1. Introduction

Optomechanical coupling between a mechanical resonator and an electromagnetic cavity via radiation pressure is the basis of the pioneering field of cavity optomechanics [1–3]. When the cavity field is driven by a strong control field, the optical response of the optomechanical system to a weak probe field can exhibit some interesting phenomena, such as optomechanically induced transparency (OMIT) [4–6], optomechanically induced absorption and amplification [7–11]. In particular, OMIT is the analog of electromagnetically induced transparency (EIT) [12,13], which has been first observed in atomic vapors and later in various solid state systems. Recently, a variety of OMIT, including reversed OMIT [14,15], nonlinear OMIT [16,17], double OMIT [18,19], higher-order OMIT and group delay [20–22], and nonreciprocal OMIT [23–25] have been under extensive investigation. These phenomena can find important applications in slow and fast light [26–28], quantum state transfer [29,30], wavelength conversion [31,32], and so on.

Recent progress in fabrication techniques allows for the coupling between mechanical resonators and other quantum systems, such as cold atoms [33], nitrogen-vacancy (NV) centers [34,35], spin qubit [36,37], superconducting qubit [38–40], and quantum dot [41,42]. By combing optomechanics and intrinsic two-level defects, the hybrid optomechanical system has been exploited to realize non-classical states of the mechanical resonator [43], photon blockade [44], double OMIT [19], phonon laser [45], and Fano resonance [46]. Cotrufo et al. proposed that the interaction between a two-level quantum emitter and the mechanical resonator can be controlled and enhanced by the optical field intensity in hybrid optomechanical systems [47]. In the microwave domain, it has been demonstrated that the addition of a two-level system can greatly strengthen the radiation pressure interaction [48] and resolve the vacuum fluctuations of an optomechanical system [49]. Most of the previous works [19,43–49] have focused on the system where the mechanical resonator is coupled to both the cavity and qubit, or the system where the cavity is coupled to both the mechanical resonator and qubit. Recently, a fully coupled hybrid optomechanical system with all mutual couplings was theoretically studied, where new quantum interference effects and correlations arose because of the interplay between different physical interactions [50].

Furthermore, optomechanical systems with additional mechanical driving field have attracted lots of attention recently. The coherent oscillation of the mechanical resonator can be generated by both the radiation pressure force and the mechanical driving field. Therefore, more complicated interference effects can occur, which results in controllable optical response of these systems [51–58]. In optomechanical experiments, mechanical driving field has been exploited to realize electro-optomechanically induced transparency [59], cascaded optical transparency [60], injection locking [61], and nonreciprocal mode conversion [62]. In addition, a classical microwave pulse was utilized to excite the mechanical resonator in a coupled resonator-qubit system, where quantum ground state and single-phonon control of the mechanical resonator have been demonstrated [40].

In this paper, we investigate how to control the optical response of the hybrid optomechanical system to the weak probe field in the simultaneous presence of a strong optical control field and a weak mechanical driving field. The hybrid system is realized by coupling a two-level system to the mechanical resonator of a generic optomechanical system [19,43–46], which doesn’t consider the direct interaction between the two-level system and the cavity. Compared with previous works without mechanical driving field, we show that phase-dependent interference effect can lead to some interesting phenomena in this system. By tuning the phase difference of the applied fields, double OMIT in [19] can be switched into perfect absorption or remarkable amplification. Moreover, perfect absorption can turn into amplification with increasing the amplitude of the mechanical driving field, which is similar to the transition between optomechanically induced absorption and amplification with a blue-detuned control field [7]. This phase-dependent effect is also evident in controlling the group delay of the transmitted probe field.

The remainder of the paper is organized as follows. In Sec. 2, we describe the theoretical model of the hybrid optomechanical system and derive the analytical expression of the probe transmission. In Sec. 3, we discuss in detail how to control the probe transmission by a series of parameters, including the phase difference, the amplitude of mechanical driving field, and the amplitude of optical control field. In Sec. 4, we study the phase-dependent group delay with respect to the amplitudes of the control field and mechanical driving field. Sec. 5 is the conclusion of our work.

2. Model and theory

The hybrid optomechanical system under consideration is schematically shown in Fig. 1. The mechanical resonator is simultaneously coupled to the optical cavity via radiation pressure and to the two-level system via the Jaynes-Cummings coupling. The interaction Hamiltonian can be described by $H_{\mathrm {om}}=-\hbar g a^{\dagger }a(b^{\dagger }+b)$ and $H_{\mathrm {JC}}=\hbar J(b^{\dagger }\sigma _{-}+b\sigma _{+})$, where $g$ is the coupling strength between the cavity mode with creation (annihilation) operator $a^{\dagger }(a)$ and the mechanical mode with creation (annihilation) operator $b^{\dagger }(b)$, $J$ is the coupling strength between the mechanical mode and the two-level system with lowering (raising) operator $\sigma _{-}(\sigma _{+})$. The optical cavity mode is driven by a strong control field at frequency $\omega _c$ and a weak probe field at frequency $\omega _p$. Furthermore, 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. The interaction Hamiltonian between the optomechanical system and the applied driving fields can be given by

 

Fig. 1. (a) Schematic diagram of the hybrid optomechanical system. One mirror of the optomechanical cavity is fixed and another vibrating mirror, treated as a mechanical resonator, is coupled to a two-level system (qubit). The cavity is driven by a strong control field at frequency $\omega _c$ and a weak probe field at frequency $\omega _p$, and $a_{\mathrm {out}}$ represents the output field of the cavity. The mechanical resonator is excited by a weak coherent mechanical driving field at frequency $\Omega =\omega _p-\omega _c$. (b) Energy-level diagram of the hybrid system where the two-level system is resonant with the mechanical resonator. A weak probe field scans the transition between $|0_a,0_m\rangle$ and $|1_a,0_m\rangle$, where the population of the mechanical mode is unchanged with $a$ and $m$ representing the cavity and mechanical modes, respectively. The coupling between the qubit and the mechanical resonator gives rise to the dressed states $|0_a,1_m+\rangle$ and $|0_a,1_m-\rangle$.

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In a rotating frame at the frequency of the control field $\omega _c$, the system Hamiltonian can be written as:

The dynamics of the system is governed by the quantum Langevin equations (QLEs) according to the Hamiltonian Eq. (2):

Subsequently, rewriting each operator as the sum of a steady-state solution and a small fluctuation with $a=\alpha +\delta a$, $b=\beta +\delta b$, $\sigma _{-}=L_0+\delta \sigma _{-}$, and $\sigma _{z}=W_0+\delta \sigma _{z}$, one can obtain the following linearized equations:

The linearized Eqs. (13)–(16) can be solved analytically by transforming all the operators into another rotating frame with $\delta o\rightarrow \delta o e^{-i\Omega t}~(o=a,b,\sigma _{-},\sigma _{z}, a_{\mathrm {in}}, b_{\mathrm {in}}, c_{-,\mathrm {in}}, c_{z,\mathrm {in}})$. Here we assume that the cavity field is driven on the red sideband (i.e., $\Delta _a\approx \omega _m$) and the system operates in the resolved sideband regime with $\omega _m\gg (\kappa ,G)$. With the rotating-wave approximation (RWA), Eqs. (13)–(16) become

3. Phase-controlled amplification

Based on the above analytical expressions, we choose the experimentally realizable parameters to demonstrate the phase-controlled amplification in the probe transmission spectrum. Specific parameters are chosen from previous works [19,43,44]: $\omega _m/2\pi =\omega _q/2\pi =100$ MHz, $\kappa /2\pi =8$ MHz, $\eta =0.45$, $\gamma _m/2\pi =2$ kHz, $\gamma _q/2\pi =0.1$ MHz, $g/2\pi =10$ MHz, and $J/2\pi =1$ MHz. In addition, we mainly study the case that the cavity field is driven on the red sideband ($\Delta _a=\omega _m$).

To see the effect of the optical and mechanical driving fields on the transmission spectrum, we plot the probe transmission $|t_p|^2$ as functions of the phase difference $\phi /\pi$ and probe-control field detuning $\Omega /\omega _m$ in Fig. 2. It is shown that probe transmission $|t_p|^2$ can be larger than unity in two symmetric regimes when $\phi$ and $\Omega$ vary. Therefore, the transmitted probe field can be amplified when $\phi$ and $\Omega$ are tuned to lie in these regimes. Furthermore, we can see that the probe transmission $|t_p|^2$ reaches the maximum value when $\Omega /\omega _m\approx 0.99$ and 1.01 for fixed $\phi =\pi$. On the other hand, if the detuning $\Omega /\omega _m$ is fixed to be $0.99$ or $1.01$, the probe transmission $|t_p|^2$ decreases monotonically when the phase difference $\phi$ deviates from $\pi$ and reaches the minimum when $\phi =0$ and $2\pi$.

 

Fig. 2. Contour plot of the probe transmission $|t_p|^2$ versus the phase difference $\phi /\pi$ and probe-control field detuning $\Omega /\omega _m$. Other parameters are $\omega _m/2\pi =\omega _q/2\pi =100$ MHz, $\kappa /2\pi =8$ MHz, $\eta =0.45$, $\gamma _m/2\pi =2$ kHz, $\gamma _q/2\pi =0.1$ MHz, $g/2\pi =10$ MHz, $J/2\pi =1$ MHz, $\Delta _a=\omega _m$, $\varepsilon _c/2\pi =10$ MHz, and $r=0.2$.

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In order to understand the phase-dependent effect, in Fig. 3, we plot $|t_1|^2$, $|t_2|^2$, and $|t_p|^2$ versus the detuning $\Omega /\omega _m$ with (a) $\phi =0$ and (b) $\phi =\pi$. $|t_1|^2$ represents the probe transmission in the absence of the mechanical driving field, and double optomechanically induced transparency is observed around $\Omega =\omega _m\pm J$, which can be explained by the radiation pressure induced interference effect between the probe field and the anti-Stokes field based on the dressed states [19]. $|t_2|^2$ is the modification of the probe transmission due to the mechanical driving field. Equations (23)–(24) show that $|t_1|^2$ and $|t_2|^2$ keeps the same when $\phi$ varies, which can also be seen from the red dashed line and blue dash-dotted line in Figs. 3(a) and 3(b). However, the phase difference $\phi$ affects the interference between $t_1$ and $t_2$, which further determines the probe transmission $|t_p|^2$. When $\phi =0$, the probe transmission $|t_p|^2$ is much smaller than both $|t_1|^2$ and $|t_2|^2$ around $\Omega =\omega _m\pm J$, which results from the destructive interference between $t_1$ and $t_2$. When $\phi =\pi$, constructive interference between $t_1$ and $t_2$ leads to the amplification of the transmitted probe field at $\Omega =\omega _m+J$ and $\Omega =\omega _m-J$. Therefore, double optomechanically induced transparency can be turned into double mechanically induced absorption or amplification by tuning the additional mechanical driving field. If the coupling strength between the resonator and the two-level system is modulated, it can be seen that one important advantage of this hybrid optomechanical system is the tunability of frequency at which the probe field can be amplified. Moreover, the probe transmission $|t_p|^2$ at $\Omega =\omega _m+J$ is plotted as a function of the phase difference $\phi$ in the inset of Fig. 3(b), which also shows that the maximum transmission is located at $\phi =\pi$ while the minimum is located at $\phi =0$ and $2\pi$.

 

Fig. 3. Plots of $|t_1|^2$, $|t_2|^2$, and $|t_p|^2$ as a function of the probe-control field detuning $\Omega /\omega _m$ when the phase difference $\phi$ equals to (a) 0 and (b) $\pi$, respectively. The inset of Fig. 3(b) shows the probe transmission $|t_p|^2$ at $\Omega =\omega _m+J$ as a function of the phase difference $\phi /\pi$. The other parameters are the same as those in Fig. 2

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We proceed to study how the amplitude $\varepsilon _m$ of the mechanical driving field influence the interference effect for different values of phase difference $\phi$. Figure 4 plots the transmission $|t_1|^2$, $|t_2|^2$, and $|t_p|^2$ at $\Omega =\omega _m+J$ versus $r=\varepsilon _m/\varepsilon _p$ for (a) $\phi =0$ and (b) $\phi =\pi$. $|t_1|^2$ is independent of the mechanical driving field and keeps constant when $r$ varies, but $|t_2|^2$ increases monotonically with enhancing $r$. As mentioned above, destructive interference exists between $t_1$ and $t_2$ if the phase difference $\phi$ is equal to 0. We can see from Fig. 4(a) that $|t_p|^2$ decreases monotonically from an initial value to zero if $r$ increases from 0 to about 0.28. Complete destructive interference occurs when $|t_1|^2=|t_2|^2$ at $r\approx 0.28$, which results in the zero transmission of the probe field. The turning point (TP) position is given by

 

Fig. 4. Plots of $|t_1|^2$, $|t_2|^2$, and $|t_p|^2$ at $\Omega =\omega _m+J$ as a function of $r=\varepsilon _m/\varepsilon _p$ for (a) $\phi =0$ and (b) $\phi =\pi$. The other parameters are the same as those in Fig. 2.

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It is well known that the optical response of the optomechanical system can be modified by the strong control field. In Fig. 5, we study the probe transmission versus the control field amplitude $\varepsilon _c/2\pi$ for different values of the mechanical driving field. Equation (23) shows that $t_1$ is independent of the mechanical driving field, and we can see from the red dashed line in Fig. 5 that the transmission $|t_1|^2$ at $\Omega =\omega _m+J$ increases from 0 to approach 1 when the control field amplitude is enhanced. Similar results can be obtained for the transmission $|t_1|^2$ at $\Omega =\omega _m-J$ when $\varepsilon _c$ is increased. Therefore, double optomechanically induced transparency can be observed in this hybrid optomechanical system in the absence of the mechanical driving field, as discussed in [19]. However, the optical response of the system can be further modified when an additional mechanical driving field is applied. For fixed $r=0.6$, transmission $|t_2|^2$ is independent of the phase difference $\phi$, and Fig. 5 shows that $|t_2|^2$ increases from 0 to the maximum value 5.7 and then gradually decreases with increasing the control field amplitude $\varepsilon _c$. If the phase difference $\phi =0$, destructive interference between $t_1$ and $t_2$ suppresses the probe transmission $|t_p|^2$, but $|t_p|^2$ is enhanced due to constructive interference for $\phi =\pi$. At the crossover point, $\varepsilon _c/2\pi \approx 21$ MHz, the probe transmission $|t_p|^2\approx 0$ due to complete destructive interference for $\phi =0$ while $|t_p|^2\approx 4|t_1|^2=4|t_2|^2$ because of complete constructive interference for $\phi =\pi$. Therefore, the transmitted probe field can be amplified at low control field amplitude due to the additional mechanical driving field, depending on the phase difference of the driving fields.

 

Fig. 5. Probe transmission $|t_1|^2$, $|t_2|^2$, and $|t_p|^2$ at $\Omega =\omega _m+J$ as a function of the control field amplitude $\varepsilon _c/2\pi$ for different values of mechanical driving field. The other parameters are the same as those in Fig. 4.

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4. Controllable slow light

Optomechanically induced transparency has been widely exploited to generate slow light effect based on the accompanied rapid phase dispersion within the transparency window. We have shown that the mechanical driving field can modify the probe transmission, and the corresponding phase dispersion can also be tuned. According to Eq. (22), the phase dispersion of the transmitted probe field is given by $\phi _t(\omega _p)=\arg [t_p(\omega _p)]$. The optical group delay is then defined as

 

Fig. 6. Group delay $\tau _g$ of the transmitted probe field as a function of the control field amplitude $\varepsilon _c/2\pi$ for different values of mechanical driving field. The other parameters are the same as those in Fig. 5.

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The group delay of the transmitted probe field can also be controlled by the amplitude and phase of the mechanical driving field for fixed optical control field. Figure 7 plots the group delay $\tau _g$ as a function of $r$ with $\phi =\pi /2,2\pi /3$, and $\pi$, respectively. For $\phi =\pi /2$ and $2\pi /3$, the group delay increases to the maximum value and then gradually decreases with increasing $r$. In these two cases, the transmission peaks around $\Omega =\omega _m+J$ and $\Omega =\omega _m-J$ are asymmetric, and here we study the group delay at $\Omega =\omega _m+J$. However, one can see that the group delay $\tau _g$ increases monotonically to the maximum value with increasing $r$ under the condition of constructive interference ($\phi =\pi$). Furthermore, the inset of Fig. 7 plots the group delay $\tau _g$ as a function of the phase difference $\phi$ for fixed $r=0.3$. The parameters are chosen to ensure that the probe transmission $|t_p|^2$ isn’t negligible. Based on the above discussions, we find that the group delay $\tau _g$ of the transmitted probe field can be flexibly controlled by the optical control field as well as the mechanical driving field.

 

Fig. 7. Group delay $\tau _g$ as a function of $r=\varepsilon _m/\varepsilon _p$ for different values of phase difference $\phi$. The inset of Fig. 7 shows the group delay $\tau _g$ versus the phase difference $\phi$ with $r=0.3$. The other parameters are the same as those in Fig. 4 except $\varepsilon _c/2\pi =4$ MHz.

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Finally, we discuss the influence of the two-level system on the transmission $|t_p|^2$ and group delay $\tau _g$ of the probe field. As pointed out in Fig. 3, one important benefit of introducing the additional two-level system is the appearance of double mechanically induced amplification, and the position of the amplification peaks can be modulated by the coupling strength $J$. In Fig. 8, we plot the probe transmission $|t_p|^2$ and group delay $\tau _g$ versus the decay rate $\gamma _q/2\pi$ of the two-level system. It can be seen that both probe transmission $|t_p|^2$ and group delay $\tau _g$ decrease monotonically with the increase of the decay rate $\gamma _q$. Therefore, minimizing the decay rate of the two-level system is beneficial for enhancing the probe transmission and the group delay. It is also worth noting that increasing the decay rate of the two-level system defect can be exploited to realize the phonon lasing in a compound cavity optomechanical system with intrinsic defect [45]. In addition, Fig. 8 shows that the influence of the coupling strength $J$ on the probe transmission and the group delay at $\Omega =\omega _m+J$ is negligible for the chosen parameters when $J$ varies. If the transition frequency $\omega _q$ of the two-level system is modulated, the transmission peaks can become asymmetric [46], which will not be discussed in detail here. Consequently, the optical response of the hybrid optomechanical system can be controlled more flexibly with the assistance of the two-level system.

 

Fig. 8. (a) Probe transmission $|t_p|^2$ and (b) Group delay $\tau _g$ at $\Omega =\omega _m+J$ as a function of the decay rate $\gamma _q/2\pi$ for different values of the coupling strength $J$. The other parameters are the same as those in Fig. 7 except $r=0.3$ and $\phi =\pi$.

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5. Conclusion

In conclusion, we have studied the transmission of a weak probe field through a hybrid optomechanical system, which consists of a cavity and mechanical resonator with a two-level system (qubit). Under the common interaction of a strong optical control field and a weak coherent mechanical driving field, the phase dependent interference effect in this hybrid system is evident. One can realize the switch between optical absorption and amplification by tuning the phase difference of the applied fields for fixed amplitude of the control field. With further increasing the amplitude of the mechanical driving field above a critical value, it is shown that only optical amplification can be observed. In addition, when the amplitude of the mechanical driving field is fixed but the control field varies, switch between optical absorption and amplification can be realized. We have also shown that the group delay of the transmitted probe field can be improved with the assistance of the mechanical driving field, which is also phase dependent.