## 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 *ω*_{b}/*n* (*ω*_{b}/*m*), where *ω*_{b} 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 *ω*_{b}/*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 [4–6], normal mode splitting [7,8], fast and slow light [9–11], optomechanical dark state [6,12,13], cooling of mechanical oscillator to the ground state of motion [14–19], frequency conversion between light and microwave [20–22], generation of quantum entanglement and squeezing of light and mechanical resonator [23–28], and so on. All of these effects also play important roles in many application fields, such as gravitational-wave detection [29,30], precision measurement [31–34], quantum computation and communication [35–37], 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 [40–42], sum and difference sideband generation [43–47], 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 [52–54]. 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 $\varepsilon _{pu}$.

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 ($\hbar =1$)

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 $\omega _{co}$):

## 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 [50,51,56–61], here the expectation value $X$ is written as the following ansatz

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 asBy 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

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.

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.

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

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 byFigure 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.

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 $\varepsilon _{pu}$.

## 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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