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.

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, Y3Fe5O12) 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.

The hybrid opto-magnomechanical system shown in Fig. 1 comprises two coupled microwave cavities with resonance frequencies ωj, 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 gab. 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 gmd. On the other hand, the magnon interacts with the magnetic field of the cavity by using magnetic dipole interaction with the coupling strength gam.24,36 The decay rates of the two cavity modes, magnon mode, and phonon modes of the mechanical resonator and YIG are denoted by κa, κm, γb, and γd, respectively.

FIG. 1.

(a) A schematic model of the system. The system consists of a membrane in cavity-1 and a YIG sphere in cavity-2. An external laser field of frequency ωl drives cavity-1 with amplitude Ωl. The YIG sphere is placed inside a microwave cavity near the maximum field of the cavity mode and simultaneously magnetized by a biased magnetic field B, which sets the magnon-photon coupling. The YIG sphere’s magnon is directly driven by a microwave magnetic field of frequency ωl with amplitude Ωm, which enhances the magnomechanical coupling. (b) The schematic representation of the coupling relationship. The two cavity modes are coupled via the coupling strength J, while the cavity-1 mode photon (â1) with the phonon mode of the membrane (b̂), the cavity-2 mode photon (â2) with the magnon mode (m̂), and the magnon mode (m̂) with the phonon mode of the YIG (d̂) are coupled via the coupling strengths gab, gam, and gmd, respectively.

FIG. 1.

(a) A schematic model of the system. The system consists of a membrane in cavity-1 and a YIG sphere in cavity-2. An external laser field of frequency ωl drives cavity-1 with amplitude Ωl. The YIG sphere is placed inside a microwave cavity near the maximum field of the cavity mode and simultaneously magnetized by a biased magnetic field B, which sets the magnon-photon coupling. The YIG sphere’s magnon is directly driven by a microwave magnetic field of frequency ωl with amplitude Ωm, which enhances the magnomechanical coupling. (b) The schematic representation of the coupling relationship. The two cavity modes are coupled via the coupling strength J, while the cavity-1 mode photon (â1) with the phonon mode of the membrane (b̂), the cavity-2 mode photon (â2) with the magnon mode (m̂), and the magnon mode (m̂) with the phonon mode of the YIG (d̂) are coupled via the coupling strengths gab, gam, and gmd, respectively.

Close modal
The total Hamiltonian of the hybrid system takes the following form: Ĥ=Ĥ0+Ĥint+Ĥd, with
Ĥ0=ω1â1â1+ω2â2â2+ωmm̂m̂+12ωb(q̂2+p̂2)+12ωd(x̂2+ŷ2),Ĥint=gabâ1â1q̂+J(â1â2+â1â2)+gmdm̂m̂x̂+gam(â2m̂+â2m̂),Ĥd=iΩl(â1eiωltâ1eiωlt)+iΩm(m̂eiωltm̂eiωlt),
(1)
where âj(âj) and m̂(m̂) are annihilation (creation) operators of the two cavities and magnon modes with frequencies ωj and ωm, respectively; (q̂,x̂) and (p̂,ŷ) are the dimensionless position operators and momentum operators of the membrane and YIG sphere with frequencies ωb and ωd, respectively; and J is the coupling strength of two cavities. The Rabi frequency Ωl=2Pκa/ωl represents the strength of the cavity driving field, which is reliant on the input power P and the frequency ωl of the drive field, as well as the decay rate (κa) of the cavity. Moreover, the Rabi frequency Ωm=54ΓNsB0 indicates the strength of the drive field of YIG sphere with frequency ωl,37 where B0 = 3.9 × 10−9 T is the amplitude of the drive field, Ns = ρVs is the total number of spins with the spin density of the YIG sphere, i.e., ρ = 4.22 × 1027 m−3, Vs represents the volume of the YIG, and Γ = 2π × 28 GHz/T is the gyromagnetic ratio. Moreover, the Rabi frequency Ωm is obtained under the fundamental assumption of low-laying excitations, i.e., m̂m̂2Nsς, where ς = 5/2 denotes the spin number of the ground state Fe3+ ion in the YIG sphere.38 The Hamiltonian that describes the system under a frame rotating with the drive field frequency ωl is given by
Ĥ=j=12Δjâjâj+Δmm̂m̂+12ωb(q̂2+p̂2)+12ωd(x̂2+y2̂)+gabâ1â1q̂+J(â1â2+â1â2)+gmdm̂m̂x̂+gam(â2m̂+â2m̂)+iΩl(â1â1)+iΩm(m̂m̂),
(2)
where Δj = ωjωl and Δm = ωmωl 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:
â˙1=κa+iΔ1â1iJâ2igabâ1q̂+Ωl+2κaâ1in,â˙2=κa+iΔ2â2iJâ1igamm̂+2κaâ2in,m̂˙=κm+iΔmigamâ2igmdm̂x̂d+Ωm+2κam̂in,x̂˙=ωdŷ,ŷ˙=ωdx̂gmdm̂m̂γdŷ+ξd,q̂˙=ωbp̂,p̂˙=ωbq̂gabâ1â1γbp̂+ξb,
(3)
where âjin, m̂in, and ξi (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 satisfy12,13,39 âjin(t)âjin(t)=nj(ωj)+1δ(tt) and m̂in(t)m̂in(t)=nm(ωm)+1δ(tt), and ξi(t)ξi(t)+ξi(t)ξi(t)/2γi[2Ni(ωi)+1]δ(tt), where a Markovian approximation has been made that is valid for a large mechanical quality factor Q = ωi/γi ≫ 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(ωw/kBT)11(w=a1,a2,m,b,d), with kB 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 Ô=Os+δÔ(Ô=â1,â2,m̂,q̂,p̂,x̂,ŷ).41,42 The steady-state solutions to Eq. (3) are obtained as
a1s=Ωlκa+iΔ2κm+iΔm+Ωlgam2ΩmJgamκa+iΔ1κa+iΔ2κm+iΔm+gam2+J2κm+iΔm,a2s=iJκm+iΔma1siΩmgamκa+iΔ2κm+iΔm+gam2,ms=Ωmigama2sκm+iΔm,xs=gmd|ms|2ωd,ys=0,qs=gab|a1s|2ωb,ps=0,
(4)
where Δ1=Δ1+gabqs and Δm=Δm+gmdxs are the effective detuning of cavity-1 and magnon modes, respectively. The quantum Langevin equations of the fluctuation operators can be obtained as24 
δâ̇1=κa+iΔ1δa1̂iJδâ2igaba1sδq̂+Ωl+2κaδâ1in,δâ̇2=κa+iΔ2δâ2iJδâ1igamδm̂+2κaδâ2in,δm̂̇=κm+iΔmδm̂igamδâ2igmdmsδx̂+Ωm+2κaδm̂in,δx̂̇=ωdδŷ,δŷ̇=ωdδx̂gmdms*δm̂+msδm̂γdδŷ+ξd,δq̂̇=ωbδp̂,δp̂̇=ωbδq̂gaba1s*δâ1+a1sδâ1γbδp̂+ξb.
(5)
Now, we present the quadrature operators for the linearized quantum Langevin equations that characterize fluctuations as follows: δx̂j=(δâj+δâj)/2,δŷj=(δâjδâj)/i2,δx̂3=(δm̂+δm̂)/2,δŷ3=(δm̂δm̂)/i2, and the equations of the quadrature fluctuation for input noise operators are δx̂jin=(δâjin+δâjin)/2,δŷjin=(δâjinδâjin)/i2,δx̂3in=(δm̂in+δm̂in)/2,δŷ3in=(δm̂inδm̂in)/i2. Then, using Eq. (5) and the quadrature expressions, the resulting evolution equations of motion for the fluctuations can be rewritten in matrix form as follows:
u̇(t)=Mu(t)+n(t),
(6)
where 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=δx̂1,δŷ1,δx̂2,δŷ2,δx̂3,δŷ3,δx̂,δŷ,δq̂,δp̂, and n(t)T=[2κaδx̂1in,2κaδŷ1in,2κaδx̂2in,2κaδŷ2in,2κmδx̂3in,2κmδŷ3in,0,ξd,0,ξb].

The drift matrix M can be given by
M=κaΔ10J0000GabI0Δ1κaJ00000GabR00JκaΔ20ga2m0000J0Δ2κaga2m00000000ga2mκmΔmGmdI00000ga2m0ΔmκmGmdR0000000000ωd000000GmdRGmdIωdγd00000000000ωbGabRGabI000000ωbγb,
(7)
where GabR=gabRe(a1s), GabI=gabIm(a1s), GmdR=gmdRe(ms), and GmdI=gmdIm(ms) are the effective coupling rates of optomechanical and magnomechanical coupling strengths, respectively.
Hence, all of the system’s information is provided by the drift matrix 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)+uj(t)ui(t)/2, which can be obtained by solving the following Lyapunov equation,40,45
MV+VMT=D,
(8)
where the diffusion matrix D=diag(κa(2na1+1),κa(2na1+1),κa(2na2+1),κa(2na2+1),κm(2nm+1),κm(2nm+1),0,γd(2nd+1),0,γb(2nb+1)) is defined through Dijδ(tt)=12ni(t)nj(t)+nj(t)ni(t).
We quantify the bipartite entanglement between two subsystems employing the logarithmic negativity (EN), which can be defined as32,46,47
EN=max0,ln(2η),
(9)
where η=212σσ24detΓ1212 is the smallest symplectic eigenvalue of the partial transposed covariance matrix of two-mode Gaussian states with σ = det A + det B − 2 det C, and we have considered the bipartite correlation matrix in a 2 × 2 block form as
Γ=ACCTB,
(10)
where A, B, and C are 2 × 2 sub-block matrices. If EN > 0, then the two subsystems are entangled.

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 4849: ωm/2π=ωa1/2π=ωa2/2π=10GHz, ωb/2π = ωd/2π = 10 MHz, κm/2π = κa/2π = 1 MHz, gam/2π=gmd/2π=0.4Hz=Gmd/2π=3.2MHz, γb/2π = γd/2π = 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 Δ1/ωd and Δ2/ωd. By employing the coupling strength J = 0.8ωd and effective magnon detuning Δm=0.9ωd (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 Δ1=1.8ωd and Δ2 = −1.57ωd. 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 Δ1=ωd and Δ2 is kept either in the red sideband regime or blue sideband regime. On the other hand, the entanglement ENdb reaches maximum near Δ1=0.5ωd and Δ2 = 0. From those results, we realized that by eventually varying both cavity detunings, we could transfer entanglement from one subsystem to another.

FIG. 2.

Density plot of distant bipartite entanglements (a) ENa1m, (b) ENa1d, (c) ENmb, and (d) ENdb as a function of normalized effective cavity-1 detuning Δ1/ωd and normalized cavity-2 detuning Δ2/ωd. In (a) and (b), the effective magnon detuning is Δm=0.9ωd, whereas in (c) and (d), Δm=0.9ωd and J = 0.8ωd.

FIG. 2.

Density plot of distant bipartite entanglements (a) ENa1m, (b) ENa1d, (c) ENmb, and (d) ENdb as a function of normalized effective cavity-1 detuning Δ1/ωd and normalized cavity-2 detuning Δ2/ωd. In (a) and (b), the effective magnon detuning is Δm=0.9ωd, whereas in (c) and (d), Δm=0.9ωd and J = 0.8ωd.

Close modal

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 Δ1/ωd and Δ2/ωd for different coupling strength J. We have used the effective magnon detuning in resonance with the blue sideband regime. Specifically, for J = 0.5ωd, the entanglement ENa1m reaches the maximum value near the resonance around Δ1=0.2ωd and Δ2 = −0.36ωd, as shown in Fig. 3(a). In addition, the bipartite entanglement ENa1d attains the maximum value near the resonance almost for Δ1 ≈ 0 and Δ2 = −0.34ωd, as observed in Fig. 3(b). It can be seen that the bipartite entanglement ENa1a2 attains the optimal value near Δ1=0.5ωd and Δ2 = −0.5ωd, as observed in Fig. 3(c). For J = 0.8ωd, 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 Δ1=1.8ωd and Δ2 = −1.57ωd or near the resonance Δ1=0.04ωd and Δ2 = −0.28ωd 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 Δ1=0.3ωd and Δ2 = −1.4ωd, as shown in Fig. 3(f). Further increasing J = ωd, 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.

FIG. 3.

Density plot of distant bipartite entanglements ENa1m in panels (a), (d), and (g), ENa1d in panels (b), (e), and (h), and ENa1a2 in panels (c), (f), and (i) as a function of normalized effective cavity-1 detuning Δ1/ωd and normalized cavity-2 detuning Δ2/ωd. For (a)–(c), the cavity coupling strength is J = 0.5ωd; for (d)–(f), the cavity coupling strength J = 0.8ωd; and for (g)–(i), the cavity coupling strength J = ωd. For each case, Δm=0.9ωd. Other parameters are consistent with those given in Fig. 2.

FIG. 3.

Density plot of distant bipartite entanglements ENa1m in panels (a), (d), and (g), ENa1d in panels (b), (e), and (h), and ENa1a2 in panels (c), (f), and (i) as a function of normalized effective cavity-1 detuning Δ1/ωd and normalized cavity-2 detuning Δ2/ωd. For (a)–(c), the cavity coupling strength is J = 0.5ωd; for (d)–(f), the cavity coupling strength J = 0.8ωd; and for (g)–(i), the cavity coupling strength J = ωd. For each case, Δm=0.9ωd. Other parameters are consistent with those given in Fig. 2.

Close modal

In Fig. 4, we discuss the impact of the cavity coupling strength J/ωd and normalized cavities detunings Δa/ωd on the distant bipartite entanglements ENa1m, ENa1d, and ENa1a2. Notably, we have studied two methods in this instance: the non-symmetric case Δa/ωd=Δ1/ωd=Δ2/ωd, which represents (a, b, c), and the symmetric case Δa/ωd=Δ1/ωd=Δ2/ωd, 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 Δa = −0.5ωd and the cavity coupling strength J varies in between 0.5ωd and 1.4ωd; after this range, the entanglement vanishes as shown in Fig. 4(a). In addition, the entanglement ENa1d reaches a maximum value around Δa = −0.3ωd and for J varying in the range of 0.5ωd to 1.4ωd as shown in Fig. 4(b). Furthermore, the bipartite entanglement ENa1a2 reaches its maximum around Δa = −1.2ωd and J is in between 0.5ωd and 1.6ωd; while, after this, it drops very quickly on slowly varying Δa/ωd and J/ωd, as observed in Fig. 4(c). For the symmetric case, the bipartite entanglement ENa1m reaches the maximum value when cavity detunings Δa are in the red sideband regime and J = 0.5ωd, while in the blue sideband, the regime reaches finite value on varying Δa/ωd and J/ωd as shown in Fig. 4(d). Moreover, the bipartite entanglement ENa1d attains the maximum value near the resonance, i.e., Δa = 0 and the value of the cavity coupling strength becomes J = ωd as shown in Fig. 4(e). Furthermore, the entanglement ENa1a2 attains the maximum value for Δa varying in the negative region, whereas the coupling strength ranges from 0.5ωd to 1.5ωd, 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.

FIG. 4.

Density plot of bipartite entanglements ENa1m in panels (a) and (d), ENa1d in panels (b) and (e), and ENa1a2 in panels (c) and (f) vs normalized cavity detunings Δa/ωd and cavity coupling strength J/ωd for Δm=0.9ωd. In (a)–(c), Δa/ωd=Δ1/ωd=Δ2/ωd, whereas in (d)–(f), Δa/ωd=Δ1/ωd=Δ2/ωd. Other parameters are consistent with those given in Fig. 2.

FIG. 4.

Density plot of bipartite entanglements ENa1m in panels (a) and (d), ENa1d in panels (b) and (e), and ENa1a2 in panels (c) and (f) vs normalized cavity detunings Δa/ωd and cavity coupling strength J/ωd for Δm=0.9ωd. In (a)–(c), Δa/ωd=Δ1/ωd=Δ2/ωd, whereas in (d)–(f), Δa/ωd=Δ1/ωd=Δ2/ωd. Other parameters are consistent with those given in Fig. 2.

Close modal

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 Δa/ωd for Δa=Δ2=Δ1 considering different coupling strengths J. For J = 0.2ωd, the entanglement ENa1a2 reaches the optimum value of 0.04 for slowly changing Δa around 0.2ωd; 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 Δa near the resonance and reaches a certain value for positive Δa/ωd. Furthermore, the entanglement ENa2d attains its maximum value of 0.23 for negative Δa/ωd, whereas almost zero with positive Δa/ωd. Similarly, the entanglement ENa2m attain its optimum value of 0.13 for negative Δa/ωd, while zero with positive Δa/ωd. 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 Δa in between −1.3ωd and 1.07ωd; after this range, it almost becomes zero, whereas the bipartite entanglement ENa1m becomes zero for both positive and negative Δa/ωd for the mentioned coupling strength. It is realized that for J = 0.2ωd, there is a considerable amount of entanglement transfer from ENmd to ENa2m and ENa2d around Δa = −1.07ωd, as shown in Fig. 5(a). For J = 0.5ωd, the bipartite entanglement ENa1a2 decreases for both positive and negative Δa/ωd, as seen in Fig. 5(b). It is observed that the bipartite entanglement ENmd attains the maximum value of 0.22 for negative Δa/ωd and maintains a certain value for positive Δa/ωd, as shown in Fig. 5(b). In addition, the bipartite entanglement ENa2m attains the maximum value of 0.12 for negative Δa/ωd, and ENa2d reaches a maximum value of 0.22 in the negative Δa/ωd, 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 Δa/ωd 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 Δa = −0.94ωd. When we increase the coupling strength J = 0.8ωd, then the entanglement ENa1a2 reaches a maximum value for both negative and positive Δa 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 Δa. Furthermore, the entanglement ENa1d attains the optimum value for positive and negative Δa in certain ranges, and the entanglement ENa1m reaches its maximum for Δa near the resonance. As observed in Fig. 5(c), the most entanglement transfer from ENmd to ENa1d and ENa1m occurs about Δa = −0.8ωd. When we further increase J = ωd, 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.

FIG. 5.

Effect of cavity coupling strength J on bipartite entanglements ENa1a2, ENmd, ENa2m, ENa2d, ENa1m, and ENa1d plotted as a function of normalized cavity detunings Δa/ωd for Δa=Δ2=Δ1 while employing normalized effective magnon detuning Δm=0.9ωd. Other parameters are consistent with those given in Fig. 2.

FIG. 5.

Effect of cavity coupling strength J on bipartite entanglements ENa1a2, ENmd, ENa2m, ENa2d, ENa1m, and ENa1d plotted as a function of normalized cavity detunings Δa/ωd for Δa=Δ2=Δ1 while employing normalized effective magnon detuning Δm=0.9ωd. Other parameters are consistent with those given in Fig. 2.

Close modal

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.

FIG. 6.

Plot of bipartite entanglements ENa1d, ENa1m, ENmb, and ENdb as a function of temperature, each considered at the optimum value of effective cavity-1 detuning Δ1, cavity-2 detuning Δ2, effective magnon detuning Δm, and cavity coupling rate J.

FIG. 6.

Plot of bipartite entanglements ENa1d, ENa1m, ENmb, and ENdb as a function of temperature, each considered at the optimum value of effective cavity-1 detuning Δ1, cavity-2 detuning Δ2, effective magnon detuning Δm, and cavity coupling rate J.

Close modal

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.

The authors have no conflicts to disclose.

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).

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

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