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Fluctuation-enhanced Kerr nonlinearity in an atom-assisted optomechanical system with atom-cavity interactions

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Abstract

We examine the effect of cavity field fluctuations on Kerr nonlinearity in an atom-assisted optomechanical system. It is found that a new self-Kerr (SK) nonlinearity term, which can greatly surpass that of a classical Λ type atomic system when the hybrid system has numerous atoms, is generated based on cavity field fluctuations by atom-cavity interactions. A strong photon–phonon cross-Kerr (CK) nonlinearity is also produced based on cavity field fluctuations. These nonlinearity features can be modified by atom-cavity and optomechanical interactions. This work may provide a new method to enhance the SK nonlinearity and generate the photon–phonon CK nonlinearity.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Kerr nonlinearity can cause nonlinear interactions of the photons; therefore, it plays a crucial role in quantum information processing and quantum computation [15]. For these applications, large third-order nonlinear susceptibilities with low light power and small absorption are desirable. It has been theoretically and experimentally demonstrated that large nonlinearity susceptibilities at low light levels can be achieved by using electromagnetically induced transparency (EIT) [610]. In the EIT schemes, the basic population transfer configuration of the atoms is achieved by using a model in which a strong (as a control field) and weak field (as a probe field) induce two transitions from an excited state to two lower energy states [11,12]. Combining EIT and cavity quantum electrodynamics (CQED) has great progress [1315] in recent years because CQED systems can provide an ideal environment for coherent manipulation of the photonic and atomic states. It also enhances the interaction between atoms and photons. These researches consider the cavity field as the probe field and set the intensity of the control field as a constant. However, when we use a cavity field as the control field, the atom-cavity interaction may cause fluctuations of the cavity field, and the control field does not remain constant. Consequently, these cavity field fluctuations may affect the Kerr nonlinearity of the EIT medium.

Quantum optomechanics is a rapidly advancing field that explores the coherent interaction of optical and mechanical degrees of freedom [16]. Quantum optomechanics provides a platform to observe the quantum mechanical behaviors of macroscopic objects and to achieve the quantum manipulation of mechanical oscillator, such as cooling of mechanical oscillators [1720], nonclassical mechanical states [2123], and phonon blockade [24,25]. On the other hand, the photon–phonon interaction induces some novel quantum optical effects, such as optomechanically induced transparency [2628], photon blockade [29,30], and optical nonreciprocity [3133]. Owing to the optomechanical interaction, the optical and mechanical fluctuations affect each other and play an important role in many optomechanical effects.

The combination of optomechanical systems with atoms gives rise to various hybrid optomechanical systems which provide more degrees of freedom for coherent manipulation. Such hybrid systems have been rapidly developed in the past few years [3440]. In an atom-assisted optomechanical system, the mechanical oscillator influences the interaction between the cavity fields and atoms, which in turn affect the EIT. Chang et al. theoretically proved the multistability of EIT in an atom-assisted cavity optomechanical system [41]. Recently, Hao et al. provided theoretical demonstration of electromagnetically and optomechanically induced transparency (EOMIT) and amplification (EOMIA) in a hybrid atom-assisted optomechanical system by considering the atom-cavity interactions [42].

Motivated by these works, we use a cavity field as the control field and investigate the influence of the fluctuations, which are caused by the atom-cavity interaction or the optomechanical interaction, on the Kerr nonlinearity via EIT in a hybrid atom-assisted optomechanical system. In this system, the self-Kerr (SK) nonlinearity contains two terms: The first SK (FSK) nonlinearity term, which is identical to that obtained in the classical three-level $\Lambda $ type atom EIT scheme [8,43]; the second self-Kerr (SSK) nonlinearity term, which depends on the cavity field fluctuation. The cavity field fluctuation is induced by the atom-cavity interaction and is proportional to the number of atoms. Thus, in the case of many atoms, the SSK nonlinearity term can be far greater than the FSK nonlinearity term, which will greatly enhance the total SK nonlinearity. We also study the effect of the control field on the SK nonlinearity without the optomechanical interaction. Furthermore, the photon and phonon states are coupled, forming the dressed states via the optomechanical interaction, which affects the cavity field fluctuation, thereby influencing the SK nonlinearity. Based on this phenomenon, we introduce a mechanical drive to drive the mechanical oscillator. The cross-Kerr (CK) nonlinearity is produced between the probe field and the mechanical mode. Then we examine the effect of the control field and the bare optomechanical coupling strength on the CK nonlinearity.

This paper is organized as follows: In Sec. 2, we introduce the hybrid optomechanical system model and derive its stable solutions. In Sec. 3, we discuss the SK nonlinearity in the atom–assisted optomechanical system. In Sec. 4, we enumerate the generation of CK nonlinearity between the probe field and mechanical mode based on the system. Finally, we summarize our work and present an outlook in Sec. 5.

2. Theoretical model

We consider an ensemble of N identical three-level atoms, which are confined inside an optical cavity with an oscillating mirror on one end, as shown in Fig. 1. Here the states ${|1 \rangle _i},$ ${|2 \rangle _i},$ and ${|3 \rangle _i}$ denote the ground, metastable, and excited states of the ith atom, respectively. The cavity field, which is driven by a pumping field with the frequency ${\omega _l}$ and driving strength ${\Omega _l}$, couples the transition $|2 \rangle \leftrightarrow |3 \rangle $ as a strong control field. A weak probe field couples the transition $|1 \rangle \leftrightarrow |3 \rangle $ with the frequency ${\omega _p}$ and Rabi frequency ${\Omega _p}.$ The total Hamiltonian for the hybrid atom-assisted optomechanical system can be written as

$$\hat{H} = {\hat{H}_0} + {\hat{H}_I},$$
where the first and second terms represent the free and interaction Hamiltonians, respectively. In the hybrid atom-assisted optomechanical system, the free Hamiltonian, ${\hat{H}_0},$ is composed of three terms corresponding to the mechanical oscillator, cavity field and three-level atomic ensemble . Its expression is given by (with $\hbar = 1$)
$${\hat{H}_0} = \Delta {\hat{a}^\dagger}\hat{a} + {\omega _m}{\hat{b}^\dagger}\hat{b} - {\Delta _p}{\hat{\sigma }_{33}} - {\Delta _{pc}}{\hat{\sigma }_{22}},$$
in the rotating frame. Here $\hat{a}$ $(\hat{b})$ and ${\hat{a}^\dagger}$ $({\hat{b}^\dagger})$ are the annihilation and creation operators, respectively, of the cavity (the mechanical) mode with frequency ${\omega _c}$ $({\omega _m}).$ The one-photon detuning $\Delta = {\omega _c} - {\omega _l}$ denotes the frequency difference between the cavity mode and the driving field. ${\Delta _p} = {\omega _p} - {\omega _{31}}$ $({\Delta _c} = {\omega _l} - {\omega _{32}})$ is also the one-photon detuning between the probe (driving) field and the corresponding atomic transition. Here ${\omega _{ij}}$ is the atomic transition frequency of $|i \rangle \leftrightarrow |j \rangle $ and ${\Delta _{pc}} = {\Delta _p} - {\Delta _c}$ is the two-photon detuning. The interaction Hamiltonian in Eq. (1) is composed of the field-oscillator term ${\hat{H}_{cm}},$ the field-atom term ${\hat{H}_{af}},$ and the driving term ${\hat{H}_{dr}}.$ In the rotating frame, their expressions are ($\hbar = 1$)
$${\hat{H}_{cm}} = g{\hat{a}^\dagger}\hat{a}({\hat{b}^\dagger} + \hat{b}),\;{\hat{H}_{af}} = {g_{ac}}({\hat{a}^\dagger}{\hat{\sigma }_{23}} + \hat{a}{\hat{\sigma }_{32}}) + {\Omega _p}({\hat{\sigma }_{31}} + {\hat{\sigma }_{13}}),\;{\hat{H}_{dr}} = i{\Omega _l}({\hat{a}^\dagger} - \hat{a}).$$
where g is the bare optomechanical coupling strength and ${g_{ac}}$ is the atom-cavity coupling strength, which is also called as the vacuum Rabi frequency. ${\hat{\sigma }_{ij}} = \sum\nolimits_{l = 1}^N {\hat{\sigma }_{ij}^{(l)}}$ denotes the collective operator of the atomic ensemble.

 figure: Fig. 1.

Fig. 1. Sketch of (a) the hybrid atom-assisted optomechanical system and (b) atomic energy-level configuration. The cavity is driven by a strong pumping field. The cavity field as a control field interacts with the atomic transition $|2 \rangle \leftrightarrow |3 \rangle $ with detuning ${\Delta _c},$ while a probe field interacts with the atomic transition $|1 \rangle \leftrightarrow |3 \rangle $ with detuning ${\Delta _p}.$

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Under the means field approximation $\left\langle {\hat{A}\hat{B}} \right\rangle = \left\langle {\hat{A}} \right\rangle \left\langle {\hat{B}} \right\rangle $ [26,44,45], we adopt the Heisenberg-Langevin formalism to derive the equations of motion for the mean values of the operators as:

$$\left\langle {\dot{\hat{a}}} \right\rangle ={-} (i\Delta + \frac{\kappa }{2})\left\langle {\hat{a}} \right\rangle - ig({\left\langle {\hat{b}} \right\rangle ^\ast } + \left\langle {\hat{b}} \right\rangle )\left\langle {\hat{a}} \right\rangle - i{g_{ac}}\left\langle {{{\hat{\sigma }}_{23}}} \right\rangle + {\Omega _l},$$
$$\left\langle {\dot{\hat{b}}} \right\rangle ={-} (i{\omega _m} + \frac{\gamma }{2})\left\langle {\hat{b}} \right\rangle - ig{\left\langle {\hat{a}} \right\rangle ^\ast }\left\langle {\hat{a}} \right\rangle ,$$
$$\left\langle {{{\dot{\hat{\sigma }}}_{31}}} \right\rangle ={-} (i{\Delta _p} + \frac{{{\gamma _{31}}}}{2})\left\langle {{{\hat{\sigma }}_{31}}} \right\rangle + i{g_{ac}}\left\langle {{{\hat{\sigma }}_{21}}} \right\rangle {\left\langle {\hat{a}} \right\rangle ^\ast } + i{\Omega _p}(\left\langle {{{\hat{\sigma }}_{11}}} \right\rangle - \left\langle {{{\hat{\sigma }}_{33}}} \right\rangle ),$$
$$\left\langle {{{\dot{\hat{\sigma }}}_{23}}} \right\rangle = (i{\Delta _c} - \frac{{{\gamma _{32}}}}{2})\left\langle {{{\hat{\sigma }}_{23}}} \right\rangle + i{g_{ac}}(\left\langle {{{\hat{\sigma }}_{33}}} \right\rangle - \left\langle {{{\hat{\sigma }}_{22}}} \right\rangle )\left\langle {\hat{a}} \right\rangle - i{\Omega _p}\left\langle {{{\hat{\sigma }}_{21}}} \right\rangle ,$$
$$\left\langle {{{\dot{\hat{\sigma }}}_{21}}} \right\rangle ={-} (i{\Delta _{pc}} + \frac{{{\gamma _{21}}}}{2})\left\langle {{{\hat{\sigma }}_{21}}} \right\rangle + i{g_{ac}}\left\langle {{{\hat{\sigma }}_{31}}} \right\rangle \left\langle {\hat{a}} \right\rangle - i{\Omega _p}\left\langle {{{\hat{\sigma }}_{23}}} \right\rangle ,$$
$$\left\langle {{{\dot{\hat{\sigma }}}_{11}}} \right\rangle = i{\Omega _p}(\left\langle {{{\hat{\sigma }}_{31}}} \right\rangle - \left\langle {{{\hat{\sigma }}_{13}}} \right\rangle ) + {\Gamma _{21}}\left\langle {{{\hat{\sigma }}_{22}}} \right\rangle + {\Gamma _{31}}\left\langle {{{\hat{\sigma }}_{33}}} \right\rangle ,$$
$$\left\langle {{{\dot{\hat{\sigma }}}_{22}}} \right\rangle = i{g_{ac}}(\left\langle {\hat{a}} \right\rangle \left\langle {{{\hat{\sigma }}_{32}}} \right\rangle - {\left\langle {\hat{a}} \right\rangle ^\ast }\left\langle {{{\hat{\sigma }}_{23}}} \right\rangle ) - {\Gamma _{21}}\left\langle {{{\hat{\sigma }}_{22}}} \right\rangle + {\Gamma _{32}}\left\langle {{{\hat{\sigma }}_{33}}} \right\rangle ,$$
$$\left\langle {{{\dot{\hat{\sigma }}}_{33}}} \right\rangle ={-} i{g_{ac}}(\left\langle {\hat{a}} \right\rangle \left\langle {{{\hat{\sigma }}_{32}}} \right\rangle - {\left\langle {\hat{a}} \right\rangle ^\ast }\left\langle {{{\hat{\sigma }}_{23}}} \right\rangle ) - i{\Omega _p}(\left\langle {{{\hat{\sigma }}_{31}}} \right\rangle - \left\langle {{{\hat{\sigma }}_{13}}} \right\rangle ) - {\Gamma _3}\left\langle {{{\hat{\sigma }}_{33}}} \right\rangle .$$
Here $\kappa $ is the decay rate of the cavity field and $\gamma $ is the damping rate of the mechanical oscillator. ${\gamma _{ij}} = {\Gamma _i} + {\Gamma _j}$ describes the coherence decay rate where ${\Gamma _i} = \sum\nolimits_{i > j} {{\Gamma _{ij}}} $ is the total decay rate of state $|i \rangle $ with ${\Gamma _{ij}}$ being the decay rate from state $|i \rangle $ to $|j \rangle .$ Considering ${\Omega _p}$ as a first-order small quantity, the weak-excitation approximation [41,43,46] can be introduced in the hybrid system. Correspondingly, the perturbation expansion of the mean values in Eq. (4) can be written as
$$\left\langle {{{\hat{\sigma }}_{ij}}} \right\rangle = \sum\limits_{l = 0}^N {c_{ij}^{(l)}} ,\;\left\langle {\hat{a}} \right\rangle = \alpha + \sum\limits_{l = 0}^N {{a^{(l)}}} ,\;\left\langle {\hat{b}} \right\rangle = \beta + \sum\limits_{l = 0}^N {{b^{(l)}}} .$$
Therefore, the electron is predominantly populated in the ground state at the initial stage, and we may set $c_{11}^{(0)} = N,$ $c_{22}^{(0)} = c_{33}^{(0)} = 0\textrm{,}$ and $c_{31}^{(0)} = c_{23}^{(0)} = c_{21}^{(0)} = 0$ [41,43,46]. Under the rotating-wave approximation, after substituting Eq. (5) into Eq. (4) we obtain the steady-state results
$$c_{31}^{(1)} = \frac{{iN{\Omega _p}}}{{{M_1}}},\;c_{21}^{(1)} = \frac{{i{g_{ac}}\alpha c_{31}^{(1)}}}{{{Y_3}}},\;c_{11}^{(1)} = c_{22}^{(1)} = c_{33}^{(1)} = {a^{(1)}} = {b^{(1)}} = 0,$$
$$c_{22}^{(2)} = i{\Omega _p}\frac{{{\Gamma _{31}}g_{ac}^2{{|\alpha |}^2}\left( {\frac{{c_{31}^{(1)}}}{{{Y_3}}} - \frac{{c_{31}^{(1) \ast }}}{{Y_3^\ast }}} \right) - (c_{31}^{(1)} - c_{31}^{(1) \ast })(2g_{ac}^2{{|\alpha |}^2} - {\Gamma _{32}}{Y_2})}}{{2g_{ac}^2{{|\alpha |}^2}({\Gamma _{21}} + {\Gamma _{31}}) - {\Gamma _{21}}{\Gamma _{32}}{Y_2}}},$$
$$c_{33}^{(2)} = i{\Omega _p}\frac{{{\Gamma _{21}}g_{ac}^2{{|\alpha |}^2}\left( {\frac{{c_{31}^{(1)}}}{{{Y_3}}} - \frac{{c_{31}^{(1) \ast }}}{{Y_3^\ast }}} \right) - (c_{31}^{(1)} - c_{31}^{(1) \ast })(2g_{ac}^2{{|\alpha |}^2} - {\Gamma _{21}}{Y_2})}}{{2g_{ac}^2{{|\alpha |}^2}({\Gamma _{21}} + {\Gamma _{31}}) - {\Gamma _{21}}{\Gamma _{32}}{Y_2}}},$$
$$c_{23}^{(2)} = \frac{{i{g_{ac}}\alpha (c_{22}^{(2)} - c_{33}^{(2)})}}{{{Y_2}}} - \frac{{i{\Omega _p}c_{21}^{(1)}}}{{{Y_2}}},\;a_{}^{(2)} ={-} \frac{{ig_{ac}^{}}}{{{P_1}}}c_{23}^{(2)},\;c_{31}^{(2)} = c_{21}^{(2)} = 0,$$
$$c_{31}^{(3)} = {u_1} + {u_2},$$
with
$${u_1} ={-} \frac{{i{\Omega _p}(2c_{33}^{(2)} + c_{22}^{(2)})}}{{\left( {{Y_1} + \frac{{g_{ac}^2{{|\alpha |}^2}}}{{{Y_3}}}} \right)}},\;{u_2} ={-} 2{\textrm{Re}}[{\alpha ^\ast }a_{}^{(2)}]\frac{{g_{ac}^2c_{31}^{(1)}}}{{{Y_3}\left( {{Y_1} + \frac{{g_{ac}^2{{|\alpha |}^2}}}{{{Y_3}}}} \right)}}.$$

Here ${M_1} = {Y_1} + {{g_{ac}^2{{|\alpha |}^2}} / {{Y_3}}},$ ${P_1} = {X_1} + {{{g^2}{{|\alpha |}^2}} / {{X_2}}},$ ${Y_2} = i{\Delta _c} - {{{\gamma _{32}}} / 2},$ ${Y_1} = i{\Delta _p} + {{{\gamma _{31}}} / 2},$ ${Y_3} = i{\Delta _{pc}} + {{{\gamma _{21}}} / 2},$ ${X_1} = i\tilde{\Delta } + {\kappa / 2},$ and ${X_2} = i{\omega _m} + {\gamma / 2}.$ The mean cavity amplitude is given by $\alpha = {{{\Omega _l}} / {(i\tilde{\Delta } + \kappa /2}})$ with the modified detuning as $\tilde{\Delta } = \Delta - 2g{\textrm{Re}}[\beta ]$ and $\beta = {{ - ig{{|\alpha |}^2}} / {(i{\omega _m} + {\gamma / 2})}}.$

It is well known that the response of the atomic medium to the probe field is determined by its polarization $P = {{{\varepsilon _0}(\chi {E_p} + {\chi ^ \ast }E_p^ \ast )} / 2}$ where ${\varepsilon _0}$ is the vacuum dielectric constant, $\chi $ is the susceptibility of the atomic medium, and ${E_p}$ is the amplitude of the probe field. Combining this equation with $P = {{NTr ({\mu _{13}}{\rho _{31}} + {\mu _{31}}{\rho _{13}})} / {2V}},$ where $\textrm{V}$ is the cavity mode volume, we can obtain the first- and third-order susceptibilities. The first-order susceptibility is given by

$${\chi ^{(1)}} = {\eta _1}\frac{{c_{31}^{(1)}}}{{N{\Omega _p}}} = {\eta _1}\frac{{i{Y_3}}}{{{Y_1}{Y_3} + \Omega _c^2}},$$
where ${\eta _1} = {{N{{|{{\mu_{13}}} |}^2}} / {{\varepsilon _0}\hbar V}}$ and $\Omega _c^{} = {g_{ac}}|\alpha |$ is the Rabi frequency of the control field. Its real and imaginary parts describe the linear dispersion and absorption of the atomic medium, respectively. The third-order susceptibility consists two terms, i.e., ${\chi ^{(3)}} = \chi _{{u_1}}^{(3)} + \chi _{{u_2}}^{(3)}.$ Its imaginary and real parts characterize the two-photon absorption and Kerr nonlinearity, respectively. The first term is given by
$$\chi _{{u_1}}^{(3)} ={-} {\eta _2}\frac{{i{\Omega _p}(2c_{33}^{(2)} + c_{22}^{(2)})}}{{{M_1}N\Omega _p^3}} ={-} i{\eta _2}\frac{{(2{\Gamma _{21}} - {\Gamma _{31}}){L_1} + 3{L_2}}}{{Q{M_1}}},$$
where ${L_1} ={-} \Omega _c^2[{{(Y_3^ \ast M_1^ \ast{+} {Y_3}{M_1})} / {{{|{{Y_3}{M_1}} |}^2}]}},$ ${L_2} = (2\Omega _c^2 - {\Gamma _{32}}{Y_2})[{{(M_1^ \ast{+} {M_1})} / {{{|{{M_1}} |}^2}}}],$ $Q = 2\Omega _c^2({\Gamma _{21}} + {\Gamma _{31}}) - {\Gamma _{21}}{\Gamma _{31}}{Y_2},$ and ${\eta _2} = {{{{|{{\mu_{13}}} |}^4}N} / {{\varepsilon _0}{\hbar ^3}}}V.$ This first term, shown in Eq. (13), is in accordance with the third-order susceptibility obtained by the classical three-level $\Lambda $ type atom EIT scheme [8,43]. The second term can be written as
$$\chi _{{u_2}}^{(3)} ={-} 2{\eta _2}{\textrm{Re}}[{\alpha ^\ast }a_{}^{(2)}]\frac{{g_{ac}^2c_{31}^{(1)}}}{{{M_1}{Y_3}N\Omega _p^3}} = 2i{\eta _2}NRe \left[ {\frac{{Q + {L_1}{Y_3}{M_1}({\Gamma _{21}} + {\Gamma _{31}})}}{{{P_1}{Y_3}{Y_2}Q{M_1}}}} \right]\frac{{g_{ac}^2\Omega _c^2}}{{{M_1}{Y_3}}}.$$
Obviously, the second term is related to the cavity field fluctuation ${a^{(2)}}.$ In general, when the probe field is incident on the atomic ensemble, it will induce the transition $|1 \rangle \leftrightarrow |3 \rangle ,$ thereby changing the populations of $|1 \rangle $ and $|3 \rangle $. Owing to the atom-cavity interaction between the transition $|2 \rangle \leftrightarrow |3 \rangle $ and the control field, the population of $|2 \rangle $ and the amplitude of the cavity field will be changed. Therefore, the cavity field fluctuation ${a^{(2)}}$ is affected by the collective effect of atoms, and thus it is proportional to the number of atoms N in the cavity. The second term $\chi _{{u_2}}^{(3)}$ is dependent on the real part of the cavity field fluctuation ${a^{(2)}}$ and $c_{31}^{(1)}$; therefore, it is proportional to the square of $N.$

3. Self-Kerr nonlinearity

3.1. New self-Kerr nonlinearity term

Here, we discuss the SK nonlinearity of the system and the effect of the system parameters, including the Rabi frequency ${\Omega _c}$ and the bare optomechanical coupling strength $g.$ To explicitly illustrate the optical features of the hybrid optomechanical system, the following experimentally realizable parameters are used [4751]: $m = 10\,\textrm{ng,}$ ${\omega _m} = 2\pi \times 10\,\textrm{MHz,}$ $\gamma = {10^{ - 5}}{\omega _m},$ $\kappa = 0.1{\omega _m},$ ${\Gamma _{32}} = {\Gamma _{31}} = 0.1{\omega _m},$ ${\Gamma _{21}} = 6 \times {10^{ - 4}}{\omega _m}.$

When the optomechanical interaction is switched off, i.e., $g = 0,$ the system becomes an atom-cavity EIT system. For this condition, Fig. 2 shows the linear absorption ${\mathop{\rm Im}\nolimits} [{\chi ^{(1)}}],$ first term ${\textrm{Re}}[\chi _{{u_1}}^{(3)}],$ and second term ${\textrm{Re}}[\chi _{{u_2}}^{(3)}]$ of the SK nonlinearity with the probe field detuning ${\Delta _p}$. The linear absorption spectrum shows a typical EIT line shape. The FSK term ${\textrm{Re}}[\chi _{{u_1}}^{(3)}]$ is zero at the linear absorption peaks ${\Delta _p} ={\pm} 0.2{\omega _m}$ or at the two-photon resonance ${\Delta _p} = {\Delta _c} = 0.$ The SSK term ${\textrm{Re}}[\chi _{{u_2}}^{(3)}]$ becomes zero at the two-photon resonance corresponding to the linear absorption dip, which is the same as the FSK term ${\textrm{Re}}[\chi _{{u_1}}^{(3)}].$ The difference is that the SSK nonlinearity term ${\textrm{Re}}[\chi _{{u_2}}^{(3)}]$ will peak at the linear absorption peaks. Moreover, it is interesting that the maximum of ${\textrm{Re}}[\chi _{{u_2}}^{(3)}]$ is approximately ${10^4}$ times greater than that of ${\textrm{Re}}[\chi _{{u_1}}^{(3)}].$ This giant self-Kerr nonlinearity can be advantageous to form bright and dark soliton [5254].

 figure: Fig. 2.

Fig. 2. The first term ${\textrm{Re}}[\chi _{{u_1}}^{^{(3)}}]$ and the second term ${\textrm{Re}}[\chi _{{u_2}}^{^{(3)}}]$ of the SK nonlinearity and the line absorption ${\mathop{\rm Im}\nolimits} [{\chi ^{(1)}}]$ vary with the atom-probe detuning ${\Delta _p}$ with $g = 0.$ Here ${\textrm{Re}}[\chi _{{u_1}}^{(3)}]$ as well as ${\textrm{Re}}[\chi _{{u_2}}^{(3)}]$ both take ${\eta _2}$ as a unit, whereas ${\mathop{\rm Im}\nolimits} [{\chi ^{(1)}}]$ takes ${\eta _1}$ as a unit. Other parameters are $\kappa = 0.1{\omega _m},$ $\gamma = {10^{ - 5}}{\omega _m},$ ${\Gamma _{32}} = {\Gamma _{31}} = 0.1{\omega _m},$ ${\Gamma _{21}} = 6 \times {10^{ - 4}}{\omega _m},$ $\Delta = {\omega _m},$ ${\Delta _c} = 0,$ ${g_{ac}} = 0.1{\omega _m},$ $|\alpha |= 2,$ ${\Omega _p} = {10^{ - 4}}{\omega _m},$ and $N = {10^5}.$

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For better understanding of the above-mentioned features of ${\textrm{Re}}[\chi _{{u_2}}^{(3)}],$ we plot the change in the populations of $|2 \rangle $ and $|3 \rangle $ and the real part of the cavity field fluctuation as functions of the detuning ${\Delta _p}$ in Fig. 3 according to the previously mentioned analysis of Eq. (14). Note that in the analytical solutions, we consider the change of the populations and the cavity amplitude contributed by the second-order populations and cavity field fluctuation and neglect the contribution of the higher order terms. The numerical results are obtained by solving Eq. (4) and using $\delta a = \left\langle {\hat{a}} \right\rangle - \alpha .$ We can see that the analytical results agree well with the numerical calculations. The trends of the atomic populations, cavity field fluctuation, and linear absorption changing with the atom-probe detuning ${\Delta _p}$ are consistent. This suggests that a higher linear absorption causes a larger change in the atomic populations, which further leads to a greater variation in the cavity field fluctuation. As shown in Fig. 3, the populations of $|2 \rangle $ and $|3 \rangle $ and the real part of the cavity field fluctuation undergo the largest and smallest changes, respectively, at the linear absorption peaks and the linear absorption dip. The cavity field fluctuation is induced by the changing atomic populations via the atom-cavity interaction, as the previous analysis of Eq. (14). Consequently, ${\textrm{Re}}[\chi _{{u_2}}^{(3)}]$ is proportional to the square of $N.$ Thus, for a large number of atoms, it will be far greater than ${\textrm{Re}}[\chi _{{u_1}}^{(3)}]$ and will vanish when the linear absorption vanishes at ${\Delta _p} = 0.$ Meanwhile, we note that the behaviors of $c_{22}^{(2)}$ and $c_{33}^{(2)}$ with ${\Delta _p}$ are symmetric while that of $a_{}^{(2)}$ is asymmetric. Thus, ${\textrm{Re}}[\chi _{{u_2}}^{(3)}]$ shows a asymmetric characteristic and ${\textrm{Re}}[\chi _{{u_1}}^{(3)}]$ shows a symmetric characteristic. According to Eq. (9), the change of $a_{}^{(2)}$ depends on the change of $c_{21}^{(1)}$ and $c_{22}^{(2)} - c_{33}^{(2)}$ when $\Delta = {\omega _m}$ and ${\Delta _c} = 0$. It indicate that the asymmetry of $a_{}^{(2)}$ is produced by $c_{21}^{(1)}$. We further plot the real and imaginary part of ${{c_{21}^{(1)}} / N}$ as a function of ${\Delta _p}$ in Fig. 4 and find that the variation curves of ${\textrm{Re}}[{{c_{21}^{(1)}} / N}]$ and ${\mathop{\rm Im}\nolimits} [{{c_{21}^{(1)}} / N}]$ are asymmetric. Therefore, the asymmetry of ${\textrm{Re}}[\chi _{{u_2}}^{(3)}]$ is produced by $c_{21}^{(1)}$.

 figure: Fig. 3.

Fig. 3. The change of the populations of $|2 \rangle $ and $|3 \rangle $ and the real part of the cavity fluctuation as functions of the detuning ${\Delta _p}.$ The diagram of curves is the analytic results and the scattering dot is the numerical results. Other parameters are the same as those given in Fig. 2.

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 figure: Fig. 4.

Fig. 4. The real and imaginary part of ${{c_{21}^{(1)}} / N}$ as functions of the detuning ${\Delta _p}.$ Other parameters are the same as those given in Fig. 2.

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3.2. Impact of the control field

Here, we will explore the effect of the control field Rabi frequency ${\Omega _c}$ on the SK nonlinearities ${\textrm{Re}}[\chi _{{u_1}}^{(3)}]$ and ${\textrm{Re}}[\chi _{{u_2}}^{(3)}].$ According to ${\Omega _c} = {g_{ac}}|\alpha |,$ we can change ${\Omega _c}$ by changing either ${g_{ac}}$ or $|\alpha |.$ Taking the relevant ${\Omega _c}$ as $0.12{\omega _m},$ $0.18{\omega _m},$ and $0.24{\omega _m},$ we plot the SK nonlinearities ${\textrm{Re}}[\chi _{{u_1}}^{(3)}]$ and ${\textrm{Re}}[\chi _{{u_2}}^{(3)}]$ as functions of ${\Delta _p}$ for different the mean cavity amplitude $|\alpha |$ and the atom-cavity coupling strength ${g_{ac}}$ in Fig. 5 and Fig. 6, respectively. From Fig. 5, it is obvious that the peak values of ${\textrm{Re}}[\chi _{{u_1}}^{(3)}]$ and ${\textrm{Re}}[\chi _{{u_2}}^{(3)}]$ decrease with increasing $|\alpha |.$ Fig. 6 indicates that the peak values of ${\textrm{Re}}[\chi _{{u_1}}^{(3)}]$ also decrease with increasing ${g_{ac}}$ whereas those of ${\textrm{Re}}[\chi _{{u_2}}^{(3)}]$ is increasing. Comparing Fig. 5 and Fig. 6, we infer that ${\textrm{Re}}[\chi _{{u_1}}^{(3)}]$ always decrease with increasing ${\Omega _c}.$ However, for ${\textrm{Re}}[\chi _{{u_2}}^{(3)}],$ adding the same value to ${\Omega _c}$ by ${g_{ac}}$ will significantly enhance ${\textrm{Re}}[\chi _{{u_2}}^{(3)}],$ whereas adding the same value to ${\Omega _c}$ by $|\alpha |$ will reduce ${\textrm{Re}}[\chi _{{u_2}}^{(3)}].$ This is because a higher interaction between the atoms and the cavity field will lead to a larger cavity fluctuation.

 figure: Fig. 5.

Fig. 5. The self-Kerr nonlinearity (a) ${\textrm{Re}}[\chi _{{u_1}}^{(3)}]$ (in ${\eta _2}$) and (b) ${\textrm{Re}}[\chi _{{u_2}}^{(3)}]$ (in ${\eta _2}$) as functions of the atom-probe detuning ${\Delta _p}$ with the atom-cavity coupling strength ${g_{ac}} = 0.03{\omega _m}$ for different mean cavity amplitudes $|\alpha |:$ $|\alpha |= 4,$ $|\alpha |= 6,$ and $|\alpha |= 8.$ Other parameters are the same as those given in Fig. 2.

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 figure: Fig. 6.

Fig. 6. The self-Kerr nonlinearity (a) ${\textrm{Re}}[\chi _{{u_1}}^{(3)}]$ (in ${\eta _2}$) and (b) ${\textrm{Re}}[\chi _{{u_2}}^{(3)}]$ (in ${\eta _2}$) as functions of the atom-probe detuning ${\Delta _p}$ with the mean cavity amplitude $|\alpha |= 3$ for different atom-cavity coupling strengths ${g_{ac}}:$ ${g_{ac}} = 0.04{\omega _m},$ ${g_{ac}} = 0.06{\omega _m},$ and ${g_{ac}} = 0.08{\omega _m}.$ Other parameters are the same as those used in Fig. 2.

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3.3. Impact of the optomechanical interaction

It is well known that the photon and phonon states will be changed by the optomechanical interaction which further affects the cavity field fluctuation. For the second order fluctuation term $a_{}^{(2)},$ we can rewrite it as:

$$\begin{array}{l} a_{}^{(2)} = \frac{{{A_ + }}}{{{\omega _m} + {x_1} - i\frac{{\kappa - \gamma }}{4}}} + \frac{{{A_ - }}}{{{\omega _m} - {x_1} - i\frac{{\kappa - \gamma }}{4}}} = a_ + ^{(2)} + a_ - ^{(2)},\\ {A_ - } ={-} \frac{{{g_{ac}}c_{23}^{(2)}[2{x_1} + i(\kappa - \gamma )]}}{{8{x_1}}},\;{A_ + } ={-} \frac{{{g_{ac}}c_{23}^{(2)}[2{x_1} - i(\kappa - \gamma )]}}{{8{x_1}}}. \end{array}$$
We can observe that two modes occur at the frequencies ${\omega _ \pm } = {\omega _m} \pm {x_1}$ with $\Delta = {\omega _m}.$ Here ${x_1} = {{\sqrt {16{g^2}{{|\alpha |}^2} - {{(\kappa - \lambda )}^2}} } / 4}$ and $a_ \pm ^{(2)}$ correspond to the contributions of the two modes ${\omega _ \pm }.$ Now we can also rewrite the SSK nonlinearity ${\textrm{Re}}[\chi _{{u_2}}^{(3)}]$ as:
$${\textrm{Re}}[\chi _{{u_2}}^{(3)}] = {\textrm{Re}}[\chi _{{u_ + }}^{(3)}] + {\textrm{Re}}[\chi _{{u_ - }}^{(3)}] = {\textrm{Re}}[{\alpha ^\ast }a_ + ^{(2)}]L + {\textrm{Re}}[{\alpha ^\ast }a_ - ^{(2)}]L,$$
where $L ={-} 2{\eta _2}{\textrm{Re}}[{{g_{ac}^2c_{31}^{(1)}} / {{Y_3}{M_1}N\Omega _p^3}}].$ We have plotted the real and imaginary parts of the complex numbers ${\omega _ \pm } - {\omega _m}$ versus g in Fig. 7. Figure 8 shows the g dependent variation of two terms ${\textrm{Re}}[\chi _{{u_ \pm }}^{(3)}]$ in Eq. (16) and ${\textrm{Re}}[\chi _{{u_2}}^{(3)}]$ with ${\Delta _p} = 0.2{\omega _m}.$

 figure: Fig. 7.

Fig. 7. The (a) real and (b) imaginary parts of complex numbers ${\omega _ \pm } - {\omega _m}$ as functions of the bare optomechanical coupling strength $g.$ Other parameters used are the same as in Fig. 2.

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 figure: Fig. 8.

Fig. 8. The self-Kerr nonlinearities (a) ${\textrm{Re}}[\chi _{{u_ \pm }}^{(3)}]$ (in ${\eta _2}$) and (b) ${\textrm{Re}}[\chi _{{u_2}}^{(3)}]$ (in ${\eta _2}$) as functions of the bare optomechanical coupling strength g with the atom-probe detuning ${\Delta _p} = 0.2{\omega _m}.$ Other parameters are the same as those given in Fig. 2.

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In the region $g < 0.0125{\omega _m},$ ${x_1}$ is a purely imaginary number as seen in Fig. 7, and thus the two modes cannot be resolved. Therefore, ${\textrm{Re}}[\chi _{{u_ + }}^{(3)}]$ is inverted relative to ${\textrm{Re}}[\chi _{{u_ - }}^{(3)}]$ which produces two opposite nonlinearities when $g < 0.0125{\omega _m}.$ They both increase with increasing g and peak at the threshold of the normal-mode splitting $(g = 0.0125{\omega _m}).$ However, the total effective ${\textrm{Re}}[\chi _{{u_2}}^{(3)}]$ shows only an ascending trend with increasing $g,$ as seen in Fig. 8(b). When $g > 0.0125{\omega _m},$ the normal-mode splitting [55,56] occurs. The magnitude of the splitting becomes large with increasing $g.$ This indicates an enlargement in ${\omega _ + }$ and a reduction in ${\omega _ - }.$ Simultaneously, the quantum destructive interference between $a_ + ^{(2)}$ and $a_ - ^{(2)}$ is weakened and ${\textrm{Re}}[\chi _{{u_ \pm }}^{(3)}]$ starts to decline from the peaks. Thus, for $g > 0.05{\omega _m},$ ${\textrm{Re}}[\chi _{u - }^{(3)}]$ increases with increasing $g,$ while the trend of ${\textrm{Re}}[\chi _{u + }^{(3)}]$ is opposite, as shown in the inset of Fig. 8(a). In this region, the total effective ${\textrm{Re}}[\chi _{{u_2}}^{(3)}]$ also increases with increasing $g.$ Experimentally, some optomechanical systems can realize strong optomechanical coupling. For example, in the zipper cavity optomechanical systems [57,58], the bare optomechanical coupling strength can be $0.03$ [57] and even $2$ [58] times the frequency of the mechanical mode. Besides, strong optomechanical coupling are possible to be achieved by using a photonic crystal membrane [59] or nanomembrane coupled quantum well [60] with the photothermal and optoelectronic forces. Because the photothermal and optoelectronic forces can be greater than the radiation pressure force approximately 1000 times [61], which can greatly enhance the optomechanical interaction.

4. Photon–phonon cross-Kerr nonlinearity

4.1. Expression of cross-Kerr nonlinearity

In this section, we consider an external weak mechanical drive acting on the mechanical oscillator with an amplitude ${\varepsilon _d}$ and frequency ${\omega _d}$ to adjust the cavity field. This means adding the term ${\varepsilon _d}{e^{ - {\omega _d}t}}$ behind $\left\langle {\dot{\hat{b}}} \right\rangle ={-} (i{\omega _m} + {\gamma / 2})\left\langle {\hat{b}} \right\rangle - ig{\left\langle {\hat{a}} \right\rangle ^\ast }\left\langle {\hat{a}} \right\rangle $ in Eq. (4), i.e.,

$$\left\langle {\dot{\hat{b}}} \right\rangle ={-} (i{\omega _m} + {\gamma / 2})\left\langle {\hat{b}} \right\rangle - ig{\left\langle {\hat{a}} \right\rangle ^\ast }\left\langle {\hat{a}} \right\rangle + {\varepsilon _d}{e^{ - {\omega _d}t}}.$$
Here the amplitude ${\varepsilon _d}$ can also be considered as a first-order small quantity. Owing to the optomechanical interaction, high-order sideband processes [40,62] will occur. Thus, we consider that the lth ($l \ge 1$) order terms of the above-mentioned perturbation expansion Eq. (5) have the following forms:
$$\begin{array}{l} {a^{(l)}} = {a^{(l)}}(0) + {a^{(l)}}({\omega _d}){e^{ - i{\omega _d}t}} + {a^{(l)}}( - {\omega _d}){e^{i{\omega _d}t}}\\ \quad \quad \; + {a^{(l)}}(2{\omega _d}){e^{ - 2i{\omega _d}t}} + {a^{(l)}}( - 2{\omega _d}){e^{2i{\omega _d}t}} +{\cdot}{\cdot} \cdot ,\\ {b^{(l)}} = {b^{(l)}}(0) + {b^{(l)}}({\omega _d}){e^{ - i{\omega _d}t}} + {b^{(l)}}( - {\omega _d}){e^{i{\omega _d}t}}\\ \quad \quad \; + {b^{(l)}}(2{\omega _d}){e^{ - 2i{\omega _d}t}} + {b^{(l)}}( - 2{\omega _d}){e^{2i{\omega _d}t}} +{\cdot}{\cdot} \cdot ,\\ c_{ij}^{(l)} = c_{ij}^{(l)}(0) + c_{ij}^{(l)}({\omega _d}){e^{ - i{\omega _d}t}} + c_{ij}^{(l)}( - {\omega _d}){e^{i{\omega _d}t}}\\ \quad \quad \; + c_{ij}^{(l)}(2{\omega _d}){e^{ - 2i{\omega _d}t}} + c_{ij}^{(l)}( - 2{\omega _d}){e^{2i{\omega _d}t}} +{\cdot}{\cdot} \cdot . \end{array}$$
Here we only focus on the zero-order sideband term in the third-order small quantities. After Eq. (17) replacing Eq. (4b), we can obtain a new Eq. (4). Substituting the perturbation expansion Eq. (5) and the ansatz Eq. (18) into this new Eq. (4), we obtain the following steady-state results
$$\begin{array}{l} \quad \quad \quad {b^{(1)}}({\omega _d}) = {F_1}{\varepsilon _d},\;a_x^{(2)}(0) ={-} \left[ {\frac{{{g^2}{{|\alpha |}^2}}}{{{X_2}{{|{{X_1}({\omega_d})} |}^2}}} + \frac{1}{{{X_1}({\omega_d})}}} \right]\frac{{{g^2}\alpha {{|{{b^{(1)}}({\omega_d})} |}^2}}}{{{P_1}}},\\ c_{31\textrm{ - }x}^{(3)}(0) ={-} \left[ {\frac{{c_{31}^{(1)}(0)}}{{{M_3}}} - \frac{{g_{ac}^2{{|\alpha |}^2}c_{31}^{(1)}(0)}}{{Y_3^2{M_2}}}} \right]\frac{{g_{ac}^2{g^2}{{|\alpha |}^2}}}{{{M_1}{{|{{X_1}({\omega_d})} |}^2}}}{|{{b^{(1)}}({\omega_d})} |^2} - 2{\textrm{Re}}[{\alpha ^\ast }a_x^{(2)}(0)]\frac{{g_{ac}^2c_{31}^{(1)}(0)}}{{{M_1}{Y_3}}}.\\ \quad \quad \quad \;\, \end{array}$$

Here ${F_1} = {[{X_2}({\omega _d}) + {{g_{}^2{{|\alpha |}^2}} / {{X_1}({\omega _d})}}]^{ - 1}},$ ${X_1}({\omega _d}) = {X_1} - i{\omega _d},$ ${X_2}({\omega _d}) = {X_2} - i{\omega _d},$ ${M_3} = {Y_3} - i{\omega _d} + {{\Omega _c^2} / {({Y_1} - i{\omega _d}}}),$ and ${M_2} = {Y_1} + i{\omega _d} + {{\Omega _c^2} / {({Y_3} + i{\omega _d})}}.$ $c_{31}^{(1)}(0)$ is the same as $c_{31}^{(1)}$ in Eq. (6). $c_{31\textrm{ - }x}^{(3)}(0)$ is relate to the phonon number of the mechanical vibration. Then we can obtain the third-order susceptibility term $\chi _{xpm}^{(3)}$ for the CK nonlinearity with respect to the phonon number, i.e., $\chi _{xpm}^{(3)}{|{{b^{(1)}}({\omega_d})} |^2}.$ The expression of $\chi _{xpm}^{(3)}$ is:

$$\chi _{xpm}^{(3)} = 2{\eta _1}i{\textrm{Re}}\left[ {\frac{1}{{{P_1}{X_1}({\omega_d})}} + \frac{{{g^2}{{|\alpha |}^2}}}{{{X_2}{P_1}{{|{{X_1}({\omega_d})} |}^2}}}} \right]\frac{{{g^2}\Omega _c^2}}{{{Y_3}M_1^2}} + i{\eta _1}\frac{{(\Omega _c^2{M_3} - Y_3^2{M_2})g_{ac}^2{g^2}{{|\alpha |}^2}}}{{Y_3^2{M_1}{M_2}{M_3}{{|{{X_1}({\omega_d})} |}^2}}}.$$

This term describes the CK interaction between photon and phonon. The CK interaction has an important effect on the bistable behavior of the mean photon number [63] and photon blockade [64].

4.2. Impact of the control field

In this section, we discuss the effect of the control field Rabi frequency ${\Omega _c}$ on the CK nonlinearity. Similar to the aforementioned discussion in Sec. 3.2, we plot the CK nonlinearity ${\textrm{Re}}[\chi _{xpm}^{(3)}]$ as a function of ${\Delta _p}$ for different $|\alpha |$ $({g_{ac}})$ with ${g_{ac}} = 0.03{\omega _m}$ $(|\alpha |= 3)$ in Fig. 9. It is found that the two peaks of the CK nonlinearity (its sites corresponding to the two linear absorption peak sites) are separated more widely from each other. Its values are enhanced by increasing ${\Omega _c}$. An increase in ${g_{ac}}$ or $|\alpha |$ produces the same magnitudes of separation and enhancement. However, the value of the CK nonlinearity at the two-photon resonance, i.e., ${\Delta _p} = {\Delta _c} = 0,$ remains almost unchanged with variations in ${\Omega _c}.$ Notably, the SK nonlinearity is zero at the two-photon resonance, which implies the existence of only the CK nonlinearity ${\textrm{Re}}[\chi _{xpm}^{(3)}]$ signal about the probe field.

 figure: Fig. 9.

Fig. 9. The CK nonlinearity ${\textrm{Re}}[\chi _{xpm}^{(3)}]$ (in ${\eta _1}$) as a function of the probe detuning ${\Delta _p}$ under different cases when $g = 5 \times {10^{ - 4}}{\omega _m}:$ first, with $|\alpha |= 3$ and different ${g_{ac}},$ i.e., ${g_{ac}} = 0.04{\omega _m},$ ${g_{ac}} = 0.06{\omega _m},$ and ${g_{ac}} = 0.08{\omega _m};$ second, with ${g_{ac}} = 0.03{\omega _m}$ and different $|\alpha |,$ i.e., $|\alpha |= 4,$ $|\alpha |= 6,$ and $|\alpha |= 8.$ Other parameters are the same as in Fig. 2.

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4.3. Impact of the optomechanical interaction

To assess the effect of g and ${\omega _d}$ on ${\textrm{Re}}[\chi _{xpm}^{(3)}],$ we plot ${\textrm{Re}}[\chi _{xpm}^{(3)}]$ versus g and ${\omega _d}$ in Fig. 10. With the enhancement of the bare optomechanical coupling strength $g,$ the CK nonlinearity ${\textrm{Re}}[\chi _{xpm}^{(3)}]$ is increased; however the phonon number ${|{{b^{(1)}}({\omega_d})} |^2}$ is decreased and splits into two resonance peaks at ${\omega _d} = {\omega _m} \pm g|\alpha |$ due to the normal-mode splitting with a strong bare optomechanical coupling strength, namely, $g > {\kappa / 4}|\alpha |.$ ${|{{b^{(1)}}({\omega_d})} |^2}$ attains its maximum value at the resonance point, and this resonance peak value in the region $g < {\kappa / 4}|\alpha |$ is higher than the splitting peak values in the region $g > {\kappa / 4}|\alpha |,$ as shown in Fig. 10(b). Therefore, ${\textrm{Re}}[\chi _{xpm}^{(3)}{|{{b^{(1)}}({\omega_d})} |^2}]$ also exhibits one peak in the region $g < {\kappa / 4}|\alpha |$ and two peaks in the region $g > {\kappa / 4}|\alpha |.$ Its maximum peak intensity appears in the region $g < {\kappa / 4}|\alpha |,$ as shown in Fig. 10(c).

 figure: Fig. 10.

Fig. 10. (a) The CK nonlinearity ${\textrm{Re}}[\chi _{xpm}^{(3)}]$ (in ${\eta _1}$), (b) ${|{b({\omega_d})} |^2},$ and (c) ${\textrm{Re}}[\chi _{xpm}^{(3)}]{|{{b^{(1)}}({\omega_d})} |^2}$ (in ${\eta _1}$) as functions of the bare optomechanical coupling strength g and the mechanical drive frequency ${\omega _d}.$ Other parameters are the same as those given in Fig. 2.

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4.4. CK coupling strength

Here, we will derive the Hamiltonian of the CK interaction between the cavity field and mechanical mode in order to estimate the magnitude of the CK coupling strength when the probe field is also a cavity field. The CK interaction Hamiltonian is derived from the expression of the time-averaged electromagnetic energy density, ${H_{em}} = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}} \right.}\!\lower0.7ex\hbox{$2$}}\int {[\vec{E}(\vec{r}) \cdot \vec{D}(\vec{r}) + \vec{H}(\vec{r}) \cdot \vec{B}(\vec{r})]} d\vec{r},$ where $\vec{H} = {{\vec{B}} / {{\mu _0}}},$ expressed as [65,66]:

$${\hat{H}_{xpm}} = \hbar {g_{nl}}{\hat{a}^\dagger}\hat{a}{\hat{b}^\dagger}\hat{b},$$
where
$$\hbar {g_{nl}} = \frac{D}{2}\frac{{\hbar {\omega _p}}}{2}{\int {d\vec{r}|{\vec{\alpha }(\vec{r})} |} ^2}\frac{{{\textrm{Re}}[\chi _{jik}^{(3)}(\vec{r})]}}{{\varepsilon (\vec{r})}} = \frac{{D\hbar {\omega _p}}}{{4{{\bar{\varepsilon }}_r}}}{\bar{\chi }_{xpm}}.$$
Here $\vec{\alpha }(\vec{r})$ is the normalized three-dimensional cavity field profile satisfying ${\int {d\vec{r}|{\vec{\alpha }(\vec{r})} |} ^2} = 1$ and D is the degeneracy. To obtain order-of-magnitude results, we assume constant values for the averaged real part of the nonlinear susceptibility ${\bar{\chi }_{xpm}}$ and relative dielectric permittivity ${\bar{\varepsilon }_r}.$ ${g_{nl}} = {{D{\omega _p}{{\bar{\chi }}_{xpm}}} / {4{{\bar{\varepsilon }}_r}}}$ is the CK coupling strength. We consider the case of two-photons resonance, ${\Delta _p} = {\Delta _c} = 0,$ where the linear absorption and dispersion of the system are approximately zero. Therefore, we can take ${\bar{\varepsilon }_r} = 1.$ The others parameters are: $D = 6,$ ${g_{ac}} = 0.03{\omega _m},$ $|\alpha |= 8,$ $g = 5 \times {10^{ - 4}}{\omega _m},$ ${\omega _p} = 2.4 \times {10^{15}}\,\textrm{Hz,}$ $\rho = {N / V} = 5 \times {10^8}\,\textrm{c}{\textrm{m}^{\textrm{ - 3}}},$ and ${\mu _{13}} = 1.269 \times {10^{ - 29}}\,\textrm{C} \cdot \textrm{m}$ [51] and then the CK coupling strength ${g_{nl}}$ is estimated to be $5.4\,\textrm{MHz}\textrm{.}$

5. Conclusion

In summary, we theoretically studied the effect of cavity field fluctuations on the Kerr nonlinearity via EIT in a hybrid optomechanical system which is composed of three-level Λ type atoms embedded in an optomechanical cavity. We found that the SK nonlinearity is composed of FSK and SSK nonlinearity terms. The former is equivalent to that obtained by the classical three-level Λ type atoms EIT scheme, and the latter is dependent on the cavity field fluctuation and linear absorption with the atom-cavity interaction. For the hybrid system with numerous atoms, the SSK nonlinearity term is greater than the first term by several orders of magnitude. Further, it can be enhanced by increasing the atom-cavity and optomechanical interactions. These results may provide a new way to enhance the SK nonlinearity.

With the extra mechanical drive on the mechanical oscillator, a giant CK nonlinearity can be generated between the probe field and mechanical mode. It can be modified by the control filed and the optomechcanical interaction. In addition, at the two-photon resonance point, only the CK nonlinearity for the probe field exists which is useful for reading the phonon number of the mechanical oscillator. Our method of producing the CK nonlinearity in a hybrid atom-assisted optomechanical cavity may be extend to other systems as well. For example, the mechanical vibrations and optomechanical interactions of the whispering-gallery-mode and photonic crystal cavity have been demonstrated experimentally [27,6771]. Moreover, a strong coupling between these cavities and nitrogen vacancy (NV) centers or quantum dots (QDs) has been achieved experimentally [7277]. Therefore, these cavities that are coupled to NV center or QDs may generate CK nonlinearity. Our results may provide a new means to produce a larger Kerr nonlinearity, which has wide applications in quantum information processing and quantum computing.

Appendix. Derivation of the steady-state solutions

Here we derive the steady-state solutions by using the perturbation method. We substitute the perturbation expansion (5) and the ansatz (18) into Eq. (4) in which Eq. (4b) has been replaced by Eq. (17). In the first-order case, we have

$$\begin{array}{l} {{\dot{a}}^{(1)}} ={-} {X_1}{a^{(1)}} - ig\alpha {b^{(1)}} - i{g_{ac}}c_{23}^{(1)},\\ {{\dot{b}}^{(1)}} ={-} {X_2}{b^{(1)}} - ig{\alpha ^ \ast }{a^{(1)}} + {\varepsilon _d}{e^{ - i{\omega _d}t}},\\ \dot{c}_{31}^{(1)} ={-} {Y_1}c_{31}^{(1)} + i{g_{ac}}{\alpha ^ \ast }c_{21}^{(1)} + i{\Omega _p}N,\\ \dot{c}_{23}^{(1)} = {Y_2}c_{23}^{(1)} + i{g_{ac}}\alpha (c_{33}^{(1)} - c_{22}^{(1)}),\\ \dot{c}_{21}^{(1)} ={-} {Y_3}c_{21}^{(1)} + i{g_{ac}}\alpha c_{31}^{(1)},\\ \dot{c}_{22}^{(1)} = i({g_{ac}}\alpha c_{32}^{(1)} - {g_{ac}}{\alpha ^ \ast }c_{23}^{(1)}) - {\Gamma _{21}}c_{22}^{(1)} + {\Gamma _{32}}c_{33}^{(1)},\\ \dot{c}_{33}^{(1)} ={-} i({g_{ac}}\alpha c_{32}^{(1)} - {g_{ac}}{\alpha ^ \ast }c_{23}^{(1)}) - {\Gamma _3}c_{33}^{(1)},\\ c_{11}^{(1)} + c_{22}^{(1)} + c_{33}^{(1)} = 0. \end{array}$$
Here ${Y_1} = i{\Delta _p} + {{{\gamma _{31}}} / 2},$ ${Y_2} = i{\Delta _c} - {{{\gamma _{32}}} / 2},$ ${Y_3} = i{\Delta _{pc}} + {{{\gamma _{21}}} / 2},$ ${X_1} = i\tilde{\Delta } + {\kappa / 2},$ and ${X_2} = i{\omega _m} + {\gamma / 2},$ with $\tilde{\Delta } = \Delta - 2g{\textrm{Re}}(\beta )$ and $\beta = {{ - ig{{|\alpha |}^2}} / {(i{\omega _m} + {\gamma / 2})}}.$ The corresponding steady-state solutions are given by
$${a^{(1)}} = {a^{(1)}}({\omega _d}){e^{ - i{\omega _d}t}} = \frac{{ - ig\alpha }}{{{X_1}({\omega _d})}}{b^{(1)}},\;{b^{(1)}} = {b^{(1)}}({\omega _d}){e^{ - i{\omega _d}t}} = {F_1}{\varepsilon _d}{e^{ - i{\omega _d}t}},\;$$
$$c_{22}^{(1)} = c_{33}^{(1)} = c_{23}^{(1)} = 0,\;c_{21}^{(1)} = c_{21}^{(1)}(0) = \frac{{i{g_{ac}}\alpha c_{31}^{(1)}(0)}}{{{Y_3}}},\;c_{31}^{(1)} = c_{31}^{(1)}(0) = \frac{{iN{\Omega _p}}}{{{M_1}}},$$
where ${F_1} = {[{X_2}({\omega _d}) + {{g_{}^2{{|\alpha |}^2}} / {{X_1}({\omega _d})}}]^{ - 1}},$ ${X_1}({\omega _d}) = {X_1} - i{\omega _d},$ ${X_2}({\omega _d}) = {X_2} - i{\omega _d},$and${M_1} = {Y_1} + {{g_{ac}^2{{|\alpha |}^2}} / {{Y_3}}}.$ In the second-order case, we obtain the following relations
$$\begin{array}{l} {{\dot{a}}^{(2)}} ={-} {X_1}{a^{(2)}} - ig\alpha {b^{(2)}} - i{g_{ac}}c_{23}^{(2)} - ig{a^{(1)}}{b^{(1)}} - ig{a^{(1)}}{b^{(1) \ast }},\\ {{\dot{b}}^{(2)}} ={-} {X_2}{b^{(2)}} - ig{\alpha ^\ast }{a^{(2)}} - ig{|{{a^{(1)}}} |^2},\\ \dot{c}_{31}^{(2)} ={-} {Y_1}c_{31}^{(2)} + i{g_{ac}}{\alpha ^\ast }c_{21}^{(2)} + i{g_{ac}}{a^{(1) \ast }}c_{21}^{(1)},\\ \dot{c}_{21}^{(2)} ={-} {Y_3}c_{21}^{(2)} + i{g_{ac}}\alpha c_{31}^{(2)} + i{g_{ac}}{a^{(1)}}c_{31}^{(1)},\\ \dot{c}_{23}^{(2)} = {Y_2}c_{23}^{(2)} + i{g_{ac}}\alpha (c_{33}^{(2)} - c_{22}^{(2)}) - i{\Omega _p}c_{21}^{(1)},\\ \dot{c}_{22}^{(2)} = i{g_{ac}}\alpha (c_{32}^{(2)} - c_{23}^{(2)}) - {\Gamma _{21}}c_{22}^{(2)} + {\Gamma _{32}}c_{33}^{(2)},\\ \dot{c}_{33}^{(2)} ={-} i{g_{ac}}\alpha (c_{32}^{(2)} - c_{23}^{(2)}) - i{\Omega _p}(c_{31}^{(1)} - c_{13}^{(1)}) - {\Gamma _3}c_{33}^{(2)},\\ c_{11}^{(2)} + c_{22}^{(2)} + c_{33}^{(3)} = 0. \end{array}$$
The solution of Eq. (26) are as following:
$$c_{33}^{(2)} = c_{33}^{(2)}(0),\;c_{22}^{(2)} = c_{22}^{(2)}(0),\;c_{23}^{(2)} = c_{23}^{(2)}(0),$$
$$c_{21}^{(2)} = c_{21}^{(2)}({\omega _d}){e^{ - i{\omega _d}t}} + c_{21}^{(2)}( - {\omega _d}){e^{i{\omega _d}t}},\;c_{31}^{(2)} = c_{31}^{(2)}({\omega _d}){e^{ - i{\omega _d}t}} + c_{31}^{(2)}( - {\omega _d}){e^{i{\omega _d}t}},$$
$${a^{(2)}} = {a^{(2)}}(0) + {a^{(2)}}(2{\omega _d}){e^{ - 2i{\omega _d}t}},$$
where
$$c_{22}^{(2)}(0) = i{\Omega _p}\frac{{{\Gamma _{31}}g_{ac}^2{{|\alpha |}^2}\left[ {\frac{{c_{31}^{(1)}(0)}}{{{Y_3}}} - \frac{{c_{31}^{(1) \ast }(0)}}{{Y_3^\ast }}} \right] - [c_{31}^{(1)}(0) - c_{31}^{(1) \ast }(0)](2g_{ac}^2{{|\alpha |}^2} - {\Gamma _{32}}{Y_2})}}{{2g_{ac}^2{{|\alpha |}^2}({\Gamma _{21}} + {\Gamma _{31}}) - {\Gamma _{21}}{\Gamma _{32}}{Y_2}}},$$
$$c_{33}^{(2)}(0) = i{\Omega _p}\frac{{{\Gamma _{21}}g_{ac}^2{{|\alpha |}^2}\left[ {\frac{{c_{31}^{(1)}(0)}}{{{Y_3}}} - \frac{{c_{31}^{(1) \ast }(0)}}{{Y_3^\ast }}} \right] - [c_{31}^{(1)}(0) - c_{31}^{(1) \ast }(0)](2g_{ac}^2{{|\alpha |}^2} - {\Gamma _{21}}{Y_2})}}{{2g_{ac}^2{{|\alpha |}^2}({\Gamma _{21}} + {\Gamma _{31}}) - {\Gamma _{21}}{\Gamma _{32}}{Y_2}}},$$
$$c_{23}^{(2)}(0) = \frac{{i{g_{ac}}\alpha [c_{22}^{(2)}(0) - c_{33}^{(2)}(0)]}}{{{Y_2}}} + \frac{{i{\Omega _p}c_{21}^{(1)}(0)}}{{{Y_2}}},$$
$$c_{21}^{(2)}({\omega _d}) = i\frac{{{g_{ac}}{a^{(1)}}({\omega _d})c_{31}^{(1)}(0)}}{{{M_3}}},\;c_{31}^{(2)}({\omega _d}) = \frac{{i{g_{ac}}{\alpha ^\ast }c_{21}^{(2)}({\omega _d})}}{{{Y_1} - i{\omega _d}}},$$
$$c_{31}^{(2)}( - {\omega _d}) = i\frac{{{g_{ac}}{a^{(1) \ast }}({\omega _d})c_{21}^{(1)}(0)}}{{{M_2}}},\;c_{21}^{(2)}( - {\omega _d}) = \frac{{i{g_{ac}}\alpha c_{31}^{(2)}( - {\omega _d})}}{{{Y_3} + i{\omega _d}}},$$
$${a^{(2)}}(0) = a_s^{(2)}(0) + a_x^{(2)}(0),$$
$$a_s^{(2)}(0) ={-} \frac{{i{g_{ac}}c_{23}^{(2)}(0)}}{{{P_1}}},\;a_x^{(2)}(0) ={-} i\frac{g}{{{P_1}}}{a^{(1)}}({\omega _d}){b^{(1) \ast }}({\omega _d}) - \frac{{{g^2}\alpha }}{{{X_2}{P_1}}}{|{{a^{(1)}}({\omega_d})} |^2},$$
$${a^{(2)}}(2{\omega _d}) ={-} i\frac{g}{{{P_1}}}{a^{(1)}}({\omega _d}){b^{(1)}}({\omega _d}).$$
Here ${M_2} = {Y_1} + i{\omega _d} + {{\Omega _c^2} / {({Y_3} + i{\omega _d})}},$ ${M_3} = {Y_3} - i{\omega _d} + {{\Omega _c^2} / {({Y_1} - i{\omega _d}}}),$ and ${P_1} = {X_1} + {{{g^2}{{|\alpha |}^2}} / {{X_2}}}.$ $a_s^{(2)}(0)$ is associated with the SSK and $a_x^{(2)}(0)$ is associated with the CK between the probe field and mechanical mode. In the third-order case, we focus on $c_{31}^{(3)}(0)$ and the corresponding equations are given by
$$\begin{array}{l} \dot{c}_{31}^{(3)} ={-} {Y_1}c_{31}^{(3)} + i{G_{ac}}c_{21}^{(3)} + i{\Omega _p}(c_{11}^{(2)} - c_{33}^{(2)}) + i{g_{ac}}c_{21}^{(1)}{a^{(2)\ast }} + i{g_{ac}}c_{21}^{(2)}{a^{(1)\ast }},\\ \dot{c}_{21}^{(3)} ={-} {Y_3}c_{21}^{(3)} + i{G_{ac}}c_{31}^{(3)} + i{g_{ac}}c_{31}^{(1)}{a^{(2)}} + i{g_{ac}}c_{31}^{(2)}{a^{(1)}} - i{\Omega _p}c_{23}^{(2)}. \end{array}$$
The third-order results are given by
$$c_{31}^{(3)}(0) = c_{31 - s}^{(3)}(0) + c_{31 - x}^{(3)}(0),$$
$$c_{31 - s}^{(3)}(0) ={-} \frac{{i{\Omega _p}[2c_{33}^{(2)}(0) + c_{22}^{(2)}(0)]}}{{{M_1}}} + 2i{\textrm{Re}}[\alpha a_s^{(2)}(0)]\frac{{g_{ac}^{}c_{21}^{(1)}(0)}}{{{M_1}}},$$
$$c_{31 - x}^{(3)}(0) = 2i{\textrm{Re}}[\alpha a_x^{(2)}(0)]\frac{{g_{ac}^{}c_{21}^{(1)}(0)}}{{{M_1}}} - [\frac{{ig_{ac}^{}\alpha c_{21}^{(1)}(0)}}{{{Y_3}{M_2}}} + \frac{{c_{31}^{(1)}(0)}}{{{M_3}}}]\frac{{g_{ac}^2\alpha }}{{{M_1}}}{|{{a^{(1)}}({\omega_d})} |^2}.$$
Here the expression of $c_{31 - s}^{(3)}(0)$ is the same as Eq. (10) and $c_{31 - s}^{(3)}(0)$ can generate the corresponding third-order susceptibility term for SK effect. $c_{31 - x}^{(3)}(0)$ is relate to the phonon number of the mechanical vibration, which can generate the corresponding third-order susceptibility term for CK effect.

Funding

National Natural Science Foundation of China (11832016, 51775471); Changsha Zhuzhou Xiangtan landmark engineering technology project (2019XK2303); Xiangtan science 19 and technology project (ZD–ZD20191007); Xiangtan University graduate research and innovation 20 project (XDCX2020B127).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

References

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References

  • View by:

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  11. K. J. Boller, A. Imamoğlu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66(20), 2593–2596 (1991).
    [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]
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    [Crossref]
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    [Crossref]
  15. J. Li, Y. Qu, R. Yu, and Y. Wu, “Generation and control of optical frequency combs using cavity electromagnetically induced transparency,” Phys. Rev. A 97(2), 023826 (2018).
    [Crossref]
  16. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  20. G. Li, W. Nie, X. Li, and A. Chen, “Dynamics of ground-state cooling and quantum entanglement in a modulated optomechanical system,” Phys. Rev. A 100(6), 063805 (2019).
    [Crossref]
  21. A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10(9), 095008 (2008).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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2020 (1)

X. Xu, Y. Zhao, H. Wang, H. Jing, and A. Chen, “Quantum nonreciprocality in quadratic optomechanics,” Photonics Res. 8(2), 143 (2020).
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2019 (9)

D. Y. Wang, C. H. Bai, S. Liu, S. Zhang, and H. F. Wang, “Distinguishing photon blockade in a PT-symmetric optomechanical system,” Phys. Rev. A 99(4), 043818 (2019).
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B. Li, R. Huang, X. Xu, A. Miranowicz, and H. Jing, “Nonreciprocal unconventional photon blockade in a spinning optomechanical system,” Photonics Res. 7(6), 630 (2019).
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C. Shang, H. Z. Shen, and X. X. Yi, “Nonreciprocity in a strongly coupled three-mode optomechanical circulatory system,” Opt. Express 27(18), 25882 (2019).
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G. Li, W. Nie, X. Li, and A. Chen, “Dynamics of ground-state cooling and quantum entanglement in a modulated optomechanical system,” Phys. Rev. A 100(6), 063805 (2019).
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L. L. Zheng, T. S. Yin, Q. Bin, X. Y. Lü, and Y. Wu, “Single-photon-induced phonon blockade in a hybrid spin-optomechanical system,” Phys. Rev. A 99(1), 013804 (2019).
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B. Chen, L. Shang, X. F. Wang, J. Bin Chen, H. Bin Xue, X. Liu, and J. Zhang, “Atom-assisted second-order sideband generation in an optomechanical system with atom-cavity-resonator coupling,” Phys. Rev. A 99(6), 063810 (2019).
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H. Hao, M. C. Kuzyk, J. Ren, F. Zhang, X. Duan, L. Zhou, T. Zhang, Q. Gong, H. Wang, and Y. Gu, “Electromagnetically and optomechanically induced transparency and amplification in an atom-assisted cavity optomechanical system,” Phys. Rev. A 100(2), 023820 (2019).
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I. M. Mirza, W. Ge, and H. Jing, “Optical nonreciprocity and slow light in coupled spinning optomechanical resonators,” Opt. Express 27(18), 25515 (2019).
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F. Zou, L. B. Fan, J. F. Huang, and J. Q. Liao, “Enhancement of few-photon optomechanical effects with cross-Kerr nonlinearity,” Phys. Rev. A 99(4), 043837 (2019).
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2018 (9)

Y. Dong, D. Wang, Y. Wang, and J. Ding, “Matching group velocity of bright and/or dark solitons via double-dark resonances,” Phys. Lett. A 382(30), 2006–2012 (2018).
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A. Barg, L. Midolo, G. Kiršanskė, P. Tighineanu, T. Pregnolato, A. İmamoǧlu, P. Lodahl, A. Schliesser, S. Stobbe, and E. S. Polzik, “Carrier-mediated optomechanical forces in semiconductor nanomembranes with coupled quantum wells,” Phys. Rev. B 98(15), 155316 (2018).
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V. Villafañe, P. Sesin, P. Soubelet, S. Anguiano, A. E. Bruchhausen, G. Rozas, C. G. Carbonell, A. Lemaître, and A. Fainstein, “Optoelectronic forces with quantum wells for cavity optomechanics in GaAs/AlAs semiconductor microcavities,” Phys. Rev. B 97(19), 195306 (2018).
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Z. X. Liu, H. Xiong, and Y. Wu, “Generation and amplification of a high-order sideband induced by two-level atoms in a hybrid optomechanical system,” Phys. Rev. A 97(1), 013801 (2018).
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X. Y. Zhang, Y. H. Zhou, Y. Q. Guo, and X. 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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J. Li, S. Gröblacher, S. Y. Zhu, and G. S. Agarwal, “Generation and detection of non-Gaussian phonon-added coherent states in optomechanical systems,” Phys. Rev. A 98(1), 011801 (2018).
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H. Xie, C. G. Liao, X. Shang, Z. H. Chen, and X. M. Lin, “Optically induced phonon blockade in an optomechanical system with second-order nonlinearity,” Phys. Rev. A 98(2), 023819 (2018).
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B. Sarma and A. K. Sarma, “Unconventional photon blockade in three-mode optomechanics,” Phys. Rev. A 98(1), 013826 (2018).
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J. Li, Y. Qu, R. Yu, and Y. Wu, “Generation and control of optical frequency combs using cavity electromagnetically induced transparency,” Phys. Rev. A 97(2), 023826 (2018).
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2017 (2)

J. B. Clark, F. Lecocq, R. W. Simmonds, J. Aumentado, and J. D. Teufel, “Sideband cooling beyond the quantum backaction limit with squeezed light,” Nature 541(7636), 191–195 (2017).
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J. Lee, H. Zamani, S. Rajgopal, C. A. Zorman, and P. X. L. Feng, “3C-SiC microdisk mechanical resonators with multimode resonances at radio frequencies,” J. Micromech. Microeng. 27(7), 074001 (2017).
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2016 (5)

M. J. Burek, J. D. Cohen, S. M. Meenehan, N. El-Sawah, C. Chia, T. Ruelle, S. Meesala, J. Rochman, H. A. Atikian, M. Markham, D. J. Twitchen, M. D. Lukin, O. Painter, and M. Lončar, “Diamond optomechanical crystals,” Optica 3(12), 1404 (2016).
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H. S. Borges and C. J. Villas-Bôas, “Quantum phase gate based on electromagnetically induced transparency in optical cavities,” Phys. Rev. A 94(5), 052337 (2016).
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Z. Y. Liu, Y. H. Chen, Y. C. Chen, H. Y. Lo, P. J. Tsai, I. A. Yu, Y. C. Chen, and Y. F. Chen, “Large Cross-Phase Modulations at the Few-Photon Level,” Phys. Rev. Lett. 117(20), 203601 (2016).
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H. M. M. Alotaibi and B. C. Sanders, “Enhanced nonlinear susceptibility via double-double electromagnetically induced transparency,” Phys. Rev. A 94(5), 053832 (2016).
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W. Xiong, D. Y. Jin, Y. Qiu, C. H. Lam, and J. Q. You, “Cross-Kerr effect on an optomechanical system,” Phys. Rev. A 93(2), 023844 (2016).
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2015 (5)

R. Leijssen and E. Verhagen, “Strong optomechanical interactions in a sliced photonic crystal nanobeam,” Sci. Rep. 5(1), 15974 (2015).
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J. Zhang, T. Zhang, A. Xuereb, D. Vitali, and J. Li, “More nonlocality with less entanglement in a tripartite atom-optomechanical system,” Ann. Phys. 527(1-2), 147–155 (2015).
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X. Chen, Y. C. Liu, P. Peng, Y. Zhi, and Y. F. Xiao, “Cooling of macroscopic mechanical resonators in hybrid atom-optomechanical systems,” Phys. Rev. A 92(3), 033841 (2015).
[Crossref]

A. Jöckel, A. Faber, T. Kampschulte, M. Korppi, M. T. Rakher, and P. Treutlein, “Sympathetic cooling of a membrane oscillator in a hybrid mechanical–atomic system,” Nat. Nanotechnol. 10(1), 55–59 (2015).
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G. Kurizki, P. Bertet, Y. Kubo, K. Mølmer, D. Petrosyan, P. Rabl, and J. Schmiedmayer, “Quantum technologies with hybrid systems,” Proc. Natl. Acad. Sci. 112(13), 3866–3873 (2015).
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2014 (5)

J. Li, S. Zhang, R. Yu, D. Zhang, and Y. Wu, “Enhanced optical nonlinearity and fiber-optical frequency comb controlled by a single atom in a whispering-gallery-mode microtoroid resonator,” Phys. Rev. A 90(5), 053832 (2014).
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Y. Chen, Z. Bai, and G. Huang, “Ultraslow optical solitons and their storage and retrieval in an ultracold ladder-type atomic system,” Phys. Rev. A 89(2), 023835 (2014).
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M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
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C. Dong, J. Zhang, V. Fiore, and H. Wang, “Optomechanically induced transparency and self-induced oscillations with Bogoliubov mechanical modes,” Optica 1(6), 425 (2014).
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C. Strelow, S. Weising, D. Bonatz, J. P. Penttinen, T. V. Hakkarainen, A. Schramm, A. Mews, and T. Kipp, “Hybrid systems of AlInP microdisks and colloidal CdSe nanocrystals showing whispering-gallery modes at room temperature,” Appl. Phys. Lett. 105(9), 091107 (2014).
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2013 (5)

B. J. M. Hausmann, B. J. Shields, Q. Quan, Y. Chu, N. P. de Leon, R. Evans, M. J. Burek, A. S. Zibrov, M. Markham, D. J. Twitchen, H. Park, M. D. Lukin, and M. Loncǎr, “Coupling of NV Centers to Photonic Crystal Nanobeams in Diamond,” Nano Lett. 13(12), 5791–5796 (2013).
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D. Woolf, P. C. Hui, E. Iwase, M. Khan, A. W. Rodriguez, P. Deotare, I. Bulu, S. G. Johnson, F. Capasso, and M. Loncar, “Optomechanical and photothermal interactions in suspended photonic crystal membranes,” Opt. Express 21(6), 7258 (2013).
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A. H. Safavi-Naeini, S. Gröblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O. Painter, “Squeezed light from a silicon micromechanical resonator,” Nature 500(7461), 185–189 (2013).
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Y. 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).
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C. B. Fu, X. B. Yan, K. H. Gu, C. L. Cui, J. H. Wu, and T. D. Fu, “Steady-state solutions of a hybrid system involving atom-light and optomechanical interactions: Beyond the weak-cavity-field approximation,” Phys. Rev. A 87(5), 053841 (2013).
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2012 (2)

S. Ferretti and D. Gerace, “Single-photon nonlinear optics with Kerr-type nanostructured materials,” Phys. Rev. B 85(3), 033303 (2012).
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A. Dantan, M. Albert, and M. Drewsen, “All-cavity electromagnetically induced transparency and optical switching: Semiclassical theory,” Phys. Rev. A 85(1), 013840 (2012).
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2011 (4)

A. H. Safavi-Naeini, T. P. M. 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 472(7341), 69–73 (2011).
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A. Szorkovszky, A. C. Doherty, G. I. Harris, and W. P. Bowen, “Mechanical Squeezing via Parametric Amplification and Weak Measurement,” Phys. Rev. Lett. 107(21), 213603 (2011).
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X. Q. Luo, D. L. Wang, Z. Q. Zhang, J. W. Ding, and W. M. Liu, “Nonlinear optical behavior of a four-level quantum well with coupled relaxation of optical and longitudinal phonons,” Phys. Rev. A 84(3), 033803 (2011).
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Y. Chang, T. Shi, Y. Liu, C. P. Sun, and F. Nori, “Multistability of electromagnetically induced transparency in atom-assisted optomechanical cavities,” Phys. Rev. A 83(6), 063826 (2011).
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2010 (3)

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).
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G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010).
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M. Nomura, N. Kumagai, S. Iwamoto, Y. Ota, and Y. Arakawa, “Laser oscillation in a strongly coupled single-quantum-dot–nanocavity system,” Nat. Phys. 6(4), 279–283 (2010).
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2009 (3)

P. E. Barclay, C. Santori, K. M. Fu, R. G. Beausoleil, and O. Painter, “Coherent interference effects in a nano-assembled diamond NV center cavity-QED system,” Opt. Express 17(10), 8081 (2009).
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K. Hammerer, M. Wallquist, C. Genes, M. Ludwig, F. Marquardt, P. Treutlein, P. Zoller, J. Ye, and H. J. Kimble, “Strong Coupling of a Mechanical Oscillator and a Single Atom,” Phys. Rev. Lett. 103(6), 063005 (2009).
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S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460(7256), 724–727 (2009).
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2008 (4)

J. M. Dobrindt, I. Wilson-Rae, and T. J. Kippenberg, “Parametric Normal-Mode Splitting in Cavity Optomechanics,” Phys. Rev. Lett. 101(26), 263602 (2008).
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C. Genes, D. Vitali, and P. Tombesi, “Emergence of atom-light-mirror entanglement inside an optical cavity,” Phys. Rev. A 77(5), 050307 (2008).
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A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10(9), 095008 (2008).
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A. Schliesser, R. Rivière, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, “Resolved-sideband cooling of a micromechanical oscillator,” Nat. Phys. 4(5), 415–419 (2008).
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2007 (4)

I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of Ground State Cooling of a Mechanical Oscillator Using Dynamical Backaction,” Phys. Rev. Lett. 99(9), 093901 (2007).
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J. L. O’Brien, “Optical Quantum Computing,” Science 318(5856), 1567–1570 (2007).
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P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79(1), 135–174 (2007).
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K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoğlu, “Quantum nature of a strongly coupled single quantum dot–cavity system,” Nature 445(7130), 896–899 (2007).
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2006 (5)

C. Ottaviani, S. Rebić, D. Vitali, and P. Tombesi, “Quantum phase-gate operation based on nonlinear optics: Full quantum analysis,” Phys. Rev. A 73(1), 010301 (2006).
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Figures (10)

Fig. 1.
Fig. 1. Sketch of (a) the hybrid atom-assisted optomechanical system and (b) atomic energy-level configuration. The cavity is driven by a strong pumping field. The cavity field as a control field interacts with the atomic transition $|2 \rangle \leftrightarrow |3 \rangle $ with detuning ${\Delta _c},$ while a probe field interacts with the atomic transition $|1 \rangle \leftrightarrow |3 \rangle $ with detuning ${\Delta _p}.$
Fig. 2.
Fig. 2. The first term ${\textrm{Re}}[\chi _{{u_1}}^{^{(3)}}]$ and the second term ${\textrm{Re}}[\chi _{{u_2}}^{^{(3)}}]$ of the SK nonlinearity and the line absorption ${\mathop{\rm Im}\nolimits} [{\chi ^{(1)}}]$ vary with the atom-probe detuning ${\Delta _p}$ with $g = 0.$ Here ${\textrm{Re}}[\chi _{{u_1}}^{(3)}]$ as well as ${\textrm{Re}}[\chi _{{u_2}}^{(3)}]$ both take ${\eta _2}$ as a unit, whereas ${\mathop{\rm Im}\nolimits} [{\chi ^{(1)}}]$ takes ${\eta _1}$ as a unit. Other parameters are $\kappa = 0.1{\omega _m},$ $\gamma = {10^{ - 5}}{\omega _m},$ ${\Gamma _{32}} = {\Gamma _{31}} = 0.1{\omega _m},$ ${\Gamma _{21}} = 6 \times {10^{ - 4}}{\omega _m},$ $\Delta = {\omega _m},$ ${\Delta _c} = 0,$ ${g_{ac}} = 0.1{\omega _m},$ $|\alpha |= 2,$ ${\Omega _p} = {10^{ - 4}}{\omega _m},$ and $N = {10^5}.$
Fig. 3.
Fig. 3. The change of the populations of $|2 \rangle $ and $|3 \rangle $ and the real part of the cavity fluctuation as functions of the detuning ${\Delta _p}.$ The diagram of curves is the analytic results and the scattering dot is the numerical results. Other parameters are the same as those given in Fig. 2.
Fig. 4.
Fig. 4. The real and imaginary part of ${{c_{21}^{(1)}} / N}$ as functions of the detuning ${\Delta _p}.$ Other parameters are the same as those given in Fig. 2.
Fig. 5.
Fig. 5. The self-Kerr nonlinearity (a) ${\textrm{Re}}[\chi _{{u_1}}^{(3)}]$ (in ${\eta _2}$) and (b) ${\textrm{Re}}[\chi _{{u_2}}^{(3)}]$ (in ${\eta _2}$) as functions of the atom-probe detuning ${\Delta _p}$ with the atom-cavity coupling strength ${g_{ac}} = 0.03{\omega _m}$ for different mean cavity amplitudes $|\alpha |:$ $|\alpha |= 4,$ $|\alpha |= 6,$ and $|\alpha |= 8.$ Other parameters are the same as those given in Fig. 2.
Fig. 6.
Fig. 6. The self-Kerr nonlinearity (a) ${\textrm{Re}}[\chi _{{u_1}}^{(3)}]$ (in ${\eta _2}$) and (b) ${\textrm{Re}}[\chi _{{u_2}}^{(3)}]$ (in ${\eta _2}$) as functions of the atom-probe detuning ${\Delta _p}$ with the mean cavity amplitude $|\alpha |= 3$ for different atom-cavity coupling strengths ${g_{ac}}:$ ${g_{ac}} = 0.04{\omega _m},$ ${g_{ac}} = 0.06{\omega _m},$ and ${g_{ac}} = 0.08{\omega _m}.$ Other parameters are the same as those used in Fig. 2.
Fig. 7.
Fig. 7. The (a) real and (b) imaginary parts of complex numbers ${\omega _ \pm } - {\omega _m}$ as functions of the bare optomechanical coupling strength $g.$ Other parameters used are the same as in Fig. 2.
Fig. 8.
Fig. 8. The self-Kerr nonlinearities (a) ${\textrm{Re}}[\chi _{{u_ \pm }}^{(3)}]$ (in ${\eta _2}$) and (b) ${\textrm{Re}}[\chi _{{u_2}}^{(3)}]$ (in ${\eta _2}$) as functions of the bare optomechanical coupling strength g with the atom-probe detuning ${\Delta _p} = 0.2{\omega _m}.$ Other parameters are the same as those given in Fig. 2.
Fig. 9.
Fig. 9. The CK nonlinearity ${\textrm{Re}}[\chi _{xpm}^{(3)}]$ (in ${\eta _1}$) as a function of the probe detuning ${\Delta _p}$ under different cases when $g = 5 \times {10^{ - 4}}{\omega _m}:$ first, with $|\alpha |= 3$ and different ${g_{ac}},$ i.e., ${g_{ac}} = 0.04{\omega _m},$ ${g_{ac}} = 0.06{\omega _m},$ and ${g_{ac}} = 0.08{\omega _m};$ second, with ${g_{ac}} = 0.03{\omega _m}$ and different $|\alpha |,$ i.e., $|\alpha |= 4,$ $|\alpha |= 6,$ and $|\alpha |= 8.$ Other parameters are the same as in Fig. 2.
Fig. 10.
Fig. 10. (a) The CK nonlinearity ${\textrm{Re}}[\chi _{xpm}^{(3)}]$ (in ${\eta _1}$), (b) ${|{b({\omega_d})} |^2},$ and (c) ${\textrm{Re}}[\chi _{xpm}^{(3)}]{|{{b^{(1)}}({\omega_d})} |^2}$ (in ${\eta _1}$) as functions of the bare optomechanical coupling strength g and the mechanical drive frequency ${\omega _d}.$ Other parameters are the same as those given in Fig. 2.

Equations (48)

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H ^ = H ^ 0 + H ^ I ,
H ^ 0 = Δ a ^ a ^ + ω m b ^ b ^ Δ p σ ^ 33 Δ p c σ ^ 22 ,
H ^ c m = g a ^ a ^ ( b ^ + b ^ ) , H ^ a f = g a c ( a ^ σ ^ 23 + a ^ σ ^ 32 ) + Ω p ( σ ^ 31 + σ ^ 13 ) , H ^ d r = i Ω l ( a ^ a ^ ) .
a ^ ˙ = ( i Δ + κ 2 ) a ^ i g ( b ^ + b ^ ) a ^ i g a c σ ^ 23 + Ω l ,
b ^ ˙ = ( i ω m + γ 2 ) b ^ i g a ^ a ^ ,
σ ^ ˙ 31 = ( i Δ p + γ 31 2 ) σ ^ 31 + i g a c σ ^ 21 a ^ + i Ω p ( σ ^ 11 σ ^ 33 ) ,
σ ^ ˙ 23 = ( i Δ c γ 32 2 ) σ ^ 23 + i g a c ( σ ^ 33 σ ^ 22 ) a ^ i Ω p σ ^ 21 ,
σ ^ ˙ 21 = ( i Δ p c + γ 21 2 ) σ ^ 21 + i g a c σ ^ 31 a ^ i Ω p σ ^ 23 ,
σ ^ ˙ 11 = i Ω p ( σ ^ 31 σ ^ 13 ) + Γ 21 σ ^ 22 + Γ 31 σ ^ 33 ,
σ ^ ˙ 22 = i g a c ( a ^ σ ^ 32 a ^ σ ^ 23 ) Γ 21 σ ^ 22 + Γ 32 σ ^ 33 ,
σ ^ ˙ 33 = i g a c ( a ^ σ ^ 32 a ^ σ ^ 23 ) i Ω p ( σ ^ 31 σ ^ 13 ) Γ 3 σ ^ 33 .
σ ^ i j = l = 0 N c i j ( l ) , a ^ = α + l = 0 N a ( l ) , b ^ = β + l = 0 N b ( l ) .
c 31 ( 1 ) = i N Ω p M 1 , c 21 ( 1 ) = i g a c α c 31 ( 1 ) Y 3 , c 11 ( 1 ) = c 22 ( 1 ) = c 33 ( 1 ) = a ( 1 ) = b ( 1 ) = 0 ,
c 22 ( 2 ) = i Ω p Γ 31 g a c 2 | α | 2 ( c 31 ( 1 ) Y 3 c 31 ( 1 ) Y 3 ) ( c 31 ( 1 ) c 31 ( 1 ) ) ( 2 g a c 2 | α | 2 Γ 32 Y 2 ) 2 g a c 2 | α | 2 ( Γ 21 + Γ 31 ) Γ 21 Γ 32 Y 2 ,
c 33 ( 2 ) = i Ω p Γ 21 g a c 2 | α | 2 ( c 31 ( 1 ) Y 3 c 31 ( 1 ) Y 3 ) ( c 31 ( 1 ) c 31 ( 1 ) ) ( 2 g a c 2 | α | 2 Γ 21 Y 2 ) 2 g a c 2 | α | 2 ( Γ 21 + Γ 31 ) Γ 21 Γ 32 Y 2 ,
c 23 ( 2 ) = i g a c α ( c 22 ( 2 ) c 33 ( 2 ) ) Y 2 i Ω p c 21 ( 1 ) Y 2 , a ( 2 ) = i g a c P 1 c 23 ( 2 ) , c 31 ( 2 ) = c 21 ( 2 ) = 0 ,
c 31 ( 3 ) = u 1 + u 2 ,
u 1 = i Ω p ( 2 c 33 ( 2 ) + c 22 ( 2 ) ) ( Y 1 + g a c 2 | α | 2 Y 3 ) , u 2 = 2 Re [ α a ( 2 ) ] g a c 2 c 31 ( 1 ) Y 3 ( Y 1 + g a c 2 | α | 2 Y 3 ) .
χ ( 1 ) = η 1 c 31 ( 1 ) N Ω p = η 1 i Y 3 Y 1 Y 3 + Ω c 2 ,
χ u 1 ( 3 ) = η 2 i Ω p ( 2 c 33 ( 2 ) + c 22 ( 2 ) ) M 1 N Ω p 3 = i η 2 ( 2 Γ 21 Γ 31 ) L 1 + 3 L 2 Q M 1 ,
χ u 2 ( 3 ) = 2 η 2 Re [ α a ( 2 ) ] g a c 2 c 31 ( 1 ) M 1 Y 3 N Ω p 3 = 2 i η 2 N R e [ Q + L 1 Y 3 M 1 ( Γ 21 + Γ 31 ) P 1 Y 3 Y 2 Q M 1 ] g a c 2 Ω c 2 M 1 Y 3 .
a ( 2 ) = A + ω m + x 1 i κ γ 4 + A ω m x 1 i κ γ 4 = a + ( 2 ) + a ( 2 ) , A = g a c c 23 ( 2 ) [ 2 x 1 + i ( κ γ ) ] 8 x 1 , A + = g a c c 23 ( 2 ) [ 2 x 1 i ( κ γ ) ] 8 x 1 .
Re [ χ u 2 ( 3 ) ] = Re [ χ u + ( 3 ) ] + Re [ χ u ( 3 ) ] = Re [ α a + ( 2 ) ] L + Re [ α a ( 2 ) ] L ,
b ^ ˙ = ( i ω m + γ / 2 ) b ^ i g a ^ a ^ + ε d e ω d t .
a ( l ) = a ( l ) ( 0 ) + a ( l ) ( ω d ) e i ω d t + a ( l ) ( ω d ) e i ω d t + a ( l ) ( 2 ω d ) e 2 i ω d t + a ( l ) ( 2 ω d ) e 2 i ω d t + , b ( l ) = b ( l ) ( 0 ) + b ( l ) ( ω d ) e i ω d t + b ( l ) ( ω d ) e i ω d t + b ( l ) ( 2 ω d ) e 2 i ω d t + b ( l ) ( 2 ω d ) e 2 i ω d t + , c i j ( l ) = c i j ( l ) ( 0 ) + c i j ( l ) ( ω d ) e i ω d t + c i j ( l ) ( ω d ) e i ω d t + c i j ( l ) ( 2 ω d ) e 2 i ω d t + c i j ( l ) ( 2 ω d ) e 2 i ω d t + .
b ( 1 ) ( ω d ) = F 1 ε d , a x ( 2 ) ( 0 ) = [ g 2 | α | 2 X 2 | X 1 ( ω d ) | 2 + 1 X 1 ( ω d ) ] g 2 α | b ( 1 ) ( ω d ) | 2 P 1 , c 31  -  x ( 3 ) ( 0 ) = [ c 31 ( 1 ) ( 0 ) M 3 g a c 2 | α | 2 c 31 ( 1 ) ( 0 ) Y 3 2 M 2 ] g a c 2 g 2 | α | 2 M 1 | X 1 ( ω d ) | 2 | b ( 1 ) ( ω d ) | 2 2 Re [ α a x ( 2 ) ( 0 ) ] g a c 2 c 31 ( 1 ) ( 0 ) M 1 Y 3 .
χ x p m ( 3 ) = 2 η 1 i Re [ 1 P 1 X 1 ( ω d ) + g 2 | α | 2 X 2 P 1 | X 1 ( ω d ) | 2 ] g 2 Ω c 2 Y 3 M 1 2 + i η 1 ( Ω c 2 M 3 Y 3 2 M 2 ) g a c 2 g 2 | α | 2 Y 3 2 M 1 M 2 M 3 | X 1 ( ω d ) | 2 .
H ^ x p m = g n l a ^ a ^ b ^ b ^ ,
g n l = D 2 ω p 2 d r | α ( r ) | 2 Re [ χ j i k ( 3 ) ( r ) ] ε ( r ) = D ω p 4 ε ¯ r χ ¯ x p m .
a ˙ ( 1 ) = X 1 a ( 1 ) i g α b ( 1 ) i g a c c 23 ( 1 ) , b ˙ ( 1 ) = X 2 b ( 1 ) i g α a ( 1 ) + ε d e i ω d t , c ˙ 31 ( 1 ) = Y 1 c 31 ( 1 ) + i g a c α c 21 ( 1 ) + i Ω p N , c ˙ 23 ( 1 ) = Y 2 c 23 ( 1 ) + i g a c α ( c 33 ( 1 ) c 22 ( 1 ) ) , c ˙ 21 ( 1 ) = Y 3 c 21 ( 1 ) + i g a c α c 31 ( 1 ) , c ˙ 22 ( 1 ) = i ( g a c α c 32 ( 1 ) g a c α c 23 ( 1 ) ) Γ 21 c 22 ( 1 ) + Γ 32 c 33 ( 1 ) , c ˙ 33 ( 1 ) = i ( g a c α c 32 ( 1 ) g a c α c 23 ( 1 ) ) Γ 3 c 33 ( 1 ) , c 11 ( 1 ) + c 22 ( 1 ) + c 33 ( 1 ) = 0.
a ( 1 ) = a ( 1 ) ( ω d ) e i ω d t = i g α X 1 ( ω d ) b ( 1 ) , b ( 1 ) = b ( 1 ) ( ω d ) e i ω d t = F 1 ε d e i ω d t ,
c 22 ( 1 ) = c 33 ( 1 ) = c 23 ( 1 ) = 0 , c 21 ( 1 ) = c 21 ( 1 ) ( 0 ) = i g a c α c 31 ( 1 ) ( 0 ) Y 3 , c 31 ( 1 ) = c 31 ( 1 ) ( 0 ) = i N Ω p M 1 ,
a ˙ ( 2 ) = X 1 a ( 2 ) i g α b ( 2 ) i g a c c 23 ( 2 ) i g a ( 1 ) b ( 1 ) i g a ( 1 ) b ( 1 ) , b ˙ ( 2 ) = X 2 b ( 2 ) i g α a ( 2 ) i g | a ( 1 ) | 2 , c ˙ 31 ( 2 ) = Y 1 c 31 ( 2 ) + i g a c α c 21 ( 2 ) + i g a c a ( 1 ) c 21 ( 1 ) , c ˙ 21 ( 2 ) = Y 3 c 21 ( 2 ) + i g a c α c 31 ( 2 ) + i g a c a ( 1 ) c 31 ( 1 ) , c ˙ 23 ( 2 ) = Y 2 c 23 ( 2 ) + i g a c α ( c 33 ( 2 ) c 22 ( 2 ) ) i Ω p c 21 ( 1 ) , c ˙ 22 ( 2 ) = i g a c α ( c 32 ( 2 ) c 23 ( 2 ) ) Γ 21 c 22 ( 2 ) + Γ 32 c 33 ( 2 ) , c ˙ 33 ( 2 ) = i g a c α ( c 32 ( 2 ) c 23 ( 2 ) ) i Ω p ( c 31 ( 1 ) c 13 ( 1 ) ) Γ 3 c 33 ( 2 ) , c 11 ( 2 ) + c 22 ( 2 ) + c 33 ( 3 ) = 0.
c 33 ( 2 ) = c 33 ( 2 ) ( 0 ) , c 22 ( 2 ) = c 22 ( 2 ) ( 0 ) , c 23 ( 2 ) = c 23 ( 2 ) ( 0 ) ,
c 21 ( 2 ) = c 21 ( 2 ) ( ω d ) e i ω d t + c 21 ( 2 ) ( ω d ) e i ω d t , c 31 ( 2 ) = c 31 ( 2 ) ( ω d ) e i ω d t + c 31 ( 2 ) ( ω d ) e i ω d t ,
a ( 2 ) = a ( 2 ) ( 0 ) + a ( 2 ) ( 2 ω d ) e 2 i ω d t ,
c 22 ( 2 ) ( 0 ) = i Ω p Γ 31 g a c 2 | α | 2 [ c 31 ( 1 ) ( 0 ) Y 3 c 31 ( 1 ) ( 0 ) Y 3 ] [ c 31 ( 1 ) ( 0 ) c 31 ( 1 ) ( 0 ) ] ( 2 g a c 2 | α | 2 Γ 32 Y 2 ) 2 g a c 2 | α | 2 ( Γ 21 + Γ 31 ) Γ 21 Γ 32 Y 2 ,
c 33 ( 2 ) ( 0 ) = i Ω p Γ 21 g a c 2 | α | 2 [ c 31 ( 1 ) ( 0 ) Y 3 c 31 ( 1 ) ( 0 ) Y 3 ] [ c 31 ( 1 ) ( 0 ) c 31 ( 1 ) ( 0 ) ] ( 2 g a c 2 | α | 2 Γ 21 Y 2 ) 2 g a c 2 | α | 2 ( Γ 21 + Γ 31 ) Γ 21 Γ 32 Y 2 ,
c 23 ( 2 ) ( 0 ) = i g a c α [ c 22 ( 2 ) ( 0 ) c 33 ( 2 ) ( 0 ) ] Y 2 + i Ω p c 21 ( 1 ) ( 0 ) Y 2 ,
c 21 ( 2 ) ( ω d ) = i g a c a ( 1 ) ( ω d ) c 31 ( 1 ) ( 0 ) M 3 , c 31 ( 2 ) ( ω d ) = i g a c α c 21 ( 2 ) ( ω d ) Y 1 i ω d ,
c 31 ( 2 ) ( ω d ) = i g a c a ( 1 ) ( ω d ) c 21 ( 1 ) ( 0 ) M 2 , c 21 ( 2 ) ( ω d ) = i g a c α c 31 ( 2 ) ( ω d ) Y 3 + i ω d ,
a ( 2 ) ( 0 ) = a s ( 2 ) ( 0 ) + a x ( 2 ) ( 0 ) ,
a s ( 2 ) ( 0 ) = i g a c c 23 ( 2 ) ( 0 ) P 1 , a x ( 2 ) ( 0 ) = i g P 1 a ( 1 ) ( ω d ) b ( 1 ) ( ω d ) g 2 α X 2 P 1 | a ( 1 ) ( ω d ) | 2 ,
a ( 2 ) ( 2 ω d ) = i g P 1 a ( 1 ) ( ω d ) b ( 1 ) ( ω d ) .
c ˙ 31 ( 3 ) = Y 1 c 31 ( 3 ) + i G a c c 21 ( 3 ) + i Ω p ( c 11 ( 2 ) c 33 ( 2 ) ) + i g a c c 21 ( 1 ) a ( 2 ) + i g a c c 21 ( 2 ) a ( 1 ) , c ˙ 21 ( 3 ) = Y 3 c 21 ( 3 ) + i G a c c 31 ( 3 ) + i g a c c 31 ( 1 ) a ( 2 ) + i g a c c 31 ( 2 ) a ( 1 ) i Ω p c 23 ( 2 ) .
c 31 ( 3 ) ( 0 ) = c 31 s ( 3 ) ( 0 ) + c 31 x ( 3 ) ( 0 ) ,
c 31 s ( 3 ) ( 0 ) = i Ω p [ 2 c 33 ( 2 ) ( 0 ) + c 22 ( 2 ) ( 0 ) ] M 1 + 2 i Re [ α a s ( 2 ) ( 0 ) ] g a c c 21 ( 1 ) ( 0 ) M 1 ,
c 31 x ( 3 ) ( 0 ) = 2 i Re [ α a x ( 2 ) ( 0 ) ] g a c c 21 ( 1 ) ( 0 ) M 1 [ i g a c α c 21 ( 1 ) ( 0 ) Y 3 M 2 + c 31 ( 1 ) ( 0 ) M 3 ] g a c 2 α M 1 | a ( 1 ) ( ω d ) | 2 .

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