1. Introduction

Cavity optomechanical systems [1–5] have attracted considerable attention in the past few years due to their potential applications in quantum information technology. Earlier researches have achieved dramatic progress in electromechanically or optomechanically induced transparency (EMIT/OMIT) [6–8], quantum mechanics in macroscopic objects [9–12], cooling [13–18], optical filters [19], quantum sensors [2,20,21], etc. Furthermore, the nanomechanical resonators can be coupled to the microwave and optical cavities simultaneously, they can therefore entangle [22,23] or coherently convert [24,25] the photons at microwave and optical regions. Recent experiments [26–41] are very encouraging to directly convert microwave and optical photons in either the classical or the quantum domain. Especially, it has been demonstrated that the cavity electro-opto-mechanical (EOM) system reaches the highest conversion efficiency of 0.47 [36]. However, it’s still challenging to meet the requirements of high conversion efficiency and low added noise. In contrast, the entanglement-based conversion [42,43], which first entangles microwave and optical photons [22,44–46], and then completes the conversion with quantum teleportation, has been theoretically demonstrated to be a more efficient and practical method due to the introduction of classical communication channels. However, the performances on either entanglement-based conversion or direct conversion are both restricted by the moderate electromechanical and optomechanical couplings.

Multimode electro-opto-mechanical systems provide a possible approach to circumvent this difficulty. Experimentally, multimode optomechanical and electromechanical systems have been employed for circulators [47–49], tunable optomechanically induced transparency [50,51], enhanced coherent entanglement or cooling [52–54], etc. By introducing another mechanical resonator, the cooperativity between the photons and phonons can be enhanced or suppressed by dissipation engineering [53,55,56] or quantum interference [50,51,57]. However, these experiments mainly concerns the indirect coupling of mechanical modes, where an optical or microwave mode serves as an intermediary. It’s still blank in the multimode mechanical couplings on the EOM systems, in which mechanical modes links the interaction between microwave and optical modes.

In this paper, we theoretically study a two-mechanical-mode EOM system, where two mechanical resonators (MRs) are phase-dependent coupled with each other. The quantum interference of two pathways of electro-opto-mechanical interaction is studied in M-O entanglement and coherent conversion. The double-channel phenomena are observed and the modulation on entanglement or conversion in each channel can be realized by tuning the couplings. These results provide an approach to optimize the parameter settings to obtain the double-channel or single-channel optimal entanglement or conversion. Additionally, our scheme could be generalized to an N-mechanical-mode EOM system or other nonlinear interaction in multimode systems such as magnon-optics and atom ensembles.

2. System

We consider a two-mechanical-mode EOM system shown in Fig. 1(a), where two mechanical resonators (MRs) with frequencies ${\omega _{m1}}$ and ${\omega _{m2}}$ are coupled with each other, and MR1 serves as an intermediary to realize indirect coupling between microwave and optical modes ${\hat{a}_{e,o}}$. By modulating the mechanical coupling ${\lambda _r}$, we can further control the indirect interaction between ${\hat{a}_e}$ and ${\hat{a}_o}$. The two-mechanical-mode configuration can be realized in many systems, such as charged MRs [58], superconducting quantum circuit [54,59] and others [60,61]. Figure 1(b) shows schematic of a potentially experimental system, in which two MRs are Coulomb coupled. MR1 plays the role of a movable plate of a capacitor in LC circuit and movable mirror of optical cavity simultaneously, and thus modulates the resonance frequency of the microwave LC circuit as well as the optical cavity. The system Hamiltonian reads as

 

Fig. 1. (a) Schematic of a two-mechanical-mode EOM system. Potentially experimental systems: (b) The two MRs are coupled with each other via Coulomb interaction, which can be controlled by the bias voltages V1 and -V2 on the MR1 and MR2, respectively. (c) Schematic of a fully coupled two-mechanical-mode EOM system. $\zeta {e^{ {\pm} i\theta }}:\zeta ({e^{ - i\theta }}{\hat{b}_1}\hat{b}_2^{\dagger} + {e^{i\theta }}\hat{b}_1^{\dagger} {\hat{b}_2})$. (d) Two MRs are capacitive coupled to a superconducting charge qubit, and the phase-dependent phonon-hopping interaction is induced and controlled by gate voltages. $C_{1,2}^x$ represents the gate capacitance, which is varied with the displacement ${x_{1,2}}$ of each MR. JJ: Josephson junction.

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First, we consider two identical resonance frequency ${\omega _1} = {\omega _2} = {\omega _m}$ of MRs for simplicity. Defining two mechanical dressed modes ${\hat{b}_ + } = {{({{\hat{b}}_1} + {{\hat{b}}_2})} / {\sqrt 2 }}$ and ${\hat{b}_ - } = {{({{\hat{b}}_1} - {{\hat{b}}_2})} / {\sqrt 2 }}$, the Hamiltonian can be derived to

 

Fig. 2. Schematic of the energy-level diagram in the four-mode EOM system. The $|{{n_e}} \rangle $, $|{{n_o}} \rangle $, $|{{m_1}} \rangle $ and $|{{m_2}} \rangle $ denote the number states of the microwave and optical cavity photon, and MR1 and MR2 phonons, respectively. $|{{n_o},{n_e},{m_1},{m_2}} \rangle $ \leftrightarrow |{{n_o},{n_e} + 1,{m_1},{m_2}} \rangle $and$|{{n_o},{n_e},{m_1},{m_2}} \rangle $ \leftrightarrow |{{n_o} + 1,{n_e},{m_1},{m_2}} \rangle $ transitions change the cavity fields.$|{{n_o},{n_e} + 1,{m_1},{m_2}} \rangle \leftrightarrow |{{n_o},{n_e},{m_1} + 1,{m_2}} \rangle $ and $|{{n_o} + 1,{n_e},{m_1},{m_2}} \rangle \leftrightarrow |{{n_o},{n_e},{m_1} + 1,{m_2}} \rangle $ transitions are caused by capacitive coupling and radiation-pressure coupling, respectively. $|{{n_o},{n_e},{m_1} + 1,{m_2}} \rangle \leftrightarrow |{{n_o},{n_e},{m_1},{m_2} + 1} \rangle $ transition is induced by the Coulomb coupling, which induces two mechanical dressed modes $|{{n_o},{n_e},{b_ + }} \rangle $ and $|{{n_o},{n_e},{b_ - }} \rangle $.

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In the following, we consider a more complete condition, where two MRs are both coupled to microwave and optical cavities simultaneously, and the MRs are coupled with each other via a phase-dependent phonon-hopping interaction shown in Fig. 1(c)–1(d). So, Hamiltonian (1) becomes

3. Entanglement and conversion

3.1 Entanglement

In this section, we discuss the effect of the two-mechanical-mode on the entanglement of itinerant microwave and optical photons. We first consider a typical three-mode EOM system, which means ${\lambda _r} = 0,{g_{j2}} = 0$. As the driving of each cavity satisfies ${\Delta _e} ={-} {\Delta _o} = {\omega _{m1}}$ (${\Delta _j} = {\omega _j} - {\omega _{d,j}}$), the Hamiltonian in the interaction picture $\hbar {\omega _e}\hat{a}_e^{\dagger} {\hat{a}_e} + \hbar {\omega _o}\hat{a}_o^{\dagger} {\hat{a}_o}$ reduces to Eq. (8) after linearization and rotation approximation.

Utilizing quantum Langevin equations, the propagating modes can be written as [62]

One of the important issues is how to discuss entanglement quantitatively. Here, entanglement degree based on average number of propagating modes as well as their correlation is utilized to deal with this issue. Denoting the temperature of EOM as ${T_E}$, we can obtain the average photon or phonon number of quantum fluctuations in Eq. (9) and Eq. (10) at respective frequencies by Planck laws, which reads

In the following, we check the performance on entanglement and transmission of two-mechanical-mode EOM system. The feasible parameter settings are listed in Table 1. In Fig. 3(a), we plot the normalized output power ${P_o}$ of the optical cavity for ${\zeta / {2\pi }} = 0,0.16\textrm{MHz},0.28\textrm{MHz}$. $\delta = \omega - {\omega _e}$ represents the detuning of itinerant microwave photons and the resonance frequency of the microwave cavity. Other parameter settings are as follows: ${G_{e2}} = 0$, ${G_{o2}} = 0$ and $\theta = 0$. It’s obvious that there exists only one window to generate quadrature entangled optical and microwave photons if MR2 is independent, while two entanglement windows exist when the two MRs are coupled with each other. Besides, the distance of two windows is determined by the coupling strength $\zeta $. The Fig. 3(b) shows double-channel phenomena under the case of ${G_{e2}} > 0$, ${G_{o2}} > 0$. By controlling the electromechanical and optomechanical couplings, the amplitude and width of two windows are unequal. Physically, the width of each window is related to the effective mechanical damping rate given by ${\gamma _{m1}} + {\Gamma _{e \pm }} - {\Gamma _{o \pm }}$, while the amplitude is more related to the effective multiphoton-phonon coupling strength. Therefore, we can select the dominant channel by modulating the phase of mechanical coupling and electromechanical (optomechanical) coupling strength between the cavity with MR2.

 

Fig. 3. M-O Entanglement of the two-mechanical-mode EOM system. (a) The normalized output optical fields. (b) The normalized output optical fields for ${{{G_{e2}}} / {2\pi }} = 99.5\textrm{kHz,}{{{G_{o2}}} / {2\pi }} = 32.7\textrm{kHz}$ (blue lines), ${{{G_{e2}}} / {2\pi }} = 63.1\textrm{kHz,}{{{G_{o2}}} / {2\pi }} = 57.8\textrm{kHz}$ (red lines). (c) The normalized logarithmic negativity ${{{E_{N \pm }}} / {{{\bar{N}}_o}}}$ as a function of $\theta $ for ${{{G_{e2}}} / {2\pi }} = 99.5\textrm{kHz,}{{{G_{o2}}} / {2\pi }} = 32.7\textrm{kHz}$ (blue), ${{{G_{e2}}} / {2\pi }} = 63.1\textrm{kHz,}{{{G_{o2}}} / {2\pi }} = 57.8\textrm{kHz}$ (red). (d) ${{{E_{N + }}} / {{{\bar{N}}_o}}}$ as a function of ${G_{e2}}$ and ${G_{o2}}$.

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Table 1. Parameter settings of the EOM system^a

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Since the most powerful entangled resources are expected to be those maximizing their quadrature correlations per photon emitted, we analyze the two-mechanical-mode EOM system in terms of the normalized logarithmic negativity ${{{E_N}} / {{{\bar{N}}_o}}}$. We first calculate the ${{{E_N}} / {{{\bar{N}}_o}}}$ as a function of $\zeta$. It’s shown that the coupling strength between MRs shows little effect on ${E_{N + }}$ and ${E_{N - }}$, but just change the distance of entanglement windows, as shown in Fig. 3(a) and 3(b). Corresponding results are listed in Appendix. D. In the following, we analyze the effect of $\theta $, ${G_{e2}}$ and ${G_{o2}}$ on entanglement under the condition of ${\zeta / {2\pi }} = 0.16\textrm{MHz}$, respectively. In Fig. 3(c), we plot the ${{{E_{N \pm }}} / {{{\bar{N}}_o}}}$ as a function of $\theta $ for different electromechanical and optomechanical couplings with MR2${{{G_{e2}}} / {2\pi }} = 99.5\textrm{kHz}$, ${{{G_{o2}}} / {2\pi }} = 32.7\textrm{kHz}$ and ${{{G_{e2}}} / {2\pi }} = 63.1\textrm{kHz,}{{{G_{o2}}} / {2\pi }} = 57.8\textrm{kHz}$. The variations of entanglement on two channels are opposite and the maximum of ${{{E_{N + }}} / {{{\bar{N}}_o}}}$ corresponds to the minimum of ${{{E_{N - }}} / {{{\bar{N}}_o}}}$, and vice versa. To further probe the effect of ${G_{e2}}$ and ${G_{o2}}$, we plot ${{{E_{N + }}} / {{{\bar{N}}_o}}}$ versus ${G_{e2}}$ and ${G_{o2}}$, which is shown in Fig. 3(d). There are two parts in the image, the stable (color region) and unstable (blank region) region. The boundary of two parts is given by Routh-Hurwitz criterion (See Appendix. E). In the color region, appropriate setting of ${G_{e2}}$ and ${G_{o2}}$ will increase the normalized entanglement and the maximum ${{{E_{N + }}} / {{{\bar{N}}_o}}}$ is obtained at certain values of ${G_{e1}}$ and ${G_{o1}}$. The blank region of the image indicates an unstable condition, which can be reduced by adjusting the couplings with MR1. Additionally, the ${{{E_{N - }}} / {{{\bar{N}}_o}}}$ versus ${G_{e2}}$ and ${G_{o2}}$ are similar with Fig. 3(d) due to the symmetry of the above parameters.

From Fig. 3, we can conclude that the M-O entanglement generated by a two-mechanical-mode EOM system is exhibited in a double-channel, and the entanglement characteristics in each channel are controllable by tuning the coupling strength and phase of phonon-phonon interaction. Physically, two pathways of electro-opto-mechanical interaction are interfered, which produces two discrete channels. In operator representation, the two mechanical modes coupled are hybridized into two decoupled modes and thus lead to a double-channel. When the cavities do not interact with MR2 directly, the two windows are equal in output fields and widths. Considering a more complicated case that two cavities are coupled with two MRs simultaneously, the amplitude and width of each entanglement window can be tuned by controlling $\zeta $, $\theta $, ${G_{e2}}$ and ${G_{o2}}$, which provides a more practical and feasible method to realize double-channel M-O entanglement.

3.1 Conversion

Coherent conversion between itinerant microwave and optical photons corresponds to the case of ${\Delta _e} = {\Delta _o} = {\omega _{m1}}$, and the Hamiltonian in the interaction picture is as follows:

In Fig. 4, we investigate the effect of two-mechanical-mode on conversion from microwave to optical photons. Figure 4(a) shows the conversion efficiency ${|{{t_ + }(\omega )} |^2}$ in the higher-frequency channel as a function of ${G_{e2}}$ and ${G_{o2}}$ when ${\zeta / {2\pi }} = 0.08\textrm{MHz,}\theta = \pi $, while Fig. 4(b) corresponds to that in the lower-frequency channel. It’s shown that ${|{{t_ \pm }(\omega )} |^2}$ is tunable by controlling the electromechanical and optomechanical couplings. The results of ${|{{t_ + }(\omega )} |^2} = 0$ mean that the higher-frequency channel disappears and there is only lower-frequency channel to convert microwave photons. When we consider the case of $\theta = 0$ with the same other parameters, the variation of ${|{{t_ - }(\omega )} |^2}$ can be shown by Fig. 4(a), while Fig. 4(b) corresponds to that of ${|{{t_ + }(\omega )} |^2}$. Besides, the distance of two channels is mainly determined by the coupling strength $\zeta $ of MRs, which is shown in Fig. 4(c). Specifically, the conversion efficiency ${|{{t_ \pm }(\omega )} |^2}$ as a function of $\theta $ in the frequency domain is shown in Fig. 4(d). In the region $0 < \theta < \pi $, the left conversion window always become much broader and more efficient while the right one becomes weaker with the increasing of $\theta $. The left window becomes completely absorbed near $\theta = \pi $, which refers to the case of ${|{{t_ + }(\omega )} |^2} = 0$ shown in Fig. 4(a). The variation in $\pi < \theta < 2\pi $ is absolutely opposite with the case of $0 < \theta < \pi $. Thus, the switchable and tunable M-O conversion can be realized by modulating the mechanical coupling strength and phase of MRs.

 

Fig. 4. M-O conversion efficiency of the two-mechanical-mode EOM system. (a) The conversion efficiency ${|{{t_ + }(\omega )} |^2}$ of the itinerant microwave photons in the higher-frequency channel as a function of ${G_{e2}}$ and ${G_{o2}}$ when ${\zeta / {2\pi }} = 0.08\textrm{MHz}$, $\theta = \pi $. (b) ${|{{t_ - }(\omega )} |^2}$ under the same parameters as (a). (c) ${|{{t_ \pm }(\omega )} |^2}$ versus $\zeta$ and $\delta $ when $\theta = {\pi / 2}$. (d) ${|{{t_ \pm }(\omega )} |^2}$ versus $\theta $ and $\delta $ when ${\zeta / {2\pi }} = 0.08\textrm{MHz}$. Other parameters are the same as Fig. 3.

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In general, any extraneous vibrations of MRs and reflected photons will introduce noise to the output fields. In the up-conversion, the former is mainly determined by mechanical resonance frequency, environmental temperature and cooperativities, while the latter, the optical noise, is much less and negligible mainly since the resonance frequency of the optical cavity is seven orders of magnitude higher than that of MRs. As Eq. (27) shows that the optomechanical couplings have less effect on the added noise caused by extraneous vibrations of MRs in up-conversion, we only probe the effect of electromechanical and mechanical couplings. The results that ${n_{add, \pm }}$ as a function of ${G_{e2}}$ and $\theta $ when ${\zeta / {2\pi }} = 0.08\textrm{MHz}$ are shown in Fig. 5. Similar to the variation of conversion efficiency, the added noise is periodically controllable. In the region $0 < \theta < \pi $ ($\pi < \theta < 2\pi $), ${n_{add, + }}$ is increased while ${n_{add, - }}$ is reduced with the increasing of $\theta $. Moreover, we get the minimum values of added noise in the higher (lower)-frequency channel when $\theta = \pi $ ($\theta = 0$), and it can be further decreased by increasing ${G_{e2}}$.

 

Fig. 5. (a), (b) Added noise ${n_{add, \pm }}$ of the two-mechanical-mode EOM system in the higher-frequency and lower-frequency channels as a function of ${G_{e2}}$ and $\theta $ when ${\zeta / {2\pi }} = 0.08\textrm{MHz}$, respectively. Other parameters are the same as Fig. 3.

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Compared with a typical three-mode EOM system, the two-mechanical-mode EOM system can not only realize double-channel M-O conversion, but also control the conversion efficiency and added noise of each channel by modulating the strength and phase of mechanical coupling between MRs. Moreover, it’s due to the fact that there are three more tunable coupling parameters than a typical three-mode EOM system, the matching conditions of coupling rates are easier to reach. For the strong interference, the distance of double channels gets wider and may exceed the linewidth of microwave cavity.

4. Tunable entanglement and conversion in an N-mechanical-mode EOM system

In this section, we extend our scheme to investigate the tunable M-O entanglement and conversion in an N-mechanical-mode EOM system, where the nearest-neighboring mechanical modes are phase-dependent phonon-phonon coupled with each other and the microwave (optical) cavity mode is coupled to ${N_e}$ (${N_o},\textrm{ for }{N_e},{N_o} \le N$) mechanical modes simultaneously. Thus, the Hamiltonian in the interaction picture can be written as

5. Conclusion

In this work, we theoretically studied a two-mechanical-mode EOM system and investigated the enhanced M-O entanglement as well as coherent conversion. It’s demonstrated that the quantum interference of two electro-opto-mechanical interaction pathways leads to the tunable double-channel phenomena. For a symmetric system where two MRs are coupled to the cavities identically, we can steer the switch from the M-O interaction with a single channel to that of the double-channel by tuning the coupling phase of two MRs, and the distance of two channels in frequency domain is linearly related with the coupling strength. For an asymmetric system, besides the tuning of phonon-phonon interaction, the entanglement and conversion characteristics of each channel can be controlled via the mismatch of electromechanical and optomechanical couplings. Specifically, the performance on one channel could be enhanced by sacrificing that of the other channel. Therefore, we can always find the optimal parameter settings to realize the best entanglement or conversion. Moreover, the matching conditions of couplings are easier to reach due to the three more tunable coupling parameters compared with a typical three-mode EOM system. Finally, we extend our scheme to investigate the M-O entanglement and conversion in an N-mechanical-mode EOM system, which shows N tunable discrete channels by appropriate parameter settings. Additionally, our scheme could be generalized to other nonlinear interaction in multimode systems such as magnon-optics and atom ensembles. The proposed multimode EOM system opens up prospects for quantum information transduction and storage with a wide bandwidth and multichannel quantum interface.

 

Fig. 6. Entanglement and conversion in a three-mechanical-mode EOM system. (a) In the entanglement generation, the output optical fields ${P_o}$ as a function of $\delta $ when ${\zeta / {2\pi }} = \textrm{0}\textrm{.08MHz},{\theta _1} = {\pi / 2},{\theta _2} = 0$ (blue solid line) and ${\zeta / {2\pi }} = \textrm{0}\textrm{.04MHz},{\theta _1} = {{3\pi } / 4},{\theta _2} = 0$ (red solid line), respectively. (b) In the conversion generation, the conversion efficiency ${|{t(\omega )} |^2}$ as a function of $\delta $ when ${\zeta / {2\pi }} = \textrm{0}\textrm{.08MHz},{\theta _1} = {\pi / 2},{\theta _2} = 0$ (blue solid line) and ${\zeta / {2\pi }} = \textrm{0}\textrm{.04MHz},{\theta _1} = {{3\pi } / 4},{\theta _2} = 0$ (red solid line), respectively. Other parameters are the same as Fig. 3.

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Appendix

A. Expressions of ${A_j}$, $B$, ${D_j}$ ($j = e,o$)

By applying quantum Langevin equations to involved cavity modes and substituting several variables as Ref. [62] did, parameters ${A_j}$, B, ${D_j}$ ($j = e,o$) can be expressed in terms of cooperativity parameter ${C_{j1}}$ ($j = e,o$) as

(33)$${A_e} = \frac{{1 - ({C_{e1}} + {C_{o1}})}}{{1 + ({C_{e1}} - {C_{o1}})}},$$

(34)$${A_o} = \frac{{1 + ({C_{e1}} + {C_{o1}})}}{{1 + ({C_{e1}} - {C_{o1}})}},$$

(35)$$B = \frac{{2\sqrt {{C_{e1}}{C_{o1}}} }}{{1 + ({C_{e1}} - {C_{o1}})}},$$

(36)$${D_e} = \frac{{2i\sqrt {{C_{e1}}} }}{{1 + ({C_{e1}} - {C_{o1}})}},$$

(37)$${D_o} = \frac{{2i\sqrt {{C_{o1}}} }}{{1 + ({C_{e1}} - {C_{o1}})}}.$$

Considering the general case of two-mechanical-mode EOM system, whose Hamiltonian and multiphoton-phonon coupling rates are expressed by Eq. (5) and Eq. (18) in the main manuscript, the cooperativity parameters for ${\hat{b}_k}$ ($k ={+} , - $) reads as

(38)$${C_{jk}} = \frac{{G_{jk}^2}}{{{\kappa _j}}}.$$

Hence, ${A_{jk}}$, ${B_k}$, ${D_{jk}}$ can be rewrite in terms of ${C_{jk}}$ like Eq. (33–37).

B. Deductions of Eqs. (12)-(14)

The covariance matrixes of fluctuation operators are expressed as

(39)$$\left\langle {\delta \hat{a}_i^{\dagger} \delta {{\hat{a}}_j}} \right\rangle = {\bar{n}_i}{\delta _{ij}},$$

(40)$$\left\langle {\delta {{\hat{a}}_i}\delta \hat{a}_j^{\dagger} } \right\rangle = ({{{\bar{n}}_i} + 1} ){\delta _{ij}},$$

(41)$$\left\langle {\hat{\delta }{a_i}\delta {{\hat{a}}_j}} \right\rangle = \left\langle {\delta \hat{a}_i^{\dagger} \delta \hat{a}_j^{\dagger} } \right\rangle = 0,$$

where $\delta {\hat{a}_i},\delta {\hat{a}_j} = \delta {\hat{a}_{e,{\mathop{\rm int}} }},\delta {\hat{a}_{o,{\mathop{\rm int}} }},\delta \hat{m}$, ${\bar{n}_i} = {\bar{n}_{e,{T_E}}},{\bar{n}_{o,{T_E}}},{\bar{n}_{m,{T_E}}}$, and ${\delta _{ij}} = \left\{ \begin{array}{cc} 1,i = j\\ &0,i \ne j \end{array} \right.$, we have

(42)$$\begin{aligned} {{\bar{N}}_M} &= \left\langle {{{\hat{E}}^{\dagger} }\hat{E}} \right\rangle \\ &= \left\langle {({A_e^ \ast \delta \hat{a}_{e,{\mathop{\rm int}} }^{\dagger} - {B^ \ast }\delta {{\hat{a}}_{o,{\mathop{\rm int}} }} - C_e^ \ast \delta \hat{m}} )({A_e^{}\delta \hat{a}_{e,{\mathop{\rm int}} }^{} - B\delta \hat{a}_{o,{\mathop{\rm int}} }^{\dagger} - C_e^{}\delta \hat{m}} )} \right\rangle \\ &= {|{A_e^{}} |^2}\left\langle {\delta \hat{a}_{e,{\mathop{\rm int}} }^{\dagger} \delta \hat{a}_{e,{\mathop{\rm int}} }^{}} \right\rangle - A_e^ \ast B\left\langle {\delta \hat{a}_{e,{\mathop{\rm int}} }^{\dagger} \delta \hat{a}_{o,{\mathop{\rm int}} }^{\dagger} } \right\rangle - A_e^ \ast C_e^{}\left\langle {\delta \hat{a}_{e,{\mathop{\rm int}} }^{\dagger} \delta \hat{m}} \right\rangle \\ &- {B^ \ast }A_e^{}\left\langle {\delta {{\hat{a}}_{o,{\mathop{\rm int}} }}\delta {{\hat{a}}_{e,{\mathop{\rm int}} }}} \right\rangle + {|B |^2}\left\langle {\delta {{\hat{a}}_{o,{\mathop{\rm int}} }}\delta \hat{a}_{o,{\mathop{\rm int}} }^{\dagger} } \right\rangle + {B^ \ast }C_e^{}\left\langle {\delta {{\hat{a}}_{o,{\mathop{\rm int}} }}\delta \hat{m}} \right\rangle \\ &- C_e^ \ast A_e^{}\left\langle {\delta {{\hat{m}}^{\dagger} }\delta {{\hat{a}}_{e,{\mathop{\rm int}} }}} \right\rangle + C_e^ \ast B\left\langle {\delta {{\hat{m}}^{\dagger} }\delta \hat{a}_{o,{\mathop{\rm int}} }^{\dagger} } \right\rangle + {|{C_e^{}} |^2}\left\langle {\delta {{\hat{m}}^{\dagger} }\delta \hat{m}} \right\rangle \\ &= {|{A_e^{}} |^2}\left\langle {\delta \hat{a}_{e,{\mathop{\rm int}} }^{\dagger} \delta \hat{a}_{e,{\mathop{\rm int}} }^{}} \right\rangle + {|B |^2}\left\langle {\delta {{\hat{a}}_{o,{\mathop{\rm int}} }}\delta \hat{a}_{o,{\mathop{\rm int}} }^{\dagger} } \right\rangle + {|{C_e^{}} |^2}\left\langle {\delta {{\hat{m}}^{\dagger} }\delta \hat{m}} \right\rangle \\ &= {|{A_e^{}} |^2}{{\bar{n}}_{e,{T_E}}} + {|B |^2}({{{\bar{n}}_{o,{T_E}}} + 1} )+ {|{C_e^{}} |^2}{{\bar{n}}_{m,{T_E}}} \end{aligned}$$

Equation (42) yields Eq. (12), and Eqs. (13)-(14) can be yielded by similar operations.

C. Calculation details of the elements of $V(\omega )$

First, the quadrature operators of the microwave and optical signals can be denoted as ${\hat{X}_j},{\hat{Y}_j}$. We define the covariance matrix of our system by the quadratures of the microwave and optical cavities’ output in the frequency domain, which can be expressed as

(43)$$\delta (\omega + \omega ^{\prime}){V_{ij}}(\omega ) = \frac{1}{2}\left\langle {{u_i}(\omega ){u_j}(\omega^{\prime}) + {u_j}(\omega^{\prime}){u_i}(\omega )} \right\rangle, $$

with ${\mathbf u} = {[{\hat{X}_e},{\hat{Y}_e},{\hat{X}_o},{\hat{Y}_o}]^T}$. Then, by using Eqs. (9), (10) and (43), we write the covariance matrix in a normal form as

(44)$${\mathbf V}(\omega ) = \left( {\begin{array}{cccc} {{V_{11}}}&0&{{V_{13}}}&0\\ 0&{{V_{11}}}&0&{ - {V_{13}}}\\ {{V_{13}}}&0&{{V_{33}}}&0\\ 0&{ - {V_{13}}}&0&{{V_{33}}} \end{array}} \right), $$

where ${V_{11}}$ and ${V_{33}}$ represent the autocorrelation of microwave and optical signal, respectively, while ${V_{13}}$ corresponds to the cross-correlation. The elements of ${\mathbf V}(\omega )$ can be figured out by Eqs. (12)-(14) and expressed as

(45)$${V_{11}} = \frac{{\left\langle {{{\hat{X}}_e}(\omega ){{\hat{X}}_e}(\omega^{\prime})} \right\rangle }}{{\delta (\omega + \omega ^{\prime})}} = {\bar{N}_e} + \frac{1}{2}, $$

(46)$${V_{33}} = \frac{{\left\langle {{{\hat{X}}_o}(\omega ){{\hat{X}}_o}(\omega^{\prime})} \right\rangle }}{{\delta (\omega + \omega ^{\prime})}} = {\bar{N}_o} + \frac{1}{2}, $$

(47)$${V_{13}} = \frac{{\left\langle {{{\hat{X}}_e}(\omega ){{\hat{X}}_o}(\omega^{\prime}) + {{\hat{X}}_o}(\omega^{\prime}){{\hat{X}}_e}(\omega )} \right\rangle }}{{\delta (\omega + \omega ^{\prime})}} = \left\langle {\hat{c}_e^{\textrm{out}}\hat{c}_o^{\textrm{out}}} \right\rangle. $$

D. Logarithmic negativity as a function of coupling strength of MRs

Table 2. The normalized logarithmic negativity as a function of coupling strength of MRs.

Table 2. Normalized logarithmic negativity as a function of coupling strength of MRs.^a

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E. Expressions of $S$

By checking the Routh-Hurwitz stability conditions, we can know whether the system is stable. It’s expressed as

(48)$${\kappa _e}{C_{em}} - {\kappa _o}{C_{om}} > K\max \left( {{\kappa_e} - {\kappa_o},\frac{{{\kappa_o}^2 - {\kappa_e}^2}}{{2{\gamma_m} + {\kappa_e} + {\kappa_o}}}} \right)$$

with $K = \frac{{{C_{em}}}}{{1 + {\kappa _o}/{\kappa _e}}} + \frac{{{C_{om}}}}{{1 + {\kappa _e}/{\kappa _o}}}$. The stability ratio is defined as Eq. (D.2) for convenience.

(49)$$S = \frac{{{\kappa _e}{C_{em}} - {\kappa _o}{C_{om}}}}{{K\max \left( {{\kappa_e} - {\kappa_o},\frac{{{\kappa_o}^2 - {\kappa_e}^2}}{{2{\gamma_m} + {\kappa_e} + {\kappa_o}}}} \right)}}$$

The system is stable only if $S > 1$. $S = 1$ represents the boundary between stable region and unstable region.

Funding

National Natural Science Foundation of China (62073338).

Acknowledgments

This work is sponsored by the National Natural Science Foundation of China under Grant Nos. 62073338.

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.

References

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