Perfect photon absorption in hybrid atom-optomechanical system

It is well known that photons as the carrier of the information, play an important role in quantum information and quantum communication. Recently, the photon absorption attracts a great deal of attention and has aroused widespread interest in recent studies [1–5]. Normally, the coherent perfect absorption corresponding to a certain frequency is determined by the intrinsic properties of the medium and the absorption of the input light was reported in refs. [6,7]. Recently, the coherent perfect photon absorption using path entanglement was also theoretically studied in ref. [5] and then was demonstrated experimentally by Faccio et al. [8]. The physical basis behind it is the destructive interference between the two input fields [7,9]. It is shown that the perfect photon absorption has potential applications in optics communications and photonic devices including transducers, modulators, and optical switches and transistors [2,4,10–14]. Therefore, some fundamental efforts have also been made on the achievement and applications of perfect photon absorption in various systems such as whispering-gallery-mode micro-resonators [15], the coupled atom-resonator-waveguide system [16], second harmonic generation [17], nanostructured graphene film [18] and strongly scattering media [19] and so on. In particular, the perfect photon absorption has also been studied not only in the cavity quantum electrodynamics [10,20] but also in the cavity optomechanical system (COM) which enables the coupling between the mechanical modes and the optical field via the radiation pressure and attracts a lot of interest due to the rich physics [21–23] and applications, e.g., ground-state cooling [24], quantum coherent state transfer [25], ponderomotive squeezing [26], quantum entanglement [27], optical Kerr nonlinearity [28], and gravitational-wave physics [29,30]. However, it is unclear whether an optomechanical interaction can be used to enhance or control the properties of coherent photon absorption afforded by two-level atoms.

In this paper, we study the perfect photon absorption as well as the optical bistability/multistability in a hybrid atom-optomechanical system which couples to an ensemble of two-level atoms. Our results show that, the perfect photon absorption can be implemented in this system but always accompanied with the bistability or multistability of the output fields. In particular, we find that in such a hybrid optomechanical system, the coupling with atomic ensemble is the sufficient and necessary condition for the perfect photon absorption, while the optomechanical coupling which enhances the nonlinearity forms the necessary condition for the optical bistability/multistability and especially can induce an additional perfect absorption point by overlap-adding the nonlinear atomic excitation. In addition, we also show that the photon absorption and optical bistability (multistability) can be modified by the interference of the two input fields, hence the bistability (multistability) and perfect absorption can be controlled by changing the relative phase between them, which provides a possible design of an optical switch.

As schematically shown in fig. 1, we consider a typical hybrid atom-optomechanical system which consists of a bare optical cavity coupled to a mechanical mode and N identical two-level atoms trapped in the cavity. The atomic transition frequency and the atomic line width are denoted by $\omega _{e}$ and Γ, respectively. In the rotating frame of the input laser frequency, the full Hamiltonian with regard for the driving and the dissipation, is given by (set $\hbar =1$ hereafter)

$$\begin{eqnarray} H &= \Delta _{a}a^{\dag }a+\delta S^{z}+\omega _{m}b^{\dag}b+g(aS^{+}+a^{\dag }S^{-}) \nonumber \\ &\quad -\,g_{0}a^{\dag }a(b^{\dag }+b)+i\sqrt{2\kappa _{1}}a_{1in}a^{\dag }+i\sqrt{2\kappa _{2}}a_{2in}a^{\dag } \qquad\nonumber\\ &\quad -\,i\sqrt{2\kappa _{1}}a_{1in}^{\dag }a-i\sqrt{2\kappa _{2}}a_{2in}^{\dag }a, \end{eqnarray} \tag{ 1 }$$

where a denotes the cavity mode, and $S^{z}=\frac{1}{2}\sum_{i=1}^{N}(\sigma _{i}^{+} \sigma _{i}^{-}-\sigma _{i}^{-}\sigma _{i}^{+})$ and $S^{\pm}=\sum_{i=1}^{N}\sigma _{i}^ {\pm }$ stand for the collective atomic operators with $\sigma _{i}^{+}=\vert e\rangle _{i}\langle g\vert,\sigma _{i}^{-}=\vert g\rangle _{i}\langle e\vert$ representing the i-th atomic raising and lowering operators. Here a_1in and a_2in are two input fields upon the cavity with the same frequency $\omega _{l}$. $\Delta _{a}=\omega _{a}- \omega _{l}$ is the laser detuning from the cavity mode and $\delta =\omega _{e}-\omega _{l}$ is the laser detuning from the atoms. Based on Hamiltonian (1) and the potential dissipation processes, one can obtain the dynamics of operators as follows:

$$\begin{eqnarray} \dot{b}&= -i\omega _{m}b-\gamma b-ig_{0}a^{\dag }a+\sqrt{2\gamma }b^{in}, \end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray} \dot{S}^{-}&= -i\delta S^{-}+2igaS^{z}-\frac{\Gamma }{2}S^{-}+\sqrt{\Gamma}S^{-in}, \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray} \dot{S}^{z}&= -\Gamma S^{z}-igaS^{+}+iga^{\dag }S^{-}-\frac{N\Gamma }{2}+\sqrt{\Gamma }S^{zin}, \end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray} \dot{a}&= -i\Delta _{a}a-(\kappa _{1}+\kappa_{2})a-igS^{-}-ig_{0}a(b^{\dag}+b)\nonumber\\ &\quad +\,\sqrt{2\kappa _{1}}a_{1in}+\sqrt{2\kappa _{2}}a_{2in}+\sqrt{2(\kappa_{1}+\kappa _{2})}a^{in}. \qquad \end{eqnarray} \tag{ 5 }$$

Here the operators bⁱⁿ, $S^{-in}$, S^zin, and aⁱⁿ denote environmental noises corresponding to the operators b, $S^{-}$, S^z and a. We assume that the mean values of the above noise operators are zero, that is, in the classical limit, we drop the quantum fluctuations and replace the operators by their expectation values. One can obtain the equations of motion of the expectation values of the operators as follows:

$$\begin{eqnarray} \left\langle \dot{b}\right\rangle&= -i\omega _{m}\left\langle b\right\rangle-\gamma \left\langle b\right\rangle-ig_{0}\left\langle a^{\dag}a\right\rangle, \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} \left\langle \dot{S}^{-}\right\rangle &= -i\delta \left\langle S^{-}\right\rangle +2ig \left\langle aS^{z}\right\rangle -\frac{\Gamma}{2}\left\langle S^{-}\right\rangle, \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray} \left\langle \dot{S}^{z}\right\rangle&= -\Gamma \left\langle S^{z}\right\rangle -ig \left\langle aS^{+}\right\rangle +ig\left\langle a^{\dag }S^{-}\right\rangle -\frac{N\Gamma} {2}, \end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray} &\left\langle \dot{a}\right\rangle = \!-i\Delta _{a}\left\langle a\right\rangle \!-\!(\kappa _{1}\!+\!\kappa _{2})\left\langle a\right\rangle \!-\!ig\left\langle S^{-}\right\rangle\nonumber\\ & -\,ig_{0}\left\langle a\right\rangle \left(\left\langle b^{\dag } \right\rangle +\,\left\langle b\right\rangle \right)+\sqrt{2\kappa _{1}}a_{1in}+\sqrt{2\kappa _{2}}a_{2in}, \qquad \end{eqnarray} \tag{ 9 }$$

where $\langle \cdots \rangle$ represents the expectation value over the steady state and $\langle oo^{\prime}\rangle =\langle o\rangle \langle o^{\prime}\rangle$ ($o,o^{\prime}$ stand for the operators b, $S^{-}$, S^z and a). Combining above equations, we can solve the steady-state solution and obtain the intra-cavity field as

$$\begin{equation} \left\langle a\right\rangle =\frac{\sqrt{2\kappa _{1}}a_{1in}+\sqrt{2\kappa _{2}}a_{2in}} {i\Delta _{a}+\kappa -\frac{g^{2}N}{\frac{N}{2\left\langle S^{z}\right\rangle } (\frac{\Gamma }{2}+i\delta )}-i\Xi \left\vert a\right\vert ^{2}}, \end{equation} \tag{ 10 }$$

where $\kappa =\kappa _{1}+\kappa _{2}$, $\langle S^{z}\rangle =-\frac{N}{2(1+\frac{2g^{2} \vert a\vert ^{2}}{\frac{\Gamma ^{2}}{4}+\delta ^{2}})}$ and $\Xi =\frac{2\omega _{m}g_{0}^{2}}{\gamma ^{2}+\omega_{m}^{2}}$. From eq. (10), one can easily get the intra-cavity intensity $\vert a\vert ^{2}$ and, hence, obtain the properties of the photon absorption.

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We first consider that the atomic ensemble is only in the low excitation regime. In this case, one can set $\langle S^{z}\rangle \approx -\frac{N}{2}$ [31], so eq. (10) can be reduced to

$$\begin{equation} \left\langle a\right\rangle =\frac{\sqrt{2\kappa _{1}}a_{1in}+\sqrt{2\kappa _{2}}a_{2in}} {i\Delta _{a}+\kappa +\frac{g^{2}N}{\frac{\Gamma }{2}+i\delta}-i\Xi \left\vert a\right\vert ^{2}}. \end{equation} \tag{ 11 }$$

It is obvious that eq. (11) includes the intra-cavity intensity $\vert a\vert ^{2}$ and demonstrates the nonlinear dependence of the intra-cavity intensity. So the system may exhibit the optical bistability under a certain parameter range. In order to find the steady states of the two output light fields, one will have to consider the following input-output relation [32]:

$$\begin{eqnarray} a_{1\textit{out}}&= \sqrt{2\kappa _{1}}a-a_{1in}, \end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray} a_{2\textit{out}}&= \sqrt{2\kappa _{2}}a-a_{2in}. \end{eqnarray} \tag{ 13 }$$

One can assume that $a_{1in}=\vert a_{in}\vert$, $a_{2in}=e^{i\varphi}\vert a_{in}\vert$, where φ denotes the relative phase of the two opposite input fields. We also consider a symmetric cavity with $\kappa _{1}=\kappa _{2}=\frac{\kappa}{2}$, then the output intensity can be calculated as

$$\begin{eqnarray} \frac{\left\vert a_{1\textit{out}}\right\vert ^{2}}{\left\vert a_{in}\right\vert ^{2}} &= \left\vert \frac{\kappa (1+e^{i\varphi })}{i\Delta _{a}+\kappa +\frac{g^{2}N} {\frac{\Gamma}{2}+i\delta }-i\Xi \left\vert a\right\vert ^{2}}-1\right\vert ^{2}, \end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray} \frac{\left\vert a_{2\textit{out}}\right\vert ^{2}}{\left\vert a_{in}\right\vert ^{2}} &= \left\vert \frac{\kappa (1+e^{i\varphi })}{i\Delta _{a}+\kappa +\frac{g^{2}N} {\frac{\Gamma}{2}+i\delta }-i\Xi \left\vert a\right\vert ^{2}}-e^{i\varphi }\right\vert ^{2}. \qquad \end{eqnarray} \tag{ 15 }$$

Let us now focus our attention on how to realize the perfect photon absorption. The perfect photon absorption implies $a_{1\textit{out}}=a_{2\textit{out}}=0$. One can find that if $\varphi \neq 2k\pi$, $k=0,\pm 1,\pm 2,\cdots$, the perfect photon absorption will not happen at any rate. So we restrict ourselves under the condition $\varphi =0$ without loss of generality. Thus we can get two specific conditions for the perfect absorption:

$$\begin{eqnarray} \frac{\kappa}{\Gamma}&= \frac{2g^{2}N}{\Gamma ^{2}+4\delta ^{2}}, \end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray} \left(\frac{\kappa \Gamma}{2}+\Delta _{a}\delta -g^{2}N\right)&= \delta \Xi \left\vert a\right\vert ^{2}. \end{eqnarray} \tag{ 17 }$$

By comparing with ref. [20], one can find that the two perfect trapping conditions presented in ref. [20] can be summarized by our eq. (16), which specifies relation between the atomic coupling rate and other atomic parameters. It indicates the importance of the participation of the atomic ensemble for the perfect absorption. However, eq. (17) is our distinguishing condition which is the result of the optomechanical interaction and specifies the relation between the atomic coupling rate and the COM parameters. It shows that no output light from the cavity is accompanied by the nonzero intra-cavity light intensity $\vert a\vert^2$. This provides the possibility to realize the perfect photon absorption by tuning the input light intensities (controlling the intra-cavity light intensity). As shown in eq. (11), the intra-cavity photon number is determined by both the atomic ensemble and the optomechanical parameters, thus we can harness them to control the optical bistable behavior and the photon absorption. For example, we can effectively control the parameters g²N and the atomic frequency to match eq. (16) and control the intra-cavity intensity to match eq. (17). One should note that it is difficult to provide the analytical way to solve eq. (11), we opt to provide the numerical calculations to characterize the perfect photon absorption and the optical bistability [10].

In fig. 2(a) and fig. 2(b) we display the intra-cavity intensity $\vert a\vert ^{2}$ and the input light intensity $\vert a_{in}\vert ^{2}$ as a function of the fitting parameter $g^{2}N/\kappa ^{2}$ under the condition of the perfect photon absorption. From fig. 2(a), one can see that, due to the nonlinear dependence of the input intensity, $\vert a\vert ^{2}$ exhibits bistability if the input intensity is large enough. In addition, even though the large input intensity can make the optomechanical system show apparent bistability, this could require a relatively strong coupling ($g^{2}N/\kappa^{2}\sim 125$ in fig. 2(a)). Figure 2(b) shows the monotonic relation between $\vert a_{in}\vert ^{2}$ and $g^{2}N/\kappa ^{2}$ to maintain the perfect photon absorption for a fixed $\Delta_a/ \kappa$. In addition, for nontrivial solutions, $\vert a_{in}\vert >0$ requires that besides the condition given in eq. (16), the perfect absorption can only occur in the following ranges: for $\Delta _{a}<0$,

$$\begin{equation} g^{2}N>\frac{\Delta _{a}^{2}+\kappa \Gamma +\sqrt{\Delta _{a}^{4}+2\kappa \Gamma \Delta _{a}^{2}-4\Delta _{a}^{2}}}{2}; \end{equation} \tag{ 18 }$$

and for $\Delta _{a}>0$,

$$\begin{equation} g^{2}N>\frac{\kappa \Gamma }{2}. \end{equation} \tag{ 19 }$$

The numerical illustration of these ranges is also given in fig. 2(b). Both the ranges indicate the strong coupling g²N. This implies that the implementation of the optical bistability is easier than the perfect absorption, since the perfect absorption needs the participation of atoms. In fact, one can see that the requirement of strong coupling can also be compensated for by adding more atoms, or weakening the leakage κ.

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In what follows, we present the result for the two output light intensities $\vert a_{1\textit{out}}\vert ^{2}$ and $\vert a_{2\textit{out}}\vert^{2}$ for the hybrid system in fig. 3(a) and fig. 3(b). One can find that the perfect photon absorption occurs solely at a particular input intensity $\vert a_{in}\vert ^{2}=\kappa \delta \Xi /(\frac{\kappa \Gamma }{2}+\Delta _{a}\delta -g^{2}N)$ with the relative phase $\varphi =2n\pi$, $n=0,\pm 1,\pm 2\cdots$ (the black "star" lines in fig. 3). Interestingly, we find that once the perfect absorption occurs, the output field intensity exhibits optical bistability where we can get three real distinct values for the output field intensity $\vert a_{\textit{out}}\vert ^{2}$ due to the nonlinear equation, eq. (11). Based on eq. (16) and eq. (17), one can find that the perfect absorption cannot occur if g = 0. Similarly, the optical bistability will vanish if $g_{0}=0$, even though the perfect absorption could still be present, which is consistent with ref. [20]. In this sense, the perfect photon absorption mainly relies on the coupling between the atomic ensemble and the cavity, whereas the nonlinear optomechanical coupling plays the dominant role in the optical bistability [21]. It is different from ref. [10] where the optical bistability relies on the nonlinear atomic excitation. In addition, the two output intensities can also be modified by the interference of the two identical input lights, which can be shown by fig. 3(a) and (b). In fig. 3, the blue "o" lines correspond to $\varphi =\pi/2$ and the red solid lines correspond to $\varphi =\pi$. One can find that the bistability disappears $(\varphi =\pi)$ or the system is driven to the complicated bistable domain $(\varphi =\pi /2)$ by the effect of the relative phase, but the perfect absorption is absent. In other words, with φ changing from π to 0, the hybrid optomechanical system is driven from the monostable domain to the complicated bistable domain. During this procedure, one particular bistability results in the perfect absorption as shown in fig. 3.

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In fig. 4, we plot the output light intensities vs. $\vert a_{in} \vert ^{2}$ for different cavity-laser detunings. One can see that the perfect photon absorption and the optical bistability are present in a large range of the atom-cavity detuning $(\Delta=\Delta_a-\delta)$, but different Δ requires different input light intensities to match the special conditions of the perfect absorption.

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Figure 3(a) and fig. 3(b) have shown that the relative phase has a deep influence on the photon absorption and the optical bistability. As mentioned above, the hybrid atom-optomechanical system could be out of the bistable domain and in particular, the perfect photon absorption does not occur due to the non-vanishing effect of the relative phase. To give an intuitive illustration, it is imperative to plot the output light intensity vs. the relative phase φ. In fig. 5 we plot the two output light intensities $\vert a_{1\textit{out}}\vert ^{2}$ and $\vert a_{2\textit{out}}\vert ^{2}$ with $\vert a_{1in}\vert^{2}=\vert a_{in}\vert ^{2}$, $\vert a_{2in}\vert^{2}=e^{i\varphi }\vert a_{in}\vert ^{2}$. One can see that the two output fields are not equal to each other except at some particular φ. These φ can be determined by solving eq. (14) and eq. (15) associated with eq. (11), and thus one can have $\sin {\varphi }=0$ or $\cos \varphi =c$, where

$$\begin{equation} c=\left(\kappa +\frac{g^{2}N\Gamma }{2(\frac{\Gamma ^{2}}{4}+\delta ^{2})}\right)^{2} \frac{\frac{\kappa \Gamma }{2}+\delta \Delta _{a}-g^{2}N}{2\kappa \Xi \delta _{0}^{2}| a_{in}|^{2}}-1. \end{equation} \tag{ 20 }$$

It is obvious that at $\sin{\varphi}=0$, the system demonstrates the perfect photon absorption, but at $\cos {\varphi }=c$ no perfect photon absorption is shown, even though equal output light intensities were present. In addition, some parameters including the driving, the detuning and so on could lead to $|c|>1$, so $\cos{\varphi}=c$ cannot hold any more. This case is illustrated in fig. 5(a) where the input intensity $\vert a_{in}\vert ^{2}$ is set below the threshold of the bistable regime, hence $\vert a_{1\textit{out}}\vert ^{2}$ and $\vert a_{2\textit{out}}\vert ^{2}$ only show the mono-stability. On the contrary, when we adjust the parameters to satisfy the condition $|c|\leq 1$, the output light intensities could show the bistability, which is plotted in fig. 5(b). In this case, the phase can be used to manipulate the bistability and obtain various complicated bistable patterns in a certain region.

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The consequences within the low-excitation limit have been studied in the previous section. According to the results mentioned above, a natural question will arise, thus, on whether stronger and more robust nonlinearity can be achieved when the atomic assemble is driven into the nonlinear excitation regime. We expect that the nonlinearity caused by the atomic nonlinear excitation can be used to enhance the cavity optical nonlinearity, or has a positive effect on the perfect photon absorption in the hybrid atom-optomechanical system.

In the following, we will give the details on the coupling between the atomic ensemble and the cavity field in the nonlinear regime. Thus, in eq. (10), we will have to directly employ $\langle S^{z}\rangle=-\frac{N}{2(1+\frac{2g^{2}\vert a\vert ^{2}} {\frac{\Gamma ^{2}}{4}+\delta ^{2}})}$ instead of $\langle S^{z}\rangle=-\frac{N}{2}$ to calculate the steady-state solution of the intra-cavity field from eqs. (6)–(9). At this moment, the $\langle S^{z}\rangle$ is a function of the intra-cavity field intensity $\vert a\vert^{2}$ which satisfies the condition $\vert a\vert ^{2}>\frac{\Gamma ^{2}}{4g^{2}}$. Therefore, two output optical intensity $\vert a_{1\textit{out}}\vert ^{\prime 2}$ and $\vert a_{2\textit{out}}\vert^{\prime2}$ can be given by

$$\begin{eqnarray} \frac{\left\vert a_{1\textit{out}}\right\vert ^{\prime 2}}{\left\vert a_{in}\right\vert ^{2}}&= \left\vert \frac{\kappa (1+e^{i\varphi })}{i\Delta_{a}+\kappa +\frac{g^{2}N} {(\frac{\Gamma}{2}+i\delta )(1+\frac{2g^{2}\left\vert a\right\vert ^{2}}{\frac{\Gamma ^{2}}{4}+\delta ^{2}})}-i\Xi \left\vert a\right\vert ^{2}}-1\right\vert ^{2}\!, \nonumber\\ \end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray} \frac{\left\vert a_{2out}\right\vert ^{\prime 2}}{\left\vert a_{in}\right\vert ^{2}} &= \left\vert \frac{\kappa (1+e^{i\varphi })}{i\Delta_{a}+\kappa +\frac{g^{2}N} {(\frac{\Gamma }{2}+i\delta )(1+\frac{2g^{2}\left\vert a\right\vert ^{2}}{\frac{\Gamma ^{2}}{4}+\delta ^{2}})}-i\Xi \left\vert a\right\vert ^{2}}-e^{i\varphi }\right\vert ^{2}\!\!.\nonumber\\ \end{eqnarray} \tag{ 22 }$$

In order to show the perfect photon absorption, we would like to first list the corresponding conditions similar to the case in low-excitation limit. With $\varphi =0$, the conditions parallel with eqs. (16) and (17) can be straightforwardly written as

$$\begin{eqnarray} \frac{\kappa \Gamma}{2}+\Delta _{a}\delta &= \frac{g^{2}N}{1+\frac{2g^{2}\left\vert a\right\vert ^{2}}{\frac{\Gamma ^{2}}{4}+\delta ^{2}}}+\delta \Xi \left\vert a\right\vert ^{2}, \end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray} \frac{\Delta _{a}\Gamma }{2}-\kappa \delta &= \frac{\Gamma \Xi }{2}\left\vert a\right\vert ^{2}. \end{eqnarray} \tag{ 24 }$$

Compared with the perfect absorption conditions of the CQED system in ref. [10], one can find that both the nonlinear cavity-mechanical coupling and the atom-cavity coupling have a certain influence on the perfect photon absorption in the nonlinear-coupling regime. In particular, the optomechanical properties are embodied in both conditions in contrast to eqs. (16) and (17). When eqs. (23) and (24) are valid, the output intensity vanishes and there is no output light from the cavity, this means that the hybrid system can be regarded as a nonlinear absorber [10].

In addition, from eq. (10), one can find that the intra-field intensity $\vert a\vert ^{2}$ satisfies a quintic equation. Let $\vert a_{in}^{1}\vert^{2}=\vert a_{in}^{2}\vert^{2}=\vert a_{in}\vert ^{2}~(\varphi =0)$ and $x:=\vert a\vert^{2}$, the quintic equation reads

$$\begin{equation} A_{5}x^{5}+A_{4}x^{4}+A_{3}x^{3}+A_{2}x^{2}+A_{1}x+A_{0}=0, \end{equation} \tag{ 25 }$$

where

$$\begin{eqnarray} A_{0}&:= -\left\vert \tilde{\Gamma}\right\vert ^{2}\left\vert a_{in}\right\vert ^{2}, \end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray} A_{1}&:= (\tilde{\kappa}\tilde{\Gamma}+g^{2}N)(\tilde{\kappa}^{\ast }\tilde{\Gamma}^{\ast} +g^{2}N)-2\left\vert \tilde{\Gamma}\right\vert ^{2}\tilde{\delta}\left\vert a_{in}\right\vert ^{2}, \end{eqnarray} \tag{ 27 }$$

$$\begin{eqnarray} A_{2} &:= \tilde{\Gamma}\tilde{\delta}(\tilde{\kappa}-i\Xi )(\tilde{\kappa}^{\ast } \tilde{\Gamma}^{\ast }+g^{2}N)\nonumber\\ &\quad +\,\tilde{\Gamma}^{\ast }\tilde{\delta}(\tilde{\kappa}^{\ast }-\tilde{\omega}_{m}^{\ast })(\tilde{\kappa}\tilde{\Gamma}+g^{2}N)-\left\vert \tilde{\Gamma}\right\vert ^{2}\tilde{\delta}^{2}\left\vert a_{in}\right\vert ^{2}, \qquad \end{eqnarray} \tag{ 28 }$$

$$\begin{eqnarray} A_{3} &:= \left\vert \tilde{\Gamma}\tilde{\delta}(\tilde{\kappa}-i\Xi)\right\vert ^{2}-i \Xi \tilde{\Gamma}\tilde{\delta}(\tilde{\kappa}^{\ast }\tilde{\Gamma}^{\ast }+g^{2}N) \nonumber\\ &\quad +\,i\Xi \tilde{\Gamma}^{\ast }\tilde{\delta}(\tilde{\kappa}\tilde{\Gamma}+g^{2}N), \end{eqnarray} \tag{ 29 }$$

$$\begin{eqnarray} A_{4}&:= i\Xi \tilde{\Gamma}^{\ast }\tilde{\delta}^{\ast }(-i\Xi -\tilde{\kappa}^{\ast}) -i\Xi \tilde{\Gamma}\tilde{\delta}(i\Xi -\tilde{\kappa}), \nonumber\\ A_{5}&:= \tilde{\delta}^{2}\left\vert i\Xi \tilde{\Gamma}\right\vert ^{2}, \end{eqnarray} \tag{ 30 }$$

$$\begin{eqnarray} \tilde{\Gamma}&:= i\delta +\frac{\Gamma }{2}, \end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray} \tilde{\kappa}&:= \kappa +i\Delta _{a}, \end{eqnarray} \tag{ 32 }$$

$$\begin{eqnarray} \tilde{\delta}&:= \frac{2g^{2}}{\frac{\Gamma ^{2}}{4}+\delta ^{2}}. \end{eqnarray} \tag{ 33 }$$

This quintic equation cannot be analytically solved, so we only solve it numerically based on the Abel-Ruffini theorem [33,34]. The numerical results are plotted in fig. 6 which reveals a remarkable behavior of the intra-cavity intensity and the output field intensity.

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In fig. 6(a), we show the optical multistability and the perfect photon absorption as well as the relation between them. We observe that with the increasing of the input intensity, the output intensity demonstrates the monostability, bistability and multistability, respectively. One can notice that the onset of multistability requires a stronger input field intensity comparing with the linear excitation regime. This result suggests that the nonlinear coupling of atomic excitation could be employed to enhance the optical nonlinearity. It is especially interesting that the perfect photon absorption occurs at two particular input intensities: one is

$$\begin{equation} \left\vert a_{in}\right\vert ^{2}=\frac{2g^{2}N\Gamma -\kappa \Gamma ^{2}-4\kappa \delta ^{2}}{8g^{2}}, \end{equation} \tag{ 34 }$$

which depends only on the atomic parameters, and the other is

$$\begin{equation} \left\vert a_{in}\right\vert ^{2}=\frac{\kappa \Gamma \Delta _{a}-2\kappa ^{2}\delta }{\Gamma \Xi }, \end{equation} \tag{ 35 }$$

which arises in conjunction with the optomechanical properties. This indicates that one can tune the input field intensity under the perfect absorption conditions (eq. (23), eq. (24)) to achieve the two perfect photon absorption points. Thus, the perfect photon absorption can be nonlinearly controlled by the input field intensity. Furthermore, one can find that the two perfect photon absorption points correspond to the optical bistability and multistability, respectively. This means that the two nonlinearities induced by the atom-cavity coupling and the cavity-mechanical coupling can be overlap-added in a certain parameters regime. In other words, the nonlinear optomechanical interaction not only enhances the optical nonlinearity but also adds an additional perfect photon absorption point.

What is more, we would like to emphasize the main difference between the current scheme and the previous work [10]. The hybrid atom-optomechanical system provides a further understanding for the corresponding relation between the optical bistability/multitability and the perfect photon absorption. This can be shown by an example in fig. 6(b). The red star line corresponds to $g_{0}/\Gamma =0$ (without the cavity-optomechanical coupling), whereas the black solid line corresponds to the standard optomechanical system without the atomic interaction. In ref. [10], we find that the nonlinearity arises from the atomic nonlinear excitation regime which plays the dominant role in the optical bistability. In our case, when the atom-optomechanical coupling is absent, the hybrid optomechanical system is reduced to the CQED system, which is just consistent with ref. [10]. Once we consider the optomechanical coupling, one can find the optical nonlinearity has been enhanced and the perfect photon absorption occurs at two special input intensities $\vert a_{in}\vert ^{2}$ as shown by the blue line in fig. 6(a). Similarly to the case of the low-excitation limit, one can find that, if g = 0 (in the absence of the atomic ensemble), both the perfect photon absorption and the optical bistability of the output intensity disappear. This can be explained as follows. The output field intensity is given by $\vert a_{\textit{out}}\vert ^{\prime 2}=\vert \frac{\kappa -i\Delta_{a}+i\Xi \vert a\vert ^{2}}{i\Delta _{a}+\kappa -i\Xi \vert a\vert ^{2}}\vert ^{2}\vert a_{in}\vert ^{2}=\vert a_{in}\vert ^{2}$ (here we set $\varphi =0$) which implies that the output field intensity still behaves with the monostable properties, even though the intra-cavity field is driven in the bistable regime. This is analogous to the optomechanically induced transparency (OMIT) [35,36]. Equivalently, one can draw the same conclusion directly from the violation of the perfect absorption conditions given in eq. (23) and eq. (24).

Before the end, we briefly discuss the experimental feasibility of our scheme. Our proposal mainly depends on the parameters $g\sqrt{N},\kappa,\Gamma$ and g₀. In experiment, $g\sqrt{N}/\Gamma = 20$ can be realized by the Rb atoms (the number of atoms $N\sim 10^{6}$) coupled with a cavity with atomic half-linewidth $\Gamma /2=2\pi \times 3\ \text{MHz}$, and cavity decay $\kappa /2\pi \approx 3\ \text{MHz}$ [37–39]. In addition, based on the current experimental parameters [40,41], the optomechanical system parameters we used can be reasonably assumed as $\omega _{m}/2\pi \approx 20\ \text{kHz}$, $Q_{m}=10^{5}$, $g_{0}/2\pi \approx 24\ \text{kHz}$. Using these parameters, the value of Ξ used in the main text can be achieved. It indicates that our proposal works in the unresolved-sideband regime, which has advantages in its easier system fabrication requirements. To sum up, one can find that all the parameter conditions required in this paper are feasible within the current experimental technology.

In summary, we have studied the optical response properties and the suppression of the output fields in the hybrid atom-optomechanical system with two identical input laser fields. It is shown that the perfect photon absorption is present in both the linear and nonlinear atomic excitation regimes. We find that in such a hybrid optomechanical system, the coupling with the atomic ensemble is the sufficient and necessary condition for the perfect photon absorption, while the optomechanical coupling, enhancing the nonlinearity, forms the necessary condition for the optical bistability/multistability and especially induces an additional perfect absorption point in the nonlinear atomic excitation regime. In other words, the existence of the perfect photon absorption in the current system only depends on the coupling between the cavity and the atomic ensemble. It is not determined by whether there exists the optomechanical coupling. But the optomechanical coupling can enhance the nonlinearity of the system, hence it not only induces the bistability in the low-excitation regime but also induces the multistability beyond the low-excitation regime and further leads to the generation of an additional perfect photon absorption. As a result, we also find that the perfect photon absorption corresponds to the optical bistability/multistability. Furthermore, one can find that the interference of the two input fields can modify the photon absorption and optical bistability (multistability), so the bistability (multistability) and perfect absorption can be controlled by changing the relative phase between them, which provides a possible design of an optical switch. Finally, one should note that all the parameters employed in the numerical procedure are taken from the practical experiments, which ensures the practical feasibility in a variety of COM systems.

This work was supported by the National Natural Science Foundation of China, under Grant No. 11375036 and 11175033, the Xinghai Scholar Cultivation Plan and the Fundamental Research Funds for the Central Universities under Grants No. DUT15LK35 and No. DUT15TD47.