Generation of single entangled photon-phonon pairs via an atom-photon-phonon interaction

TL;DR

This paper proposes a method to generate entangled photon-phonon pairs using an atom-photon-phonon interaction in a hybrid optomechanical system, with potential applications in quantum networks.

Contribution

It introduces a novel approach to produce single entangled photon-phonon pairs through tripartite interactions, observable in both weak and strong coupling regimes.

Findings

Photon and phonon blockade observed

Photon-phonon entanglement demonstrated

Potential for quantum network applications

Abstract

Quantum blockade and entanglement play important roles in quantum information and quantum communication as quantum blockade is an effective mechanism to generate single photons (phonons) and entanglement is a crucial resource for quantum information processing. In this work, we propose a method to generate single entangled photon-phonon pairs in a hybrid optomechanical system. We show that photon blockade, phonon blockade, and photon-phonon correlation and entanglement can be observed via the atom-photon-phonon (tripartite) interaction, under the resonant atomic driving. The correlated and entangled single photons and single phonons, i.e., single entangled photon-phonon pairs, can be generated in both the weak and strong tripartite interaction regimes. Our results may have important applications in the development of highly complex quantum networks.

Equations

$$H_{\mathrm{int}}=g\left(x\right)\left({\sigma }_{+}a+{\sigma }_{-}{a}^{†}\right).$$

$$g\left(x\right)=g\left(0\right)+J\left(b^{†}+b\right),$$

$$H_{\mathrm{int}}=J\left(b^{†}+b\right)\left({\sigma }_{+}a+{\sigma }_{-}{a}^{†}\right).$$

$$H_{\mathrm{int}}=J\left({\sigma }_{+}ab+{\sigma }_{-}{a}^{†}{b}^{†}\right),$$

$$H=\Delta {\sigma }_{+}{\sigma }_{-}+\Delta a^{†}a+J\left({\sigma }_{+}ab+{\sigma }_{-}{a}^{†}{b}^{†}\right)+\Omega {\sigma }_x,$$

$$g_n^{\left(2\right)}\left(0\right)\equiv \frac{⟨a^{†}{a}^{†}aa⟩}{⟨n{⟩}^2},$$

$$g_m^{\left(2\right)}\left(0\right)\equiv \frac{⟨b^{†}{b}^{†}bb⟩}{⟨m{⟩}^2},$$

$$g_{nm}^{\left(2\right)}\left(0\right)\equiv \frac{⟨a^{†}{b}^{†}ba⟩}{⟨n⟩⟨m⟩},$$

$$\frac{\partial \rho }{\partial t}$$

$$⟨n⟩\approx {\rho }_{5,5}+{\rho }_{4,4},$$

$$⟨m⟩\approx {\rho }_{5,5}+{\rho }_{3,3},$$

$$g_{nm}^{(2)}(0)\approx \frac{{\rho }_{5,5}}{({\rho }_{5,5}+{\rho }_{3,3})({\rho }_{5,5}+{\rho }_{4,4})},$$

$${\rho }_{3,3}\approx \frac{{\gamma }_c}{{\gamma }_m}{\rho }_{5,5},$$

$${\rho }_{4,4}\approx \frac{{\gamma }_m}{{\gamma }_c}{\rho }_{5,5}.$$

$$g_{nm}^{(2)}(0)\approx \frac{1}{2⟨n⟩}.$$

$$E_N={\log }_2{\Vert {\rho }_{AB}^{T_A}\Vert }_1,$$

Full text

I Introduction

Optomechanical systems with parametric coupling between optical and mechanical modes provide us a perfect platform for manipulating the states of photons and phonons AspelmeyerARX13 . As an important application, photon (phonon) blockade ImamogluPRL97 ; BirnbaumNat05 ; YXLiuPRA10 , that only allows single photon (phonon) excitation in the optical (mechanical) mode, based on optomechanical interaction has attracted significant interest in the past few years. A number of designs based on diverse mechanisms are proposed to demonstrate photon (phonon) blockade in optomechanical systems, such as photon (phonon) blockade based on strong optomechanical couplings RablPRL11 ; NunnenkampPRL11 ; StannigelPRL12 ; XWXuPRA13a ; KronwaldPRA13 ; JQLiaoPRA13 ; XYLuPRL15 ; DHuPRA15 ; HXiePRA16 ; SeokPRA17 ; HXiePRA17 and photon (phonon) blockade in weak nonlinear regime induced by quantum interference XWXuJPB13 ; SavonaArx13 ; HQShiSR18 ; MWangPRA19 ; BJLiPR19 .

In a recent experiment RiedingerNat16 , the non-classical correlations between single photons and phonons from a nanomechanical resonator was reported by driving the nanomechanical photonic crystal cavity with blue-detuned optical pulse. After that, we studied the photon and phonon statistics in a quadratically coupled optomechanical system, and show that both photon blockade and phonon blockade can be observed in the same parameter regime, and more important, the single photons and single phonons are strongly anticorrelated XWXuPRA18 . Here, we will do a further study and propose a method to generate correlated single photons and single phonons under the constant atomic driving. Even more interestingly, we will show that the correlated single photons and single phonons are entangled with each other, i.e., they are single entangled photon-phonon pairs.

Entangled states have great significance of both fundamental physics study and applications in quantum information processing and quantum communication. The optomechanical entanglement has already been proposed theoretically VitaliPRL07 ; HartmannPRL08 ; BorkjePRL11 ; BarzanjehPRL12 ; YDWangPRL12 ; LTianPRL12 and demonstrated experimentally PalomakiSci13 ; RiedingerNat18 ; Ockeloen-KorppiNat18 ; MarinkovicPRL18 . Optomechanical systems provide a perfect platform to generate both bipartite WJNiePRA12 ; XWXuPRA13 ; JQLiaoPRA14 ; XYLuPRA18 and multipartite PaternostroPRL07 ; GenesPRA08 ; HTTanPRA11 ; XuerebPRA12 ; JLiPRL18 entanglement. However, there are substantial differences between the entanglement we will discuss in this paper and entanglement proposed before. One striking difference is the entanglement we proposed here is for single photons and phonons, which is non-Gaussian, so that the generally adopted method, i.e., the linearization of the optomechanical interaction, is no longer applicable.

Inspired by a recent experiment FTianArx19 , in which the coupling between an optomechanical resonator with two-level emitters was realized, here we consider a hybrid system which enables a tripartite interaction between a two-level atom, an optical mode, and a mechanical mode YChangJPB09 . We study the generation of single entangled photon-phonon pairs, which are uesful for quantum information and quantum communication. Such atom-photon-phonon interactions were proposed to provide an optically controllable interaction between a two-level atom and a macroscopic mechanical oscillator CotrufoPRL17 ; MWangPRA19 by driving the optical mode strongly. Nevertheless, in this paper we drive the two-level atom coherently and show that single entangled photon-phonon pairs can be generated in the hybrid optomechanical system. The single entangled photon-phonon pairs have potential application in the development of highly complex quantum networks.

The remainder of this paper is organized as follows. In Sec. II, we introduce the theoretical model of a hybrid optomechanical system, and show the simple derivation of the atom-photon-phonon interaction and the energy spectrum of the Hamiltonian. In Sec. III, the photon and phonon statistics, and the quantum correlation between the photons and phonons are discussed numerically. Finally, a summary is given in Sec. IV.

II Model and Hamiltonian

We study a hybrid system with a two-level atom ($\sigma_{\pm}$ being the ladder operators) of transition frequency $\omega_{0}$ and a mechanical resonator $b$ of resonance frequency $\omega_{m}$ in an optical cavity $a$ of resonance frequency $\omega_{c}$, as shown in Fig. 1(a) and (b). Here, we consider a special situation in which the mechanical displacement $x$ induces a variation of the spatial distribution of the cavity field CotrufoPRL17 , while the mechanical effect on the optical frequency $\omega_{c}$ can be neglected. Thus, the coupling strength $g\left(x\right)$ between the two-level atom and the optical mode depends on the position of the mechiancal resonator $x$, which is described by the interaction Hamiltonian under the rotating-wave approximation as ($\hbar=1$)

[TABLE]

Typically the mechanical displacement $x$ is very small, and $g\left(x\right)$ can be expanded to the first order in $x$,

[TABLE]

where $J\equiv\left(\partial g/\partial x\right)|_{x=0}x_{\mathrm{zpf}}$ is the tripartite atom-photon-phonon interaction strength. In the specific condition that the two-level atom is placed at the node of the optical mode with mechanical resonator in equilibrium, i.e., $g\left(0\right)=0$, and the only possible interaction between them is the atom-photon-phonon interaction as

[TABLE]

Under particular resonant conditions, the tripartite interaction allows swapping the excitation between the three quantum systems. Under the conditions $\omega_{0}=\omega_{c}+\omega_{m}$ and $\min\{\omega_{0},\omega_{c}\}\gg\omega_{m}\gg J$, the Hamiltonian of the resonant interaction reads

[TABLE]

which describes the simultaneous generation of a photon and a phonon with the two-level atom jumping from the excited state to its ground state and the reverse process. This tripartite interaction provides us an effective way to generate photon-phonon pairs. Such a hybrid system can be implemented in the electromechanical systems TeufelNat11 ; MasselNC12 ; PalomakiSci13 ; SuhSci14 with artificial atom at the node or in a Fabry-Pérot cavity with a membrane containing two-level atoms in the node of the cavity mode ThompsonNat08 ; Flowers-JacobsAPL12 ; HXuNat16 ; HXuNC17 .

Next, we consider the case that the two-level atom is pumped by a coherent field (strength $\Omega$, frequency $\omega_{p}$), and the total Hamiltonian for the hybrid system in the rotating frame with respect to $R\left(t\right)=\exp\left[i\omega_{p}\sigma_{+}\sigma_{-}t+i\left(\omega_{p}-\omega_{m}\right)a^{{\dagger}}at+i\omega_{m}b^{{\dagger}}bt\right]$ reads

[TABLE]

where we introduce the detuning $\Delta\equiv\omega_{0}-\omega_{p}=\omega_{c}+\omega_{m}-\omega_{p}$.

The energy spectrum of the Hamiltonian in Eq. (5) for hybrid optomechanical system is shown in Figs. 1(c) and 1(d). In the non-coupling basis [Fig. 1(c)], $\left|e\right\rangle$ ($\left|g\right\rangle$) denotes the excited (ground) state of the two-level atom, and $\left|n,m\right\rangle$ represents the Fock state with $n$ photons in the optical mode and $m$ phonons in the mechanical mode. In Fig. 1(d), we denote the eigenstates in the diagonal basis as $\left|0,0\right\rangle_{0}\equiv\left|g\right\rangle\left|0,0\right\rangle$, $\left|1,0\right\rangle_{0}\equiv\left|g\right\rangle\left|1,0\right\rangle$, $\left|0,1\right\rangle_{0}\equiv\left|g\right\rangle\left|0,1\right\rangle$, $\left|1,1\right\rangle_{\pm}\equiv(\left|g\right\rangle\left|1,1\right\rangle\pm\left|e\right\rangle\left|0,0\right\rangle)/\sqrt{2}$ with eigenvalues [math], $\omega_{c}$, $\omega_{m}$, $\omega_{0}\pm J$, respectively.

III Correlation and entanglement

To quantify the statistics of the phonons and photons in the system, we consider the equal-time second-order correlation functions $g^{(2)}_{n}(0)$ and $g^{(2)}_{m}(0)$, and cross-correlation function $g^{(2)}_{nm}(0)$ in the steady state ($t\rightarrow\infty$) defined by

[TABLE]

[TABLE]

[TABLE]

where $\langle n\rangle\equiv\left\langle a^{{\dagger}}a\right\rangle$ and $\langle m\rangle\equiv\left\langle b^{{\dagger}}b\right\rangle$ are the mean photon and phonon numbers. The dynamic behavior of the total open system is described by the master equation for the density matrix $\rho$ of the system Carmichael93

[TABLE]

where $L[o]\rho=o\rho o^{{\dagger}}-\left(o^{{\dagger}}o\rho+\rho o^{{\dagger}}o\right)/2$ denotes a Lindbland term for an operator $o$; $\kappa$ is the damping rate of the two-level atom and $\gamma_{c}$ ($\gamma_{m}$) is the damping rate of the optical (mechanical) mode; $m_{\mathrm{th}}$ is the mean thermal phonon number. We assume that the frequencies of the two-level atom and the optical mode are so high that the thermal effect can be neglected.

The equal-time second-order correlation functions [$g^{(2)}_{n}(0)$ and $g^{(2)}_{m}(0)$] and cross-correlation function $g^{(2)}_{nm}(0)$ are plotted as functions of the detuning $\Delta/\kappa$ in Fig. 2 under both weak-coupling condition [(a) $J=0.1\kappa$] and strong-coupling condition [(c) $J=100\kappa$]. It is clear that photon blockade and phonon blockade, i.e., $g^{(2)}_{n}(0)=g^{(2)}_{m}(0)<1$, appear simultaneously around $|\Delta|=J$ for the same parameters. Simultaneously, the single photons and single phonons generated by photon blockade and phonon blockade are strongly correlated with each other, i.e., $g^{(2)}_{nm}(0)\gg 1$. The optimal detuning $\Delta$ for correlated photon blockade and phonon blockade depends on the coupling strength $J$: $\Delta=0$ for weak coupling and $|\Delta|\approx J$ for strong coupling. Moreover, the mean photon (phonon) number $\langle n\rangle=\langle m\rangle$ in the weak-coupling case is much smaller than the one in the strong-coupling case.

Physically, the single photon and phonon pairs are generated one by one with the two-level atom jumping from the excited state to its ground state. In the weak-coupling regime ($J\ll\kappa$), the system is driven resonantly with detuning $\Delta=0$ because the states $\left|1,1\right\rangle_{+}$ and $\left|1,1\right\rangle_{-}$ are not resolved. In the strong-coupling regime ($J\gg\kappa$), the system should be investigated by the dressed states as shown in Fig. 1(d), and the single photon and phonon pairs are generated with detuning $\Delta=\pm J$ for resonant pumping.

In order to understand the behavior of the cross-correlation function $g^{(2)}_{nm}(0)$, we can give the expression of $g^{(2)}_{nm}(0)$ approximately. Under the weak-exciting condition, i.e., $\max\{\langle n\rangle,\langle m\rangle\}\ll 1$, we have mean photon (phonon) number

[TABLE]

[TABLE]

and the cross-correlation function $g^{(2)}_{nm}(0)$

[TABLE]

where $\rho_{3,3}=\langle g|\langle 0,1|\rho|g\rangle|0,1\rangle$, $\rho_{4,4}=\langle g|\langle 1,0|\rho|g\rangle|1,0\rangle$, and $\rho_{5,5}=\langle g|\langle 1,1|\rho|g\rangle|1,1\rangle$, and they satisfy the relations

[TABLE]

[TABLE]

If we set $\gamma_{c}=\gamma_{m}$, then we have $\rho_{3,3}\approx\rho_{4,4}\approx\rho_{5,5}$, $\langle n\rangle=\langle m\rangle\approx 2\rho_{5,5}$, and

[TABLE]

Under the resonant conditions at the detuning $|\Delta|=J$, we have maximum $\langle n\rangle=\langle m\rangle$, and thus minimum cross-correlation function $g^{(2)}_{nm}(0)$, corresponding to the dips around the detuning $|\Delta|=J$.

It’s not hard to guess that the strongly correlated single photons and single phonons generated by photon blockade and phonon blockade are entangled with each other. The entanglement between the optical and mechanical modes can be characterized by the logarithmic negativity VidalPRA02

[TABLE]

where the symbol $\left\|\cdot\right\|_{1}$ denotes the trace norm, and $\rho_{AB}^{T_{A}}$ is the partial transpose of the reduced density matrix $\rho_{AB}$ of the optical and mechanical modes. It is worth mentioning that the entangled state for the single photons and single phonons obtained here is non-Gaussian. Thus the logarithmic negativity for Gaussian states AdessoPRA04 widely used in the previous works  VitaliPRL07 ; HartmannPRL08 ; BorkjePRL11 ; BarzanjehPRL12 ; YDWangPRL12 ; LTianPRL12 cannot be used to accurately describe the entangled state here.

The logarithmic negativity $E_{N}$ is shown in Figs. 2(c) and 2(f). Obviously, the strongly correlated single photons and single phonons generated by photon blockade and phonon blockade are entangled with each other in both the weak ($J<\kappa$) and strong ($J>\kappa$) coupling regimes. In the weak-coupling regime as shown in Fig. 2(c), there is a dip around the detuning $\Delta=0$, which is induced by the quantum interferences between two routes: (a) the direct transition channel $\left|g\right\rangle\left|0,0\right\rangle\overset{\Omega}{\rightarrow}\left|e\right\rangle\left|0,0\right\rangle\overset{J}{\rightarrow}\left|g\right\rangle\left|1,1\right\rangle$; (b) the indirect transition channel $\left|g\right\rangle\left|0,0\right\rangle\overset{\Omega}{\rightarrow}\left|e\right\rangle\left|0,0\right\rangle\overset{\Omega}{\rightarrow}\left|g\right\rangle\left|0,0\right\rangle\overset{\Omega}{\rightarrow}\left|e\right\rangle\left|0,0\right\rangle\overset{J}{\rightarrow}\left|g\right\rangle\left|1,1\right\rangle$ (or higher-order variants). Thus the width of the dip depends on the driving strength $\Omega$, as shown in Fig. 3(a). Similar mechanism can induce transparency in lambda-type three-level atoms HarrisPT97 ; FleischhauerRMP05 and optomechanical systems AgarwalPRA10 ; WeisSci10 ; SafaviNaeiniNat11 . Differently, in Fig. 2(f), there are two peaks around the detunings $\Delta=\pm J$ in the strong-coupling regime. This phenomenon can be understood by analyzing the energy spectrum shown in Fig. 1(d): the transition process $\left|0,0\right\rangle_{0}\rightarrow\left|1,1\right\rangle_{\pm}$ is resonantly enhanced with detunings $\Delta=\pm J$. As a consequence, we can shift the optimal value of the detuning for entanglement by tuning the coupling strength $J$ as shown in Fig. 3(b).

Figure 4 shows the second-order correlation functions [$g^{(2)}_{n}(0)$ and $g^{(2)}_{m}(0)$] and cross-correlation function $g^{(2)}_{nm}(0)$ with the coupling strength $J$ from weak to strong. The mean photon (phonon) number [$\langle n\rangle=\langle m\rangle$] and logarithmic negativity $E_{N}$ increase with the enhancing of the coupling strength $J$. As shown in Fig. 4(b), the cross-correlation function $g^{(2)}_{nm}(0)$ decreases with the increasing of the mean photon (phonon) number [$\langle n\rangle=\langle m\rangle$], and the numerical results (red dashed curve) agrees well with the analytical results given in Eq. (15) (blue short-dashed curve). The second-order correlation functions [$g^{(2)}_{n}(0)$ and $g^{(2)}_{m}(0)$] increases first with the mean photon (phonon) number, and then decreases with the coupling strength $J$, as the excitations of states $\left|2,1\right\rangle_{\pm}\equiv(\left|g\right\rangle\left|2,1\right\rangle\pm\left|e\right\rangle\left|1,0\right\rangle)/\sqrt{2}$ and $\left|1,2\right\rangle_{\pm}\equiv(\left|g\right\rangle\left|1,2\right\rangle\pm\left|e\right\rangle\left|0,1\right\rangle)/\sqrt{2}$ are suppressed for the enhancement of the effective damping rates with the coupling strength $J$. The suitable coupling strength $J$ for observing correlated single photons and single phonons, i.e., $g^{(2)}_{n}(0)=g^{(2)}_{m}(0)\ll 1$ and $g^{(2)}_{nm}(0)\gg 1$, is $J\ll\kappa$ or $J\gg\kappa$.

Generally, the damping rate of the mechanical mode is much smaller than the damping rate of the optical mode. However, the effective damping of the mechanical mode can be controlled and significantly enhanced by coupling the mechanical mode to an auxiliary optical mode WilsonRaePRL07 ; MarquardtPRL07 ; LiYPRB08 ; XWXuPRA15 . In addition, the phonon statistics can be observed indirectly by measuring statistics of the photons output from the auxiliary optical mode DidierPRB11 ; RamosPRL13 ; CohenNat15 ; XWXuPRA16 . The dependence of the second-order correlation functions [$g^{(2)}_{n}(0)$ and $g^{(2)}_{m}(0)$] and cross-correlation function $g^{(2)}_{nm}(0)$ on the mechanical damping rate $\gamma_{m}$ is shown in Fig. 5. In the weak-coupling case [Figs. 5(a) and 5(b)], the correlation and cross-correlation functions change monotonically with the mechanical damping rate. In the strong-coupling case [Figs. 5(c) and 5(d)], the correlation and cross-correlation functions changing non-monotonously with the mechanical damping rate. The mean phonon number $\langle m\rangle$ decreases rapidly with the mechanical damping rate in both weak- and strong-coupling regimes and the mean photon number $\langle n\rangle$ decreases monotonously for weak coupling ($J\ll\kappa$). While $\langle n\rangle$ increases first and then decreases with the mechanical damping rate in the strong coupling regime ($J\gg\kappa$), i.e., we can enhance photon emission by increasing the mechanical damping rate when $\gamma_{m}<\kappa$. Moreover, there is a optimal mechanical damping rate $\gamma_{m}$ for entanglement $E_{N}$ around the point $\gamma_{m}\approx 1.32\kappa$ ($\gamma_{m}\approx 3.47\kappa$) in the case of $J=0.1\kappa$ ($J=100\kappa$).

These interesting phenomena can be understood by the probability distribution in the bare states as shown in Fig. 6, where $\rho_{1,1}=\langle g|\langle 0,0|\rho|g\rangle|0,0\rangle$, $\rho_{2,2}=\langle e|\langle 0,0|\rho|e\rangle|0,0\rangle$, $\rho_{3,3}=\langle g|\langle 0,1|\rho|g\rangle|0,1\rangle$, $\rho_{4,4}=\langle g|\langle 1,0|\rho|g\rangle|1,0\rangle$, $\rho_{5,5}=\langle g|\langle 1,1|\rho|g\rangle|1,1\rangle$, $\rho_{1,5}=\langle g|\langle 0,0|\rho|g\rangle|1,1\rangle$, $\rho_{2,5}=\langle e|\langle 0,0|\rho|g\rangle|1,1\rangle$. It is clear that we have $\rho_{3,3}\approx\rho_{4,4}\approx\rho_{5,5}$ around $\gamma_{m}=\gamma_{c}$, which is agree with Eqs. (13) and (14). In the weak-coupling regime ($J\ll\kappa$) as shown in Fig. 6(a), most of the probability is distributed in the states $|g\rangle|0,0\rangle$ and $|e\rangle|0,0\rangle$; the probability in states $|g\rangle|1,1\rangle$ (as well as the off-diagonal elements $|\rho_{1,4}|$ and $|\rho_{1,5}|$, which determine the entanglement $E_{N}$ between the photons and phonons) increases slowly in the regime of $\gamma_{m}<\kappa$, and decreases rapidly when $\gamma_{m}>\kappa$; the probability in single phonon state $|g\rangle|0,1\rangle$ (single photon state $|g\rangle|1,0\rangle$) decreases (increases) in the regime of $\gamma_{m}<\gamma_{c}$, and decreases rapidly when $\gamma_{m}>\gamma_{c}$. Differently, in the strong-coupling regime ($J\gg\kappa$) as shown in Fig. 6(b), most of the probability ($87.5\%$) is distributed in the single phonon state $|g\rangle|0,1\rangle$ when $\gamma_{m}\ll\kappa$, as the damping rate of the state $|g\rangle|0,1\rangle$ is much smaller than the other states; the probability in the ground state $|g\rangle|0,0\rangle$ increase monotonously with the mechanical damping rate; the probability in states $|e\rangle|0,0\rangle$ and $|g\rangle|1,1\rangle$ are almost the same, i.e., $\rho_{2,2}\approx\rho_{5,5}$, and they (as well as the off-diagonal elements $|\rho_{1,4}|$ and $|\rho_{1,5}|$) increases first and then decreases with the mechanical damping rate, which is corresponding to the phenomena of photon emission and entanglement enhancing by increasing the mechanical damping rate when $\gamma_{m}<\kappa$.

The thermal effect of the mechanical mode on the statistic properties of the generated photons and phonons are shown in Fig. 7. It is clear that the thermal phonons have a significant effect on the statistic properties of the generated photons and phonons. As the mean phonon number is much larger in the strong-coupling regime than the one in the weak-coupling regime, the correlated and entangled photon blockade and phonon blockade in the strong-coupling regime is more robust against the thermal noise than the one in the weak-coupling regime.

IV Conclusions

In summary, we have studied the photon and phonon statistics, and the quantum correlation between photons and phonons in a hybrid optomechanical system including an atom-photon-phonon (tripartite) interaction. We have shown that both the photon and phonon blockade can be observed in the same parameter area, and the generated single photons and single phonons are correlated and entangled with each other. Moreover, the single entangled photon-phonon pairs can be observed in both the weak and strong tripartite interaction regime. The phonons with low-loss can be used for quantum memories, and photons are suitable for the transmission of quantum information. The generated single entangled photon-phonon pairs will have applications in quantum communication and the hybrid optomechanical system can serve as quantum transducers in building hybrid quantum networks. In addition, the basic mechanism of this work can be generalized to a nondegenerate two-photon Jaynes-Cummings model SCGouPRA89 ; AshrafPRA92 , to generate entangled photon pairs with different frequency, such as entangled microwave-optical photon pairs CZhongARX19 .

Acknowledgement

We thank Yan-Jun Zhao, Hui Wang, and Qiang Zheng for helpful discussions. X.-W.X. was supported by the National Natural Science Foundation of China (NSFC) under Grant No. 11604096. A.-X.C. is supported by NSFC under Grant No. 11775190. J.-Q.L. is supported in part by National Natural Science Foundation of China under Grants No. 11822501 and No. 11774087, and Natural Science Foundation of Hunan Province, China under Grant No. 2017JJ1021.