Abstract

We theoretically investigate the transmission and group delay of a probe field incident on a hybrid optomechanical system, which consists of a mechanical resonator simultaneously coupled to an optical cavity and a two-level system (qubit). The cavity field is driven by a strong red-detuned control field, and a weak coherent mechanical driving field is applied to the mechanical resonator. With the assistance of additional mechanical driving field, it is shown that double optomechanically induced transparency can be switched into absorption due to destructive interference or amplification because of constructive interference, which depends on the phase difference of the applied fields. We study in detail how to control the probe transmission by tuning the parameters of the optical and mechanical driving fields. Furthermore, we find that the group delay of the transmitted probe field can be prolonged by the tuning the amplitude and phase of the mechanical driving field.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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 [13]. 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) [46], optomechanically induced absorption and amplification [711]. 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 [2022], and nonreciprocal OMIT [2325] have been under extensive investigation. These phenomena can find important applications in slow and fast light [2628], 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 [3840], 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 [19,4349] 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 [5158]. 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 [19,4346], 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

$$\begin{aligned}H_{\mathrm{dr}} = &i\hbar\varepsilon_c(a^{{\dagger}}e^{{-}i\omega_c t-i\phi_c}-a e^{i\omega_c t+i\phi_c})+i\hbar\varepsilon_p(a^{{\dagger}}e^{{-}i\omega_p t-i\phi_p}-a e^{i\omega_p t +i\phi_p})\\ &+i\hbar\varepsilon_m(b^{{\dagger}}e^{{-}i\Omega t-i\phi_m}-b e^{i\Omega t+i\phi_m}), \end{aligned}$$
where $\varepsilon _c(\varepsilon _p)$ and $\phi _c(\phi _p)$ represent the amplitude and phase of the control (probe) fields.

 figure: Fig. 1.

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:

$$\begin{aligned} H = &\hbar\Delta_a a^{{\dagger}}a+\hbar\omega_m b^{{\dagger}}b+\frac{\hbar}{2}\omega_q\sigma_z+H_{\mathrm{om}}+H_{\mathrm{JC}}+i\hbar\varepsilon_c (a^{{\dagger}}-a)\\ &+i\hbar\varepsilon_p(a^{{\dagger}}e^{{-}i\Omega t-i\phi_{pc}}-a e^{i\Omega t +i\phi_{pc}})+i\hbar\varepsilon_m(b^{{\dagger}}e^{{-}i\Omega t-i\phi_m}-b e^{i\Omega t+i\phi_m}), \end{aligned}$$
where $\Delta _a=\omega _a-\omega _c$ is the detuning between the cavity mode and the control field, $\omega _m$ is the resonance frequency of the mechanical mode, and $\omega _q$ is the transition frequency of the two-level system. Here $\sigma _z=|e\rangle \langle e|-|g\rangle \langle g|$ is the Pauli operator with $|g\rangle$ and $|e\rangle$ representing the ground state and excited state of the two-level system, and $\phi _{pc}=\phi _p-\phi _c$ is the phase difference between the probe field and control field.

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

$$\begin{aligned} &\dot{a}={-}\left(\kappa/2+i\Delta_a\right)a+ig(b^{{\dagger}}+b)a+\varepsilon_c+\varepsilon_p e^{{-}i\Omega t-i\phi_{pc}}+\sqrt{\kappa}a_{\mathrm{in}}(t),\end{aligned}$$
$$\begin{aligned} &\dot{b}={-}\left(\gamma_m/2+i\omega_m\right)b+iga^{{\dagger}}a-iJ\sigma_{-}+\varepsilon_m e^{{-}i\Omega t-i\phi_m}+\sqrt{\gamma_m}b_{\mathrm{in}}(t),\end{aligned}$$
$$\begin{aligned} &\dot{\sigma_{-}}={-}\left(\gamma_q/2+i\omega_q\right)\sigma_{-}+iJb\sigma_{z}+\sqrt{\gamma_q}c_{-,\mathrm{in}}(t),\end{aligned}$$
$$\begin{aligned} &\dot{\sigma_{z}}={-}\gamma_q(\sigma_z+1)-2iJ(b\sigma_{+}-b^{{\dagger}}\sigma_{-})+\sqrt{\gamma_q}c_{z,\mathrm{in}}(t),\end{aligned}$$
where the terms corresponding to the cavity decay rate $\kappa$, mechanical damping rate $\gamma _m$, and decay rate $\gamma _q$ of the two-level system have been added phenomenologically. The operators $a_{\mathrm {in}}(t)$, $b_{\mathrm {in}}(t)$, $c_{-,\mathrm {in}}(t)$, and $c_{z,\mathrm {in}}(t)$ represent the noises corresponding to the operators $a$, $b$, $\sigma _{-}$, and $\sigma _z$ with $\langle a_{\mathrm {in}}(t)\rangle =\langle b_{\mathrm {in}}(t)\rangle =\langle c_{-,\mathrm {in}}(t)\rangle =\langle c_{z,\mathrm {in}}(t)\rangle =0$ [19]. The steady-state solutions can be obtained by setting all the time derivatives in Eqs. (3)–(6) to be zero, which are given by
$$\begin{aligned} &\alpha=\langle a\rangle_{s}=\frac{\varepsilon_c}{\kappa/2+i\Delta_a'}, \end{aligned}$$
$$\begin{aligned} &\beta=\langle b\rangle_{s}=\frac{ig|\alpha|^2-iJL_0}{\gamma_m/2+i\omega_m}, \end{aligned}$$
$$\begin{aligned} &L_0=\langle\sigma_{-}\rangle_s=\frac{iJ\beta W_0}{\gamma_q/2+i\omega_q}, \end{aligned}$$
$$\begin{aligned} &W_0=\langle\sigma_{z}\rangle_s={-}\frac{\gamma_q^2+4\omega_q^2}{\gamma_q^2+4\omega_q^2+8J^2\beta^2}, \end{aligned}$$
where $\Delta _a'=\Delta _a-g(\beta +\beta ^*)$ is the effective cavity-control field detuning. Therefore, the steady-state photon number $|\alpha |^2$ and phonon number $|\beta |^2$ satisfy the following coupled equations:
$$|\alpha|^2\left\{\left(\frac{\kappa}{2}\right)^2+\left[\Delta_a-\frac{2g^2|\alpha|^2(2\omega_q\epsilon_1 +\gamma_q\epsilon_2)}{\epsilon_1^2 +\epsilon_2^2} \right]^2\right\}=\varepsilon_c^2,$$
$$|\beta|^2(\epsilon_1^2+\epsilon_2^2)^2=g^2 |\alpha|^4\left[(2\omega_q\epsilon_1+\gamma_q\epsilon_2)^2+(\gamma_q\epsilon_1-2\omega_q\epsilon_2)^2 \right],$$
where $\epsilon _1=2\omega _m\omega _q-\gamma _q\gamma _m/2+2J^2W_0,\epsilon _2=\omega _m\gamma _q+\gamma _m\omega _q$, and $W_0=-\frac {\gamma _q^2+4\omega _q^2}{\gamma _q^2+4\omega _q^2+8J^2|\beta |^2}$.

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:

$$\begin{aligned} \dot{\delta a}={-}\left(\kappa/2+i\Delta_a'\right)\delta a+iG(\delta b^{{\dagger}}+\delta b)+\varepsilon_p e^{{-}i\Omega t-i\phi_{pc}}+\sqrt{\kappa}a_{\mathrm{in}}(t),\end{aligned}$$
$$\begin{aligned} \dot{\delta b}={-}\left(\gamma_m/2+i\omega_m\right)\delta b+i\left(G^*\delta a+G\delta a^{{\dagger}}\right)-iJ\delta\sigma_{-}+\varepsilon_m e^{{-}i\Omega t-i\phi_m}+\sqrt{\gamma_m}b_{\mathrm{in}}(t),\end{aligned}$$
$$\begin{aligned} \dot{\delta\sigma_{-}}={-}\left(\gamma_q/2+i\omega_q\right)\delta\sigma_{-}+iJ(\beta\delta\sigma_z +W_0\delta b)+\sqrt{\gamma_q}c_{-,\mathrm{in}}(t),\end{aligned}$$
$$\begin{aligned} \dot{\delta\sigma_z}={-}\gamma_q\delta\sigma_z-2iJ(\beta\delta\sigma_{+}+L_0^*\delta b-\beta^*\delta\sigma_{-}-L_0\delta b^{{\dagger}})+\sqrt{\gamma_q}c_{z,\mathrm{in}}(t),\end{aligned}$$
where $G=g\alpha$ is the effective optomechanical coupling strength, and the nonlinear terms such as $\delta a\delta b$ and $\delta a^{\dagger }\delta a$ have been neglected.

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

$$\begin{aligned} &\dot{\delta a}={-}\Gamma_a\delta a+iG\delta b+\varepsilon_p e^{{-}i\phi_{pc}}+\sqrt{\kappa}a_{\mathrm{in}}(t),\end{aligned}$$
$$\begin{aligned} &\dot{\delta b}={-}\Gamma_m\delta b+iG^*\delta a-iJ\delta\sigma_{-}+\varepsilon_m e^{{-}i\phi_m}+\sqrt{\gamma_m}b_{\mathrm{in}}(t),\end{aligned}$$
$$\begin{aligned} &\dot{\delta \sigma_{-}}={-}\Gamma_{-}\delta\sigma_{-}+iJ(\beta\delta\sigma_z+W_0\delta b)+\sqrt{\gamma_q}c_{-,\mathrm{in}}(t),\end{aligned}$$
$$\begin{aligned} &\dot{\delta\sigma_z}={-}\Gamma_q\delta\sigma_z-2iJ(L_0^*\delta b-\beta^*\delta\sigma_{-})+\sqrt{\gamma_q}c_{z,\mathrm{in}}(t),\end{aligned}$$
where $\Gamma _a=\kappa /2+i(\Delta _a'-\Omega )$, $\Gamma _m=\gamma _m/2+i(\omega _m-\Omega )$, $\Gamma _{-}=\gamma _q/2+i(\omega _q-\Omega )$, and $\Gamma _q=\gamma _q-i\Omega$. According to Eqs. (17)–(20) and under the steady-state condition $\langle \dot {\delta a}\rangle =\langle \dot {\delta b}\rangle =\langle \dot {\delta \sigma _{-}}\rangle =\langle \dot {\delta \sigma _z}\rangle =0$, we can obtain
$$\langle\delta a\rangle=\frac{(\Gamma_m\Theta+2i\beta L_0^*J^3-W_0\Gamma_q J^2)\varepsilon_p e^{{-}i\phi_{pc}}+iG\Theta\varepsilon_m e^{{-}i\phi_m}}{\Gamma_a(\Gamma_m\Theta+2i\beta L_0^* J^3-W_0\Gamma_q J^2)+|G|^2\Theta},$$
where $\Theta =\Gamma _{-}\Gamma _q+2|\beta |^2J^2$. Similar methods have been widely employed in previous works [51,5658]. The noise terms have been neglected actually since we focus on the mean response of the system to the probe field and the mean values of the noise operators are zero. Based on the standard input-output relation $\langle a_{\mathrm {out}}\rangle +\varepsilon _c+\varepsilon _p e^{-i\Omega t-i\phi _{pc}}=\kappa _e\langle a\rangle$ [63] with $\kappa _e=\eta \kappa$ representing the external decay rate of the cavity, the output field of the cavity can be obtained. The transmission coefficient of the probe field is defined as the ratio of the output and input field amplitudes at the probe frequency, which is given by
$$t_p=\frac{\kappa_e\langle\delta a\rangle-\varepsilon_p e^{{-}i\phi_{pc}}}{\varepsilon_p e^{{-}i\phi_{pc}}}=t_1+t_2$$
with
$$\begin{aligned} &t_1=\frac{(\Gamma_m\Theta+2i\beta L_0^* J^3-W_0\Gamma_q J^2)\kappa_e}{\Gamma_a(\Gamma_m\Theta+2i\beta L_0^* J^3-W_0\Gamma_q J^2)+|G|^2\Theta}-1,\end{aligned}$$
$$\begin{aligned} &t_2=\frac{iG\Theta\kappa_e r e^{{-}i\phi}}{\Gamma_a(\Gamma_m\Theta+2i\beta L_0^* J^3-W_0\Gamma_q J^2)+|G|^2\Theta},\end{aligned}$$
where $r=\varepsilon _m/\varepsilon _p$ and $\phi =\phi _m-\phi _{pc}$ is the phase difference between the mechanical driving field and optical driving fields. Equations (22)–(24) show that the transmission coefficient can be divided into two parts, where $t_1$ corresponds to the contribution from the optical control field and $t_2$ is the modification of probe transmission due to the mechanical driving field. Interference between $t_1$ and $t_2$ determines the actual probe transmission, where the phase difference $\phi$ can be exploited to control the transmission spectrum.

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 $2\pi$.

 figure: Fig. 2.

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 $2\pi$.

 figure: Fig. 3.

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

$$r|_{\mathrm{TP}}=\left|\frac{(\kappa_e-\Gamma_a)(\Gamma_m\Theta+2i\beta L_0^*J^3-W_0\Gamma_qJ^2)-|G|^2\Theta}{G\Theta\kappa_e}\right|,$$
which corresponds to $r|_{\mathrm {TP}}\approx 0.28$ for the chosen parameters, as shown in Fig. 4. However, with further increasing $r$, the probe transmission $|t_p|^2$ starts to increase monotonically and can be larger than unity, which is because $|t_2|^2$ is much larger than $|t_1|^2$ and the interference effect between $t_1$ and $t_2$ becomes weak. This result is similar to optomechanically induced absorption with a blue-detuned control field [7]. If the phase difference is tuned to be $\pi$, constructive interference cause the probe transmission $|t_p|^2$ being larger than both $|t_1|^2$ and $|t_2|^2$. With the increase of the mechanical driving amplitude, $|t_p|^2$ becomes increasingly larger. Note that $|t_p|^2\approx 4|t_1|^2=4|t_2|^2$ at $r\approx 0.28$, where complete constructive interference occurs.

 figure: Fig. 4.

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.

 figure: Fig. 5.

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

$$\tau_g=\frac{d\phi_t(\omega_p)}{d\omega_p}=\frac{d\{\mathrm{arg}[t_p(\omega_p)]\}}{d\omega_p}.$$
Fig. 6 plots the group delay $\tau _g$ as a function of the control field amplitude $\varepsilon _c/2\pi$ for different values of mechanical driving field. In the absence of the mechanical resonator, i.e., $r=0$, the maximum group delay $\tau _g\approx 2.5$ $\mu$s when the control field amplitude $\varepsilon _c/2\pi \approx 3.6$ MHz. If the additional mechanical driving field is applied, the group delay $\tau _g$ will be changed, which is dependent on the phase difference $\phi$. When we choose $r=0.6$ and $\phi =0$, the maximum group delay $\tau _g$ will be prolonged to be about 6.1 $\mu s$ at $\varepsilon _c/2\pi \approx 1$ MHz. In addition, the group delay $\tau _g$ is negative around $\varepsilon _c/2\pi =21$ MHz, which represents the appearance of fast light effect. However, it is shown in Fig. 5 that the probe transmission around $\varepsilon _c/2\pi \approx 21$ MHz is approximately equal to zero due to complete destructive interference. Therefore, the indication of fast light in this regime is meaningless. If the phase difference $\phi$ is tuned to be $\pi$, the constructive interference between $t_1$ and $t_2$ leads to the amplification of the transmitted probe field. The group delay $\tau _g$ of the amplified probe field with respect to the control field amplitude $\varepsilon _c/2\pi$ can be seen from the blue dash-dotted line in Fig. 6. Only slow light effect is observed and the maximum group delay $\tau _g\approx 5.3$ $\mu$s around $\varepsilon _c/2\pi \approx 1.2$ MHz. For the chosen parameters, it is evident that the maximum group delay can be prolonged with the assistance of the additional mechanical driving field.

 figure: Fig. 6.

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.

 figure: Fig. 7.

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.

 figure: Fig. 8.

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 $\phi =\pi$.

Download Full Size | PPT Slide | PDF

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.

Funding

National Natural Science Foundation of China (11874170); China Postdoctoral Science Foundation (2017M620593); Natural Science Research of Jiangsu Higher Education Institutions of China (18KJA140001, 19KJA150011); Qinglan Project of Jiangsu Province of China.

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References

  • View by:

  1. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
    [Crossref]
  2. F. Marquardt and S. M. Girvin, “Optomechanics,” Physics 2, 40 (2009).
    [Crossref]
  3. H. Xiong, L. G. Si, X. Y. Lv, X. X. Yang, and Y. Wu, “Review of cavity optomechanics in the weak-coupling regime: from linearization to intrinsic nonlinear interactions,” Sci. China: Phys., Mech. Astron. 58(5), 1–13 (2015).
    [Crossref]
  4. G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010).
    [Crossref]
  5. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
    [Crossref]
  6. J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature (London) 471(7337), 204–208 (2011).
    [Crossref]
  7. F. Hocke, X. Zhou, A. Schliesser, T. J. Kippenberg, H. Huebl, and R. Gross, “Electromechanically induced absorption in a circuit nano-electromechanical system,” New J. Phys. 14(12), 123037 (2012).
    [Crossref]
  8. V. Singh, S. J. Bosman, B. H. Schneider, Y. M. Blanter, A. Castellanos-Gomez, and G. A. Steele, “Optomechanical coupling between a multilayer graphene mechanical resonator and a superconducting microwave cavity,” Nat. Nanotechnol. 9(10), 820–824 (2014).
    [Crossref]
  9. F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. J. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature (London) 480(7377), 351–354 (2011).
    [Crossref]
  10. C. F. Ockeloen-Korppi, E. Damskägg, J.-M. Pirkkalainen, T. T. Heikkilä, F. Massel, and M. A. Sillanpää, “Low-noise amplification and frequency conversion with a multiport microwave optomechanical device,” Phys. Rev. X 6(4), 041024 (2016).
    [Crossref]
  11. L. D. Tóth, N. R. Bernier, A. Nunnenkamp, A. K. Feofanov, and T. J. Kippenberg, “A dissipative quantum reservoir for microwave light using a mechanical oscillator,” Nat. Phys. 13(8), 787–793 (2017).
    [Crossref]
  12. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005).
    [Crossref]
  13. Y. Wu and X. X. Yang, “Electromagnetically induced transparency in $V-$V−, $\Lambda -$Λ−, and cascade-type schemes beyond steady-state analysis,” Phys. Rev. A 71(5), 053806 (2005).
    [Crossref]
  14. H. Jing, Ş. K. Özdemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanicallyinduced transparency in parity-time-symmetric microresonators,” Sci. Rep. 5(1), 9663 (2015).
    [Crossref]
  15. H. Lü, C. Q. Wang, L. Yang, and H. Jing, “Optomechanically induced transparency at exceptional points,” Phys. Rev. Appl. 10(1), 014006 (2018).
    [Crossref]
  16. H. Xiong, L. G. Si, A. S. Zheng, X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86(1), 013815 (2012).
    [Crossref]
  17. Y. C. Liu, Y. F. Xiao, Y. L. Chen, X. C. Yu, and Q. Gong, “Parametric down-conversion and polariton pair generation in optomechanical systems,” Phys. Rev. Lett. 111(8), 083601 (2013).
    [Crossref]
  18. P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90(4), 043825 (2014).
    [Crossref]
  19. H. Wang, X. Gu, Y. X. Liu, A. Miranowicz, and F. Nori, “Optomechanical analog of two-color electromagnetically induced transparency: Photon transmission through an optomechanical device with a two-level system,” Phys. Rev. A 90(2), 023817 (2014).
    [Crossref]
  20. Y. Jiao, H. Lü, J. Qian, Y. Li, and H. Jing, “Nonlinear optomechanics with gain and loss: amplifying higher-order sideband and group delay,” New J. Phys. 18(8), 083034 (2016).
    [Crossref]
  21. H. Zhang, F. Saif, Y. Jiao, and H. Jing, “Loss-induced transparency in optomechanics,” Opt. Express 26(19), 25199–25210 (2018).
    [Crossref]
  22. Y.-F. Jiao, T.-X. Lu, and H. Jing, “Optomechanical second-order sidebands and group delays in a Kerr resonator,” Phys. Rev. A 97(1), 013843 (2018).
    [Crossref]
  23. Z. Shen, Y. L. Zhang, Y. Chen, C. L. Zou, Y. F. Xiao, X. B. Zou, F. W. Sun, G. C. Guo, and C. H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10(10), 657–661 (2016).
    [Crossref]
  24. K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, “Generalized nonreciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13(5), 465–471 (2017).
    [Crossref]
  25. H. Lü, Y. Jiang, Y. Z. Wang, and H. Jing, “Optomechanically induced transparency in a spinning resonator,” Photonics Res. 5(4), 367 (2017).
    [Crossref]
  26. A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature (London) 472(7341), 69–73 (2011).
    [Crossref]
  27. C. Jiang, H. X. Liu, Y. S. Cui, X. W. Li, G. B. Chen, and B. Chen, “Electromagnetically induced transparency and slow light in two-mode optomechanics,” Opt. Express 21(10), 12165–12173 (2013).
    [Crossref]
  28. M. J. Akram, M. M. Khan, and F. Saif, “Tunable fast and slow light in a hybrid optomechanical system,” Phys. Rev. A 92(2), 023846 (2015).
    [Crossref]
  29. Y. D. Wang and A. A. Clerk, “Using interference for high fidelity quantum state transfer in optomechanics,” Phys. Rev. Lett. 108(15), 153603 (2012).
    [Crossref]
  30. L. Tian, “Adiabatic state conversion and pulse transmission in optomechanical systems,” Phys. Rev. Lett. 108(15), 153604 (2012).
    [Crossref]
  31. J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3(1), 1196 (2012).
    [Crossref]
  32. Y. X. Liu, M. Davanço, V. Aksyuk, and K. Srinivasan, “Electromagnetically induced transparency and wideband wavelength conversion in silicon nitride microdisk optomechanical resonators,” Phys. Rev. Lett. 110(22), 223603 (2013).
    [Crossref]
  33. P. Treutlein, D. Hunger, S. Camerer, T. W. Hänsch, and J. Reichel, “Bose-Einstein Condensate coupled to a nanomechanical resonator on an atom chip,” Phys. Rev. Lett. 99(14), 140403 (2007).
    [Crossref]
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2019 (2)

S. C. Wu, L. G. Qin, J. Lu, and Z. Y. Wang, “Phase-dependent double optomechanically induced transparency in a hybrid optomechanical cavity system with coherently mechanical driving,” Chin. Phys. B 28(7), 074204 (2019).
[Crossref]

T. X. Lu, Y. F. Jiao, H. L. Zhang, F. Saif, and H. Jing, “Selective optomechanically-induced amplification with driven oscillators,” Phys. Rev. A 100(1), 013813 (2019).
[Crossref]

2018 (5)

D. B. Sohn, S. Kim, and G. Bahl, “Time-reversal symmetry breaking with acoustic pumping of nanophotonic circuits,” Nat. Photonics 12(2), 91–97 (2018).
[Crossref]

C. S. Muñoz, A. Lara, J. Puebla, and F. Nori, “Hybrid systems for the generation of nonclassical mechanical states via quadratic interactions,” Phys. Rev. Lett. 121(12), 123604 (2018).
[Crossref]

H. Lü, C. Q. Wang, L. Yang, and H. Jing, “Optomechanically induced transparency at exceptional points,” Phys. Rev. Appl. 10(1), 014006 (2018).
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H. Zhang, F. Saif, Y. Jiao, and H. Jing, “Loss-induced transparency in optomechanics,” Opt. Express 26(19), 25199–25210 (2018).
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Y.-F. Jiao, T.-X. Lu, and H. Jing, “Optomechanical second-order sidebands and group delays in a Kerr resonator,” Phys. Rev. A 97(1), 013843 (2018).
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2017 (10)

K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, “Generalized nonreciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13(5), 465–471 (2017).
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H. Lü, Y. Jiang, Y. Z. Wang, and H. Jing, “Optomechanically induced transparency in a spinning resonator,” Photonics Res. 5(4), 367 (2017).
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L. D. Tóth, N. R. Bernier, A. Nunnenkamp, A. K. Feofanov, and T. J. Kippenberg, “A dissipative quantum reservoir for microwave light using a mechanical oscillator,” Nat. Phys. 13(8), 787–793 (2017).
[Crossref]

H. Lü, S. K. Özdemir, L.-M. Kuang, F. Nori, and H. Jing, “Exceptional points in random-defect phonon lasers,” Phys. Rev. Appl. 8(4), 044020 (2017).
[Crossref]

C. Jiang, L. Jiang, H. L. Yu, Y. S. Cui, X. W. Li, and G. B. Chen, “Fano resonance and slow light in hybrid optomechanics mediated by a two-level system,” Phys. Rev. A 96(5), 053821 (2017).
[Crossref]

M. Cotrufo, A. Fiore, and E. Verhagen, “Coherent atom-phonon interaction through mode field coupling in hybrid optomechanical systems,” Phys. Rev. Lett. 118(13), 133603 (2017).
[Crossref]

J. Restrepo, I. Favero, and C. Ciuti, “Fully coupled hybrid cavity optomechanics: Quantum interferences and correlations,” Phys. Rev. A 95(2), 023832 (2017).
[Crossref]

C. Bekker, R. Kalra, C. Baker, and W. P. Bowen, “Injection locking of an electro-optomechanical device,” Optica 4(10), 1196–1204 (2017).
[Crossref]

L. G. Si, H. Xiong, M. Zubairy, and Y. Wu, “Optomechanically induced opacity and amplification in a quadratically coupled optomechanical system,” Phys. Rev. A 95(3), 033803 (2017).
[Crossref]

Y. Li, Y. Y. Huang, X. Z. Zhang, and L. Tian, “Optical directional amplification in a three-mode optomechanical system,” Opt. Express 25(16), 18907 (2017).
[Crossref]

2016 (4)

C. Jiang, Y. S. Cui, X. T. Bian, F. Zuo, H. L. Yu, and G. B. Chen, “Phase-dependent multiple optomechanically induced absorption in multimode optomechanical systems with mechanical driving,” Phys. Rev. A 94(2), 023837 (2016).
[Crossref]

C. F. Ockeloen-Korppi, E. Damskägg, J.-M. Pirkkalainen, T. T. Heikkilä, F. Massel, and M. A. Sillanpää, “Low-noise amplification and frequency conversion with a multiport microwave optomechanical device,” Phys. Rev. X 6(4), 041024 (2016).
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Z. Shen, Y. L. Zhang, Y. Chen, C. L. Zou, Y. F. Xiao, X. B. Zou, F. W. Sun, G. C. Guo, and C. H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10(10), 657–661 (2016).
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Y. Jiao, H. Lü, J. Qian, Y. Li, and H. Jing, “Nonlinear optomechanics with gain and loss: amplifying higher-order sideband and group delay,” New J. Phys. 18(8), 083034 (2016).
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2015 (10)

M. J. Akram, M. M. Khan, and F. Saif, “Tunable fast and slow light in a hybrid optomechanical system,” Phys. Rev. A 92(2), 023846 (2015).
[Crossref]

H. Jing, Ş. K. Özdemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanicallyinduced transparency in parity-time-symmetric microresonators,” Sci. Rep. 5(1), 9663 (2015).
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H. Xiong, L. G. Si, X. Y. Lv, X. X. Yang, and Y. Wu, “Review of cavity optomechanics in the weak-coupling regime: from linearization to intrinsic nonlinear interactions,” Sci. China: Phys., Mech. Astron. 58(5), 1–13 (2015).
[Crossref]

L. Fan, K. Y. Fong, M. Poot, and H. X. Tang, “Cascaded optical transparency in multimode-cavity optomechanical systems,” Nat. Commun. 6(1), 5850 (2015).
[Crossref]

W. Z. Jia, L. F. Wei, Y. Li, and Y. X. Liu, “Phase-dependent optical response properties in an optomechanical system by coherently driving the mechanical resonator,” Phys. Rev. A 91(4), 043843 (2015).
[Crossref]

J. Y. Ma, C. You, L. G. Si, H. Xiong, J. H. Li, X. X. Yang, and Y. Wu, “Optomechanically induced transparency in the presence of an external time-harmonic-driving force,” Sci. Rep. 5(1), 11278 (2015).
[Crossref]

X. W. Xu and Y. Li, “Controllable optical output fields from an optomechanical system with mechanical driving,” Phys. Rev. A 92(2), 023855 (2015).
[Crossref]

J.-M. Pirkkalainen, S. U. Cho, F. Massel, J. Tuorila, T. T. Heikkilä, P. J. Hakonen, and M. A. Sillanpää, “Cavity optomechanics mediated by a quantum two-level system,” Nat. Commun. 6(1), 6981 (2015).
[Crossref]

F. Lecocq, J. D. Teufel, J. Aumentado, and R. W. Simmonds, “Resolving the vacuum fluctuations of an optomechanical system using an artificial atom,” Nat. Phys. 11(8), 635–639 (2015).
[Crossref]

H. Wang, X. Gu, Y. X. Liu, A. Miranowicz, and F. Nori, “Tunable photon blockade in a hybrid system consisting of an optomechanical device coupled to a two-level system,” Phys. Rev. A 92(3), 033806 (2015).
[Crossref]

2014 (5)

I. Yeo, P-L. de Assis, A. Gloppe, E. Dupont-Ferrier, P. Verlot, N. S. Malik, E. Dupuy, J. Claudon, J-M. Gérard, A. Auffèves, G. Nogues, S. Seidelin, J-Ph. Poizat, O. Arcizet, and M. Richard, “Strain-mediated coupling in a quantum dot-mechanical oscillator hybrid system,” Nat. Nanotechnol. 9(2), 106–110 (2014).
[Crossref]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
[Crossref]

V. Singh, S. J. Bosman, B. H. Schneider, Y. M. Blanter, A. Castellanos-Gomez, and G. A. Steele, “Optomechanical coupling between a multilayer graphene mechanical resonator and a superconducting microwave cavity,” Nat. Nanotechnol. 9(10), 820–824 (2014).
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P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90(4), 043825 (2014).
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H. Wang, X. Gu, Y. X. Liu, A. Miranowicz, and F. Nori, “Optomechanical analog of two-color electromagnetically induced transparency: Photon transmission through an optomechanical device with a two-level system,” Phys. Rev. A 90(2), 023817 (2014).
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2013 (5)

Y. C. Liu, Y. F. Xiao, Y. L. Chen, X. C. Yu, and Q. Gong, “Parametric down-conversion and polariton pair generation in optomechanical systems,” Phys. Rev. Lett. 111(8), 083601 (2013).
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Y. X. Liu, M. Davanço, V. Aksyuk, and K. Srinivasan, “Electromagnetically induced transparency and wideband wavelength conversion in silicon nitride microdisk optomechanical resonators,” Phys. Rev. Lett. 110(22), 223603 (2013).
[Crossref]

C. Jiang, H. X. Liu, Y. S. Cui, X. W. Li, G. B. Chen, and B. Chen, “Electromagnetically induced transparency and slow light in two-mode optomechanics,” Opt. Express 21(10), 12165–12173 (2013).
[Crossref]

T. Ramos, V. Sudhir, K. Stannigel, P. Zoller, and T. J. Kippenberg, “Nonlinear quantum optomechanics via individual intrinsic two-level defects,” Phys. Rev. Lett. 110(19), 193602 (2013).
[Crossref]

J. Bochmann, A. Vainsencher, D. D. Awschalom, and A. N. Cleland, “Nanomechanical coupling between microwave and optical photons,” Nat. Phys. 9(11), 712–716 (2013).
[Crossref]

2012 (6)

S. Kolkowitz, A. C. B. Jayich, Q. P. Unterreithmeier, S. D. Bennett, P. Rabl, J. G. E. Harris, and M. D. Lukin, “Coherent sensing of a mechanical resonator with a single-spin qubit,” Science 335(6076), 1603–1606 (2012).
[Crossref]

Y. D. Wang and A. A. Clerk, “Using interference for high fidelity quantum state transfer in optomechanics,” Phys. Rev. Lett. 108(15), 153603 (2012).
[Crossref]

L. Tian, “Adiabatic state conversion and pulse transmission in optomechanical systems,” Phys. Rev. Lett. 108(15), 153604 (2012).
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J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3(1), 1196 (2012).
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H. Xiong, L. G. Si, A. S. Zheng, X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86(1), 013815 (2012).
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F. Hocke, X. Zhou, A. Schliesser, T. J. Kippenberg, H. Huebl, and R. Gross, “Electromechanically induced absorption in a circuit nano-electromechanical system,” New J. Phys. 14(12), 123037 (2012).
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2011 (4)

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature (London) 471(7337), 204–208 (2011).
[Crossref]

F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. J. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature (London) 480(7377), 351–354 (2011).
[Crossref]

O. Arcizet, V. Jacques, A. Siria, P. Poncharal, P. Vincent, and S. Seidelin, “A single nitrogen-vacancy defect coupled to a nanomechanical oscillator,” Nat. Phys. 7(11), 879–883 (2011).
[Crossref]

A. H. Safavi-Naeini, T. P. Mayer Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature (London) 472(7341), 69–73 (2011).
[Crossref]

2010 (3)

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010).
[Crossref]

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref]

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature (London) 464(7289), 697–703 (2010).
[Crossref]

2009 (3)

M. D. LaHaye, J. Suh, P. M. Echternach, K. C. Schwab, and M. L. Roukes, “Nanomechanical measurements of a superconducting qubit,” Nature (London) 459(7249), 960–964 (2009).
[Crossref]

F. Marquardt and S. M. Girvin, “Optomechanics,” Physics 2, 40 (2009).
[Crossref]

P. Rabl, P. Cappellaro, M. V. Gurudev Dutt, L. Jiang, J. R. Maze, and M. D. Lukin, “Strong magnetic coupling between an electronic spin qubit and a mechanical resonator,” Phys. Rev. B 79(4), 041302 (2009).
[Crossref]

2007 (1)

P. Treutlein, D. Hunger, S. Camerer, T. W. Hänsch, and J. Reichel, “Bose-Einstein Condensate coupled to a nanomechanical resonator on an atom chip,” Phys. Rev. Lett. 99(14), 140403 (2007).
[Crossref]

2005 (2)

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005).
[Crossref]

Y. Wu and X. X. Yang, “Electromagnetically induced transparency in $V-$V−, $\Lambda -$Λ−, and cascade-type schemes beyond steady-state analysis,” Phys. Rev. A 71(5), 053806 (2005).
[Crossref]

2004 (1)

I. Wilson-Rae, P. Zoller, and A. Imamoglu, “Laser Cooling of a nanomechanical resonator mode to its quantum ground state,” Phys. Rev. Lett. 92(7), 075507 (2004).
[Crossref]

2002 (1)

A. D. Armour, M. P. Blencowe, and K. C. Schwab, “Entanglement and decoherence of a micromechanical resonator via coupling to a Cooper-Pair Box,” Phys. Rev. Lett. 88(14), 148301 (2002).
[Crossref]

Agarwal, G. S.

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010).
[Crossref]

Akram, M. J.

M. J. Akram, M. M. Khan, and F. Saif, “Tunable fast and slow light in a hybrid optomechanical system,” Phys. Rev. A 92(2), 023846 (2015).
[Crossref]

Aksyuk, V.

Y. X. Liu, M. Davanço, V. Aksyuk, and K. Srinivasan, “Electromagnetically induced transparency and wideband wavelength conversion in silicon nitride microdisk optomechanical resonators,” Phys. Rev. Lett. 110(22), 223603 (2013).
[Crossref]

Allman, M. S.

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature (London) 471(7337), 204–208 (2011).
[Crossref]

Ansmann, M.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature (London) 464(7289), 697–703 (2010).
[Crossref]

Arcizet, O.

I. Yeo, P-L. de Assis, A. Gloppe, E. Dupont-Ferrier, P. Verlot, N. S. Malik, E. Dupuy, J. Claudon, J-M. Gérard, A. Auffèves, G. Nogues, S. Seidelin, J-Ph. Poizat, O. Arcizet, and M. Richard, “Strain-mediated coupling in a quantum dot-mechanical oscillator hybrid system,” Nat. Nanotechnol. 9(2), 106–110 (2014).
[Crossref]

O. Arcizet, V. Jacques, A. Siria, P. Poncharal, P. Vincent, and S. Seidelin, “A single nitrogen-vacancy defect coupled to a nanomechanical oscillator,” Nat. Phys. 7(11), 879–883 (2011).
[Crossref]

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref]

Armour, A. D.

A. D. Armour, M. P. Blencowe, and K. C. Schwab, “Entanglement and decoherence of a micromechanical resonator via coupling to a Cooper-Pair Box,” Phys. Rev. Lett. 88(14), 148301 (2002).
[Crossref]

Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
[Crossref]

Auffèves, A.

I. Yeo, P-L. de Assis, A. Gloppe, E. Dupont-Ferrier, P. Verlot, N. S. Malik, E. Dupuy, J. Claudon, J-M. Gérard, A. Auffèves, G. Nogues, S. Seidelin, J-Ph. Poizat, O. Arcizet, and M. Richard, “Strain-mediated coupling in a quantum dot-mechanical oscillator hybrid system,” Nat. Nanotechnol. 9(2), 106–110 (2014).
[Crossref]

Aumentado, J.

F. Lecocq, J. D. Teufel, J. Aumentado, and R. W. Simmonds, “Resolving the vacuum fluctuations of an optomechanical system using an artificial atom,” Nat. Phys. 11(8), 635–639 (2015).
[Crossref]

Awschalom, D. D.

J. Bochmann, A. Vainsencher, D. D. Awschalom, and A. N. Cleland, “Nanomechanical coupling between microwave and optical photons,” Nat. Phys. 9(11), 712–716 (2013).
[Crossref]

Bahl, G.

D. B. Sohn, S. Kim, and G. Bahl, “Time-reversal symmetry breaking with acoustic pumping of nanophotonic circuits,” Nat. Photonics 12(2), 91–97 (2018).
[Crossref]

Baker, C.

Bekker, C.

Bennett, S. D.

S. Kolkowitz, A. C. B. Jayich, Q. P. Unterreithmeier, S. D. Bennett, P. Rabl, J. G. E. Harris, and M. D. Lukin, “Coherent sensing of a mechanical resonator with a single-spin qubit,” Science 335(6076), 1603–1606 (2012).
[Crossref]

Bernier, N. R.

L. D. Tóth, N. R. Bernier, A. Nunnenkamp, A. K. Feofanov, and T. J. Kippenberg, “A dissipative quantum reservoir for microwave light using a mechanical oscillator,” Nat. Phys. 13(8), 787–793 (2017).
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Figures (8)

Fig. 1.
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 $|0_a,1_m-\rangle$.
Fig. 2.
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 $r=0.2$.
Fig. 3.
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
Fig. 4.
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.
Fig. 5.
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.
Fig. 6.
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.
Fig. 7.
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.
Fig. 8.
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 $\phi =\pi$.

Equations (26)

Equations on this page are rendered with MathJax. Learn more.

H d r = i ε c ( a e i ω c t i ϕ c a e i ω c t + i ϕ c ) + i ε p ( a e i ω p t i ϕ p a e i ω p t + i ϕ p ) + i ε m ( b e i Ω t i ϕ m b e i Ω t + i ϕ m ) ,
H = Δ a a a + ω m b b + 2 ω q σ z + H o m + H J C + i ε c ( a a ) + i ε p ( a e i Ω t i ϕ p c a e i Ω t + i ϕ p c ) + i ε m ( b e i Ω t i ϕ m b e i Ω t + i ϕ m ) ,
a ˙ = ( κ / 2 + i Δ a ) a + i g ( b + b ) a + ε c + ε p e i Ω t i ϕ p c + κ a i n ( t ) ,
b ˙ = ( γ m / 2 + i ω m ) b + i g a a i J σ + ε m e i Ω t i ϕ m + γ m b i n ( t ) ,
σ ˙ = ( γ q / 2 + i ω q ) σ + i J b σ z + γ q c , i n ( t ) ,
σ z ˙ = γ q ( σ z + 1 ) 2 i J ( b σ + b σ ) + γ q c z , i n ( t ) ,
α = a s = ε c κ / 2 + i Δ a ,
β = b s = i g | α | 2 i J L 0 γ m / 2 + i ω m ,
L 0 = σ s = i J β W 0 γ q / 2 + i ω q ,
W 0 = σ z s = γ q 2 + 4 ω q 2 γ q 2 + 4 ω q 2 + 8 J 2 β 2 ,
| α | 2 { ( κ 2 ) 2 + [ Δ a 2 g 2 | α | 2 ( 2 ω q ϵ 1 + γ q ϵ 2 ) ϵ 1 2 + ϵ 2 2 ] 2 } = ε c 2 ,
| β | 2 ( ϵ 1 2 + ϵ 2 2 ) 2 = g 2 | α | 4 [ ( 2 ω q ϵ 1 + γ q ϵ 2 ) 2 + ( γ q ϵ 1 2 ω q ϵ 2 ) 2 ] ,
δ a ˙ = ( κ / 2 + i Δ a ) δ a + i G ( δ b + δ b ) + ε p e i Ω t i ϕ p c + κ a i n ( t ) ,
δ b ˙ = ( γ m / 2 + i ω m ) δ b + i ( G δ a + G δ a ) i J δ σ + ε m e i Ω t i ϕ m + γ m b i n ( t ) ,
δ σ ˙ = ( γ q / 2 + i ω q ) δ σ + i J ( β δ σ z + W 0 δ b ) + γ q c , i n ( t ) ,
δ σ z ˙ = γ q δ σ z 2 i J ( β δ σ + + L 0 δ b β δ σ L 0 δ b ) + γ q c z , i n ( t ) ,
δ a ˙ = Γ a δ a + i G δ b + ε p e i ϕ p c + κ a i n ( t ) ,
δ b ˙ = Γ m δ b + i G δ a i J δ σ + ε m e i ϕ m + γ m b i n ( t ) ,
δ σ ˙ = Γ δ σ + i J ( β δ σ z + W 0 δ b ) + γ q c , i n ( t ) ,
δ σ z ˙ = Γ q δ σ z 2 i J ( L 0 δ b β δ σ ) + γ q c z , i n ( t ) ,
δ a = ( Γ m Θ + 2 i β L 0 J 3 W 0 Γ q J 2 ) ε p e i ϕ p c + i G Θ ε m e i ϕ m Γ a ( Γ m Θ + 2 i β L 0 J 3 W 0 Γ q J 2 ) + | G | 2 Θ ,
t p = κ e δ a ε p e i ϕ p c ε p e i ϕ p c = t 1 + t 2
t 1 = ( Γ m Θ + 2 i β L 0 J 3 W 0 Γ q J 2 ) κ e Γ a ( Γ m Θ + 2 i β L 0 J 3 W 0 Γ q J 2 ) + | G | 2 Θ 1 ,
t 2 = i G Θ κ e r e i ϕ Γ a ( Γ m Θ + 2 i β L 0 J 3 W 0 Γ q J 2 ) + | G | 2 Θ ,
r | T P = | ( κ e Γ a ) ( Γ m Θ + 2 i β L 0 J 3 W 0 Γ q J 2 ) | G | 2 Θ G Θ κ e | ,
τ g = d ϕ t ( ω p ) d ω p = d { a r g [ t p ( ω p ) ] } d ω p .

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