δ_{a}/*ω*_{d} and maintains a certain value for positive Δ_{a}/*ω*_{d}, as shown in Fig. 5(b). In addition, the bipartite entanglement $ENa2m$ attains the maximum value of 0.12 for negative Δ_{a}/*ω*_{d}, and $ENa2d$ reaches a maximum value of 0.22 in the negative Δ_{a}/*ω*_{d}, whereas both entanglements have nearly zero for positive detunings. However, the bipartite entanglement $ENa1m$ increases to 0.02 and $ENa1d$ reaches a maximum value for both positive and negative Δ_{a}/*ω*_{d} in a certain range. The results for entanglements $ENa1a2$ and $ENa1m$ are comparable with the result reported by Wang,^{22} who used the optical cavity as an auxiliary cavity to enhance the magnon-photon entanglement in coupled microwave cavities. Moreover, in this case, we realize a significant entanglement transfer from $ENmd$ to $ENa1d$ and $ENa1m$ for Δ_{a} = −0.94*ω*_{d}. When we increase the coupling strength *J* = 0.8*ω*_{d}, then the entanglement $ENa1a2$ reaches a maximum value for both negative and positive Δ_{a} in certain ranges as shown in Fig. 5(c). In addition, the bipartite entanglements $ENa2d$ nearly reaches its optimum value of 0.2 and $ENa2m$ attains a maximum of 0.09 for negative Δ_{a}. Furthermore, the entanglement $ENa1d$ attains the optimum value for positive and negative Δ_{a} in certain ranges, and the entanglement $ENa1m$ reaches its maximum for Δ_{a} near the resonance. As observed in Fig. 5(c), the most entanglement transfer from $ENmd$ to $ENa1d$ and $ENa1m$ occurs about Δ_{a} = −0.8*ω*_{d}. When we further increase *J* = *ω*_{d}, the bipartite entanglements $Ena1a2$, $ENa2d$, and $ENa2m$ decrease, while the entanglements $ENa1d$ and $ENa1m$ increase to certain value, as seen in Figs. 5(a)–5(d). It can also be seen that for anti-symmetric cavities, we get both Stokes and anti-Stokes processes; hence, it gives weak bipartite entanglement between the two cavity photons. Therefore, we realize optimum entanglement transfer from directly coupled magnon-phonon modes to indirectly coupled cavity-2-magnon modes and cavity-2-phonon modes of YIG, as shown in Fig. 5.

Figure 6 shows the distant bipartite entanglement’s robustness as a function of environmental temperature. Specifically, we find that the entanglement between a phonon of YIG and a phonon of a membrane $(ENdb)$ remains constant with the environmental temperature until about *T* = 0.072 K. After that, the entanglement decreases rapidly and can only persist for temperatures up to 0.121 K. Similarly, the entanglement between the magnon and phonon of a membrane $(ENmb)$ does not vary with the environmental temperature until about *T* = 0.072 K, after which it rapidly decreases and only survives temperatures up to 0.142 K. Thus, mechanical cooling is required to get entanglement in the subsystem, which reduces thermal noise. On the other hand, the entanglement between cavity-1 photon and magnon $(ENa1m)$ does not vary much with the environmental temperature until about 0.075 K, but beyond that, it decreases rapidly and can withstand temperatures as high as 0.155 K. Likewise, the entanglement between cavity-1 photon and phonon of YIG $(ENa1d)$ can maintain entanglement at temperatures as high as 0.178 K. In general, the entanglement decreases as the temperature increases because thermal noise in the environment causes decoherence.

## IV. CONCLUSION

We investigated the generation of distant bipartite entanglements through a hybrid opto-magnomechanical system. The hybrid system consists of two cavity photons, a phonon of YIG, and a phonon of membrane. More specifically, one of the cavities contains a membrane, and the other is composed of a YIG sphere, where the two cavities are coupled through the optical fiber with a coupling strength denoted as *J*. We use logarithmic negativity to quantify the distant bipartite entanglements between subsystems. Our results show considerable bipartite entanglements between indirectly coupled subsystems in coupled microwave cavities for a suitable set of parameters. In addition, we obtain the proper detuning regimes of the magnon and the two cavities for bipartite entanglements of different subsystems. Furthermore, we found that the amount of entanglement and its transfer between subsystems, coupled directly and indirectly, were significantly dependent on the cavities’ coupling strength. Moreover, the distant bipartite entanglements are resilient to the environmental temperature. Hopefully, our proposed approach to generate distant bipartite entanglement offers a promising platform to realize continuous variables applicable to quantum information processing.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Mulugeta Tadesse**: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **Tesfay Gebremariam Tesfahannes**: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Methodology (equal); Resources (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **Tewodros Yirgashewa Darge**: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Project administration (equal); Resources (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). **Muhdin Abdo Wodado**: Conceptualization (equal); Data curation (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **Habtamu Dagnaw Mekonnen**: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

*Quantum Noise*

*Springer Series in Synergetics Vol. 56*