1. Introduction

Benefiting from the rapid development of nanotechnology in recent years, cavity optomechanics as the cross field of nanophysics and quantum optics has received extensive attention and undergone rapid growth. The field of cavity optomechanics, which studies the nonlinear coupling interaction between the mechanical vibration and the optical cavity field via radiation pressure [1], has a wide range of applications in gravitational-wave detection [2], preparation of macroscopic-scale quantum entanglement [3,4], and monitoring of mechanical motion [5]. As is well known, in a coupled optomechanical system how the mechanical vibration interacts to the optical cavity field depends on the position of the mechanical resonator (MR) in the optical cavity field [6]. When the position of the MR is at the nodes of the cavity modes, the coupling between the mechanical vibration and the optical cavity field is linear [7–10]. Such the so-called linearly coupled optomechanical system has been extensively studied both experimentally [11–13] and theoretically [14–18] over the past few years, in which many interesting phenomena and effects such as optomechanically induced transparency (OMIT) [19–23], slow light [24–28], cooling of mechanical resonators [29–31] and $\mathcal {PT}$-symmetry optomechanics [32–35] have been investigated. Conversely, when the position of the MR is placed at the antinodes of the cavity modes in an optomechanical system, the optical cavity field could be quadratically coupled to the mechanical vibration [36–39], which is the so-called quadratically coupled optomechanical system [6]. While the underlying physical processes in a quadratically coupled optomechanical system implies two-phonon processes, which are different from that in the linearly coupled optomechanical system in which the underlying physical processes involves one-phonon processes [15,40]. Using the similar procedure to deal with the light-mechanical interaction in the linearly coupled optomechanical system, many interesting phenomena and effects in a quadratically coupled optomechanical system are also verified, with examples such as photon blockade [41], higher-order sidebands generation [42,43] and optical amplification [44,45].

Recently, it is shown that in a linearly coupled optomechanical system driven by one strong control field and two weak probe fields, the output fields at the sum or difference sideband can be generated [46] when the intrinsic nonlinearity arising from optomechanical interactions are taken into account. Subsequently, the potential applications of difference sideband generation to achieve measurement of electric charge (or other weak forces) with higher precision is proposed [47]. Motivated by these works and in consideration of the underlying physical processes in the quadratically coupled optomechanical system being different from that in the linearly coupled optomechanical system, a question arises naturally: Could sum sideband generation (SSG) be obtained when the quadratically coupled optomechanical system is driven by one strong control field and two weak probe fields and the nonlinear terms of the intrinsic nonlinear optomechanical interactions are considered?

On the other hand, it has been reported that the optomechanical interaction can be enhanced in the linearly coupled optomechanical system assisted by an optical parametric amplifier (OPA), which can be used to improve the cooling of the MR [48], the normal mode splitting [49] and the strong mechanical squeezing [50]. Based on these reasons mentioned above, in this work, we propose a scheme to realize the generation and enhancement of sum sideband in a quadratically coupled optomechanical system with parametric interactions. Using experimentally achievable parameters, our results illustrate that not only the output fields at the sum sideband with frequencies $\pm \Omega _{+}$ (in a frame rotating at frequency $\omega _{c}$) can be generated, but also the nonlinear gain of OPA permits us to enhance the efficiencies of SSG. As we expected, even under the condition of weak driving, the existence of OPA makes the efficiencies of SSG significantly enhanced. It is worth noting that in the presence of OPA, the matching conditions of upper sum sideband generation (USSG) are modified. When the nonlinear gain of OPA increases, the linewidth and peak value of the sum sideband window become wider and larger, respectively. In addition, via adjusting the nonlinear gain of OPA, the efficiency of lower sum sideband generation (LSSG) can be enhanced by several orders of magnitude. From the viewpoint of application, enhancement of optomechanically induced SSG may be used for measurement of electric charge (or other weak forces) with higher precision [51] and on-chip manipulation of light propagation. Moreover, the existing mechanism of SSG may be applied to other similar systems, such as artificial molecules [52] and quantum well systems [53].

This paper is structured as follows: In Sec. 2, under the condition of considering the nonlinear terms of the optomechanical dynamics, the Langevin equations and the steady-state solutions of the system are derived. In addition, the amplitude of SSG is given by further analytic calculation with experimentally achievable parameters. In Sec. 3, we analyze the characteristics of SSG and discuss in detail the influence of the system parameters on SSG, especially the effect of the nonlinear gain of OPA on SSG. Finally, a conclusion is summarized in Sec. 4.

2. Physical model and dynamical equation

We consider such a model, a quadratically coupled optomechanical system with a membrane (treated as a MR) and an OPA inserted between two fixed mirrors. And the system is driven by two relatively weak probe fields with the frequencies $\omega _{1}$, $\omega _{2}$ and a strong control field with frequency $\omega _{c}$, as is schematically shown in Fig. 1(a). Generally, we suppose that $\Omega _{m}$ and $R$ are the angular frequency and limited reflectivity of the MR with the effective mass $m$ and the decay rate $\Gamma _{m}$. And $Q=\Omega _{m}/\Gamma _{m}$ is the mechanical quality factor. Moreover, the MR interacts with the environment at the thermal equilibrium temperature $T$. The resonant frequency of the cavity is $\omega _{o}$ which will be obtained when the MR in a balance position in the absence of additional excitation [1].

 

Fig. 1. (a) Schematic diagram of a quadratic optomechanical system consisting of an OPA and a MR inserted between two fixed mirrors. The system is excited by two relatively weak probe fields and a strong control field. The two probe fields are represented by probe field 1 with frequency $\omega _{1}$ and probe field 2 with frequency $\omega _{2}$, respectively. And $\omega _{c}$ is the frequency of the control field. The MR and cavity field are quadratically coupled via the radiation pressure. (b) Frequency spectrogram of sum sideband generation in the quadratically coupled optomechanical system with a control field and two probe fields driven.

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In a frame rotating at frequency $\omega _{c}$, the Hamiltonian of our scheme can therefore be written as [57].

The Heisenberg-Langevin equations of our scheme with Hamiltonian (1) can therefore be written as follows [14]:

Similar to the linearly coupled optomechanical system, we mainly study the mean response of the system to the probe fields in the quadratically coupled optomechanical system. However, we can obtain from Eqs. (2) that the mean values of $\hat {x}$ and $\hat {p}$ are zero which is the difference between the linearly coupled optomechanical system and the quadratically coupled optomechanical system. By further analyzing the evolution equations of the MR displacement and the cavity field amplitude in Eqs. (2), we can know that the MR displacement is decoupled from the cavity field amplitude in steady state. In other words, the mean value of the MR displacement has no effect on the output field. The reason for this feature is that the underlying physical process of a quadratically coupled optomechanical system is a two-phonon process [57]. Consequently, in the following works, we concentrate on calculating the evolution of the mean values of $\hat {a}$, $\hat {x}^2$, $\hat {p}^2$, and $\hat {x}\hat {p}+\hat {p}\hat {x}$, viz., $a(t)\equiv \langle \hat {a}(t)\rangle$, $X(t)\equiv \langle \hat {x}^2(t)\rangle$, $P(t)\equiv \langle \hat {p}^2(t)\rangle$, and $Q(t)\equiv \langle \hat {x}(t)\hat {p}(t)+\hat {p}(t)\hat {x}(t)\rangle$.

Using a so-called mean-field approximation, viz., $\langle abc\rangle =\langle a\rangle \langle b\rangle \langle c\rangle$ [1], the mean value equations can be written in the following forms:

Considering that the process of SSG is a second order process, thus we can get the following equations:

(1) The first group represents the process of the probe field with frequency $\omega _{1}$ ($\omega _{2}$), where we ignore the nonlinear terms of the optomechanical dynamics,

Further, according to the input-output relation [37], we can get the output fields (in a frame rotating at frequency $\omega _{c}$) of the system as follows:

3. Results and discussion

In this section, we will numerically discuss the efficiency of SSG in a quadratically coupled optomechanical system with parametric interactions. In addition, in order to understand the characteristics of SSG more systematically, we also study the effect of system parameters on SSG, especially the effect of the nonlinear gain of OPA on SSG. The amplitude of the input probe field 1 is $\varepsilon _{1}$ , while the amplitude of the output field with the upper (lower) sum sideband is $|\sqrt {\eta \kappa } A_{+s}|$ ($|\sqrt {\eta \kappa }A_{-s}|$). We define $\eta _{s}^+=|\sqrt {\eta \kappa } A_{+s}/\varepsilon _{1}|$ ($\eta _{s}^-=|\sqrt {\eta \kappa }A_{-s}/{\varepsilon _{1}}|$) [46] which is dimensionless as the efficiency of generating the upper (lower) sum sideband. In this work, the parameters we used are similar to those in [37]. The specific parameters are as follows: $m=100$ pg, $\Omega _{m}=2\pi \times 0.1$ MHz, $R$=0.8, $\kappa =2\pi \times 75$ KHz, $T=50$ K, $Q=\Omega _{m}/\Gamma _{m}=3.14\times 10^4$, $L$=67 mm, $\lambda _{c}=2\pi c/\omega _{c}$=532 nm.

Firstly, we consider the effects of the power of the control field and the frequency detuning of the probe field 1 on SSG. Then we plot the efficiencies (in logarithmic form) of USSG and LSSG as a function of the power of the control field $P_{c}$ and the frequency detuning of the probe field 1 $\Omega _{1}$ in Fig. 2. It is shown that from Fig. 2 the peak structure appears for some special values of $\Omega _{1}$ in the efficiencies of SSG. This means that when the appropriate $\Omega _{1}$ is selected, the efficiencies of SSG will be enormously reinforced. It should be noted from Fig. 2 that the efficiencies of both USSG and LSSG increase monotonously with the increases of the power of the control field. The reason for this phenomenon is that the frequency difference between the two scattering paths is non-zero, so there is no interference between the two scattering paths. Accordingly, the efficiencies of SSG increase speedy with the increases of the power of the control field at first, and then the increase rate gradually slows down. From Fig. 2, we can also find that the efficiencies of SSG are quite small, which is caused by the intrinsic weak nonlinearity arising from optomechanical interaction. In addition to the phenomena mentioned above, we also find that the frequency detuning of the probe field 1 has a significant influence on SSG. From Fig. 2(a), we can see that there is only one peak in USSG, and the position of the point corresponding to the peak value is $\Omega _{1} = 2\Omega _{m}$. Unlike this, as can be seen from Fig. 2(b), there are two peaks in LSSG, and the positions of the point corresponding to the peak values are $\Omega _{1} = 2\Omega _{m}$ and $\Omega _{1} = 1.96\Omega _{m}$, respectively. The special value of $\Omega _{1} = 1.96\Omega _{m}$ is attributed to $\Omega _{1} + \Omega _{2} = 2\Omega _{m}$ with $\Omega _{2} = 0.02\times 2\Omega _{m}$. We also confirm that $\Omega _{1}$ varies with $\Omega _{2}$, which still satisfies the relation $\Omega _{1} + \Omega _{2} = 2\Omega _{m}$. The special value of $\Omega _{1}$ ($\Omega _{2}$) corresponding to these peak structures is called the matching conditions.

 

Fig. 2. (a) The efficiency ${\log _{10}\eta _{s}^+}$ (in logarithmic form) of a USSG and (b) the efficiency ${\log _{10}\eta _{s}^-}$ (in logarithmic form) of a LSSG as a function of the control field power $P_{c}$ and the probe field 1 detuning $\Omega _{1}$ for $\Omega _{2}=0.02\times 2\Omega _{m}$. The specific parameters are as follows: $m=100$ pg, $\Omega _{m}=2\pi \times 0.1$ MHz, $R$=0.8, $\kappa =2\pi \times 75$ KHz, $T=50$ K, $Q=\Omega _{m}/\Gamma _{m}=3.14\times 10^4$, $L$=67 mm, $\lambda _{c}=2\pi c/\omega _{c}$=532 nm, $P_{1}=P_{2}$=0.1 nW, $G=0$.

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Secondly, in order to understand the effect of OPA on SSG, the efficiencies (in logarithmic form) of SSG as a function of $\Omega _{1}$ for different values of $G$ are plotted in Fig. 3. We can clearly see from Fig. 3 that the efficiencies of SSG are greatly enhanced in the presence of OPA. To simplify, we assume that the phase $\theta$ of the field driving OPA is zero. From Fig. 3(a), we can see clearly that when $G=0$ which corresponds to the black line, there is only one peak in USSG. Interestingly, when $G\neq 0$ which corresponds to the blue (green, red) line in Fig. 3(a), it is shown that one peak appears at $\Omega _{1} = 2.02\Omega _{m}$ in USSG. And we also find that when OPA exists with $G=0.6\kappa$, the efficiency of USSG at $\Omega _{1} = 2.02\Omega _{m}$ is about 100 times higher than that under the condition without OPA. Moreover, the additional new peak makes the matching conditions of USSG modified. As can be seen from Fig. 3(b), whether $G$ exists or not, the peak structure of LSSG does not change. From Fig. 3(b) we also find that when OPA exists with $G=0.6\kappa$, the efficiency of LSSG at $\Omega _{1} = 2\Omega _{m}$ is about $0.1\%$ , while when OPA does not exist, the efficiency of LSSG is about $1\times 10^{-5}\%$. It is worth emphasizing that the efficiency of LSSG can be enhanced by several orders of magnitude with the increase of $G$. In other words, the existence of $G$ has a significant effect on the efficiencies of SSG. Associating Figs. 3(a) with 3(b), one can find that the efficiency of USSG is higher than that of LSSG.

 

Fig. 3. (a) The efficiency ${\log _{10}\eta _{s}^+}$ (in logarithmic form) of a USSG and (b) the efficiency ${\log _{10}\eta _{s}^-}$ (in logarithmic form) of a LSSG as a function of the probe field 1 detuning $\Omega _{1}$ for different values of $G$ with $P_c$=1 nW, $\Omega _{2}=-0.01\times 2\Omega _{m}$ . Other parameters are the same as those in Fig. 2.

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Finally, for the purpose of a more systematic understanding of SSG, we plot the efficiencies (in logarithmic form) of SSG as a function of detuning $\Omega _{1}$ and $\Omega _{2}$ in Fig. 4. We can clearly see from Figs. 4(a) or 4(b) the so-called matching conditions of SSG. And we also find that the efficiencies of USSG and LSSG exhibit peaks at some special values of $\Omega _{1}$ and $\Omega _{2}$. From Fig. 4(a), we can see that when $\Omega _{1}\rightarrow 2\Omega _{m}$, USSG becomes obvious, that is to say, the efficiency of USSG is enhanced at $\Omega _{1}=2\Omega _{m}$. In addition, we also find that USSG has another local maximum at $\Omega _{1}=-2\Omega _{m}$. It is worth noting that Fig. 4(a) is almost symmetrical to $\Omega _{1}$ and $\Omega _{2}$, which is caused by the fact that $\Omega _{1}$ and $\Omega _{2}$ play the same role in USSG. Therefore, USSG also has a local maximum at $\Omega _{2}=\pm 2\Omega _{m}$. What is more interesting is that the efficiency of USSG near the point of $(\Omega _{1}, \Omega _{2}) = (2\Omega _{m}, 2\Omega _{m})$ is much greater than that near the point of $(\Omega _{1}, \Omega _{2}) = (-2\Omega _{m}, -2\Omega _{m})$. The underlying physical explanation for this interesting phenomenon is that $(\Omega _{1}, \Omega _{2}) = (2\Omega _{m}, 2\Omega _{m})$ is close to the resonant condition of the cavity field $\bar {\Delta }=\Delta +g\bar {X}=2\Omega _{m}$. Through the above analysis, we conclude that the efficiency of USSG has local maximum values at $\Omega _{1} = \pm 2\Omega _{m}$ and $\Omega _{2} = \pm 2\Omega _{m}$. In other words, the matching condition for USSG is $\Omega _{1} = \pm 2\Omega _{m}$ and $\Omega _{2} = \pm 2\Omega _{m}$. As can be seen from Fig. 4(b), the efficiency of LSSG has local maximum not only at $\Omega _{1}=\pm 2\Omega _{m}$ and $\Omega _{2}=\pm 2\Omega _{m}$, but also local maximum at $\Omega _{1}+\Omega _{2}=\pm 2\Omega _{m}$. Therefore, compared with the matching conditions of USSG, the matching conditions of LSSG include $\Omega _{1}+\Omega _{2}=\pm 2\Omega _{m}$ in addition to $\Omega _{1}=\pm 2\Omega _{m}$ and $\Omega _{2}=\pm 2\Omega _{m}$. This additional matching condition is associated with two diagonals in Fig. 4(b). It is worth noting that in the intersection region the efficiency of LSSG is reinforced. It is precisely because of the existence of matching condition $\Omega _{1}+\Omega _{2}=\pm 2\Omega _{m}$ that there are two sidebands for lower sum sideband. On the contrary, there is only one sideband for upper sum sideband. The physical insight of matching conditions can be explained as follows: the special matching condition $\Omega _{1}+\Omega _{2}=\pm 2\Omega _{m}$ which can be understood as the interaction between the two probe fields generates a radiation pressure force oscillating at the mechanical resonance frequency, hence the mechanical mode starts to oscillate coherently. It is worth noting that the above underlying physical process in the quadratically coupled optomechanical system is a two-phonon process. Similarly, the common matching conditions $\Omega _{1}=\pm 2\Omega _{m}$ and $\Omega _{1}=\pm 2\Omega _{m}$ can also be explained as the interaction between the control field and one of the probe fields induced the radiation pressure force oscillating with frequency $\Omega _{1}$ and $\Omega _{2}$. If $\Omega _{1}$ and $\Omega _{2}$ are close to the mechanical resonance frequency, the significant mechanical oscillation will happen. Thus, the remarkable signals at the sum sideband can be generated via the Stokes- and anti-Stokes scattering of the cavity field. By comparing Figs. 4(a) with 4(c), we find that when $G\neq 0$, this means that when OPA exists, the matching conditions of USSG are modified. On the basis of matching conditions $\Omega _{1} = \pm 2\Omega _{m}$ and $\Omega _{2} = \pm 2\Omega _{m}$, the other matching condition $\Omega _{1}+\Omega _{2}=\pm 2\Omega _{m}$ appears, which corresponds to the two diagonals in Fig. 4(c). By combining Figs. 4(b) with 4(d), we know that the matching conditions of LSSG have not been modified when OPA exists. In addition, we also find that the existence of OPA increases the efficiency of LSSG by several orders of magnitude.

 

Fig. 4. (a), (c) The efficiency ${\log _{10}\eta _{s}^+}$ (in logarithmic form) of a USSG and (b), (d) the efficiency ${\log _{10}\eta _{s}^-}$ (in logarithmic form) of a LSSG as a function of the probe field 1 detuning $\Omega _{1}$ and the probe field 2 detuning $\Omega _{2}$ for $P_{c}$=1 nW. For the cases of (a), (b) we use $G=0$, and (c), (d) we use $G=0.6\kappa$. Other parameters are the same as those in Fig. 2.

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4. Conclusion

In summary, we have analyzed SSG in a quadratically coupled optomechanical system containing an OPA. The system is driven by one control field with frequency $\omega _{c}$ and two relatively weak probe fields with frequencies $\omega _{1}$ and $\omega _{2}$. We find that such a system can induce SSG when the nonlinear terms of the optomechanical dynamics are considered. By analytic calculation, we reveal that the amplitude of sun sideband can be well tuned by the power of the control field, the frequency detuning of the probe fields and the nonlinear gain of OPA. In particular, the SSG can be obtained even under weak driven fields and greatly enhanced via meeting the matching conditions. More interestingly, when $G\neq 0$, the efficiencies of SSG are greatly enhanced, especially the efficiency of LSSG can be enhanced by several orders of magnitude. However, it is worth noting that the matching conditions of USSG have been modified. This investigation may deepen the understanding of the nonlinear quadratically coupled optomechanical interactions within the perturbative regime and provide potential applications to the precision measurement and optical communications.