1. Introduction

In the past decades, quantum entanglement and cooling as an important source of quantum information processing has been widely studied in many quantum systems, such as trapped ions [1,2], atomic ensembles [3–5], superconducting qubits [6,7], and cavity optomechanical systems [8–26], and significant progress has been made. Very recently, the emerging cavity magnomechanical system has opened up a new way for the study of quantum entanglement by virtue of its many superior properties, such as high spin density, low dissipation rate, and strong coupling. In a pioneering theoretical scheme, bipartite and tripartite steady-state entanglement has been demonstrated for a standard cavity magnomechanical system [27]. Later, magnomechanical entanglement has also been reported in multimode coupled-cavity quantum systems [28–33].

However, the preparation of great quantum entanglement for continuous variable systems is always a direction that we need to strive for. It is shown that the entanglement can be enhanced when the Kerr nonlinear effect is considered in the hybrid cavity magnomechanics [34,35]. Yu $et$ $al$. improved entanglement by injecting a two-mode squeezed microwave field into a double cavity-magnon system [36]. The bipartite and tripartite entanglement of standard cavity magnomechanical systems can be simultaneously enhanced via using the squeezing characteristics of optical parametric amplifiers [37]. Subsequent studies of our group discussed the entanglement between a microwave field and a macroscopic mechanical oscillator achievable by reservoir engineering approach can beyond the limit of entanglement generated via the coherent parametric interaction [38]. Like other quantum phenomena, quantum entanglement is usually susceptible to environmental temperature, which leads to the experimental preparation of quantum entanglement still faces some challenges. Thus, it has been an important research topic to improve the resistance of entanglement to thermal noises of high temperature. The robustness of entanglement to temperature can be improved by placing a degenerate parametric amplifier in the cavity optomechanical (or cavity magnomechanical) system [37,39,40]. Moreover, the capability of entanglement resistance to thermal noise can also be improved via using the cooperative effect of a periodically amplitude-modulated laser with a parametric amplifier [41]. By employing the phenomenon of exceptional points in $\mathcal {PT}$-symmetric systems, the sudden death of the entanglement affected by noises can be delayed [42]. The steady-state entanglement can survive at a higher thermal temperature via using the reservoir engineering method [43–45]. A constructive work is presented to obtain thermal noise-tolerant quantum entanglement via a synthetic magnetism mechanism [46].

It is well known that cooling systems to their quantum ground states have been intensively studied in the last decades. For example, several typical cooling schemes have been proposed in cavity optomechanical systems, sideband cooling [47,48], auxiliary-cavity cooling [49,50], and feedback-aided cooling [51–61]. Recently, some interesting work on cooling mechanisms has also been reported, such as quantum interference techniques [62–64], domino effect [65,66], modulated pulses methods [67,68], $\mathcal {PT}$-symmetric mechanisms [69], strong coupling principles [70,71], nonreciprocal ground-state cooling [72], auxiliary feedback loop [73], and so on. However, the study of cooling in cavity magnomechanical systems is currently rare [74–80]. Therefore, we focus on enhancing magnomechanical cooling and entanglement and improving the robustness of entanglement against temperature.

In this paper, we investigate how to simultaneously enhance cooling and entanglement in an auxiliary-microwave-cavity-assisted (AMCA) magnomechanical system. We connect an auxiliary system to a main system via photon tunneling. By optimizing the parameter regime, we reveal that the magnomechanical cooling of the main system can be significantly enhanced. Remarkably, the maximum cooling enhancement rate obtained in our work is $98.53{\%}$, which far exceeds that obtained in the cavity optomechanical system [50]. Furthermore, a significant enhancement for dual-mode entanglement can be obtained based on the AMCA method compared to the case without AMCA. We numerically show that the magnon-phonon entanglement $E_N^{m_1b}$ can be enhanced roughly eleven times with experimentally achievable parameters, which outperforms other entanglement enhancement methods such as the optical parametric amplifier mechanism [37], and the Kerr nonlinear effect [34,35]. We also show that the robustness of the entanglement against temperature can be increased in the presence of the AMCA, which provides a way to preserve fragile quantum resources. Apart from the above-mentioned enhancements, the AMCA has also been used to establish a wider variety of indirect coupling-entanglement.

The rest of this paper is arranged as follows: In Sec. 2, we explain our physical model and the Hamiltonian. In Sec. 3, we obtain the linearized Hamiltonian through the quantum Langevin equations and the linearization process, and describe the steady-state dynamics of the system via the covariance matrix. In Sec. 4, we study the AMCA magnomechanical cooling. In Sec. 5, we discuss the steady-state entanglement mediated by AMCA. Finally, the conclusion is summarized in Sec. 6.

2. System model and Hamiltonian

We consider a hybrid multimode AMCA magnomechanical system, which consists of a magnomechanical microwave cavity mode $c_{1}$, a auxiliary microwave cavity mode $c_{2}$, two magnon modes $m_{1}$ and $m_{2}$, and a mechanical deformation mode $b$, as illustrated in Fig. 1. The deformation mode results from the varying magnetization induced by the magnon excitation inside the YIG sphere. The Hamiltonian of the system is given by ($\hbar =1$)

 

Fig. 1. Sketch of a coupled AMCA magnomechanical model with two yttrium-iron-garnet (YIG) spheres. Each YIG sphere is placed in an uniform bias magnetic field and near the maximum magnetic field of the magnomechanical cavity, which establishes the coupling between the photon mode $c_{1}$ and the two magnon modes ($m_{1}$ and $m_{2}$). The sphere on the left is driven directly by an external microwave field (not shown), which induces the magnon-phonon ($m_{1}$ and $b$) coupling. The magnomechanical cavity $c_{1}$ is coupled to the auxiliary cavity $c_{2}$ via photon hopping with the rate $J$.

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Here $H_{fr}$ is the free Hamiltonian with $c_{1}$, $c_{2}$, $m_{j}$ ($j=1,2$), and $b$ being the annihilation operators for the magnomechanical cavity (frequency $\omega _{c}^{1}$), the auxiliary cavity (frequency $\omega _{c}^{2}$), $j$th magnon (frequency $\omega _{m}^{j}$), and the vibrational (frequency $\omega _{b}$) modes, respectively, and the frequency $\omega _{m}^{j}$ of the $j$th magnon mode is determined by the bias magnetic field $H_{j}$ of the $j$th YIG sphere and the gyromagnetic ratio $\eta$, i.e., $\omega _{m}^{j}=\eta H_{j}$. $H_{cm}$ represents the magnetic dipole interaction between the microwave field $c_{1}$ and the $j$th magnon mode $m_{j}$, where $G_{j}$ is the corresponding coupling strength. $H_{mb}$ describes the interaction between the magnetostatic mode $m_{1}$ and the mechanical mode $b$ mediated by the magnetostrictive effect with the weak single-magnon coupling rate $G_{0}$ ($G_{0}\ll \kappa _{m}^{1}$, $\kappa _{m}^{1}$ is the dissipation rate of the magnon mode $m_{1}$). $H_{cc}$ symbolizes the beam-splitter-like interaction between the magnomechanical cavity mode $c_{1}$ and the auxiliary cavity mode $c_{2}$ with the tunneling coupling strength $J$. The last term $H_{dr}$ means that the magnon mode $m_{1}$ is directly driven by a microwave source with frequency $\omega _{0}$ and driving strength $E$.

Then the total Hamiltonian (1), in the rotating frame with respect to the laser driving frequency $\omega _0$, can be expressed as

3. Quantum Langevin equations and steady-state dynamics

In this section, we study the steady-state dynamics of the system via deducing the quantum Langevin equation and linearizing the Hamiltonian (Eq. (2)). When the dissipation and input noises are phenomenally added to the Heisenberg equation of motion, the quantum Langevin equations (QLEs) of the system can be derived as

To investigate the dynamic behavior of the system, we consider the strong-driving regime for the left sphere in Fig. 1, so that we can rewrite the Heisenberg operators as $c_{j}=\alpha _{j}+\delta c_{j}$, $m_{j}=\varepsilon _{j}+\delta m_{j}$, and $b=\beta +\delta b$, where $\alpha _{j}$, $\varepsilon _{j}$, and $\beta$ are classical steady-state values, and $\delta c_{j}$, $\delta m_{j}$, and $\delta b$ are the small fluctuation operators with zero mean values. In this case, $|\alpha _{j}|$, $|\varepsilon _{j}|$, $|\beta |$ $\gg 1$, which allows us to linearize our physical system. Substituting these new Heisenberg operators into Eq. (3), we then obtain

In essence, we can carefully discuss the quantum properties of the hybrid system according to Eq. (7). However, in order to more intuitively embody the dynamic evolution of our physical model, we introduce a set of quadrature coordinate operators and quadrature momentum operators as follows ($j=1,2$)

4. Accelerated magnomechanical cooling via AMCA

In this section, we focus on the AMCA magnomechanical cooling performance, which can be described via mean particle number. By using the relationship between the quadrature operator and the fluctuation operator defined in Eq. (9), we obtain the steady-state average photon, magnon, and phonon number as follows

To reflect the positive effect of the AMCA mechanism on the cavity magnomechanical cooling phenomenon, we plotted the mean particle number of the four bosonic modes as a function of detunings $\Delta _c^2/\omega _b$ and $\Delta _m^2/\omega _b$ as illustrated in Fig. 2. We choose the parameters $\Delta _c^1=-\omega _b$, $\Delta _m^1=\omega _b$ in Fig. 2, which ensure that the magnomechanical cooling interaction $G(\delta m_{1}^{\dagger }\delta b+\delta m_{1}\delta b^{\dagger })$ in Eq. (8) can be safely preserved during the rotating-wave approximation. This is the primary condition for the magnomechanical cooling effect. By comparing the top row with the bottom row in Fig. 2, we find that the average magnon numbers $W_m^1$ and $W_m^2$, the average photon number $W_c^1$, and the average phonon number $W_b$ are significantly reduced in the case with AMCA. For example, when $J=0.8\omega _b$, the mean phonon number $W_b$ for the mechanical mode and the mean magnon number $W_m^1$ for the magnetic mode can reach $W_b\simeq 0.08$, $W_m^1\simeq 0.07$ [see Figs. 2(e) and 2(g)], while the mean photon number $W_c^1$ for the optical mode and the mean magnon number $W_m^2$ for the magnetic mode can even reach $W_c^1\simeq 0.005$, $W_m^2\simeq 0.0003$ [see Figs. 2(f) and 2(h)], which is smaller than the case without AMCA $W_b\simeq 0.10$, $W_m^1\simeq 0.11$, $W_c^1\simeq 0.038$, and $W_m^2\simeq 0.001$ [see Figs. 2(a)–2(d)], respectively. This suggests that our proposed AMCA mechanism makes the cavity magnomechanical system closer to the quantum ground state (i.e., $W\ll 1$). Note that the cooling results obtained through our scheme for the mechanical mode $\delta b$ are comparable with the other schemes [74–76,78–80], but the cooling effects of the optical mode $\delta c_1$ and the magnon mode $\delta m_2$ are far superior to the ones of the mechanical mode $\delta b$ in our system. More interestingly, in the absence of AMCA, the mechanical modes $\delta b$, magnetic modes $\delta m_1$ and $\delta m_2$, and optical mode $\delta c_1$ cannot be cooled around $\Delta _c^2=-\omega _b$ (the average particle numbers $W_b$, $W_m^1$, $W_m^2$, and $W_c^1$ have maximum values) [see Figs. 2(a)–2(d)], while this phenomenon is broken in the presence of AMCA [see Figs. 2(e)–2(h)]. This means that the AMCA effect provides a wider range of detuning parameters for magnomechanical cooling.

 

Fig. 2. Density plot of average particle number versus dimensionless detunings $\Delta _c^2/\omega _b$ and $\Delta _m^2/\omega _b$ in the case without AMCA ($J=0$) for (a)-(d) and with AMCA ($J=0.8\omega _b$) for (e)-(h). The parameters are $G=G_j=0.32\omega _b$, $\kappa _c^j=\kappa _m^j=0.1\omega _b$ ($j=1,2$), $\gamma _b=10^{-5}\omega _b$, $\Delta _c^1=-\omega _b$, and $\Delta _m^1=\omega _b$. Here we temporarily disregard the effect of ambient thermal noise, i.e., $\overline {n}_{b}=0$ and $\overline {n}_{c}^{j}=\overline {n}_{m}^{j}=0$.

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In Fig. 3, we show the average phonon number $W_b$ (red solid line) of the mechanical oscillator, the average magnon numbers $W_m^1$ (green dashed line) and $W_m^2$ (blue dashed line) of the YIG spheres $m_1$ and $m_2$, and the average photon number $W_c^1$ (magenta solid line) of the microwave cavity versus the cavity-cavity tunneling coupling $J/\omega _b$. We can see that at the initial $J=0$ (without AMCA), these four average particle numbers $W_b$, $W_c^1$, $W_m^1$, and $W_m^2$ are at their maximum values, while when we consider the tunneling coupling $J$ (with AMCA), $W_b$, $W_c^1$, $W_m^1$, and $W_m^2$ start to decrease. Moreover, the magnomechanical cooling is significantly enhanced as the photon hopping rate $J$ increases, which is due to the fact that the rotating-wave approximation condition $2\omega _b\gg J$ is no longer well satisfied as $J$ increases, so much so that the beam-splitter interaction term $J(\delta c_{1}^{\dagger }\delta c_{2}+\delta c_{1}\delta c_{2}^{\dagger })$ in Eq. (8) cannot be neglected. Here, the auxiliary cavity $\delta c_2$ can be treated as an additional cold reservoir which can effectively cool the hybrid cavity magnomechanical system by the interaction $J$. Finally, with further increase of $J$, the mean particle numbers $W_b$, $W_c^1$, $W_m^1$, and $W_m^2$ all decrease to a limit and remain almost constant ($W_b\simeq W_m^1\approx 0.09$, $W_c^1\simeq W_m^2\approx 0.001$), which is imposed by the stability condition of the continuous variable system. That is, a coupling strength $J$ that is too large will result in the system being in an unstable state according to the Routh-Hurwitz criterion [81–86], which limits further cooling of the system. Also, $W_b$ (red solid line) and $W_m^1$ (green dashed line) are always larger than $W_c^1$ (magenta solid line) and $W_m^2$ (blue dashed line) in Fig. 3, which is due to the parametric amplification interaction $G(\delta m_{1}\delta b+\delta m_{1}^{\dagger }\delta b^{\dagger })$ between the mechanical mode $\delta b$ and the magnetic mode $\delta m_1$ (see Eq. (8)) leading to a mutual heating process between them.

 

Fig. 3. The average particle number $W$ as a function of the tunneling coupling strength $J/\omega _b$ with $\Delta _c^2=\omega _b$ and $\Delta _m^2=-\omega _b$. The other parameters are the same as those in Fig. 2.

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In Fig. 5, we show four cooling enhancement rates as a function of the dimensionless detunings $\Delta _c^2/\omega _b$ and $\Delta _m^2/\omega _b$ for $J=1.66\omega _b$, respectively. We find that there is a cooling suppression ($\Gamma _m^2>0$) region $0\lesssim \Delta _c^2/\omega _b\lesssim 2$, $-2\lesssim \Delta _m^2/\omega _b\lesssim 0$ for the magnon mode $\delta m_2$, but outside this region the cooling of $\delta m_2$ can be significantly amplified ($\Gamma _m^2<0$) assisted by an auxiliary microwave cavity [see Fig. 5(d)]. Strikingly, the cooling of phonon mode $\delta b$, photon mode $\delta c_1$, and magneton mode $\delta m_1$ can all be significantly enhanced via AMCA in the full map [see Figs. 5(a)-(c)], which is quite interesting. In the presence of AMCA, the cooling enhancement region of the system is wide, which is an important feature from the experimental point of view.

To see more intuitively how large the magnomechanical cooling can be enhanced, we define here the cooling enhancement rate as follows:

A similar definition has been previously considered in cavity optomechanics [50]. From Eq. (16) we can easily find that $\Gamma <0$ indicates that the magnomechanical cooling is enhanced and conversely $\Gamma >0$ indicates that the magnomechanical cooling is suppressed. In Fig. 4, the cooling enhancement rates $\Gamma$ are plotted as a function of the dimensionless detuning $\Delta _m^2/\omega _b$. First of all, under the AMCA physical mechanism, the cooling of the system can be significantly enhanced, in particular, the cooling enhancement rates $\Gamma _b$, $\Gamma _c^1$, $\Gamma _m^1$, and $\Gamma _m^2$ reach a maximum around the parameter $\Delta _m^2/\omega _b=-1$, as shown in Figs. 4(a) and 4(b). Moreover, under the strong coupling AMCA parameter condition $J=1.66\omega _b$, the maximum cooling enhancement rates of the four bosonic modes $\delta b$, $\delta c_1$, $\delta m_1$, and $\delta m_2$ are $\Gamma _b=67.75{\%}$, $\Gamma _c^1=98.13{\%}$, $\Gamma _m^1=65.97{\%}$, and $\Gamma _m^2=98.53{\%}$ [see Fig. 4(b)], respectively, which are much larger than in the weak coupling $J=0.8\omega _b$ case [see Fig. 4(a)]. Note that the cooling enhancement rates $67.75{\%}$, $98.13{\%}$, $65.97{\%}$, $98.53{\%}\gg 40{\%}$, where $40{\%}$ is obtained in a three-mode auxiliary cavity optomechanical system [50]. More importantly, in Fig. 4(b), the cooling of any of the four bosonic modes can always be enhanced when the detuning $\Delta _m^2/\omega _b$ is varied from $-2$ to $2$. However, this is no longer fulfilled in Fig. 4(a), for example the cooling of the phonon mode $\delta b$ (red line) and the magnon mode $\delta m_2$ (blue line) is suppressed in a specific detuning parameter regime.

 

Fig. 4. Cooling enhancement rates $\Gamma _b$ (red line), $\Gamma _c^1$ (magenta line), $\Gamma _m^1$ (green line), and $\Gamma _m^2$ (blue line) as a function of detuning $\Delta _m^2$ for (a) $J=0.8\omega _b$ and (b) $J=1.66\omega _b$. Here $\Delta _c^2=\omega _b$ and the other parameters are the same as those in Fig. 2.

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Fig. 5. Cooling enhancement rates $\Gamma _b$ (a), $\Gamma _c^1$ (b), $\Gamma _m^1$ (c), and $\Gamma _m^2$ (d) versus dimensionless detunings $\Delta _c^2/\omega _b$ and $\Delta _m^2/\omega _b$ with $J=1.66\omega _b$. The rest of the parameters are the same as in Fig. 2.

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5. AMCA-induced strong quantum entanglement and its robustness against temperature

In the previous section, we confirmed that the cooling of the system can be effectively improved by the AMCA mechanism. Now, we explore whether AMCA can also significantly enhance magnomechanical entanglement via calculating the logarithmic negativity.

Because Eq. (12) is a linear equation and can be solved directly, it is convenient to use $\zeta$ to calculate the bipartite entanglement between any pair of bosonic modes of the hybrid magnomechanical system. For example, the entanglement between the cavity mode $\delta c_{1}$ and the magnon mode $\delta m_{2}$ can be readily computed from the reduced $4\times 4$ covariance matrix $\zeta ^{*}$, which is extracted from the full $10\times 10$ covariance matrix $\zeta$ by keeping the components in the $k$th rows and $l$th columns ($k,l\in \{1,2,7,8\}$). If the bipartite covariance submatrix $\zeta ^{*}$ is given in the following form:

Since the cooling of the cavity magnomechanical system can be effectively enhanced by the AMCA method as analyzed in Sec. 4, intuitively, the quantum entanglement of the system should also be enhanced. To verify this idea, we plot the two-mode entanglement $E_N^{m_2b}$ and $E_N^{m_1b}$ as a function of the detunings $\Delta _c^2/\omega _b$ and $\Delta _m^2/\omega _b$ when the system operates in both the with AMCA and without AMCA regimes as demonstrated in Fig. 6. As expected, both the magnon-phonon entanglement $E_N^{m_2b}$ and $E_N^{m_1b}$ can be significantly enhanced via AMCA, which can be clearly seen by comparing Fig. 6(a) (Fig. 6(c)) with 6(b) (6(d)). Further, when there is no AMCA, the two-mode entanglement $E_N^{m_1b}$ is very small ($E_N^{m_1b}\approx 0.04$) around $\Delta _c^2/\omega _b=-1$ [see Fig. 6(c)], which is due to the fact that neither the magnon mode $\delta m_1$ nor the phonon mode $\delta b$ can be effectively cooled at $\Delta _c^2\approx -\omega _b$ as described in Figs. 2(a) and 2(c). However, in the presence of AMCA, the bipartite entanglement $E_N^{m_1b}$ breaks this limit, i.e., $E_N^{m_1b}$ is quite large around $\Delta _c^2/\omega _b=-1$ in Fig. 6(d), which is closely related to the fact that the bosonic modes $\delta m_1$ and $\delta b$ can be efficiently cooled by AMCA as demonstrated in Figs. 2(e) and 2(g). Physically, this makes the scheme more feasible toward experimental realization.

 

Fig. 6. Density plot of steady-state entanglement $E_N^{m_2b}$ between magnon and phonon modes versus dimensionless detunings $\Delta _c^2/\omega _b$ and $\Delta _m^2/\omega _b$ for (a) $J=0$ and (b) $J=0.32\omega _b$. Density plot of steady-state entanglement $E_N^{m_1b}$ between sphere $m_1$ and mechanical oscillator $b$ versus dimensionless detunings $\Delta _c^2/\omega _b$ and $\Delta _m^2/\omega _b$ for (c) $J=0$ and (d) $J=1.66\omega _b$. The rest of the parameters are the same as in Fig. 2 .

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From Figs. 6(b) and 6(d) we can find the optimal set of detunings corresponding to entanglement $E_N^{m_2b}$ and $E_N^{m_1b}$ as $\Delta _c^2=-\omega _b$, $\Delta _m^2=-0.7\omega _b$ and $\Delta _c^2=-\omega _b$, $\Delta _m^2=-\omega _b$, respectively. In order to obtain a larger entanglement, another important task is to find the optimal dissipation rate $\kappa _c^2$ for the auxiliary cavity $\delta c_2$. In Fig. 7, the two-mode entanglement $E_N^{m_1b}$ and $E_N^{m_2b}$ are plotted versus the decay ratio $\kappa _c^2/\kappa _c^1$ between the auxiliary cavity $\delta c_2$ and the magnomechanical cavity $\delta c_1$. The bipartite entanglement $E_N^{m_1b}$ increases as the dissipation ratio $\kappa _c^2/\kappa _c^1$ increases and can saturate at a maximum value, and then $E_N^{m_1b}$ decreases when $\kappa _c^2/\kappa _c^1$ increases further, where the optimal decay ratio corresponding to the maximum entanglement is $\kappa _c^2/\kappa _c^1=2$ (see the red curve in Fig. 7). In contrast, the steady-state entanglement $E_N^{m_2b}$ decreases monotonically as $\kappa _c^2/\kappa _c^1$ varies from 0 to 10 (see the green curve in Fig. 7). Thus, dissipation engineering would be a promising avenue for enhancing entanglement in cavity-magnetic systems, which has been extensively studied based on hybrid cavity quantum electrodynamic systems [87–90].

 

Fig. 7. The steady-state entanglement $E_N$ as a function of the dissipation ratio $\kappa _c^2/\kappa _c^1$. The red curve corresponds to parameters $\Delta _c^2=-\omega _b$, $\Delta _m^2=-\omega _b$, $J=1.66\omega _b$. The green curve corresponds to parameters $\Delta _c^2=-\omega _b$, $\Delta _m^2=-0.7\omega _b$, $J=0.32\omega _b$. The rest of the parameters are the same as in Fig. 2.

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By choosing an optimized parameter $\kappa _c^2/\kappa _c^1=0.01$ according to Fig. 7, the maximum value of the entanglement between the magnon $\delta m_2$ and the oscillator $\delta b$ is clearly amplified compared to the case of $\kappa _c^2/\kappa _c^1=1$, which can be seen in Fig. 8(a). To clearly demonstrate the magnitude of the steady-state entanglement enhanced via AMCA, we introduce here an entanglement amplification factor

 

Fig. 8. (a) The steady-state magnon-phonon entanglement $E_N^{m_2b}$ and (b) the entanglement amplification factor $\Upsilon _{m_2b}$ versus dimensionless detuning $\Delta _c^2/\omega _b$ for $J=0.32\omega _b$ and $\Delta _m^2=-0.7\omega _b$. The entanglement amplification factor $\Upsilon _{m_1b}$ versus dimensionless detunings $\Delta _c^2/\omega _b$ and $\Delta _m^2/\omega _b$ with $J=1.66\omega _b$ for (c) $\kappa _c^2/\kappa _c^1=10$ and (d) $\kappa _c^2/\kappa _c^1=2$. The rest of the parameters are the same as in Fig. 2.

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In Figs. 8(c) and 8(d), the entanglement amplification factor $\Upsilon _{m_1b}$ is plotted as a function of the dimensionless detunings $\Delta _c^2/\omega _b$ and $\Delta _m^2/\omega _b$ when the auxiliary cavity $\delta c_2$ has a high decay rate $\kappa _c^2/\kappa _c^1=10$ and a low decay rate $\kappa _c^2/\kappa _c^1=2$, respectively. Interestingly, the entanglement amplification factor $\Upsilon _{m_1b}>1$ occurs in the full maps in Figs. 8(c) and 8(d), which implies that the two-mode entanglement $E_N^{m_1b}$ can be enhanced via the AMCA mechanism over the entire parameter space. However, bipartite entanglement can be enhanced only in the blue detuned driving parameter regime through the auxiliary cavity optomechanical scheme [50]. This makes our work more feasible for the generation and detection of entanglement experimentally. Another important result is that the magnon-vibrator entanglement $E_N^{m_1b}$ can be enhanced by a factor of about 11 ($\Upsilon _{m_1b}\approx 11$) at detuning $\Delta _c^2\approx -\omega _b$ via the AMCA approach under the optimal dissipation relation $\kappa _c^2/\kappa _c^1=2$ [see Fig. 8(d)]. This is difficult to realize for other enhanced quantum entanglement schemes in magnomechanics [34,35,37].

In Fig. 9(a), we show the steady-state entanglement $E_N^{m_1b}$ and the dual-mode entanglement between the optical mode $\delta c_2$ of the auxiliary system and the four bosonic modes $\delta m_1$, $\delta m_2$, $\delta c_1$, and $\delta b$ of the main system, respectively, versus the tunneling coupling strength $J/\omega _b$, where we fix the decay ratio between the two cavities to be $\kappa _c^2/\kappa _c^1=1$. As expected, there is no entanglement (i.e., $E_N^{m_1c_2}=E_N^{m_2c_2}=E_N^{c_1c_2}=E_N^{bc_2}=0$) between the auxiliary system and the main system when there is no tunneling coupling $J$. This is due to the fact that the initial magnon-phonon entanglement $E_N^{m_1b}$ of the main system cannot be transferred to the auxiliary system when the state-swap interaction $J=0$ between the two microwave cavities $\delta c_1$ and $\delta c_2$. Then, as $J$ increases, $E_N^{m_1b}$ decreases, but the dual-mode entanglements $E_N^{m_1c_2}$, $E_N^{m_2c_2}$, $E_N^{c_1c_2}$, and $E_N^{bc_2}$ increase and gradually reach their maximum values, that is due to the initial entanglement $E_N^{m_1b}$ being distributed to other two-mode subsystems. Therefore, we can find the optimal couplings corresponding to the four maximum entanglements are $J=0.35\omega _b$, $J=0.79\omega _b$, $J=1.24\omega _b$, and $J=0.53\omega _b$, respectively. In a way, the AMCA mechanism also increases the diversity of entanglement. Then, as the coupling $J$ moves away from the optimum, the noninitial entanglement gradually annihilates while the initial entanglement recovers to its maximum value. Counter-intuitively, the direct coupling entanglement $E_N^{c_1c_2}$ appears later than the indirect coupling entanglement $E_N^{m_1c_2}$, $E_N^{m_2c_2}$, and $E_N^{bc_2}$ for $J$. This is due to the fact that the cavity-cavity direct coupling first transfers the initial entanglement directly to the other dual-mode subspaces, and the cavity-cavity subsystem do not get entangled until $J$ is large enough. This interesting entanglement transfer mechanism can be applied to the distribution of quantum information. In Fig. 9(b), we plot the dual-mode entanglements $E_N^{m_1c_2}$, $E_N^{m_2c_2}$, $E_N^{c_1c_2}$, and $E_N^{bc_2}$ as a function of the decay ratio $\kappa _c^2/\kappa _c^1$ under their respective optimal coupling parameters $J$. Remarkably, when the dissipation in the auxiliary cavity $\delta c_2$ is much smaller than that in the magnomechanical cavity $\delta c_1$ (i.e., $\kappa _c^2\ll \kappa _c^1$), the entanglements $E_N^{m_1c_2}$, $E_N^{m_2c_2}$, $E_N^{c_1c_2}$, and $E_N^{bc_2}$ are large. This can be understood from the point of view of the statistical distribution of photons, i.e., if the cavity $\delta c_2$ has a large decay $\kappa _c^2$ then there are very few photons distributed in $\delta c_2$, and thus it is difficult to create entanglement between the auxiliary system $\delta c_2$ and the main system. Moreover, the entanglement between the two modes of the indirect coupling is obvious, especially the magnon-photon and phonon-photon entanglements can reach $E_N^{m_1c_2}\simeq 0.26$ (see the red curve in Fig. 9(b)) and $E_N^{bc_2}\simeq 0.24$ (see the green curve in Fig. 9(b)), respectively, which are comparable to the case of the direct coupling ($E_N^{m_1b}\simeq 0.3$). This is due to the combined effect of two-mode squeezing and beam-splitter interactions effectively transferring the initial magnon-phonon entanglement to the indirectly coupled dual-mode subsystem.

 

Fig. 9. (a) Steady-state dual-mode entanglement $E_N^{m_1c_2}$ (red line) with $\Delta _c^2=-\omega _b$ and $\Delta _m^2=\omega _b$, $E_N^{m_2c_2}$ (magenta line) with $\Delta _c^2=0.7\omega _b$ and $\Delta _m^2=-1.2\omega _b$, $E_N^{c_1c_2}$ (blue line) with $\Delta _c^2=-1.5\omega _b$ and $\Delta _m^2=-2.4\omega _b$, $E_N^{bc_2}$ (green line) with $\Delta _c^2=-\omega _b$ and $\Delta _m^2=0.7\omega _b$, and $E_N^{m_1b}$ (black line) with $\Delta _c^2=-1.5\omega _b$ and $\Delta _m^2=-\omega _b$ as a function of the tunneling coupling $J/\omega _b$. (b) steady-state dual-mode entanglement $E_N^{m_1c_2}$ with $J=0.35\omega _b$, $E_N^{m_2c_2}$ with $J=0.79\omega _b$, $E_N^{c_1c_2}$ with $J=1.24\omega _b$, and $E_N^{bc_2}$ with $J=0.53\omega _b$ as a function of the decay ratio $\kappa _c^2/\kappa _c^1$. The rest of the parameters are the same as in Fig. 2.

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In Fig. 10, we plot the steady-state entanglement of the system as a function of the ambient temperature $T$. As can be summarized by comparing the blue and red lines in Figs. 10(a)–10(f), in addition to increasing the degree of entanglement, AMCA also significantly enhances the robustness of entanglement against temperature. Furthermore, the bipartite entanglement and its robustness against temperature can be further enhanced under an optimized value of the decay ratio $\kappa _c^2/\kappa _c^1$, as demonstrated in Figs. 10(a)–10(f). Here we consider a practical set of parameters $\omega _b/2\pi =10$ MHz, $\omega _{m}^{j}=\omega _{c}^{j}=2\pi \times 14.09$ GHz, $G=G_{j}=2\pi \times 3.2$ MHz, $\kappa _c^j=\kappa _m^j=2\pi \times 1$ MHz, and $\gamma _b/2\pi =100$ Hz, which are experimentally achievable [27,91,92]. These results suggest that the AMCA mechanism provides a feasible avenue for generating temperature-tolerant quantum entanglement, which is an important feature from the experimental point of view. Recently, some typical optomechanical experiments have demonstrated that both ground-state cooling of mechanical oscillators [93–95] and quantum entanglement [96,97] are obtainable. Moreover, strong [98–101] and superstrong [102,103] couplings between magnon modes and mechanical modes have also been reported experimentally. Therefore, analogous to cavity optomechanical systems, experiments on cavity magnomechanical cooling and entanglement are likely to be realized, which may also become a dominant trend for future experiments in the field of hybrid quantum optics.

 

Fig. 10. Steady-state dual-mode entanglement (a) $E_N^{m_2b}$ with $\Delta _c^2=-\omega _b$ and $\Delta _m^2=-0.7\omega _b$, (b) $E_N^{m_1b}$ with $\Delta _c^2=-\omega _b$ and $\Delta _m^2=-\omega _b$, (c) $E_N^{m_1c_2}$ with $\Delta _c^2=-\omega _b$ and $\Delta _m^2=\omega _b$, (d) $E_N^{m_2c_2}$ with $\Delta _c^2=0.7\omega _b$ and $\Delta _m^2=-1.2\omega _b$, (e) $E_N^{c_1c_2}$ with $\Delta _c^2=-1.5\omega _b$ and $\Delta _m^2=-2.4\omega _b$, and (f) $E_N^{bc_2}$ with $\Delta _c^2=-\omega _b$ and $\Delta _m^2=0.7\omega _b$ versus temperature. See text for the details of the other parameters.

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6. Conclusion

In summary, we have shown how to enhance the cooling and entanglement of the system by designing a hybrid AMCA magnomechanical model. We have found that the magnomechanical cooling of the system is significantly enhanced via AMCA under optimal parameter conditions, so much so that both the photon mode $\delta c_1$ and the magnon mode $\delta m_2$ are very close to their quantum ground states (i.e., $W_c^1=W_m^2\approx 0.01\ll 1$). Moreover, the presence of the AMCA also significantly enhances the steady-state entanglement, in particular the maximum entanglement amplification factor between the magnon $\delta m_1$ and the oscillator $\delta b$ can reach $\Upsilon _{m_1b}\approx 11$. Another important finding is that the AMCA entanglement has a much stronger robustness against temperature in comparison with the case without AMCA. Interestingly, the AMCA mechanism also extends the diversity of bipartite entanglement. Our work provides a potential scheme for preparing magnomechanical entanglement experimentally with reasonable parameters, and can be further used to generate temperature-resistant quantum resources.