1. Introduction

In recent years, ferrimagnetic materials including the yttrium iron garnet (YIG) sphere have drawn extensive focus and witnessed tremendous progress due to its high spin density and low damping rate [1,2], making it much easier to accomplish quantum information processing [3]. Magnons in the YIG represent the collective excitations of spin waves and couple to phonons from the vibration of deformation of the sphere caused by the magnetostrictive force [4], which is usually reckoned as the prerequisite for generating quantum states in cavity magnomechanics. In parallel with cavity optomechanics [5], cavity magnomechanics as a hybrid system consisting of photons, magnons, and phonons, provides an alternative platform to create quantum effects at a macroscopic scale. Specially, many recent works have already reported magnon-induced nonreciprocity [6], ground state cooling of magnomechanical resonator [7], quadrature squeezing [8,9], magnon blockade [10–12], and entanglement [4]. Besides, hybrid quantum systems involving magnons have committed to implementing high-order sideband generation [13], ultrasensitive magnetometer [14], magnon-assisted photon-phonon conversion [15], exceptional points [16], and storage and retrieval of quantum states [17]. These studies encourage us to further explore novel nonclassical effects based on the macroscopic quantum medium of photons, magnons, and phonons.

Entanglement, characterized by quantum mechanics, has become a significant resource for quantum information science and thus raised widespread interest in various physical branches and has been realized in many kinds of systems at both the microscopic level [18] and macroscopic level [19,20]. In optomechanics, the entanglement between the cavity and macroscopic vibrating mirror can be generated by means of radiation pressure [21–24]. Subsequently, the approach of engineering reservoir [25,26] was proposed to enhance entanglement level over the bound on the maximum stationary entanglement limited by a coherent two-mode squeezing interaction. Moreover, magnon-photon-phonon [4] and atom-cavity-mirror [27,28] tripartite entanglement have been observed. It is worth noting that in the cavity magnomechanical system, the enhanced magnon-magnon entanglement by improving the cavity-magnon coupling [29], introducing the Kerr nonlinearity [30], utilizing magnetostrictive interaction [31], and applying correlated quantum microwave fields [32], has been achieved.

As a kind of quantum correlation stronger than entanglement but weaker than Bell nonlocality [33,34], Einstein-Podolsky-Rosen (EPR) steering [35], originally proposed by Schödinger to answer the famous EPR paradox [36], features one’s capability to control the state of the remote body via local measurements on its body entangled with the remote one. Being of fundamental interest, quantum steering is a hot research, including steady-state light-mechanical quantum steerable correlations in cavity optomechanics [37], phase control of entanglement and quantum steering in a three-mode optomechanical system [38], more nonlocality with less entanglement in a tripartite atom-optomechanical system [27], and manipulation and enhancement of asymmetric steering via interference effects induced by closed-loop coupling [39]. Furthermore, stationary one-way steering via thermal effects [40] and time-modulated amplitude [41] in optomechanics have also been realized. Meanwhile, genuine photon-magnon-phonon EPR steerable nonlocality has been realized in a continuously-monitored cavity magnomechanical system [42]. In addition, the enhanced entanglement and steering in PT-symmetric cavity magnomechanics [43], the enhanced entanglement and asymmetric EPR steering between magnons [44], and EPR entanglement and asymmetric steering between distant macroscopic mechanical and magnonic systems [45] have also been investigated.

In this paper, we present a scheme to generate quantum entanglement and one-way steering in cavity magnomechanics via a weak squeezed vaccum field. Physically, the generation of entanglement and steering originates from the squeezed cavity field, produced by a flux-driven Josephson parametric amplifier (JPA), which leads to the driven cavity acting effectively as a bath whose force noise is squeezed. Under the resonance condition, the squeezing from the cavity mode is transferred to the magnon mode, which accounts for the photon-phonon entanglement by the delocalized Bogoliubov mode. When considering the red-detuned driving, we find that steady-state entanglement and steering can be achieved in the weak coupling regime. Moreover, the ratio of two coupling rates contributes to the maximum values of entanglement and steering. Besides, with the much weaker magnomechanical coupling, we show that photon-magnon entanglement and steering can be achieved, where only the smaller dissipation rate can steer the other one, i.e., one-way steering. Furthermore, we reveal that the entanglement and steering between any mode pairs are more robust to the temperature by appropriately selecting squeezing parameter. Finally, we present that the blue-detuned driving requires strong coupling rate condition in comparing with the red-detuned driving.

The paper is organized as follows: In Sec. 2, we introduce the general cavity magnomechanical system involving cavity, magnon, and phonon modes. Next, we describe the stationary three-mode Gaussian states of the system in the red-detuned regime by the standard Langevin formalism, and finally quantify the measure for quantum entanglement and steering. In Sec. 3, we numerically study the steady-state entanglement and one-way steering between distinct mode pairs, explore the physical mechanism for generating photon-phonon entanglement and steering by means of the mean square fluctuations, and verify the stronger robustness to the temperature. In Sec. 4, we clarify the advantages of our scheme by studying how the detunings affect the generation of entanglement and steering. Finally, a conclusion is given in Sec. 5.

 

Fig. 1. Sketch of the system. One macroscopic YIG is placed inside the cavity near the maximum magnetic field of the cavity mode, and simultaneously in a uniform bias magnetic field which mediates the coupling between the cavity mode and the magnon mode. The cavity is driven by a weak squeezed vacuum field produced by a flux-driven JPA. The bias magnetic field ($z$ direction), the driven magnetic field ($y$ direction), and the magnetic field ($x$ direction) of the cavity mode are mutually perpendicular at the site of the sphere.

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2. System and Hamiltonian

As schematically shown in Fig. 1, we consider a general cavity magnomechanical system including cavity, magnon, and phonon modes, where the YIG supporting magnons and phonons is adjusted in the suitable position to activate the magnon-phonon coupling, which can be considerably enhanced via applying a driven laser to the YIG. The Hamiltonian of the whole system is described by ($\hbar =1$)

In the following, we consider the weak coupling parameter regime of $\omega _{b}$ $\gg$ $G_{mb}$, $g$, $\kappa _{a}$, $\kappa _{m}$, $\gamma _{b}$, thus the above RWA can be performed. When the dissipation and noise terms are involved, the quantum Langevin equations (QLEs) describing the system dynamics are given by

Since the system dynamics of fluctuations is governed by a linearized Hamiltonian, and the noises remain Gaussian in the steady state, the system dynamics can be completely characterized by the $6 \times 6$ covariance matrix (CM) $\mathcal {V}$ whose components are $\mathcal {V}_{kl}=\langle U_{k}U_{l}+U_{l}U_{k}\rangle /2$ with $U_{k}$ being the $k$th element of the vector $U$ of quadratures. When the system is in the steady state, one can determine the matrix $\mathcal {V}$ via solving the Lyapunov equation:

For arbitrary targeted two-mode Gaussian state among the three modes, we consider the reduced CM $\tilde {\mathcal {V}}_{12}$ ($4 \times 4$) involving the two modes, in the form $\tilde {\mathcal {V}}_{12}=[\mathcal {V}_{1}, \mathcal {V}_{c}; \mathcal {V}_{c}^{\mathrm {T}}, \mathcal {V}_{2}]$, where $\mathcal {V}_{1}$, $\mathcal {V}_{2}$, and $\mathcal {V}_{c}$ are $2 \times 2$ subblock matrices respectively associated with modes 1, 2, and their correlation. To measure the entanglement, we adopt the logarithmic negativity $E_{N}=$ max [0, $-$ln2$\eta ^{-}$], where $\eta ^{-}=\sqrt {\Sigma -[\Sigma ^{2}-4 \mathrm {det} \tilde {\mathcal {V}}_{12}]^{1/2}}/\sqrt {2}$, with $\Sigma =$det $\mathcal {V}_{1}$+det $\mathcal {V}_{2}$-2det $\mathcal {V}_{c}$.

It is well-known that the quantum steering features its asymmetric information between the parties. Here, the steering from the mode $1$ to the mode $2$ ($1$-to-$2$ steering) is quantified by the measure

3. Entanglement and one-way steering

At the very beginning, we study entanglement and steering between modes $a$ and $b$. Inspired from Ref. [9], the magnon mode can be generated in the squeezed state due to the squeezed cavity field and photon-magnon beam-splitter interaction. Additionally, resulted from the magnon-phonon beam-splitter interaction of the effective Hamiltonian Eq. (3), considerable quantum entanglement and steering between modes $a$ and $b$ can be generated by activating the magnetostrictive force, instead of modes $a$ and $m$, and the detailed physical mechanism will be illustrated as follows.

In Fig. 2(a), we plot the entanglement $E_{ab}$ and steering ( $\mathcal {G}_{b\rightarrow a}$, $\mathcal {G}_{a\rightarrow b}$) between two modes $a$ and $b$ as a function of effective magnomechanical coupling $G_{mb}$. The experimentally realizable parameters are chosen as: $\kappa _{a}/2\pi =\kappa _{m}/2\pi =$1 MHz, $\gamma _{b}/2\pi =10^{2}$ Hz, $r=1.5$, and $g=0.5\kappa _{a}$. All the results are guaranteed in the steady state by all the negative eigenvalues of real parts of the drift matrix $M$. It is shown that there exist two different optimal $G_{mb}$ for the maximum $E_{ab}$ and $\mathcal {G}_{b\rightarrow a}$, respectively, and $\mathcal {G}_{a\rightarrow b}$ is zero in the whole parameter interval, which means that one-way steering can be achieved by the unequal dissipations. Moreover, it also shows that only with sufficient magnomechanical coupling strength, the entanglement $E_{ab}$ and steering $\mathcal {G}_{b\rightarrow a}$ can appear, which implies that large enough $G_{mb}$ is necessary for the squeezed state transfer from the magnon mode to the phonon mode. In Fig. 2(b), we plot the entanglement $E_{ab}$ and steering ( $\mathcal {G}_{b\rightarrow a}$, $\mathcal {G}_{a\rightarrow b}$) as a function of the ratio $g/G_{mb}$. We find that the cooperative effect of $g$ and $G_{mb}$ is essential to the maximum entanglement $E_{ab}$ and steering $\mathcal {G}_{b\rightarrow a}$. The coupling $G_{mb}/2\pi$ = 2 MHz corresponds to the drive magnetic field $B_{0}\simeq 1.75\times 10^{-5}$ T and hence the drive power $P\simeq 1.79$ mW [4,9] for $\eta /2\pi = 0.125$ Hz and an optimal coupling $g/2\pi = 0.5$ MHz. Further, these two maximums originate from the competition of the two opposing tendencies. Indeed, the enhancement of $g/G_{mb}$ increases the squeezing parameter and thus squeezing associated with the vacuum of the delocalized mode [25,26], while reduces the effective coupling strength between the magnon mode and the delocalized mode (specific definitions will be given below). As a consequence, the optimum entanglement and steering are a tradeoff between the two competing factors.

 

Fig. 2. Stationary entanglement $E_{ab}$, steering $\mathcal {G}_{b\rightarrow a}$, and steering $\mathcal {G}_{a\rightarrow b}$ between two modes $a$ and $b$ as a function of (a) $G_{mb}/\kappa _{a}$ for $\kappa _{a}/2\pi =$1 MHz and $g=0.5\kappa _{a}$, and (b) $g/G_{mb}$ for $\kappa _{a}/2\pi =$1 MHz and $G_{mb}=2\kappa _{a}$. The other parameters are: $\kappa _{m}/2\pi =$1 MHz, $\gamma _{b}/2\pi =10^{2}$ Hz, $r=1.5$, $\theta =0$, and $n_{th}=0$.

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To give the physical mechanism of generating entanglement $E_{ab}$ and steering $\mathcal {G}_{b\rightarrow a}$, we plot the mean square fluctuations (variances) of cavity mode $\langle \delta Y_{a}^{2}\rangle$, magnon mode $\langle \delta X_{m}^{2}\rangle$, and phonon mode $\langle \delta Y_{b}^{2}\rangle$ as a function of cavity-magnon coupling $g$ in Fig. 3. It is found that the squeezed state from cavity mode significantly transfers to magnon mode and further to phonon mode as cavity-magnon coupling $g$ increases. To see the insight, we recover the two-mode Hamiltonian $H_{\mathrm {lin}}^{'}=\omega _{b}(\delta m^{\dagger}\delta m+\beta _{1}^{\dagger}\beta _{1}+\beta _{2}^{\dagger}\beta _{2})+J(\beta _{1}\delta m^{\dagger}+\beta _{1}^{\dagger}\delta m^{\dagger}+\mathrm {H.c.})$ from Hamiltonian (2) by introducing the delocalized Bogoliubov mode $\beta _{1}=\delta b\ \mathrm {cosh}\ \lambda +\delta a^{\dagger}\ \mathrm {sinh}\,\lambda$ and $\beta _{2}=\delta a\ \mathrm {cosh}\,\lambda +\delta b^{\dagger}\ \mathrm {sinh}\,\lambda$, with $\lambda =\mathrm {arctanh}(g/G_{mb})$ being the squeezing parameter, and $J=\sqrt {G_{mb}^{2}-g^{2}}$ being the effective coupling strength between the magnon mode $\delta m$ and the delocalized mode $\beta _{1}$. At the same time, we assume that $g, G_{mb}$ $\ll$ $2\omega _{b}$ and $g<G_{mb}$, which allows one to apply the RWA and rewrite the linearized Hamiltonian (2) as $H_{\mathrm {lin}}^{''}=J(\beta _{1}\delta m^{\dagger}+\beta _{1}^{\dagger}\delta m)$. The beam-splitter Hamiltonian $H_{\mathrm {lin}}^{''}$ describes the state transfer process between the magnon mode $\delta m$ and the hybridized mode $\beta _{1}$, which makes one to effectively realize the squeezed state of the hybridized mode $\beta _{1}$ if the magnon mode $\delta m$ is initially squeezed. As clearly visible from Fig. 3, the magnon mode is subsequently squeezed because of the squeezed cavity field and beam-splitter cavity-magnon coupling, thus one can achieve steady-state entanglement between the photon and phonon modes by the reservoir engineering [25,26].

 

Fig. 3. The mean square fluctuations of cavity mode $\langle \delta Y_{a}^{2}\rangle$, magnon mode $\langle \delta X_{m}^{2}\rangle$, and phonon mode $\langle \delta Y_{b}^{2}\rangle$ as a function of cavity-magnon coupling $g$. The parameters are the same as given in Fig. 2.

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We further investigate the entanglement $E_{ab}$ and steering $\mathcal {G}_{b\rightarrow a}$ as functions of squeezing parameter $r$ and cavity-magnon coupling $g$ in Figs. 4(a) and 4(b), and as functions of squeezing parameter $r$ and environment temperature $T$ in Figs. 4(c) and 4(d). From Figs. 4(a) and 4(b), it can be seen that the maximum values $E_{ab}$ and $\mathcal {G}_{b\rightarrow a}$ cannot be obtained at the largest squeezing parameter $r$, while the larger $r$ conversely reduces the ranges of $g$ for $E_{ab}$ and $\mathcal {G}_{b\rightarrow a}$. Therefore, one can employ appropriate squeezing parameter to ensure the desired cavity-magnon coupling ranges of $g$ for $E_{ab}$ and $\mathcal {G}_{b\rightarrow a}$. Besides, the entanglement $E_{ab}$ and steering $\mathcal {G}_{b\rightarrow a}$ are quite robust to the environment temperature, as shown in Figs. 4(c) and 4(d). The moderate entanglement can be achieved even at $T=$ 0.5 K for a certain range of $r$. For steering $\mathcal {G}_{b\rightarrow a}$, the largest $r$ does not imply the optimal steering value, and the robustness to the temperature is improved when decreasing $r$. On the other hand, entanglement and steering start to decrease after reaching their maximum. Here the explanation is that, in this period, the photon number becomes important in the cavity field, and consequently, the thermal noise entering the cavity becomes more aggressive, bring the quantum correlations degradation.

 

Fig. 4. (a) Stationary entanglement $E_{ab}$ and (b) steering $\mathcal {G}_{b\rightarrow a}$ between two modes $a$ and $b$ as functions of squeezing parameter $r$ and cavity-magnon coupling $g$. (c) $E_{ab}$ and (d) $\mathcal {G}_{b\rightarrow a}$ as functions of squeezing parameter $r$ and temperature $T$. We take $T=0$ in (a) and (b), $g=0.5\kappa _{a}$ in (c) and (d), $G_{mb}=2\kappa _{a}$ and $\theta =0$ for all the plots. The other parameters are the same as given in Fig. 2.

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Note that the effective magnomechanical coupling $G_{mb}=2\kappa _{a}$ in the above is only suitable for generating quantum entanglement and steering between the modes $a$ and $b$, whereas no entanglement (entangled states are steerable but not necessarily vice versa) occurs for $E_{am}$ between the modes $a$ and $m$. Indeed, $G_{mb}=2\kappa _a$ is too large to create $E_{am}$, which causes the descending of reversible state exchange between the cavity photons and magnons, and further leads to reduction of cooling rate of magnon mode [48]. Therefore, next we diminish the effective magnomechanical coupling to investigate the entanglement and steering between the modes $a$ and $m$.

In the following, we plot the entanglement $E_{am}$ and steering ( $\mathcal {G}_{m\rightarrow a}$, $\mathcal {G}_{a\rightarrow m}$) between two modes $a$ and $m$ as a function of the ratio $\kappa _{a}/\kappa _{m}$ of the dissipation rate of the cavity to magnon mode in Fig. 5(a). We set $G_{mb}$ as $0.02\kappa _{a}$ much smaller in contrast to $G_{mb}=2\kappa _{a}$. The value $\kappa _{a}/\kappa _{m}$ determines asymmetric steering resulted from the asymmetry of the system. From Fig. 5(a) it can be found that the entanglement $E_{am}$ at first increases and then slowly decreases with the increase of $\kappa _{a}/\kappa _{m}$, which implies that once beyond the optimal value of $\kappa _{a}/\kappa _{m}$, the continuous enhancement of $\kappa _{a}/\kappa _{m}$ has a destructive effect on the entanglement $E_{am}$. For the one-way steering $\mathcal {G}_{m\rightarrow a}$ and $\mathcal {G}_{a\rightarrow m}$ appearing at different ranges, we give the condition for achieving the magnon-to-photon steering is $\kappa _{a}>1.088\kappa _{m}$, and the condition for the reverse steering is $\kappa _{a}<0.938\kappa _{m}$. Thus we conclude that only the mode with the larger dissipation rate can be steered by the other one. This is because that the mode with larger dissipation rate possesses a smaller average number of population and thus smaller quantum fluctuations of the quadrature operators [e.g. $\langle X_{a}^{2}\rangle =\langle a^{\dagger }a \rangle +1/2$, $\langle a^{\dagger }a \rangle =\mathcal {V}_{11}/2+\mathcal {V}_{22}/2-1/2$]. Here $\mathcal {V}_{11}$ and $\mathcal {V}_{22}$ are the first and second diagonal elements of the covariance matrix. Compared to the mode with smaller dissipation, the mode possessing smaller quantum fluctuations is more easily steered by the other mode with lager fluctuations. In Fig. 5(b), the entanglement $E_{am}$ and steering ($\mathcal {G}_{m\rightarrow a}$, $\mathcal {G}_{a\rightarrow m}$) are plotted as a function of the ratio $g/\kappa _{a}$. It can be observed that there exist the optimal ratio that determine the maximum entanglement and steering. In addition, too large cavity-magnon coupling leads to the disappearance of the one-way steering.

 

Fig. 5. Stationary entanglement $E_{am}$, steering $\mathcal {G}_{m\rightarrow a}$, and steering $\mathcal {G}_{a\rightarrow m}$ between two modes $a$ and $m$ as a function of (a) $\kappa _{a}/\kappa _{m}$ for $\kappa _{m}/2\pi =1$ MHz and $g=0.06\kappa _{a}$, and (b) $g/\kappa _{a}$ for $\kappa _{a}/2\pi =5\kappa _{m}/2\pi =5$ MHz. The other parameters are: $\gamma _{b}/2\pi =10^{2}$ Hz, $r=1.5$, $\theta =0$, $n_{th}=0$, and $G_{mb}=0.02\kappa _{a}$.

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Finally, we investigate the entanglement $E_{am}$ and steering $\mathcal {G}_{m\rightarrow a}$ as functions of squeezing parameter $r$ and cavity-magnon coupling $g$ in Figs. 6(a) and 6(b), and as functions of squeezing parameter $r$ and environment temperature $T$ in Figs. 6(c) and 6(d). From the trend of the contour lines in Figs. 6(a) and 6(b), we find that $E_{am}$ and $\mathcal {G}_{m\rightarrow a}$ can be achieved over a wider parameter range of $g$ with the smaller $r$, while flexibly enhancing $r$ can evidently improve the values of $E_{am}$ and $\mathcal {G}_{m\rightarrow a}$. From Fig. 6(c) it can be found that the larger $r$ is, the stronger robustness to temperature $E_{am}$ is against, where the moderate entanglement can be generated even at $T=3$ K. Different from the entanglement, the steering first gets a peak value beyond which it drops. This is because that the continuously increased $r$ induces more thermal noise injecting to the cavity, which has a negative influence on the quantum correlation. Interestingly, the steering still exists nearly at $T=3$ K by appropriately selecting $r$, as shown in Fig. 6(d).

 

Fig. 6. (a) Stationary entanglement $E_{am}$ and (b) steering $\mathcal {G}_{m\rightarrow a}$ between two modes $a$ and $m$ as functions of squeezing parameter $r$ and cavity-magnon coupling $g$. (c) $E_{am}$ and (d) $\mathcal {G}_{m\rightarrow a}$ as functions of squeezing parameter $r$ and temperature $T$. We take $T=0$ in (a) and (b), $g=0.06\kappa _{a}$ in (c) and (d), $G_{mb}=0.02\kappa _{a}$ and $\theta =0$ for all the plots. The other parameters are the same as given in Fig. 5.

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4. Discussions

In Section 3, we study the generation of entanglement and steering between distinct mode pairs in the weak coupling regime, i.e. $G_{mb}$, $g$ $\ll \omega _{b}$, in which the numerical results are obtained according to Hamiltonian (3) derived from the red-detuned driving and RWA. Then how the detunings in Hamiltonian (2) affect the generation of entanglement and steering deserves to be studied, in which case the detunings are unselected and the RWA is not made. If calculating the system QLEs based on Hamiltonian (2), the drift matrix $M$ is replaced by the following matrix:

In the same way, we plot the entanglement $E_{ab}$, steering $\mathcal {G}_{b\rightarrow a}$, entanglement $E_{am}$, and steering $\mathcal {G}_{m\rightarrow a}$ as functions of $\Delta _{a}$ and $\Delta _{m}$ in the steady state in Figs. 7(a)–7(d), respectively. Note that only when strong magnomechanical coupling and cavity-magnon coupling are applied, relatively larger quantum entanglement and one-way steering can be generated. Moreover, it is necessary to apply a vacuum field driving the cavity field for the generation of entanglement and steering. Otherwise, no entanglement and steering can be created (not shown here). In comparison with the selected red-detuned driving, the strong coupling regime should be satisfied to achieve larger entanglement and steering occurring around the blue-detuned driving. At the beginning, therefore, we apply the squeezed vacuum field to drive the cavity field, and choose red-detuned driving and equal detunings, such that the modest entanglement and steering can be generated in the weak coupling regime.

 

Fig. 7. (a) Stationary entanglement $E_{ab}$, (b) steering $\mathcal {G}_{b\rightarrow a}$, (c) entanglement $E_{am}$, and (d) steering $\mathcal {G}_{m\rightarrow a}$ as functions of $\Delta _{a}$ and $\Delta _{m}$. We choose $\omega _{b}=10\kappa _{a}$, $r=0$, and $G_{mb}=g=4\kappa _{a}$. $\kappa _{a}/2\pi =\kappa _{m}/2\pi =1$ MHz in (a) and (b), and $\kappa _{a}/2\pi =5\kappa _{m}/2\pi =5$ MHz in (c) and (d). The white area represents the unstable region of the system. The other parameters are the same as given in Fig. 2.

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5. Conclusions

In conclusion, we have investigated the generation of quantum entanglement and one-way steering between distinct mode pairs in a generic cavity magnomechanical system via a weak squeezed vacuum field. We first investigate the steady-state entanglement and steering between the cavity and phonon modes. We find that when the system is driven by the red-detuned laser, the optimal entanglement and one-way steering occur, depending on the ratio of two coupling rates with the RWA in the weak coupling regime. By introducing the delocalized Bogoliubov modes, we explain detailedly the physical origin of generating the entanglement and steering between cavity and phonon modes. Moreover, it is shown that quantum entanglement and one-way steering occur between the cavity and magnon modes in a much weaker magnomechanical coupling regime, where the relative dissipation strength of the cavity to the magnon mode fully determines the steering direction. In comparison with the red-detuned driving, much larger coupling strengths are required to achieve considerable entanglement and one-way steering with the blue-detuned driving. Furthermore, it is revealed that the entanglement and steering of photon-phonon and photon-magnon can still be achievable even at higher temperatures, benefiting from the choice of appropriate squeezing parameters. Our work may pave an alternative way to generate entanglements and steerings at the macroscopic scale based on cavity magnomechanics, and will have potential applications in quantum information processing and communication.