Perfect photon absorption in hybrid atom-optomechanical system

TL;DR

This paper investigates perfect photon absorption in a hybrid atom-optomechanical system, demonstrating its persistence beyond low-excitation limits and its potential for optical switching applications.

Contribution

It introduces a scheme for perfect photon absorption in a hybrid system, analyzing its behavior both below and beyond the low-excitation limit, including the relation to optical multistability.

Findings

Perfect photon absorption persists in both linear and nonlinear regimes.

Optical nonlinearity from optomechanical and atomic interactions can be combined.

Switching from optical bistability to multistability is achievable by increasing input intensity.

Abstract

Recently, the photon absorption attracts lots of interest and plays an important role in a variety of applications. Here, we propose a valuable scheme to investigate the perfect photon absorption in a hybrid atom-optomechanical system both under and beyond the low-excitation limit. The perfect photon absorption persists both in the linear atomic excitation regime and nonlinear atomic excitation regime, below the threshold of the optical bistability/multistability, respectively. We also show that the optical nonlinearity raised by the nonlinear optomechanical interaction and nonlinear atomic excitation can be overlap-added, there presents a perfect corresponding relation between perfect photon absorption and the optical multistability beyond the low-excitation limit, the optical bistability can be switched to the optical multistability by increasing the input intensity. The combination of the perfect photon absorption and optical bistability/mutistability is useful for the photon switch application. We believe that this study will provide a possible design of an optical switch.

Equations

$$H$$

$$⟨\dot{b}⟩=-i{\omega }_m⟨b⟩-\gamma ⟨b⟩-i{g}_0⟨a^{†}a⟩,$$

$$⟨{\dot{S}}^{-}⟩=-i\delta ⟨S^{-}⟩+2ig⟨a{S}^z⟩-\frac{\Gamma }{2}⟨S^{-}⟩,$$

$$⟨{\dot{S}}^z⟩=-\Gamma ⟨S^z⟩-ig⟨a{S}^{+}⟩+ig⟨a^{†}{S}^{-}⟩-\frac{N\Gamma }{2},$$

$$⟨\dot{a}⟩$$

$$⟨a⟩=\frac{\sqrt{2{\kappa }_1}{a}_{1in}+\sqrt{2{\kappa }_2}{a}_{2in}}{i{\Delta }_a+\kappa -\frac{g^{2}N}{\frac{N}{2⟨S^z⟩}(\frac{\Gamma }{2}+i\delta )}-i\Xi {\left|a\right|}^2},$$

$$⟨a⟩=\frac{\sqrt{2{\kappa }_1}{a}_{1in}+\sqrt{2{\kappa }_2}{a}_{2in}}{i{\Delta }_a+\kappa +\frac{g^{2}N}{\frac{\Gamma }{2}+i\delta }-i\Xi {\left|a\right|}^2}.$$

$$a_{1out}=\sqrt{2{\kappa }_1}a-a_{1in},$$

$$a_{2out}=\sqrt{2{\kappa }_2}a-a_{2in}.$$

$$\frac{{\left|a_{1out}\right|}^2}{{\left|a_{in}\right|}^2}$$

$$\frac{{\left|a_{2out}\right|}^2}{{\left|a_{in}\right|}^2}$$

$$\frac{\kappa }{\Gamma }=\frac{2g^{2}N}{{\Gamma }^2+4{\delta }^2},$$

$$(\frac{\kappa \Gamma }{2}+{\Delta }_a\delta -g^{2}N)=\delta \Xi {\left|a\right|}^2.$$

$$\frac{{\left|a_{1out}\right|}^{\prime 2}}{{\left|a_{in}\right|}^2}={\left|\frac{\kappa (1+e^{i\phi })}{i{\Delta }_a+\kappa +\frac{g^{2}N}{(\frac{\Gamma }{2}+i\delta )(1+\frac{2g^2{|a|}^2}{\frac{{\Gamma }^2}{4}+{\delta }^2})}-i\Xi {\left|a\right|}^2}-1\right|}^2,$$

$$\frac{{\left|a_{2out}\right|}^{\prime 2}}{{\left|a_{in}\right|}^2}={\left|\frac{\kappa (1+e^{i\phi })}{i{\Delta }_a+\kappa +\frac{g^{2}N}{(\frac{\Gamma }{2}+i\delta )(1+\frac{2g^2{|a|}^2}{\frac{{\Gamma }^2}{4}+{\delta }^2})}-i\Xi {\left|a\right|}^2}-e^{i\phi }\right|}^2.$$

$$\frac{\kappa \Gamma }{2}+{\Delta }_a\delta$$

$$\frac{{\Delta }_a\Gamma }{2}-\kappa \delta$$

$$A_5{x}^5+A_4{x}^4+A_3{x}^3+A_2{x}^2+A_{1}x+A_0=0,$$

$$A_0:=-{\left|\tilde{\Gamma }\right|}^2{\left|a_{in}\right|}^2,$$

$$A_1:=(\tilde{\kappa }\tilde{\Gamma }+g^{2}N)({\tilde{\kappa }}^{\ast }{\tilde{\Gamma }}^{\ast }+g^{2}N)-2{\left|\tilde{\Gamma }\right|}^2\tilde{\delta }{\left|a_{in}\right|}^2,$$

$$A_2$$

$$A_3$$

$$A_4:=i\Xi {\tilde{\Gamma }}^{\ast }{\tilde{\delta }}^{\ast }(-i\Xi -{\tilde{\kappa }}^{\ast })-i\Xi \tilde{\Gamma }\tilde{\delta }(i\Xi -\tilde{\kappa }),$$

$$A_5:={\tilde{\delta }}^2{\left|i\Xi \tilde{\Gamma }\right|}^2,$$

$$\tilde{\Gamma }:=i\delta +\frac{\Gamma }{2},$$

$$\tilde{\kappa }:=\kappa +i{\Delta }_a,$$

$$\tilde{\delta }:=\frac{2g^2}{\frac{{\Gamma }^2}{4}+{\delta }^2}.$$

Full text

I Introduction

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 wan ; stone ; MK ; huang . 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 Ref. cupta1 ; cupta . Recently, the coherent perfect photon absorption using path entanglement was also theoretically studied in Ref. huang and then was demonstrated experimentally by Faccio et al Faccio . The physical basis behind it is the destructive interference between the two input fields stefano ; cupta . 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 Agarwal-arxiv ; stone ; 12-19. 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 chow , the coupled atom-resonator-waveguide system fan , second harmonic generation chen , nanostructured graphene film jinfa and strongly scattering media stone2 and so on. In particular, the perfect photon absorption has also been studied not only in the cavity quantum electrodynamics Agarwal-PRA ; Agarwal-arxiv 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 vaha ; modern and applications, e.g., ground state cooling cooling , quantum coherent state transfer Palo , ponderomotive squeezing safa , quantum entanglement vita , optical Kerr nonlinearity rabl , and gravitational wave physics ligo ; abbott . However, is the optomechanical coupling necessary in the perfect photon absorption in COM? or what’s the essential role of the optomechancial coupling in the perfect photon absorption?

In this paper, we study the perfect photon absorption as well we 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. while the optical intensity onset the bistability. In difference with Re Agarwal-arxiv , where the optical nonlinearity relies on the nonlinear atomic excitation, here the optical nonlinearity used is arisen from the radiation pressure interaction between moveable mirror and the cavity field, meanwhile the atomic ensemble linearily coupled with the cavity under the low excitation limit. Moreover, the photon absorption and optical bistability can be modified by the interference between the two input fields. In addition, we also employ both the atomic nonlinear excitation and nonlinear optomechanical interaction, the two nonlinearity can be overlap-add, both the intra-field and output field intensity can exhibit the optical multistability, the optical bistability switches to the optical multistability by increasing the input field intensity. Furthermore, the perfect absorption point also can be increased with the help of the optomechanical interaction. Finally, the demonstration of realizable within the current experimental technology on our scheme is also been discussed.

II The model

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 $\Gamma$, 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)

[TABLE]

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 $\sigma_{i}^{\pm}=\left|e\right\rangle_{i}\left\langle g\right|$ 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 (II) and the potential dissipation processes, one can derive the dynamics of the system from the Heisenberg equations of the operators. In the classical limit, namely, 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.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $\left\langle\cdots\right\rangle$ represents the expectation value over the steady state. Combining above equations, we can solve the steady-state solution and obtain the intra-cavity field as

[TABLE]

where $\left\langle S^{z}\right\rangle=-\frac{N}{2(1+\frac{2g^{2}\left|a\right|^{2}}{\frac{\Gamma^{2}}{4}+\delta^{2}})}$ and $\Xi=\frac{2\omega_{m}g_{0}^{2}}{\gamma^{2}+\omega_{m}^{2}}$. From Eq. (6), one can easily get the intra-cavity intensity $\left|a\right|^{2}$ and, hence, obtain the properties of the photon absorption.

III The photon absorption

III.1 Perfect photon absorption in the low-excitation limit

We first consider that the atomic ensemble is only in the low excitation regime. In this case, one can set $\left\langle S^{z}\right\rangle\approx-\frac{N}{2}$, so Eq. (6) can be reduced to

[TABLE]

It is obvious that Eq. (7) includes the intra-cavity intensity $\left|a\right|^{2}$ and demonstrates the nonlinear dependence of the intra-cavity intensity. So the system may exhibit the optical bistability under the 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 wall

[TABLE]

[TABLE]

One can assume that $a_{1in}=\left|a_{in}\right|$, $a_{2in}=e^{i\varphi}\left|a_{in}\right|$, where $\varphi$ 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

[TABLE]

Let us now focus our attention on how to realize the perfect photon absorption. The perfect photon absorption implies $a_{1out}=a_{2out}=0$. 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:

[TABLE]

[TABLE]

In terms of Eqs. (12) and (13), we discuss some pertinent results about the perfect absorption and the bistable behaviors in the linear atomic excitation regime. By comparing with Ref. Agarwal-PRA , one can find that the two perfect trapping conditions presented in Ref. Agarwal-PRA can be summarized by our Eq. (12). However, Eq. (13) is our distinguishing condition which is the result of the optomechanical interaction. The nonlinear interaction of the cavity-mechanical plays the dominant role in the optical bistability, and the hybrid atom-optomechanical system is driven into the bistable domain when the input light intensity is above the threshold for the onset of the optical bistability jiangcheng . As shown in Eq. (7), 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 $g^{2}N$ and the atomic frequency to match Eq. (12) and control the intra-cavity intensity to match Eq. (13).

In Fig. 2 (a) and Fig. 2 (b) we display the intra-cavity intensity $\left|a\right|^{2}$ and the input light intensity $\left|a_{in}\right|^{2}$ 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 the $\left|a\right|^{2}$ exhibits the bistability if the input intensity is large enough. In addition, even though the large input intensity can make the optomechanical system show the apparent bistability, this could require a relatively strong coupling $g^{2}N/\kappa^{2}$, which is also illustrated by the monotonic relation between $\left|a_{in}\right|^{2}$ and $g^{2}N/\kappa^{2}$ in Fig. 2 (b). It is also shown in Fig. 2 that the requirement of strong coupling can actually be compensated for by adding more atoms, or weakening the leakage $\kappa$ or reducing the detuning $\Delta_{a}/\kappa$. In addition, for nontrivial solutions, $\left|a_{in}\right|>0$ requires that besides the condition given in Eq. (12), the perfect absorption should only occur in the following range of $g^{2}N$, that is, (1) $g^{2}N>\frac{\Delta_{a}^{2}+\kappa\Gamma+\sqrt{\Delta_{a}^{4}+2\kappa\Gamma\Delta_{a}^{2}-4\Delta_{a}^{2}}}{2}$ for $\Delta_{a}<0$; (2) $g^{2}N>\frac{\kappa\Gamma}{2}$ for $\Delta_{a}>0$. The numerical illustration of these ranges is given in Fig. 2 (b). Both the ranges imply the strong coupling $g^{2}N$. In addition, one can also find that the large atomic frequency detuning $\Delta_{a}<0$ will lead to the stronger coupling.

In what follows, we present the result for the two output light intensities $\left|a_{1out}\right|^{2}$ and $\left|a_{2out}\right|^{2}$ 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 $\left|a_{in}\right|^{2}=\kappa\delta\Xi/(\frac{\kappa\Gamma}{2}+\Delta_{a}\delta-g^{2}N)$ with the relative phase $\varphi=2n\pi$, $n=0,\pm 1,\pm 2\cdots$ (the black lines in the Fig. 3). Interestingly, we find that once the perfect absorption occurs, the output field intensity behaves the optical bistability where we can get three real distinct values for the output field intensity $\left|a_{out}\right|^{2}$ due to the nonlinear equation Eq. (7). Based on the Eq. (12) and Eq. (13), one can find that the perfect absorption can’t 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. Agarwal-PRA . 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 jiangcheng . It is different from Ref. Agarwal-arxiv 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 lines correspond to $\varphi=\pi/2$ and the red 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 $\varphi$ changing from $\pi$ to [math], 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.

In Fig. 4, we plot the output light intensities versus $\left|a_{in}\right|^{2}$ for the different cavity-laser detunings. One can see that the different values of the atom-cavity detuning ($\Delta=\Delta_{a}-\delta$) don’t take the edge off the perfect photon absorption and the optical bistability. The different $\Delta$ requires the different input light intensities to match the special conditions of the perfect absorption. With the small $\Delta$, the system can be easily driven to the nonlinear regime and achieve the optical bistability, meanwhile, the perfect photon absorption can also happen with the relatively weak input light intensity.

Fig. 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 versus the relative phase $\varphi$. In Fig. 5 we plot the two output light intensities $\left|a_{1out}\right|^{2}$ and $\left|a_{2out}\right|^{2}$ with $\left|a_{1in}\right|^{2}=\left|a_{in}\right|^{2}$, $\left|a_{2in}\right|^{2}=e^{i\varphi}\left|a_{in}\right|^{2}$. One can see that the two output fields are not equal to each other except at some particular $\varphi$. These $\varphi$ can be determined by solving Eq. (10) and Eq. (11) associated with Eq. (7), and thus one can have $\sin{\varphi}=0$ or $\cos\varphi=c=(\kappa+\frac{g^{2}N\Gamma}{2(\frac{\Gamma^{2}}{4}+\delta^{2})})^{2}(\frac{\kappa\Gamma}{2}+\delta\Delta_{a}-g^{2}N)/(2\kappa\Xi\delta_{0}^{2}|a_{in}|^{2})-1$. 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 the equal output light intensities have been present. In addition, some parameters including the driving, the detuning and so on could lead to $|c|>1$, so $\cos{\varphi}=c$ can’t hold any more. This case is illustrated in Fig. 5 (a) where the input intensity $\left|a_{in}\right|^{2}$ is set below the threshold of the bistable regime, hence $\left|a_{1out}\right|^{2}$ and $\left|a_{2out}\right|^{2}$ 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.

III.2 Perfect photon absorption beyond the low-excitation limit

The consequences within the low-excitation limit have been studied in the previous section. Drowning on the findings above mentioned, a natural question will be arisen, thus, 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. (6), we will have to directly employ $\left\langle S^{z}\right\rangle=-\frac{N}{2(1+\frac{2g^{2}\left|a\right|^{2}}{\frac{\Gamma^{2}}{4}+\delta^{2}})}$ instead of $\left\langle S^{z}\right\rangle=-\frac{N}{2}$ to calculate the steady-state solution of the intra-cavity field from the Eq (2)- Eq (5). At this moment, the $\left\langle S^{z}\right\rangle$ is a function of the intra-cavity field intensity $\left|a\right|^{2}$ which satisfies the condition $\left|a\right|^{2}>\frac{\Gamma^{2}}{4g^{2}}$. Therefore, two output optical intensity $\left|a_{1out}\right|^{\prime 2}$ and $\left|a_{2out}\right|^{{}^{\prime}2}$ can be given by

[TABLE]

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 $\phi=0$, the conditions parallel with Eqs. (12) and (13) can be straightforwardly written as

[TABLE]

Compared with the perfect absorption conditions of the CQED system in Ref. Agarwal-arxiv , 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 addition, from the Eq. (6), one can find that the intra-field intensity $\left|a\right|^{2}$ satisfies a quintic equation. Let $\left|a_{in}^{1}\right|^{2}=\left|a_{in}^{2}\right|^{2}=\left|a_{in}\right|^{2}$ $(\varphi=0)$ and $x:=$ $\left|a\right|^{2}$, the quintic equation reads

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

This quintic equation can not be analytically solved, so we only solve it numerically based on the Abel–Ruffini theorem jjj ; zhang . The numerical results are plotted in Fig. 6 which reveals a remarkable behavior of the intra-cavity intensity and the output field intensity.

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 $\left|a_{in}\right|^{2}=\frac{2g^{2}N\Gamma-\kappa\Gamma^{2}-4\kappa\delta^{2}}{8g^{2}}$ and the other is $\left|a_{in}\right|^{2}=\frac{\kappa\Gamma\Delta_{a}-2\kappa^{2}\delta}{\Gamma\Xi}$. This indicates that one can tune the input field intensity under the perfect absorption conditions (Eq.(16),Eq.(17)) 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’s more, we would like to emphasize the main difference between the current scheme and the previous work Agarwal-arxiv . 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 line corresponds to $g_{0}/\Gamma=0$ (without the cavity-optomechanical coupling), whereas the black line corresponds to the standard optomechanical system without the atomic interaction. In Ref. Agarwal-arxiv , we find that the nonlinearity arises from the atomic nonlinear excitation regime which plays the dominate 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 consist with Ref. Agarwal-arxiv . 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 $\left|a_{in}\right|^{2}$ as shown by the blue line in Fig. 6 (a). Similar to the case of the low-excitation limit, one can find that, if $g=0$ (in absence of the atom 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 $\left|a_{out}\right|^{\prime 2}=\left|\frac{\kappa-i\Delta_{a}+i\Xi\left|a\right|^{2}}{i\Delta_{a}+\kappa-i\Xi\left|a\right|^{2}}\right|^{2}\left|a_{in}\right|^{2}=\left|a_{in}\right|^{2}$ (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) omit ; science330 . Equivalently, one can draw the same conclusion directly from the violation of the perfect absorption conditions given in Eq. (16) and Eq. (17).

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 $g_{0}$. In experiment, $g\sqrt{N}/\Gamma$=20 can be realized by the ultra cold atomic ensemble coupled with the cavity with the atomic half-linewidth $\Gamma$= $2\pi\times 3M$ Hz Bren ; omshiyan ; shiyan1 . In addition, the optomechanical parameters we used can be realized in various optomechanical systems, for example, $\omega_{m}/2\pi=4.2\times 10^{4}$ Hz, $g_{0}/2\pi=6\times 10^{5}$ Hz, $\kappa/2\pi=6.6\times 10^{5}$ Hz have been reported in Ref. modern , so the values of $\Xi$ used in the main text can be easily achieved. To sum up, one can find that all the conditions required in this paper are realizable within the current experimental technology.

IV Conclusions and discussion

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 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. 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 controled 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.

Acknowledgements.

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.