Abstract

Tunable high-order sideband generation has important applications in the realization of the optical frequency comb with a varying spectral region (corresponding to the sideband range) and frequency resolution (corresponding to the sideband interval). In this paper, we propose a theoretical scheme to tune both the range and the interval of the high-order sidebands in a coupled double-cavity optomechanical system, which consists of an optomechanical cavity and an auxiliary cavity. Our proposal can be realized by driving the optomechanical cavity with a control field and a probe field simultaneously, driving the auxiliary cavity with a pump field. Furthermore, we assume that the frequency detuning between the control field and the probe field (the pump field) equals ωb/n (ωb/m), where ωb is the mechanical frequency, m and n are integers. When n = m = 1, we find that the sideband range can be effectively enlarged by increasing the pump amplitude or the photon-hopping coupling rate, or by decreasing the auxiliary cavity damping rate. When n = 1 and m > 1, the output spectrum consists of a series of integer-order sidebands, fraction-order sidebands, and the sum and difference sidebands, and the sideband interval becomes ωb/m and can be diminished by simultaneously increasing m and the pump amplitude.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Optomechanics investigates the radiation-pressure induced interaction between light and mechanical oscillation [1,2]. In general, such a kind of optomechanical interaction is intrinsic nonlinear and it is very hard to give an exact analytical solution [3]. However, by using the linearization approximation, great progresses have been made in this research area over the past few decades, and a lot of effective and fruitful works based on the linearized optomechanical systems have been reported, for instance, generation of optomechanically induced transparency [46], normal mode splitting [7,8], fast and slow light [911], optomechanical dark state [6,12,13], cooling of mechanical oscillator to the ground state of motion [1419], frequency conversion between light and microwave [2022], generation of quantum entanglement and squeezing of light and mechanical resonator [2328], and so on. All of these effects also play important roles in many application fields, such as gravitational-wave detection [29,30], precision measurement [3134], quantum computation and communication [3537], as well as the fundamental test of quantum mechanics [38,39].

Recently, more attentions have been attracted into the nonlinear behaviors of optomechanical systems. Studying the nonlinear interactions enables us a deeper and broader understanding of optomechanics. Many novel nonlinear phenomena have been predicted and observed, for example, second-order sideband generation [4042], sum and difference sideband generation [4347], optomechanical chaos [48,49]. Besides those, an important effect named high-order sideband generation (HSG) is of great interest for its potential application in the realization of optical frequency comb (OFC) [50,51]. As we know, OFC is the most accurate "ruler" on the earth and has important applications in many engineering fields [5254]. To support a broader application space, OFCs have experienced rapid changes over the past twenty years to enable the coverage at different spectral regions, and variable frequency resolutions [55]. Hence it is quite important to achieve the tunable high-order sideband generation. In previous works, people have demonstrated that many physical effects and systems, such as photonic molecule optomechanical system [56], Coulomb effect [57], atomic ensemble [58], Casimir effect [59], two-cavity optomechanical system with modulated photon-hopping interaction [60], and parity-time symmetry structure [61], can be used to enlarge the frequency range of high-order sideband. However, the research on tuning another important factor, i.e., the frequency interval of high-order sideband, is still few. More recently, we propose a scheme to generate a new kind of sideband, i.e., the fraction-order sideband, which can be used to diminish the sideband interval [62].

In this paper, we present a proposal to tune both the range and interval of the high-order sidebands output from an optomechanical system by using a pumped auxiliary cavity. Our system model is shown in Fig. 1, an optomechanical microtoroid cavity is simultaneously driven by a control field ($\varepsilon _{co}$, $\omega _{co}$) and a probe field ($\varepsilon _{pr}$, $\omega _{pr}$), the auxiliary cavity is driven by a pump field ($\varepsilon _{pu}$, $\omega _{pu}$). The frequency detuning between the control field and the probe field (pump field) is $\delta _{pr}=\omega _{b}/n$ ($\delta _{pu}=\omega _{b}/m$), in which $\omega _{b}$ is the mechanical frequency, $m$ and $n$ are integers. By setting $m=n=1$, we show that the sideband range can be effectively enlarged by increasing the pump amplitude $\varepsilon _{pu}$ or the photon-hopping coupling rate $G$, or by decreasing the auxiliary cavity damping rate $\eta$. When $n=1$ and $m >1$, the output spectrum consists of a series of integer-order sidebands, fraction-order sidebands, as well as the sum and difference sidebands between the integer- and fraction-order sidebands. Moreover, the sideband interval becomes $\omega _{b}/m$ and can be diminished by increasing $m$ and the pump amplitude $\varepsilon _{pu}$.

 figure: Fig. 1.

Fig. 1. The schematic diagram of our proposed system model, which consists of an optomechanical microtoroid cavity and an auxiliary cavity. The driving fields couple with the cavity fields by the optical waveguides.

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The paper is organized as follows. In Section 2., we present the Hamiltonian of our system and give the derivation of the Heisenberg-Langevin equations. In Section 3., we give the theoretical analyses and numerical methods of the high-order sideband generation. In Section 4 and Section 5., we exhibit how to enlarge the sideband range and to diminish the sideband interval by using the pumped auxiliary cavity, respectively. Finally, a brief conclusion is summarized in Section 6.

2. Hamiltonian and Heisenberg-Langevin equations

As shown in Fig. 1, we consider an optomechanical microtoroid cavity where the cavity mode $a$ couples with the mechanical mode $b$ via radiation pressure. The auxiliary cavity mode $c$ couples with the optomechanical cavity mode $a$ via photon-hopping interaction. The optomechanical cavity is driven by a control field $\varepsilon _{co}$ and a probe field $\varepsilon _{pr}$. In addition, the auxiliary cavity is pumped by a laser field $\varepsilon _{pu}$. Hence we can write the total Hamiltonian as ($\hbar =1$)

$$\begin{aligned} H=&\ \omega_{a}a^{\dagger }a+\omega _{b}b^{\dagger }b+\omega_{c}c^{\dagger }c+ga^{\dagger }a(b^{\dagger }+b)\\ &+G(a^{{\dagger}}c+c^{{\dagger}}a)+i(c^{{\dagger}}\varepsilon_{pu}e^{{-}i\omega_{pu}t}-H.c.)\\ &+i[\hat{a}^{{\dagger}}(\varepsilon_{co}e^{{-}i\omega_{co}t}+\varepsilon_{pr}e^{{-}i\omega_{pr}t})-H.c.], \end{aligned}$$
where $a$ ($c$) and $b$ are the bosonic annihilation operators for the optomechanical (auxiliary) cavity mode and the mechanical mode with frequencies $\omega _{a}$ ($\omega _{c}$) and $\omega _{b}$, respectively. $g$ $=$ $x_{zpf} \omega _{a}/L$ is the single-photon optomechanical coupling strength where $x_{zpf}=\sqrt {\hbar /2M\omega _{b}}$ is the zero-point fluctuation, $M$ is the effective mass of the mechanical mode and $L$ is the length of the cavity. $G$ represents the photon-hopping coupling rate and can be efficiently tuned by changing the distance between the two cavities. The amplitudes of the driving fields are $\varepsilon _{x}=\sqrt {2\kappa P_{x}/\hbar \omega _{x}}$ ($x=co, pr$) and $\varepsilon _{pu}=\sqrt {2\eta P_{pu}/\hbar \omega _{pu}}$, respectively, in which $\kappa$ ($\eta$) denotes the optomechanical (auxiliary) cavity damping rate and $P_{y}$ ($y=co, pr, pu$) refers to the corresponding input power.

In this paper, we focus on the mean response of the system to the input laser fields. In the semi-classical limit, the system operators are reduced to their expectation values, i.e., $\left \langle a\right \rangle \equiv \alpha$, $\langle b\rangle \equiv \beta$, $\left \langle c\right \rangle \equiv \chi$, and the quantum and thermal noises can be dropped because their expectation values are zero. By using the mean-field approximation, i.e., $\langle ab\rangle =\left \langle a\right \rangle \langle b\rangle$, the evolution of the system can be described by the following Heisenberg-Langevin equations (in a frame rotating at the frequency $\omega _{co}$):

$$\begin{aligned} \frac{d\alpha}{dt}=&-(i\Delta _{a}+\kappa )\alpha-ig\alpha(\beta+\beta^{*})\\ &-iG\chi +\varepsilon _{co}+\varepsilon_{pr}e^{{-}i\delta_{pr}t}, \end{aligned}$$
$$\frac{d\chi}{dt}=-(i\Delta _{c}+\eta )\chi-iG\alpha+\varepsilon_{pu}e^{{-}i\delta_{pu}t},$$
$$\frac{d\beta}{dt}=-(i\omega_{b}+\gamma)\beta-ig\left|\alpha \right|^{2},$$
where $\Delta _{a}$ $=$ $\omega _{a}-\omega _{co}$ ($\Delta _{c}$ $=$ $\omega _{c}-\omega _{co}$) is the frequency detuning between the control field and the optomechanical (auxiliary) cavity field. $\delta _{pr}$ $=$ $\omega _{pr}-\omega _{co}$ ($\delta _{pu}$ $=$ $\omega _{pu}-\omega _{co}$) is the frequency detuning between the control field and the probe (pump) field. Moreover, $\gamma$ is the damping rate of the mechanical mode.

3. Solving methods

The governing Eqs. (2)–(4) of the system are nonlinear due to the coupling between the optomechanical cavity mode and mechanical mode (corresponding to the terms $-ig\left |\alpha \right |^{2}$ and $-ig\alpha (\beta ^{* }+\beta )$), and it is very difficult to obtain an analytical solution. In previous works, the common method is to use the linearization approximation. When the probe field and pump field are much weaker than the control field (i.e., $\varepsilon _{pr}/\varepsilon _{co}\ll 1, \varepsilon _{pu}/\varepsilon _{co}\ll 1$), and the two frequency detunings satisfy $\delta _{pr}=\delta _{pu}=\omega _{b}$, the expectation value $X$ ($X=\alpha ,\beta , \chi$) can be written as $X=X_{0}+X_{+}^{(1)}\exp {(-i\omega _{b} t)}+X_{-}^{(1)}\exp {(i\omega _{b} t)}$. By using this linearization method, some important physical effects, such as ground-state cooling of the mechanical resonator [63], three-pathway electromagnetically induced transparency [64], quantum backaction and noise interference [65] have been demonstrated in the similar two-cavity optomechanical systems.

In the presented paper, we care about the high-order sideband generation when the probe field and pump field are stronger than the control field (i.e., $\varepsilon _{pr}> \varepsilon _{co}$, $\varepsilon _{pu}> \varepsilon _{co}$), and the frequency detunings satisfy $\delta _{pr}=\omega _{b}/n$, $\delta _{pu}=\omega _{b}/m$ ($m$ and $n$ are integers). In this regime, the linearization method is not applicable, and the optomechanical nonlinearity induced high-order terms must be taken into account. Different from the previous works [50,51,5661], here the expectation value $X$ is written as the following ansatz

$$X=X_{0}+X_{pr}+ X_{pu}+X_{sum}+ X_{dif},$$
where $X_{0}$ represents the steady-state solution when there is no probe field and pump field (i.e., $\varepsilon _{pr}=\varepsilon _{pu}=0$). Due to the optomechanical nonlinearity, many new photons with different frequencies will be generated in the system, the driving energy will be transferred to a series of sidebands, which can be expressed as
$$X_{pr}=\sum_{j=1}^{\mathcal{J}}X_{pr+}^{(j)}e^{{-}ij\frac{\omega_{b}}{n}t}+X_{pr-}^{(j)}e^{ij\frac{\omega_{b}}{n}t},$$
$$X_{pu}=\sum_{k=1}^{\mathcal{K}}X_{pu+}^{(k)}e^{{-}ik\frac{\omega_{b}}{m}t}+X_{pu-}^{(k)}e^{ik\frac{\omega_{b}}{m}t},$$
where $j$ and $k$ are positive integers. Furthermore, a series of sum and difference sidebands will be generated, which can be written as
$$X_{sum}=\sum_{j=1}^{\mathcal{J}}\sum_{k=1}^{\mathcal{K}}X_{sum+}^{(j,k)}e^{{-}i(\frac{j}{n}+\frac{k}{m})\omega_{b}t}+X_{sum-}^{(j,k)}e^{i(\frac{j}{n}+\frac{k}{m})\omega_{b}t},$$
$$X_{dif}=\sum_{j=1}^{\mathcal{J}}\sum_{k=1}^{\mathcal{K}}X_{dif+}^{(j,k)}e^{{-}i(\frac{j}{n}-\frac{k}{m})\omega_{b}t}+X_{dif-}^{(j,k)}e^{i(\frac{j}{n}-\frac{k}{m})\omega_{b}t}.$$

By substituting the ansatz Eqs. (5)–(9) into the governing Eqs. (2)–(4) and using the input-output relation [66] $S_{out}=S_{in}-\sqrt {2\kappa }\alpha$, where $S_{in}=\varepsilon _{co}+\varepsilon _{pr}e^{-i\delta _{pr}t}$, we can finally obtain the sideband spectrum output from the optomechanical system.

From the above theoretical analyses, we can see that it is very hard to give the solutions of the sidebands for every orders. To exhibit all the generated sidebands, we can use the Runge-Kutta method [50] to solve the differential Eqs. (2)–(4), and the output spectrum can be obtained by using the fast Fourier transform (FFT). The parameters we adopted are chosen as [67,68]: $\omega _{b}/2\pi =10$ MHz and $\gamma /2\pi =100$ Hz, $M=10$ ng, $\Delta _{a}=\Delta _{c}=\omega _{b}$, $\kappa /2\pi =1$ MHz, the wavelength of the control field $\lambda _{co}=795$ nm.

4. Enlarging the sideband range by using the pumped auxiliary cavity

In this section, we will show the remarkable impact of the auxiliary cavity on the sideband spectrum output from the optomechanical cavity, and we assume that the two detunings satisfy: $\delta _{pr}=\omega _{b}$, $\delta _{pu}=\omega _{b}$ (i.e., $n=m=1$). Under these conditions the output spectrum can be expressed as

$$S_{out}=S_{0}+\sum_{l=1}^{\mathcal{L}}S_{-}^{(l)}e^{{-}il\omega_{b}t}+S_{-}^{(l)}e^{il\omega_{b}t},$$
where $S_{0}=\varepsilon _{co}-\sqrt {2\kappa }\alpha _{0}$ is the zeroth-order sideband which corresponds to the control field $\omega _{co}$. Moreover, we will see a series of integer-order sidebands appearing at $\omega =\pm l\omega _{b}$ ($l$ is a positive integer and represents the order of sideband), the amplitudes can be obtained from the governing equations and the input-output relation. For instance, the 1st-order sideband $S_{-}^{(1)}=\varepsilon _{pr}-\sqrt {2\kappa }(\alpha _{pr+}^{(1)}+\alpha _{pu+}^{(1)}+\alpha _{d+}^{(2,1)}+\dots )$. By using the numerical computation, the amplitudes of every order of sidebands can be clearly exhibited in the output spectra. Before showing the numerical result, we need to point out that the spectra obtained have shifted the frequency $\omega _{co}$, since the Heisenberg-Langevin Eqs. (2)–(4) describes the time evolution of the system in a frame rotating at the frequency of the control field.

As a contrast, let us firstly show the influence of the non-pumped auxiliary cavity on the sideband generation of the optomechanical cavity. In Fig. 2, we plot the output spectra for different photon-hopping coupling rate $G$. It can be seen that when $G=0$, a series of integer-order sidebands will appear in the output spectrum, as Fig. 2(a) shows, the positive and negative sidebands end up at the orders of 7th and −6th, respectively. By increasing $G$, the range and amplitude of the sidebands are both decreasing rapidly. When $G=10\kappa$, only the $-2$nd-order to 3rd-order sidebands can be obviously seen. It should be noted that the similar results have also been exhibited in some previous works [56,60]. Here we give the physical interpretations: the non-pumped auxiliary cavity can be actually regarded as a bare resonator, the driving energy will flow from the optomechanical cavity to the auxiliary cavity after they have interaction. With the increase of $G$, more photons will be transferred to the auxiliary cavity, which results in less photons output from the optomechanical cavity. Therefore, the influence of the non-pumped auxiliary cavity is just weakening the sideband generation and reducing the sideband range.

 figure: Fig. 2.

Fig. 2. The integer-order sideband spectra output from the optomechanical system with the non-pumped auxiliary cavity for different $G$: (a) $G=0$; (b) $G=6\kappa$; (c) $G=10\kappa$. The other parameters are: $\varepsilon _{co}=0.1$ THz, $\varepsilon _{pr}/\varepsilon _{co}=1$, $\varepsilon _{pu}/\varepsilon _{co}=0$, $\eta /\kappa =1$, and the rest parameters are stated in the text.

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Below we demonstrate the effects of the pumped auxiliary cavity on the sideband generation for some important parameters, including the pump amplitude $\varepsilon _{pu}$, the photon-hopping coupling rate $G$, and the auxiliary cavity damping rate $\eta$. Here we note that the above parameters will change the sideband spectra in a same way at the positive frequency and negative frequency parts. For simplicity, we only show and discuss the positive frequency parts in the following.

Figure 3(a) displays the output spectra for different $\varepsilon _{pu}$. It is shown that the cutoff-order number and the amplitudes of the sidebands can both be significantly enhanced by increasing $\varepsilon _{pu}$. When $\varepsilon _{pu}/\varepsilon _{co}=0.5$, the order of the sideband visibly ends up at the order of $7$th (as the green circles show). With the increase of $\varepsilon _{pu}$, more photons will be injected to the auxiliary cavity and be transferred to the optomechanical cavity, when $\varepsilon _{pu}/\varepsilon _{co}=5$, the cutoff-order number is raised to 10th and the amplitude of each sideband is also distinctly enhanced. Moreover, for higher order of the sideband, the amplitude is smaller. To further enhance the sideband generation, we continue to increase $\varepsilon _{pu}=30\varepsilon _{co}$, and the cutoff-order number is raised to 19th. In addition, the output spectrum expresses some new features, for example, the amplitude of 4th-order sideband is larger than that of 3rd-order sideband, a plateau appears (from 2nd-order to 10th-order) where all the sidebands have almost the same amplitudes.

 figure: Fig. 3.

Fig. 3. The positions and amplitudes of the integer-order sidebands output from the optomechanical system with pumped auxiliary cavity for: (a) different $\varepsilon _{pu}$, $G/\kappa =1$; (b) different $G$, $\varepsilon _{pu}/\varepsilon _{co}=10$; (c) different $\eta$, $G/\kappa =1$ and $\varepsilon _{pu}/\varepsilon _{co}=10$. The other parameters are the same as those in Fig. (2).

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In Fig. 3(b), we plot the output spectra for different $G$ when there exists the pump field $\varepsilon _{pu}$. The cutoff-order number and the amplitudes of the sidebands will both increase with $G$. When $G=0.1\kappa$, there are only eight sidebands in the output spectrum. If we increase $G$ to $0.6\kappa$, the cutoff-order number can be raised to $10$th and the corresponding amplitudes also become larger. By further increasing $G$ and when $G=3\kappa$, the sidebands will end up at the order of $14$th. It can be seen that the effect of the photon-hopping coupling on the sideband generation is just the opposite compared with that in Fig. 2. The physical picture of such a result can be explained as follows: when the auxiliary cavity and the optomechanical cavity have the same damping rates ($\eta =\kappa$), but the auxiliary cavity is more strongly driven than the optomechanical cavity (i.e., $\varepsilon _{pu}>\varepsilon _{pr}$), the auxiliary cavity will contain more photons than the optomechanical cavity. After their interaction, the photons will flow from the more-photon cavity to the less-photon cavity, hence there is a net-photon flow from the auxiliary cavity to the optomechanical cavity. By increasing $G$, more photons will be transferred to the optomechanical cavity, and consequently the output sideband spectra will get enhanced.

In what follows, we discuss the effect of the auxiliary cavity damping on the sideband generation, and we plot the output spectra for different $\eta$ in Fig. 3(c). It can be evidently seen that the cutoff-order number and the amplitude will increase with the decrease of $\eta$. When $\eta /\kappa =20$, there are only seven sidebands in the output spectrum (as the gray pentagons show). When we decrease $\eta$, less photons can leak out from the auxiliary cavity, and consequently more photons will be transferred to the optomechanical cavity. When $\eta /\kappa =1$, the output sidebands end up at the order of 10th (as the green circles show). When $\eta /\kappa =0.2$, the cutoff-order number is drastically raised to 29th (as the orange rectangles show). Moreover, a wide-range plateau appears (from 2nd-order to 15th-order).

5. Diminishing the sideband interval by using the pumped auxiliary cavity

From the above discussions, we have shown that the range of the sideband spectrum output from the optomechanical system can be effectively enlarged by using the pumped auxiliary cavity. However, the sideband interval keeps unchanged and equals the mechanical frequency $\omega _{b}$, which is usually fixed in a practical optomechanical system. In other words, the sideband interval is untunable and has a minimum frequency limitation $\omega _{b}$, and this will limit the precision of the sideband comb. Next we will show that how to diminish the sideband interval by using the pumped auxiliary cavity. We set the detuning conditions as $\delta _{pr}=\omega _{b}$, $\delta _{pu}=\omega _{b}/m$ ($n=1$, $m>1$), so the role of the probe field is to generate the integer-order sidebands, $m$ can be used to tune the sideband interval, the role of the pump field is to generate the fraction-order sidebands. Moreover, the sum and difference sidebands between the integer-order and fraction-order sidebands can also be generated. Hence, the output spectrum becomes

$$S_{out}=S_{0}+S_{pr}+ S_{pu}+S_{sum}+S_{dif},$$
in which the first term $S_{0}$ is the zeroth-order sideband. The second and third terms represent a series of integer-order sidebands and fraction-order sidebands, respectively, which are given by
$$S_{pr}=\sum_{j=1}^{\mathcal{J}}S_{pr+}^{(j)}e^{{-}ij\omega_{b}t}+S_{pr-}^{(j)}e^{ij\omega_{b}t},$$
$$S_{pu}=\sum_{k=1}^{\mathcal{K}}S_{pu+}^{(k)}e^{{-}i\frac{k}{m}\omega_{b}t}+S_{pu-}^{(k)}e^{i\frac{k}{m}\omega_{b}t},$$
where $S_{pr\pm }^{(j)}$ and $S_{pu\pm }^{(k)}$ are called as the $\pm j$th-order and $\pm \frac {k}{m}$th-order sideband, respectively. The $1$st-order and $\frac {1}{m}$th-order sidebands are in the same colors with the probe and pump field, respectively. In addition, the last two terms in Eq. (11) denote the output sum and difference sidebands, respectively, which are expressed as
$$S_{sum}=\sum_{j=1}^{\mathcal{J}}\sum_{k=1}^{\mathcal{K}}S_{sum+}^{(j,k)}e^{{-}i(j+\frac{k}{m})\omega_{b}t}+S_{sum-}^{(j,k)}e^{i(j+\frac{k}{m})\omega_{b}t},$$
$$S_{dif}=\sum_{j=1}^{\mathcal{J}}\sum_{k=1}^{\mathcal{K}}S_{dif+}^{(j,k)}e^{{-}i(j-\frac{k}{m})\omega_{b}t}+S_{dif-}^{(j,k)}e^{i(j-\frac{k}{m})\omega_{b}t},$$
and we will see a lot of new fraction-order sidebands appear at $\omega =\pm (j\pm \frac {k}{m})\omega _{b}$ in the output spectrum, in which we call $S_{s\pm }^{(j,k)}$ and $S_{d\pm }^{(j,k)}$ as the $\pm \frac {jm+ k}{m}$th-order and $\pm \frac {jm- k}{m}$th-order sideband, respectively.

Figure 4 exhibits the numerical result of the sideband spectrum output from the optomechanical system with the pumped auxiliary cavity when $\delta _{pr}=\omega _{b}$, $\delta _{pu}=\omega _{b}/5$ (i.e., $n=1$, $m=5$). It can be seen that, the sideband generation is quite different from that in section 4. We can clearly see some integer-order sidebands (as the orange rectangles show), for example, the $1$st-order, $5$th-order sideband, and so on. Besides those, there are two fraction-order sidebands (the $\frac {1}{5}$th-order and $\frac {2}{5}$th-order sidebands) appear in the output spectrum (as the purple stars show). In addition, the sum sidebands between the fraction-order sidebands and the integer-order sidebands come out (as the blue triangles show), such as the $\frac {6}{5}$th-order, $\frac {7}{5}$th-order, $\frac {12}{5}$th-order, $\frac {26}{5}$th-order sidebands, and so on. Furthermore, the different sidebands between the fraction-order sidebands and the integer-order sidebands is also shown in the output spectrum, such as the $\frac {4}{5}$th-order, $\frac {9}{5}$th-order, $\frac {24}{5}$th-order sidebands, and so on. We point out that we only exhibit the sideband spectrum in the positive frequency part, and that is also similar in the negative frequency part.

 figure: Fig. 4.

Fig. 4. The integer-order, fraction-order, sum, and difference sidebands output from the optomechanical system with the pumped auxiliary cavity. The used parameters are: $\delta _{pr}/\omega _{b}=1$, $\delta _{pu}/\omega _{b}=0.2$, $\varepsilon _{pr}/\varepsilon _{co}=0.8$, $\varepsilon _{pu}/\varepsilon _{co}=1.6$, $\varepsilon _{co}=0.5$ THz, $G/\kappa =\eta /\kappa =1$, the other parameters can be found in the text.

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In Fig. 5, we show that how to diminish the sideband interval by tuning the parameter configurations of the pumped auxiliary cavity. In Fig. 5(a), we increase the pump amplitude $\varepsilon _{pu}$ to $6\varepsilon _{co}$. Compared with Fig. 4, more fraction-order, sum, and different sidebands appear in the output spectrum, all the generated sidebands are equidistant, and the sideband interval becomes one fifth of the mechanical frequency, i.e., $\omega _{b}/5$. In order to further decrease the sideband interval, we can continue to increase $m$ and the pump amplitude $\varepsilon _{pu}$ simultaneously. When $m=10$ and $\varepsilon _{pu}=7.5\varepsilon _{co}$, we can see that the sideband interval is decreased by one order of magnitude and becomes $\omega _{b}/10$, as shown in Fig. 5(b). When $m=20$ and $\varepsilon _{pu}=10\varepsilon _{co}$, the sideband interval is $\omega _{b}/20$, which can be clearly seen in Fig. 5(c). To sum up, with the pumped auxiliary cavity, the interval of the sidebands output from the optomechanical system becomes $\omega _{b}/m$ and can be decreased by increasing $m$ and the pump amplitude $\varepsilon _{pu}$.

 figure: Fig. 5.

Fig. 5. The sideband spectra output from the optomechanical system with the pumped auxiliary cavity for different pump parameters: (a) $m=5$, $\varepsilon _{pu}/\varepsilon _{co}=6$; (b) $m=10$, $\varepsilon _{pu}/\varepsilon _{co}=7.5$; (c) $m=20$, $\varepsilon _{pu}/\varepsilon _{co}=10$. The other parameters are the same as those in Fig. 4.

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6. Conclusion

In summary, we have studied the sideband generation in an optomechanical system assisted by a pumped auxiliary cavity. By driving the optomechanical cavity with a control field $\varepsilon _{co}$ and a probe field $\varepsilon _{pr}$, simultaneously, driving the auxiliary cavity with a pump field $\varepsilon _{pu}$, and by setting their frequency detunings as $\delta _{pr}=\omega _{b}/n$, $\delta _{pu}=\omega _{b}/m$, $m$ and $n$ are integers, we show that when $n=m=1$, the sideband range can be effectively enlarged by increasing the pump amplitude $\varepsilon _{pu}$, or the photon-hopping coupling rate $G$, or by decreasing the auxiliary cavity damping rate $\eta$. When $n=1$, $m >1$, the output spectrum consists of a series of integer-order sidebands, fraction-order sidebands, as well as the sum and difference sidebands between the integer- and fraction-order sidebands. The sideband interval becomes $\omega _{b}/m$ and can be diminished by increasing $m$ and the pump amplitude $\varepsilon _{pu}$. Our work may provide further understanding of nonlinear optomechanical interaction and can help realizing tunable optical frequency comb.

Funding

Natural Science Foundation of Guangdong Province (2018A030310109); National Natural Science Foundation of China (11574092, 12047520, 61378012, 61771205, 61775062, 61941501, 62071186, 91121023).

Disclosures

The authors declare no conflicts of interest.

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14. G. A. T. Pender, P. F. Barker, F. Marquardt, J. Millen, and T. S. Monteiro, “Optomechanical cooling of levitated spheres with doubly resonant fields,” Phys. Rev. A 85(2), 021802 (2012). [CrossRef]  

15. W. J. Gu and G. X. Li, “Quantum interference effects on ground-state optomechanical cooling,” Phys. Rev. A 87(2), 025804 (2013). [CrossRef]  

16. Y. C. Liu, Y. W. Hu, C. W. Wong, and Y. F. Xiao, “Review of cavity optomechanical cooling,” Chin. Phys. B 22(11), 114213 (2013). [CrossRef]  

17. N. T. Otterstrom, R. O. Behunin, E. A. Kittlaus, and P. T. Rakich, “Optomechanical cooling in a continuous system,” Phys. Rev. X 8(4), 041034 (2018). [CrossRef]  

18. C. Wang, Q. Lin, and B. He, “Breaking the optomechanical cooling limit by two drive fields on a membrane-in-the-middle system,” Phys. Rev. A 99(2), 023829 (2019). [CrossRef]  

19. D. G. Lai, J. F. Huang, X. L. Yin, B. P. Hou, W. Li, D. Vitali, F. Nori, and J. Q. Liao, “Nonreciprocal ground-state cooling of multiple mechanical resonators,” Phys. Rev. A 102(1), 011502 (2020). [CrossRef]  

20. Y. Liu, M. Davanco, V. Aksyuk, and K. Srinivasan, “Electromagnetically induced transparency and wideband wavelength conversion in silicon nitride microdisk optomechanical resonators,” Phys. Rev. Lett. 110(22), 223603 (2013). [CrossRef]  

21. L. Fan, C. L. Zou, M. Poot, R. Cheng, X. Guo, X. Han, and H. X. Tang, “Integrated optomechanical single-photon frequency shifter,” Nat. Photonics 10(12), 766–770 (2016). [CrossRef]  

22. M. Wu, E. Zeuthen, K. C. Balram, and K. Srinivasan, “Microwave-to-optical transduction using a mechanical supermode for coupling piezoelectric and optomechanical resonators,” Phys. Rev. Appl. 13(1), 014027 (2020). [CrossRef]  

23. M. Abdi, S. Barzanjeh, P. Tombesi, and D. Vitali, “Effect of phase noise on the generation of stationary entanglement in cavity optomechanics,” Phys. Rev. A 84(3), 032325 (2011). [CrossRef]  

24. T. P. Purdy, P. L. Yu, R. W. Peterson, N. S. Kampel, and C. A. Regal, “Strong optomechanical squeezing of light,” Phys. Rev. X 3(3), 031012 (2013). [CrossRef]  

25. M. Wang, X. Y. Lü, Y. D. Wang, J. Q. You, and Y. Wu, “Macroscopic quantum entanglement in modulated optomechanics,” Phys. Rev. A 94(5), 053807 (2016). [CrossRef]  

26. C. H. Bai, D. Y. Wang, S. Zhang, S. Liu, and H. F. Wang, “Engineering of strong mechanical squeezing via the joint effect between duffing nonlinearity and parametric pump driving,” Photonics Res. 7(11), 1229–1239 (2019). [CrossRef]  

27. L. Liu, B. P. Hou, X. H. Zhao, and B. Tang, “Squeezing transfer of light in a two-mode optomechanical system,” Opt. Express 27(6), 8361–8374 (2019). [CrossRef]  

28. X. Zhao, “Macroscopic entanglement in optomechanical system induced by non-Markovian environment,” Opt. Express 27(20), 29082–29097 (2019). [CrossRef]  

29. C. M. Caves, “Quantum-mechanical radiation-pressure fluctuations in an interferometer,” Phys. Rev. Lett. 45(2), 75–79 (1980). [CrossRef]  

30. A. Buonanno and Y. Chen, “Scaling law in signal recycled laser-interferometer gravitational-wave detectors,” Phys. Rev. D 67(6), 062002 (2003). [CrossRef]  

31. J. Q. Zhang, Y. Li, M. Feng, and Y. Xu, “Precision measurement of electrical charge with optomechanically induced transparency,” Phys. Rev. A 86(5), 053806 (2012). [CrossRef]  

32. M. H. Wimmer, D. Steinmeyer, K. Hammerer, and M. Heurs, “Coherent cancellation of backaction noise in optomechanical force measurements,” Phys. Rev. A 89(5), 053836 (2014). [CrossRef]  

33. Z. X. Liu, B. Wang, C. Kong, L. G. Si, H. Xiong, and Y. Wu, “A proposed method to measure weak magnetic field based on a hybrid optomechanical system,” Sci. Rep. 7(1), 12521 (2017). [CrossRef]  

34. G. Q. Qin, M. Wang, J. W. Wen, D. Ruan, and G. L. Long, “Brillouin cavity optomechanics sensing with enhanced dynamical backaction,” Photonics Res. 7(12), 1440–1446 (2019). [CrossRef]  

35. K. Stannigel, P. Rabl, A. S. Sørensen, P. Zoller, and M. D. Lukin, “Optomechanical transducers for long-distance quantum communication,” Phys. Rev. Lett. 105(22), 220501 (2010). [CrossRef]  

36. W. Z. Zhang, J. Cheng, and L. Zhou, “Quantum control gate in cavity optomechanical system,” J. Phys. B: At., Mol. Opt. Phys. 48(1), 015502 (2015). [CrossRef]  

37. J. Li, A. Wallucks, R. Benevides, N. Fiaschi, B. Hensen, T. P. M. Alegre, and S. Gröblacher, “Proposal for optomechanical quantum teleportation,” Phys. Rev. A 102(3), 032402 (2020). [CrossRef]  

38. D. Kleckner, I. Pikovski, E. Jeffrey, L. Ament, E. Eliel, J. Van Den Brink, and D. Bouwmeester, “Creating and verifying a quantum superposition in a micro-optomechanical system,” New J. Phys. 10(9), 095020 (2008). [CrossRef]  

39. O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Large quantum superpositions and interference of massive nanometer-sized objects,” Phys. Rev. Lett. 107(2), 020405 (2011). [CrossRef]  

40. H. Suzuki, E. Brown, and R. Sterling, “Nonlinear dynamics of an optomechanical system with a coherent mechanical pump: Second-order sideband generation,” Phys. Rev. A 92(3), 033823 (2015). [CrossRef]  

41. B. Chen, L. Shang, X. F. Wang, J. B. Chen, H. B. Xue, X. Liu, and J. Zhang, “Atom-assisted second-order sideband generation in an optomechanical system with atom-cavity-resonator coupling,” Phys. Rev. A 99(6), 063810 (2019). [CrossRef]  

42. Y. F. Jiao, T. X. Lu, and H. Jing, “Optomechanical second-order sidebands and group delays in a Kerr resonator,” Phys. Rev. A 97(1), 013843 (2018). [CrossRef]  

43. H. Xiong, L. G. Si, X. Y. Lü, and Y. Wu, “Optomechanically induced sum sideband generation,” Opt. Express 24(6), 5773–5783 (2016). [CrossRef]  

44. H. Xiong, Y. W. Fan, X. Yang, and Y. Wu, “Radiation pressure induced difference-sideband generation beyond linearized description,” Appl. Phys. Lett. 109(6), 061108 (2016). [CrossRef]  

45. S. Liu, B. Liu, and W. X. Yang, “Highly sensitive mass detection based on nonlinear sum-sideband in a dispersive optomechanical system,” Opt. Express 27(4), 3909–3919 (2019). [CrossRef]  

46. W. A. Li and G. Y. Huang, “Enhancement of optomechanically induced sum sidebands using parametric interactions,” Phys. Rev. A 100(2), 023838 (2019). [CrossRef]  

47. X. Y. Wang, L. G. Si, X. H. Lu, and Y. Wu, “Generation and enhancement of sum sideband in a quadratically coupled optomechanical system with parametric interactions,” Opt. Express 27(20), 29297–29308 (2019). [CrossRef]  

48. J. Ma, C. You, L. G. Si, H. Xiong, J. Li, X. Yang, and Y. Wu, “Formation and manipulation of optomechanical chaos via a bichromatic driving,” Phys. Rev. A 90(4), 043839 (2014). [CrossRef]  

49. X. Y. Lü, H. Jing, J. Y. Ma, and Y. Wu, “PT-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114(25), 253601 (2015). [CrossRef]  

50. H. Xiong, L. G. Si, X. Y. Lü, X. X. Yang, and Y. Wu, “Carrier-envelope phase-dependent effect of high-order sideband generation in ultrafast driven optomechanical system,” Opt. Lett. 38(3), 353–355 (2013). [CrossRef]  

51. H. Xiong, L. G. Si, X. Y. Lü, X. X. Yang, and Y. Wu, “Nanosecond-pulse-controlled higher-order sideband comb in a GaAs optomechanical disk resonator in the non-perturbative regime,” Ann. Phys. 349, 43–54 (2014). [CrossRef]  

52. K. Minoshima and H. Matsumoto, “High-accuracy measurement of 240m distance in an optical tunnel by use of a compact femtosecond laser,” Appl. Opt. 39(30), 5512–5517 (2000). [CrossRef]  

53. J. L. Hall, Nobel Lecture, “Defining and measuring optical frequencies,” Rev. Mod. Phys. 78(4), 1279–1295 (2006). [CrossRef]  

54. N. G. Pavlov, G. Lihachev, S. Koptyaev, E. Lucas, M. Karpov, N. M. Kondratiev, I. A. Bilenko, T. J. Kippenberg, and M. L. Gorodetsky, “Soliton dual frequency combs in crystalline microresonators,” Opt. Lett. 42(3), 514–517 (2017). [CrossRef]  

55. T. Fortier and E. Baumann, “20 years of developments in optical frequency comb technology and applications,” Commun. Phys. 2(1), 153 (2019). [CrossRef]  

56. C. Cao, S. C. Mi, Y. P. Gao, L. Y. He, D. Yang, T. J. Wang, R. Zhang, and C. Wang, “Tunable high-order sideband spectra generation using a photonic molecule optomechanical system,” Sci. Rep. 6(1), 22920 (2016). [CrossRef]  

57. C. Kong, H. Xiong, and Y. Wu, “Coulomb-interaction-dependent effect of high-order sideband generation in an optomechanical system,” Phys. Rev. A 95(3), 033820 (2017). [CrossRef]  

58. Z. X. Liu, H. Xiong, and Y. Wu, “Generation and amplification of a high-order sideband induced by two-level atoms in a hybrid optomechanical system,” Phys. Rev. A 97(1), 013801 (2018). [CrossRef]  

59. J. Yao, Y. F. Yu, and Z. M. Zhang, “Effects of Casimir force on high-order sideband generation in an optomechanical system,” Chin. Opt. Lett. 16(11), 111201 (2018). [CrossRef]  

60. C. Kong, S. W. Bin, B. Wang, Z. X. Liu, H. Xiong, and Y. Wu, “High-order sideband generation in a two-cavity optomechanical system with modulated photon-hopping interaction,” Laser Phys. Lett. 15(11), 115401 (2018). [CrossRef]  

61. L. Y. He, “Parity-time-symmetry–enhanced sideband generation in an optomechanical system,” Phys. Rev. A 99(3), 033843 (2019). [CrossRef]  

62. J. H. Liu, G. He, Q. Wu, Y. F. Yu, J. D. Wang, and Z. M. Zhang, “Fraction-order sideband generation in an optomechanical system,” Opt. Lett. 45(18), 5169–5172 (2020). [CrossRef]  

63. Y. Guo, K. Li, W. Nie, and Y. Li, “Electromagnetically-induced-transparency-like ground-state cooling in a double-cavity optomechanical system,” Phys. Rev. A 90(5), 053841 (2014). [CrossRef]  

64. F. C. Lei, M. Gao, C. Du, Q. L. Jing, and G. L. Long, “Three-pathway electromagnetically induced transparency in coupled-cavity optomechanical system,” Opt. Express 23(9), 11508–11517 (2015). [CrossRef]  

65. Y. Yanay, J. C. Sankey, and A. A. Clerk, “Quantum backaction and noise interference in asymmetric two-cavity optomechanical systems,” Phys. Rev. A 93(6), 063809 (2016). [CrossRef]  

66. C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Phys. Rev. A 31(6), 3761–3774 (1985). [CrossRef]  

67. C. Genens, D. Vitali, and P. Tombesi, “Emergence of atom-light- mirror entanglement inside an optical cavity,” Phys. Rev. A 77(5), 050307 (2008). [CrossRef]  

68. C. B. Fu, X. B. Yan, K. H. Gu, C. L. Cui, J. H. Wu, and T. D. Fu, “Steady-state solutions of a hybrid system involving atom-light and optomechanical interactions: Beyond the weak-cavity-field approximation,” Phys. Rev. A 87(5), 053841 (2013). [CrossRef]  

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  45. S. Liu, B. Liu, and W. X. Yang, “Highly sensitive mass detection based on nonlinear sum-sideband in a dispersive optomechanical system,” Opt. Express 27(4), 3909–3919 (2019).
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    [Crossref]
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  50. H. Xiong, L. G. Si, X. Y. Lü, X. X. Yang, and Y. Wu, “Carrier-envelope phase-dependent effect of high-order sideband generation in ultrafast driven optomechanical system,” Opt. Lett. 38(3), 353–355 (2013).
    [Crossref]
  51. H. Xiong, L. G. Si, X. Y. Lü, X. X. Yang, and Y. Wu, “Nanosecond-pulse-controlled higher-order sideband comb in a GaAs optomechanical disk resonator in the non-perturbative regime,” Ann. Phys. 349, 43–54 (2014).
    [Crossref]
  52. K. Minoshima and H. Matsumoto, “High-accuracy measurement of 240m distance in an optical tunnel by use of a compact femtosecond laser,” Appl. Opt. 39(30), 5512–5517 (2000).
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  58. Z. X. Liu, H. Xiong, and Y. Wu, “Generation and amplification of a high-order sideband induced by two-level atoms in a hybrid optomechanical system,” Phys. Rev. A 97(1), 013801 (2018).
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  59. J. Yao, Y. F. Yu, and Z. M. Zhang, “Effects of Casimir force on high-order sideband generation in an optomechanical system,” Chin. Opt. Lett. 16(11), 111201 (2018).
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    [Crossref]
  61. L. Y. He, “Parity-time-symmetry–enhanced sideband generation in an optomechanical system,” Phys. Rev. A 99(3), 033843 (2019).
    [Crossref]
  62. J. H. Liu, G. He, Q. Wu, Y. F. Yu, J. D. Wang, and Z. M. Zhang, “Fraction-order sideband generation in an optomechanical system,” Opt. Lett. 45(18), 5169–5172 (2020).
    [Crossref]
  63. Y. Guo, K. Li, W. Nie, and Y. Li, “Electromagnetically-induced-transparency-like ground-state cooling in a double-cavity optomechanical system,” Phys. Rev. A 90(5), 053841 (2014).
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  64. F. C. Lei, M. Gao, C. Du, Q. L. Jing, and G. L. Long, “Three-pathway electromagnetically induced transparency in coupled-cavity optomechanical system,” Opt. Express 23(9), 11508–11517 (2015).
    [Crossref]
  65. Y. Yanay, J. C. Sankey, and A. A. Clerk, “Quantum backaction and noise interference in asymmetric two-cavity optomechanical systems,” Phys. Rev. A 93(6), 063809 (2016).
    [Crossref]
  66. C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Phys. Rev. A 31(6), 3761–3774 (1985).
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  67. C. Genens, D. Vitali, and P. Tombesi, “Emergence of atom-light- mirror entanglement inside an optical cavity,” Phys. Rev. A 77(5), 050307 (2008).
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  68. C. B. Fu, X. B. Yan, K. H. Gu, C. L. Cui, J. H. Wu, and T. D. Fu, “Steady-state solutions of a hybrid system involving atom-light and optomechanical interactions: Beyond the weak-cavity-field approximation,” Phys. Rev. A 87(5), 053841 (2013).
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2020 (7)

D. G. Lai, X. Wang, W. Qin, B. P. Hou, F. Nori, and J. Q. Liao, “Tunable optomechanically induced transparency by controlling the dark-mode effect,” Phys. Rev. A 102(2), 023707 (2020).
[Crossref]

Q. Liao, X. Xiao, W. Nie, and N. Zhou, “Transparency and tunable slow-fast light in a hybrid cavity optomechanical system,” Opt. Express 28(4), 5288–5305 (2020).
[Crossref]

D. P. Lake, M. Mitchell, B. C. Sanders, and P. E. Barclay, “Two-colour interferometry and switching through optomechanical dark mode excitation,” Nat. Commun. 11(1), 2208 (2020).
[Crossref]

D. G. Lai, J. F. Huang, X. L. Yin, B. P. Hou, W. Li, D. Vitali, F. Nori, and J. Q. Liao, “Nonreciprocal ground-state cooling of multiple mechanical resonators,” Phys. Rev. A 102(1), 011502 (2020).
[Crossref]

M. Wu, E. Zeuthen, K. C. Balram, and K. Srinivasan, “Microwave-to-optical transduction using a mechanical supermode for coupling piezoelectric and optomechanical resonators,” Phys. Rev. Appl. 13(1), 014027 (2020).
[Crossref]

J. Li, A. Wallucks, R. Benevides, N. Fiaschi, B. Hensen, T. P. M. Alegre, and S. Gröblacher, “Proposal for optomechanical quantum teleportation,” Phys. Rev. A 102(3), 032402 (2020).
[Crossref]

J. H. Liu, G. He, Q. Wu, Y. F. Yu, J. D. Wang, and Z. M. Zhang, “Fraction-order sideband generation in an optomechanical system,” Opt. Lett. 45(18), 5169–5172 (2020).
[Crossref]

2019 (12)

L. Y. He, “Parity-time-symmetry–enhanced sideband generation in an optomechanical system,” Phys. Rev. A 99(3), 033843 (2019).
[Crossref]

T. Fortier and E. Baumann, “20 years of developments in optical frequency comb technology and applications,” Commun. Phys. 2(1), 153 (2019).
[Crossref]

G. Q. Qin, M. Wang, J. W. Wen, D. Ruan, and G. L. Long, “Brillouin cavity optomechanics sensing with enhanced dynamical backaction,” Photonics Res. 7(12), 1440–1446 (2019).
[Crossref]

B. Chen, L. Shang, X. F. Wang, J. B. Chen, H. B. Xue, X. Liu, and J. Zhang, “Atom-assisted second-order sideband generation in an optomechanical system with atom-cavity-resonator coupling,” Phys. Rev. A 99(6), 063810 (2019).
[Crossref]

S. Liu, B. Liu, and W. X. Yang, “Highly sensitive mass detection based on nonlinear sum-sideband in a dispersive optomechanical system,” Opt. Express 27(4), 3909–3919 (2019).
[Crossref]

W. A. Li and G. Y. Huang, “Enhancement of optomechanically induced sum sidebands using parametric interactions,” Phys. Rev. A 100(2), 023838 (2019).
[Crossref]

X. Y. Wang, L. G. Si, X. H. Lu, and Y. Wu, “Generation and enhancement of sum sideband in a quadratically coupled optomechanical system with parametric interactions,” Opt. Express 27(20), 29297–29308 (2019).
[Crossref]

C. Wang, Q. Lin, and B. He, “Breaking the optomechanical cooling limit by two drive fields on a membrane-in-the-middle system,” Phys. Rev. A 99(2), 023829 (2019).
[Crossref]

C. H. Bai, D. Y. Wang, S. Zhang, S. Liu, and H. F. Wang, “Engineering of strong mechanical squeezing via the joint effect between duffing nonlinearity and parametric pump driving,” Photonics Res. 7(11), 1229–1239 (2019).
[Crossref]

L. Liu, B. P. Hou, X. H. Zhao, and B. Tang, “Squeezing transfer of light in a two-mode optomechanical system,” Opt. Express 27(6), 8361–8374 (2019).
[Crossref]

X. Zhao, “Macroscopic entanglement in optomechanical system induced by non-Markovian environment,” Opt. Express 27(20), 29082–29097 (2019).
[Crossref]

M. H. Zheng, T. Wang, D. Y. Wang, C. H. Bai, S. Zhang, C. S. An, and H. F. Wang, “Manipulation of multi-transparency windows and fast-slow light transitions in a hybrid cavity optomechanical system,” Sci. China: Phys., Mech. Astron. 62(5), 950311 (2019).
[Crossref]

2018 (6)

M. Rossi, N. Kralj, S. Zippilli, R. Natali, A. Borrielli, G. Pandraud, E. Serra, G. D. Giuseppe, and D. Vitali, “Normal-mode splitting in a weakly coupled optomechanical system,” Phys. Rev. Lett. 120(7), 073601 (2018).
[Crossref]

N. T. Otterstrom, R. O. Behunin, E. A. Kittlaus, and P. T. Rakich, “Optomechanical cooling in a continuous system,” Phys. Rev. X 8(4), 041034 (2018).
[Crossref]

Y. F. Jiao, T. X. Lu, and H. Jing, “Optomechanical second-order sidebands and group delays in a Kerr resonator,” Phys. Rev. A 97(1), 013843 (2018).
[Crossref]

Z. X. Liu, H. Xiong, and Y. Wu, “Generation and amplification of a high-order sideband induced by two-level atoms in a hybrid optomechanical system,” Phys. Rev. A 97(1), 013801 (2018).
[Crossref]

J. Yao, Y. F. Yu, and Z. M. Zhang, “Effects of Casimir force on high-order sideband generation in an optomechanical system,” Chin. Opt. Lett. 16(11), 111201 (2018).
[Crossref]

C. Kong, S. W. Bin, B. Wang, Z. X. Liu, H. Xiong, and Y. Wu, “High-order sideband generation in a two-cavity optomechanical system with modulated photon-hopping interaction,” Laser Phys. Lett. 15(11), 115401 (2018).
[Crossref]

2017 (3)

C. Kong, H. Xiong, and Y. Wu, “Coulomb-interaction-dependent effect of high-order sideband generation in an optomechanical system,” Phys. Rev. A 95(3), 033820 (2017).
[Crossref]

N. G. Pavlov, G. Lihachev, S. Koptyaev, E. Lucas, M. Karpov, N. M. Kondratiev, I. A. Bilenko, T. J. Kippenberg, and M. L. Gorodetsky, “Soliton dual frequency combs in crystalline microresonators,” Opt. Lett. 42(3), 514–517 (2017).
[Crossref]

Z. X. Liu, B. Wang, C. Kong, L. G. Si, H. Xiong, and Y. Wu, “A proposed method to measure weak magnetic field based on a hybrid optomechanical system,” Sci. Rep. 7(1), 12521 (2017).
[Crossref]

2016 (6)

H. Xiong, L. G. Si, X. Y. Lü, and Y. Wu, “Optomechanically induced sum sideband generation,” Opt. Express 24(6), 5773–5783 (2016).
[Crossref]

H. Xiong, Y. W. Fan, X. Yang, and Y. Wu, “Radiation pressure induced difference-sideband generation beyond linearized description,” Appl. Phys. Lett. 109(6), 061108 (2016).
[Crossref]

C. Cao, S. C. Mi, Y. P. Gao, L. Y. He, D. Yang, T. J. Wang, R. Zhang, and C. Wang, “Tunable high-order sideband spectra generation using a photonic molecule optomechanical system,” Sci. Rep. 6(1), 22920 (2016).
[Crossref]

Y. Yanay, J. C. Sankey, and A. A. Clerk, “Quantum backaction and noise interference in asymmetric two-cavity optomechanical systems,” Phys. Rev. A 93(6), 063809 (2016).
[Crossref]

L. Fan, C. L. Zou, M. Poot, R. Cheng, X. Guo, X. Han, and H. X. Tang, “Integrated optomechanical single-photon frequency shifter,” Nat. Photonics 10(12), 766–770 (2016).
[Crossref]

M. Wang, X. Y. Lü, Y. D. Wang, J. Q. You, and Y. Wu, “Macroscopic quantum entanglement in modulated optomechanics,” Phys. Rev. A 94(5), 053807 (2016).
[Crossref]

2015 (6)

M. J. Akram, M. M. Khan, and F. Saif, “Tunable fast and slow light in a hybrid optomechanical system,” Phys. Rev. A 92(2), 023846 (2015).
[Crossref]

H. Xiong, L. G. Si, X. Y. Lü, X. X. Yang, and Y. Wu, “Review of cavity optomechanics in the weak-coupling regime: From linearization to intrinsic nonlinear interactions,” Sci. China: Phys., Mech. Astron. 58(5), 1–13 (2015).
[Crossref]

X. Y. Lü, H. Jing, J. Y. Ma, and Y. Wu, “PT-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114(25), 253601 (2015).
[Crossref]

W. Z. Zhang, J. Cheng, and L. Zhou, “Quantum control gate in cavity optomechanical system,” J. Phys. B: At., Mol. Opt. Phys. 48(1), 015502 (2015).
[Crossref]

H. Suzuki, E. Brown, and R. Sterling, “Nonlinear dynamics of an optomechanical system with a coherent mechanical pump: Second-order sideband generation,” Phys. Rev. A 92(3), 033823 (2015).
[Crossref]

F. C. Lei, M. Gao, C. Du, Q. L. Jing, and G. L. Long, “Three-pathway electromagnetically induced transparency in coupled-cavity optomechanical system,” Opt. Express 23(9), 11508–11517 (2015).
[Crossref]

2014 (6)

J. Ma, C. You, L. G. Si, H. Xiong, J. Li, X. Yang, and Y. Wu, “Formation and manipulation of optomechanical chaos via a bichromatic driving,” Phys. Rev. A 90(4), 043839 (2014).
[Crossref]

H. Xiong, L. G. Si, X. Y. Lü, X. X. Yang, and Y. Wu, “Nanosecond-pulse-controlled higher-order sideband comb in a GaAs optomechanical disk resonator in the non-perturbative regime,” Ann. Phys. 349, 43–54 (2014).
[Crossref]

Y. Guo, K. Li, W. Nie, and Y. Li, “Electromagnetically-induced-transparency-like ground-state cooling in a double-cavity optomechanical system,” Phys. Rev. A 90(5), 053841 (2014).
[Crossref]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
[Crossref]

P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90(4), 043825 (2014).
[Crossref]

M. H. Wimmer, D. Steinmeyer, K. Hammerer, and M. Heurs, “Coherent cancellation of backaction noise in optomechanical force measurements,” Phys. Rev. A 89(5), 053836 (2014).
[Crossref]

2013 (6)

T. P. Purdy, P. L. Yu, R. W. Peterson, N. S. Kampel, and C. A. Regal, “Strong optomechanical squeezing of light,” Phys. Rev. X 3(3), 031012 (2013).
[Crossref]

Y. Liu, M. Davanco, V. Aksyuk, and K. Srinivasan, “Electromagnetically induced transparency and wideband wavelength conversion in silicon nitride microdisk optomechanical resonators,” Phys. Rev. Lett. 110(22), 223603 (2013).
[Crossref]

W. J. Gu and G. X. Li, “Quantum interference effects on ground-state optomechanical cooling,” Phys. Rev. A 87(2), 025804 (2013).
[Crossref]

Y. C. Liu, Y. W. Hu, C. W. Wong, and Y. F. Xiao, “Review of cavity optomechanical cooling,” Chin. Phys. B 22(11), 114213 (2013).
[Crossref]

H. Xiong, L. G. Si, X. Y. Lü, X. X. Yang, and Y. Wu, “Carrier-envelope phase-dependent effect of high-order sideband generation in ultrafast driven optomechanical system,” Opt. Lett. 38(3), 353–355 (2013).
[Crossref]

C. B. Fu, X. B. Yan, K. H. Gu, C. L. Cui, J. H. Wu, and T. D. Fu, “Steady-state solutions of a hybrid system involving atom-light and optomechanical interactions: Beyond the weak-cavity-field approximation,” Phys. Rev. A 87(5), 053841 (2013).
[Crossref]

2012 (4)

G. A. T. Pender, P. F. Barker, F. Marquardt, J. Millen, and T. S. Monteiro, “Optomechanical cooling of levitated spheres with doubly resonant fields,” Phys. Rev. A 85(2), 021802 (2012).
[Crossref]

C. Dong, V. Fiore, M. C. Kuzyk, and H. Wang, “Optomechanical dark mode,” Science 338(6114), 1609–1613 (2012).
[Crossref]

M. Aspelmeyer, P. Meystre, and K. Schwab, “Quantum optomechanics,” Phys. Today 65(7), 29–35 (2012).
[Crossref]

J. Q. Zhang, Y. Li, M. Feng, and Y. Xu, “Precision measurement of electrical charge with optomechanically induced transparency,” Phys. Rev. A 86(5), 053806 (2012).
[Crossref]

2011 (2)

M. Abdi, S. Barzanjeh, P. Tombesi, and D. Vitali, “Effect of phase noise on the generation of stationary entanglement in cavity optomechanics,” Phys. Rev. A 84(3), 032325 (2011).
[Crossref]

O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Large quantum superpositions and interference of massive nanometer-sized objects,” Phys. Rev. Lett. 107(2), 020405 (2011).
[Crossref]

2010 (2)

K. Stannigel, P. Rabl, A. S. Sørensen, P. Zoller, and M. D. Lukin, “Optomechanical transducers for long-distance quantum communication,” Phys. Rev. Lett. 105(22), 220501 (2010).
[Crossref]

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref]

2008 (3)

J. M. Dobrindt, I. Wilson-Rae, and T. J. Kippenberg, “Parametric normal-mode splitting in cavity optomechanics,” Phys. Rev. Lett. 101(26), 263602 (2008).
[Crossref]

D. Kleckner, I. Pikovski, E. Jeffrey, L. Ament, E. Eliel, J. Van Den Brink, and D. Bouwmeester, “Creating and verifying a quantum superposition in a micro-optomechanical system,” New J. Phys. 10(9), 095020 (2008).
[Crossref]

C. Genens, D. Vitali, and P. Tombesi, “Emergence of atom-light- mirror entanglement inside an optical cavity,” Phys. Rev. A 77(5), 050307 (2008).
[Crossref]

2006 (1)

J. L. Hall, Nobel Lecture, “Defining and measuring optical frequencies,” Rev. Mod. Phys. 78(4), 1279–1295 (2006).
[Crossref]

2003 (1)

A. Buonanno and Y. Chen, “Scaling law in signal recycled laser-interferometer gravitational-wave detectors,” Phys. Rev. D 67(6), 062002 (2003).
[Crossref]

2000 (1)

1985 (1)

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Phys. Rev. A 31(6), 3761–3774 (1985).
[Crossref]

1980 (1)

C. M. Caves, “Quantum-mechanical radiation-pressure fluctuations in an interferometer,” Phys. Rev. Lett. 45(2), 75–79 (1980).
[Crossref]

Abdi, M.

M. Abdi, S. Barzanjeh, P. Tombesi, and D. Vitali, “Effect of phase noise on the generation of stationary entanglement in cavity optomechanics,” Phys. Rev. A 84(3), 032325 (2011).
[Crossref]

Akram, M. J.

M. J. Akram, M. M. Khan, and F. Saif, “Tunable fast and slow light in a hybrid optomechanical system,” Phys. Rev. A 92(2), 023846 (2015).
[Crossref]

Aksyuk, V.

Y. Liu, M. Davanco, V. Aksyuk, and K. Srinivasan, “Electromagnetically induced transparency and wideband wavelength conversion in silicon nitride microdisk optomechanical resonators,” Phys. Rev. Lett. 110(22), 223603 (2013).
[Crossref]

Alegre, T. P. M.

J. Li, A. Wallucks, R. Benevides, N. Fiaschi, B. Hensen, T. P. M. Alegre, and S. Gröblacher, “Proposal for optomechanical quantum teleportation,” Phys. Rev. A 102(3), 032402 (2020).
[Crossref]

Ament, L.

D. Kleckner, I. Pikovski, E. Jeffrey, L. Ament, E. Eliel, J. Van Den Brink, and D. Bouwmeester, “Creating and verifying a quantum superposition in a micro-optomechanical system,” New J. Phys. 10(9), 095020 (2008).
[Crossref]

An, C. S.

M. H. Zheng, T. Wang, D. Y. Wang, C. H. Bai, S. Zhang, C. S. An, and H. F. Wang, “Manipulation of multi-transparency windows and fast-slow light transitions in a hybrid cavity optomechanical system,” Sci. China: Phys., Mech. Astron. 62(5), 950311 (2019).
[Crossref]

Arcizet, O.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref]

Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
[Crossref]

M. Aspelmeyer, P. Meystre, and K. Schwab, “Quantum optomechanics,” Phys. Today 65(7), 29–35 (2012).
[Crossref]

O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Large quantum superpositions and interference of massive nanometer-sized objects,” Phys. Rev. Lett. 107(2), 020405 (2011).
[Crossref]

Bai, C. H.

M. H. Zheng, T. Wang, D. Y. Wang, C. H. Bai, S. Zhang, C. S. An, and H. F. Wang, “Manipulation of multi-transparency windows and fast-slow light transitions in a hybrid cavity optomechanical system,” Sci. China: Phys., Mech. Astron. 62(5), 950311 (2019).
[Crossref]

C. H. Bai, D. Y. Wang, S. Zhang, S. Liu, and H. F. Wang, “Engineering of strong mechanical squeezing via the joint effect between duffing nonlinearity and parametric pump driving,” Photonics Res. 7(11), 1229–1239 (2019).
[Crossref]

Balram, K. C.

M. Wu, E. Zeuthen, K. C. Balram, and K. Srinivasan, “Microwave-to-optical transduction using a mechanical supermode for coupling piezoelectric and optomechanical resonators,” Phys. Rev. Appl. 13(1), 014027 (2020).
[Crossref]

Barclay, P. E.

D. P. Lake, M. Mitchell, B. C. Sanders, and P. E. Barclay, “Two-colour interferometry and switching through optomechanical dark mode excitation,” Nat. Commun. 11(1), 2208 (2020).
[Crossref]

Barker, P. F.

G. A. T. Pender, P. F. Barker, F. Marquardt, J. Millen, and T. S. Monteiro, “Optomechanical cooling of levitated spheres with doubly resonant fields,” Phys. Rev. A 85(2), 021802 (2012).
[Crossref]

Barzanjeh, S.

M. Abdi, S. Barzanjeh, P. Tombesi, and D. Vitali, “Effect of phase noise on the generation of stationary entanglement in cavity optomechanics,” Phys. Rev. A 84(3), 032325 (2011).
[Crossref]

Baumann, E.

T. Fortier and E. Baumann, “20 years of developments in optical frequency comb technology and applications,” Commun. Phys. 2(1), 153 (2019).
[Crossref]

Behunin, R. O.

N. T. Otterstrom, R. O. Behunin, E. A. Kittlaus, and P. T. Rakich, “Optomechanical cooling in a continuous system,” Phys. Rev. X 8(4), 041034 (2018).
[Crossref]

Benevides, R.

J. Li, A. Wallucks, R. Benevides, N. Fiaschi, B. Hensen, T. P. M. Alegre, and S. Gröblacher, “Proposal for optomechanical quantum teleportation,” Phys. Rev. A 102(3), 032402 (2020).
[Crossref]

Bilenko, I. A.

Bin, S. W.

C. Kong, S. W. Bin, B. Wang, Z. X. Liu, H. Xiong, and Y. Wu, “High-order sideband generation in a two-cavity optomechanical system with modulated photon-hopping interaction,” Laser Phys. Lett. 15(11), 115401 (2018).
[Crossref]

Blaser, F.

O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Large quantum superpositions and interference of massive nanometer-sized objects,” Phys. Rev. Lett. 107(2), 020405 (2011).
[Crossref]

Borrielli, A.

M. Rossi, N. Kralj, S. Zippilli, R. Natali, A. Borrielli, G. Pandraud, E. Serra, G. D. Giuseppe, and D. Vitali, “Normal-mode splitting in a weakly coupled optomechanical system,” Phys. Rev. Lett. 120(7), 073601 (2018).
[Crossref]

Bouwmeester, D.

D. Kleckner, I. Pikovski, E. Jeffrey, L. Ament, E. Eliel, J. Van Den Brink, and D. Bouwmeester, “Creating and verifying a quantum superposition in a micro-optomechanical system,” New J. Phys. 10(9), 095020 (2008).
[Crossref]

Brown, E.

H. Suzuki, E. Brown, and R. Sterling, “Nonlinear dynamics of an optomechanical system with a coherent mechanical pump: Second-order sideband generation,” Phys. Rev. A 92(3), 033823 (2015).
[Crossref]

Buonanno, A.

A. Buonanno and Y. Chen, “Scaling law in signal recycled laser-interferometer gravitational-wave detectors,” Phys. Rev. D 67(6), 062002 (2003).
[Crossref]

Cao, C.

C. Cao, S. C. Mi, Y. P. Gao, L. Y. He, D. Yang, T. J. Wang, R. Zhang, and C. Wang, “Tunable high-order sideband spectra generation using a photonic molecule optomechanical system,” Sci. Rep. 6(1), 22920 (2016).
[Crossref]

Caves, C. M.

C. M. Caves, “Quantum-mechanical radiation-pressure fluctuations in an interferometer,” Phys. Rev. Lett. 45(2), 75–79 (1980).
[Crossref]

Chen, B.

B. Chen, L. Shang, X. F. Wang, J. B. Chen, H. B. Xue, X. Liu, and J. Zhang, “Atom-assisted second-order sideband generation in an optomechanical system with atom-cavity-resonator coupling,” Phys. Rev. A 99(6), 063810 (2019).
[Crossref]

Chen, J. B.

B. Chen, L. Shang, X. F. Wang, J. B. Chen, H. B. Xue, X. Liu, and J. Zhang, “Atom-assisted second-order sideband generation in an optomechanical system with atom-cavity-resonator coupling,” Phys. Rev. A 99(6), 063810 (2019).
[Crossref]

Chen, Y.

A. Buonanno and Y. Chen, “Scaling law in signal recycled laser-interferometer gravitational-wave detectors,” Phys. Rev. D 67(6), 062002 (2003).
[Crossref]

Cheng, J.

W. Z. Zhang, J. Cheng, and L. Zhou, “Quantum control gate in cavity optomechanical system,” J. Phys. B: At., Mol. Opt. Phys. 48(1), 015502 (2015).
[Crossref]

Cheng, R.

L. Fan, C. L. Zou, M. Poot, R. Cheng, X. Guo, X. Han, and H. X. Tang, “Integrated optomechanical single-photon frequency shifter,” Nat. Photonics 10(12), 766–770 (2016).
[Crossref]

Cirac, J. I.

O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, and J. I. Cirac, “Large quantum superpositions and interference of massive nanometer-sized objects,” Phys. Rev. Lett. 107(2), 020405 (2011).
[Crossref]

Clerk, A. A.

Y. Yanay, J. C. Sankey, and A. A. Clerk, “Quantum backaction and noise interference in asymmetric two-cavity optomechanical systems,” Phys. Rev. A 93(6), 063809 (2016).
[Crossref]

Collett, M. J.

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation,” Phys. Rev. A 31(6), 3761–3774 (1985).
[Crossref]

Cui, C. L.

C. B. Fu, X. B. Yan, K. H. Gu, C. L. Cui, J. H. Wu, and T. D. Fu, “Steady-state solutions of a hybrid system involving atom-light and optomechanical interactions: Beyond the weak-cavity-field approximation,” Phys. Rev. A 87(5), 053841 (2013).
[Crossref]

Davanco, M.

Y. Liu, M. Davanco, V. Aksyuk, and K. Srinivasan, “Electromagnetically induced transparency and wideband wavelength conversion in silicon nitride microdisk optomechanical resonators,” Phys. Rev. Lett. 110(22), 223603 (2013).
[Crossref]

Deléglise, S.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref]

Dobrindt, J. M.

J. M. Dobrindt, I. Wilson-Rae, and T. J. Kippenberg, “Parametric normal-mode splitting in cavity optomechanics,” Phys. Rev. Lett. 101(26), 263602 (2008).
[Crossref]

Dong, C.

C. Dong, V. Fiore, M. C. Kuzyk, and H. Wang, “Optomechanical dark mode,” Science 338(6114), 1609–1613 (2012).
[Crossref]

Du, C.

Eliel, E.

D. Kleckner, I. Pikovski, E. Jeffrey, L. Ament, E. Eliel, J. Van Den Brink, and D. Bouwmeester, “Creating and verifying a quantum superposition in a micro-optomechanical system,” New J. Phys. 10(9), 095020 (2008).
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[Crossref]

Z. X. Liu, B. Wang, C. Kong, L. G. Si, H. Xiong, and Y. Wu, “A proposed method to measure weak magnetic field based on a hybrid optomechanical system,” Sci. Rep. 7(1), 12521 (2017).
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H. Xiong, Y. W. Fan, X. Yang, and Y. Wu, “Radiation pressure induced difference-sideband generation beyond linearized description,” Appl. Phys. Lett. 109(6), 061108 (2016).
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[Crossref]

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Yang, W. X.

Yang, X.

H. Xiong, Y. W. Fan, X. Yang, and Y. Wu, “Radiation pressure induced difference-sideband generation beyond linearized description,” Appl. Phys. Lett. 109(6), 061108 (2016).
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J. Ma, C. You, L. G. Si, H. Xiong, J. Li, X. Yang, and Y. Wu, “Formation and manipulation of optomechanical chaos via a bichromatic driving,” Phys. Rev. A 90(4), 043839 (2014).
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Yin, X. L.

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M. Wang, X. Y. Lü, Y. D. Wang, J. Q. You, and Y. Wu, “Macroscopic quantum entanglement in modulated optomechanics,” Phys. Rev. A 94(5), 053807 (2016).
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T. P. Purdy, P. L. Yu, R. W. Peterson, N. S. Kampel, and C. A. Regal, “Strong optomechanical squeezing of light,” Phys. Rev. X 3(3), 031012 (2013).
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Yu, Y. F.

Zeuthen, E.

M. Wu, E. Zeuthen, K. C. Balram, and K. Srinivasan, “Microwave-to-optical transduction using a mechanical supermode for coupling piezoelectric and optomechanical resonators,” Phys. Rev. Appl. 13(1), 014027 (2020).
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Figures (5)

Fig. 1.
Fig. 1. The schematic diagram of our proposed system model, which consists of an optomechanical microtoroid cavity and an auxiliary cavity. The driving fields couple with the cavity fields by the optical waveguides.
Fig. 2.
Fig. 2. The integer-order sideband spectra output from the optomechanical system with the non-pumped auxiliary cavity for different $G$: (a) $G=0$; (b) $G=6\kappa$; (c) $G=10\kappa$. The other parameters are: $\varepsilon _{co}=0.1$ THz, $\varepsilon _{pr}/\varepsilon _{co}=1$, $\varepsilon _{pu}/\varepsilon _{co}=0$, $\eta /\kappa =1$, and the rest parameters are stated in the text.
Fig. 3.
Fig. 3. The positions and amplitudes of the integer-order sidebands output from the optomechanical system with pumped auxiliary cavity for: (a) different $\varepsilon _{pu}$, $G/\kappa =1$; (b) different $G$, $\varepsilon _{pu}/\varepsilon _{co}=10$; (c) different $\eta$, $G/\kappa =1$ and $\varepsilon _{pu}/\varepsilon _{co}=10$. The other parameters are the same as those in Fig. (2).
Fig. 4.
Fig. 4. The integer-order, fraction-order, sum, and difference sidebands output from the optomechanical system with the pumped auxiliary cavity. The used parameters are: $\delta _{pr}/\omega _{b}=1$, $\delta _{pu}/\omega _{b}=0.2$, $\varepsilon _{pr}/\varepsilon _{co}=0.8$, $\varepsilon _{pu}/\varepsilon _{co}=1.6$, $\varepsilon _{co}=0.5$ THz, $G/\kappa =\eta /\kappa =1$, the other parameters can be found in the text.
Fig. 5.
Fig. 5. The sideband spectra output from the optomechanical system with the pumped auxiliary cavity for different pump parameters: (a) $m=5$, $\varepsilon _{pu}/\varepsilon _{co}=6$; (b) $m=10$, $\varepsilon _{pu}/\varepsilon _{co}=7.5$; (c) $m=20$, $\varepsilon _{pu}/\varepsilon _{co}=10$. The other parameters are the same as those in Fig. 4.

Equations (15)

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H =   ω a a a + ω b b b + ω c c c + g a a ( b + b ) + G ( a c + c a ) + i ( c ε p u e i ω p u t H . c . ) + i [ a ^ ( ε c o e i ω c o t + ε p r e i ω p r t ) H . c . ] ,
d α d t = ( i Δ a + κ ) α i g α ( β + β ) i G χ + ε c o + ε p r e i δ p r t ,
d χ d t = ( i Δ c + η ) χ i G α + ε p u e i δ p u t ,
d β d t = ( i ω b + γ ) β i g | α | 2 ,
X = X 0 + X p r + X p u + X s u m + X d i f ,
X p r = j = 1 J X p r + ( j ) e i j ω b n t + X p r ( j ) e i j ω b n t ,
X p u = k = 1 K X p u + ( k ) e i k ω b m t + X p u ( k ) e i k ω b m t ,
X s u m = j = 1 J k = 1 K X s u m + ( j , k ) e i ( j n + k m ) ω b t + X s u m ( j , k ) e i ( j n + k m ) ω b t ,
X d i f = j = 1 J k = 1 K X d i f + ( j , k ) e i ( j n k m ) ω b t + X d i f ( j , k ) e i ( j n k m ) ω b t .
S o u t = S 0 + l = 1 L S ( l ) e i l ω b t + S ( l ) e i l ω b t ,
S o u t = S 0 + S p r + S p u + S s u m + S d i f ,
S p r = j = 1 J S p r + ( j ) e i j ω b t + S p r ( j ) e i j ω b t ,
S p u = k = 1 K S p u + ( k ) e i k m ω b t + S p u ( k ) e i k m ω b t ,
S s u m = j = 1 J k = 1 K S s u m + ( j , k ) e i ( j + k m ) ω b t + S s u m ( j , k ) e i ( j + k m ) ω b t ,
S d i f = j = 1 J k = 1 K S d i f + ( j , k ) e i ( j k m ) ω b t + S d i f ( j , k ) e i ( j k m ) ω b t ,

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