1. Introduction

The magnons, as the quanta of collective spin excitations in yttrium iron garnet (Y$_3$Fe$_5$O$_{12}$, YIG), have attracted considerable attention due to their high spin density, low damping rate and great tunability. Many experiments have realized strong [1–6] or even ultra-strong [7,8] cavity-magnon coupling in YIG sphere, producing abundant phenomena. Besides, hybrid magnonic systems provide a new platform in macroscopic quantum interface. Cavity magnomechanics [9,10] demonstrates the simultaneous coupling of magnons to cavity photons and phonons from the sphere’s vibration. Cavity optomagnonics [11–15] realizes that a YIG sphere concurrently supports magnetostatic modes and optical whispering gallery modes (WGMs). Quantum magnonics [16–20] describes the coherent coupling between the magnetostatic modes and the superconducting qubit by the virtual photon excitation in a microwave cavity. To date, many novel phenomena have been explored theoretically or experimentally, ranging from magnon dark modes [21,22], exceptional points [23,24], bistability of cavity-magnon polaritons [25], cooling of magnons [26], and nonreciprocity and unidirectional invisibility [27], to magnon quantum blockade [28–30], single-shot detection of a single magnon [20], cavity-magnon dissipative couplings [31–33], and remote asymmetric quantum steering of magnons [34].

Entanglement based on magnons belongs to the macroscopic nonclassical effects [35–40], which has great use for testing the validity of quantum mechanics [41,42], and probing decoherence theories of large mass scales [43,44,45]. The realization of the macroscopic entangled states have potential applications in quantum technologies [46], such as quantum cryptography [47], quantum computation [48–50], and quantum nonlocality test [51]. Recently, Lachance-Quirion $et$ $al.$ [20] reported the detection of a single magnon in a millimeter-sized ferromagnetic crystal in experiment, which is based on the entanglement between a magnetostatic mode and a superconducting qubit. Theoretically, Li $et$ $al.$ [10] achieved the tripartite entanglement between magnons, microwave photons, and phonons via the nonlinear magnetostrictive interaction in cavity magnomechanics. Yuan $et$ $al.$ [52] showed that the magnon and cavity photon can form a high-fidelity Bell state with maximum entanglemnet in the parity-time broken phase. However, these schemes are confined to the entanglement of the magnons and microwave media, which is very detrimental for building a scalable quantum network.

Solid-state magnons working in the microwave domain are perhaps the one of promising candidates for quantum computation due to significant advantages [53]. In the distributed quantum communication, a long-pursued goal is that distant quantum nodes are connected through low-loss optical transmission channel [54–58]. In order to achieve this goal, an essential part is to generate quantum entangled states between the quantum nodes and the optical channels [59], which are in quite different frequency ranges, e.g., magnons at $\sim$10 GHz and optical photons at $\sim$400 THz. Although cavity optomagnonics has been designed to generate the entanglement between the magnons and the optical photons [60], their interaction strength is still weak and the quality of magnon and whispering gallery cavity is highly required, which is a critical challenge in practical application. This inspires us to further explore novel mechanism to generate controllable entanglement between magnons in the microwave domain and systems in the optical domain.

Here we proposed a scheme for using microwave photon-mediated Raman interaction to generate magnon-atom-optical photon tripartite entanglement. In this system, a three-level atomic ensemble is dispersive coupled to both an optical cavity and a microwave cavity. Meanwhile, the magnons in a macroscopic ferromagnet are coupled nonresonantly to the common microwave cavity. The indirect interaction of the magnons with the atomic ensemble is created via virtual photon exchange due to their interactions with the common microwave cavity. Once the condition of Raman magnon-photon resonance is satisfied, the nonlinear interaction of the magnon, atom, and optical photon can be established by adiabatically eliminating the atomic auxiliary level. In consequence, the magnon-atom entanglement, the magnon-optical photon entanglement, and even the genuine magnon-atom-optical photon tripartite entanglement can be generated simultaneously. It’s worth noting that the magnons with inbuilt merits are good candidates for scalable quantum computation [53], the atomic ensemble is an ideal node for storing local quantum information [61,62], and the optical photons are robust long-distance quantum bus [55]. Therefore, this quantum interface bridging the magnons, the atomic ensemble, and the optical photons, may provide a promising building block for hybrid quantum networks [63].

The remaining part of this article is organized as follows. In Sec. II, we describe the hybrid model that consists of a three-level atomic ensemble, a YIG sphere, a dispersive microwave cavity and a dispersive optical cavity, and then derive the effective Hamiltonian via microwave photon-mediated Raman interaction. In Sec. III, we present the numerical results of the magnon-atom-optical photon tripartite entanglement, discuss the classical field-driven case, and then analyze the feasibility of the experimental scheme. This article ends with a conclusion in Sec. IV.

2. Microwave photon-mediated Raman interaction

The hybrid system consists of an ensemble of $\mathcal {N}_a$ three-level atoms, a YIG sphere, a microwave cavity, and an optical cavity. As shown in Fig. 1, an atomic ensemble is located at the central intersection of a microwave cavity and an optical cavity. A YIG sphere is placed in the same microwave cavity and simultaneously in a uniform bias magnetic field. The optical cavity mode with frequency $\omega _c$ interacts dispersively with the electric-dipole allowed transition $\left |{g} \right \rangle \leftrightarrow \left |{i} \right \rangle$. The ground state $\left |{g} \right \rangle$ and the excited state $\left |{e} \right \rangle$ are separate with an optical frequency $\omega _{eg}$. The microwave cavity mode with frequency $\omega _b$ is coupled dispersively with the electric-dipole forbidden transition $\left |{e} \right \rangle \leftrightarrow \left |{i} \right \rangle$ and, at the same time, with the magnon mode of a YIG sphere with frequency $\omega _m$. The whole Hamiltonian of the composite system can be written as

 

Fig. 1. (a) Sketch of the hybrid system. An atomic ensemble locates at the central intersection of a microwave cavity and an optical cavity. A YIG sphere is installed in the same microwave cavity and simultaneously in a uniform bias magnetic field. The optical cavity and the magnon mode are driven by an optical field and a microwave source respectively (not shown). The drive magnetic field ($x$ direction), the magnetic field of the microwave cavity ($y$ direction) and the bias magnetic field ($z$ direction) are mutually perpendicular at the site of the YIG sphere. (b) Left: Dispersion interactions of the three-level atomic ensemble and two cavity fields. The atomic ensemble consists of a ground state $\left |{g} \right \rangle$, an excited state $\left |{e} \right \rangle$ and an auxiliary state $\left |{i} \right \rangle$. The microwave cavity mode $b$ and the optical cavity mode $c$ couples dispersively the transition $\left |{e} \right \rangle \leftrightarrow \left |{i} \right \rangle$ and $\left |{g} \right \rangle \leftrightarrow \left |{i} \right \rangle$ respectively. $a$ is the annihilation operator of the atomic spin wave after adiabatically eliminating the auxiliary energy level $\left |{i} \right \rangle$. Right: Dispersion interaction of the magnon mode $m$ and the microwave cavity mode $b$. The corresponding detunings are $\Delta _b =\omega _{ie}-\omega _{b}$, $\Delta _c=\omega _{ig}-\omega _{c}$, and $\Delta _m=\omega _{m}-\omega _{b}$.

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Assuming the cavity-atom detunings and the cavity-magnon detuning are much larger than the coupling strength between them, i.e., $\Delta _b =\omega _{ie}-\omega _{b}\gg g_b$, $\Delta _c=\omega _{ig}-\omega _{c}\gg g_c$, and $\Delta _m=\omega _{m}-\omega _{b}\gg g_m$, the system operates in the dispersive regime. Using a unitary transformation $H_{\textrm {eff}}=e^{-\lambda X}He^{\lambda X}$ [64–66], we can obtain the effective magnon-atom-optical photon coupling. The $\lambda$ is introduced to show the orders in the perturbation expansion, and would be set to 1 after the calculations. The anti-Hermitian operator $X$ is introduced with the form

3. Magnon-atom-optical photon tripartite entanglement

3.1 Numerical results

For the generation of magnon-atom-optical photon tripartite entanglement, the optical cavity and the magnon mode are driven resonantly by an optical field with Rabi frequency $\Omega _c$ and a microwave source with $\Omega _m$, respectively. The effective Hamiltonian of this hybrid magnon-atom-optical photon system in the appropriate rotating frame can be written as

The linearized Hamiltonian consists of three parts: one parametric interaction between the magnon mode and the atomic mode, two beam-splitter interactions of the optical mode with the magnon mode and the atomic mode. These interactions are alternately cascaded in a closed triangular contour, as shown in Fig. 2. It is well known that the parametric interaction is responsible for entanglement while the beam-splitter interaction leads to the quantum state transfer. Obviously, the entanglement between the magnon mode and the atomic mode is firstly established via the parametric interaction, and then the quantum state of the atomic (magnon) mode is transferred to the optical mode. As a consequence, the magnon (atomic) mode and the optical mode are prepared in the entangled state, and in the meantime the genuine magnon-atom-optical photon tripartite entanglement is generated.

 

Fig. 2. Schematic diagram of the fluctuating parts of the magnon-atom-photon system with a closed coupling loop. Modes $\delta a$ and $\delta m$ interact effectively via a parametric interaction (for entangled states) with strength $\tilde {g}_{c}$; at the same time, they are coupled to the mode $\delta c$ via a beam-splitter interaction (for state transfer) with strengths $\tilde {g}_{m}$ and $\tilde {g}_{a}$, respectively.

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The linearized QLEs describing the quadrature fluctuations with $\delta X_O=(\delta O+\delta O^{\dagger})/\sqrt {2}$, $\delta Y_O=i(\delta O^{\dagger}-\delta O)/\sqrt {2}$ ($O=a,c,m$), can be written as

To investigate bipartite and tripartite entanglement of the hybrid system, we adopt quantitative measures of the logarithmic negativity $E_N$ [72,73,74] and the residual contangle $\mathcal {R}_{min}$ [75,76], respectively. The logarithmic negativity for Gaussian states is defined as

In Fig. 3, we show the density plot of bipartite entanglement and tripartite entanglement versus $g/\kappa _m$ and $\Omega _{c}/\Omega _m$ ($\kappa _m$, $\Omega _m$ is fixed) by taking the experimentally feasible parameters $\omega _c/2\pi =446$THz, $\omega _m/2\pi =10$GHz, $\gamma _a/2\pi =\kappa _c/2\pi =\kappa _m/2\pi =5$MHz, $\Omega _m/2\pi =100$MHz, $T=0.05$K. All results are in the steady state guaranteed by the negative eigenvalues (real parts) of the drift matrix $A$. The complementary distribution of the entanglement in Figs. 3(b) and 3(a), 3(c) indicates that the initial magnon-atom entanglement is partially transferred to the magnon-optical photon and atom-optical photon subsystems, and this efficiency is determined by the collective coupling strength and the intensity ratio of the drive fields. It can be found from Fig. 3(b) that $E_{ma}$ is always remaining zero when $\Omega _c=0$, no matter how $g$ changes. Therefore, no entanglement is transferred to the magnon-optical photon and atom-optical photon subsystems, which causes both bipartite and tripartite entanglement to disappear. This phenomenon can be easily explained by the linearized Hamiltonian in Eq. (11): when $\Omega _c=0$, the linear coupling rate $\tilde {g}_{c}$ is zero, the parametric interaction vanishes. Similarly, when the coupling strength $g=0$, all entanglements also disappear. With the increase of the driving strength $\Omega _c$, even if the coupling strength $g$ is small, a strong magnon-atom entanglement can be obtained, as shown in Fig. 3(b). This could be attributed to the significant enhancement of the effective linear coupling rates. By comparing Figs. 3(a) and 3(c), we can find that $E_{cm}$ is much stronger than $E_{ca}$. The reason is that $\tilde {g}_{m}$ is always higher than $\tilde {g}_{a}$ since the driving fields are added directly to the magnon and optical cavity, and only a few atoms are excited in the dispersion situation. As a result, the transfer process $\delta m\stackrel {\tilde {g}_{c}}{\leftrightsquigarrow } \delta a\stackrel {\tilde {g}_{m}}{\leftrightarrow } \delta c$ to generate magnon-optical photon entanglement is more dominant than the transfer process $\delta a\stackrel {\tilde {g}_{c}}{\leftrightsquigarrow } \delta m\stackrel {\tilde {g}_{a}}{\leftrightarrow } \delta c$ to generate atom-optical photon entanglement. On the other hand, if $\tilde {g}_{a}$ and $\tilde {g}_{m}$ are large enough, the established magnon-optical photon and atom-optical photon entanglements are transferred immediately back to the magnon-atom entanglement. In this case, only the magnon-atom entanglement exists, which can be seen from the lower right area of Figs. 3(a), 3(b), and 3(c). It should be noted that the obtained atom-optical photon entanglement is far smaller than other bipartite entanglements. Our scheme focuses on the generation of entanglements based on magnons since the entanglement between atoms and optical photons has been extensively studied in other systems [78,79,82]. In addition to the presence of all bipartite entanglements, the steady state of the system is also a genuinely tripartite entangled state, which is considerable under the appropriate parameters, as shown in Fig. 3(d).

 

Fig. 3. Density plot of bipartite entanglement (a) $E_{cm}$, (b) $E_{ma}$, (c) $E_{ca}$, and tripartite entanglement (d) $\mathcal {R}_{min}$ versus $g/\kappa _m$ and $\Omega _c/\Omega _m$ ($\kappa _m$, $\Omega _m$ is fixed). The parameters are chosen as $\omega _c/2\pi =446$THz, $\omega _m/2\pi =10$GHz, $\gamma _a/2\pi =\kappa _c/2\pi =\kappa _m/2\pi =5$MHz, $\Omega _m/2\pi =100$MHz, and $T=0.05$K.

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Moreover, it is interesting to discuss the effects of the decay rates of the optical cavity and the atoms on hybrid bipartite and tripartite entanglement. In Fig. 4, we plot the entanglement as a function of $\Omega _{c}/\Omega _m$ by choosing $\kappa _c/\kappa _m$=0.1 (red solid line), 1 (blue dashed line), 10 (green dot dash line). It is seen from Figs. 4(a) and 4(c) that the maximum values of $E_{cm}$ and $E_{ca}$ decrease with the increase of cavity decay rate, and $E_{cm}$ has better robustness than $E_{ca}$. Figure 4(b) shows that the damage of entanglement $E_{ma}$ due to the large dissipation can be compensated by increasing $\Omega _{c}/\Omega _m$. When $\Omega _{c}/\Omega _m$ is large enough, the magnon-atom entanglement $E_{ma}$ will reach saturation. In Fig. 4(d), the existence region of genuine tripartite entanglement $\mathcal {R}_{min}$ becomes more extensive with the increase of the cavity decay rate, and the optimal tripartite entanglement can be found when the magnons, the atoms, and the optical cavity have almost the same dissipation rates. Similarly, Fig. 5 shows the entanglements versus the ratio $\Omega _{c}/\Omega _m$ under the different atomic decay rates. This is clearly shown in Fig. 5, the achieved maximum values of the bipartite entanglement and genuine tripartite entanglement first increases and then decreases with an increase of atomic decay rate, and $E_{ma}$ will reach saturation when $\Omega _{c}/\Omega _m$ is large enough. The reason is that the effective coupling strengths $\tilde {g}_{c}$ and $\tilde {g}_{m}$ increase rapidly and $\tilde {g}_{a}$ decreases slowly with the rise of the atomic damping, which will enhance the quantum correlation among atoms, magnons, and optical photons. However, when the atomic decay is too large, the enhancement of quantum correlation will be destroyed due to the decoherence induced by the environmental noise, so the entanglement will first increase and then decrease with the increase of the atomic decay. From the above results, it is clearly seen that the present scheme can realize the considerable magnon-atom entanglement, magnon-optical photon entanglement, and genuine tripartite entanglement by taking the experimentally feasible dissipation rates $\gamma _a/2\pi =\kappa _c/2\pi =\kappa _m/2\pi =$5MHz (blue dashed line).

 

Fig. 4. Bipartite entanglement (a) $E_{cm}$, (b) $E_{ma}$, (c) $E_{ca}$, and tripartite entanglement (d) $\mathcal {R}_{min}$ versus $\Omega _c/\Omega _m$ for $g/\kappa _m=0.1$ in (a), 0.01 in (b), 0.3 in (c), 0.02 in (d), and $\kappa _c/\kappa _m$=0.1 (red solid line), 1 (blue dashed line), 10 (green dot dash line). The other parameters are the same as in Fig. 3.

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Fig. 5. Bipartite entanglements (a) $E_{cm}$, (b) $E_{ma}$, (c) $E_{ca}$, and tripartite entanglement (d) $\mathcal {R}_{min}$ versus $\Omega _c/\Omega _m$ for $g/\kappa _m=0.1$ in (a), 0.01 in (b), 0.3 in (c), 0.02 in (d), and $\gamma _a/\kappa _m$=1 (red solid line), 5 (blue dashed line), 15 (green dot dash line) in (a), (c); $\gamma _a/\kappa _m$=0.1 (red solid line), 1 (blue dashed line), 10 (green dot dash line) in (b), (d). The other parameters are the same as in Fig. 3.

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Finally, the influence of the ambient temperature on the bipartite and tripartite entanglements is shown in Fig. 6. All entanglements are robust against environmental temperature, and $E_{cm}$, $E_{ma}$, $E_{ca}$, and $\mathcal {R}_{min}$ can survive up to about $0.32$K, 0.73K, 0.13K and 0.25K, respectively, when the parameters are chosen as those at the maximum entanglements in Fig. 3.

 

Fig. 6. Bipartite entanglement (a) $E_{cm}$, (b) $E_{ma}$, (c) $E_{ca}$, and tripartite entanglement (d) $\mathcal {R}_{min}$ versus temperature $T$. The other parameters are chosen as those at the maximum entanglements in Fig. 3.

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3.2 Discussions

So far, the magnon-atom, magnon-optical photon bipartite entanglements and the genuine magnon-atom-optical photon tripartite entanglement have been achieved by using the microwave photon-mediated Raman process. In fact, it is much easier to obtain the magnon-atom bipartite entanglement as the quantum cavity field $c$ is replaced by the classical driving field with Rabi frequency $\Omega _d$ and frequency $\omega _d$. We still consider that the system operates in the dispersive regime, i.e., $\Delta _b =\omega _{ie}-\omega _{b}\gg g_b$, $\Delta _m=\omega _{m}-\omega _{b}\gg g_m$, and $\Delta _d=\omega _{ig}-\omega _d\gg \Omega _d$, and the frequency of the classical driving field satisfies the Raman magnon-photon resonances, i.e., $\omega _d\simeq \omega _{eg}+\omega _m$. Following the similar derivation in Sec. 2., a new effective Hamiltonian can be obtained in the interaction picture,

Using the same method as in Sec. 3.1, we adopt the logarithmic negativity to quantify the magnon-atom entanglement, and the stability condition $g'^{2}<\kappa _m \gamma _a$ can be obtained from the Routh-Hurwitz criterion [69]. Figure 7 plots the stationary state magnon-atom entanglement versus the collective coupling strength and the environmental temperature. It is seen from Fig. 7(a) that the magnon-atom entanglement $E_{ma}$ increases with $g'$, and reaches a maximum about ln2 $\approx$ 0.7 as $g'/\kappa _m\rightarrow 1$ ($\gamma _a=\kappa _m$). In addition, the magnon-atom entanglement is robust against temperature and survives up to about $0.7$K, which is shown in Fig. 7(b).

 

Fig. 7. The magnon-atom entanglement $E_{ma}$ versus (a) the effective coupling strength $g'$ for $T=0.1K$ and (b) temperature $T$ for $g'=0.99\kappa _m$. The other parameters are the same as those in Fig. 3.

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3.3 Experimental implementation

In this section, we briefly discuss the feasibility of the experimental scheme. Proposed as in Fig. 1(a), our scheme contains a undoped single-crystal YIG sphere, an atomic ensemble, a optical cavity and a microwave cavity. Here, we can chosen a coplanar waveguide resonantor as the microwave cavity [80,81], which can apply the microwave quantum field. An atomic ensemble and a YIG sphere are placed inside the microwave cavity, and at the wave crest of the microwave field. Simultaneously, the atomic ensemble is coupled to an optical cavity. In this scheme, the frequencies of the optical cavity and the microwave cavity are both far-detuned from the atoms and magnon, and the large decay rate of the microwave cavity is needed. On the other hand, a great number of atomic structures can be used as candidates for the present scheme. For example, the $^{6}$Li atoms on the $2S_{1/2}\leftrightarrow 2P_{1/2},2P_{3/2}$ transition hyperfine structure is a good candidate [83]. The designated states in Fig. 1 (b) can be chosen as follows: $\left |{g} \right \rangle =\left |{2^{2}S_{1/2},F=1/2}\right \rangle$, $\left |{e} \right \rangle =\left |{2^{2}P_{1/2},F=1/2} \right \rangle$, and $\left |{i} \right \rangle =\left |{2^{2}P_{3/2},F=1/2} \right \rangle$. The transition $\left |{e} \right \rangle =\left |{2^{2}P_{1/2},F=1/2} \right \rangle \leftrightarrow \left |{i} \right \rangle =\left |{2^{2}P_{3/2},F=1/2} \right \rangle$ with microwave frequency 10.05 GHz is electric dipole-forbidden, which can be coupled with a microwave cavity field via magnetic-dipole interaction. Meanwhile, the transition $\left |{g} \right \rangle =\left |{2^{2}S_{1/2},F=1/2} \right \rangle \leftrightarrow \left |{i} \right \rangle =\left |{2^{2}P_{3/2},F=1/2} \right \rangle$ is electric dipole-allowed, which can be interacted with an optical cavity field by electric-dipole interaction. The atoms are assumed to be cooled by the the magneto-optical trapping technique and trapped at the central intersection of the optical cavity and the microwave cavity.

Note that the above results are valid only when the atom excitation number $\left \langle a^{\dagger} a\right \rangle \ll \mathcal {N}_a$, and the magnon excitation number $\left \langle m^{\dagger} m\right \rangle \ll 2\mathcal {N}_ms=5\mathcal {N}_m$, where $\mathcal {N}_m$ is the total number of spins and $s=5/2$ is the spin number of the ground state Fe$^{3+}$ ion in the YIG sphere. For a 0.5-mm-diameter YIG sphere, the number of spins $\mathcal {N}_m\simeq 2.8\times 10^{17}$ [16]. Taking the parameters $g=0.05\kappa _m=2\pi \times 0.25$MHz and $\Omega _c=2\Omega _m=2\pi \times 200$MHz, corresponds to the good entanglements, leading to $\left \langle m^{\dagger} m\right \rangle \simeq 950\ll 5\mathcal {N}_m=1.4\times 10^{18}$, which is well fulfill the low magnon excitation limit. Now $^{6}$Li atoms can be cooled to about 50$\mu K$ with $5\times 10^{7}$ atom numbers by Zeeman slower and the standard magneto-optical trapping technique [84]. Under the above parameter conditions, our atom excitation number is $\left \langle a^{\dagger} a\right \rangle \simeq 330\ll \mathcal {N}_a=5\times 10^{7}$, which nicely conforms to the low atomic excitation limit. To sum up, low excitation approximation in our scheme is perfectly valid under experimentally feasible parameters.

Finally, the generated entanglements can be verified by measuring the corresponding CMs [85,86]. The atoms and magnons states can be read out by coupling to the weak optical and microwave probe fields via beam-splitter interaction, respectively. The states of optical photons can be measured by the output field. By homodyning the outputs of the probe fields and measuring the corresponding CMs, one can verify the entanglements.

4. Conclusion

In conclusion, we have presented a scheme to generate not only the hybrid magnon-atom, magnon-optical photon bipartite entanglement, but also the genuine magnon-atom-optical photon tripartite entanglement. Instead of using the optical whispering gallery cavity, the magnons, atoms, and optical photons entangle with each other via the microwave photon-mediated Raman interaction. We also show that both bipartite and tripartite entanglements strongly depend on the relative strength of two driving fields, which provides a convenient method to control hybrid entanglement. Additionally, the generated entanglement is robust against the environment temperature and system decoherence. Our scheme paves the way for exploring hybrid quantum interface based on different quantum information carriers, which may find potential applications in scalable quantum network.

Appendix: the validity of the effective Hamiltonian

In order to verify the validity of the adiabatic approximation, we plot the time evolution of the system governed by the effective Hamiltonian (7) and the original Hamiltonian (1) with the strong resonant driving fields in Fig. 8, where we don’t consider the dissipation of the system. We observe from Fig. 8 that our results based on the effective Hamiltonian coincide with the numerical results based on the original Hamiltonian with driving fields very well. This confirms that the adiabatic approximation is valid under the strong driving fields when the Raman magnon-photon resonance is satisfied. Therefore, the numerical analysis based on the effective Hamiltonian Eq. (8) is reasonable.

 

Fig. 8. The average quantum numbers of the optical photon (red), the magnon (blue), and the probability of the atom in excited state (green) are plotted as functions of the evolution time t/T. The solid line, dashed line, and dot dash line are the corresponding results of the effective Hamiltonian (8), the empty quadrates, circles, and triangles are the corresponding results of the original Hamiltonian (1) plus the resonant driving terms. The parameters are set as $g_b=g_c=10g$, $g_m=100g$, $\Omega _c=\Omega _m=200g$, $\omega _m=2\times 10^{4}g$, $\omega _c=8.92\times 10^{8}g$, $\Delta _m=\Delta _c=1\times 10^{3}g$, $\Delta _b=2\times 10^{3}g$, $N_a=1\times 10^{4}$ and $T=g^{-1}$. We assume the system is prepared in the state $\left |{\psi (0)} \right \rangle =\left |{1} \right \rangle _c\left |{0} \right \rangle _m\left |{g} \right \rangle _a$ initially.

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Funding

National Natural Science Foundation of China (Grant No. 61875067).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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