Abstract
We theoretically investigate the nonlinear optical phenomena including optical bistability and fourwave mixing (FWM) process in a composite photonicmolecule cavity optomechanical system. The photonicmolecule cavity consisted of two whispering gallery mode (WGM) microcavities, where one WGM cavity is an optomechanical cavity with highcavity dissipation κ and the other WGM cavity is an auxiliary ordinary optical cavity with highquality factor (Q). Controlling the parameters of the system, such as the coupling strength J between the two cavities, the decay rate ratio δ of the two cavities, and the pump power P, the optical bistability can be controlled. Furthermore, the FWM process which presents the normal modesplitting is also investigated in the FWM spectrum under different parameter regimes. Our study may provide a further insight of nonlinear phenomena in the composite photonicmolecule optomechanic systems.
Background
Optomechanic systems (OMS) [1], consisting of optical cavities coupled to mechanical resonators and exploring radiation pressureinduced coherent photonphonon interactions, have recently attracted much attention because they offer a platform to manipulate mechanical resonators and electromagnetic fields, and pave the way for potential applications of optomechanical devices, such as phonon laser [2, 3], sensing [4], phonon squeezing [5], the realization of squeezed light [6–8], groundstate cooling [9–11], and optomechanically induced transparency (OMIT) [12–15]induced store light in solidstate devices [16, 17]. Although most attention has been paid to the single OMS, to realize compound OMS by integrating more optical or mechanical modes such as one mechanical mode coupled to two optical modes via radiation pressure [18, 19] and the phononic interaction between two mechanical resonators [20, 21] become a tendency for further investigating the OMS and their potential applications in quantum information processing. Based on the hybrid compound OMS, the transfer of a quantum state [22], OMITlike phonon cooling [23], optomechanical dark mode [24], and phononmediated electromagnetically induced absorption [25] has been researched widely. In the numerous compound OMS, as a natural extension of the generic OMS, two directly coupled whispering gallery mode (WGM) microcavities termed photonicmolecule [26, 27] with optomechanical effect in one WGM microcavity have attracted much attention. There are two kinds of interplay in the compound photonicmolecule optomechanical system: the first one is the optomechanical interaction induced by the radiation pressure and the other one is cavitycavity coupling via tunable photon tunneling. The two interactions together give rise to several interesting phenomena including phonon lasing [2, 3], chaos [28], groundstate cooling [23], and coherent control of light transmission [25, 29, 30].
On the other hand, OMS also provide a platform to investigate the nonlinear effect of lightmatter interaction. Among all the nonlinear phenomena in OMS, optical bistability and fourwave mixing (FWM) are typical nonlinear optical phenomenon focusing on researchers’ interest. In recent year, the bistable behavior of the mean intracavity photon number has been extensively studied in various OMS, such as BoseEinstein condensate cavity optomechanical system [31, 32], OMS with a quantum well [33], ultracold atoms [34, 35], and other hybrid OMS [36, 37]. In addition, FWM can be described as the cavity driven by a strong pump laser with frequency ω_{p} and a weak probe laser frequency ω_{s}, and then, two pump photons would mix with a probe photon via the mechanical mode to yield an idler photon at frequency 2ω_{p}−ω_{s} in OMS, and it is also investigated in previous works, such as the modesplitting in strong coupling optomechanical system [38], coherent mechanical driving OMS [39, 40], and a twomode cavity optomechanical system [41]. However, optical bistability and FWM have been seldom studied in composite photonicmolecule OMS, where the coupling strength represented by J of the two cavities play a key role affecting these nonlinear optical phenomena.
In the present work, we consider a composite photonicmolecule cavity optomechanical system, consisted of two WGM microcavities, where one WGM cavity is an optomechanical cavity with highcavity dissipation κ, and the other WGM cavity is an auxiliary ordinary optical cavity with highquality factor (Q) [42]. As Liu et al. [43] demonstrated, it remains difficult to achieve high Q factor and small mode volume (V) simultaneously for the same type of resonator. In the photonicmolecule optomechanics, by coupling the originally optomechanical cavity c with highcavity dissipation κ (without high Q) to an auxiliary cavity mode a with high Q but large V, the requirement for high Q and small V for the same cavity can be removed. We introduce a ratio parameter δ=κ_{c}/κ_{a}, where κ_{c}=ω_{c}/Q_{c} and κ_{a}=ω_{a}/Q_{a} are the decay rates of cavity modes c and a (ω_{c} and ω_{a} are the frequencies of cavity c and a) to investigate the nonlinear effect in the photonicmolecule optomechanics. Here, the optomechanical cavity c is driven by the pump laser while the auxiliary cavity a is driven by the probe laser. The cavity c is coupled to cavity a via evanescent field, and the coupling strength J between the two cavities can be controlled by varying the separation between the two WGM cavities [26]. We investigate the optical bistability and FWM based on the composite photonicmolecule OMS by varying the coupling strength J between the cavity resonators, and a tunable and controllable optical bistability and FWM can be achieved with manipulating the coupling strength J between the two cavities. Further, with adjusting the parameter δ and the pump power P, the FWM process can be controlled.
Model and Theory
The photonicmolecule optomechanics is shown in Fig. 1. The first cavity supports an optical mode c with the frequency ω_{c} driven by the pump laser with frequency ω_{p} and the amplitude \(\varepsilon _{p}= \sqrt {P/\hbar \omega _{p}}\). The radiation pressure induces a mechanical mode b with the mechanical resonator frequency ω_{m}, and the singlephoton optomechanical coupling rate is g=g_{0}x_{0} (g_{0}=ω_{c}/R and R is the radius of cavity c), and the zeropoint fluctuation of the mechanical oscillator’s position is \(x_{0}=\sqrt {\hbar /2M\omega _{m}} \) [13]. Then, the Hamiltonian of optomechanics c is [13]
where Δ_{c}=ω_{c}−ω_{p} is the detuning of the pump field and cavity c. c and c^{†} represent the bosonic annihilation and creation operators of the cavity mode c, and b^{†}(b) is the creation (annihilation) operator of mechanical mode. The auxiliary cavity only supports an optical mode a driven by the probe laser with frequency ω_{s}, and its amplitude ε_{s} is \(\varepsilon _{s}=\sqrt { P_{s}/\hbar \omega _{s}}\). We introduce the annihilation and creation operators a and a^{†} to describe the cavity a, and its Hamiltonian is [13]
where Δ_{a}=ω_{a}−ω_{p} is the detuning of the pump field and cavity a, and Ω=ω_{s}−ω_{p} is the pumpprobe detuning. We use two tapered fibers to excite the cavity mode a and cavity mode c as the optical waveguide with the coupling rate κ_{ae} and κ_{ce}. The optomechanical cavity c couples to cavity a through an evanescent field, and the cavitycavity coupling rate J can be efficiently tuned by changing the distance between them [26]. When the coupling strength J is weak in between the two cavities, then the energy from cavity c cannot transfer easily to cavity a. Conversely, if the coupling strength J increases with decreasing the distance between the two cavities, then the energy can easily flow from the two cavities. The linearly coupled interaction between the two cavities is described by [26] \(\hbar J\left (a^{\dag }c+ac^{\dag }\right)\). Then, the total Hamiltonian in the rotating wave frame of pump frequency ω_{c} can be written [3, 13, 23]
The decay rate of the two cavities mode κ=κ_{c}=κ_{a}=κ_{ex}+κ_{0} with the intrinsic photon loss rate κ_{0}, and κ_{ex} describes the rate at which energy leaves the optical cavity into propagating fields [13]. Here, for simplicity, we only consider the condition of κ_{ex}=κ_{0}=κ_{ae}=κ_{ce}, and we consider ω_{c}=ω_{a}.
We use the Heisenberg equation of motion \(i\hbar \partial _{t}O=[O,H]\) (O=a,c,X) and introduce corresponding damping and noise operators, and we obtain the quantum Langevin equations as follows [44]:
where X=b^{†}+b is the position operator and γ_{m} is the decay rate of the resonator. a_{in} and c_{in} describing the Langevin noises follow the relations [45]
The resonator mode is influenced by stochastic force process with the following correlation function [46]
where k_{B} is Boltzmann constant and T indicates the reservoir temperature.
When the optomechanical cavity c is driven by a strong pump laser, the Heisenberg operator can be divided into two parts, i.e., steadystate mean value O_{0}, and small fluctuation δO with zero mean value 〈δO〉=0. The steadystate values determine the intracavity photon numbers (n_{a}=a_{s}^{2}and n_{c}=c_{s}^{2}) determined by
where \(\Delta ^{^{\prime }}=\Delta _{c}2g^{2}n_{c}/\omega _{m}\). This form of coupled equations are characteristic of the optical bistability. In the following section, we will discuss the parameters such as the pump power P, the cavitycavity coupling strength J, and the ratio parameter δ that affect the optical bistability. Keeping only the linear terms of the fluctuation operators and making the ansatz [47] 〈δa〉=a_{+}e^{−iΩt}+a_{−}e^{iΩt}, 〈δc〉=c_{+}e^{−iΩt}+c_{−}e^{iΩt}, 〈δX〉=X_{+}e^{−iΩt}+X_{−}e^{iΩt}, we then obtain
where \(\Lambda _{1}=igc_{s}^{2}\eta ^{\ast }J^{2}\varepsilon _{s}\sqrt { \kappa _{ae}}\), Λ_{2}=(iΔ_{a2}+κ_{a})(iΔ_{2}+κ_{c})[(iΔ_{1}−κ_{c})(iΔ_{a1}−κ_{a})−J^{2}], \(\Lambda _{3}=g^{2}\eta ^{\ast 2}n_{c}^{2}(i\Delta _{a1}\kappa _{a})(i\Delta _{a2}+\kappa _{a})\), Δ_{a1}=Δ_{a}−Ω, Δ_{a2}=Δ_{a}+Ω, \(\Delta _{1}=\Delta ^{^{\prime }}\Omega +g\eta n_{c}\), \(\Delta _{2}=\Delta ^{^{\prime }}+\Omega +g\eta ^{\ast }n_{c}\), and \(\eta =2g\omega _{m}/(\omega _{m}^{2}i\gamma _{m}\Omega \Omega ^{2})\). Using the standard inputoutput relation [45] \(a_{\text {out}}(t)=a_{\text {in}}(t)\sqrt {2\kappa _{a}}a(t)\), where a_{out}(t) is the output field operator, and obtain the expectation value of the output fields:
where a_{out}(t) is the output field operator. Equation (13) shows that the output field consists of three terms. The first term corresponds to the output field at driving field with amplitude ε_{p} and frequency ω_{p}. The second term corresponds to the probe field with frequency ω_{s} related to the antiStokes field resulting in OMIT, which has been investigated in various optomechanical systems [12–15, 48]. The last one corresponds to the output field with frequency 2 ω_{p}−ω_{s} related to the stoke field displaying the FWM. In the FWM process, the two photons of the driving field interact with a single photon of the probe field each with frequencies ω_{p} and ω_{s} born a new photon of frequency 2 ω_{p}−ω_{s}. The FWM intensity in terms of the probe field can be defined as [49]
which is determined by the optomechanical coupling strength g, the pump power P, the cavitycavity coupling strength J, and the decay rate ratio δ of the two cavities.
Numerical Results and Discussions
In this section, we first investigate the bistable behavior of the steadystate photon number n_{c} and n_{a} of the two cavities according to Eqs. (10) and (11). Because it is too cumbersome to give the analytical expression of the bistability condition, here we will present the numerical results. We choose the parameters similar to those in Ref. [13, 26] : the parameters of cavity c as [13]: g_{0}=12 GHz/nm, γ_{m}=41 kHz, ω_{m}=51.8 MHz, κ_{c}=5 MHz, m=20 ng, λ=750 nm, and Q=1500, and the order of magnitude of the pump power is milliwatt (1 mW =10^{−3} W). For cavity a, we consider ω_{a}=ω_{c} and κ_{c}=κ_{a}. The coupling strength J between the two cavity modes plays a key role and can affect the bistable behavior and FWM. It has been reported experimentally that the coupling strength J depends on the distance between cavity c and cavity a [26] (also the coupling strength decreases exponentially with increasing the distance of the two cavities). Here, we expect the coupling strength \( J\sim \sqrt {\kappa _{c}\kappa _{a}}\).
Equations (10) and (11) giving the intracavity photon numbers of optomechanical cavity c and ordinary cavity a are coupled cubic equations, which exhibit bistable behavior. We first consider the condition of J=0, i.e., only a single optomechanical cavity c, and Fig. 2a plots the mean intracavity photon number n_{c} of optomechanical cavity c as a function of the cavitypump detuning Δ_{c}=ω_{c}−ω_{p} with three pump powers. When the pump power is less than P=0.4 mW (such as P=0.1 mW), the curve is nearly Lorentzian. With increasing the power P to a critical value, the optomechanical cavity c exhibits bistable behavior, as shown in the curves for P=0.4 mW to P=0.8 mW, where the initially Lorentzian resonance curve becomes asymmetric. The mean intracavity photon number n_{c} has three real roots (Eq. (10)), and the largest and smallest roots are stable, and the middle one is unstable, which is represented in an oval in Fig. 2a. However, when we consider the optical cavity a, i.e., J≠0 such as J=1.0 κ_{a}, the bistable behavior is broken in some ways as shown in Fig. 2b. That is because when optomechanical cavity c coupled to optical cavity a, parts of intracavity photon number n_{c} of optomechanical cavity c will coupled into optical cavity a, and therefore, intracavity photon number n_{c} will decrease and then result in a destroyed bistable behavior. Figure 2c shows the mean intracavity photon number n_{c} of optomechanical cavity c as a function of the cavitycavity coupling strength J with three pump powers. Obviously, the mean intracavity photon number n_{c} depends on the pump power P, and the intracavity photon number n_{c} is always decreasing with the increasing coupling strength J because parts of photon number are coupled into optical cavity a. Further, larger cavitypump detuning is beneficial to observe the optical bistable behavior with increasing pump power P. Figure 2d plots the mean intracavity photon number n_{c} versus the pump power P with cavity a at red sidebands (Δ_{a}=ω_{m}) and blue sidebands (Δ_{a}=−ω_{m}), respectively, and the bistability presents the hysteresis loop behavior [50]. However, our results are different from the previous work of twomode optomechanical system without considering the cavitycavity coupling J. Therefore, the coupling strength J plays an important role in the bistability.
We further investigate bistable behavior of optical cavity a with Eq. (11). Figure 3a gives the intracavity photon number n_{a} of ordinary cavity a as a function of the cavitypump detuning Δ_{a}=ω_{a}−ω_{p} with pump powers P=0.1 mW, P=1.0 mW, and P=10 mW at J=1.0 κ_{a}. It is obvious that optical cavity a cannot behave as bistable behavior due to intracavity photon number n_{a} of cavity a from cavity c cannot maintain bistability in lowpump power. Actually, only highpump power P can cavity a present bistable behavior, because only highpump power driven optomechanical cavity c, much more photon number can couple into optical cavity a. We also plot the mean intracavity photon number n_{a} of optical cavity a as a function of the coupling strength J under three pump powers as shown in Fig. 3b. It is clear that when J=0, n_{a}=0, because there is no coupling between the two cavities at J=0, and at this condition, no photon couples into optical cavity a. With increasing the coupling strength J (decreasing the distance of the two cavities [26]), the intracavity photon numbers n_{a} of ordinary optical cavity a increase but not always. There is an optimum coupling strength J for the maximum value of n_{a} under different pump power, and then, n_{a} will decrease with the increasing J. It is a remarkable fact that the coupling strength J between the two cavities can be adjusted [26].
In addition, we consider a ratio parameter δ=κ_{c}/κ_{a} (κ_{c}=ω_{c}/Q_{c} and κ_{a}=ω_{a}/Q_{a}) to investigate the parameters of the two cavities that influence bistable behavior. κ is the decay rate of the cavity mode, which is related to the frequency and quality factor of the cavity. As we know, it is difficult to achieve high Q and small V simultaneously for a cavity mode due to the diffraction limit. For an optical cavity, a smaller V corresponding to a larger radiative decay rate results in a lower Q. Although different types of cavities possess their own unique properties, the weigh between high Q and small V still exists. However, when by coupling the originally OMS c with highcavity dissipation to an auxiliary cavity mode a with high Q but large V, the bistable behavior will change significantly. Figure 3c shows the mean intracavity photon number n_{c} of optomechanical cavity c as a function of Δ_{a} under several different δ=κ_{c}/κ_{a} with an unchanged coupling strength J=1.0 κ_{a}. We can find that the bistable behavior can appear, but the intracavity photon number n_{c} is small at δ=0.1 with J=2 κ_{a}, i.e., κ_{c}=0.1 κ_{a} which means Q_{c}>Q_{a}. When increasing the ratio δ from δ=1.0 to δ=2.0, the intracavity photon number n_{c} experiences the change from bistable behavior to nearly Lorentzian line profile. That is to say when Q_{c}<Q_{a}, the bistable behavior will be broken, but there is an optimal condition, i.e., Q_{c}=Q_{a}. In Fig. 3d, we give the intracavity photon number n_{c} as a function of δ with two different J, and obviously, in increasing the ratio parameter δ, the intracavity photon numbers n_{c} increase. When it reaches an optimum value for a given J, then n_{c} decrease. Therefore, controlling the cavity parameters, like the decay rate κ or the quality factor of the cavities, the bistable behavior can be controlled.
On the other hand, as a typical nonlinear optical phenomenon, we also investigate the FWM process with Eq. (14) in the photonicmolecule optomechanical system. Figure 4 plots the FWM spectrum as a function of the probecavity a detuning Δ_{s}=ω_{s}−ω_{a} at Δ_{a}=Δ_{c}=0 under different parameter regimes. Figure 4a–d displays the FWM spectra evolution under different pump power P at J=1.0 κ_{a}. It is clear that the FWM spectra present three peaks, where a Lorentzian peak near Δ_{s}=0 and two modesplitting peaks locate at ±ω_{m}, and the FWM intensity decreases with increasing the pump power. Figure 4e–h shows the change of FWM spectra from J=0.5 κ_{a} to J=2.0 κ_{a} at pump power P=1.0 mW. With increasing the coupling strength J from J=0.5 κ_{a} to J=2.0 κ_{a}, the FWM spectra change significantly. The phenomena can be explained with a dressedstate picture which has demonstrated in single cavity optomechanical system [51].
We then investigate the FWM spectra at Δ_{a}=Δ_{c}≠0. Figure 5a–d gives the FWM spectra at the red sideband, i.e., Δ_{a}=Δ_{c}=ω_{m} under an unchanged J=1.0 κ_{a} with increasing the pump power from P=1.0 to P=10 mW. Two normal modesplitting peaks appear in the FWM spectra locating at ±ω_{m} respectively, and the FWM intensity decreases with increasing the pump power. Figure 5e–h shows the FWM spectra at the red sideband, i.e., Δ_{a}=Δ_{c}=ω_{m} under a fixed pump power P=2.0 mW with increasing the coupling strength J from J=0.5 κ_{a} to J=2.0 κ_{a}. Obviously, the FWM intensity increases with increasing the coupling strength J, and the bigger J means more photon numbers coupled into optical cavity a. When changing the detuning Δ_{a} and Δ_{c} from the red sideband to the blue sideband, i.e., Δ_{a}=Δ_{c}=−ω_{m}, the evolution of the FWM spectra change prominently. Figure 5i–l displays the FWM spectra at blue sideband under four different pump powers, and the FWM intensity decreases with increasing the pump power even at the blue sideband. Except two normal modesplitting peaks locating at ±ω_{m}, there are also two sharp sideband peaks appear in the FWM spectra and their location are related to the pump power. In Fig. 5m–p, we also discuss the coupling strength J that affect the FWM spectra under the blue sideband. Whether other sharp sideband peaks appear in the FWM spectra depend on the the coupling strength J.
Further, since the ratio parameter δ=κ_{c}/κ_{a} can influence the intracavity photon number in the composite photonicmolecule OMS, the FWM spectra can be manipulated with controlling the parameter δ. Figure 6a–h presents the FWM spectra at unchanged parameters J=2.0 κ_{a} and P=10 mW under the red sideband with increasing the ratio δ from δ=0.05 to δ=3.0, and the FWM intensity decreases with increasing the ratio δ. While in the blue sideband, other sharp sideband peaks will appear in the FWM spectra as shown in Fig. 6i–p, and the FWM intensity also decreases with increasing the ratio δ. Therefore, with controlling the cavity parameters, like the decay rate κ or the Q of the cavities, the FWM can achieve straightforward in the composite photonicmolecule OMS.
Conclusion
We have investigated the optical bistability and fourwave mixing in a composite WGM cavity photonicmolecule optomechanical system, which includes an optomechanical cavity with highcavity dissipation coupled to an auxiliary cavity with highquality factor. We investigate the optical bistability under different parameter regimes such as the coupling strength J between the two cavities and the decay rate ratio δ of the two cavities in the system. The optical bistability can be adjusted by the pump field driving the optomechanical cavity, and the intracavity photon number in the two cavities is determined by the coupling strength J. Further, we have also demonstrated how to control the FWM process in the photonicmolecule optomechanical system under different driving conditions (the red sideband and the blue sideband) and different parameter conditions (the coupling strength J and the ratio δ). Numerical results show that the FWM process can be controlled with such parameters. These results are beneficial for better understanding the nonlinear phenomena in the composite photonicmolecule optomechanical system.
Abbreviations
 COMS:

Cavity optomechanics systems
 FWM:

Fourwave mixing
 OMS:

Optomechanics systems
 OMIT:

Optomechanically induced transparency
 Q:

Quality
 V:

Volume
 WGM:

Whispering gallery mode
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Acknowledgements
The authors gratefully acknowledge support from the National Natural Science Foundation of China (Nos:11647001 and 11804004) and Anhui Provincial Natural Science Foundation (No:1708085QA11). We acknowledge Lan Yang at Washington University for helpful suggestion in a meeting of “Microcavity Photonics”.
Funding
HuaJun Chen is supported by the National Natural Science Foundation of China (Nos:11647001 and 11804004) and Anhui Provincial Natural Science Foundation (No:1708085QA11).
Availability of data and materials
The photonicmolecule optomechanical cavity system is demonstrated by Yang in Ref. [26], and the optomechanical model and parameters are investigated by Weis in Ref. [13].
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HJC finished the main work of this paper, including conceiving of the idea, deducing the formulas, plotting the figures, and drafting the manuscript. HWW, XCL, JYY, YJS, and YP participated in the discussion and provided some useful suggestion. All authors are involved in revising the manuscript. All authors read and approved the final manuscript.
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Chen, HJ., Wu, HW., Yang, JY. et al. Controllable Optical Bistability and FourWave Mixing in a PhotonicMolecule Optomechanics. Nanoscale Res Lett 14, 73 (2019). https://doi.org/10.1186/s1167101928932
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DOI: https://doi.org/10.1186/s1167101928932
Keywords
 Photonicmolecule optomechanics
 Optical bistability
 Fourwave mixing