1. Introduction

Quantum correlation is a fundamental aspect of quantum mechanics and serves as the crucial link between quantum and classical physics. When quantum correlations between subsystems for a system reach a certain threshold, the system is entangled, which has diverse applications in quantum information processing [1,2]. Substantial progress has been achieved for the generation of macroscopic entanglement in different optomechanical systems, including bipartite [3–5] and tripartite entanglement [6,7]. It is noted that tripartite entanglement refers to entangled states that exist in three parties, while it only exist within two for bipartite ones. Nowadays, tripartite entanglement has been proved to be of great importance in quantum communications, such as secure quantum key distribution protocols [8–10], quantum error correction [11], and quantum state transfer [12]. Besides, over the past few decades, researchers have also extensively investigated different quantum phenomena with cavity optomechanics, the platform mainly studying the interaction between confined light fields and phonons [13,14], such as cooling [15], induced transparency [16–18], induced absorption [19,20], induced amplification [16–18] and Faraday effect [21]. Similar to cavity optomechanics, cavity magnomechanics utilizing yttrium iron garnet (YIG) as a base material, which can offer a significant platform for investigating the interaction between magnons and photons (or phonons, et.al.) [22,23], has recently attracted a lot of interesting and much significant advancement has been made with it. The reason is mainly due to exceptional capability of collective excitations (i.e. magnons) in these ferromagnetic materials being able to interact with many platforms and holding low dissipation rates. On the basis of this platform, tripartite entanglement [23–25] of magnon-photon-phonon has been proposed, which can be transferred to other subsystems through coupling.

In addition to quantum entanglement, the concept of quantum steering [26–28], a subset of entanglement, has been extensively studied. It originated in 1935 when Einstein, Podolsky, and Rosen (EPR) questioned the completeness of quantum mechanics [29,30]. In their paper, they described how measuring one of two entangled states could steer the state of the other distant particle. Unlike quantum entanglement [23,31] and Bell nonlocality [32,33], quantum steering has certain asymmetric features between the two bodies (Alice and Bob) involved. For example, in one-way steering, Alice can steer Bob, but not vice versa. So far, one-way quantum steering has been observed in experiment [26,34,35]. This asymmetric quantum steering has practical applications in quantum subchannel discrimination [36], high-fidelity transmission [37–39], quantum cryptography [40], and quantum tele-amplification [41].

Moreover, non-Hermitian parity-time-(PT-) symmetry [42] has been used to enhance some quantum phenomena. Such as optical nonreciprocity in PT-symmetric whispering-gallery microcavities (WGM) [43], detection sensitivity of weak mechanical motion [44], strengthening optics nonlinearity [45], non-reciprocal light propagation [46] and so on [47–50]. PT-symmetry requires a strict balance between loss and gain, but strong gain can break system stability and is difficult to achieve experimentally due to environmental disturbances. However, PT-symmetric-like systems that do not require strict balancing can still function as PT-symmetric systems and have attracted considerable attention [51,52]. In PT-symmetric-like systems, similar to PT-symmetric systems, the Hamiltonian eigenvalues have one or two distinct values for the real and imaginary parts at different coupling strengths. This signifies the existence of a transition point between the unbroken and broken PT-like ranges. All investigations in this paper focus on values within the unbroken PT-like range.

Lately, a newly proposed opto-magnomechanical system that involves magnon, phonon and photon has captured the attention of researchers [53–57]. Building upon this system, we will introduce an approach that connects optomechanics and cavity magnomechanics to realize one-way quantum steering. Specifically, we use a four-mode PT-symmetric-like system containing a common Fabry-Perot cavity [58] in optomechanics and microwave cavities in cavity magnomechanics. By adding an auxiliary active gain to the microwave cavity, bipartite and tripartite entanglements between optical cavity photons, phonon, magnons and microwave cavity photons are enhanced [24,59], and a one-way EPR steering is achieved by determining the appropriate microwave cavity gain. Observation of entanglement and steering transfer phenomena within the system was also achieved. Additionally, we find that the entanglement is more robust against thermal excitations in the presence of microwave cavity gain, which helps to produce strong entanglement at higher temperatures.

This paper is organized as follows: The second section presents the system model and the Hamiltonian. We demonstrate numerical simulation and discussion in Sec. 3. Finally, in the fourth section, we make a conclude.

2. System model and Hamiltonian

As shown in Fig. 1(a), we propose to build a PT-symmetric-like cavity-opto-magnomechanical hybrid setup. It consists of a microwave cavity, an optical cavity, a magnon mode (kittel mode [60]) in a YIG crystal, and a vibrational mode due to deformation of the YIG crystal which is tightly attached to a highly reflective mirror. In this case, we can consider the two phonon modes separatedly induced by the mirror and the deformation of YIG as a whole, i.e. there is only one phonon mode in the system. The magnon mode in the YIG crystal is activated by applying a uniform bias magnetic field and a tunable microwave driving field perpendicular to the bias field. It is coupled to the microwave field by magnetic dipole interaction [22,61]. The optical cavity is directly coupled to phonon mode via the motion of the mirror [57,62,63]. It is noted that the mirror is minimized in size to avoid relative motion between YIG and the mirror. In other words, the deformation of the YIG crystal and the motion of the mirror can only induce one phonon mode. The magnetostrictive interaction of the ferromagnet results in a dispersive coupling between magnon and phonon [22]. For the clearity, we draw Fig. 1(b) to illustrate the corresponding modes and their interaction. Thus, the Hamiltonian for the whole system is given by

 

Fig. 1. (a) Schematic diagram of the active-passive cavity-opto-magnomechanical hybrid system. A bias magnetic field H along the z-axis is used to realize magnon-photon coupling by magnetic dipole interaction. The phonon couples to the magnon excitations in a YIG crystal via the dispersive magnetostriction interaction, and to an optical cavity via the radiation-pressure interaction. The directions of the bias magnetic field (z-axis), the drive magnetic field (x-axis) and the cavity mode magnetic field (y-axis) are perpendicular to each other. (b) Interaction and equivalent mode coupling model between the subsystems. The optical cavity-mechanical mode and microwave cavity-magnon coupling via linear beam splitter interaction. The magnon-mechanical mode coupling is created via magnomechanical parametric down-conversion interaction.

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The Hamiltonian of the whole system under the rotating-wave approximation can be written as

The quantum Langevin equations, which accounts for the relevant dissipations and noises of the operators in the system, can be obtained as

The microwave cavity’s effective damping rate $\kappa _{a1}$ is determined by subtracting the gain of the microwave cavity mode $g$ from the intrinsic damping rate $\kappa _{a}$. This can be expressed as $\kappa _{a1}=\kappa _{a}-g$. By defining the gain coefficient as $\eta =\kappa _{a1}/\kappa _{c}$, we can classify the microwave cavity a as a gain cavity when $\eta <0$ and as a passive cavity when $\eta >0$. The dissipation rates of the magnon mode, optical cavity mode, and phonon mode are denoted as $\eta >0$. $\kappa _{m}, \kappa _{c}$ and $\gamma _{b}$ respectively. The input noise correlation functions [65] are shown as: $\langle a_{i n}(t)a_{i n}^{\dagger}(t^{\prime})\rangle=[N_{a}(\omega_{a})+1]\delta(t-t^{\prime}), \langle a_{i n}^{\dagger}(t)a_{i n}(t^{\prime})\rangle=N_{a}(\omega_{a})\delta(t-t^{\prime}), \langle m_{i n}(t)m_{i n}^{\dagger}(t^{\prime})\rangle=[N_{m}(\omega_{m})+1]\delta(t-t^{\prime}), \langle m_{i n}^{\dagger}(t)m_{i n}(t^{\prime})\rangle=N_{m}(\omega_{m})\delta(t-t^{\prime}), \langle c_{i n}(t)c_{i n}^{\dagger}(t^{\prime})\rangle=[N_{c}(\omega_{c})+1]\delta(t-t^{\prime}), \langle c_{i n}^{\dagger}(t)c_{i n}(t^{\prime})\rangle=N_{c}(\omega_{c})\delta(t-t^{\prime})$. The noise operator associated with the gain in the microwave cavity $a$ is [66] $\langle a_{in}^{(g)}(t)a_{in}^{(g)\dagger }(t')\rangle =N_{a}(\omega _{a})\delta (t-t'),\langle a_{in}^{(g)\dagger }(t)a_{in}^{(g)}(t')\rangle =[N_{a}(\omega _{a})+1]\delta (t-t').$ Under the large mechanical quality factor $Q_m=\omega _b/\gamma _{b}\gg 1$ [67,68], the Langevin force operator $\xi$ is simplified to a $\delta$-correlated function with a Markovian approximation function, i.e. $\langle \xi (t)\xi (t')+\xi (t')\xi (t)\rangle /2\simeq \gamma _b[2N_b(\omega _b)+1]\delta (t-t')$ and $N_j(\omega _j)=[\exp {(\hbar \omega _j/k_BT)}-1]^{-1}$ $(j=a,b,c,m)$ are the mean thermal excitation number at environment temperature $T$ and $k_B$ is the Boltzmann constant.

Since the magnon and cavity modes are strongly driven, resulting in large steady-state amplitudes $|\langle m\rangle |\gg 1$, $|\langle a\rangle |\gg 1$. We can linearize the dynamics of the system around the steady-state values by writing any operators as $O=\langle O\rangle +\delta O$ $(O=a,b,c,m)$ [69] and neglecting second-order and higher-order fluctuation terms. As a result, the quantum Langevin equations are decomposed into two sets of equations for the steady-state value and the quantum fluctuation. By solving the equations for the classical average in the steady state, we obtain

So the drift matrix $A$ is given by

In addition, we use the logarithmic negativity (LN) $E_N$ to quantify the Gaussian bipartite entanglement [71–74]

$\Sigma \equiv \mathcal {R}_{1}+\mathcal {R}_{2}-2\mathcal {R}_{3}$, where $\mathcal {R}_{1}=\operatorname *{det}\mathcal {V}_{1}$, $\mathcal {R}_{2}=\operatorname *{det}\mathcal {V}_{2}$, $\mathcal {R}_{3}=\operatorname *{det}\mathcal {V}_{3}$ and $\mathcal {R}=\operatorname *{det}{\mathcal {V}}_{m}$. While the tripartite entanglement of microwave cavity/ optical cavity-magnon-phonon modes are quantified by introducing the minimum residual contangle given by [75,76]

Here, the notation $C_{i|j}$ represents the contangle of subsystems $i$ and $j$ ($j$ contains one or two modes). For the presence of genuine tripartite entanglement in the system, there must be nonzero minimum residual contangle $R_{\tau }^{min}>0$.

Furthermore, the Gaussian quantum steering is given by

To check the steering asymmetry of the two-mode Gaussian state, we introduce the difference [26]

3. Numerical simulation and discussion

Before our numerious simulations for the system, we first show our aim of the presented proposal and provide most of parameters we will use. In our cavity-opto-magnomechanical hybrid system, we can generate distant entanglement between the microwave cavity and the optical cavity (i.e. $E_{ac}>0$) by introducing a microwave cavity $a$ to the opto-magnomechanical system. Then by introducing gain into the microwave cavity $a$ (i.e. $\eta =\kappa _{a1}/\kappa _c <0$ ), the distant entanglement for the optical and microwave cavities can be notably enhanced and quantum steering can also be realized. It is noted experimentally feasible parameters are chosen, such as [13,22,53,54,63]: $\omega _{a}/2\pi =10 GHz$, $\omega _{m}/2\pi =10 GHz$, $\omega _{c}/2\pi =50 GHz$, $\omega _{b}/2\pi =40 GHz$, $\kappa _{a}/2\pi =2 MHz$, $\kappa _{c}/2\pi =2 MHz$, $\kappa _{m}/2\pi =1.5 MHz$, $\gamma _{b}/2\pi =100 Hz$, $g_{am}=4.3\times 2\pi$ $GHz$, $G_{mb}/2\pi =2MHz$, $G_{cb}/2\pi =8 MHz$, $T=10 mK$. In addition, All results in the article are obtained when the system is in a steady state, guaranteed by the negative eigenvalues (real part) of the drift matrix $\mathcal {A}$ [77].

Now, let us begin simulations and analysis. To ensure the realization of entanglement and steering in the system to a last degree, we have to select the optimal microwave cavity gain rate and optimal detuning. For this aim, we plot LN of the entanglement $E_{ac}$ between the two cavity modes versus ${\Delta }_{m1}/{\Delta }_{c1}$ and $\eta$ within the parameter range for the stability of the system in Fig. 2(a/b) with $\Delta _{c1} = \omega _b/\Delta _{m1} = -\omega _b$, $\Delta _{a} = \Delta _{m1}$ and the other paramters shown above. Our findings indicate that an optimal entanglement between microwave cavity and optical cavities can be achived (i.e. $E_{ac}$ is obtained to be a maximum value) at about ${\Delta }_{c1}=\omega _{b}$ and ${\Delta }_{m1}=-\omega _{b}$. Additionally, $E_{ac}$ gradually increases as the microwave cavity gain rate $\eta$ decreases, particularly when $\eta =\kappa _{a1}/\kappa _{c}<0$. In this case, we obtain the maximum value of $E_{ac}=0.44$ at $\eta =-0.25$. This suggests that the incoherent gain process in the PT-like symmetric structure can enhance the entanglement resulted from coherent nonlinear coupling. However, it is crucial to maintain the system’s stability by ensuring that the gain does not become too large. In other words, it must satisfy an ideal gain-loss balance, i.e. the system should be in a PT-symmetric-like state.

 

Fig. 2. Logarithmic negativity of entanglement. (a) Graph of $E_{ac}$ between microwave and optical cavity mode versus microwave cavity gain rate $\eta$ and the effective optical cavity detuning ${\Delta }_{m1}$ with $\Delta _{c1}=\omega _b$. (b) Graph of $E_{ac}$ versus $\eta$ and magnon detuning ${\Delta }_{c1}$ with $\Delta _{m1}=-\omega _b$, $\Delta _a=\Delta _{m1}$, where $\eta =\kappa _{a1}/\kappa _c$ is the microwave cavity gain rate and the environment temperature is $10mK$. See the third part of the paper for other parameters.

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Based on the above analysis, in the following passages, let us further investigate LN for the magnon-and-optical-photon entanglement ($E_{mc}$) and for the microwave-photon-and-phonon entanglement ($E_{ab}$), which is displayed in Fig. 3 with $E_{ac}$ (solid line), $E_{mc}$ (dashed line), and $E_{ab}$ (dotted line) as a function of the dimensionless detuning ${\Delta }_{m1}/\omega _b$ and ${\Delta }_{c1}/\omega _b$, where the red line represents the absence of microwave-cavity gain ($\eta >0$) and the blue line denotes the system with a microwave cavity gain rate ($\eta <0$). For the sake of simplicity and with the consideration of system’s stability, we choose $\eta = 1$ when $\eta >0$ and $\eta = -0.25$ when $\eta <0$. From Fig. 3, we observe that the microwave-photon-and-optical-photon entanglement is also significantly enhanced (i.e. $E_{ac}$ reaches the most value) when ${\Delta }_{m1}/\omega _b \approx -1$ and ${\Delta }_{c1}/\omega _b \approx 1$. Its value increases from 0.23 to 0.44, which corresponds to a two-fold enhancement in entanglement. But different from $E_{ac}$, we find from Fig. 3(a) that $E_{mc}$ increases from 0.12 to 0.19 and $E_{ab}$ from 0.14 to 0.21 at two non-resonant frequencies (${\Delta }_{m1}=-1.2\omega _b$ and $-0.8\omega _b$). These enhancements are quite substantial. Similarly, if we set $\Delta _{m1} = -\omega _{b}$, we can obtain from Fig. 3(b) that $E_{ab}$ increases from 0.14 to 0.43, and $E_{mc}$ is also enhanced. This enhancement may be due to the injection of gain into the microwave cavity, which increases the average photon number $|\langle a\rangle |^{{2}}$ in the cavity $a$. Consequently, energy can be transferred to the magnon via coupling, resulting in a larger average magnon number $|\langle m\rangle |^{{2}}$. This increase in the effective magnomechanical coupling $G_{mb}$ enhances LN of the entanglement $E_{mc}$ and $E_{ab}$. Notably, the enhancement of the distant entanglement between cavity $a$ and cavity $c$ is primarily due to the introduction of gain in the microwave cavity, which alters the dissipation of the subsystem [78].

 

Fig. 3. (a)-(b) Microwave cavity-optical cavity entanglement $E_{ac}$ (solid line), magnon-optical cavity entanglement $E_{mc}$ (dashed line) and microwave cavity-phonon entanglement $E_{ab}$ (dot-dashed line) as a function of the dimensionless detuning ${\Delta }_{m1}/\omega _b$ and ${\Delta }_{c1}/\omega _b$ for various microwave cavity gain rate $\eta$, the environment temperature is $10mK$. All other parameters are the same as those in Fig. 2.

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Based on the above two paragraphs, we have discuss the changing of the bipartite entanglement for $a$ and $c$, $a$ and $b$, $m$ and $c$. Now, let us discuss the generation mechanism for some of them, i.e. entanglement transfer, which is crucial for quantum information processing and transmission. Figure 4 illustrates LN of the entanglement between microwave cavity and optical cavity $E_{ac}$, magnon and optical cavity $E_{mc}$, microwave cavity and phonon mode $E_{ab}$, and magnon and phonon mode $E_{mb}$ versus the coupling rate $g_{am}$ between microwave cavity mode and magnon mode set at $\eta = -0.25$. In the stable region, it is evident that $E_{mb}$ and $E_{mc}$ increase rapidly as the microwave cavity-magnon coupling rate $g_{am}$ increases. $E_{mb}$ reaches its peak at $g_{am}/2\pi = 3.5 MHz$. Subsequently, as $g_{am}$ continues to increase, $E_{ac}$ begin to be larger than zero, while $E_{mb}$ decreases along with LN of the entanglement $E_{ab}$. $E_{ac}$ peaks at $g_{am}/2\pi = 4.3 MHz$, and $E_{ab}$ peaks at $g_{am}/2\pi = 6MHz$. Additionally, the participation of magnon-optical cavity entanglement (i.e. $E_{mc}>0$) is observed, with $E_{mc}$ reaching its peak at $g_{am}/2\pi = 6.2 MHz$. This adjustment of the coupling strength $g_{am}$ leading one entanglement to be weakened but the other to be strengthened, which can be considered as a transfer between different entanglements. Entanglement transfer, as a significant quantum phenomenon, has diverse physical applications and enables the realization of quantum communication [79], quantum computing [80], quantum key distribution [40,81], and quantum sensing [82].

 

Fig. 4. Plot of the $E_{ac}$ (black solid line), $E_{mc}$ (red dashed line), $E_{ab}$ (blue dot-dashed line), and the entanglement between magnon the phonon $E_{mb}$ (green dashed line), as a function of the microwave cavity-magnon coupling rate $g_{am}$. The microwave cavity gain rate is set to $\eta =-0.25$, and the optimal effective detuning values are $\Delta _{c1}=-\Delta _{m1}=\omega _b$ and $\Delta _a={\Delta }_{m1}$. The remaining parameters are consistent with those in Fig. 2.

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With the presence of microwave-cavity gain, the robustness of entanglement against environment temperature in our system is significantly enhanced, as depicted in Fig. 5. We present graphs of LN of the distant entanglement $E_{ac}$ with the microwave cavity gain rate $\eta$ and the environment temperature $T$ in Fig. 5(a), where the optimal detuning ${\Delta }_{c1}=\omega _{b}$ and ${\Delta }_{m1}=-\omega _{b}$ are obtained from Fig. 2. This also proves that the maximum LN of distant entanglement $E_{ac}$ can be obtained at $\eta = -0.25$. To study the detailed relationship between entanglement and temperature, we plot LN of the distant entanglement $E_{ac}$ as a function of temperature $T$ in Fig. 5(b). The black solid line shows that the maximum LN of entanglement $E_{ac} = 0.23$ disappears at $T = 0.33 K$ when no microwave cavity gain is present. In the entanglement graph plotted by the green line for input microwave cavity gain $\eta =-0.25$ at the same temperature, the entanglement value is $E_{ac} = 0.44$ and the entanglement persists until $T = 0.9 K$. The red line and blue line also show this phenomenon at the same temperature, but it is not as pronounced as the green line. The graphs of LN of near entanglement $E_{mc}$ and $E_{ab}$ with temperature plotted in Fig. 5(c) and (d) also exhibit this trend. Therefore, the input of microwave cavity gain enhances the strength of entanglement at higher temperatures. Besides, if we take the interaction of the two phonons induced by the mirror and deformation of YIG into account, then we can still use dressed-state method and consider the symmetry to combine the two phonon modes to be a whole to interact with magnons and photons. In this case, we can increase the phonon decay rate $\gamma _b$ to discuss its effect on LN of the entanglement. After a simple simulation for LN of the bipartite entanglement versus $\gamma _b$ (not shown in the paper), we find this decay rate can scarcely weaken the entanglement.

 

Fig. 5. (a) Graphs of distant entanglement $E_{ac}$, as a function of the microwave cavity gain rate $\eta$, and emvironment temperature $T$. (b)-(d) Optimal entanglement $E_{ac}$, $E_{mc}$ and $E_{ab}$ as a function of temperature for different microwave cavity-magnon coupling strengths $g_{am}$, where $g_{am}=4.3\times 2\pi$ $GHz$ in (a) and (b), $g_{am}=6.2\times 2\pi$ $GHz$ in (c), and $g_{am}=6\times 2\pi$ $GHz$ in (d). The optimal effective detuning is $\Delta _{c1}=-\Delta _{m1}=\omega _b$ and $\Delta _a={\Delta }_{m1}$. Other parameters remain the same as in Fig. 2.

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Now, let us turn to study tripartite entanglement. When the magnon frequency resonates with the red sideband and the optical cavity frequency resonates with the blue sideband, tripartite entanglement $R_{\tau, amb}^{\min }$ (microwave cavity, magnon, and phonon mode) and $R_{\tau, mbc}^{\min }$ (optical cavity, phonon mode, and magnon) can be observed in the system. In Fig. 6, we have plotted the minimum residual contangle as a function of microwave cavity detuning $\Delta _a/\omega _b$ to analyze the impact of microwave cavity gain on $R_{\tau, amb}^{\min }$ and $R_{\tau, mbc}^{\min }$. The green dashed line represents $R_{\tau, amb}^{\min }$ without microwave cavity gain, while the blue solid line indicates $R_{\tau, amb}^{\min }$ at a microwave cavity gain of $\eta = -0.25$. It is evident that although the peak value of $R_{\tau, amb}^{\min }$ decreases, it increases in the overall region. For the tripartite entanglement $R_{\tau, mbc}^{\min }$ between the optical cavity, phonon, and magnon, the maximum value at microwave cavity gain (black solid line) rises from 0.009 to 0.02 compared to the tripartite entanglement $R_{\tau, mbc}^{\min }$ in a normal dissipative system (red dashed line).

 

Fig. 6. The tripartite entanglement $R_{\tau, amb}^{\min }$ (microwave cavity, magnon, and phonon mode and $R_{\tau, mbc}^{\min }$ (optical cavity, phonon mode, and magnon) are plotted in subplots (a) and (b), respectively, as a function of the microwave cavity detuning $\Delta _a/\omega _b$. The green dashed line represents the tripartite entanglement $R_{\tau, amb}^{\min }$ ($\eta = 1$) and the blue solid line represents the tripartite entanglement $R_{\tau, amb}^{\min }$ ($\eta = -0.25$). The red dashed line represents the tripartite entanglement $R_{\tau, mbc}^{\min }$($\eta = 1$) and the black solid line represents the tripartite entanglement $R_{\tau, mbc}^{\min }$ ($\eta = -0.25$). The optimal effective detuning is set as $\Delta _{c1}=-\Delta _{m1}=\omega _b$, and the remaining parameters are consistent with those in Fig. 2.

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In this passage, let us continue to study quantum steering for bipartite subsystems. In this tunable cavity-opto-magnomechanical hybrid system, asymmetric quantum EPR steering can also be achieved. Figure 7 illustrates the quantum steering as a function of the microwave cavity-magnon coupling rate $g_{am}$ with $\eta =-0.25$. In Fig. 7(a), it can be observed that one-way steering from the microwave cavity $a$ to the optical cavity $c$ is achieved between $g_{am} = 1.8G_{mb}$ and $g_{am} = 2.7G_{mb}$. The maximum value of the one-way steering $\zeta ^{a\rightarrow c}=0.11$ occurs around $g_{am} = 2.2G_{mb}$, corresponding to the maximum entanglement $E_{ac}$ at this point. This indicates that $a \rightarrow c$ one-way controllable entangled states can be achieved under specific conditions. $a \rightarrow c$ one-way steering implies that Alice can convince Bob that their shared state is entangled, while the reverse is not true [26,83]. This has potential applications in providing security in one-sided device-independent quantum key distribution, where only one side of the measurement device is untrustworthy. In Fig. 7(b), one-way steering from microwave cavity $a$ to phonon mode $b$ is achieved between $g_{am} = 2.6G_{mb}$ and $g_{am} = 3.2G_{mb}$. Two-way asymmetric steering between $a$ and $b$ is observed on the right side of $g_{am}/G_{mb} = 3.2$. Figure 7(c) demonstrates the one-way steering from the magnon $m$ to the optical cavity $c$, achieved between $g_{am} = 3.1 G_{mb}$ and $g_{am} = 4.3 G_{mb}$. This process enables the transfer of distant (microwave cavity-optical cavity) and near (microwave cavity-phonon mode and magnon-optical cavity) directional steering. Therefore, by tuning the relevant parameters, this adjustable scheme allows for the transfer and exchange of entanglement and steering between optional modes in a steady state.

 

Fig. 7. (a) The distant steering between the microwave cavity and optical cavity ($\zeta ^{a\to c}$ and $\zeta ^{c\to a}$) is examined as a function of the coupling rate $g_{am}/G_{mb}$ in the system. (b) Similarly, the near steeringing between the microwave cavity and phonon mode ($\zeta ^{a\to b}$ and $\zeta ^{b\to a}$) is investigated as a function of the coupling rate $g_{am}/G_{mb}$. In addition, the relationship between the magnon-optical cavity near steering ($\zeta ^{m\to c}$ and $\zeta ^{c\to m}$) is explored as a function of the coupling rate $g_{am}/G_{mb}$. Here, $G_{mb}$ is held constant at $G_{mb}/2\pi =2MHz$. The optimal effective detuning is $\Delta _{c1}=-\Delta _{m1}=\omega _b$ and $\Delta _a={\Delta }_{m1}$, while the other parameters remain the same as in Fig. 2.

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Finally, we discuss the feasibility of the cavity-magnon-optomechanical hybrid system used in this paper. The parameters employed in this study fall within the experimentally feasible range [13,22,53,54,63]. The effective magnomechanical coupling strength $G_{mb}$, and the microwave cavity-magnon coupling strength $g_{am}$, within the system can be altered by manipulating the position of the YIG crystal and the orientation of the bias magnetic field [22]. Additionally, the effective optomechanical coupling strength $G_{cb}$ can be adjusted by modifying the position of the YIG crystal and mirror [13,62,63]. This PT-symmetric coupling system, comprising gain introduced into the microwave cavity, has been extensively investigated both theoretically [84,85] and experimentally [43,86]. The magnitudes of these drive strengths are adjustable [87,88]. Li et al., employed a red-detuned driving field (${\Delta }_{m}\simeq \omega _{b}$) to drive the magnon mode $m$. Activation of magnomechanical anti-Stokes scattering and enhancement magnomechanical coupling to produce entanglement [23]. In our scheme, we employ a blue-detuned driving field (${\Delta }_{m1}\simeq -\omega _{b}$) to drive the magnon $m$, effectively suppressing anti-Stokes scattering and facilitating the cooling of the phonon mode. Furthermore, a red-detuned driving field (${\Delta }_{c1}\simeq \omega _{b}$) is utilized to drive the optical cavity mode $c$, activating the optomechanical anti-Stokes scattering and greatly enhancing the optomechanical coupling, thereby enabling the generation of entanglement.

4. Conclusion

We have investigated a PT-symmetric-like cavity-opto-magnomechanical system that consists of active-passive double cavities with experimentally feasible parameters. Our results indicate that the presence of microwave cavity gain significantly enhances the bipartite entanglement between the microwave cavity-optical cavity, magnon-optical cavity and microwave cavity-phonon mode subsystems, which can be transferred to other subsystems. Furthermore, we have observed that the system’s bipartite entanglement is robust to changes in environmental temperature, which is essential for generating strong distant entanglement at high temperatures. Moreover, we have observed some enhancement in the tripartite entanglement of the system when considering the feasibility parameters. Finally, we have found that the gain of the cavity mode breaks the system’s balance, resulting in one-way quantum steering of the microwave cavity-optical cavity, one-way quantum steering of the magnon-optical cavity, one-way steering of the microwave cavity-phonon mode, and asymmetric two-way steering. Our scheme establishes a fundamental connection between optomechanical and cavity magnomechanical, yielding crucial applications in fields including quantum key distribution, quantum teleportation, quantum optical devices and quantum information processing.