1. Introduction

Cavity optomechanical system [1–4] studies the interaction between mechanical oscillators and cavity photon modes, and is used for studies of quantum mechanics foundations [5], quantum precision measurement [6], and quantum information processing [7,8]. Due to the inherent nonlinearity of optomechanical interactions, many interesting phenomena and effects have been revealed, such as cooling of mechanical resonators [9,10], slow light [11,12], photon blockade [13–15], high-order sideband generation [16–21], chaos [22,23], and optomechanically induced transparency (OMIT) [11,24–26]. OMIT is a kind of induced transparency enabled by the radiation-pressure coupling in the case of a single-probe-field-driven cavity optomechanical system, which can be explained by linearizing the Heisenberg-Langevin equations. Further, if a double- or multiple-probe-field-driven is considered applying to the optomechanical systems, there will be more interesting phenomena such as sum- and difference-sideband generation [27,28].

The sum- and difference-sideband generation are the output spectral components of the higher-order sideband [24] generated by the nonlinear optomechanical interactions, which can be obtained by the perturbation method. However, different from one of the higher-order sidebands, the second-order sideband [29,30], which reveals the nonlinear quantum nature of the optomechanical interactions in an optomechanical system with a single-probe-field-driven, the sum- and difference-sideband generation originate from the nonlinear process in a double-probe-field-driven optomechanical system. Recently, the nonlinear characteristics of double-probe-field-driven optomechanical systems have been discussed, and the spectral signals of sum- and difference-sideband generation have been analytically proved, which may provide measurement with higher precision for determination of parameters including coupling rate [6], weak force [31,32], mass [33,34], displacement [35,36], electrical charges [37,38] and phonon number [39,40] of optomechanical systems.

In recent years, we have witnessed the development of cavity optomechanical systems from single-cavity to double-cavity, which can enhance the degree of control of the system. In general, we can classify the coupling between a mechanical resonator and two cavity fields into two categories. In the former case, the mechanical resonator is coupled to one of two cavities, which are in turn tunnel-coupled to each other. In the latter case, the mechanical resonator with reflective coatings on both sides serves as a double-face mirror that is coupled to both the left-side cavity and the right-side cavity. Double-cavity optomechanical systems may be used to realize potentially richer behavior, such as quantum entanglement [41,42], OMIT [43,44], squeezing [45], optical bistability [46], nonreciprocity [47,48], and higher-order sideband generation [19,49], the efficiency of which is very weak and could not be easily detected and utilized in experiments. With the attention paid to the problem, a number of approaches have been proposed to improve and control the efficiency of sideband generation. For example, Laguerre-Gaussian optical sum-sideband generation via orbital angular momentum exchange has been theoretically studied [50]. Also, the radiation pressure induced sum-sideband generation in an optomechanical system with an optical parametric amplifier (OPA) has been proposed to enhance the sideband generation [51,52]. Moreover, both parity-time (PT) -symmetric cavity optomechanical system and mechanical PT-symmetric system have been structured to improve the efficiency of sideband generation by making use of exceptional points (EPs) [53,54].

In the present work, we propose to study the difference-sideband generation in a double-cavity optomechanical system with nonreciprocal coupling. The double-cavity optomechanical system is driven by a strong control field with the frequency $\omega _c$ and two relatively weak probe fields with different frequencies ($\omega _{p.1}$ and $\omega _{p.2}$, respectively). According to the general nonlinear analysis methods proposed previously [28], we show that there are output spectral components at the difference-sideband (with frequency $\pm$($\delta _1$ - $\delta _2$) in a frame rotating at $\omega _c$) in such a special optomechanical system. Although the amplitudes of difference-sideband generation are often small due to the weak optomechanical interactions, the efficiencies of difference-sideband generation can be enhanced significantly when the suitable matching conditions [28] are met. Compared with previous studies of sum- and difference-sideband generation, we discuss the difference-sideband generation in a unique double-cavity optomechanical system with nonreciprocal coupling, which can provide new control degrees of freedom for the difference-sideband generation. This allows more flexibility in regulating the generation of the difference-sideband and exploring more optimal configurations of the parameters for obtaining high efficiencies of difference-sideband generation at low power.

Here, we summarise the main features of this work. First, the high efficiency of difference-sideband generation can be obtained at low power, which is comparable to the difference-sideband generation in some systems with gain. Second, by regulating the nonreciprocal coupling of two cavities, different efficiencies of difference-sideband generation can be obtained, which highlights the tunability of the nonreciprocal coupling strength between two cavities. Finally, we investigate the optimal matching conditions of upper and lower difference-sideband generation in the certain two-cavity coupling strengths. We find that when the matching conditions are met, the efficiencies of the difference-sideband generation are greatly improved, and moreover, new matching conditions are discovered to strengthen the sideband efficiency, the emergence of which is attributed to the strong oscillation of the mechanical oscillator when the pump power increases to a certain value. The enhancement of difference-sideband signals in the double-cavity optomechanical system with nonreciprocal coupling provides a more efficient platform for optical signal processing, precision measurement, and experimental research. Additionally, this system is also applied to the control of the sum-sideband generation.

The structure of this paper is as follows. In Sec. 2, a double-cavity optomechanical system with nonreciprocal coupling is described and the evolution of the Heisenberg-Langevin equations of motion for the system is given. In Sec. 3, we discuss a series of parameters that affect the difference-sideband generation, including the pump power, two-cavity coupling strength and frequency detuning of the probe field, and give the physical interpretation of the phenomena in detail. Finally, in Sec. 4, a conclusion of the results is summarized.

2. Model and equations in a double-cavity optomechanical system with nonreciprocal coupling

The double-cavity optomechanical system we propose consists of an empty cavity $a_2$ and an optomechanical cavity $a_1$, which is formed by a fixed end mirror and a moving one regarded as a mechanical oscillator shown schematically in Fig. 1. The two optical cavities are coupled together through the tunneling effect and the coupling strengths between two cavities are nonreciprocal ($\lambda _1\neq \lambda _2$), which can be achieved by making use of an imaginary gauge field [55], impurity [56], or an auxiliary nonreciprocal transition device [57]. The double-cavity optomechanical system is driven from the cavity $a_2$, and the input driving fields include a strong control field with frequency $\omega _c$ and two relatively weak probe fields (with frequencies $\omega _{p.1}$ and $\omega _{p.2}$). In addition, the dissipation rates of two cavities are $\kappa _1$ and $\kappa _2$, respectively. The vibration frequency and the dissipation rate of the mechanical oscillator are $\omega _m$ and $\gamma _m$ respectively. In a frame rotating at $\omega _c$, when the control field and two probe fields are applied to the double-cavity system, there are output fields with frequencies $\pm \Omega$. Here $\Omega$ represents the difference between the frequency detuning of the two probe fields and the control field, i.e., $\delta _1 - \delta _2$. The Hamiltonian formulation of this system can be written as follows [58]:

 

Fig. 1. (a) Schematic diagram of a double-cavity optomechanical system with nonreciprocal coupling. The system consists of two optical cavities and a mechanical oscillator, where the coupling strengths between the two cavities are $\lambda _1$ and $\lambda _2$. The system is driven by a strong control field with frequency $\omega _c$ and two relatively weak probe fields with frequencies $\omega _{p.1}$ and $\omega _{p.2}$, respectively. (b) The frequency spectrogram of the difference-sideband generation in the optomechanical cavity $a_1$. The control field (blue line) is detuned by $\Delta _c$ from cavity field $a_1$. When the system is in a frame rotating at frequency $\omega _c$, there are output fields with frequency $\pm \Omega$.

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In a frame rotating at $\omega _c$, introducing the dissipation and noise to the above Hamiltonian, the Heisenberg-Langevin equations of the double-cavity optomechanical system can be written as follows:

In this work, we are interested in the mean response, so the operators can be simplified to their expected values, viz. $\langle \hat {a}_{1}\rangle =a_{1}$, $\langle \hat {a}_{2}\rangle =a_{2}$, $\langle \hat {b}\rangle =b$. In this case we reduce the operator equations to the mean value equations, the quantum noise and the thermal noise terms can be dropped safely because their expectation values, viz. $\langle \hat {a}_{\mathrm {1.in}}(t)\rangle$, $\langle \hat {a}_{\mathrm {2.in}}(t)\rangle$, $\langle \hat {b}_{\mathrm {in}}(t)\rangle$ are zero in the semiclassical approximation. Therefore, the resulting mean value equations are as follows:

To verify this conjecture, we plot Fig. 2 to show that photon number in cavity $a_1$ varies with the pump power of the control field for different coupling strengths $\lambda _1$ and $\lambda _2$. There are three branches in Fig. 2, which can be distinguished by the extremum points obtained by taking the derivative of Eqs. (8–10). When the power increases slowly from 0, the system changes slowly along the lower branch of $PQ$. Once the power increases to point $Q$, the system state will transition from point $Q$ to point $Q^{'}$ overleaping the middle curve, and then evolves in the upper branch. Similarly, if the system starts in the upper branch $Q^{'}\;P^{'}$, as the power slowly decreases to point $P^{'}$, the system will jump from point $P^{'}$ to point $P$, and then evolves along the lower branch. The phenomenon that this state changes not along the original route but along the $PQQ^{'}\;P^{'}\;P$ route is called hysteresis phenomenon [59].

 

Fig. 2. Photon number in cavity $a_1$ as a function of the pump power $P_c$ for different coupling strengths $\lambda _{1,2}$. (a) $\lambda _2= 0.8\kappa _1$, and (b) $\lambda _1=0.8\kappa _1$. The other parameters are $g/2\pi =25.2$ KHz, $\gamma _m/2\pi =41$ KHz, $\kappa _1/2\pi = 15$ MHz, $\omega _m/2\pi = 51.8$ MHz, $\Delta _1=\omega _m$, $\Delta _2=\Delta _1$, $\kappa _2=\kappa _1$, $\varepsilon _1=\varepsilon _2=0.05\varepsilon _c$, which are adopted from recent relevant studies [9,24,49] and are available in the experiments.

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Although there are three branches, the upper and lower branches [seeing the solid lines] are always stable for the system, but the middle branch is unstable [seeing the green dotted lines], so it is often called bistable [59]. To better describe the difference-sideband generation, we need the parameter region with only one solution, and in the following work, the power we take is much smaller than the milliwatt level. Figures 2(a) and 2(b) respectively show the influence of different coupling strengths $\lambda _1$ and $\lambda _2$ on bistable curves. We find that with the increase of $\lambda _1$, the photon number in cavity $a_1$ increases, while with the increase of $\lambda _2$, the photon number decreases, showing a completely opposite state, and it agrees with the evolution of Eqs. (8–10). Moreover, this is also consistent with the distribution of photon number in two optical cavities obtained by our physical analysis, which satisfies the law of conservation of energy. In addition, an interesting phenomenon is that for the same value of $\lambda _1$ and $\lambda _2$, we get different number of photons. For example, when the pump power is 3 mW and $\lambda _1=\lambda _2$, the number of photons presented is significantly different, $2.622\times 10^{6}$ and $2.475\times 10^{6}$ respectively. The main reason for generating different photon numbers is the presence of the nonlinear term $\hbar g{\hat {a}_1}^{\dagger }\hat {a}_1(\hat {b}^{\dagger }+\hat {b})$ in the Hamiltonian of the system, as well as the two-cavity coupling terms $\hbar \lambda _1{\hat {a}_1}^{\dagger }\hat {a}_2$ and $\hbar \lambda _2\hat {a}_1{\hat {a}_2}^{\dagger }$. And at the same time, we can find through Eqs. (5–7) that the coupling strengths $\lambda _1$ and $\lambda _2$ have different effects on the photon number of the system. Therefore by changing the two non-reciprocal coupling strengths $\lambda _1$ and $\lambda _2$, different photon numbers can be obtained. This reflects that the essentially nonreciprocal coupling strength of the system brings about different sideband effects. The double-cavity photomechanical system we consider here provides new adjustable variables for the bistable behavior of the intracavity photon number, where $\lambda _1$ is more tunable than $\lambda _2$ by comparing Figs. 2(a) and 2(b).

We solve the evolution equations of the double-cavity optomechanical system by the perturbation method, and write the operators as follows: $a_i=\bar {a}_i+\delta a_i$, $a_i^*=\bar {a}_i^*+\delta a_i^*\quad (i=1,2)$, $b=\bar {b}+\delta b$, $b^*=\bar {b}^*+\delta b^*$, denoting the steady state and perturbation solutions respectively. Substituting the ansatz into Eqs. (5–7), the Heisenberg-Langevin equations become:

3. Results and analysis of upper and lower difference-sideband generation

In contrast to the normal double-cavity optomechanical systems, the unique feature of our proposed double-cavity optomechanical system is that the coupling between the two cavities is nonreciprocal, which gives the system new characteristics. By modulating the coupling strengths $\lambda _{1}$ and $\lambda _{2}$, we find some interesting phenomena in the difference-sideband generation. One is the nonreciprocal coupling strength leading to different efficiencies of difference-sideband generation at low power. The other is that we find new matching conditions, under which the efficiencies of the upper difference-sideband can be effectively increased.

A high dependence of the efficiencies of difference-sideband generation on the pump power is observed, as shown in Fig. 3. Figures 3(a) and 3(b) show the efficiencies (in logarithmic form) of upper and lower difference-sideband generation as a function of the frequency detuning of the first probe field $\delta _1$ and the pump power $P_c$. There is an obvious pointed structure in Fig. 3(a), as well as two very sharp structures appear in Fig. 3(b), and the reason for the phenomena is that the generation of difference-sidebands satisfies the matching conditions [28], which can improve the efficiency of the difference-sideband generation. In addition, due to the weak optomechanical nonlinearity in the system, the efficiency of difference-sideband generation is not very high as a whole.

 

Fig. 3. (a) and (b) The efficiencies (in logarithmic form) of the upper and lower difference-sideband generation vs pump power $P_c$ and frequency detuning $\delta _1$ of the first probe field respectively. (c) and (d) The efficiencies (in logarithmic form) of the upper and lower difference-sideband generation for different pump powers versus the frequency detuning $\delta _1$ with $\lambda _1 = \lambda _2 = \kappa _1$, $\delta _2 = 0.05\omega _m$ respectively. (e) The efficiencies (in logarithmic form) of the upper and lower difference-sideband generation versus the pump power $P_c$ with $\delta _1 = 1.05\omega _m$, $\delta _2 = 0.05\omega _m$. Other parameters are the same as in Fig. 2.

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In order to clearly observe the matching conditions corresponding to these protruding structures, we further show the efficiencies of upper and lower difference-sideband generation as a function of the frequency detuning of the first probe field $\delta _1$ for different pump powers $P_c$ of the control field in Figs. 3(c) and 3(d). As the power increases in Fig. 3(c), the efficiency of difference-sideband generation increases monotonically, except at $\delta _1=1.05\omega _m$, which is as a result of the weak interference between multiple scattering paths. For example, the solid blue line (corresponding to the pump power of 10 $\mu$W) is located above the dashed red line (corresponding to the pump power of 14 $\mu$W) at the resonance point $\delta _1=1.05\omega _m$. When $\delta _1=1.05\omega _m$, there are the sharp peaks, because the parameter choice of $\delta _1$ and $\delta _2$ satisfies the relation $\mid \delta _1-\delta _2\mid =\omega _m$, which is one of the matching conditions to achieve the maximum upper difference-sideband generation [28]. Moreover, the linewidth near at $\delta _1=1.05\omega _m$ also widens monotonically with increasing the pump power. The lower difference-sideband generation in Fig. 3(d) shows the curves similar to that in Fig. 3(c). The only difference is that at point $\delta _1=\omega _m$, the change of the difference-sideband generation is more the same as that at point $\delta _1=1.05\omega _m$, because this is another matching condition for the strong lower difference-sideband generation.

Figure 3(e) shows the calculation results of the efficiencies of the upper (solid red line) and lower (solid blue line) difference-sideband generation for different pump powers $P_c$ at $\delta _1 = 1.05\omega _m$ and $\delta _2 = 0.05\omega _m$. The difference $\mid \delta _1-\delta _2\mid =\omega _m$ satisfies the matching condition for the maximum difference-sideband generation. We can intuitively observe that the efficiencies of both the upper and lower difference-sideband generation do not monotonically increase with the increase of power, but reach their maximum value near at $P_c=10\mu$W. Theoretically, this can be explained by the system reaching saturation due to an increase in pump power to a certain value [24].

In addition to the pump power of the control field, the coupling strengths $\lambda _{1}$ and $\lambda _{2}$ between the two cavities also affect the difference-sideband generation. Due to the non-reciprocity of two coupling strengths, we find an interesting phenomenon that the difference-sideband generation regulated by $\lambda _1$ and $\lambda _2$ is different, and the sideband effects they modulate are exactly opposite, by comparing Fig. 4 and Fig. 5. Figure 4 and Fig. 5 respectively show the efficiencies (in logarithmic form) of the upper and lower difference-sideband generation versus $\delta _1$ with different coupling strengths $\lambda _1$ and $\lambda _2$. As shown in Fig. 4, the efficiencies of difference-sideband generation monotonously increase with the increase of the $\lambda _1$. According to the matching condition $\mid \delta _1-\delta _2\mid =\omega _m$, it can be seen one apparent peak located at $\lambda _1=1.05\omega _m$, but there is other small peak located at $\delta _1=\omega _m$, which can be improved by the pump power in Figs. 4(a) and 4(c), as we have illustrated in Figs. 3(a) and 3(c). For the lower difference-sideband generation in Figs. 4(b) and 4(d), in addition to the matching condition mentioned above, $\delta _1=\omega _m$ is another matching condition to maximize the efficiency of sideband generation, so we can observe two obvious peaks, located at $\delta _1=\omega _m$ and $\delta _1=1.05\omega _m$ respectively. It is worth noting that at the resonance point $\delta _1=1.05\omega _m$, the efficiencies of difference-sideband generation do not monotonically increase with the increase of coupling strength $\lambda _1$, such as the green dot line (corresponding to the coupling strength of 1.2$\kappa _1$) located below the solid red line (corresponding to the coupling strength of $\kappa _1$), and we find that under this experimental parameters, when $\lambda _1=\kappa _1$, the efficiency of difference-sideband generation reaches the best, which means that the sideband efficiency can be adjusted by the coupling strengths between the two cavities.

 

Fig. 4. (a) and (b) The efficiencies (in logarithmic form) of the upper and lower difference-sideband generation vs coupling strength $\lambda _1$ and frequency detuning $\delta _1$ of the first probe field respectively. (c) and (d) The efficiencies (in logarithmic form) of the upper and lower difference-sideband generation for different coupling strengths $\lambda _1$ versus the frequency detuning $\delta _1$ with $P_c = 10\mu$W, $\lambda _2 = \kappa _1$, $\delta _2 = 0.05\omega _m$ respectively. Other parameters are the same as in Fig. 2.

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Fig. 5. (a) and (b) The efficiencies (in logarithmic form) of the upper and lower difference-sideband generation vs coupling strength $\lambda _2$ and frequency detuning $\delta _1$ of the first probe field respectively. (c) and (d) The efficiencies (in logarithmic form) of the upper and lower difference-sideband generation for different coupling strengths $\lambda _2$ versus the frequency detuning $\delta _1$ with $P_c = 10\mu$W, $\lambda _1 = \kappa _1$, $\delta _2 = 0.05\omega _m$ respectively. Other parameters are the same as in Fig. 2.

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Similarly, we show the efficiencies (in logarithmic form) of the upper and lower difference-sideband generation for different coupling strengths $\lambda _2$ in Fig. 5. Different from Fig. 4, we find that the efficiencies of difference-sideband generation relatively get smaller and the linewidths narrow monotonically with increasing the $\lambda _2$ except for at a resonance point $\delta _1=1.05\omega _m$ in Fig. 5. More interestingly, at the resonance point, the efficiencies of difference-sideband generation increase with the increase of the $\lambda _2$, as shown the partially enlarged part in Fig. 5(d), which is the exactly opposite of the change in sideband efficiency at other points. When $\lambda _2=0.8\kappa _1$, the sideband efficiency reaches a local maximum at the peak structure in Figs. 5(c) and 5(d). In addition, the upper difference-sideband generation in Fig. 5(c) shows a local depression at $\delta _1 = \omega _m$ when $\lambda _2 = 0.2\kappa _1$, which will present a new small peak structure as the $\lambda _2$ increases. Based on Fig. 4 and Fig. 5, we can conclude that the nonreciprocal couplings $\lambda _1$ and $\lambda _2$ bring about different efficiencies of difference-sideband generation, and the sideband effect is completely opposite. Moreover, when the matching condition is satisfied, that is at the resonance point $\delta _1=1.05\omega _m$ for the upper difference-sideband, and at the resonance points $\delta _1=\omega _m$ and $\delta _1=1.05\omega _m$ for the lower difference-sideband, we can maximize the efficiency of difference-sideband generation by jointly adjusting $\lambda _1$ to $\kappa _1$ and $\lambda _2$ to $0.8\kappa _1$.

For the sake of more intuitively showing the adjustability and difference of the two coupling strengths ($\lambda _1$ and $\lambda _2$) on the efficiencies of upper and lower difference-sideband generation, we further plot Fig. 6. Figure 6(a) shows that when the difference-sideband generation changes with $\lambda _1$, then $\lambda _2$ is $0.8\kappa _1$, and when the difference-sideband generation changes with $\lambda _2$, then $\lambda _1$ is $\kappa _1$. We find that as shown the blue lines (including solid and dashed lines), with the $\lambda _1$ increasing, both the efficiencies of upper and lower difference-sideband generation monotonically increase until they reach their respective maximum values. For example, when $\lambda _1$ is taken as 0.9$\kappa _1$, its corresponding maximum efficiency of upper difference-sideband (bule solid line) is −1.039. Then, increasing the value of $\lambda _1$ further, the efficiencies of upper and lower difference-sideband generation monotonically decrease. The result is in complete agreement with Fig. 4, that is, at the matching condition $\mid \delta _1-\delta _2\mid =1.05\omega _m-0.05\omega _m=\omega _m$, the value of $\lambda _1$ does not monotonically affect the difference-sideband generation, and there is an optimal value that maximizes the efficiencies of upper and lower difference-sideband generation. By comparing the red curve and the blue curve, we can find that on the whole, the adjustment results of $\lambda _2$ to the difference-sideband generation and $\lambda _1$ to the difference-sideband generation are different. With the increase of $\lambda _2$, the efficiencies of the difference-sideband generation change very slowly and tend to weaken, while with the increase of $\lambda _1$, the efficiencies of the sidebands are first sharply enhanced and then level off. In the sideband process of coupling strength regulation, we select the best coupling strength to make the effect of difference-sideband generation the best. The state of the red curve evolving with $\lambda _2$ is also consistent with the curve change in Fig. 5.

 

Fig. 6. The efficiencies of upper and lower difference-sideband generation as a function of coupling strengths $\lambda _1$ and $\lambda _2$. We use (a) $P_c = 10\mu$W, $\lambda _1 =\kappa _1$, $\lambda _2 =0.8\kappa _1$ and (b) $P_c = 20\mu$W, $\lambda _1=\lambda _2 =\kappa _1$. $\delta _1 = 1.05\omega _m$, $\delta _2 = 0.05\omega _m$. Other parameters are the same as in Fig. 2.

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According to Fig. 6(a), we know that the highest efficiency of difference-sideband generation can be obtained by adjusting the coupling strengths $\lambda _1$ and $\lambda _2$, and the effects of $\lambda _1$ and $\lambda _2$ on sideband regulation are different. To further explore the properties of the system, we take the condition of $\lambda _1=\lambda _2$, and then observe the evolution of difference-sideband generation with coupling strengths $\lambda _1$ and $\lambda _2$ respectively in Fig. 6(b). We find that except for a few points, the nonreciprocal coupling strength leads to different efficiencies of the difference-sideband generation, especially within the range of coupling strengths ($\lambda _{1,2}<\kappa _1$), at which it greatly highlights the different responses of $\lambda _1$ and $\lambda _2$ to sideband regulation. Compared to Fig. 6(a), we increase the pump power to 20 $\mu$W in Fig. 6(b), which changes the value of $\lambda _1$ for optimal difference-sideband effect, but the overall curve evolution does not change much. By observing the generation of the difference-sidebands modulated by $\lambda _2$, it can be found that the red curve becomes very flat, that is, the adjustment of $\lambda _2$ to the sideband has been significantly weakened. This is consistent with the results in Fig. 2(b), that is to say, compared to $\lambda _1$, the adjustability of $\lambda _2$ is relatively weak.

Finally, in order to understand more fully the process of difference-sideband generation in our double-cavity optomechanical system with nonreciprocal coupling, we plot the efficiencies (in logarithmic form) of the upper and lower difference-sideband generation versus the frequency detunings $\delta _1$ and $\delta _2$ of the probe fields in Fig. 7 and Fig. 8 separately. The efficiencies of the upper difference-sideband generation are shown in Fig. 7(a), which exhibits an “ancient goblet" structure, corresponding to four bright lines in Fig. 7(b). Especially in the range of $\delta _1$ greater than 0, the sideband efficiency is higher. At $\delta _2=\pm \omega _m$, which is the matching condition [28] of the upper difference-sideband generation with high efficiency, two bright lines occur. The other two brighter lines are located at $\mid \delta _1-\delta _2\mid =\omega _m$, which is also a matching condition. In addition, when the two matching conditions are superimposed, such as at $\delta _2=\omega _m$ and $\delta _1=0$, we find that the efficiencies of difference-sideband generation get greatly enhanced relative to nearby points, which is the result of the coherent superposition of the two resonant paths. Besides, there are two avoid crossing blue lines, which represents the absorption peaks at these locations. The two lines vary in distance from far to near, but do not intersect, reaching the closest distance and then recrossing away. The anticrossing blue lines appear due to the strong coupling between two cavities, which are affected by the nonreciprocal coupling strength. The efficiencies of upper difference-sideband generation vary with $\delta _1$ for different pump powers shown in Figs. 7(c) and 7(d). Since one of the matching conditions ($\delta _2=-\omega _m$) is satisfied in Fig. 7(c), its curve does not change much and the sideband efficiency basically saturates when the power increases to a certain value (10$\mu$W). Similar to the results of previous work [28], when $\mid \delta _1-\delta _2\mid =\omega _m$, the efficiencies of upper difference-sideband generation are strengthened. As shown in Fig. 7(d), there are two peaks (peak 2 and peak 4) at $\delta _1=-0.5\omega _m$ and $\delta _1=1.5\omega _m$. Moreover, we find a new phenomenon that with increasing power, new peaks (peak 1 and peak 3) occur at $\delta _1=\pm \omega _m$, especially at $\delta _1=\omega _m$, which can be understood as the enhancement of sideband efficiency by the strong resonance of the mechanical oscillation, from the original trough to the local maximum.

 

Fig. 7. (a) Three-dimensional plot of the efficiencies (in logarithmic form) of the upper difference-sideband generation versus the frequency detunings $\delta _1$ and $\delta _2$ of the probe fields. (b) Top view of the efficiencies (in logarithmic form) of the upper difference-sideband generation versus the frequency detunings $\delta _1$ and $\delta _2$. (c) and (d) Two-dimensional plot of the efficiencies of the upper difference-sideband generation varies with frequency detuning $\delta _1$ for different pump powers $P_c$ with $\delta _2 = -\omega _m$ and $\delta _2 = 0.5\omega _m$ respectively. Here we set $\lambda _1=\kappa _1$, $\lambda _2 = 0.8\kappa _1$, and other parameters are the same as in Fig. 2.

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Fig. 8. (a) Three-dimensional plot of the efficiencies (in logarithmic form) of the lower difference-sideband generation versus the frequency detunings $\delta _1$ and $\delta _2$ of the probe fields. (b) Top view of the efficiencies (in logarithmic form) of the lower difference-sideband generation versus the frequency detunings $\delta _1$ and $\delta _2$. (c)and (d) Two-dimensional plot of the efficiencies of the lower difference-sideband generation varies with frequency detuning $\delta _1$ for different pump powers $P_c$ with $\delta _2 = 1.5\omega _m$ and $\delta _2 = 0.5\omega _m$ respectively. Here we set $\lambda _1=\kappa _1$, $\lambda _2 = 0.8\kappa _1$, and other parameters are the same as in Fig. 2.

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The efficiencies of the lower difference-sideband generation are shown in Fig. 8(a). Compared to Fig. 7(a), Fig. 8(a) exhibits a "goblet" structure with a different opening direction, which is due to the different effects of $\delta _1$ and $\delta _2$ on the efficiencies of the upper and lower difference-sideband generation. Immediately afterwards, the bright area observed in the top view Fig. 8(a) is also shifted, mainly concentrated in the upper part of $\delta _2=0$. When one of matching conditions is met, at the mechanical resonance point $\delta _1=\pm \omega _m$ that the beating between the control field and one of the probe fields has a resonance with the mechanical eigenfrequency, the mechanical oscillation of the torsional pendulum becomes significant. Then the efficiencies of the lower difference-sideband generation can get enhanced, so we can observe the peaks of the two tips (peak 1 and peak 3) in Figs. 8(c) and 8(d). As with the upper difference-sideband, the another matching condition is $\mid \delta _1-\delta _2\mid =\omega _m$, where we find two peaks (peak 2 and peak 4) at $\delta _1=-0.5\omega _m$ and $\delta _1=1.5\omega _m$ in Fig. 8(d), but one peak (peak 2) at $\delta _1=0.5\omega _m$ in Fig. 8(c). There should also be another peak in Fig. 8(c), but according to the matching condition $\mid \delta _1-\delta _2\mid =\omega _m$, the location of the peak is at $\delta _1=-2.5\omega _m$, which is out of parameter range $\delta _1\in (-2\omega _m,2\omega _m)$. In contrast to Fig. 7(d), the efficiency of the difference-sideband generation is higher at the resonance points $\delta _1=\pm \omega _m$ in Fig. 8(d), because $\delta _1=\pm \omega _m$ is the matching condition for maximizing the efficiency of lower difference-sideband generation.

4. Conclusions

In summary, we have studied the upper and lower difference-sideband generation in a double-cavity optomechanical system with nonreciprocal coupling between two cavities. A derivation of the process and specific expressions for the difference-sideband generation have been given. Based on the steady-state equations, we show the tunability of the nonreciprocal coupling strengths ($\lambda _1$ and $\lambda _2$ ) to the bistable state of the system. To obtain the high sideband efficiency, we have discussed the experimentally available parameters on the difference-sideband generation, including the coupling strength, pump power and frequency detuning of the probe field. We find that the nonreciprocal coupling strengths lead to different efficiencies of difference-sideband generation. Moreover, when the matching conditions are met, the difference-sideband generation can be remarkable even at low power. In addition to the matching conditions $\delta _2=\pm \omega _m$ and $\mid \delta _1-\delta _2\mid =\omega _m$ applied to the upper difference-sideband, and $\delta _1=\pm \omega _m$ and $\mid \delta _1-\delta _2\mid =\omega _m$ applied to the lower difference-sideband proposed by previous studies, we further find new matching conditions $\delta _1=\pm \omega _m$, which can enhance the efficiency of upper difference-sideband generation, when the pump power reaches a certain value. This investigation may provide further insight into the understanding of the optomechanical nonlinear interaction with nonreciprocal coupling and find applications in nonreciprocal optical frequency combs and communications, precision measurements, on-chip manipulation of light. In the current paper, as an example, we have only investigated the difference-sideband generation, this special cavity optomechanical system is also suitable for the study of sum-sideband generation.