Selective optomechanically-induced amplification with driven oscillators

TL;DR

This paper investigates how selective acoustic mode driving in an optomechanical system can amplify and control transparency and signal delay, enabling advanced optical modulation and sensing applications.

Contribution

It introduces a method for selective acoustic control in OMIT systems to achieve tunable amplification and signal delay effects.

Findings

Selective driving amplifies OMIT peaks.

Achieves transition from signal advance to delay.

Enables multi-band optical modulation.

Abstract

We study optomechanically-induced transparency (OMIT) in a compound system consisting of an optical cavity and an acoustic molecule, which features not only double OMIT peaks but also light advance. We find that by selectively driving one of the acoustic modes, OMIT peaks can be amplified either symmetrically or asymmetrically, accompanied by either significantly enhanced advance or a transition from advance to delay of the signal light. The sensitive impacts of the mechanical driving fields on the optical properties, including the signal transition and its high-order sidebands, are also revealed. Our results confirm that selective acoustic control of OMIT devices provides a versatile route to achieve multi-band optical modulations, weak-signal sensing, and coherent communications of light.

Tables

Table 1. Table 1: Experimental parameters of the mechanical resonators

Reference           Material         Mechanical frequency ${\omega }_1\left({\omega }_2\right)$     Damping rate ${\gamma }_1\left({\gamma }_2\right)$             Coupling form

L. Fan et al. Fan6        AlN        $6.87\mathrm{GHz}\left(456.5\mathrm{MHz}\right)$               $105.5\mathrm{kHz}\left(8.478\mathrm{kHz}\right)$            nonlinearly

Q. Lin et al. Lin4        ${\mathrm{Si}}_3{N}_4$ / ${\mathrm{SiO}}_2$       $8.3\mathrm{MHz}\left(13.6\mathrm{MHz}\right)$         $2.1\mathrm{MHz}\left(0.11\mathrm{MHz}\right)$          linearly

H. Okamoto et al. Yamaguchi9          GaAs              $1.845\mathrm{MHz}\left(1.848\mathrm{MHz}\right)$          $131.9\mathrm{Hz}\left(131.9\mathrm{Hz}\right)$            linearly

M. J. Weaver et al. weaver         ${\mathrm{Si}}_3{N}_4$              $297\mathrm{kHz}\left(659\mathrm{kHz}\right)$             $9.42\mathrm{Hz}\left(6.28\mathrm{Hz}\right)$                 linearly

Equations

$$H_{\text{coul}}$$

$${\mathcal{H}}_{\text{coul}}$$

$$\lambda =\frac{k_e{C}_1{V}_1{C}_2{V}_2}{r_0^3}\sqrt{\frac{\hslash }{m_1{m}_2{\omega }_{m,1}{\omega }_{m,2}}},$$

$$H$$

$$H_{\text{0}}$$

$$H_{\text{int}}$$

$$H_{\text{dr}}$$

$$+i\hslash {\epsilon }_p{c}^{†}{e}^{-i\xi t-i{\varphi }_{pl}}-\text{H}.\text{c}.,$$

$$\dot{c}$$

$${\dot{b}}_1$$

$${\dot{b}}_2$$

$$c_s$$

$$b_{1,s}$$

$$b_{2,s}$$

$$\delta c$$

$$\delta b_1$$

$$\delta b_2$$

$$h_1^{+}{A}_1^{-}$$

$$h_1^{-}{A}_1^{+\ast }$$

$$h_2^{+}{B}_1^{-}$$

$$h_2^{-}{B}_1^{+\ast }$$

$$h_3^{+}{D}_1^{-}$$

$$h_3^{-}{D}_1^{+\ast }$$

$$h_4^{+}{A}_2^{-}$$

$$h_4^{-}{A}_2^{+\ast }$$

$$h_5^{+}{B}_2^{-}$$

$$h_5^{-}{B}_2^{+\ast }$$

$$h_6^{+}{D}_2^{-}$$

$$h_6^{-}{D}_2^{+\ast }$$

$$h_1^{\pm }=$$

$$h_3^{\pm }=$$

$$h_5^{\pm }=$$

$$A_1^{-}=\left[\frac{(h_1^{-}{U}_1^{+}{U}_1^{-}+|G{|}^2\Pi ){\epsilon }_p}{h_1^{+}{h}_1^{-}{U}_1^{+}{U}_1^{-}+2i\Delta |G{|}^2\Pi }+\frac{iG{h}_1^{-}{h}_3^{+}{U}_1^{-}{\epsilon }_1{e}^{-i{\Phi }_1}}{h_1^{+}{h}_1^{-}{U}_1^{+}{U}_1^{-}+2i\Delta |G{|}^2\Pi }+\frac{G{h}_1^{-}{U}_1^{-}\lambda {\epsilon }_2{e}^{-i{\Phi }_2}}{h_1^{+}{h}_1^{-}{U}_1^{+}{U}_1^{-}+2i\Delta |G{|}^2\Pi }\right]e^{-i{\varphi }_{pl}},$$

$$A_2^{-}=\frac{-i\xi gG\Gamma -g{h}_4^{-}{U}_2^{+}{U}_2^{-}(h_1^{-}/G^{\ast })}{h_4^{+}{h}_4^{-}{U}_2^{+}{U}_2^{-}+2i\Delta {\left|G\right|}^2\Gamma }{A}_1^{-}{A}_1^{+\ast }$$

$${\Phi }_i=$$

$$U_1^{\pm }=$$

$$U_2^{\pm }=$$

$$c^{\text{out}}=c^{\text{in}}-\sqrt{{\eta }_c\kappa }{A}_1^{-},$$

$$|t({\omega }_p){|}^2={\left|1-\frac{{\eta }_c\kappa A_1^{-}}{{\epsilon }_p{e}^{-i{\varphi }_{pl}}}\right|}^2.$$

$${\left(\frac{{\epsilon }_1}{{\epsilon }_p}\right)}_{\text{TP}}$$

$$\alpha =2h_1^{+}{h}_1^{-}{h}_2^{+}{h}_2^{-}+|G{|}^2\kappa {\gamma }_1-4\Delta {\omega }_{m,1}|G{|}^2,$$

Full text

I Introduction

Cavity optomechanics (COM), focusing on the interplay of optical lasers and mechanical devices, provides unprecedented opportunities to explore both fundamental issues of quantum mechanics Aspelmeyer86 ; Metcalfe1 and practical quantum control of light and sound Bagci507 ; Teufel116 ; Korppi6 ; Gavartin7 ; Kampel7 ; Wollman349 ; Sillanpaa115 ; Lei117 ; Yamaguchi110 ; Grudinin104 ; Jing113 ; Jing8 ; ha1 ; ha2 . A prominent example, which is closely related to our present work, is optomechanically-induced transparency (OMIT) wwu5 ; Kronwald111 ; Agarwal81 ; Weis330 . Playing a key role in COM-based coherent control of light, OMIT has been experimentally demonstrated with microtoroid resonators Weis330 , diamond crystals diamond , microwave circuits Teufel471 , nanobeam or membrane devices Safavi472 ; Karuza88 , and nonlinear resonators Dong1 ; Shen41 . In recent works, more exotic properties of OMIT devices have been revealed, such as cascaded OMIT Fan6 , nonreciprocal OMIT Shen10 ; Fang13 ; Shen ; H5 , reversed OMIT Jing5 ; Jing014006 ; Huilai , vector OMIT yingv , nonlinear OMIT Xiong86 ; Jiao18 ; Liu111 , two-color OMIT Wang90 , and sub-Hertz OMIT arxiv10184 . These devices provide a powerful platform to realize, for examples, quantum memory Safavi472 ; Fiore107 ; Hill3 , signal sensing Zhang86 ; wang91 ; Xiong42 ; Xiong95 ; Xiong110 , and phononic engineering Guo90 ; Ojanen90 ; Liu58 ; Zhang92 .

Very recently, COM devices fabricated with optical dimers (i.e., coupled optical resonators) Tomita98 ; Grudinin104 ; zhangjing12 ; Xiaomin20 ; Peng101 or acoustic dimers Fan6 ; Yamaguchi9 ; weaver ; Lin4 ; Riedinger ; Ockeloen , have been utilized to achieve, for examples, COM-based phonon lasing Grudinin104 ; Jing113 ; Jing8 , unconventional photon blockade Li464 ; xiuminlin93 ; xiuminlin96 ; wu99 , and topological COM control zhangjing12 ; harris ; Yamaguchi1708 . In particular, by using multi-mode mechanical elements, experimentalists have demonstrated phonon-phonon entanglement Riedinger ; Ockeloen , two-mode phonon laser Yamaguchi110 , optomechanical Ising dynamics Yamaguchi2 , mechanical synchronization or multi-wave phonon mixing Lin4 ; Yamaguchi9 ; Colombano1810 , and coherent phonon transfer arxiv09369 . Appealing predictions for this system also include acoustic Josephson junctions BJJ , COM superradiance Agarwal ; wuu99 , parity-time symmetry acoustics XWXu92 , and phononic crystal shield arxiv00561 .

In this paper, we focus on the role of selective mechanical pump in OMIT with two coupled mechanical resonators (MRs). In experiments, this three-mode COM system has been demonstrated with a double-microdisk resonator, a zipper nanobeam photonic crystal, or a microwave device with two micromechanical beams Massel3 ; Lin4 . Strong mechanical driving has also been utilized to achieve hybrid quantum spin-phonon devices driving11 or ultra-strong exciton-phonon coupling driving9 . In the absence of the mechanical driving, such a system features double OMIT spectrum, i.e., the appearance of two symmetric transparent peaks around an absorption dip at the cavity resonance (which is otherwise a transparent peak for COM with a single mechanical oscillator Weis330 ; Safavi472 ). Here we find that, by selectively driving the mechanical resonators, the OMIT peaks and the accompanied optical group delays can be significantly altered. In comparison with the case of only a single driven oscillator Jia91 ; Xu92 ; liyong25 ; Si95 ; Suzuki92 ; JIANG ; wu5 , in our system, we can achieve symmetric or asymmetric suppressions or amplifications of double OMIT peaks, which is accompanied by either significantly enhanced advance or a transition from advance to delay of the signal light. Our results confirm that multi-mode OMIT devices with selective acoustic control, provide a versatile route to realize coherent multi-band modulations, switchable signal amplifications, and COM based light communications.

II THEORETICAL MODEL

We consider a three-mode COM system composed of an optical resonator with two MRs (see Fig. 1). The MR1 couples not only with the cavity field (via radiation pressure force), but also with the MR2 (through the position-position coupling). As shown in experiments, the position-position coupling can be realized by e.g., using a piezoelectric transducerYamaguchi9 , or applying an electrostatic force between the MRs Tian ; Hensinger . For two MRs coupled by the Coulomb force Tian ; Hensinger ; Ma90 , the interaction between them is written in the simple level as

[TABLE]

where $k_{e}$ is the electrostatic constant, $r_{0}$ is the equilibrium separation of the two charged oscillators in absence of any interaction between them, and $m_{i}$ $(i=1,2)$ or $q_{i}=C_{i}V_{i}$ is the effective mass or the charge of the MRi, with $C_{i}$ and $V_{i}$ being the capacitance and the voltage of the bias gate, respectively. $x_{i}$ $(i=1,2)$ is the small oscillations of the MRi from their equilibrium position. In the case of $r_{0}\gg\{x_{1},x_{2}\}$, with the second-order expansion, one can expand

[TABLE]

here the constant term and the linear term which can be absorbed into the definition of the equilibrium positions, and the quadratic term includes a renormalization of the oscillation frequency for the two MRs. Moreover, the small oscillations of the two mechanical oscillators from their equilibrium positions can be represented $x_{i}=\sqrt{\hbar/2m_{i}\omega_{m,i}}(b_{i}+b_{i}^{\dagger})$ $(i=1,2)$. Therefore, the effective Coulomb interaction can be simplified as

[TABLE]

here $\lambda$ is the Coulomb interaction strength

[TABLE]

for the typical experimental parameters $r_{0}=2\,\mathrm{mm}$, $C_{1}=C_{2}$ = 27.5$\,\mathrm{nF}$, and $V_{1}=V_{2}=1\,\mathrm{V}$ Tian ; Hensinger , we find $\lambda\simeq 0.1\,\mathrm{MHz}$. Table 1 shows more relevant parameters of experimentally achieved coupled MRs. The cavity is driven by a pump field and a weak probe field. Meanwhile, as also shown in experiments Fan6 ; Bochmann ; bowenli , mechanical driving fields with frequency $\omega_{i}$ and phase $\phi_{i}$ $(i=1,2)$ can be applied to selectively pump the MRs. We note that in the simplest two-mode COM (i.e., without the MR2), pumping the MR1 leads to a closed-loop $\Delta$-type energy-level structure [see Fig. 1(b)], under which optical properties of the system become highly sensitive to the mechanical pump parameters Jia91 ; Xu92 ; liyong25 ; Si95 ; Suzuki92 ; JIANG . In the presence of coupled two MRs, as shown in a recent experiment Fan6 , the effective phonon-phonon coupling also can be enhanced by the mechanical pump. Inspired by these works, here we show that by selectively driving the MRs, significantly different OMIT properties can be revealed, which offers flexible ways to control light in practice.

In the rotating frame at the pump frequency, the total Hamiltonian of the system can be written at the simplest level as

[TABLE]

where $c$ or $b_{i}$ $(i=1,2)$ is the annihilation operator of the cavity field with frequency $\omega_{c}$ or the MRi with frequency $\omega_{m,i}$, respectively. $\Delta_{c}=\omega_{c}-\omega_{l}$, $\phi_{pl}=\phi_{p}-\phi_{l}$, and $g$ denotes the optomechanical coupling coefficient. In the following, we focus on the features of OMIT by selectively driving the MRs, including the signal transmission, group delay, and its higher-order sidebands.

For this purpose, the equations of motion (EOM) of this system are

[TABLE]

$\kappa$ and $\gamma_{i}$ $(i=1,2)$ are the optical and mechanical decay rates, respectively. In the case of $\{\varepsilon_{p},\varepsilon_{i}\}\ll\varepsilon_{l}$, we express the dynamical variables as the sum of their steady-state values and small fluctuations, i.e., $c=c_{s}+\delta c$ and $b=b_{i,s}+\delta b_{i}$ $(i=1,2)$. The steady-state values of the dynamical variables are

[TABLE]

here $\Delta=\Delta_{c}-g(b_{1,s}^{\ast}+b_{1,s})$. To calculate the amplitudes of the first-order and second-order sidebands, we assume that the fluctuation terms $\delta a$ and $\delta b_{i}$ $(i=1,2)$ have the following forms Xiong86

[TABLE]

Substituting Eq. (8) into Eq. (6) leads to twelve equations. We can simplify these twelve equations into two groups Xiong86 : one group describes the process of the first-order sideband which corresponds to the linear case

[TABLE]

and the other group describes the the second-order sideband

[TABLE]

here $G=gc_{s}$ and

[TABLE]

By solving Eq. (9) and Eq. (10) leads to

[TABLE]

and

[TABLE]

here

[TABLE]

With these at hand, by using the input-output relation Collett

[TABLE]

where $c^{\textrm{in}}$ and $c^{\textrm{out}}$ are the input and output field operators, respectively, we can obtain the transmission rate of the probe as

[TABLE]

In Eq. (11), the first term is the contribution from the standard OMIT process due to the destructive interference of the probe absorption Weis330 ; Agarwal81 . The second and third term are the contribution from the phonon-photon parametric process Jia91 and the phonon-phonon parametric process Fan6 , induced by driving the MR1 and MR2. Clearly, these parametric process can modify and control the transmission of the signal field by adjusting the amplitude $\varepsilon_{i}$ and the photon-phonon mixing phase $\Phi_{i}$ $(i=1,2)$.

III RESULTS AND DISCUSSION

III.1 Linear OMIT spectrum

In our numerical simulations, to demonstrate that the observation of the signal transmission is within current experimental reach, we calculate Eq. (13) and (17) with parameters from Ref. Hill3 ; weaver ; Tian ; Hensinger ; Ma90 : $\omega_{m,i}/2\pi=947\,\mathrm{kHz}$ $(i=1,2)$, $m_{i}=145\,\mathrm{ng}$, $\gamma_{i}=\omega_{m,i}/Q$, $\kappa/2\pi=215\,\mathrm{kHz}$, $Q=6700$, $\lambda_{l}=1064\,\mathrm{nm}$, $L=25\,\mathrm{mm}$, $\lambda=0.1\,\mathrm{MHz},$ and $P_{L}=3\,\mathrm{mW}$. We have confirmed that for the pump power $P_{L}<7\,\mathrm{mW}$, single stable solution exists and the compound system has no bistability (see stability analysis in Appendix A).

Figure 2 shows the transmission rate $|t(\omega_{p})|^{2}$ as a function of the optical detuning $\Delta_{p}/\omega_{m}=(\xi-\omega_{m})/\omega_{m}$ and the phase $\Phi_{1}$. For comparisons, we first consider the single-mirror case ($\lambda=0$). As in standard COM system (without any mechanical driving), a standard single transparency window emerges around the resonance point $\Delta_{p}=0$ [see the blue solid line in Fig. 2(a)], as a result of the destructive interference of two absorption channels of the probe photons (by the cavity or by the phonon mode) Agarwal81 ; Weis330 . When a mechanical driving field is applied to the MR1, there are three coupling pathways of this system. The transitions $|m,n_{1},n_{2}\rangle$ $\leftrightarrow$ $|m+1,n_{1},n_{2}\rangle$, $|m,n_{1}+1,n_{2}\rangle$ $\leftrightarrow$ $|m+1,n_{1},n_{2}\rangle$, and $|m,n_{1},n_{2}\rangle$ $\leftrightarrow$ $|m,n_{1}+1,n_{2}\rangle$ can be achieved by applying a probe field, an optical pump field, and a mechanical pump field. Clearly, the three couplings result in a closed-loop $\Delta$-type transition strcture, leading to the phase-sensitive optical behaviors of the OMIT system Jia91 ; Xu92 ; liyong25 . As shown in Fig. 2, the transmission rate at $\Delta_{p}=0$ can be firstly suppressed and then amplified by increasing the mechanical driving strength due to the interference between the OMIT process and the phonon-photon parametric process [represented by the first and the second terms in Eq. (11), respectively]. By setting $|t(\omega_{p})|^{2}=0$, the turning point (TP) position turns out to be

[TABLE]

with

[TABLE]

which, for the parameter values chosen here, corresponds to $({\varepsilon_{1}}/{\varepsilon_{p}})_{\textrm{TP}}\simeq 0.45$.

For $\lambda\neq 0$, in the absence of any mechanical driving, double OMIT spectrum is known to appear in the two-mirror system [see the blue solid line in Fig. 2(b)], i.e., the purely mechanical coupling splits the original single-mirror OMIT peak into two wang91 ; Ma90 . Now we study the new features of double OMIT with switchable mechanical driving applied to either MR1 or MR2.

By driving the MR1, both effective optoemchanical coupling and phonon-phonon coupling can be enhanced  [27] and a closed-loop $\Delta$-type energy-level transitions configuration is formed in this system (for similar systems, see Refs. Jia91 ; Xu92 ; liyong25 ). This leads to symmetric suppressions ($\Phi_{1}=0$) or amplifications ($\Phi_{1}=\pi$) for both OMIT peaks [see Fig. 2(b)], with a resonant absorption dip at $\Delta_{p}=0$ [Fig. 2(b) and the blue dashed line in Fig. 2(d)]. In contrast, by driving the MR2, with the enhanced phonon-phonon coupling Fan6 , highly asymmetric Fano-like OMIT spectrum appears due to the competition between the OMIT process and the phonon-phonon coupling process, corresponding to the first and the third terms in Eq. (11), respectively, as shown in Fig. 2(c). The physics of these features can be explained as follows: In such a system, the MR1 couples not only with the cavity field, but also with the MR2. By driving the MR1, both effective optoemchanical coupling and phonon-phonon coupling can be enhanced (see e.g., Ref. Fan6 for similar results). However, by driving the MR2, only effective phonon-phonon coupling can be enhanced. Thus, as expected, asymmetric amplifications of the signal light can be achieved by selectively driving the MR2. These intuitive pictures agree well with our numerical results as shown in Fig. 2.

Interestingly, for single mirror case, the transmission of the probe light changes periodically with the phase $\Phi_{1}$ [see Fig. 3(a)]. For example, with $\varepsilon_{1}/\varepsilon_{p}=0.45$, the transmission rate changes from strong absorption to amplification by tuning the phase $\Phi_{1}$ from 0 to $\pi$. Also, for two mirrors case, periodic changes of the optical transmission rate can be found by varying the phase of the mechanical driving field [see Fig. 3(b) and Fig. 3(c)]. Hence, more flexible OMIT control of the signal light becomes accessible by selective driving the MRs, e.g., the signal can be completely blockaded or greatly amplified by driving the MR1 or MR2, at the resonance point $\Delta_{p}=0$. This ability of selectively switching and amplifying the input weak signal could be highly desirable in practical optical communications Jia91 ; Xu92 ; liyong25 ; Si95 ; Nunnenk .

III.2 Optical group delay

The group delay of the transmitted light is given by

[TABLE]

Accompanying with the standard single-mirror OMIT, slow light [see the blue dashed line in Fig. 4] can emerge due to the abnormal dispersion Safavi472 . In contrast, by introducing active gain into the system, fast light can be observed in experiments Jing5 ; Xu92 ; wanglijun . A merit of our system is the ability to selectively achieve either slow light or fast light by controlling the mechanical parameters. Figure 4(b) shows that by driving the MR1, significant enhancement of the light advance can be observed in comparison with the case without any mechanical pump [see the blue dashed line in Fig. 4(b)]. However, by driving the MR2, a tunable switch from fast to slow light can be achieved [see the blue solid line in Fig. 4(b)]. For $P_{L}=3.5\,\mathrm{mW}$, in comparison with the single-mirror system, $\sim 5$ times enhancement can be observed for the group delay by using the two-mirror device. This is useful for achieving a multi-functional amplifier with the extra ability to selectively tune the optical group velocities.

III.3 Nonlinear higher-order sidebands

As defined in Ref. Xiong86 , the efficiency of the second-order sideband process is

[TABLE]

Due to nonlinear optoemchanical interactions, in the OMIT process, output fields with frequencies $\omega_{l}\pm n\xi$ can emerge, where $n$ is an integer representing the order of the sidebands Xiong86 . The output fields with frequencies $\omega_{l}+2\xi$ is the second order upper sideband, while $\omega_{l}-2\xi$ is the lower sideband. In this work, we only consider the second-order upper sideband. For the single-mirror case, in the absence of any mechanical driving, the second-order sideband is subdued when the OMIT occurs, which results in a local minimum between the two sideband peaks around $\Delta_{p}=0$ Xiong86 . The efficiency of second-order sideband $\eta$ is, however, extremely small in conventional COM systems, e.g., $1\%-3\%$ Xiong86 .

As shown in Fig. 5, by driving the MR1, i.e., $\eta$ remains almost unchanged at the resonance [see Fig. 5(a) and the blue dashed line in Fig. 5(c)], which is similar to the linear OMIT spectrum [see the blue dashed line in Fig. 2(d)]. In contrast, by driving the MR2, giant enhancement of the second-order sideband can be observed at the resonance [see the red solid line in Fig. 5(c)]. For example, for $\varepsilon_{2}/\varepsilon_{p}=0.7$, the efficiency $\eta$ is about $25\%$ [see the purple solid line in Fig. 5(b)], which is in sharp contrast to the corresponding result $\eta\approx 0$ by driving MR1. This giant enhancement of second-order sidebands, with much narrower bandwidth, can be used in precision measurement of very weak signals, e.g., single-charge detections Xiong42 ; Xiong95 .

IV Conclusion

In conclusion, we have studied the mechanically controlled optical amplification and tunable group delay in a compound system composed of an optical resonator and two coupled mechanical resonators. We find that by driving one of the mechanical modes, both OMIT peaks can be symmetrically suppressed or amplified, which is accompanied by significantly enhanced light advance. In contrast, by driving the other mechanical mode, the OMIT spectrum becomes highly asymmetric, accompanied by a transition from fast light to slow light. In addition, periodic changes of both the linear OMIT spectrum and the higher-order sidebands can be observed by tuning the phases of the mechanical driving fields. These features of selective OMIT amplifications and switchable group delays of light provide more flexible ways in practical applications ranging from optical storage or modulations to multi-band optical communications. In future works, it will be also of interests to study the role of selective mechanical driving in enhancing or steering, for examples, light-sound entanglement Vitali98 ; Nunnenkamp107 , photon-phonon mutual blockade Rempe14 , precision measurement Xiong42 ; Xiong95 , and switchable amplification of light or sound.

Note added. After completing this work, we became aware of a preprint also on OMIT utilizing an acoustic dimer, but with only a fixed mechanical pump arxiv1812 .

V ACKNOWLEDGMENTS

This work is supported by the National Natural Science Foundation of China (NSFC) under Grants No. 11474087 and No. 11774086, and the HuNU Program for Talented Youth.

Appendix A Stability analysis

Considering photon damping and the Brownian noise from the cavity and the environment, the EOM are given by

[TABLE]

where $c_{\textrm{in}}\left(t\right)$ is the input noise operator with zero mean value, and $\xi_{i}\left(t\right)$ $(i=1,2)$ is the Brownian noise operators associated with the damping of the MRi. Under the Markov approximation, two-time correlation functions of these input noise operators are

[TABLE]

here $n_{\textrm{th}}=\left(e^{\hbar\omega/k_{B}T}-1\right)^{-1}$, with $k_{B}$ is the Boltzmann constant and $T$ is the bath temperature. By setting all the time derivatives to zero of Eq. (18), the steady-state value of $c$ is

[TABLE]

where $\Delta=\Delta_{c}-g(b_{1,s}^{{\dagger}}+b_{1,s})$ is the effective detuning, including the effects of radiation pressure and Coulomb interaction. We now study the steady-state behavior of the mean photon number $\left|c_{s}\right|^{2}$ . In this case, using Eq. (20), it is straightforward to show that $\left|c_{s}\right|^{2}$ satisfies

[TABLE]

We provide a direct and efficient estimation on how many positive solutions exist in Eq. (21) according to the Descartes rule. Eq. (21) can be recast as

[TABLE]

where we define $x=\left|c_{s}\right|^{2}$, and the coefficients are

[TABLE]

with

[TABLE]

here all parameters $g$, $\kappa$, $\lambda$, $\omega_{m,1}$, $\omega_{m,2}$, $\gamma_{1}$, $\gamma_{2}$, and $\varepsilon_{l}$ in Eq. (23) are positive, we have $a_{0}<0$, $a_{1}>0$, $a_{2}<0$ and $a_{3}>0$, corresponding to the following unique sign sequence:

[TABLE]

According to the Descartes rule, Eq. (22) has three real solutions at most, two of which are dynamically stable. We also have checked numerically that the parameters we chosen in this paper satisfy the stability condition. When the cavity is driven on its red sideband, Figure 6 shows the mean intracavity photon number $\left|c_{s}\right|^{2}$ as a function of pump power $P_{L}$ with $\lambda=0.1\,\mathrm{MHz}$. It can be seen that the mean photon number exhibits the standard S-shaped bistability. As the pump power $P_{L}$ increases from zero, there is only single stable solution of Eq. (22) at the beginning. However, when $P_{L}$ is larger than a critical value, there are three real solutions. The largest and smallest solutions are stable, and the middle one is unstable.

Below we determine the stability of the steady states of our system using the Routh-Hurwitz criterion RH . The fluctuation terms of the EOM are

[TABLE]

here $G=gc_{s}$, In a compact matrix form, Eq. (26) can be recast as

[TABLE]

where vectors $\textbf{u}=(\delta c,\delta c^{{\dagger}},\delta b_{1},\delta b_{1}^{{\dagger}},\delta b_{2},\delta b_{2}^{{\dagger}})^{\textrm{T}}$ and $\delta\textbf{v}_{\text{in}}=\sqrt{2}(\sqrt{\kappa}\,\delta\hat{c}_{in},\sqrt{\kappa}\delta\hat{c}_{in}{}^{{\dagger}},\sqrt{\gamma_{1}}\,\delta\xi_{1},\sqrt{\gamma_{1}}\delta\xi_{1}^{{\dagger}},\sqrt{\gamma_{2}}\,\delta\xi_{2},\sqrt{\gamma_{2}}\delta\xi_{2}^{{\dagger}})^{\text{T}}$, in which T denotes the transpose of a matrix. The matrix C is given by

[TABLE]

The characteristic equation $\left|\textbf{C}-\Upsilon\textbf{I}\right|=0$ can be reduced to $\Upsilon^{6}+C_{1}\Upsilon^{5}+C_{2}\Upsilon^{4}+C_{3}\Upsilon^{3}+C_{4}\Upsilon^{2}+C_{5}\Upsilon+C_{6}=0,$ where the coefficients can be derived using straightforward but tedious algebra. From the Routh-Hurwitz criterion RH , a solution is stable only if the real part of the corresponding eigenvalue $\Upsilon$ is negative and the stability conditions can then be obtained as

[TABLE]

Through these analyses, we have confirmed that the experimentally accessible parameters in the main manuscript can keep the compound system in a stable zone.