1. Introduction

Cavity optomechanics [1] has attracted great interest as an emerging cross-discipline between nanoscience and quantum optics, which focuses on exploring the nonlinear coupling between optical cavities and mechanical motion under optical radiation pressure. In recent years, optomechanical systems [2] have also received considerable theoretical discussion [3–8], and its research has become a cutting-edge topic and a popular issue nowadays. Its numerous potential applications have been discovered in research from a range of physical fields, including optical mechanical dark modes [9], optomechanically induced transparency (OMIT) [10–15], gravitational wave detection [16–18], cooling of mechanical oscillators [19–23], precision force sensing [24–26], and optical frequency comb generation [27–30]. OMIT is an interesting phenomenon caused by the radiation pressure between the optical and mechanical modes, which is a derivative of electromagnetically induced transparency. Based on OMIT, there are numerous novel research phenomena which have been identified, such as higher-order OMIT [31], reverse OMIT [32,12], and multiple OMIT [33,34], which provides a new possibility to realize optical storage and optical delay.

Recently, nonlinear sideband effects in OMIT have aroused much attention. Such as second-order sidebands [35–40], higher-order sidebands [25,41–45], and chaos [46]. Especially, the existence of sum [47] or difference sidebands [48] are revealed for the first time in an optomechanical system driven by multiple probe fields. These may provide a new experimental solution for precision measurements of charge [49,50] and mass detection [51]. However, the nonlinear interactions arising from weak coupling are not easily detected. Many theoretical works have been completed in order to enhance the efficiency of sum sideband generation (SSG). For example, it was demonstrated that an enhancement of several orders of magnitude was achieved for the efficiency of lower sum sideband generation (LSSG) by nonlinear optomechanical interactions via two-phonon processes [52]. In addition, Wu et al. have proposed a method to generate Laguerre-Gaussian optics sum sideband by exchanging orbital angular momentum in Laguerre-Gaussian rotating systems [53].

The above works were completed in an optomechanical system driven by one pump field and two probe fields. Furthermore, it was shown that the quantum coherence process due to dual driving can completely restrain the decoherence caused by dissipation of the MR, and this dual driving device can achieve optical signal amplification [54]. According to the above reasons, we present a double-cavity system driven by red and blue detuned pump fields. The results show that the efficiency of the SSG enhances significantly with increases of the ratio between the dissipation rate of the cavity and the decay rate of the MR. Furthermore, the amplitudes of USSG and LSSG tend to the identical maximum in the case of the threshold condition is satisfied. It is worth noting that the amplitude of the LSSG can be superior to that of the USSG in the unresolved sideband regime. The enhanced sum sideband effect can be applied to quantum information processing and optical communications, and so on.

The paper is organized as follows. In Sec. 2, we give a theoretical description of the double-cavity system and provide a derivation of the Heisenberg-Langevin motion equation under double driving, where the steady-state solution of the system and the amplitude of sum sideband generation are obtained. In Sec. 3, we concretely discuss the effect of system parameters on the sum sideband and further analyze the characteristics of sum sideband generation in both the resolved and unresolved sideband regime. Finally, the conclusions are presented in Sec. 4.

2. Theoretical model and formulations

As shown in Fig. 1(a), the optomechanical system considered here contains a double-cavity field, which inserts a totally reflecting MR between two fixed partially transmitting mirrors. The eigen-frequencies and decay rate of MR are ${\omega _m}$ and ${\gamma _m}$. Two identical optical cavities with length L and frequency ${\omega _0}$ will be obtained when MR is at rest. Although the two optical cavities are respectively coupled to the MR, the tunneling coupling between the two cavities is weak and therefore negligible [54]. The Hamiltonian of this system can be written as below (in a frame rotating at ${\omega _c}$):

 

Fig. 1. (a) Schematic diagram of a double-cavity optomechanical system formed by a MR embedded in the middle of two fixed lenses. The opposite sides of the double-cavity optomechanical system are driven by a pump field 1 with frequency ${\omega _\textrm{c}}$ and a pump field 2 with frequency ${\omega _d}$, respectively. Two relatively weak probe fields with frequencies ${\omega _1}$ and ${\omega _2}$ are injected into the left optical cavity. Both cavities have the same cavity length L and frequency ${\omega _0}$ under static, while coupling with MR separately. (b) Frequency spectrogram of sum sideband generation in a double-cavity optomechanical system driven by dual pump fields and dual probe fields. ${\Delta _1} = {\omega _0} - {\omega _\textrm{c}}$, ${\Delta _2} = {\omega _0} - {\omega _d}$, ${\delta _1} = {\omega _1} - {\omega _\textrm{c}}$, ${\delta _2} = {\omega _2} - {\omega _\textrm{c}}$.

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Assuming that the input pump laser field is much more intense than the probe field. The perturbation method can be applied to these equations by factorization of mean field approximation, i.e., $\left\langle {Qc} \right\rangle = \left\langle Q \right\rangle \left\langle c \right\rangle $ [47]. The overall solution of intracavity field and mechanical displacement in the dual pump field and dual probe field can be written as $O = \bar{O} + \delta O({O = {a_1},{a_2},x} )$. The steady state value of the system can be obtained:

Now we consider the perturbation caused by the double probe field. The Langevin equation of the quantum fluctuation is as follows:

We must consider the nonlinear terms $ig\delta x\delta {a_1}$, $ig\delta x\delta {a_2}$ and $\hbar g({\delta a_2^ \ast \delta {a_2} - \delta a_1^ \ast \delta {a_1}} )$ since they play an important role in the sum sideband generation. In order to calculate the amplitude of the sum sideband, we deal with SSG in the perturbed state by introducing the following equations that neglect second-order and higher-order terms [47]:

The first group describes the linear response of the probe field. The solution of Eq. (6) is obtained as follows:

The second group corresponds to the SSG process. The output amplitude of the frequency components at sum sideband and the amplitude of the MR are as follows:

Furthermore, based on the input-output relation [56], the output field of this double-cavity system can be given as follows:

The term $\left( {{\varepsilon_c} - \sqrt {\eta \kappa } {{\bar{a}}_1}} \right)$ represents the frequency spectra of the pump field with ${\omega _c}$. $- \sqrt {\eta \kappa } A_j^ - {e^{i{\delta _j}t}}$ and $\left( {{\varepsilon_j} - \sqrt {\eta \kappa } A_j^ + } \right){e^{ - i{\delta _j}t}}({j = 1,2} )$ express the Stokes and anti-Stokes fields, respectively. Furthermore, the terms $- \sqrt {\eta \kappa } A_s^ + {e^{ - i{\Omega _ + }t}}$ and $- \sqrt {\eta \kappa } A_s^ - {e^{i{\Omega _ + }t}}$ describe the output fields with frequencies ${\omega _c} + {\Omega _ + }$ and ${\omega _c} - {\Omega _ + }$, which are correlated with the upper and lower sum sideband separately. As shown in Fig. 1(b), the upper sum sideband can be generated when the pump and probe fields satisfy certain detuning conditions ${\Delta _1} = {\omega _m}$, ${\Delta _2} ={-} {\omega _m}$ and ${\delta _1} + {\delta _2} = {\omega _m}$ with the cavity field. And the lower sum sideband is generated at ${\delta _1} + {\delta _2} ={-} {\omega _m}$.

3. Results and discussion

Here, we only consider sum sideband effects. The efficiency of the upper (lower) sum sideband can be given by $\eta _s^ +{=} \left|{{{ - \sqrt {\eta \kappa } A_s^ + } / {{\varepsilon_1}}}} \right|\left( {\eta_s^ -{=} \left|{{{ - \sqrt {\eta \kappa } A_s^ - } / {{\varepsilon_1}}}} \right|} \right)$, which represents the proportion between the amplitudes of the upper (lower) sum sideband and the first probe field. The specific parameters are as follows [47]: $m = 20ng $, ${G / {2\mathrm{\pi }}} = {{ - 12.0\textrm{GHz}} / {\textrm{nm}}}$, ${{{\gamma _m}} / {2\mathrm{\pi }}} = 41.0\textrm{kHz}$, ${\kappa / {\mathrm{2\pi }}} = 15.0\textrm{MHz}$, ${{{\omega _m}} / {\mathrm{2\pi }}} = 51.8\textrm{MHz}$, ${\textrm{P}_\textrm{1}} = {\textrm{P}_\textrm{2}} = \textrm{1}\;{\mathrm{\mu} \mathrm{W}}$, ${\lambda _c} = 532\textrm{nm}$.

In order to illustrate the properties of SSG under double radiation pressure, Figs. 2(a) and 2(b) interpret the first probe field frequency ${\delta _1}$ and red detuned pump field power ${P_\textrm{c}}$ versus the efficiency (in logarithmic form) of upper (lower) sum sideband generation. It is shown that from Fig. 2 the efficiencies of both USSG and LSSG increases with the enhancement of the pump field power ${P_c}$. It can be clearly seen that in this double-cavity system, the efficiency of SSG is several orders of magnitude higher than that in general optomechanical system [47]. As displayed in Fig. 2(a), there is only one peak in the USSG, which reaches its maximum only at ${\delta _1} = {\omega _m}$ and generates a new absorption peak corresponding to the position of ${\delta _1} = 1.05{\omega _m}$. The physical explanation of this phenomenon is that the anti-Stokes scattering of the strongly red detuned pump field leads to its transparency around the cavity resonance [57]. In contrast, as depicted in Fig. 2(b), there are two peaks in the LSSG and the point positions corresponding to the peaks are ${\delta _1} = {\omega _m}$ and ${\delta _1} = 1.05{\omega _m}$, respectively. And the peak value at ${\delta _1} = {\omega _m}$ is significantly higher than that at ${\delta _1} = 1.05{\omega _m}$. The physical reason for their different behavior is that the latter is more separated from the detuning conditions of the cavity ${\Delta _1} = {\omega _m}$ [47]. The particular value of ${\delta _1} = 1.05{\omega _m}$ is attributed to ${\delta _1} + {\delta _2} = {\omega _m}$, which is called the matching condition. The efficiencies of USSG and LSSG as a function of the detuning frequency of the first probe field are shown in Fig. 2(c). It can be seen that a transparent window appearing in the USSG corresponds to the absorption peak in Fig. 2(a). Figure 2(d) depicts the amplitude of the MR at the sum sideband versus the detuning frequencies ${\delta _1}$ and ${\delta _2}$. It has been shown that ${x_s}$ becomes prominent on the two lines of ${\delta _1} + {\delta _2} ={\pm} {\omega _m}$ and at two points of $({{\delta_1},{\delta_2}} )= ({0, \pm {\omega_m}} )$. In particular, the amplitude of MR ${x_s}$ is especially remarkable at the four points $({{\delta_1},{\delta_2}} )= ({ \pm {\omega_m},0} )$, $({{\delta_1},{\delta_2}} )= ({ - {\omega_m},{\omega_m}} )$ and $({{\delta_1},{\delta_2}} )= ({{\omega_m}, - {\omega_m}} )$, which is about 7 fm. In Fig. 2, the amplitude of the USSG in this double-cavity system is approximately two orders of magnitude than a typical optomechanical system, while the LSSG is increased by around three orders of magnitude.

 

Fig. 2. The diagram of the efficiencies (in logarithmic form) of (a) USSG and (b) LSSG versus pump power ${P_\textrm{c}}$ and detuning frequency ${\delta _1}$ for ${\delta _2} ={-} 0.05{\omega _m}$. (c) Plot of the efficiency of upper (lower) sum sideband generation vs ${\delta _1}$ with ${P_\textrm{c}} = 20{\mathrm{\mu} \mathrm{W}}$. (d) Color graph of the amplitude of the MR (in femtometers) as a function of ${\delta _1}$ and ${\delta _2}$ at sum sideband. The other parameters are $m = 20\textrm{ng}$, ${G / {2\pi }} ={-} 12.0\textrm{GHz/nm}$, ${{{\gamma _m}} / {2\pi }} = 41.0\textrm{kHz}$, ${\kappa / {2\pi }} = 15.0\textrm{MHz}$, ${{{\omega _m}} / {2\pi }} = 51.8\textrm{MHz}$, ${P_1} = {P_2} = 1{\mathrm{\mu} \mathrm{W}}$, ${\lambda _c} = 532\textrm{nm}$, ${\Delta _1} ={-} {\omega _m}$, ${\Delta _2} = {\omega _m}$.

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The dissipation rate of the optomechanical cavity is an important parameter in the cavity optomechanical system. Then we discuss the properties of the upper (lower) sum sideband in the resolved $({\kappa < {\omega_m}} )$ and unresolved $({\kappa > {\omega_m}} )$ sideband regime. Figure 3 plots the efficiency of SSG for different ${\kappa / {{\omega _m}}}$ versus ${\delta _1}$ in the unresolved sideband regime. We can see from Figs. 3(a)–3(c) that the efficiencies of USSG and LSSG increase as the ratio of ${\kappa / {{\omega _m}}}$ rises in a certain range $({{{1 < \kappa } / {{\omega_m} < 5}}} )$. And the USSG and LSSG have the same peak at ${\delta _1} = {\omega _m}$ when ${\kappa / {{\omega _m}}} = 5$. As shown in Fig. 3(d), the amplitude of LSSG is slightly higher than that of USSG at ${\delta _1} = {\omega _m}$ in the case of ${\kappa / {{\omega _m}}} = 20$. Besides, LSSG changed from bimodal to unimodal and the absorption peak of USSG disappeared. The result shows that the efficiency of SSG can be improved in the unresolved sideband regime by choosing an appropriate detuning frequency ${\delta _1}$.

 

Fig. 3. The efficiency curves of the USSG and LSSG in the unresolved sideband regime vary with ${\delta _1}$, where ${{{\omega _m}} / {2\pi }} = 51.8\textrm{MHz}$. The ratio of the decay rate of the cavity fields versus the frequency of the MR for (a), (b), (c) and (d) is ${\kappa / {{\omega _m}}} = 1$, ${\kappa / {{\omega _m}}} = 2$, ${\kappa / {{\omega _m}}} = 5$ and ${\kappa / {{\omega _m}}} = 20$, respectively. Other parameters are the same as those in Fig. 2.

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In Fig. 4, the efficiency of SSG for different ratios of ${\kappa / {{\omega _m}}}$ varies with ${\delta _1}$ in the resolved sideband regime. It can be seen from Fig. 4(a), the amplitude of USSG is about two orders of magnitude higher than that of LSSG at ${\delta _1} = {\omega _m}$. However, the LSSG is more efficient at ${\delta _1} = 1.05{\omega _m}$. Figure 4(b) clearly shows that the efficiency of USSG is approximately three orders of magnitude greater than that of LSSG at ${\delta _1} = {\omega _m}$, and the amplitude difference between the USSG and LSSG is decreasing with the ratio of ${\kappa / {{\omega _m}}}$ at ${\delta _1} = 1.05{\omega _m}$. The efficiency difference between USSG and LSSG reaches five orders of magnitude when ${\delta _1} = {\omega _m}$, as shown in Fig. 4(c). In addition, we find that the amplitudes of USSG and LSSG are equal in the case of ${\delta _1} = 1.05{\omega _m}$. The results indicate that the efficiency of the USSG is consistently superior to that of the LSSG when the detuning frequency ${\delta _1} = {\omega _m}$ is satisfied. In parallel, the amplitude difference between USSG and LSSG decreases at ${\delta _1} = {\omega _m}$ and increases at ${\delta _1} = 1.05{\omega _m}$ with the reduce of ${\kappa / {{\omega _m}}}$. With the decrease of the ratio of ${\kappa / {{\omega _m}}}$, the amplitudes of USSG and LSSG are also reducing in the resolved sideband regime, as shown in Figs. 4(a)–4(c). It is notable that the efficiency of SSG decreases significantly in the resolved sideband regime, while the peak structures of USSG and LSSG are almost unchanged.

 

Fig. 4. The efficiency diagrams of USSG and LSSG in resolved sideband regimes with ${\delta _1}$. Other parameters are the same as those in Fig. 3.

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To describe the SSG more specifically, Fig. 5 shows the dependence of the efficiency of SSG on the detuning frequency ${\delta _1}({{\delta_2}} )$ of the probe field. As shown in Figs. 5(a) and 5(b), the 3D images show the peak structure of the USSG and LSSG. From Fig. 5(a), we can see that two reinforcement belts similar to mountain range are produced in the USSG. As expected, the USSG is enhanced at ${\delta _1} ={\pm} {\omega _m}$, which corresponds to the matching condition in Fig. 2(d). It is clear from Fig. 5(c) that excluding the four points $({ \pm {\omega_m},0} )$ and $({0, \pm {\omega_m}} )$, the efficiency of USSG is significantly increased at the four straight lines ${\delta _1} ={\pm} {\omega _m}$ and ${\delta _2} ={\pm} {\omega _m}$, which is different from the matching condition of SSG in general optomechanical systems. In comparison, ten sharp peaks are generated for the LSSG in Fig. 5(b), which indicates that the LSSG can be dramatically reinforced at these ten points. The matching condition for the LSSG is ${\delta _1} ={\pm} {\omega _m}$, ${\delta _2} ={\pm} {\omega _m}$, ${\delta _1} + {\delta _2} ={-} {\omega _m}$ and ${\delta _1} + {\delta _2} = {\omega _m}$ as shown in Fig. 5(d). It is worth noting that the maximum efficiency of LSSG can be achieved at these four points $({ - {\omega_m},{\omega_m}} )$, $({{\omega_m}, - {\omega_m}} )$ and $({ \pm {\omega_m},0} )$. The physical interpretation of these phenomena is that these four points are close to the resonance conditions for the maximum amplitude of the MR.

 

Fig. 5. The efficiencies (in logarithmic form) of USSG and LSSG as a function of detuning frequency ${\delta _1}$ and ${\delta _2}$. Other parameters are the same as those in Fig. 2.

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4. Conclusion

In summary, we have studied the characteristics of sum sideband generation via double radiation pressure. The results demonstrate that the efficiency of the LSSG is enhanced with more than three orders of magnitude compared to that in the general optomechanical system. Meanwhile, the USSG is improved for two orders of magnitude and its matching conditions are also altered. We also investigate the optimal matching conditions for the SSG in the resolved and unresolved sideband regimes. A threshold condition is determined. Under this condition, the efficiencies of USSG and LSSG tend to the same maximum value, and the maximum amplitude of the LSSG can be increased by approximately five orders of magnitude. Furthermore, the peak structures of USSG and LSSG can be changed under specific parameter conditions, and the LSSG is more efficient than the USSG. This research will deepen the understanding of nonlinear optomechanical interactions in resolved and unresolved sideband regimes and provide a method for quantum information processing and precision measurements.