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Abstract
We investigate the enhanced four-wave mixing (FWM) process in a parity-time ($\mathcal{PT}$)-symmetric optomechanical system, where an active cavity is coupled to a passive cavity supporting a mechanical mode. The passive cavity is optically driven by a strong control field and a weak probe field, and the mechanical mode is excited by a weak coherent driving field. By tuning the coupling strength between the two cavities with balanced gain and loss, we find that the FWM intensity can be significantly enhanced near the exceptional points (EPs) at low control power, which is about 12 orders of magnitude higher than that of the single-cavity case. Due to the interference effect induced by the optical and mechanical driving field, it is shown that the FWM intensity can be further enhanced or suppressed by tuning the amplitude and phase of the mechanical driving field. Moreover, the dependence of the FWM intensity on the frequency and power of the control field is also discussed. Our work provides a route to enhance the four-wave mixing process in a flexible way.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The nonlinear coupling between optical and mechanical modes via radiation pressure is the basis of cavity optomechanical systems [1–3], which has been exploited to realize ground state cooling of a mechanical oscillator [4,5], preparation of nonclassical mechanical states [6], generation of squeezed light [7,8], optomechanically induced transparency (OMIT) [9–11], and so on. Moreover, four-wave mixing (FWM) is an important third-order nonlinear optical process in a variety of media including semiconductor-doped glasses [12], atomic systems [13,14], and photonic cystal fibers [15], which can find a number of applications in optical telecommunications. Recently, radiation-pressure-induced FWM has been theoretically studied in several types of optomechanical sytems [16–20]. However, the FWM intensity is typically weak in these systems, which makes it difficult to be observed in experiments.
In 1998, Bender and Boettcher proposed that non-Hermitian Hamiltonian having parity-time ($\mathcal {PT}$)-symmetry can possess entirely real spectra [21]. $\mathcal {PT}$-symmetric systems can exhibit a phase transition between the unbroken- and broken-symmetry phases at the exceptional points (EPs), where the eigenvalues and the corresponding eigenvectors simultaneously coalesce [22]. This phase transition has been demonstrated in various physical systems and can lead to many intriguing phenomena, such as loss-induced transparency [23], topological energy transfer [24], enhanced sensitivity [25,26], controllable matching conditions for the FWM [27], and nonlinearity enhancement [28–30]. In addition, Zhang et al. have recently demonstrated that symmetry-breaking-induced second-order nonlinearity at a microcavity surface can be improved by 14 orders of magnitude higher than that of the non-enhancement case [31].
More recently, $\mathcal {PT}$-symmetric optomechanical systems have been under extensive exploration in various aspects, including phonon laser [32,33], chaos [34], optomechanically induced transparency [35,36], optomechanically induced absorption [37], enhanced ground state cooling [38,39], enhanced high-order sideband generation [40,41], enhanced sensitivity [42], and so on. However, third-order nonlinear effect has seldom been studied in this system. In the present paper, we mainly focus on the four-wave mixing process in $\mathcal {PT}$-symmetric optomechanical systems in the presence of a strong optical control field, a weak optical probe field, and a weak coherent mechanical driving field. We demonstrate that the FWM intensity is significantly enhanced near the phase transition point between the unbroken- and broken-symmetry. Moreover, the FWM intensity can be effectively controlled by the frequency and power of the optical control field as well as the amplitude and phase of the mechanical driving field. It is worth pointing out that mechanical driving field has been exploited in experiments to realize virtual exceptional points [43], injection locking [44], electro-optomechanically induced transparency [45], cascaded optical transparency [46], nonreciprocal mode conversion [47], and phase-sensitive parametric amplifier [48]. Theoretical studies also show that the additional mechanical driving field can be employed to control the optical response of the optomechanical systems due to the more complicated interference effects [49–54]. We note that mechanical driving field may also play a vital role in tuning the optomechanical systems and in achieving one-way optical flow [55–57]. Different from the four-wave mixing in a $\mathcal {PT}$-symmetric coupler in Ref. [27], the FWM process in $\mathcal {PT}$-symmetric optomechanical systems is induced by the radiation pressure, where two control photons at frequency $\omega _c$ can mix with a probe photon at frequency $\omega _p$ via the mechanical mode to yield an idler photon at frequency $2\omega _c-\omega _p$ [16–20]. With the assistance of the mechanical driving field, the FWM process in this optomechanical systems can be controlled more flexibly.
This paper is organized as follows. In Sec. 2, we present the theoretical model of the $\mathcal {PT}$-symmetric optomechanical system and derive the analytical expression of the FWM intensity. In Sec. 3, we numerically study the dependence of FWM intensity on various parameters, including the cavity-cavity coupling strength, the gain-loss ratio, the amplitude and phase of the mechanical driving field, and the frequency and power of the optical control field. Finally, we summarize our work in Sec. 4.
2. Model and theory
We consider the $\mathcal {PT}$-symmetric optomechanical system schematically shown in Fig. 1, which consists of two directly coupled whispering-gallery-mode (WGM) microtoroidal cavities with coupling strength $J$. One of the cavities is passive with resonance frequency $\omega _1$ and loss rate $\kappa _1$, which supports a mechanical radial breathing mode with effective mass $m$, resonance frequency $\omega _m$ and damping rate $\gamma _m$. Another cavity is active with resonance frequency $\omega _2$ and tunable gain rate $\kappa _2$, which can be fabricated from Er$^{3+}$-doped silica and can emit photons in the 1550 nm band [28,29]. The passive cavity is simultaneously driven by a strong control field and a weak probe field with amplitudes $\varepsilon _{i}$, frequency $\omega _{i}$, and phase $\phi _{i}$ ($i=c,p$), respectively. In addition, a weak coherent mechanical driving field with amplitude $\varepsilon _m$, frequency $\Omega =\omega _p-\omega _c$, and phase $\phi _m$ is applied to excite the mechanical mode. It should be pointed out that mechanical driving of the radial breathing mode has been experimentally realized in a microtoroid optomechanical cavity with an integrated electrical interface [44], where an inertial force was applied to the microtoroid in the radial direction. Moreover, optomechanical interaction can also be realized by the evanescent coupling of nanobeams and nanomembranes to WGM resonators [58,59]. In this way, it is possible to realize mechanical mixing of different mechanical resonators [54,58].
Fig. 1. Schematic illustration of the $\mathcal {PT}$-symmetric optomechanical system. The passive cavity with loss rate $\kappa _1$ interacts with a mechanical mode with resonance frequency $\omega _m$ and damping rate $\gamma _m$ via radiation pressure. The active cavity with tunable gain rate $\kappa _2$ is coupled to the passive cavity, and the coupling strength $J$ can be adjusted by the distance between them. In addition, the passive cavity is driven by a strong control field at frequency $\omega _c$ and a weak probe field at frequency $\omega _p$, and the mechanical resonator is excited by a weak coherent mechanical driving field at frequency $\Omega =\omega _p-\omega _c$.
In a rotating frame at the frequency $\omega _c$ of the control field, the Hamiltonian of this system is given by
According to Eq. (1) and Heisenberg equations of motion, we can derive the time evolution of the system operators as follows:
By setting all the time derivatives in Eqs. (2)–(5) to be zero, we can obtain the steady-state solutions of the system, which are given as follows:
3. Enhanced four-wave mixing
In previous works, the properties of the transmitted probe field have been extensively investigated in $\mathcal {PT}$-symmetric optomechanical systems [49–54]. In this section, we mainly study the effects of $\mathcal {PT}$-symmetry and mechanical driving field on the properties of the four-wave-mixing field. In the numerical simulations, we choose the parameters from recent experiments [29,61]: $\omega _1=2\pi c/\lambda$ with $\lambda =1550$ nm, the radius of the passive cavity $R=15$ $\mu$m, $g_1=\omega _1/R$, $\kappa _1/2\pi =6$ MHz, $m=6.2$ ng, $\omega _m/2\pi =78$ MHz, $\gamma _m/2\pi =12$ kHz, and $\eta _c=0.4$.
In Fig. 2, we first investigate the influence of the cavity-cavity coupling strength $J$ on the four-wave mixing intensity. Figure 2(a) plots the FWM intensity $\log _{10}|\mathrm {FWM_p}|^2$ versus $(\Omega -\omega _m)/\gamma _m$ for different values of coupling strength $J$. It is shown that the FWM intensity reaches the maximum at $\Omega =\omega _m$, where optomechanically induced transparency can occur. In the absence of the direct coupling between the two cavities, the FWM intensity is very weak, which is about $10^{-4.5}$. However, if the cavity-cavity coupling is turned on, such as $J=0.24(\kappa _1+\kappa _2)$, the FWM intensity can be improved about 4 orders of magnitude. Most importantly, if the coupling strength $J$ is tuned to approach the exceptional points with $J=0.251(\kappa _1+\kappa _2)$, the FWM intensity can be greatly enhanced, which is about 8 orders of magnitude higher than that of the single cavity case. In addition, when the cavity-control field detuning $\Delta _1^{'}=\Delta _2=0,$ the FWM intensity at $\Omega =\omega _m$ as a function of the coupling strength $J$ is plotted in Fig. 2(b). The maximum FWM intensity at the exceptional points is enhanced by over 12 orders of magnitude compared with the case that the coupling between the passive cavity and the active cavity is turned off. Therefore, the FWM intensity in $\mathcal {PT}$-symmetric optomechanical systems can be significantly enhanced around the exceptional points. This can be explained intuitively as follows. If the coupling strength $J$ is weak, the energy is strongly localized in the active cavity and the $\mathcal {PT}$-symmetry is broken. If the coupling strength is increased to surpass the exceptional points, the energy in the active cavity can flow into the passive one to compensate the dissipation, and the two supermodes have different frequencies but with the same damping rate. With balanced gain and loss, the two supermodes will be lossless in the $\mathcal {PT}$-symmetric regime. At the exceptional points, the two supermodes are degenerate with zero loss. Consequently, the FWM intensity is significantly enhanced due to field localization and negligible loss in the vicinity of the exceptional points [29,30].
Fig. 2. FWM intensitiy $\log _{10}|\mathrm {FWM_p}|^2$ as a function of (a) $(\Omega -\omega _m)/\gamma _m$ for different values of coupling strength $J$ and (b) $J/(\kappa _1+\kappa _2)$ with $\Omega =\omega _m$. Other parameters are $\Delta _1^{'}=\Delta _2=0$, $\omega _1=2\pi c/\lambda$ with $\lambda =1550$ nm, $R=15$ $\mu$m, $g_1=\omega _1/R$, $\kappa _1/2\pi =6$ MHz, $\kappa _2=0.99\kappa _1$, $m=6.2$ ng, $\omega _m/2\pi =78$ MHz, $\gamma _m/2\pi =12$ kHz, $P_c=50~$nW, $\eta _c=0.4$, $\varepsilon _p=\varepsilon _c/1000$, and $\varepsilon _m=0.$
Moreover, the optical response of the system can be further modified by the additional mechanical driving field. In Fig. 3, we plot the FWM intensity $|\mathrm {FWM_p}|^2$ at $\Omega =\omega _m$ as functions of the mechanical driving amplitude $\varepsilon _m$ and phase difference $\phi$ with (a) $\Delta _1^{'}=\Delta _2=0$ and (b) $\Delta _1^{'}=\Delta _2=0.5\omega _m$. With the increase of the driving amplitude $\varepsilon _m$, Fig. 3(a) shows that the dependence of the FWM intensity on the phase difference is evident. The maximum FWM intensity is obtained when the phase difference $\phi =\pi /2$, but the minimum value locates around $\phi =3\pi /2$. However, if the cavity-control field detuning is tuned to be $\Delta _1^{'}=\Delta _2=0.5\omega _m$, it can be seen from Fig. 3(b) that the FWM intensity is enhanced monotonically with increasing the amplitude $\varepsilon _m$, where the influence of the phase difference $\phi$ is negligible for fixed $\varepsilon _m$. In the following, we will focus on the case that the phase difference plays an important role in controlling the FWM intensity.
Fig. 3. Contour plots of FWM intensity $|\mathrm {FWM_p}|^2$ at $\Omega =\omega _m$ as functions of the amplitude $\varepsilon _m$ and phase difference $\phi /\pi$ with (a) $\Delta _1^{'}=\Delta _2=0$ and (b) $\Delta _1^{'}=\Delta _2=0.5\omega _m$. The other parameters are the same as those in Fig. 2 except $J=0.251(\kappa _1+\kappa _2)$.
To explain the physical origin of the above phase-dependent effect induced by the mechanical driving field, we show $|\mathrm {FWM_1}|^2$, $|\mathrm {FWM_2}|^2$, and $|\mathrm {FWM_p}|^2$ as functions of the mechanical driving amplitude $\varepsilon _m$ in Fig. 4. It is shown in Eq. (18) that $\mathrm {FWM_p}$ can be divided into $\mathrm {FWM_1}$ and $\mathrm {FWM_2}$, where $|\mathrm {FWM_1}|$ is independent on $\varepsilon _m$ but $|\mathrm {FWM_2}|$ increases monotonically with $\varepsilon _m$. When the phase difference $\phi =\pi /2$, there is constructive interference between $\mathrm {FWM_1}$ and $\mathrm {FWM_2}$. We can see from Fig. 4(a) that perfect constructive interfere occurs if $|\mathrm {FWM_1}|^2=|\mathrm {FWM_2}|^2$ at $\varepsilon _m\approx 2.7$ pN, where $|\mathrm {FWM_p}|^2=4|\mathrm {FWM_1}|^2$. However, if the phase difference is tuned to be $\phi =3\pi /2$, there is destructive interference between $\mathrm {FWM_1}$ and $\mathrm {FWM_2}$. Figure 4(c) shows that $|\mathrm {FWM_p}|^2$ can be completely suppressed at $\varepsilon _m\approx 2.7$ pN due to perfect destructive interference. With further increasing the amplitude $\varepsilon _m$, the FWM intensity $|\mathrm {FWM_p}|^2$ will keep growing because of the mechanical driving. Figures 4(b) and (d) show that the FWM intensity can be significantly enhanced for both $\phi =\pi /2$ and $\phi =3\pi /2$ if the mechanical driving amplitude is strong enough, where the interference effect is weak since $|\mathrm {FWM_1}|^2\gg |\mathrm {FWM_2}|^2$. However, if the mechanical driving field is too strong, the fluctuation $\delta a$ becomes larger and the second-order sideband effects should be considered [40,41]. For the chosen parameters, we find that $|a_{1+}|$= $|a_{1-}|\approx 0.017 a_{1s}$ at $\varepsilon _m=50$ pN. Therefore, the it is still reasonable to use the linearized method and neglect the quadratic term in $\delta a$. In addition, the mechanical driving field has no influence on the steady-state solution $a_{1s}$ and the effective optomechanical coupling strength $g_1 a_{1s}$, which will not result in the shift of the exceptional points.
Fig. 4. Plots of $|\mathrm {FWM}_1|^2$, $|\mathrm {FWM}_2|^2$, and $|\mathrm {FWM_p}|^2$ at $\Omega =\omega _m$ as functions of the mechanical driving amplitude $\varepsilon _m$ with $\phi =\pi /2$ in (a) and (b) and $\phi =3\pi /2$ in (c) and (d). Here we choose $\Delta _1^{'}=\Delta _2=0$ and $J=0.251(\kappa _1+\kappa _2)$. The other parameters are the same as those in Fig. 2.
The gain rate of the active cavity $a_2$ can be controlled in experiments [28,29]. In Fig. 5, we plot the FWM intensity $\log _{10}|\mathrm {FWM_p}|^2$ versus the gain-loss ratio $\kappa _2/\kappa _1$ with $J=0.251(\kappa _1+\kappa _2)$. In the regime $\kappa _2/\kappa _1<0$, the cavity $a_2$ corresponds to a passive cavity, and the FWM intensity decreases with the increase of the gain-loss ratio $\kappa _2/\kappa _1$. The minimum value locates at $\kappa _2=0$. When the cavity $a_2$ becomes active, the FWM intensity starts to strengthen again, and the maximum value locates at $\kappa _2/\kappa _1\approx 1$ for $\varepsilon _m=0$. Therefore, the FWM intensity can be greatly enhanced near the exceptional points. In the presence of the additional mechanical driving field, the FWM intensity can be further strengthened and the influence of the phase difference is obvious in the vicinity of $\kappa _2/\kappa _1=1$. With balanced gain and loss, the FWM intensity is enhanced for $\phi =\pi /2$ and suppressed for $\phi =3\pi /2$ when the amplitude $\varepsilon _m$ equals to 4 pN.
Fig. 5. FWM intensity $\log _{10}|\mathrm {FWM_p}|^2$ at $\Omega =\omega _m$ as a function of gain-loss ratio $\kappa _2/\kappa _1$ for different values of $\varepsilon _m$ and $\phi$. The other parameters are the same as those in Fig. 4.
We proceed to study the dependence of the FWM intensity on the frequency of the control field. In Fig. 6, the FWM intensity $\log _{10}{|\mathrm {FWM_p}|^2}$ is plotted versus the cavity-control field detuning $\Delta _1^{'}/\omega _m$ with $\Delta _1^{'}=\Delta _2$ for different values of the mechanical driving amplitude $\varepsilon _m$ and phase difference $\phi$. In the absence of the mechanical driving field, it can be seen that the FWM intensity reaches the maximum when the control field is nearly resonant with the passive cavity. Furthermore, the FWM intensity can be greatly strengthened by the additional mechanical driving field. It should be noted that the phase-dependent effect is only evident around $\Delta _1^{'}=0$ and $\Delta _1^{'}=\omega _m$, as shown in the insets of Fig. 6. For the chosen parameters, the FWM intensity can be enhanced for $\phi =\pi /2$ and suppressed for $\phi =3\pi /2$, which is consistent with the results in Fig. 3.
Fig. 6. FWM intensity $\log _{10}|\mathrm {FWM_p}|^2$ at $\Omega =\omega _m$ as a function of $\Delta _{1}^{'}/\omega _m$ for different values of $\varepsilon _m$ and $\phi$. Here we fix $\Delta _2=\Delta _1^{'}$, and the other parameters are the same as those in Fig. 4.
Finally, Fig. 7 shows the FWM intensity $\log _{10}|\mathrm {FWM_p}^2|$ versus $(\Omega -\omega _m)/\gamma _m$ for different values of the control power $P_c$. With the increase of the control power $P_c$, the FWM intensity can be enhanced, as shown in Fig. 7. Furthermore, the inset of Fig. 7 plots the FWM intensity at $\Omega =\omega _m$ as a function of the control power $P_c$. It can be seen that the maximum FWM intensity locates at $P_c\approx 51.1$ nW, which is the critical value between the stable and unstable regimes due to the gain of the cavity $a_2$. With further increasing the control power $P_c$, the system becomes unstable and the FWM intensity decreases monotonically. Therefore, the FWM intensity can be greatly strengthened by increasing the power of the control field as long as the $\mathcal {PT}$-symmetric optomechanical system operates in the stable regime.
Fig. 7. FWM intensity $\log _{10}|\mathrm {FWM_p}|^2$ as a function of $(\Omega -\omega _m)/\gamma _m$ for different control power $P_c$. The inset shows the FWM intensity $\log _{10}|\mathrm {FWM_p}|^2$ at $\Omega =\omega _m$ versus the control power $P_c$. The other parameters are the same as those in Fig. 4 except $\Delta _1^{'}=\Delta _2=\omega _m$, $\varepsilon _m=4$ pN and $\phi =\pi /2$.
4. Conclusion
In summary, we have explored the enhancement of four-wave mixing process based on a $\mathcal {PT}$-symmetric optomechanical system driven by a strong optical control field, a weak optical probe field, and a weak coherent mechanical driving field. We study the dependence of the FWM intensity on $\mathcal {PT}$-symmetry by tuning the cavity-cavity coupling strength and the gain-loss ratio, and it is found that the FWM intensity is greatly enhanced near the exceptional points, which can be $10^{12}$ orders of magnitude higher than that of the case without the active cavity. We also demonstrate that the FWM intensity can be further enhanced by controlling the amplitude and phase of the mechanical driving field. Moreover, it is shown that the FWM intensity increases monotonically with the power of the control field in the stable regime of this optomechanical system. The results obtained herein may be useful for gaining further insight into the properties of $\mathcal {PT}$-symmetric optomechanical system and find potential applications in optical wavelength converter and optical switch.
Funding
National Natural Science Foundation of China (11874170); China Postdoctoral Science Foundation (2017M620593); Qinglan Project of Jiangsu Province of China; Natural Science Research of Jiangsu Higher Education Institutions of China (17KJA460004, 19KJA150011); Practice Innovation Training Program Projects for Jiangsu College Students (201910323021Z).
Disclosures
The authors declare no conflicts of interest.
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