Parametric controllable one-way quantum steering induced by four-wave mixing in cavity magnonics

Abstract

Quantum steering plays a crucial role in quantum communication and one-way quantum computation. Here, we study quantum steering between two magnon modes in a cavity-magnonics system by applying a two-photon drive field to the microwave cavity. The two magnon modes are entangled and the one-way steering can be implemented when the four-wave mixing is triggered. Different from most schemes that use the dissipation of the system to control quantum steering, in our scheme the one-way steering can be modulated on demand by adjusting the coherent coupling ratio between the two magnons and the cavity, which provides a flexible and feasible way in experiments. We also reveal that the directionality of quantum steering is related to the mode populations, i.e., the mode with a larger population dominants the steering. Our study has high manipulability and may provide a promising platform for one-way quantum computing and communications based on macroscopic entanglement state.

1 Introduction

The light-spin interaction has been studied extensively in recent years which has promoted the rapid development of cavity-magnonics. Magnon, the quanta of collective excitation of spin ensembles in magnetic materials, has many unique characteristics and has attracted much attentions in the fields of quantum information processing (QIP). For instance, the higher spin density enabled strong or even ultra-strong coupling between the magnons and photons, namely, a cavity-magnon polaron can be easily formed [1–10], the lower dissipation rate at room temperature makes the cavity-magnon polaron possess good coherence, and the high tunability allows us to observe more phenomena over a wide range of parameters. Due to these advantages the cavity-magnonics has been observed with many interesting applications and phenomena, such as quantum memory [11], quantum transducer [12–17], exceptional point [18, 19] and bistability [20]. The coherent coupling between the photons and magnons plays a crucial role in the construction of QIP platform.

As the key resource of quantum technologies and the most fundamental tests of quantum mechanics [21–23], quantum entanglement plays a crucial role in carrying out the QIP tasks, such as quantum teleportation [24–26], quantum metrology [27], and quantum computing [28–31]. As a strict subset of quantum entanglement, quantum steering was proposed by Schrödinger in respond to the famous Einstein-Podolsky-Rosen (EPR) paradox in 1935 [32, 33]. For such “spooky action-at-a-distance” phenomenon, the correct explanation is that two observers share pairs of entangled particles, but have completely different abilities to adjust (steer) that particle held by the other one only based on unilateral local measurements. The method of locally measuring one’s own particles to deduce the state of another side is called quantum steering or EPR steering. It is a type of nonclassical correlation that exhibits a strength greater than quantum entanglement [34], yet weaker than Bell nonlocality [35]. Furthermore, the steering has strict directionality and asymmetry, which is particularly evident in one-way steering, i.e., only one side can steer to another side, but not vice versa. The one-way steering effectively breaks the symmetry of controllability, and can be used as a efficient method in quantum manipulation and quantum communications, e.g., one-sided device-independent quantum cryptography [36], quantum secret sharing [37–39], subchannel discrimination [40] and quantum teleportation [41–43].

So far, quantum entanglement and one-way steering have been studied in cavity magnonics by exploiting various methods. The bipartite entanglement and a genuine tripartite entanglement are produced in a magnon-photon-phonon hybrid system by activating the nonlinear magnetostrictive interaction in ferrimagnet, which is accompanied by a cooling process of the phonon mode while the magnon mode is resonant with the anti-Stokes sideband [44–47]. By activating the Kerr nonlinearity of the magnetic material, cavity-magnon entanglement can be generated and then partially transferred to the two magnon modes [48]. Meanwhile, the entanglement caused by the mode overlap between Kittel mode [49] and higher-order magnetostatic mode has also been proposed theoretically [50] by simultaneously activating the self- and cross-Kerr nonlinear effects as well as the four-wave mixing used. Furthermore, the enhanced asymmetric EPR steering and entanglement between two types of magnons on two sublattices have been proposed [51, 52] by introducing a cavity to cool down the magnon modes. More recently, Yang and Guan et al. propose the one-way steering and entanglement in cavity-magnon system and demonstrate that the directivity of EPR steering can be changed by adjusting the ratio of coherent information exchange frequencies [53], and by utilizing the cooperative effect of coherent coupling and dissipative coupling [54], respectively. Inspired by these important advances, here we construct a hybrid cavity-magnonic system to generate bipartite entanglement and one-way steering between two magnons via a process of four-wave mixing. In this system both magnons are coupled to the microwave photons via magnetic dipole interaction, and the microwave cavity is driven by a two-photon drive field (the so-called squeezed drive field). The quantum correlation arises from a four-wave mixing process [55], wherein the two driving photons are absorbed, while the magnon and cavity modes are excited simultaneously. When the four-wave mixing is triggered by manipulating the detuning of two magnon modes to satisfy the frequency matching condition, the bipartite entanglement of subsystem and one-way steering can be realized. Under certain conditions the system parameters are clearly divided into three distinct regions in which the quantum steering between two magnon modes shows completely different unidirectivity, respectively. In particular, the direction of one-way steering can be regulated not only by the dissipation ratio of the magnons, but also by the coupling strength ratio of two magnon modes to the cavity. This is different from most current contributions in which quantum steering are controlled only by dissipation rates of the magnon modes [56], which cannot be easily adjusted in experimental operation due to the surface roughness, impurities and defects of materials. Hence using the coupling strength to regulate the unidirectivity of quantum steering is more feasible in experimental operations. By analyzing the populations of magnon modes we find that the steering direction is closely dependent on the populations of magnon modes, i.e., the mode with larger population dominates the steering.

The paper is organized as follows. In Sect. 2, the cavity-magnonics system is constructed and the system dynamics are described by quantum Langevin equations. In Sect. 3, the entanglements between different subsystems are generated based on four-wave mixing, and the characteristics of entanglements are discussed. Also, the physical mechanism of entanglement is analysed by introducing the Bogoliubov modes. In Sect. 4, the implementation of quantum one-way steering is proposed as a controllable manner by regulating the coupling strength ratio between two magnons and the cavity or designing the dissipation ratio of two magnon modes. The connection between one-way steering and the mode populations is revealed. Meanwhile, we demonstrate that the bipartite entanglement and steering are robust to ambient temperature. The feasibility of this system is discussed in Sect. 5, and the conclusions are given in Sect. 6.

2 Model and dynamics

As shown in Fig. 1, we consider a cavity-magnonics system formed by a microwave cavity containing two ferromagnetic yttrium iron garnet (YIG) spheres, where a two-photon drive field is applied to drive the microwave cavity, which results in the squeezing of the cavity field. The two magnon modes are embodied by the collective excitations of a large number of spins in the macroscopic YIG spheres. Furthermore, the strong coupling between the magnon modes and the microwave photons is mediated by the magnetic dipole interaction. We assume that the size of two YIG spheres is smaller enough than the microwave wavelength so that the radiation pressure caused by microwave photons can be well ignored [44, 45, 57, 58]. Under the rotating-wave approximation, the Hamiltonian of the system can be written as ($\hbar =1$)

$$\begin{aligned} H=\omega _{a}a^{\dagger}a+\sum^{2}_{j=1} \bigl[\omega _{j}m_{j}^{ \dagger}m_{j}+g_{j} \bigl(am_{j}^{\dagger}+a^{\dagger}m_{j} \bigr) \bigr]+{\Omega _{0}} \bigl({{a^{\dagger}}^{2}}e^{-2i{\omega _{l}}t}+{a^{2}}e^{2i{ \omega _{l}}t} \bigr), \end{aligned}$$

(1)

where O ($O^{\dagger}$) ($O= a,m_{j}$) denotes the annihilation (creation) operators of the cavity mode and jth magnon mode with the corresponding resonance frequency $\omega _{a}$ and $\omega _{j}$, with $[O, O^{\dagger}]=1$. The frequency of magnon mode is determined by the adjustable external bias magnetic field $H_{j}$, i.e., $\omega _{j}=\gamma H_{j}$ with the gyromagnetic ratio $\gamma /2\pi = 28\text{ GHz/T}$. The parameter $g_{j}$ represents the linear coupling rate between the cavity mode and the jth magnon mode which can be adjusted by varying the direction of the bias field or the position of the YIG spheres inside the microwave cavity [59]. The Rabi frequency $\Omega _{0}$ represents the coupling strength between the two-photon drive field with frequency $2\omega _{l}$ and the microwave cavity mode. In the frame rotating at frequency $\omega _{l}$, the quantum Langevin equations (QLEs) are given by

$$\begin{aligned}& \dot{a} = -(i\Delta _{a}+\kappa _{a})a-i\sum ^{2}_{j=1}g_{j}m_{j}-2i \Omega _{0}a^{\dagger}+\sqrt{2\kappa _{a}}a^{\mathrm{in}}, \\& \dot{m}_{j} = -(i\Delta _{j}+\kappa _{j})m_{j}-ig_{j}a+ \sqrt{2 \kappa _{j}}m_{j}^{\mathrm{in}}, \end{aligned}$$

(2)

where $\Delta _{a} = \omega _{a}-\omega _{l}$, $\Delta _{j} = \omega _{j}-\omega _{l}$ ($j=1,2$), $\kappa _{a}$ and $\kappa _{j}$ denote the dissipation rate of the cavity and magnon mode, respectively, and $O^{\mathrm{in}}(t)$ ($O = a, m_{j}$) denotes the input noise operator with zero mean of the corresponding mode characterized by the following correlation functions

$$\begin{aligned}& \bigl\langle {O^{\mathrm{in}}(t)O^{\mathrm{in}\dagger}\bigl(t'\bigr)}\bigr\rangle = (n_{O} + 1) \delta \bigl(t-t'\bigr), \\& \bigl\langle {O^{\mathrm{in}\dagger}(t)O^{\mathrm{in}}\bigl(t'\bigr)}\bigr\rangle = n_{O} \delta \bigl(t-t'\bigr), \end{aligned}$$

(3)

for the equilibrium mean thermal number of the corresponding mode $n_{O} = [\exp (\hbar \omega _{O}/ K_{B}T)-1]^{-1}$ with the ambient temperature T and Boltzmann constant $K_{B}$. Since the microwave cavity is strongly driven by a two-photon drive field, the photon amplitude $\vert \langle{a}\rangle \vert \gg 1$. Meanwhile, the magnon modes also have large amplitudes due to the beamsplitter interactions between the cavity photons and the magnon modes, i.e., $\vert \langle{m_{j}}\rangle \vert \gg 1$. By linearizing the dynamics of the system around the steady-state values, each operator can be written as $O=\langle{O}\rangle +\delta O$. Thus the linearized QLEs of the quantum fluctuations are obtained as

$$\begin{aligned}& \delta \dot{a} = -(i\Delta _{a}+\kappa _{a}) \delta a-i \sum^{2}_{j=1}g_{j} \delta m_{j}-2i\Omega _{0} \delta a^{\dagger}+\sqrt{2\kappa _{a}} \delta a^{\mathrm{in}}, \\& \delta \dot{m}_{j} = -(i\Delta _{j}+\kappa _{j}) \delta m_{j}-ig_{j} \delta a+\sqrt{2\kappa _{j}} \delta m_{j}^{\mathrm{in}}. \end{aligned}$$

(4)

By defining the fluctuations of quadratures $\delta X_{a}= (\delta a+\delta a^{\dagger})/\sqrt{2}$, $\delta Y_{a}= i(\delta a ^{\dagger}-\delta a)/\sqrt{2}$, $\delta X_{j}= (\delta m_{j}+\delta m_{j}^{\dagger})/\sqrt{2}$, $\delta Y_{j}= i(\delta m_{j} ^{\dagger}-\delta m_{j})/\sqrt{2}$, the linearized QLEs can be rewritten in a compact matrix form

$$\begin{aligned} \dot{u} (t) = Au(t)+n(t), \end{aligned}$$

(5)

with $u(t)=[\delta X_{a}(t), \delta Y_{a}(t), \delta X_{1}(t), \delta Y_{1}(t), \delta X_{2}(t), \delta Y_{2}(t)]^{\mathrm{T}}$, and $n(t)= [\sqrt{2\kappa _{a}}X_{a}^{\mathrm{in}}(t), \sqrt{2\kappa _{a}}Y_{a}^{\mathrm{in}}(t), \sqrt{2\kappa _{1}}X_{1}^{ \mathrm{in}}(t), \sqrt{2\kappa _{1}}Y_{1}^{\mathrm{in}}(t), \sqrt{2\kappa _{2}}X_{2}^{ \mathrm{in}}(t), \sqrt{2\kappa _{2}}Y_{2}^{\mathrm{in}}(t) ]^{\mathrm{T}}$ is the vector of input noises and the superscript ‘T’ represents the transpose. The drift matrix A is written as

$$\begin{aligned} A= \begin{pmatrix} -\kappa _{a} & \Delta _{a}-2\Omega _{0} & 0 & g_{1} & 0 & g_{2} \\ -\Delta _{a}-2\Omega _{0} & -\kappa _{a} & -g_{1} & 0 & -g_{2} & 0 \\ 0 & g_{1} & -\kappa _{1} & \Delta _{1} & 0 & 0 \\ -g_{1} & 0 & -\Delta _{1} & -\kappa _{1} & 0 & 0 \\ 0 & g_{2} & 0 & 0 & -\kappa _{2} & \Delta _{2} \\ -g_{2} & 0 & 0 & 0 & -\Delta _{2} & -\kappa _{2} \end{pmatrix} . \end{aligned}$$

(6)

Owing to the linearized dynamics and the Gaussian nature of quantum noises, quantum fluctuation of a steady-state system is a continuous variable three-mode Gaussian state, which can be completely characterized by a 6 × 6 covariance matrix (CM) $\mathcal{V}$ defined as $2\mathcal{V}_{ij}(t,t^{\prime})=\langle u_{i}(t)u_{j}(t^{\prime})+u_{j}(t^{ \prime})u_{i}(t)\rangle$. The stationary CM $\mathcal{V}$ can be obtained by solving the Lyapunov equation [60]

$$\begin{aligned} A\mathcal{V}+\mathcal{V} A^{\mathrm{T}}=-D, \end{aligned}$$

(7)

where $D = {\mathrm{diag}}[\kappa _{a}(2n_{a}+1), \kappa _{a}(2n_{a}+1), \kappa _{1}(2n_{m_{1}}+1), \kappa _{1}(2n_{m_{1}}+1), \kappa _{2}(2n_{m_{2}}+1), \kappa _{2}(2n_{m_{2}}+1)]$ is the diffusion matrix defined by $2D_{ij}\delta (t-t^{\prime})=\langle n_{i}(t)n_{j}(t^{\prime})+n_{j}(t^{ \prime})n_{i}(t)\rangle$.

Figure 1

(a) Schematic diagram of cavity magnonics system. Two YIG spheres are placed in a microwave cavity with frequency $\omega _{a}$ and linewidth $\kappa _{a}$ which is driven by a two-photon driving field with frequency $2\omega _{l}$, the magnon mode $m_{1(2)}$ with frequency $\omega _{1(2)}$ is modulated by a bias magnetic field $H_{1(2)}$ along the z direction, and the magnetic field of the cavity is along the y direction. (b) Spectrum diagram of four-wave mixing. The four-wave mixing occurs when the frequencies of cavity and the magnon modes satisfy the matching condition $\omega _{a} + \omega _{1(2)} = 2\omega _{l}$, i.e., $\Delta _{a} = -\Delta _{1(2)}$, where the energy of driving photon pair is absorbed, and the cavity mode and the magnon mode are simultaneously excited

Next we study the entanglement and quantum steering of the two magnon modes by using the CM $\mathcal{V}$.

3 Bipartite entanglement generated via four-wave mixing

We adopt the logarithmic negativity $E_{N}$ to quantify the quantum entanglement [61–64], which can be computed from the reduced 4 × 4 CM $\tilde{\mathcal{V}}$ for any two modes

$$\begin{aligned} \tilde{\mathcal{V}}= \begin{pmatrix} \mathcal{V}_{1} & \mathcal{V}_{3} \\ \mathcal{V}_{3}^{\mathrm{T}} & \mathcal{V}_{2} \end{pmatrix} , \end{aligned}$$

(8)

where $\mathcal{V}_{1}$, $\mathcal{V}_{2}$ and $\mathcal{V}_{3}$ are 2 × 2 sub-block of matrices of $\tilde{\mathcal{V}}$. The logarithmic negativity for entanglement of two modes is defined as

$$\begin{aligned} E_{N} \equiv \max [0, -\ln 2\nu ], \end{aligned}$$

(9)

where $\nu =\sqrt{\mathcal{E}-(\mathcal{E}^{2}-4\mathcal{R})^{1/2}}/\sqrt{2}$ with $\mathcal{E} = \mathcal{R}_{1}+\mathcal{R}_{2}-2\mathcal{R}_{3}$, $\mathcal{R}_{1}=\det \mathcal{V}_{1}$, $\mathcal{R}_{2} = \det \mathcal{V}_{2}$, $\mathcal{R}_{3} =\det \mathcal{V}_{3}$ and $\mathcal{R}=\det \tilde{\mathcal{V}}$ being symplectic invariants.

The entanglement characteristics of the system in steady state is guaranteed by the eigenvalues (negative real parts) of the drift matrix A (i.e., $\arrowvert A-\lambda \mathbf{I} \arrowvert =0$). On this premise, the following feasible experimental parameters in Ref. [50] can be used: $\Delta _{a}/2\pi = 200\text{ MHz}$, $g_{1}/2\pi = g_{2}/2\pi = 42\text{ MHz}$, $\kappa _{a}/2\pi = 6\text{ MHz}$, $\kappa _{1}/2\pi = \kappa _{2}/2\pi = 12\text{ MHz}$ and $\Omega _{0}/2\pi = 30\text{ MHz}$. The cavity-magnon entanglements $E_{N}^{am_{1}}$ and $E_{N}^{am_{2}}$ are shown in Figs. 2(a) and 2(b), where only one magnon mode $m_{1}$ or $m_{2}$ is considered, i.e., $g_{1}\neq 0$, $g_{2}=0$ in the former and $g_{1}=0$, $g_{2}\neq 0$ in the latter. We can see that the maximum entanglement occurs at $\Delta _{1} = -\Delta _{a}$ for $E_{N}^{am_{1}}$ and at $\Delta _{2} = - \Delta _{a}$ for $E_{N}^{am_{2}}$, exactly corresponding to the condition of four-wave mixing $\omega _{a} + \omega _{1(2)} = 2\omega _{l}$. Due to the two-photon drive field, the energy of one photon of the driving photon pair is absorbed to excite the magnon mode with detuning $\Delta _{1}/2\pi = -200\text{ MHz}$ (or $\Delta _{2}/2\pi = -200\text{ MHz}$), and the remaining energy is simultaneously scattered into the cavity resonance with detuning $\Delta _{a}/2\pi = 200\text{ MHz}$. The above frequency matching condition is the key for generating entanglements. When the two magnon modes are simultaneously coupled to the cavity mode, i.e., $g_{1}\neq 0$ and $g_{2}\neq 0$, the bipartite entanglement of the hybrid cavity magnonic system presents a different phenomenon, as shown in the Figs. 2(d) and 2(e). A breaking point at $\Delta _{1} =-\Delta _{a}=-\Delta _{2}$ appears for entanglement $E_{N}^{am_{1}}$ in Fig. 2(d), and $\Delta _{1}=\Delta _{a}=-\Delta _{2}$ appears for $E_{N}^{am_{2}}$ in Fig. 2(e). Meanwhile, the magnon-magnon entanglement degree $E_{N}^{m_{1}m_{2}}$ increases significantly in these two breaking points as seen in the Fig. 2(c). This type of phenomenon is also observed in the magnetostriction-induced stationary entanglement between two microwave fields [46], and can be enhanced by using an optical parametric amplifier [65]. By comparing the Fig. 2(c) with 2(d) and 2(e), we can clearly see that the cavity-magnon entanglement is transferred to that between the two magnon modes via the beam splitter interactions. In the Fig. 2(f), magnon-magnon entanglement $E_{N}^{m_{1}m_{2}}$ is shown as a function of the cavity-magnon coupling ratio $g_{2}/g_{1}$ and the dissipation ratio of magnon modes $\kappa _{2}/\kappa _{1}$. We can see that the bipartite entanglement $E_{N}^{m_{1}m_{2}}$ increases gradually to its maximum value as the increasing of the coupling ratio $g_{2}/g_{1}$ for a fixed dissipation ratio $\kappa _{2}/\kappa _{1}$. On the contrary, it decreases gradually as the magnon dissipation ratio $\kappa _{2}/\kappa _{1}$ increases when the coupling ratio $g_{2}/g_{1}$ is fixed. This indicates the bipartite entanglement between two magnons can be controlled with either the dissipation ratio of two magnons or coupling ratio between two magnon-cavity interactions.

Figure 2

Bipartite entanglement between the cavity and one magnon versus $\Delta _{1}$ and $\Delta _{2}$ (in unit of cavity detuning $\Delta _{a}$) when another is decoupled with the cavity, i.e., $E_{N}^{a m_{1}}$ for $g_{2}=0\text{ MHz}$ in (a) and $E_{N}^{a m_{2}}$ for $g_{1}=0\text{ MHz}$ in (b). When the two magnons are simultaneously coupled with the cavity, the cavity-magnon entanglement $E_{N}^{a m_{1}}$ is depicted in (d), $E_{N}^{a m_{2}}$ in (e) and the magnon-magnon entanglement $E_{N}^{m_{1} m_{2}}$ in (c). Meanwhile, the magnon-magnon entanglement $E_{N}^{m_{1} m_{2}}$ is depicted versus the coupling strength ratio $g_{2}/g_{1}$ and the dissipation ratio $\kappa _{2}/\kappa _{1}$ in (f) for $\Delta _{1}=-\Delta _{2}=\Delta _{a}$. See text for details of the other parameters

To better understand the physical mechanism of entanglement between the subsystems, we diagonalize the Hamiltonian by introducing Bogoliubov modes [66, 67] $\alpha = S^{\dagger}(r)aS(r) = a\cosh r + a^{\dagger}\sinh r$, $\alpha ^{\dagger}=S^{\dagger}(r)a^{\dagger}S(r) = a^{\dagger}\cosh r + a\sinh r$, where r denotes the squeezing coefficient, $\tanh 2r=2\Omega _{0}/\Delta _{a}$, and α and $\alpha ^{\dagger}$ satisfy the bosonic-commutation relation. In the frame rotating at frequency $\omega _{l}$, the Hamiltonian Eq. (1) becomes

$$\begin{aligned} \mathcal{H} = \Delta _{\alpha}\alpha ^{\dagger}\alpha + \sum ^{2}_{j=1} \bigl[\Delta _{j}m_{j}^{\dagger}m_{j} + g_{j}^{cos}\bigl(\alpha m_{j}^{ \dagger} + \alpha ^{\dagger}m_{j}\bigr) + g_{j}^{sin} \bigl(\alpha ^{\dagger} m_{j}^{ \dagger} + \alpha m_{j}\bigr) \bigr], \end{aligned}$$

(10)

with $\Delta _{\alpha}=\sqrt{\Delta _{a}^{2}-4\Omega _{0}^{2}}$, $g_{j}^{cos} = g_{j}\cosh r$ and $g_{j}^{sin}=-g_{j}\sinh r$. The Eq. (10) shows that $\Delta _{\alpha}=-\Delta _{j}$ is the optimal detuning for the cavity-magnon entanglement due to the cavity-magnon squeezing term $(\alpha ^{\dagger} m_{j}^{\dagger} + \alpha m_{j})$, which corresponds exactly to the frequency matching condition of four-wave mixing in Fig. 1(b). Meanwhile, the state-swap interaction between the cavity and magnon modes is also enhanced due to the term $g_{j}^{cos} > g_{j}$ for $r>0$. The cavity-magnon entanglement is originated from the parametric-type interaction (two-mode-squeezing term) caused by two-photon drive field. While the magnon-magnon entanglement results from the entanglement transfer caused by the state-swap interaction, as shown in the Fig. 2.

4 Parametric controllable one-way steering

Now we turn to the generation and manipulation of one-way steering. The Gaussian steering can be obtained through Gaussian measurements on one of the two magnon modes, which is given by [68]

$$\begin{aligned}& \mathcal{G}_{1\rightarrow 2} = \max \biggl[0, \frac{1}{2}\ln \frac{\mathcal{R}_{1}}{4\mathcal{R}} \biggr], \\& \mathcal{G}_{2\rightarrow 1} = \max \biggl[0, \frac{1}{2}\ln \frac{\mathcal{R}_{2}}{4\mathcal{R}} \biggr]. \end{aligned}$$

(11)

To assess the degree of asymmetry in the steerability of the two magnon modes, we define the Gaussian steering asymmetry as

$$\begin{aligned} \mathcal{G}_{as}= \vert \mathcal{G}_{1\rightarrow 2} - \mathcal{G}_{2 \rightarrow 1} \vert . \end{aligned}$$

(12)

To better explain the directionality of the one-way steering between two magnon modes, we introduce the mode populations, i.e., the final mean magnon numbers, which can be obtained from the relation

$$\begin{aligned} N_{m_{j}}=\frac{1}{2} \bigl[\bigl\langle \delta X_{j}^{2} \bigr\rangle + \bigl\langle \delta Y_{j}^{2} \bigr\rangle -1 \bigr], \end{aligned}$$

(13)

with $j = 1, 2$ corresponding to the two magnons, respectively.

In most current studies, the magnon dissipation rate is often used as a main means to regulate the direction of one-way steering [56], which will not only introduce additional dissipation or noise, reducing the entanglement and steering degree, but also increase the difficulty of experimental operations. Since the dissipation of magnon mode is only related to the surface roughness, impurities and defects of YIG sphere, it cannot be flexibly adjusted in operations. However, as we will see later that in our current study the one-way steering can be regulated on demand with the coherence coupling strength ratio between the two magnons and the cavity, which can be performed by changing the position of the YIG spheres in the cavity field. This provides great convenience for experimental operations.

From the Eq. (9) and Eqs. (11)–(13), the bipartite entanglement, quantum steering, steering asymmetry and populations of the magnon modes are described as the functions of dissipation ratio $\kappa _{2}/\kappa _{1}$ or coupling strength ratio $g_{2}/g_{1}$ of the two magnon modes in Fig. 3. In order to illustrate the controllability of quantum steering more intuitively, we have marked the one-way steering region of $\mathcal{G}_{1\rightarrow 2}$ ($\mathcal{G}_{2\rightarrow 1}$) in blue (purple), the two-way steering region in white, and steering-free region in grey. As shown in Fig. 3(a), since the interactions between the cavity and two magnon modes are nearly balanced for equivalent cavity-magnon coupling strengths $g_{2}/g_{1}=1$ and dissipation rates $\kappa _{2}/\kappa _{1}=1$, a narrow steering-free region appears in the range $0.8<\kappa _{2}/\kappa _{1}<1.2$ centered on this balance point. On both sides of this region, the balance of cavity-magnon interactions is broken due to the different dissipation rates of the two magnon modes, thus the one-way steering appears, i.e., on the left side ($\kappa _{2}/ \kappa _{1}< 0.8$) $\mathcal{G}_{2\rightarrow 1}\neq 0$, $\mathcal{G}_{1\rightarrow 2}=0$ and reversely on the right side ($\kappa _{2}/\kappa _{1}>1.2$). We can observe that on the left side the one-way steering $\mathcal{G}_{2\rightarrow 1}$ (purple line in purple area) decreases with the increase of dissipation ratio $\kappa _{2}/\kappa _{1}$ until vanishes at $\kappa _{2}/\kappa _{1}\simeq 0.8$, while on the right side $\mathcal{G}_{1\rightarrow 2}$ begin to increase gradually from the point $\kappa _{2}/\kappa _{1}\simeq 1.2$. Obviously, the steering-free region is a transition zone between two different directions of one-way steering, within which the two magnons is always entangled. Therefore, the dissipation ratio of the two magnon modes can be used as a “double-throw switch” of quantum steering under a fixed coupling ratio $g_{2}/g_{1}$, i.e., it switches the one-way steering between $\mathcal{G}_{2\rightarrow 1}$ and $\mathcal{G}_{1\rightarrow 2}$, and turn off in the steering-free region.

Figure 3

The magnon-magnon entanglement $E_{N}^{m_{1}m_{2}}$, quantum steering $\mathcal{G}_{1\rightarrow 2}$ and $\mathcal{G}_{2\rightarrow 1}$, steering asymmetry $\mathcal{G}_{as}$ and populations of magnon modes versus $\kappa _{2}/\kappa _{1}$ for $g_{1}=g_{2}$ in (a,c), and versus $g_{2}/g_{1}$ for $\kappa _{2}/2\pi = 1.5\kappa _{1}/2\pi = 6\text{ MHz}$ in (b,d). The cavity detuning and magnon detuning satisfy the relation $\Delta _{1}=\Delta _{a}$ ($\Delta _{2}=-\Delta _{a}$), and the other parameters are the same as in Fig. 2

However, as mentioned above that the dissipation ratio of the magnons cannot be conveniently adjusted in operations. While the cavity-magnon coupling strength can be flexibly controlled in the cavity-magnonics system by adjusting the external bias magnetic field or arranging the position of YIG spheres in the cavity [53]. When the dissipation ratio is fixed at an appropriate value, e.g., $\kappa _{2}/\kappa _{1}=1.5$ as shown in Fig. 3(b), the steering can be performed in a controllable manner by adjusting the coupling strength ratio. We can see that steering $\mathcal{G}_{1\rightarrow 2}$ (blue line in blue region) gradually increases as the increase of $g_{2}/g_{1}$. Nevertheless, after reaching a peak around $g_{2}/g_{1}\approx$1.5, the steering $\mathcal{G}_{2\rightarrow 1}$ increases rapidly and creates a competitive relationship with $\mathcal{G}_{1\rightarrow 2}$ in the white region. In other words, the two-way steering region $1.5< g_{2}/g_{1}<1.9$ appears, which is a competition zone of two directions of quantum steering. With the increase of $g_{2}/g_{1}$, $\mathcal{G}_{2\rightarrow 1}$ gradually dominants over $\mathcal{G}_{1\rightarrow 2}$, until the transition to the one-way steering region of $\mathcal{G}_{2\rightarrow 1}$. From the perspective of cavity-magnon couplings, the cavity magnonics system is completely symmetric when both the dissipation ratio and the coupling strength ratio are close to 1, thus no quantum steering exists between the two magnons despite entanglement. However, two ways can be used to break the balance of interaction, via the dissipation ratio or coupling strength ratio, and thus asymmetric steering can be realized. A mode with a higher dissipation rate requires a greater coupling strength to maintain this balance of interactions, i.e., more frequent coherent information exchange with the cavity than a mode with a lower dissipation rate. Therefore, in case of a fixed coupling strength ratio, the party with higher dissipation rate is passive in steerability. Meanwhile, in case of a fixed dissipation ratio, the party with the greater coherent coupling with the cavity is dominant over the other one in steerability.

The directionality of quantum steering is analyzed in terms of the mode populations in Figs. 3(c) and 3(d). Compare 3(a) with 3(c), and 3(b) with 3(d), the correlation is clearly shown between the steering direction and the mode populations. The purple area for $N_{m_{2}}>N_{m_{1}}$ corresponds to one-way steering region of $\mathcal{G}_{2\rightarrow 1}$, and the blue area for $N_{m_{1}}>N_{m_{2}}$ corresponds to the one-way steering region of $\mathcal{G}_{1\rightarrow 2}$. In the steering-free and competition region the populations of two magnon modes are nearly equal. One mode with a larger population has dominance in steering over the other one.

The above results show that the directionality of one-way quantum steering can be performed in a parametric controllable manner. Especially, the magnon populations can be changed by modulating the cavity-magnon coupling ratio under an appropriate dissipation ratio of two magnon modes, which consequently causes the change of steering direction. Undoubtedly, this provides another flexible path to manipulate quantum steering.

In Figs. 4(a) and 4(b), the quantum steering $\mathcal{G}_{1\rightarrow 2}$ and $\mathcal{G}_{2\rightarrow 1}$ are depicted, respectively, as a function of the magnon dissipation rates in unit of the cavity decay rate. As we can see in the Fig. 4(a), the steering $\mathcal{G}_{1\rightarrow 2}$ reaches its maximum at $\kappa _{1}=0$, and then decreases with the increase of $\kappa _{1}$ for a fixed $\kappa _{2}$. Similarly, in the Fig. 4(b) the steering $\mathcal{G}_{2\rightarrow 1}$ reaches its maximum at $\kappa _{2}=0$. Then $\mathcal{G}_{2\rightarrow 1}$ decreases with the increase of $\kappa _{2}$ for a fixed $\kappa _{1}$. The Fig. 4(a) and 4(b) clearly show that the steering in opposite directions $\mathcal{G}_{2\rightarrow 1}$ and $\mathcal{G}_{1\rightarrow 2}$ appear in completely different parameter areas.

Figure 4

The quantum steering $\mathcal{G}_{1\rightarrow 2}$ in (a) and $\mathcal{G}_{2\rightarrow 1}$ in (b) versus $\kappa _{1}$ and $\kappa _{2}$ in unit of $\kappa _{a}$ for $g_{1}/2\pi = g_{2}/2\pi = 42\text{ MHz}$ and $\Omega _{0} = 25\text{ MHz}$, and versus $\Omega _{0}/\kappa _{a}$ and $g_{2}/g_{1}$ in (c) and (d) for $\kappa _{2}/2\pi = 1.5\kappa _{1}/2\pi = 6\text{ MHz}$. The other parameters are the same as in Fig. 2

In Figs. 4(c) and 4(d), $\mathcal{G}_{1\rightarrow 2}$ and $\mathcal{G}_{2\rightarrow 1}$ are described, respectively, versus the drive strength $\Omega _{0}$ (in unit $\kappa _{a}$) and the coherent coupling ratio $g_{2}/g_{1}$. We can see that the steering degree increases gradually as the drive strength $\Omega _{0}$ increases. However, the directionality of steering can be changed by increasing the cavity-magnon coupling strength ratio, i.e., with the increasing of $g_{2}/g_{1}$, the steering between two magnons will transits from one-way steering region of $\mathcal{G}_{1\rightarrow 2}$ to two-way steering region and then to one-way steering region of $\mathcal{G}_{2\rightarrow 1}$. This is consistent with the description in Fig. 3.

Finally, the maximum values of $\mathcal{G}_{1\rightarrow 2}$ and $\mathcal{G}_{2\rightarrow 1}$ in Fig. 3(b) as well as the entanglement $E_{N}^{m_{1}m_{2}}$ are described for the ambient temperature T as shown in Fig. 5(a) and 5(b). It is evident that the entanglement and one-way steering between two magnons exhibit robustness within a certain temperature range. Specifically, for the taken coupling strength ratios $g_{2}/g_{1}=1.5$, the entanglement exists until 1.6 K, and the one-way steering $\mathcal{G}_{1\rightarrow 2}$ remains below 0.6 K, as shown in Fig. 5(a). Whereas the increasing in $g_{2}/g_{1}$ results in the enhancement of the robustness of entanglement, as shown in Fig. 5(b), where the entanglement exists until 2 K for $g_{2}/g_{1}=2.2$. When magnon coupling strength ratio and dissipation ratio are fixed both the entanglement and quantum steering decrease with the increase of ambient temperature T.

Figure 5

The magnon-magnon entanglement $E_{N}^{m_{1}m_{2}}$, quantum steering versus ambient temperature T for $g_{2}/g_{1}=1.5$ in (a) and $g_{2}/g_{1}=2.2$ in (b). The detunings and dissipation are taken as the same as in the Fig. 3(b)

5 Discussions

In our present system two YIG spheres are mounted in a microwave cavity driven by a two-photon drive field. The purpose of using spherical objects is to avoid the influences of inhomogeneous demagnetization field. Meanwhile, a nearly uniform microwave field is used to avoid the interaction of YIG spheres with other unnecessary magnetostatic modes [18]. Typically, the fabrication of single-crystal YIG sphere is limited to a diameter range of 200–1000 μm [5]. However, the volume of YIG sphere is positively correlated with the cavity-magnon coupling strength, while negatively correlated with the Kerr nonlinear coefficient [69]. Therefore, in order to optimize the coupling strength while minimizing nonlinearity, we select a 250 μm diameter YIG sphere [44]. The construction of four-wave mixing and the generation of optimal magnon-magnon entanglement can be achieved by modulating the bias magnetic field, wherein the frequency of the cavity $\mathrm{TE} _{120}$ mode is fixed at $\omega _{a}/2\pi = 10\text{ GHz}$. When the frequency of one of the two magnon modes is taken as $\omega _{j}/2\pi = 9.6\text{ GHz}$ (corresponding to $\Delta _{j} = -\Delta _{a}$ and $H_{j} = 0.343\text{ T}$), and the frequency of two-photon drive field satisfies $2\omega _{l}/2\pi = 19.6\text{ GHz}$, the four-wave mixing occurs. This indicates that four-wave mixing process can be triggered by either of the two magnon modes that satisfy the frequency matching condition $2\omega _{l} = \omega _{a} + \omega _{j}$. If the detuning of magnon modes satisfies the relation $\Delta _{1} = \Delta _{a}$ & $\Delta _{2} = -\Delta _{a}$ by tuning the bias fields as $H_{1}(H_{2}) = 0.357\text{ T}$ (0.343 T), or the relation $\Delta _{1} =-\Delta _{a}$ & $\Delta _{2} =\Delta _{a}$ by adjusting the bias fields as $H_{1}(H_{2}) = 0.343\text{ T}$ (0.357 T), the maximum magnon-magnon entanglement can be generated, corresponding to the situation in Fig. 2(c).

In order to regulate the direction of one-way steering on demand with the cavity-magnon coupling strength ratio, the relation between cavity-magnon coupling rates and the location of YIG spheres in the cavity can be used according to the methods in Refs. [11, 18]. In detail, one of the two YIG spheres is firstly fixed, the other one is glued on a thin wooden rod to change its position in microwave magnetic field of the $\mathrm{TE} _{120}$ mode along the y direction by using a position adjustment stage. This method is more flexible in operations than the method using the dissipation ratio of the magnons. Once a YIG sphere has been prepared its dissipation rate is determined and cannot be easily adjusted during the experimental operations. But the detection of the one-way steering is considered as a method to determine the dissipation rate of the magnon mode [52]. The bipartite entanglement and quantum steering can be demonstrated by measuring the covariance matrix via balanced homodyne detection [60]. Coupling magnon mode to an additional weak microwave field with a dissipation rate greater than that of the magnon modes via a beamsplitter-type interaction, the quadratures of magnon mode can be measured. By probing the output of the auxiliary cavity, the magnon state can be read out [70, 71]. These experimentally feasible technologies can support our current scheme.

6 Conclusion

We have constructed a cavity-magnonics system by placing two identical YIG spheres in a microwave cavity to implement quantum entanglement and one-way steering between two magnons by using a two-photon drive field to trigger a four-wave mixing. The entanglement of subsystem is originated from the nonlinearity caused by the four-wave mixing process. When the beamsplitter interaction between two magnons and the cavity is balanced, although the two magnon modes are entangled, no quantum steering is generated between them. However, the one-way steering occurs once this balance is broken. It can be completed not only by designing the dissipation ratio between the two magnons but also by regulating the coherent coupling strength ratio between the two magnons and the cavity. Especially, adjusting the coupling strength ratio is a method more flexible and convenient in operations for realizing the optimal one-way steering in different directions. We find that the steering directionality is related with the mode populations, i.e., the mode with a higher population always dominants quantum steering. Both the quantum entanglement and one-way steering are robust to ambient temperature. Our results demonstrate a feasible way for coherently manipulating entanglement and one-way steering in cavity-magnonics, which provides a macroscopic QIP platform for quantum key distribution, quantum secret sharing, one-way quantum computing.