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Transparency and tunable slow-fast light in a hybrid cavity optomechanical system

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Abstract

We theoretically investigate the optomechanically induced transparency (OMIT) phenomenon in a hybrid optomechanical system composing of an optomechanical cavity and a traditional one. A Kerr medium is inserted in the optomechanical cavity and the other traps the atomic ensemble. We demonstrate the appearance of electromagnetically and optomechanically induced transparency when there is only Kerr medium or atoms in the system. We give an explicit explanation for the mechanism of the transparency. Moreover, we set up new scheme for the measurement of Kerr coefficient and the single atom-photon coupling strength. It is shown that Kerr nonlinearity can inhibit the normal mode splitting (NMS) when the tunnel strength is strong coupling. Furthermore, in the output field, slow light and fast light are converted to realize the tunable switch from slow light to fast light. This study has some important guiding significance in the fields of the high precision measurement and quantum information processing.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

A typical optomechanical system comprises a Fabry-Pérot cavity [1] and mechanical oscillator, which can be either a suspended mirror [2] or a membrane [3]. An optomechanical system (OMS) describes the interaction between the optical field in the cavity and the movable oscillator by the radiation pressure of the system [46]. Optomechanical systems have potential applications in many fields, such as cooling of mechanical resonators [7,8], the entanglement between the macroscopic oscillator and the cavity field [912], mechanical memory [13], the researches of quantum entanglement [1417], Kerr nonlinear [18], coherent manipulation of light [19,20], self-induced oscillations [21], OMIT [22,23] , and electromagnetically induced transparency (EIT) [24]. The system also provides a new way for practical application, such as gravitational wave detection [25,26] and quantum limited displacement sensing [27], and so forth.

It is well-known that EIT is a quantum interference effect occurring in three-level atoms. When the auxiliary strong control field exists, the resonance absorption peak of the three-level atomic ensemble is completely suppressed and a transparent window appears. This phenomenon is caused by the quantum interference between the two different pathways of excitation in atomic system [2830], which has been extensively studied both theoretically [3133] and experimentally [28,34]. EIT has important application value in various applications such as quantum memory [3537], slow light [38], optical switch [39], vibrational cooling [40] and so on. Moreover, in an optomechanical system, the EIT analogs due to optomechanical interaction are called OMIT [41] and optomechanically induced absorption (OMIA) [42], which was the first prediction in theory [43,44] and then experimentally demonstrated in [43,45,46]. The physical mechanism of OMIT is the destructive interference between the anti-Stokes scattering field and the probe field. Up to now, the multiple-OMIT have been studied theoretically in multiple-resonators optomechanical system [47], atomic-media assisted optomechanical system [48,49], triple optomechanical system [50]. On the other hand, Safavi-Naeini et al. [45] have successfully observed an optically tunable delay of $50\textrm{ns}$ with near-unity optical transparency and superluminal light with a $1.4\mu s$ signal advance in an optomechanical crystal device. Boyd and Gauthier [51] have realized a high degree of control over the velocity at which light pulses pass through material media. They also described some applications (e.g., telecommunication, microwave photonics, interferometry) of slow light. Magnus Albert et al. [52] demonstrated for the first time EIT as well as all-optical EIT-based light switching using ion Coulomb crystals situated in an optical cavity.

The effect of nonlinear media on the dynamics of optomechanical systems was extensively studied [5357]. Nonlinear phenomena at single-photon energies were observed in a design of cavity optomechanics in the microwave frequency regime involving a Josephson junction qubit [54]. Furthermore, it was showed that the displacement spectrum of the micromirror could serve as a tool to detect the photon blocking effect [56]. The authors of [57] considered an optomechanical system containing a Kerr-down-conversion nonlinearity. They demonstrated that in the presence of Kerr-down-conversion nonlinearity, one can effectively control the width and the spectral positions of the OMIT. The effects of Kerr media and optical parametric amplifier (OPA) on the stability, cooling and entanglement of optomechanical systems were studied in [58,59]. They found that the non-linear medium can be used as a new means to coherently control the entanglement between the vibration modes of the micromirror in the optical mechanical cavity system.

As discussed above, it is meaningful to investigate the impacts of Kerr effect and atomic assistant cavity on OMIT in stable optomechanical systems. Two cavities are coupled to a common mechanical resonator. This setup has been realized in several experiments [6062]. Giant optical Kerr nonlinearities are obtained by placing a ${\chi ^{\left ( 3 \right )}}$ medium inside a cavity [53,56]. Using a three-dimensional circuit quantum electrodynamic architecture, Gerhard Kirchmair et al. [63,64] engineered an artificial Kerr medium that enters this regime. The collapse and revival of a coherent state can be observed. In addition, the nonlinearity of atom-optomechanical system has been discussed in [54]. The authors of [64] recently proposed an experimentally feasible method for enhancing the atom-field coupling as well as the ratio between this coupling and dissipation in atom-assisted optomechanical systems.

Motivated by these developments, we consider an optomechanical cavity, driven by a strong driving field and a weak probe field, which contains a Kerr medium. We shall add an atomic assistant cavity to couple the optomechanical cavity. Compared to the simple OMIT with a single transparency window [4345] and double OMIT [41,42], our system owns some favorable features: (i) The two output lights with different frequencies are controlled by a single driving light. (ii) Compared with [65], the broad frequency position depends linearly on the Kerr coefficient, and Kerr nonlinearity inhibits the NMS when tunneling strength is a strong coupling. (iii) We note that the separation between two transparent windows varies linearly with the single atom-photon coupling strength. This characteristic indicates that our research has practical application in precisely measuring the single atom-photon coupling strength. In this regard, we must emphasize that our proposal is fundamentally different from previous ones [66,67], where a two-level system is directly coupled to the mechanical resonator. In contrast, our study uses an auxiliary cavity to trap atomic ensemble, which provides more flexibility and operability for the transition between absorption and transparency.

The organization of this paper is as follows. In Sec. II, the theoretical model is introduced and the Hamiltonian of the optomechanical system is described. In Sec. III, the quantum Langevin equation is linearized and the dynamics of the quantum fluctuations around the steady-state mean values are obtained. Further, we analyze the Lyapunov exponents and derive the analytic expression for the output field by the steady-state solutions of the first-order sidebands. In Sec. IV, we discuss in detail the OMIT in the double passive Kerr optical mechanical system. The multi-transparency windows of the output field are realized and the influences of different system’s parameters on the output field are discussed in Sec. V. Also, the fast light and slow light of the output field are studied in Sec. VI. The summary is given in Sec. VII.

2. System model and Hamiltonian

As shown in Fig. 1, we consider a system comprises an optomechanical cavity $A$ and a traditional one $C$, which are directly coupled each other with a coupling strength $J$. The optomechanical cavity $A$ with a one-end oscillating mirror, contains a nonlinear Kerr medium with a Kerr coefficient of $\chi$. Here the cavity $A$ is passive and the loss rate and the resonance frequency of the cavity are, respectively, ${\kappa _a}$ and ${\omega _{c1}}$; ${\kappa _c}$ and ${\omega _{c2}}$ are the decay rate and the resonance frequency of the cavity $C$. It has two-end fixed mirrors and traps the atomic ensemble, $N$ identical two-level atoms (with transition frequency ${\omega _{at}}$ and decay rate ${\gamma _a}$). The interaction between cavity $C$ and atomic ensemble can be represented by Jaynes-Cummings model with a single atom-photon coupling constant of ${g_{at}}$.

 figure: Fig. 1.

Fig. 1. Schematic of the hybrid optomechanical system. An optomechanical cavity, driven by a strong driving field and a weak probe field, contains a Kerr medium, and an atomic assistant cavity couples to the optomechanical cavity.

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In addition, the oscillating mirror (with resonance frequency ${\omega _m}$, mass $m$ and damping rate ${\gamma _m}$) interacts with the cavity $A$ with optomechanical coupling strength ${g_{om}}$. The cavity $A$ is driven by a strong driving field with amplitude ${\Omega _d} = \sqrt {2{\kappa _a}{p_d}/\hbar {\omega _d}}$ with frequency ${\omega _d}$, and a weak probe field ${\varepsilon _p} = \sqrt {\left ( {2{\kappa _a}{p_p}} \right )/\hbar {\omega _p}}$ with frequency ${\omega _p}$. ${P_d}$ and ${P_p}$ are the powers of the strong driving field and probe field, respectively. The total Hamiltonian of the system in the rotating frame at the frequency ${\omega _d}$ can be written as the following form:

$$\begin{aligned} H&=\hbar {\Delta _a}{a^{\dagger}}a + \hbar {\Delta _c}{c^{\dagger}}c + \frac{1}{2}\hbar {\Delta _{at}}{\sigma _z} + \left( {\frac{{{p^2}}}{{2m}} + \frac{1}{2}m\omega _m^2{x^2}} \right)\\ &\quad{}- \hbar {g_{om}}{a^{\dagger}}ax + \hbar J\left( {{a^{\dagger}}c + a{c^{\dagger}}} \right) + \hbar {g_{at}}\left( {{c^{\dagger}}{\sigma _ - } + c{\sigma _ + }} \right) + \hbar \chi {a^{\dagger 2}}{a^2}\\ &\quad{}+ i\hbar \left( {{a^{\dagger}}{\Omega _d} + a\Omega _d^ * } \right) + i\hbar \left( {{a^{\dagger}}{\varepsilon _p}{e^{ - i\delta t}} - a\varepsilon _p^ * {e^{i\delta t}}} \right), \end{aligned}$$
where ${\Delta _a} = {\omega _a} - {\omega _d}$ and ${\Delta _c} = {\omega _c} - {\omega _d}$ are the detunings of the two cavity modes from the driving field; ${\Delta _{at}} = {\omega _{at}} - {\omega _d}$ represents the detuning between atomic ensemble and the driving field; $\delta = {\omega _p} - {\omega _d}$ is the detuning caused by the probe and the driving field. $p\left ( x \right )$ is the momentum (position) operator of the oscillating mirror with commutation relation $\left [ {x,p} \right ] = i\hbar$. Furthermore, $a$ and $c$ denote the annihilation operators of cavities $A$ and $C$, respectively. The two-level system can be characterized by the atomic ensemble operators ${\sigma _ + },\;{\sigma _ - },$ and ${\sigma _z}$ [68]. Here,
$${\sigma _ + } = \sum_{i = 1}^N {\sigma _ + ^{\left( i \right)}} ,\;{\sigma _ - } = \sum_{i = 1}^N {\sigma _ - ^{\left( i \right)}} ,\;{\sigma _z} = \sum_{i = 1}^N {\sigma _z^{\left( i \right)}} ,$$
where ($j = + , - ,z$ and $i = 1, \ldots ,N$) denotes the ${\sigma _j}$ operator of the $i\textrm{th}$ atom, and they satisfy the commutation relations $\left [ {\sigma _ + ^{\left ( i \right )},\sigma _ - ^{\left ( i \right )}} \right ] = \sigma _z^{\left ( i \right )}$ and $\left [ {\sigma _z^{\left ( i \right )},\sigma _ \pm ^{\left ( i \right )}} \right ] = \pm 2\sigma _ \pm ^{\left ( i \right )}$. The atomic ensemble is arranged in a thin layer whose size in the direction of the cavity axis is much smaller than the wavelength of the cavity field, thus all the atoms have the same coupling strength with the cavity field [69,70]. Experimental advances [71,72] have demonstrated many atoms in a cavity with each atom identically and strongly coupled to the cavity. The proposed scheme might be realizable. The first four terms denote the free energies of the two cavities, atomic ensemble and the mechanical oscillator. The next term depicts the optomechanical coupling between the cavity $A$ and the mechanical oscillator. The next two terms represent the coupling between two passive cavities with coupling strength $J$ and the interaction between atomic ensemble and cavity $A$, respectively. The Hamiltonian of the Kerr medium can be characterized by the term $\hbar \chi {a^{\dagger 2}}{a^2}$, and the last two terms depict the interaction of the cavity field with the driving field and that of the cavity field with the probe field.

3. Quantum dynamics and fluctuations

Based on the Hamiltonian in Eq. (1), the quantum dynamics of the system can be described by the following Heisenberg-Langevin equation:

$$\begin{aligned} \dot x &= \frac{p}{m}, \\ \dot p &= \hbar {g_{om}}{a^{\dagger}}a - m\omega _m^2x - {\gamma _m}p + \xi \left( t \right), \\ \dot a &= - \left[ {{\kappa _a} + i\left( {{\Delta _a} - {g_{om}}x + 2\chi {a^{\dagger}}a} \right)} \right]a - iJc + {\Omega _d} + {\varepsilon _p}{e^{ - i\delta t}} + \sqrt {2{\kappa _a}} {a_{in}}\left( t \right), \\ \dot c &= - \left( {{\kappa _c} + i{\Delta _c}} \right)c - i{g_{at}}{\sigma _ - } - iJa + \sqrt {2{\kappa _c}} {c_{in}}\left( t \right), \\ {{\dot \sigma }_ - } &= - \left( {{\gamma _{at}} + i{\Delta _{at}}} \right) + i{g_{at}}c{\sigma _z} + \sqrt {2{\gamma _{at}}} {c_{in}}\left( t \right), \end{aligned}$$
where ${\gamma _m},{\gamma _{at}}$, and ${\kappa _a}\left ( {{\kappa _c}} \right )$ represent the damping rates of the quantum mechanical oscillator, the atomic system, and cavity $A\left ( C \right )$, respectively. $\xi \left ( t \right )$ is quantum Brownian stochastic noise with zero mean value and its correlation function [73] is $\left \langle {\xi \left ( t \right )\xi \left ( {t'} \right )} \right \rangle = \frac {{{\gamma _m}}}{{{\omega _m}}}\int {\frac {{\textrm{d}\omega }}{{2\pi }}} {e^{ - i\omega \left ( {t - t'} \right )}}\omega \left ( {1 + \coth \frac {{\hbar \omega }}{{2{k_B}T}}} \right )$, where ${k_B}$ is the Boltzmann constant and $T$ is the temperature of the reservoir of the movable mirror. ${a_{in}}\left ( t \right )$, ${c_{in}}\left ( t \right )$ are the input vacuum noise operators in the cavity with zero mean value [74]. In particular, we consider that the number of the atoms is very large, i.e. $N \gg 1$. At the same time, the atomic ensemble is presumed to be in low excitation condition. Therefore, it can be assumed that $\left \langle {{\sigma _z}} \right \rangle = \left \langle {\sum _{i = 1}^N {\sigma _z^{\left ( i \right )}} } \right \rangle \simeq - N$ [69,75], $N$ is the number of two-level atom. We mainly study the mean response of the coupled system to the probe field, in which the quantum noise and thermal noise of the cavity field are neglected in Eq. (3). If the system is asymptotically stable and keeps a long evolution time, an OMIT phenomenon can be obtained. So, the stability of the coupling system needs to be studied in detail. Considering the perturbation of the probe field, we can regard each operator as the sum of its steady-state value and quantum fluctuation, i.e., $O = {O_s} + \delta O$. It is further divided into real part and imaginary part, i.e., $a = {\textrm{Re}} \left [ {{a_s}} \right ] + i{\textrm{Im}} \left [ {{a_s}} \right ] + \delta {\textrm{Re}} \left [ a \right ] + i\delta {\textrm{Im}} \left [ a \right ]$. Inserting the above hypothesis into Eq. (3), and all higher order terms and the probe term ${\varepsilon _p}{e^{ - i\delta t}}$ are ignored in Eq. (3). Then, the linearized dynamics of the system can be written in the following form: $\dot f\left ( t \right ) = Mf\left ( t \right )$, where ${f^T}\left ( t \right ) = \left ( {\delta x\left ( t \right ),\delta p\left ( t \right ),\delta {\textrm{Re}} \left [ a \right ]\left ( t \right ),\delta {\textrm{Im}} \left [ a \right ]\left ( t \right ),\delta {\textrm{Re}} \left [ c \right ]\left ( t \right ),\delta {\textrm{Im}} \left [ c \right ]\left ( t \right ),\delta {\textrm{Re}} \left [ {{\sigma _ - }} \right ]\left ( t \right ),\delta {\textrm{Im}} \left [ {{\sigma _ - }} \right ]\left ( t \right )} \right )$ is the column vector of the fluctuation operator. The Jacobian matrix $M$ can be used to describe the stability or randomness of the system.
$$M = \left( {\begin{array}{cccccccc} 0 & {\frac{1}{m}} & 0 & 0 & 0 & 0 & 0 & 0\\ { - m\omega _m^2} & { - {\gamma _m}} & {2\hbar {g_{om}}{\textrm{Re}} \left[ {{a_s}} \right]} & {2\hbar {g_{om}}{\textrm{Im}} \left[ {{a_s}} \right]} & 0 & 0 & 0 & 0\\ { - {g_{om}}{\textrm{Im}} \left[ a \right]} & 0 & {4\chi {{\left| {{a_s}} \right|}^2} - {\kappa _a}} & {{\Delta _a} - {g_{om}}{x_s}} & 0 & J & 0 & 0\\ {{g_{om}}{\textrm{Re}} \left[ a \right]} & 0 & {{g_{om}}{x_s} - {\Delta _a}} & { - \left( {4\chi {{\left| {{a_s}} \right|}^2} + {\kappa _a}} \right)} & { - J} & 0 & 0 & 0\\ 0 & 0 & 0 & J & { - {\kappa _c}} & {{\Delta _c}} & 0 & {{g_{at}}}\\ 0 & 0 & { - J} & 0 & { - {\Delta _c}} & { - {\kappa _c}} & { - {g_{at}}} & 0\\ 0 & 0 & 0 & 0 & 0 & {{g_{at}}N} & { - {\gamma _{at}}} & {{\Delta _{at}}}\\ 0 & 0 & 0 & 0 & { - {g_{at}}N} & 0 & { - {\Delta _{at}}} & { - {\gamma _{at}}} \end{array}} \right).$$
The real part of the eigenvalue of the above matrix $M$ is also called Lyapunov exponent of the nonlinear dynamic system [76]. Only when all Lyapunov exponents are negative, the mean trajectory of the system tends to a fixed point in phase space [77]. The maximum Lyapunov exponent of the system can be used as an index to distinguish the stable and unstable states of the system. By comparing the maximum Lyapunov exponent of the system with zero, we can judge whether the system is stable or not. This stability condition can also be reformulated by the Routh-Hurwitz criterion [78], but it is still very complicated. Here, we can verify the stability of the steady-state solution by numerical calculation. All the external parameters selected in this paper satisfy that the real part of the eigenvalue of the matrix must be negative. That is to say, all parameters satisfy the stability condition of the system. The steady-state solution of the Eq. (3) can be expanded into multiple Fourier components, and the higher order terms are neglected under the limitation of weak probe field. Then, each operator can have the following form [44,47,79]:
$$O=O_s+{}^{+}O\varepsilon _{p}\ e^{-i\delta t}+{}^{-}O\varepsilon _{p}^{*}\ e^{i\delta t},$$
$$\dot{O}=-i\delta{}^{+}O\varepsilon _{p}\ e^{-i\delta t}+i\delta{}^{-}O\varepsilon _{p}^{*}\ e^{i\delta t}.$$
Using the above transformation, we can get the following steady-state solutions and the analytic expression for the output field, i.e.,
$$ {x_s} = \frac{{\hbar {g_{om}}}}{{m\omega _m^2}}{\left| {{a_s}} \right|^2}, {a_s} = \frac{{{\Omega _d}}}{{{\kappa _a} + i\Delta + \frac{{{J^2}}}{{{\kappa _c} + i{\Delta _c} + \frac{{g_{at}^2N}}{{{\gamma _{at}} + i{\Delta _{at}}}}}}}}, $$
and
$$ {}^ + a = \frac{{Y + {\beta _1}\left( {\frac{1}{B} - \frac{{4\chi }}{{\hbar g_{om}^2}}} \right)}}{{\left[ {{\kappa _a} + i\left( {{\Delta _1} - \delta + Q} \right) - {\beta _1}\left( {\frac{1}{B} - \frac{{4\chi }}{{\hbar g_{om}^2}}} \right)} \right]\left[ {Y + {\beta _1}\left( {\frac{1}{B} - \frac{{4\chi }}{{\hbar g_{om}^2}}} \right)} \right] + {{\left[ {{\beta _1}\left( {\frac{{2\chi }}{{\hbar g_{om}^2}} - \frac{1}{B}} \right)} \right]}^2}}}, $$
where the undefined variables are $\Delta = {\Delta _a} + \left ( {2\chi - \frac {{\hbar g_{om}^2}}{{m\omega _m^2}}} \right ){\left | {{a_s}} \right |^2}$, ${\Delta _1} = {\Delta _a} - {g_{om}}{x_s}$, $B = m\left ( {\omega _m^2 - {\delta ^2} - i\delta {\gamma _m}} \right )$, $E = - \frac {{g_{at}^2N}}{{\left ( {{\Delta _{at}} - \delta } \right ) - i{\gamma _{at}}}}$, $F = - \frac {{g_{at}^2N}}{{\left ( {{\Delta _{at}} + \delta } \right ) + i{\gamma _{at}}}}$, ${\beta _1} = im{g_{om}}\omega _m^2{x_s},$ $R = - \frac {{{J^2}}}{{\delta + {\Delta _c} + F + i{\kappa _c}}},$ $Q = \frac {{{J^2}}}{{i{\kappa _c} - \left ( {{\Delta _c} - \delta + E} \right )}},$ $Y = {\kappa _a} - i\left ( {{\Delta _1} + \delta + R} \right )$.

Further, based on the input-output relation [80] of the cavity $A$,

$${\varepsilon _{out}}\left( t \right) + {\varepsilon _p}{e^{ - i\delta t}} + {\Omega _d} = 2{\kappa _a}a.$$
Then,
$${}^ + {\varepsilon _{out}} = 2{\kappa _a}{}^ + a - 1.$$
In order to study the optical response of hybrid cavity optomechanical system to weak detection field, the amplitude of the rescaled output field corresponding to the weak probe can be defined as ${\varepsilon _T} = {}^ + {\varepsilon _{out}} + 1 = 2{\kappa _a}{}^ + a$; we all know that absorption and dispersion of the probe field can be evaluated by real and imaginary parts of ${\varepsilon _T}$ individually.

4. The Kerr effect of OMIT for the optomechanical system

Now we study the absorption of the probe field versus the normalized detuning $\delta /{\omega _m}$ with different tunneling strengths in the double-passive optomechanical system, as shown in Fig. 2(a). Under the weak tunnel coupling condition $J\;<\;\left | {{\kappa _a} + {\kappa _c}} \right |/2$, we plot magenta, black, red, and blue for J = 0 $J = 0.5{\kappa _a},$ $J = 1.2{\kappa _a},$ and $J = 2.5{\kappa _a}$, respectively. We can see in Fig. 2(a) that the absorption doublet decreases with the increasing of tunnel strength $J$, thus forming a shallower transparency window. The absorption spectrum of probe field is similar to the OMIT characteristic of a single-mode OMS. At this time, the normal modes are not split. The effect of tunneling coupling is neglected and the absorption spectrum of the probe field is mainly affected by the optomechanical interaction in the cavity $A$. When tunnel coupling is adjusted to a strong coupling region with $J\;>\;\left | {{\kappa _a} + {\kappa _c}} \right |/2$, i.e., $J = 3{\kappa _a}$, we can find that these two absorption peaks are still decreasing with the increase of $J$. If tunnel coupling increases further, i.e., $J = 5{\kappa _a}$ and $J = 7{\kappa _a}$, they are represented by red line and blue line in Fig. 2(b), respectively. In addition, the two peaks of absorption appear respectively on the two sides of $\delta = {\omega _m}$, which are distorted outwards and more widely separated from each other. The coupling between the two cavities will affect the absorption spectrum. This distortion indicates that the strong coupling between the two cavities leads to the existence of NMS. This distorted spectrum is superimposed by the OMIT spectra resulted from the optomechanical interaction and the normal mode splitting caused by strong tunnel coupling.

 figure: Fig. 2.

Fig. 2. The real part of the output probe field as a function of the probe normalized frequency $\delta /{\omega _m}$ for different values of the tunnel coupling. (a) The weak tunnel coupling $\left ( {J\;<\;\left | {{\kappa _a} + {\kappa _c}} \right |/2} \right )$ is set as different values: $\chi=0$ (magenta, dashed curve); $J = 0.5{\kappa _a}$ (black, solid curve); $J = 1.2{\kappa _a}$ (red, dashed curve); $J = 2.5{\kappa _a}$ (blue, dash-dotted curve). (b) The values of the strong tunnel coupling $\left ( {J\;>\;\left | {{\kappa _a} + {\kappa _c}} \right |/2} \right )$ are given by: $J = 3{\kappa _a}$ (black, solid curve); $J = 5{\kappa _a}$ (red, dashed curve); $J = 7{\kappa _a}$ (blue, dash-dotted curve). The other parameters are ${\Omega _d} = \pi \times {10^3}\textrm{MHz}$, ${\omega _m} = 2\pi \times 100\textrm{MHz}$, $m = 10\textrm{ng}$, $N = {10^6}$, ${\kappa _a} = 2\pi \times 4\textrm{MHz}$, ${\kappa _c} = 5{\kappa _a}$, ${\gamma _m} = 2\pi \times 147\textrm{Hz}$, ${g_{om}} = 2\pi \times 2\textrm{MHz}$, $\chi = 0$, ${g_{at}} = 0$, ${\Delta _a} = {\Delta _c} = {\Delta _{at}} = {\omega _m}$, ${\gamma _{at}} = 2\pi , \times 0.01\textrm{MHz}$.

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When the tunneling coupling strength is zero, we analyze the effect of Kerr effect on probe absorption in optomechanical system. It is shown in Fig. 3(a) that it is an asymmetric OMIT spectrum with one broad absorption peak on the right side of $\delta = {\omega _m}$ and another absorption line at the resonance point $\delta = {\omega _m}$. The results show that with the increase of Kerr coefficient, the left absorption line becomes thinner and the right absorption peak is further away from the resonance point. The absorption peak on the right side moves obviously compared with the absorption line with the change of Kerr coefficient. It becomes apparent that the frequency position of the broad absorption peak is mainly determined by the Kerr interaction. In addition, these three absorption profiles have the same minimum or zero at $\delta = {\omega _m}$. Next we plot the frequency position of the right absorption peak as a function of the Kerr coefficient of $\chi$ as shown in Fig. 3(b). It is obvious that the frequency position depends linearly on the Kerr coefficient of $\chi$, this further proves that the asymmetry of absorption spectrum is caused by the Kerr effects. Therefore, the Kerr coefficient of $\chi$ can be easily detected by measuring the position of the right absorption peak. It provides an effective and accurate method for measuring Kerr coefficient.

 figure: Fig. 3.

Fig. 3. (a) The real part of the output probe field as a function of the probe detuning $\delta /{\omega _m}$ for different values of Kerr coefficient of $\chi$, $\chi=0$ (magenta, dashed curve); $\chi = 0.05{g_{om}}$ (black, solid curve); $\chi = 0.08{g_{om}}$ (red, dashed curve); $\chi = 0.10{g_{om}}$ (blue, dash-dotted curve). (b) The frequency position of the right absorption peak as a function of Kerr coefficient of $\chi$. The other parameter values are the same as in Fig. 2 except for $J = 0$.

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In the above analysis, we consider the effects of tunnel coupling or Kerr medium interaction on the probe absorption individually. Here, we show the optical properties of the system under the simultaneous action of tunnel coupling and Kerr medium interaction. As shown in Fig. 4(a), the absorption spectrum is similar to Fig. 3(a) except that the absorption line becomes narrower and the right absorption peak turns taller with the increase of the Kerr coefficient. When the tunneling strength is a strong coupling $\left ( {J = 7{\kappa _a}} \right )$, we can find that one of the peak increases while the other peak decreases with increasing Kerr nonlinearity in Fig. 4(b). The NMS slowly becomes less prominent. It indicates Kerr nonlinearity inhibits the NMS. Under the fixed Kerr coefficient of $\chi = 0.005{g_{om}}$, we display the change of the absorption spectrum of the probe with tunnel coupling. In the weak tunnel coupling condition $\left ( {J\;<\;\left | {{\kappa _a} + {\kappa _c}} \right |/2} \right )$, it is found from Fig. 5(a) that the positions of the broader peak around resonance and the absorption line caused by Kerr effect do not move, but their heights decrease with the increase of $J$. When the tunneling coupling strength $J$ approaches the strong region with $J\;>\;\left | {{\kappa _a} + {\kappa _c}} \right |/2$. According to Fig. 5(b), the absorption line on the right side becomes broader and the wide peak near the resonance point splits into two peaks, which is due to the normal mode splitting caused by the strong tunnel coupling between the two cavities in the presence of the Kerr medium. This can be proved by Eq. (3).

 figure: Fig. 4.

Fig. 4. (a) The real part of the output probe field as a function of the probe detuning $\delta /{\omega _m}$ with $J = 2.5{\kappa _a}$ for different values of Kerr coefficient of $\chi$, $\chi = 0.05{g_{om}}$ (black, solid curve); $\chi = 0.08{g_{om}}$ (red, dashed curve); $\chi = 0.10{g_{om}}$ (blue, dash-dotted curve). (b) The value of the strong tunnel coupling is given by $J = 7{\kappa _a}$. The other parameter values are the same as in Fig. 3(a).

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 figure: Fig. 5.

Fig. 5. The real part of the output probe field as a function of the normalized frequency $\delta /{\omega _m}$ with $\chi = 0.005{g_{om}}$ for different values of the tunnel coupling. (a) The weak tunnel coupling $\left ( {J\;<\;\left | {{\kappa _a} + {\kappa _c}} \right |/2} \right )$ is set as different values: $J = 1{\kappa _a}$ (black, solid curve); $J = 1.6{\kappa _a}$ (red, dashed curve); $J = 2{\kappa _a}$ (blue, dash-dotted curve). (b) The strong tunnel coupling $\left ( {J\;>\;\left | {{\kappa _a} + {\kappa _c}} \right |/2} \right )$, $J$ gives different values as: $J = 3{\kappa _a}$ (black, solid curve); $J = 3.6{\kappa _a}$ (red, dashed curve); $J = 5{\kappa _a}$ (blue, dash-dotted curve). The other parameter values are the same as in Fig. 2 except for ${\Delta _a} = {\Delta _c} = 0.8{\omega _m}$.

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5. The multi-transparency windows of the output field

Based on the hybrid cavity optomechanical system, the properties of the output field for the system is studied by using the available experimental parameters [3,81,82]: ${\omega _m} = 2\pi \times 100\textrm{MHz}$, $m = 10\textrm{ng}$, ${\gamma _m} = 2\pi \times 147\textrm{Hz}$, ${g_{om}} = 2\pi \times 2\textrm{MHz}$, ${\kappa _a} = 2\pi \times 4\textrm{MHz}$, ${\Delta _a} = {\Delta _c} = {\Delta _{at}} = {\omega _m}$, ${g_{at}} = 2\pi \times {10^4}\textrm{Hz}$, ${\gamma _{at}} = 2\pi \times 0.01\textrm{MHz}$. We place an atomic ensemble with $N = {10^6}$ in cavity $C$, the cavity $A$ is driven by a strong driving field with ${\Omega _d} = 2\pi \times {10^3}\textrm{MHz}$, and ${\kappa _c} = 5{\kappa _a}$. If we ignore Kerr medium ($\chi = 0$) in the cavity $A$, the term ${\varepsilon _T}$ becomes

$${\varepsilon _T} = \frac{{2{\kappa _a}}}{{\left[ {{\kappa _a} + i\left( {{\Delta _1} - \delta + Q} \right) - D} \right] + \frac{{{D^2}}}{{{\kappa _a} - i\left( {{\Delta _1} + \delta + R} \right) + D}}}},$$
where the defined variables are $D = \frac {{{\beta _1}}}{B}$.

Firstly, assuming that there is no coupling in the system and the system has only weak probe field, the Lorentz line pattern appears in the output field, as shown in the green line in Fig. 6(a). At this moment the system has transited from state $\left | {{n_{a1}},{n_{a2}},{n_m},{n_c}} \right \rangle$ to state $\left | {{n_{a1}} + 1,{n_{a2}},{n_m},{n_c}} \right \rangle$. Once a strong driving field is introduced, it means that there is optomechanical coupling in the system. It will transit from state $\left | {{n_{a1}} + 1,{n_{a2}},{n_m},{n_c}} \right \rangle$ to state $\left | {{n_{a1}} + 1,{n_{a2}},{n_m},{n_c}} \right \rangle$, as depicted in the cyan line in Fig. 6(a). Owing to our system works in the discrimination sideband regime $\left ( {{\kappa _a}\;<\;{\omega _a}} \right )$ and the red detuning driving regime $\left ( {{\Delta _a} = {\omega _m}} \right )$. Stokes scattering is strongly suppressed owing to its high off-resonant with the optical cavity. Therefore, we can assume that only the anti-Stokes field with frequency ${\omega _p} = {\omega _d} + {\omega _m}$ is established in the cavity. This field is formed by the degeneration of the probe field sent into the cavity and the two fields interfere destructively, leading to the occurrence of OMIT. If we further consider the tunnel coupling $J$ (see the magenta line in Fig. 6(a)) between two cavities. The path is from the state $\left | {{n_{a1}} + 1,{n_{a2}},{n_m},{n_c}} \right \rangle$ to $\left | {{n_{a1}},{n_{a2}} + 1,{n_m},{n_c}} \right \rangle$. The single transparency window split into two transparency windows owing to the normal mode splitting effect in the two cavities. It is found from the blue line in Fig. 6(a) that adding atomic ensemble to the system will cause destructive interference, if we properly select the transition frequency of atoms. The system has transited from state $\left | {{n_{a1}},{n_{a2}} + 1,{n_m},{n_c}} \right \rangle$ to state $\left | {{n_{a1}},{n_{a2}},{n_m},{n_c} + 1} \right \rangle$. The third transparent window can be generated, EIT and OMIT can exist in the system at the same time.

 figure: Fig. 6.

Fig. 6. The absorption as a function of $\delta /{\omega _m}$ for different opticalmechanical systems. Label a, without any interaction $\left ( {{\Omega _d} = 0,\;J = 0,\;{g_{at}} = 0} \right )$. Label b, existing the optomechanical coupling (${\Omega _d} = 2\pi \times {10^3}\textrm{MHz}$). Label c, adding the tunnelling coupling $\left ( {J = 3{\kappa _a},\;{\Delta _c} = 0.8{\omega _m}} \right )$ between the two cavities. Label d, including the single atom-photon coupling (${g_{at}} = 2\pi \times {10^4}\textrm{Hz}$, ${\Delta _a} = {\Delta _c} = {\Delta _{at}} = {\omega _m}$). The other parameter values are the same as in Fig. 2. (b) The energy-level diagram of the system. $\left | {{n_{a1}},{n_{a2}},m,{n_c}} \right \rangle$ represents the state of the entire system with ${n_{a1}},\;{n_{a2}}$ photons in the two optical modes, $m$ phonons in the mechanical mode, and ${n_c}$ atomic excitations.

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The optomechanical interaction between the mechanical oscillator and the cavity mode results in an extra interference channel. Accordingly, OMIT can be produced. In Fig. 7, the real part and imaginary part of the output probe field is given as a function of $\delta /\omega _{m}$ with different optomechanical coupling strengths ${g_{om}}$. It is found that in the case of optomechanical coupling, the transparent window generated by optomechanical interaction always appears at position of $\delta /\omega _{m}=1$. According to Figs. 7(b)–7(d) that, two absorption peaks are added near $\delta /\omega _{m}=1$. In addition, the distance between the two absorption peaks becomes wider with the increase of coupling strength ${g_{om}}$. Therefore, the resonant absorption can be controlled and the optical properties of the system can be significantly adjusted by changing the interaction between mechanical oscillator and optical system.

 figure: Fig. 7.

Fig. 7. The real part (blue line) and imaginary part (purple dashed line) of the output probe field as a function of the probe detuning $\delta /\omega _{m}$ for different optomechanical coupling strengths ${g_{om}}$. The other parameter values are the same as in Fig. 6.

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Under the action of optomechanical coupling, we studied the effect of tunneling strength $J$ on the absorption peak of the probe field. As shown in Fig. 8, it is found that the transparent window caused by optomechanical coupling always appears at $\delta /\omega _{m}=1$. The position of transparent windows on the left and right sides does not change with $J$, but the width of transparent windows obviously broadens with $J$ increasing. Furthermore, the depths of the dips in absorption also change slightly, become deeper. This shows that greater tunnel coupling strength $J$ has better transparency effect on the response of the system output probe field.

 figure: Fig. 8.

Fig. 8. The real part of the output probe field as a function of the probe detuning $\delta /\omega _{m}$ for different values of the tunnel coupling. $J = 1.5{\kappa _a}$ (blue, solid curve); $J = 2.5{\kappa _a}$ (magenta, dashed curve); $J = 3{\kappa _a}$ (green, dash-dotted curve). The other parameter values are the same as in Fig. 6.

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In Fig. 9(a), we plot the absorption ${\textrm{Re}} \left [ {{\varepsilon _T}} \right ]$ of the probe field as a function of the normalized detuning $\delta /{\omega _m}$ for different the single atom-photon coupling strengths. It clearly shows that the transparent window on the left (right) moves leftward (rightward), and the two transparency dips deeper, but the values of the peaks of absorption decrease, with the increase of ${g_{at}}$. Thus, the distance between the left and right transparency windows increases significantly with increasing the coupling strength ${g_{at}}$. Moreover, the real part of ${\varepsilon _T}$ describes the absorption of the probe field, which can be measured via, e.g., homodyne detection [80]. By using a similar heterodyne detection scheme as in Ref. [83], measuring the position of the optomechanically induced transparency windows has been realized experimentally in the Ref. [84]. This shows that it is feasible to measure the coupling strength by the distance between two transparency windows. The right and left two minimums (${{\textrm{Re}} \left [ {{\varepsilon _T}} \right ]_{\min }^{R,L}}$) of absorption spectra in Fig. 9(a) can be described by

$$\frac{{d{\textrm{Re}} \left[ {{\varepsilon _T}} \right]}}{{d\delta }}\left| {\begin{array}{c} {}\\ {\delta = {\delta _R}} \end{array}} \right. = 0,\;\frac{{d{\textrm{Re}} \left[ {{\varepsilon _T}} \right]}}{{d\delta }}\left| {\begin{array}{c} {}\\ {\delta = {\delta _L}} \end{array}} \right. = 0.$$

 figure: Fig. 9.

Fig. 9. (a) The real part of the output probe field as a function of the probe detuning $\delta /\omega _{m}$ for different values of the single atom-photon coupling constant of ${g_{at}}$. ${g_{at}} = 2.5\pi \times {10^4}\textrm{Hz}$ (blue, solid curve); ${g_{at}} = 3\pi \times {10^4}\textrm{Hz}$ (magenta, dashed curve); ${g_{at}} = 3.5\pi \times {10^4}\textrm{Hz}$ (green, dash-dotted curve), $J = 1.8{\kappa _a}$, other parameter values are the same as in Fig. 6. (b) The separation $d$ between the left and right two transparency windows in the absorption spectrum as a function of the coupling strength ${g_{at}}$ (units of ${g_0} = \pi \times {10^4}\textrm{Hz}$).

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Where ${\delta _R}$ $\left ( {{\delta _L}} \right )$ is the detuning corresponding to the right (left) transparency windows. Therefore, the separation of the transparency windows is $d = \left | {{\delta _R} - {\delta _L}} \right |$, In Fig. 9(b), we plot the separation $d$ between the left and right two transparency windows in the absorption spectrum as a function of the coupling strength ${g_{at}}$. Further, we find from Fig. 9(b) that the separation varies linearly with the coupling strength ${g_{at}}$. Therefore, the coupling strength can be obtained by simply measuring the distance between two transparent windows in the probe absorption spectrum.

In order to study the effect of cavity decay rate on the output spectrum, we plot the absorption ${\textrm{Re}} \left [ {{\varepsilon _T}} \right ]$ of the probe field as a function of the normalized detuning $\delta /{\omega _m}$ with different cavity decay rates, as shown in Fig. 10(a). With the increase of ${\kappa _c}$, the depths of the dips on the left and right windows become shallower, and the values of the peaks of absorption decrease, but the intermediate transparent window produced by optomechanical interaction is not affected by the cavity decay rate. The effect of atomic decay rates on the output field is similar to the decay rate of cavity $C$, as shown in Fig. 10(b). This indicates that the better transparency effect can be achieved by appropriately reducing the decay rates of cavity and atom.

 figure: Fig. 10.

Fig. 10. (a) The real part of the output probe field as a function of the probe detuning $\delta /\omega _{m}$ with different cavity decay rates. ${\kappa _c} = 0.08{\kappa _a}$ (blue, solid curve); ${\kappa _c} = 0.8{\kappa _a}$ (magenta, dashed curve); ${\kappa _c} = 1.6{\kappa _a}$ (green, dash-dotted curve). (b) The real part of the output probe field as a function of the probe detuning $\delta /\omega _{m}$ with different atomic decay rates. ${\gamma _{at}} = 2\pi \times {10^4}\textrm{Hz}$ (blue, solid curve); ${\gamma _{at}} = 2\pi \times {10^6}\textrm{Hz}$ (red, dashed curve); ${\gamma _{at}} = 8\pi \times {10^6}\textrm{Hz}$ (green, dash-dotted curve). Other parameter values are the same as in Fig. 6.

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We summarize the influence of each component on the optical response properties. The transparent window generated by optomechanical interaction always appears at position of $\delta / {\omega _m}=1$. Kerr medium destroys the symmetry of absorption spectrum and changes the position of the absorption spectrum. The normal mode splitting caused by strong tunnel coupling will produce distorted spectrum. Due to the existence of atomic ensemble, three transparent windows can be generated. The distance between the left and right transparency windows is affected by the single atom-photon coupling constant. The location, width, and absorption of each transparency window can be flexibly manipulated by varying the appropriate parameters.

6. Slow-fast light effect of the output field

Finally, the fast and slow light effects in the weak probe field are studied. It is well known that the probe field in a transparent window usually has a rapid phase dispersion, which leads to a sharp decrease in the group velocity, and the optical response of the system to the weak probe field can be described by group delay. The group delay of transmitted light is defined as:

$$\tau = \frac{{\partial \phi }}{{\partial {\omega _p}}} = {\textrm{Im}} \left[ {\frac{1}{{{\varepsilon _T}}}\frac{{\partial {\varepsilon _T}}}{{\partial {\omega _p}}}} \right] = {\textrm{Im}} \left[ {\frac{1}{{{\varepsilon _T}}}\frac{{\partial {\varepsilon _T}}}{{\partial \delta }}} \right].$$
Where $\tau$ represent the group delay of the output field. If the group delay is greater than zero, the system will show a slow light phenomenon, and if the group delay is less than zero, it will be a fast light phenomenon. From the previous analysis, since the output spectrum of the hybrid optomechanical system has three transparent windows, we can expect three group delay peaks.

Now we study the effect of tunnel coupling strength on the group delay. According to Fig. 11, Blue solid line, red dashed curve and dash-dotted curve correspond to the group delay $\tau$ in three cases: $J = 3{\kappa _a}$, $J = 5{\kappa _a}$, $J = 7{\kappa _a}$. A positive group delay of the output field at $\delta \approx 0.9{\omega _m}$ or $\delta \approx 1.1{\omega _m}$ is obtained, which corresponds to the slow light effect of the output probe field. And the maximum group delay increases significantly with the increase of $J$, which is due to the enhancement of optical transparency caused by the tunneling coupling between the cavity modes. The group delay $\tau$ of transmitted light near $\delta = {\omega _m}$ is negative value, therefore the output detection field contains fast light phenomenon, as shown the inset in Fig. 11. Moreover, with the increase of $J$, group delay will move away from the point $\delta = {\omega _m}$, tunneling coupling strength influences the displacement of the mechanical vibrator by changing the number of photons in the cavity. Therefore, we can enhance the slow light effect and change the frequency position and group delay of fast light phenomena by changing tunnel coupling $J$.

 figure: Fig. 11.

Fig. 11. Group delay $\tau$ as a function of the probe detuning $\delta /{\omega _m}$ for different values of the tunnel coupling strength $J$. The inset is the enlarged drawing of the group delay near $\delta /{\omega _m}$. Other parameters are the same as in Fig. 6.

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Finally, we investigate the influences of the driving strength ${\Omega _d}$ on the output field group delay. In Fig. 12, it becomes apparent that the group delay peaks increase and broaden with the increasing of driving strength. The group delay $\tau$ of transmitted light near $\delta = {\omega _m}$ changes rapidly from negative to positive, therefore the output detection field contains fast light and slow light phenomenon, as shown the inset in Fig. 12. When ${\Omega}_\textrm{d} = 1.3\pi \times 10^3\textrm{MHz}$, we can see that the minimum group delay is about $- 20\mu s$ (blue, soild curve), and ${\Omega}_\textrm{d} = 1.5\pi \times 10^3\textrm{MHz}$, the minimum group delay is about $- 70\mu s$ (red, dashed curve), while the minimum group delay is about $- 110\mu s$ for ${\Omega}_\textrm{d} = 2\pi \times 10^3\textrm{MHz}$ (black, dash-dotted curve). We find that the group delay dip becomes deeper and wider with the increase of the driving strength of the system, and moves father from the resonant point. The effective optomechanical coupling between the mechanical oscillator and the dissipative cavity can be significantly enhanced. OMIT can cause dramatic changes in the absorption and dispersion characteristics of the system, thereby enhancing the non-linear sensitivity, which can be used to control the group velocity of light. The tunability of group delay is caused by quantum interference between the probe field and the anti-Stokes field. This indicates that the group delay can be dynamically adjusted and the modulation of fast light and slow light can be achieved by controlling the driving strength [85].

 figure: Fig. 12.

Fig. 12. Group delay $\tau$ as a function of the probe detuning $\delta /{\omega _m}$ for different values of driving strength ${\Omega _d}$. The inset is the enlarged drawing of the group delay near $\delta /{\omega _m}$. The other parameter values are the same as in Fig. 6.

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7. Conclusions

In this study, we have proposed a hybrid optomechanical system. We showed that the existence of Kerr medium destroys the symmetry of absorption spectrum. Further, it was clear that the Kerr coefficient of $\chi$ can be easily confirmed by measuring the peak position of the right absorption peak. The effects of the coupling between two cavity modes on the light response characteristics of the output field and the fast and slow light were discussed in detail. We also found that Kerr nonlinearity can inhibit NMS when the tunnel strength is strong coupling. Moreover, we found that the separation of left and right transparent windows and the single atom-photon coupling strength change approximately linearly. The conversion between slow light and fast light occurs by changing the relevant parameters in the output field. These carefully analyzed optical response characteristics provide potential application prospects for more effective manipulation of light transmission. This scheme provides new possibilities for the measurement of Kerr coefficient and the single atom-photon coupling strength, the tunable switch from slow light to fast light and the coherent control of light pulses.

Funding

National Natural Science Foundation of China (61368002); Foundation for Distinguished Young Scientists of Jiangxi Province (20162BCB23009); Research Foundation of the Education Department of Jiangxi Province (GJJ13051); Open Project Program of CAS Key Laboratory of Quantum Information (KQI201704); Open Research Fund Program of the State Key Laboratory of Low-Dimensional Quantum Physics (KF201711); Graduate Innovation Special Fund of Jiangxi Province (YC2019-S102).

Acknowledgments

X. X. sincerely thanks for Xi-Yun Li for valuable discussions. The authors would like to thank the Quantum Laboratory of the School of Information Engineering of Nanchang University for the discussion of the physics mechanism and effective methods of this paper.

Disclosures

The authors declare no conflicts of interest.

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

55. G. Agarwal, K. Di, L. Wang, and Y. Zhu, “Perfect photon absorption in the nonlinear regime of cavity quantum electrodynamics,” Phys. Rev. A 93(6), 063805 (2016). [CrossRef]  

56. T. Kumar and A. B. Bhattacherjee, “Dynamics of a movable micromirror in a nonlinear optical cavity,” Phys. Rev. A 81(1), 013835 (2010). [CrossRef]  

57. S. Shahidani, M. Naderi, and M. Soltanolkotabi, “Control and manipulation of electromagnetically induced transparency in a nonlinear optomechanical system with two movable mirrors,” Phys. Rev. A 88(5), 053813 (2013). [CrossRef]  

58. S. Shahidani, M. Naderi, M. Soltanolkotabi, and S. Barzanjeh, “Steady-state entanglement, cooling, and tristability in a nonlinear optomechanical cavity,” J. Opt. Soc. Am. B 31(5), 1087–1095 (2014). [CrossRef]  

59. J. Li, B. Hou, Y. Zhao, and L. Wei, “Enhanced entanglement between two movable mirrors in an optomechanical system with nonlinear media,” EPL 110(6), 64004 (2015). [CrossRef]  

60. C. Dong, V. Fiore, M. C. Kuzyk, and H. Wang, “Optomechanical dark mode,” Science 338(6114), 1609–1613 (2012). [CrossRef]  

61. 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]  

62. R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, “Bidirectional and efficient conversion between microwave and optical light,” Nat. Phys. 10(4), 321–326 (2014). [CrossRef]  

63. G. Kirchmair, B. Vlastakis, Z. Leghtas, S. E. Nigg, H. Paik, E. Ginossar, M. Mirrahimi, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, “Observation of quantum state collapse and revival due to the single-photon kerr effect,” Nature 495(7440), 205–209 (2013). [CrossRef]  

64. W. Qin, A. Miranowicz, P.-B. Li, X.-Y. Lü, J.-Q. You, and F. Nori, “Exponentially enhanced light-matter interaction, cooperativities, and steady-state entanglement using parametric amplification,” Phys. Rev. Lett. 120(9), 093601 (2018). [CrossRef]  

65. X. Zhao, B. Hou, L. Liu, Y. Zhao, and M. Zhao, “Cross-Kerr effect in a parity-time symmetric optomechanical system,” Opt. Express 26(14), 18043–18054 (2018). [CrossRef]  

66. S. Huang, “Double electromagnetically induced transparency and narrowing of probe absorption in a ring cavity with nanomechanical mirrors,” J. Phys. B: At., Mol. Opt. Phys. 47(5), 055504 (2014). [CrossRef]  

67. 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]  

68. W. J. Nie, A. X. Chen, Y. H. Lan, Q. H. Liao, and S. Y. Zhu, “Enhancing steady-state entanglement via vacuum-induced emitter–mirror coupling in a hybrid optomechanical system,” J. Phys. B: At., Mol. Opt. Phys. 49(2), 025501 (2016). [CrossRef]  

69. C. P. Sun, Y. Li, and X. F. Liu, “Quasi-spin-wave quantum memories with a dynamical symmetry,” Phys. Rev. Lett. 91(14), 147903 (2003). [CrossRef]  

70. Y. Xiao, Y.-F. Yu, and Z.-M. Zhang, “Controllable optomechanically induced transparency and ponderomotive squeezing in an optomechanical system assisted by an atomic ensemble,” Opt. Express 22(15), 17979–17989 (2014). [CrossRef]  

71. Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom–field coupling for Bose–Einstein condensates in an optical cavity on a chip,” Nature 450(7167), 272–276 (2007). [CrossRef]  

72. F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, “Cavity QED with a Bose–Einstein condensate,” Nature 450(7167), 268–271 (2007). [CrossRef]  

73. V. Giovannetti and D. Vitali, “Phase-noise measurement in a cavity with a movable mirror undergoing quantum brownian motion,” Phys. Rev. A 63(2), 023812 (2001). [CrossRef]  

74. C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, “Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes,” Phys. Rev. A 77(3), 033804 (2008). [CrossRef]  

75. C. Genes, H. Ritsch, M. Drewsen, and A. Dantan, “Atom-membrane cooling and entanglement using cavity electromagnetically induced transparency,” Phys. Rev. A 84(5), 051801 (2011). [CrossRef]  

76. L. Lü, C. Li, S. Liu, Z. Wang, J. Tian, and J. Gu, “The signal synchronization transmission among uncertain discrete networks with different nodes,” Nonlinear Dyn. 81(1-2), 801–809 (2015). [CrossRef]  

77. C. Jiang, L. Song, and Y. Li, “Directional amplifier in an optomechanical system with optical gain,” Phys. Rev. A 97(5), 053812 (2018). [CrossRef]  

78. E. X. DeJesus and C. Kaufman, “Routh-Hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations,” Phys. Rev. A 35(12), 5288–5290 (1987). [CrossRef]  

79. W. Li, Y. Jiang, C. Li, and H. Song, “Parity-time-symmetry enhanced optomechanically-induced-transparency,” Sci. Rep. 6(1), 31095 (2016). [CrossRef]  

80. D. F. Walls and G. J. Milburn, Quantum optics (Springer Science & Business Media, 2007).

81. O. Astafiev, K. Inomata, A. Niskanen, T. Yamamoto, Y. A. Pashkin, Y. Nakamura, and J. S. Tsai, “Single artificial-atom lasing,” Nature 449(7162), 588–590 (2007). [CrossRef]  

82. R. Schoelkopf and S. Girvin, “Wiring up quantum systems,” Nature 451(7179), 664–669 (2008). [CrossRef]  

83. A. Butsch, J. Koehler, R. Noskov, and P. S. J. Russell, “Cw-pumped single-pass frequency comb generation by resonant optomechanical nonlinearity in dual-nanoweb fiber,” Optica 1(3), 158–164 (2014). [CrossRef]  

84. F. Buters, F. Luna, M. Weaver, H. Eerkens, K. Heeck, S. De Man, and D. Bouwmeester, “Straightforward method to measure optomechanically induced transparency,” Opt. Express 25(11), 12935–12943 (2017). [CrossRef]  

85. H. Xiong and Y. Wu, “Fundamentals and applications of optomechanically induced transparency,” Appl. Phys. Rev. 5(3), 031305 (2018). [CrossRef]  

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2019 (2)

Y.-H. Chen, W. Qin, and F. Nori, “Fast and high-fidelity generation of steady-state entanglement using pulse modulation and parametric amplification,” Phys. Rev. A 100(1), 012339 (2019).
[Crossref]

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2018 (9)

X. Y. Li, W. J. Nie, A. X. Chen, and Y. H. Lan, “Effect of the mechanical oscillator on the optical-response properties of an optical trimer system,” Phys. Rev. A 98(5), 053848 (2018).
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X. Zhang, Y. Zhou, Y. Guo, and X. Yi, “Double optomechanically induced transparency and absorption in parity-time-symmetric optomechanical systems,” Phys. Rev. A 98(3), 033832 (2018).
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H. Xiong and Y. Wu, “Fundamentals and applications of optomechanically induced transparency,” Appl. Phys. Rev. 5(3), 031305 (2018).
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2017 (4)

2016 (5)

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2015 (8)

L. Lü, C. Li, S. Liu, Z. Wang, J. Tian, and J. Gu, “The signal synchronization transmission among uncertain discrete networks with different nodes,” Nonlinear Dyn. 81(1-2), 801–809 (2015).
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J. Li, B. Hou, Y. Zhao, and L. Wei, “Enhanced entanglement between two movable mirrors in an optomechanical system with nonlinear media,” EPL 110(6), 64004 (2015).
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2014 (10)

S. Huang, “Double electromagnetically induced transparency and narrowing of probe absorption in a ring cavity with nanomechanical mirrors,” J. Phys. B: At., Mol. Opt. Phys. 47(5), 055504 (2014).
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Y. J. Guo, K. Li, W. J. Nie, and Y. Li, “Electromagnetically-induced-transparency-like ground-state cooling in a double-cavity optomechanical system,” Phys. Rev. A 90(5), 053841 (2014).
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2013 (6)

P. Meystre, “A short walk through quantum optomechanics,” Ann. Phys. 525(3), 215–233 (2013).
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2012 (3)

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2011 (6)

M. Bagheri, M. Poot, M. Li, W. P. Pernice, and H. X. Tang, “Dynamic manipulation of nanomechanical resonators in the high-amplitude regime and non-volatile mechanical memory operation,” Nat. Nanotechnol. 6(11), 726–732 (2011).
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2010 (5)

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2009 (3)

A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photonics 3(12), 706–714 (2009).
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2008 (4)

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2007 (5)

O. Astafiev, K. Inomata, A. Niskanen, T. Yamamoto, Y. A. Pashkin, Y. Nakamura, and J. S. Tsai, “Single artificial-atom lasing,” Nature 449(7162), 588–590 (2007).
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2006 (1)

J. Q. Shen and S. He, “Dimension-sensitive optical responses of electromagnetically induced transparency vapor in a waveguide,” Phys. Rev. A 74(6), 063831 (2006).
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Y. Wu and X. Yang, “Electromagnetically induced transparency in v-, λ-, and cascade-type schemes beyond steady-state analysis,” Phys. Rev. A 71(5), 053806 (2005).
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D. Rugar, R. Budakian, H. Mamin, and B. Chui, “Single spin detection by magnetic resonance force microscopy,” Nature 430(6997), 329–332 (2004).
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C. P. Sun, Y. Li, and X. F. Liu, “Quasi-spin-wave quantum memories with a dynamical symmetry,” Phys. Rev. Lett. 91(14), 147903 (2003).
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Figures (12)

Fig. 1.
Fig. 1. Schematic of the hybrid optomechanical system. An optomechanical cavity, driven by a strong driving field and a weak probe field, contains a Kerr medium, and an atomic assistant cavity couples to the optomechanical cavity.
Fig. 2.
Fig. 2. The real part of the output probe field as a function of the probe normalized frequency $\delta /{\omega _m}$ for different values of the tunnel coupling. (a) The weak tunnel coupling $\left ( {J\;<\;\left | {{\kappa _a} + {\kappa _c}} \right |/2} \right )$ is set as different values: $\chi=0$ (magenta, dashed curve); $J = 0.5{\kappa _a}$ (black, solid curve); $J = 1.2{\kappa _a}$ (red, dashed curve); $J = 2.5{\kappa _a}$ (blue, dash-dotted curve). (b) The values of the strong tunnel coupling $\left ( {J\;>\;\left | {{\kappa _a} + {\kappa _c}} \right |/2} \right )$ are given by: $J = 3{\kappa _a}$ (black, solid curve); $J = 5{\kappa _a}$ (red, dashed curve); $J = 7{\kappa _a}$ (blue, dash-dotted curve). The other parameters are ${\Omega _d} = \pi \times {10^3}\textrm{MHz}$ , ${\omega _m} = 2\pi \times 100\textrm{MHz}$ , $m = 10\textrm{ng}$ , $N = {10^6}$ , ${\kappa _a} = 2\pi \times 4\textrm{MHz}$ , ${\kappa _c} = 5{\kappa _a}$ , ${\gamma _m} = 2\pi \times 147\textrm{Hz}$ , ${g_{om}} = 2\pi \times 2\textrm{MHz}$ , $\chi = 0$ , ${g_{at}} = 0$ , ${\Delta _a} = {\Delta _c} = {\Delta _{at}} = {\omega _m}$ , ${\gamma _{at}} = 2\pi , \times 0.01\textrm{MHz}$ .
Fig. 3.
Fig. 3. (a) The real part of the output probe field as a function of the probe detuning $\delta /{\omega _m}$ for different values of Kerr coefficient of $\chi$ , $\chi=0$ (magenta, dashed curve); $\chi = 0.05{g_{om}}$ (black, solid curve); $\chi = 0.08{g_{om}}$ (red, dashed curve); $\chi = 0.10{g_{om}}$ (blue, dash-dotted curve). (b) The frequency position of the right absorption peak as a function of Kerr coefficient of $\chi$ . The other parameter values are the same as in Fig. 2 except for $J = 0$ .
Fig. 4.
Fig. 4. (a) The real part of the output probe field as a function of the probe detuning $\delta /{\omega _m}$ with $J = 2.5{\kappa _a}$ for different values of Kerr coefficient of $\chi$ , $\chi = 0.05{g_{om}}$ (black, solid curve); $\chi = 0.08{g_{om}}$ (red, dashed curve); $\chi = 0.10{g_{om}}$ (blue, dash-dotted curve). (b) The value of the strong tunnel coupling is given by $J = 7{\kappa _a}$ . The other parameter values are the same as in Fig. 3(a).
Fig. 5.
Fig. 5. The real part of the output probe field as a function of the normalized frequency $\delta /{\omega _m}$ with $\chi = 0.005{g_{om}}$ for different values of the tunnel coupling. (a) The weak tunnel coupling $\left ( {J\;<\;\left | {{\kappa _a} + {\kappa _c}} \right |/2} \right )$ is set as different values: $J = 1{\kappa _a}$ (black, solid curve); $J = 1.6{\kappa _a}$ (red, dashed curve); $J = 2{\kappa _a}$ (blue, dash-dotted curve). (b) The strong tunnel coupling $\left ( {J\;>\;\left | {{\kappa _a} + {\kappa _c}} \right |/2} \right )$ , $J$ gives different values as: $J = 3{\kappa _a}$ (black, solid curve); $J = 3.6{\kappa _a}$ (red, dashed curve); $J = 5{\kappa _a}$ (blue, dash-dotted curve). The other parameter values are the same as in Fig. 2 except for ${\Delta _a} = {\Delta _c} = 0.8{\omega _m}$ .
Fig. 6.
Fig. 6. The absorption as a function of $\delta /{\omega _m}$ for different opticalmechanical systems. Label a, without any interaction $\left ( {{\Omega _d} = 0,\;J = 0,\;{g_{at}} = 0} \right )$ . Label b, existing the optomechanical coupling ( ${\Omega _d} = 2\pi \times {10^3}\textrm{MHz}$ ). Label c, adding the tunnelling coupling $\left ( {J = 3{\kappa _a},\;{\Delta _c} = 0.8{\omega _m}} \right )$ between the two cavities. Label d, including the single atom-photon coupling ( ${g_{at}} = 2\pi \times {10^4}\textrm{Hz}$ , ${\Delta _a} = {\Delta _c} = {\Delta _{at}} = {\omega _m}$ ). The other parameter values are the same as in Fig. 2. (b) The energy-level diagram of the system. $\left | {{n_{a1}},{n_{a2}},m,{n_c}} \right \rangle$ represents the state of the entire system with ${n_{a1}},\;{n_{a2}}$ photons in the two optical modes, $m$ phonons in the mechanical mode, and ${n_c}$ atomic excitations.
Fig. 7.
Fig. 7. The real part (blue line) and imaginary part (purple dashed line) of the output probe field as a function of the probe detuning $\delta /\omega _{m}$ for different optomechanical coupling strengths ${g_{om}}$ . The other parameter values are the same as in Fig. 6.
Fig. 8.
Fig. 8. The real part of the output probe field as a function of the probe detuning $\delta /\omega _{m}$ for different values of the tunnel coupling. $J = 1.5{\kappa _a}$ (blue, solid curve); $J = 2.5{\kappa _a}$ (magenta, dashed curve); $J = 3{\kappa _a}$ (green, dash-dotted curve). The other parameter values are the same as in Fig. 6.
Fig. 9.
Fig. 9. (a) The real part of the output probe field as a function of the probe detuning $\delta /\omega _{m}$ for different values of the single atom-photon coupling constant of ${g_{at}}$ . ${g_{at}} = 2.5\pi \times {10^4}\textrm{Hz}$ (blue, solid curve); ${g_{at}} = 3\pi \times {10^4}\textrm{Hz}$ (magenta, dashed curve); ${g_{at}} = 3.5\pi \times {10^4}\textrm{Hz}$ (green, dash-dotted curve), $J = 1.8{\kappa _a}$ , other parameter values are the same as in Fig. 6. (b) The separation $d$ between the left and right two transparency windows in the absorption spectrum as a function of the coupling strength ${g_{at}}$ (units of ${g_0} = \pi \times {10^4}\textrm{Hz}$ ).
Fig. 10.
Fig. 10. (a) The real part of the output probe field as a function of the probe detuning $\delta /\omega _{m}$ with different cavity decay rates. ${\kappa _c} = 0.08{\kappa _a}$ (blue, solid curve); ${\kappa _c} = 0.8{\kappa _a}$ (magenta, dashed curve); ${\kappa _c} = 1.6{\kappa _a}$ (green, dash-dotted curve). (b) The real part of the output probe field as a function of the probe detuning $\delta /\omega _{m}$ with different atomic decay rates. ${\gamma _{at}} = 2\pi \times {10^4}\textrm{Hz}$ (blue, solid curve); ${\gamma _{at}} = 2\pi \times {10^6}\textrm{Hz}$ (red, dashed curve); ${\gamma _{at}} = 8\pi \times {10^6}\textrm{Hz}$ (green, dash-dotted curve). Other parameter values are the same as in Fig. 6.
Fig. 11.
Fig. 11. Group delay $\tau$ as a function of the probe detuning $\delta /{\omega _m}$ for different values of the tunnel coupling strength $J$ . The inset is the enlarged drawing of the group delay near $\delta /{\omega _m}$ . Other parameters are the same as in Fig. 6.
Fig. 12.
Fig. 12. Group delay $\tau$ as a function of the probe detuning $\delta /{\omega _m}$ for different values of driving strength ${\Omega _d}$ . The inset is the enlarged drawing of the group delay near $\delta /{\omega _m}$ . The other parameter values are the same as in Fig. 6.

Equations (13)

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H = Δ a a a + Δ c c c + 1 2 Δ a t σ z + ( p 2 2 m + 1 2 m ω m 2 x 2 ) g o m a a x + J ( a c + a c ) + g a t ( c σ + c σ + ) + χ a 2 a 2 + i ( a Ω d + a Ω d ) + i ( a ε p e i δ t a ε p e i δ t ) ,
σ + = i = 1 N σ + ( i ) , σ = i = 1 N σ ( i ) , σ z = i = 1 N σ z ( i ) ,
x ˙ = p m , p ˙ = g o m a a m ω m 2 x γ m p + ξ ( t ) , a ˙ = [ κ a + i ( Δ a g o m x + 2 χ a a ) ] a i J c + Ω d + ε p e i δ t + 2 κ a a i n ( t ) , c ˙ = ( κ c + i Δ c ) c i g a t σ i J a + 2 κ c c i n ( t ) , σ ˙ = ( γ a t + i Δ a t ) + i g a t c σ z + 2 γ a t c i n ( t ) ,
M = ( 0 1 m 0 0 0 0 0 0 m ω m 2 γ m 2 g o m Re [ a s ] 2 g o m Im [ a s ] 0 0 0 0 g o m Im [ a ] 0 4 χ | a s | 2 κ a Δ a g o m x s 0 J 0 0 g o m Re [ a ] 0 g o m x s Δ a ( 4 χ | a s | 2 + κ a ) J 0 0 0 0 0 0 J κ c Δ c 0 g a t 0 0 J 0 Δ c κ c g a t 0 0 0 0 0 0 g a t N γ a t Δ a t 0 0 0 0 g a t N 0 Δ a t γ a t ) .
O = O s + + O ε p   e i δ t + O ε p   e i δ t ,
O ˙ = i δ + O ε p   e i δ t + i δ O ε p   e i δ t .
x s = g o m m ω m 2 | a s | 2 , a s = Ω d κ a + i Δ + J 2 κ c + i Δ c + g a t 2 N γ a t + i Δ a t ,
+ a = Y + β 1 ( 1 B 4 χ g o m 2 ) [ κ a + i ( Δ 1 δ + Q ) β 1 ( 1 B 4 χ g o m 2 ) ] [ Y + β 1 ( 1 B 4 χ g o m 2 ) ] + [ β 1 ( 2 χ g o m 2 1 B ) ] 2 ,
ε o u t ( t ) + ε p e i δ t + Ω d = 2 κ a a .
+ ε o u t = 2 κ a + a 1.
ε T = 2 κ a [ κ a + i ( Δ 1 δ + Q ) D ] + D 2 κ a i ( Δ 1 + δ + R ) + D ,
d Re [ ε T ] d δ | δ = δ R = 0 , d Re [ ε T ] d δ | δ = δ L = 0.
τ = ϕ ω p = Im [ 1 ε T ε T ω p ] = Im [ 1 ε T ε T δ ] .

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