Rotational mirror–mirror entanglement via dissipative atomic reservoir in a double-Laguerre–Gaussian-cavity system

In recent years, the interaction between structured light beams and matter has been paid continuous attention in quantum optics and quantum information. Particularly, the Laguerre–Gaussian (LG) beam, as a typical type of structured light, can be obtained by solving the paraxial wave equation in cylindrical coordinates, possessing a helical wavefront and a doughnut-shaped intensity distribution with a hollow at the beam center [1, 2]. The LG beam usually carries an orbital angular momentum (OAM) of $l\hbar$ per photon with its phase of $e^{il\phi}$, where l is topological charge value and φ represents azimuthal angle [3]. In experiment, the typical methods to generate LG light are spatial light modulators [4, 5], spiral phase plate or mirror [6, 7] and computer-generated holograms [8, 9]. It is demonstrated that the topological charge value l can reach as high as 1000 by using the spiral phase plate [10]. Interestingly, the LG beams can exert a torque on objects due to the exchange of OAM [11–13], with which it is possible to trap and cool the rotational mirrors [14]. So far, a large number of quantum optics phenomena have been reported based on the interaction of LG light with matter, including the vortex light information storage [15], quantum memory [16], spatially dependent electromagnetically induced transparency [17], transfer of optical vortices via the multi-wave mixing [18–21], optomechanical induced transparency [22], ground-state cooling [23], the detection of OAM [24], the generation of higher order sideband [25, 26], and the high-dimensional quantum entanglement [27–34] etc.

On the other hand, the macroscopic entanglement is of great importance both in verifying the fundamental quantum theory [35] and realizing realistic applications such as remote quantum communication, reliable quantum computation, precision measurement and quantum sensing [36, 37] etc. In experiment, the entanglement between two macroscopic objects is demonstrated by engineering the dissipation with laser and magnetic fields [38]. The mechanical resonator at the micrometer scale, an ideal candidate for investigating macroscopic entanglement, has attracted much attention in last years because it locates at the interface of the quantum-to-classical transition [39–44]. It is explored that the macroscopic oscillators can be entangled by using the radiation pressure, behind which both the parametric interaction and beam splitter interaction are hidden [45]. Successively, various literatures focus on the methods to entangle two massive, movable cavity mirrors at steady state [46–49]. As a profitable model, the bipartite and multipartite entanglement between mirrors and cavity modes are also acquired under the effect of radiation pressure force [50, 51]. Interestingly, it provides a useful way to control the macroscopic entanglement by atomic coherence effects [52–56], in which the parametric interaction between mechanical oscillators is established by the coherence-controlled evolution processes. A lot of recent studies show that the hybrid quantum systems have developed an extensive platform to generate macroscopic entanglement between distant objects [57–60], which may find potential applications in remote quantum communications.

Notably, in a rotational optomechanical system, Bhattacharya opens up fascinating possibilities, in their pioneering works, to entangle a LG cavity mode and a rotating mirror [61], or generate entanglement between the rovibrational modes of a macroscopic mirror using radiation pressure [62]. Motivated by this device, Chen et al report the macroscopic entanglement between two rotating mirrors through the exchange of OAM with the same LG cavity [63]. Later, by placing a yttrium iron garnet (YIG) sphere into the LG cavity, the tripartite entanglement are generated between cavity mode, mirror and magnon [64]. While the YIG sphere is replaced by a two-level atomic ensemble, the cavity–mirror entanglement and quantum coherence are enhanced by choosing suitable parameters [65], which is attributed to the fact that the effective coupling of the parametric interaction is enhanced by injecting the atoms. Nevertheless, the quantum coherence would be spoiled when the large detuning limit is not satisfied. In addition, they point out that the quantum coherence are also significantly suppressed by the large atomic decay.

In this paper, we present a different way to establish macroscopic entanglement between two spatially separate rotating mirror via a single pathway of Bogoliubov dissipation. The main results we find in the present scheme are listed as follows. First, in dressed-state picture, it is seen that the two original modes constitute a pair of Bogoliubov modes, one of which mediates into the interaction while the other 'dark mode' is decoupled with the system. When the dissipation process is dominant over the amplification process, the Bogoliubov mode would evolve into a squeezed vacuum state, causing the generation of entanglement. This entangled state is successively transferred to two rotating mirrors via the beam splitter interaction. Obviously, the macroscopic entanglement is essentially originated from dissipation rather than radiation torque. Second, we find that the optimal macroscopic entanglement can be obtained when the driving field is nearly resonant with the atomic transition. There exists two extreme cases where the entanglement is vanished. At the exact resonant condition, the entanglement disappears since the dissipation rate is negligible when the dressed-state populations are identical. For the large detuning cases, the entanglement is also significantly suppressed due to the reduction squeezing parameter. Therefore, the best entanglement is obtained at an appropriate value of driving detuning. Third, we note that the macroscopic entanglement can be obtained when the damping rates of the atoms are much larger than the cavity losses, i.e. $\gamma\gg\kappa$. As a result, the atomic system can be viewed as a reservoir and the treatment of adiabatically eliminating atomic variables is valid. We extract the atomic contribution of the dissipative reservoir effects clearly and the quantum noises from atoms are included to calculate the quantum entanglement. Our results explore that the atomic damping rates have positive effects on macroscopic entanglement, which is completely different from the schemes in [65]. Finally, comparing with previous schemes [61–65], the macroscopic entanglement can be simply controlled by the strength and detuning of the driving field on atoms, which may provide conveniences for experimental implement.

The remaining part of the present paper is organized as follows. In section 2, we describe the system model of the double-Laguerre–Gaussian-cavity (DLGC) system. The master equation and Heisenberg–Langevin equations for the cavity modes and rotating mirrors are derived. In section 3 we present the physical mechanisms and discuss the numerical results of the remote mirror–mirror entanglement with two different methods. Finally, the conclusion is given in section 4.

The rotational DLGC system under consideration is presented in figure 1(a), in which the system consists of two fixed mirrors (FM$_{1,2}$) and two rotational mirrors (RM$_{1,2}$). We consider that a two-level atomic ensemble is placed at the cross site of the two cavities. The RM mirrors are mounted on the support points S$_{1,2}$ and can rotate around the axis Z$_{1,2}$, respectively. Without loss of generality, we assume that the fixed mirrors FM$_{1,2}$ are partially transparent while RM$_{1,2}$ have perfect reflection. When the LG beams G$_{1,2}$ with 0 charge are incident on FM$_{1,2}$, we only consider the transmitted beams with unchanged 0 charge since the reflected components do not interact with the atomic ensemble. The two 0 charge beams reflected from the RM$_{1,2}$ can get charged to $+2l_{1,2}$. It has been explored in experiment that LG beams can exert a torque on microscopic particles, in which the optical vortices are generated [13]. In addition, Bhattacharya and Meystre et al use OAM transfer from a LG beam to trap and cool the rotational motion of a macroscopic mirror made of a perfectly reflecting spiral phase element [14]. Assuming both RMs have the same mass m and radius r₀, their moments of inertia about the axis Z$_{1,2}$ passing through the center are $I_{1,2} = mr_{0}^{2}/2$. The RM$_{1,2}$ oscillate like two pendulums with angular frequencies $\omega_{\phi_{1,2}}$. Notably, the RM$_{1,2}$ act as two harmonic oscillators for the angular deviations $\phi_{1,2}\ll2 \pi$ and they have the equilibrium position $\phi_{10,20} = 0$. Apart from the internal spin angular moment ($\ll\hbar$), photons in LG beams carry integral OAM numbers $\hbar l_{1,2}$, respectively [2]. Therefore, when the LG beams are incident on the cavity, they can transfer torques $\hbar\xi_{\phi_{j}} = cl_{j}\hbar/L_{j}(j = 1,2)$ per photon to the RM$_{1,2}$, in which c represents the velocity of light and $L_{1,2}$ are the length of the two cavities. Then the mechanical effects of the LG beams interacting with the macroscopic rotating mirrors should be considered. The Hamiltonians of the DLGC system plus the atomic ensemble are written as $(\hbar = 1)$

$$\begin{align} H_{0}&=\omega_{21}\sigma_{22}- \frac{\Omega}{2}(\sigma_{12}\mathrm{e}^{i\omega_{0} t} + \sigma_{21}\mathrm{e}^{-i\omega_{0} t}), \end{align} \tag{ 1 }$$

$$\begin{align} H_{1}&=\sum_{j=1}^{2}\left[ \nu_{j}a_{j}^{\dagger}a_{j}+(g_{j}a_{j}\sigma_{21}+g_{j}a_{j}^{\dagger}\sigma_{12})\right], \end{align} \tag{ 2 }$$

$$\begin{align} H_{2}&=\sum_{j=1}^{2}\left[ \frac{L_{z_{j}}^{2}}{2I_{j}}+ \frac{1}{2}I_{j}\omega_{\phi_{j}}^{2}\phi_{j}^{2} -\xi_{\phi_{j}}a_{j}^{\dagger}a_{j}\phi_{j} +i(\varepsilon_{j}a_{j}^{\dagger}\mathrm{e}^{-i\omega_{L_{j}}}-\varepsilon_{j}^{*}a_{j}\mathrm{e}^{i\omega_{L_{j}}})\right], \end{align} \tag{ 3 }$$

wherein H₀ represents the system Hamiltonian for the two-level atoms driven by a strong field with frequency ω₀ and Rabi frequency Ω. The atomic transition frequency is denoted by ω₂₁. $\sigma_{lm} = \sum_{\mu = 1}^{N}|l_{\mu}\rangle\langle m_{\mu}|(l,m = 1,2)$ are the projection operators of N independent atoms for l = m and the flip operators for l ≠ m. H₁ describes the free Hamiltonian of the two cavity modes plus the interaction Hamiltonian between the atoms and cavity modes with coupling constants $g_{j}(j = 1,2)$. a_j ($a_{j}^{\dagger}$) are the annihilation (creation) operators for the two cavity modes with the frequencies ν_j . H₂ denotes the free Hamiltonians of the two rotational cavity mirrors, the interaction between the cavity modes and two rotating mirrors, the driving Hamiltonian to the cavity modes by the external laser fields with frequencies $\omega_{L_{j}}$ and the amplitudes of $|\varepsilon_{j}| = \sqrt{2P_j\kappa_{j}/\hbar\omega_{L_j}}$, wherein κ_j are the cavity damping rates and P_j input power of the laser. $L_{z_{j}}$ are the angular momentum of RM about the axis with the commutation relation [$L_{z_{j}}$, φ_k ] = $-i\hbar \delta_{jk}(j,k = 1,2)$. According to [14], we can define

$$\begin{align} \phi_{j}&=\sqrt{\frac{\hbar}{2I_j\omega_{\phi_j}}}(b_{j}^{\dagger}+b_{j}),\\ L_{z_{j}}&=i\sqrt{\frac{\hbar I_j\omega_{\phi_j}}{2}}(b_{j}^{\dagger}-b_{j}),\nonumber \end{align} \tag{ 4 }$$

in which the annihilation and creation operators for the rotating mirrors are denoted by b_j and $b_{j}^{\dagger}$, respectively. Substituting the above expression into equation (3), making a unitary transform with $U =$exp$(-iH^{^{^{\prime}}}t)$ and $H^{^{^{\prime}}} = \omega_{0}(a_{1}^{\dagger}a_{1}+a_{2}^{\dagger}a_{2}+\sigma_{22})$, we can rewrite the Hamiltonian H₀ and H₁ as

$$\begin{align} \tilde{H}_{0}&=\Delta\sigma_{22}- \frac{\Omega}{2}(\sigma_{12} + \sigma_{21}), \end{align} \tag{ 5 }$$

$$\begin{align} \tilde{H}_{1}&=\sum_{j=1}^{2}[\Delta_{c_{j}}a_{j}^{\dagger}a_{j}+(g_{j}a_{j}\sigma_{21}+g_{j}a_{j}^{\dagger}\sigma_{12})], \end{align} \tag{ 6 }$$

$$\begin{align} \tilde{H}_{2}&=\sum_{j=1}^{2}[\omega_{\phi_{j}}b_{j}^{\dagger}b_{j}-G_{j}a_{j}^{\dagger}a_{j}(b_{j}^{\dagger}+b_{j})+i\varepsilon_{j} (a_{j}^{\dagger} \mathrm{e}^{-i\delta_{j}t}-a_{j} \mathrm{e}^{i\delta_{j}t})], \end{align} \tag{ 7 }$$

where $\Delta_{c_{j}} = \nu_{j}-\omega_{0}$, $\Delta = \omega_{21}-\omega_{0}$, $\delta_{j} = \omega_{L_j}-\omega_{0}$ are cavity detunings, atomic detunings and driving detunings, respectively. The coupling parameters G_j are defined as $G_{j} = \frac{cl_{j}}{L_{j}}\sqrt{\frac{\hbar}{2I\omega_{\phi_{j}}}} (j = 1,2)$.

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The atomic relaxation, the cavity losses and the intrinsic damping rates of rotating mirrors are taken the forms as

$$\begin{align} \mathcal{L}_{a}{\rho}&=\frac{\gamma}{2}(2\sigma_{12}\rho\sigma_{21}-\sigma_{21}\sigma_{12}\rho-\rho\sigma_{21}\sigma_{12}), \end{align} \tag{ 8 }$$

$$\begin{align} \mathcal{L}_{c}{\rho}&=\sum_{j=1,2}\frac{\kappa_{j}}{2}(2a_{j}\rho a_{j}^{\dagger}-\rho a_{j}^{\dagger}a_{j}-a_{j}^{\dagger}a_{j}\rho), \end{align} \tag{ 9 }$$

$$\begin{align} \mathcal{L}_{b}{\rho}&=\sum_{j=1,2}\frac{\gamma_{\phi_{j}}}{2}(2b_{j}\rho b_{j}^{\dagger}-\rho b_{j}^{\dagger}b_{j}-b_{j}^{\dagger}b_{j}\rho). \end{align} \tag{ 10 }$$

To obtain the reduced master equation of cavity modes, as proposed in [66], we resort to the dressed atomic picture by diagonalizing the Hamiltonian $\tilde{H}_{0}$ under the conditions of $|\Omega|\gg \gamma, \kappa_{j}, g_{j}$. The dressed atomic states are expressed in terms of bare states as [67]

$$\begin{align} |\tilde{1}\rangle&=\cos\theta|1\rangle-\sin\theta|2\rangle,\\ |\tilde{2}\rangle&=\sin\theta|1\rangle+\cos\theta|2\rangle,\nonumber \end{align} \tag{ 11 }$$

in which $\cos\theta = \sqrt{\frac{1}{2} + \frac{d}{2 \sqrt{1 + d^2}}}$ and $\sin\theta = \sqrt{\frac{1}{2} - \frac{d}{2 \sqrt{1 + d^2}}}$ with normalized detuning $d = \frac{\Delta}{\Omega}$. The dressed states $|\tilde{1}\rangle$ and $|\tilde{2}\rangle$ have their eigenvalues $\lambda_{1,2} = \frac{1}{2} (\Delta \mp \tilde{\Omega})$ with $\tilde{\Omega} = \sqrt{\Delta^2 + \Omega^2}$, respectively. Now the system Hamiltonian of $\tilde{H}_{0}$ is rewritten in the dressed-state picture as $H_{d} = \sum_{j}\lambda_{j}\sigma_{\tilde{j}\tilde{j}} (j = 1,2)$. By transforming the bare atomic relaxation term (8) into the dressed-state picture and neglecting the quantized mods temporarily, the steady-state populations of the dressed states $N_{j} = \langle\sigma_{\tilde{j}\tilde{j}}\rangle (j = 1,2)$ are obtained as

$$\begin{align} N_{1}=\frac{N\cos^4{\theta}}{\cos^4{\theta} + \sin^4{\theta}}, \quad N_{2}= \frac{N\sin^4{\theta}}{\cos^4{\theta} + \sin^4{\theta}}. \end{align} \tag{ 12 }$$

Clearly, for $\Delta = 0$, we have $\sin\theta = \cos\theta = \frac{1}{2}$ and $N_{1} = N_{2} = \frac{N}{2}$, namely, the populations of the two dressed states are identical at the exact resonant condition. For $\Delta\neq0$, we have $N_{1}\gt N_{2}$ (or $N_{1}\lt N_{2})$ when $\Delta\gt0$ (or $\Delta\lt0)$. The equilibrium is broken and then the dissipative atomic reservoir effect is possible [68]. Making a further unitary transformation with $U^{^{^{\prime}}} = \exp(-iH_{d}t)$ on the Hamiltonian $\tilde{H}_{1}$, the effective Hamiltonian is derived as

$$\begin{align} H_{\textrm{eff}}= (g_1\cos^2{\theta}a_1-g_2\sin^2{\theta}a_2^\dagger)\sigma_{\tilde{2} \tilde{1}}+\textrm{H.c.}, \end{align} \tag{ 13 }$$

in which we take the conditions of $\Delta_{c_{1}} = -\Delta_{c_{2}} = \tilde{\Omega}$, i.e. the cavities are tuned to be resonant with the Rabi sidebands. If the cavity detunings are tuned to be comparable to or larger than the level spacing $\tilde{\Omega}$, the effective Hamiltonian is no longer valid. Under the good-cavity limit of $\gamma\gg\kappa_{{j}}$($j = 1,2$), the atomic variable can be adiabatically eliminated using the standard quantum optics techniques [69, 70]

$$\begin{align} \dot{\rho}_c = -i\textrm{Tr}_a{[H_{\textrm{eff}}, \rho_{c}]}+\mathcal{L}_c \rho_c. \end{align} \tag{ 14 }$$

Finally, the reduced master equation of the cavity modes takes the form

$$\begin{align} \dot{\rho}_c =&\ A_{1}(a_{1}\rho_{c}a_{1}^{\dagger}-\rho_{c}a_{1}^{\dagger}a_{1}) +A_{2}(a_{2}\rho_{c}a_{2}^{\dagger}-\rho_{c}a_{2}^{\dagger}a_{2})\\ &\ +B_{1}(a_{1}^{\dagger}\rho_{c}a_{1}-\rho_{c}a_{1}a_{1}^{\dagger}) +B_{2}(a_{2}^{\dagger}\rho_{c}a_{2}-\rho_{c}a_{2}a_{2}^{\dagger})\nonumber\\ &\ -C_{1}(a_{1}\rho_{c}a_{2}-\rho_{c}a_{2}a_{1})-C_{2}(a_{2}\rho_{c}a_{1}-\rho_{c}a_{1}a_{2}) \nonumber\\ &\ -C_{1}(a_{2}^{\dagger}\rho_{c}a_{1}^{\dagger}-\rho_{c}a_{1}^{\dagger}a_{2}^{\dagger}) -C_{2}(a_{1}^{\dagger}\rho_{c}a_{2}^{\dagger}-\rho_{c}a_{1}^{\dagger}a_{2}^{\dagger})+\textrm{H.c.},\nonumber \end{align} \tag{ 15 }$$

where the parameters are $A_{j} = (n_{j}+1)\kappa_{j}+N_{j} \xi_{j}$, $B_{j} = n_{j}\kappa_{j}+N_{3-j}\xi_{j}$, and $C_{j} = N_{j}\sqrt{\xi_{1}\xi_{2}}$ ($j = 1,2$) with the coefficients $\xi_{1} = 2g_{1}^{2}\cos^{4}{\theta}/(\gamma+\gamma_{c})$, $\xi_{2} = 2g_{2}^{2} \sin^{4}{\theta}/(\gamma +\gamma_{c})$, $\gamma_{c} = \gamma\sin^{2}{2\theta}/2$. Here n_j represent the mean thermal photon number of the cavities.

In present scheme, we focus on investigating macroscopic entanglement between two rotating mirrors. To do so, we make use of reduced master equation (15) and the Hamiltonian $\tilde{H}_{2}$ to obtain the dynamics equation as

$$\begin{align} \dot{a}_1&=-(\kappa_{a_{1}}-i\omega_1)a_{1}+\chi a_{2}^{\dagger}+\varepsilon_{1}\mathrm{e}^{-i\delta_1 t}+F_{1}, \nonumber \\ \dot{a}_2&=-(\kappa_{a_{2}}-i\omega_2)a_{2}-\chi a_{1}^{\dagger}+\varepsilon_{2}\mathrm{e}^{-i\delta_2 t}+F_{2},\\ \dot{b}_j&=-(\gamma_{\phi_j}+i\omega_{\phi_j})b_j+iG_j a_{j}^{\dagger}a_j+\sqrt{2\gamma_{\phi_{j}}}F_{b_{j}}.\nonumber \end{align} \tag{ 16 }$$

For simplicity, we define $\kappa_{a_{1}} = \kappa_{1}-(N_2-N_1)\xi_1$, $\kappa_{a_{2}} = \kappa_{2}+(N_2-N_1)\xi_2$, $\omega_j = G_j\langle b_j+b_j^\dagger \rangle (j = 1,2)$, $\chi = C_1-C_2$. F_j are noise operators including the cavity losses and the atomic contribution part. $F_{b_{j}}$ represent the mechanical noise operators coupling to the rotating mirrors from the thermal environment. Without loss of generality, we have vanishing mean $\langle F_{x}(t)\rangle = 0$ and the nonzero second order correlation terms $\langle F_{x}(t)F_{y}(t^{^{\prime}})\rangle = 2D_{xy}\delta(t-t^{^{\prime}})$ as

$$\begin{align} \langle F_{j}(t)F_{j}^{\dagger}(t^{^{\prime}})\rangle&=2A_{j} \delta{(t-t^{^{\prime}})}, \quad \langle F_{j}^{\dagger}(t)F_{j}(t^{^{\prime}})\rangle=2B_{j} \delta{(t-t^{^{\prime}})},\\ \langle F_{1}(t)F_{2}(t^{^{\prime}})\rangle&=2C_{1}\delta{(t-t^{^{\prime}})}, \quad \langle F_{2}(t)F_{1}(t^{^{\prime}})\rangle=2C_{2}\delta{(t-t^{^{\prime}})},\nonumber\\ \langle F^{\dagger}_{b_{j}}(t^{^{\prime}})F_{b_{j}}(t)\rangle&=n_{\phi_j} \delta{(t-t^{^{\prime}})}, \quad \langle F_{b_{j}}(t)F^{\dagger}_{b_{j}}(t^{^{\prime}})\rangle=(n_{\phi_j}+1) \delta{(t-t^{^{\prime}})},\nonumber \end{align} \tag{ 17 }$$

wherein $n_{\phi_{j}} = [\textrm{exp}\frac{\hbar\omega_{\phi_{j}}}{/k_\mathrm{B}T_{j}}-1]^{-1}$ is the mean occupation number, $k_\mathrm{B}$ is the Boltzmann constant and T_j the environmental temperature of the mechanical resonator.

According to equation (16), by defining $a_j = \tilde{a}_{j}\mathrm{e}^{-i\delta_j t}$, we can neglect the the highly oscillating terms for $\delta_1+\delta_2\gg1$ and then the steady-state solutions for $\langle \tilde{a}_{j}\rangle$ and $\langle b_{j}\rangle$ are derived as

$$\begin{align} \langle\tilde{a}_{j}\rangle&=\frac{\varepsilon_{j}}{\kappa_{a_j}-i (\delta_j+\omega_{j})}, \nonumber \\ \langle b_j\rangle&=\frac{(\omega_{\phi_j}+i\gamma_{\phi_j})G_j}{\gamma_{\phi_j}^2 +\omega_{\phi_j}^2}\langle a_{j}^{\dagger} a_{j}\rangle. \end{align} \tag{ 18 }$$

The equation for the intracavity mean photon numbers is given by

$$\begin{align} I_{a_j}[\kappa_{a_j}^{2}+(\delta_j+\eta_{j}I_{a_j})^{2}] = |\varepsilon_j|^2, \end{align} \tag{ 19 }$$

in which we define $\eta_{j} = 2\omega_{\phi_j}G_j/(\gamma_{\phi_j}^2+\omega_{\phi_j}^{2})$, $I_{a_j} = \langle a_{j}^{\dagger}a_{j} \rangle = \langle \tilde{a}_{j}^{\dagger} \tilde{a}_{j} \rangle$. It is clear that the two rotating mirrors will exhibit optical bistable behavior and experience strong nonlinearities, which can be controlled by the detuning and strength of the driving field on atoms.

To explore the macroscopic entanglement between two rotating mirrors, we can linearize the equation (16) around the semiclassical state corresponding to a working point in the stable range, i.e. $\hat{o} = \langle \hat{o}_{s}\rangle+\delta \hat{o}$. The steady state solutions $\langle \hat{o}_{s}\rangle$ are given in equation (18) and $\delta \hat{o}$ represents the quantum fluctuation around the mean value. Furthermore, we introduce the slowly varying fluctuation operators $\delta{a_j} = \delta{\tilde{a}_j}\mathrm{e}^{-i\delta_j t}$ and $b_j = \tilde{b}_j \mathrm{e}^{-i\omega_{\phi_j}t}$. By dropping the high-frequency oscillating terms $\exp[-i(\delta_{1}+\delta_{2})t]$ at $\delta_j = \omega_{\phi_j}$, the corresponding linear quantum Langevin equations for the quantum fluctuations are obtained as

$$\begin{align} \delta{\dot{a}_{1}} &= -(\kappa_{a_{1}}-i\omega_{1})\delta{a_1}+\chi\delta{a_{2}^{\dagger}} + i\tilde{g}_{1}\delta{\tilde{b}_1}+F_1,\\ \delta{\dot{a}_{2}} &= -(\kappa_{a_{2}}-i\omega_{2})\delta{a_2}-\chi\delta{a_{1}^{\dagger}}+ i\tilde{g}_{2}\delta{\tilde{b}_2}+F_2,\nonumber\\ \delta{\dot{\tilde{b}}_j}&=-\gamma_{\phi_j} \delta{\tilde{b}_j}+i\tilde{g}_{j}^{*}\delta{a_j}+\tilde{F}_{b_{j}},\nonumber \end{align} \tag{ 20 }$$

wherein $\tilde{F}_{b_{j}} = \sqrt{2 \gamma_{\phi_j}}F_{b_{j}} \mathrm{e}^{i \omega_{\phi_j}t}$, $\tilde{g}_{j} = G_j \langle \tilde{a}_{j}\rangle (j = 1,2)$. Then the quantum entanglement between the two mirrors can be calculated and discussed according to the above equations, which will be presented in the following section.

In this section, we would like to elucidate the mechanism for the generation of rotational optomechanical entanglement. Next, the numerical results of mirror–mirror entanglement are presented by nonadiabatical eliminating and adiabatical eliminating of cavities, respectively. The possible experimental parameters are briefly discussed in the last subsection.

3.1. Physical mechanism analysis

3.1.1. Dissipation of cavity fields in light of Bogoliubov modes

In order to describe the internal mechanisms for the generation of quantum entanglement more clearly, we define a pair of Bogoliubov modes in equation (13) for the cavity fields: $c_{1} = a_{1}\cosh r-a_{2}^{\dagger}\sinh r$, $c_{2} = a_{2}\cosh r-a_{1}^{\dagger}\sinh r$ with the squeezing parameter $\tanh r = \tan^{2}\theta$ for $\Delta\gt0$ and $\tanh r = \cot^{2}\theta$ for $\Delta\lt0$. The interaction between the dressed atoms and the Bogoliubov mode is shown in figure 1(b). Correspondingly, the system Hamiltonian of equation (13) is simply rewritten as

$$\begin{align} \tilde{H}_{\textrm{eff}}&=G_{\textrm{eff}}(c_{1}\sigma_{\tilde{2}\tilde{1}}+\sigma_{\tilde{1}\tilde{2}}c_{1}^{\dagger})\quad \textrm{for} \quad \Delta>0,\\ \tilde{H}_{\textrm{eff}}&=-G_{\textrm{eff}}(c_{2}^{\dagger}\sigma_{\tilde{2}\tilde{1}}+\sigma_{\tilde{1}\tilde{2}}c_{2})\quad \textrm{for} \quad \Delta<0,\nonumber \end{align} \tag{ 21 }$$

where the effective coupling constant $G_{\textrm{eff}} = g\sqrt{|\cos2\theta|}$ by assuming $g_{1} = g_{2} = g$. Note that only the collective mode c₁ (or c₂) mediates into the interaction while the other mode c₂ (or c₁) is decoupled with the system for the two cases. This is termed as a single pathway of Bogoliubov dissipation, which can lead to the occurrence of quantum entanglement [66, 68]. Under the adiabatic elimination conditions of $\gamma\gg \kappa_{1,2}$, we can obtain the atomic contribution part of the reduced master equation for the Bogoliubov mode $c_{1,2}$ as

$$\begin{align} \dot{\mathcal{\rho}}_{c_{1}}&=\mathcal{A}_{1}(2c_{1}\mathcal{\rho}c_{1}^{\dagger}-c_{1}^{\dagger}c_{1}\mathcal{\rho}-\mathcal{\rho}c_{1}^{\dagger}c_{1})+ \mathcal{B}_{1}(2c_{1}^{\dagger}\mathcal{\rho}c_{1}-c_{1}c_{1}^{\dagger}\mathcal{\rho}-\mathcal{\rho}c_{1}c_{1}^{\dagger})\quad \textrm{for} \quad \Delta>0,\\ \dot{\mathcal{\rho}}_{c_{2}}&=\mathcal{A}_{2}(2c_{2}\mathcal{\rho}c_{2}^{\dagger}-c_{2}^{\dagger}c_{2}\mathcal{\rho}-\mathcal{\rho}c_{2}^{\dagger}c_{2})+ \mathcal{B}_{2}(2c_{2}^{\dagger}\mathcal{\rho}c_{2}-c_{2}c_{2}^{\dagger}\mathcal{\rho}-\mathcal{\rho}c_{2}c_{2}^{\dagger})\quad \textrm{for} \quad \Delta<0,\nonumber \end{align} \tag{ 22 }$$

in which $\mathcal{A}_{1} = \mathcal{B}_{2} = 2G_{\textrm{eff}}^{2}N_{1}/\Gamma$ and $\mathcal{A}_{2} = \mathcal{B}_{1} = 2G_{\textrm{eff}}^{2}N_{2}/\Gamma$ with $\Gamma = \frac{\gamma}{2}+\frac{\gamma}{4}\sin^{2}2\theta$. It is seen that the $\mathcal{A}$ terms represent the dissipation and the $\mathcal{B}$ terms denote the excitation of Bogouliubov modes, respectively. When dissipation rate $\mathcal{R} = \mathcal{A}-\mathcal{B}\gg\kappa$, the Bogoliubov mode c₁ (or c₂) will evolve into the squeezed vacuum state while the other mode c₂ (or c₁) is decoupled with the system. Generally, the larger the dissipation rate is, the stronger the entanglement will be. For $\Delta = 0$, we have $N_{1} = N_{2}$ and $\mathcal{R} = 0$, leading to the absence of quantum entanglement, which is verified by the following numerical calculations. However, when the normalized detuning d deviates slightly away from the resonant conditions, we have $\mathcal{R}\gg\kappa$ in the positive ($\Delta\gt0$) and negative frequency regions ($\Delta\lt0$) due to $N_{1}\neq N_{2}$, thus resulting in the appearance of entanglement. For example, at d = 0.3, we have $N_{1}/N = 0.77$, $N_{2}/N = 0.23$ and $\mathcal{R} = 1.7\gamma\gg\kappa$, meaning that the adiabatical elimination of atomic variables and the single-pathway dissipation of Bogoliubov are valid. On the other hand, the entanglement is simultaneously determined by the squeezing parameter r. When d is increased, the squeezing parameter r is inversely decreased accompanying by the increasing of the dissipation rate $\mathcal{R}$. For example, the squeezing parameter ($\tanh r$ = $\tan^{2}\theta$) is r = 1.15 at d = 0.1, r = 0.41 at d = 0.5, and r = 0.17 at d = 1. As a consequence, the best entanglement appears at the near-resonant conditions when the dissipation rate and the squeezing parameter r have a compatible value.

3.1.2. Entangled state transfer from cavity fields to rotating mirrors

Next, we would like to clarify the frequency arrangement of the present system. As shown in figure 2(a), on the one hand, the cavity fields $a_{1,2}$ are tuned to be resonant with the sidebands, i.e. $\nu_{j} = \omega_{0}\pm\tilde{\Omega}$. For the cavity detunings $\Delta_{c_{1}} = -\Delta_{c_{1}} = \tilde{\Omega}$, we have $\nu_{j}-\Delta_{c_{j}} = \omega_{0}$. On the other hand, it is seen that the frequencies of the rotating mirrors $\omega_{\phi_{j}}$ and the frequencies of driving fields on cavities $\omega_{L_{j}}(j = 1,2)$ satisfy the relations of $\omega_{L_{j}}-\omega_{\phi_{j}} = \omega_{0}$. Due to the driving detunings $\delta_{j} = \omega_{L_{j}}-\omega_{0}$, we have the relations of $\delta_{j} = \omega_{\phi_{j}}$, giving rise to that the high-frequency oscillating terms $\textrm{exp}[-i(\delta_{1}+\delta_{2})]$ in equation (20) should be dropped. Obviously, these terms with the form of $\tilde{g}_{j}(\delta a_{j}\delta b_{j}+\delta a_{j}^{\dagger}\delta b_{j}^{\dagger})$ describing parametric amplification are absent while the terms of $\tilde{g}_{j}(\delta a_{j}\delta b_{j}^{\dagger}+\delta a_{j}^{\dagger}\delta b_{j})(j = 1,2)$ being responsible for quantum state transfer are existent [71]. In addition, as shown in figure 2(b), the internal physical mechanisms for the generation of macroscopic entanglement are depicted in detail. For $\Delta\gt0$, two original modes $a_{1,2}$ constitute a pair of collective modes $c_{1,2}$, in which the 'bright mode' c₁ would dissipate into the atomic reservoir for $\mathcal{R}\gt0$ while the other 'dark mode' c₂ (not shown) is decoupled with the system. As a result, the bipartite photon–photon entanglement is essentially established by the atomic reservoir effects and then it is transferred to two macroscopic rotating mirrors at $\delta_{j} = \omega_{\phi_{j}}$ via the two transfer processes $a_{j}\leftrightsquigarrow b_{j}$. Finally, it is worthwhile to point out that the present system is completely different from the previous schemes proposed in [65]. In their works, the two-level atomic ensemble or magnon is placed into the cavity to enhance the cavity–mirror entanglement or to realize the tripartite entanglement based on the parametric interaction.

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3.2. Rotating mirror–mirror entanglement without adiabatic elimination of cavities

In the first of place, we assume that the cavity modes are not adiabatically eliminated. To numerically calculate the quantum correlations between the two rotating mirrors, two pairs of quadrature operators are defined as $\delta{x_j} = (\delta{a_j}+\delta{a_{j}^{\dagger}})/\sqrt{2}$, $\delta{p_j} = -i(\delta{a_j}-\delta{a_{j}^{\dagger}})/\sqrt{2}$, $\delta{\phi_j} = (\delta{\widetilde{b}_j}+\delta{\widetilde{b}_{j}^{\dagger}})/\sqrt{2}$, $\delta{L_j} = -i(\delta{\widetilde{b}_j}-\delta{\widetilde{b}_{j}^{\dagger}})/\sqrt{2}$. Then we can write the dynamical equations of quantum fluctuations in a concise form

$$\begin{align} \dot{\vec{u}} = M\vec{u}(t) + \vec{F}(t), \end{align} \tag{ 23 }$$

where $\vec{u}(t) = (\delta{x_1},\delta{p_1},\delta{x_2},\delta{p_2},\delta{\phi_1},\delta{L_1},\delta{\phi_2},\delta{L_2})^T$, $\vec{F}(t) = (F_{x_1},F_{p_1},F_{x_2},F_{p_2},F_{\phi_1},F_{L_1},F_{\phi_2},F_{L_2})^T$. The corresponding composite noise operators are written as $F_{x_j} = (F_j + F_{j}^{\dagger})/\sqrt{2}$, $F_{p_j} = -i(F_j - F_{j}^{\dagger})/\sqrt{2}$, $F_{\phi_j} = (\tilde{F}_{b_{j}}+\tilde{F}_{b_{j}}^{\dagger})/\sqrt{2}$, $F_{L_j} = -i(\tilde{F}_{b_{j}}-\tilde{F}_{b_{j}}^{\dagger})/\sqrt{2}$. The drift matrix M takes the form

$$\begin{align} \left( \begin{array}{cccccccc} -\kappa_{a_{1}} &-\omega_{1} & \chi & 0 & -\mu_{1} & \mu_{2} & 0 & 0 \\ \omega_{1} & -\kappa_{a_{1}} & 0 & -\chi & \mu_{2} & -\mu_{1} & 0 & 0 \\ -\chi & 0 & -\kappa_{a_{2}} & -\omega_{2} & 0 & 0 & -\mu_{3} & \mu_{4} \\ 0 & \chi & \omega_{2} & -\kappa_{a_{2}} & 0 & 0 & \mu_{4} & -\mu_{3} \\ -\mu_{1} & \mu_{2} & 0 & 0 & -\gamma_{\phi_{1}} & 0 & 0 & 0 \\ \mu_{2} & -\mu_{1} & 0 & 0 & 0 & -\gamma_{\phi_{1}} & 0 & 0 \\ 0 & 0 & -\mu_{3} & \mu_{4} & 0 & 0 & -\gamma_{\phi_{2}} & 0 \\ 0 & 0 & \mu_{4} & -\mu_{3} & 0 & 0 & 0 & -\gamma_{\phi_{2}} \\ \end{array} \right), \end{align} \tag{ 24 }$$

where in $\mu_{1} = \textrm{Im}\tilde{g}_{1}$, $\mu_{2} = \textrm{Re}\tilde{g}_{1}$, $\mu_{3} = \textrm{Im}\tilde{g}_{2}$, $\mu_{4} = \textrm{Re}\tilde{g}_{2}$. Generally, the system would be stable if the real parts of all eigenvalues of M are negative, which can be judged based on numerical calculation in the present scheme [72]. Throughout this paper, we always guarantee that the system is stable via choosing appropriate system parameters. When the stability conditions are fulfilled, the steady-state Lyapunov equation is given by [50]

$$\begin{align} MV + MV^T = -D, \end{align} \tag{ 25 }$$

in which D denotes the diffusion matrix and V is a $8\times 8$ covariance matrix (CM) with the matrix elements of $V_{ij} = \frac{1}{2}[\langle u_{i}(\infty)u_{j}(\infty)\rangle+\langle u_{j}(\infty)u_{i}(\infty)\rangle]$. The CM of two modes is taken the form as

$$\begin{equation} V=\left( \begin{array}{cc} V_{1} & V_{3} \\ V^{T}_{3} & V_{2} \\ \end{array} \right), \end{equation} \tag{ 26 }$$

wherein the matrix V₁, V₂ and V₃ are the $2\times2$ submatrices. The diffusion matrix D are defined as $D_{ij}\delta(t-t^{^{\prime}}) = \langle F_{i}(t)F_{j}(t^{^{\prime}})+F_{j}(t^{^{\prime}})F_{i}(t)\rangle/2$ and the nonzero diffusion coefficients are $D_{11} = D_{22} = A_{1}+B_{1}$, $D_{33} = D_{44} = A_{2}+B_{2}$, $D_{55} = D_{66} = \gamma_{\phi_{1}}(2n_{\phi_{1}}+1)$, $D_{77} = D_{88} = \gamma_{\phi_{2}}(2n_{\phi_{2}}+1)$, $D_{13} = D_{31} = -D_{24} =$ $-D_{42} = C_{1}+C_{2}$.

Once the CM of the system is achieved, one can calculate the degree of the mirror-mirror entanglement. We adopt a reliable logarithmic negativity criterion to study continuous variable quantum entanglement for Gaussian states [73, 74]. The definition of $E_\mathrm{N}$ is given by

$$\begin{equation} E_\mathrm{N} = \max{[0,-\ln{2\Lambda}]}, \end{equation} \tag{ 27 }$$

where $\Lambda = 2^{-1/2} [\Sigma - \sqrt{\Sigma^2 - 4\det{V}}]^{1/2}$ with $\Sigma = \det{V_1} + \det{V_2} - 2 \det{V_{3}}$.

In the following numerical calculations, the possible experimental parameters are listed in table 1 as proposed in [14, 61, 62]. In figure 3, the evolution of logarithmic negativity $E_\mathrm{N}$ is plotted versus the normalized detuning d and quality factor Q of cavities by choosing the OAM as $l_{1} = l_{2} = 40$. The angular frequencies of rotating mirrors are $\omega_{\phi_{1}} = 2\pi\times1$ MHz and $\omega_{\phi_{2}} = 2\pi\times10$ MHz. The cavity-atom coupling constants are always set as $g_{1} = g_{2} = 2\pi\times8.8$ MHz. It is found that the entanglement first increases to a maximal value and then decreases slowly to zero in the positive frequency region. At this time, the entanglement disappears in the negative frequency domain because the stability condition is not satisfied when $\omega_{\phi_{1}}\lt\omega_{\phi_{2}}$. On the contrary, the stability condition would be satisfied in the region of d < 0 for $\omega_{\phi_{1}}\gt\omega_{\phi_{2}}$. Specially, at d = 0, as shown in equation (22), the gain coefficients $\mathcal{B}_{j}$ are equal to the absorption coefficients $\mathcal{A}_{j}(j = 1,2)$ for $N_{1} = N_{2}$, yielding the disappearance of mirror–mirror entanglement together with the single-pathway dissipation. Differently, with the increasing of normalized detuning d, the best entanglement happens at an appropriate value of d. Besides, we notice that the entanglement is remarkably enhanced and the entanglement region becomes wide with high quality factor Q.

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In figure 4, the logarithmic negativity $E_\mathrm{N}$ is plotted as a function of the angular frequency ratio of $r_{\phi} = \omega_{\phi_{1}}/\omega_{\phi_{2}}$ for two cases: (a) $l_{1} = l_{2} = 40$; (b) $l_{1} = 40, l_{2} = 46$ by choosing different quality factor $Q = 1\times10^{7}$ (solid line), $Q = 2\times10^{7}$ (dashed line), $Q = 3\times10^{7}$ (dotted line), respectively. The normalized detuning d is chosen as d = 0.012 and the input powers are $P_{1} = P_{2} = 100$ mW. From these figures, it is found that the variation trend of $E_\mathrm{N}$ is similar to that in figure 3. Notably, at $r_{\phi} = 1$, the entanglement vanishes for $l_{1} = l_{2}$ while occurs for $l_{1}\neq l_{2}$, which can be attributed to that there is no exchange of total orbital angular moment between the cavity modes and rotating mirrors since the system is totally symmetrical [63]. Such a balance is broken for $l_{1}\neq l_{2}$, leading to the generation of quantum entanglement at $r_{\phi} = 1$.

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Next, we plot the entanglement evolution of $E_\mathrm{N}$ versus the input power $P = P_{1} = P_{2}$. The other parameters are chosen as $l_{1} = l_{2} = 40$, $\omega_{\phi_{1}} = 2\pi\times10$ MHz, $\omega_{\phi_{2}} = 2\pi\times 6$ MHz. In figure 5(a), logarithmic negativity $E_\mathrm{N}$ is plotted by choosing different quality factor $Q = 2\times 10^{6}$ (solid line), $4\times 10^{6}$ (dashed line), $6\times 10^{6}$ (dotted line) and the influence of the rotating mirror masses on quantum entanglement is shown in figure 5(b) by choosing $m_{1} = m_{2} = m = 50$ ng (solid line), 75 ng (dashed line), 100 ng (dotted line) for $Q = 2\times 10^{6}$, respectively. As seen from figure 5(a), we have $P_{\textrm{th}} = 274$ mW at $Q = 2\times 10^{6}$, $P_{\textrm{th}} = 135$ mW at $Q = 4\times 10^{6}$ and $P_{\textrm{th}} = 91$ mW at $Q = 6\times 10^{6}$. Obviously, the larger the quality factor is, the smaller the threshold power P_th will be. From figure 5(b), the threshold power P_th is changed from 136 mW, 205 mW to 271 mW when the mirror masses are increased from 50 ng to $100\,$ng, demonstrating that the macroscopic quantum effects are modified by the system parameters including mirror mass m, cavity length L and the mirror radius r₀ etc.

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The density plot of logarithmic negativity $E_\mathrm{N}$ versus the OAM l and the normalized detuning d is shown in figure 6(a) and the two-dimensional curves are plotted in figure 6(b) by setting $l = l_{1} = l_{2}$. The parameters are chosen as $\omega_{\phi_{1}} = 2\pi\times10$ MHz, $\omega_{\phi_{2}} = 2\pi\times6$ MHz, $\gamma = 2\pi \times 20$ MHz, $Q = 2\times10^{7}$ and the mirror masses are chosen as $m_{1} = m_{2} = 50$ ng. As shown in figure 6(a), when the absolute value of $|d|$ is increased, the entanglement appears and the region for $E_\mathrm{N}\gt0$ becomes wide. To show this characteristic more clearly, the two-dimensional curves of the logarithmic negativity $E_\mathrm{N}$ are plotted in figure 6(b) by choosing different normalized detuning d. It is seen that the best entanglement appears at an appropriate value of l, which can be used to detect topological charge value via the measurement of quantum entanglement.

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Figure 7 shows the evolution of logarithmic negativity $E_\mathrm{N}$ as a function of $\kappa/\gamma$ by choosing different OAM numbers of l for $\kappa_{1} = \kappa_{2} = \kappa$. It is clear that the good entanglement is obtained under the conditions of $\gamma\gg\kappa$. With the increasing of $\kappa/\gamma$, the values of logarithmic negativity $E_\mathrm{N}$ are monotonously decreased to zero at a specific value of $\kappa/\gamma$. By changing l from 200 to 500, the entanglement disappears at $\kappa/\gamma = 0.2$, 0.088, 0.05, 0.03, respectively. When the cavity losses are comparable to the atomic damping rates, the entanglement would be vanished, which is different from the results presented in [65]. What's more, the effects of thermal noise on quantum entanglement are also discussed in figure 8. The parameters are the same as those in figure 6(b) except for $d = -0.014$. As shown in figures 8(a) and (b), we find that the entanglement is relatively robust against environmental noise although it becomes worse as the thermal occupation numbers of $n_{\phi_{1,2}}$ and $n_{1,2}$ increase.

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3.3. Rotating mirror–mirror entanglement with adiabatic elimination of cavities

Usually, the damping rates of the rotating mirrors are much smaller than the cavity losses, i.e. $\gamma_{\phi} \ll \kappa_{j}$. Then we derive the fluctuations dynamical equations for the two rotating mirrors by adiabatically eliminating cavities as

$$\begin{align} \delta\dot{\tilde{b}}_1=-\alpha_1\delta{\tilde{b}_1}+\beta_1\delta{\tilde{b}_2^\dagger}+F^{^{\prime}}_{b_1},\\ \delta\dot{\tilde{b}}_2=-\alpha_2\delta{\tilde{b}_2}-\beta_2\delta{\tilde{b}_1^\dagger}+F^{^{\prime}}_{b_2},\nonumber \end{align} \tag{ 28 }$$

where the parameters $\alpha_1 = \gamma_{\phi_1}-|\tilde{g}_{1}|^{2}(\kappa_{a_{2}}-i\tilde{g}_2)/\Theta^*$, $\alpha_2 = \gamma_{\phi_2}-|\tilde{g}_{2}|^{2}(\kappa_{a_{1}}-i\omega_1)/\Theta$, $\beta_1 = \beta_2^{*} = \tilde{g}_{1}\tilde{g}_{2}\xi_{12}/\Theta^*$ with $\Theta = (\kappa_{a_{1}} - i\omega_1)(\kappa_{a_{2}}+i\omega_{2})+\xi_{12}^2$, $\xi_{12} = \xi_{1}-\xi_{2}$. The noise operators are given by

$$\begin{align} F^{^{\prime}}_{b_1}&=-u_{11}F_1^\dagger-u_{12}F_2 + \sqrt{2\gamma_{\phi_{1}}}\tilde{F}_{b_1}\\ F^{^{\prime}}_{b_2}&=u_{21}F_1 -u_{22}F_2^\dagger + \sqrt{2\gamma_{\phi_{2}}}\tilde{F}_{b_2}, \nonumber \end{align} \tag{ 29 }$$

in which we have the coefficients $u_{11} = -i\tilde{g}_{1}(\kappa_{a_{2}}-i \omega_2)/\Theta^*$, $u_{12} = i\tilde{g}_{1}\xi_{12}/\Theta^*$, $u_{21} = i\tilde{g}_{2}\xi_{12}/\Theta$ and $u_{22} = -i\tilde{g}_{2}(\kappa_{a_{1}}-i \omega_1)/\Theta$. From the equation (28), it is seen that the parametric interaction is hidden behind the effective Hamiltonian of the two rotating mirrors. This can be used to explain the physical origin of the remote mirror–mirror entanglement. It is worthwhile to point out that the parametric interaction arises from the two-photon process in dissipative atomic reservoir effects rather than the exchange of OAM. Interestingly, as show in equation (29), we find that the dissipation effects of the present system are determined by the following four factors: the atomic decay rate γ, the thermal noise of the mechanical oscillators $\gamma_{\phi_{j}}$, the cavity dissipation κ_j and the coherent-controlled dissipative atomic reservoir effect. Being different from the previous work [52–54], we find that the dissipative atomic reservoir plays an important role in generating rotating mirror–mirror entanglement, which may find potential applications in remote quantum communications.

In order to verify the validity of the adiabatic elimination approach, we follow the same procedure as in the preceding subsection to investigate the mirror–mirror entanglement with the same criterion. The dynamical equations of the quantum fluctuations of the rotating mirrors are written in a concise form as

$$\begin{align} \dot{\vec{u^{^{\prime}}}}(t)=M^{^{\prime}} \vec{u^{^{\prime}}}(t)+\vec{F^{^{\prime}}}(t), \end{align} \tag{ 30 }$$

in which the column vectors $\vec{u^{^{\prime}}}(t) = (\delta{\phi_1}, \delta L_1, \delta \phi_2, \delta L_2)^{T}$, $\vec{F^{^{\prime}}}(t) = (F^{^{\prime}}_{\phi_1}, F^{^{\prime}}_{L_1}, F^{^{\prime}}_{\phi_2}, F^{^{\prime}}_{L_2})^{T}$ and the drift matrix M^ʹ is derived as

$$\begin{align} M^{^{\prime}}&=\left( \begin{array}{cccc} -\alpha_{11}&\alpha_{12}&\beta_{11}&\beta_{12}\\ -\alpha_{12}&-\alpha_{11}&\beta_{12}&-\beta_{11}\\ -\beta_{22}&-\beta_{21}&-\alpha_{22}&\alpha_{21}\\ -\beta_{21}&\beta_{22}&-\alpha_{21}&-\alpha_{22} \end{array} \right), \end{align} \tag{ 31 }$$

where $\alpha_{11} = \textrm{Re}{\alpha_1}$, $\alpha_{22} = \textrm{Re}{\alpha_2}$, $\alpha_{12} = \textrm{Im}{\alpha_1}$, $\alpha_{21} = \textrm{Im}{\alpha_2}$, $\beta_{11} = \textrm{Re}{\beta_1}$, $\beta_{22} = \textrm{Re}{\beta_2}$, $\beta_{12} = \textrm{Im}{\beta_1}$ and $\beta_{21} = \textrm{Im}{\beta_2}$. The quadrature noise operators are defined as $F^{^{\prime}}_{\phi_j} = (F_{b^{^{\prime}}_j}+F_{b^{^{\prime}}_j}^{\dagger})/\sqrt{2}$ and $F^{^{\prime}}_{L_j} = (F_{b^{^{\prime}}_j}-F_{b^{^{\prime}}_j}^{\dagger})/{\sqrt{2}i}$. According to Lyapunov equation, we can calculate the quantum correlations between two mirrors numerically. Notably, the nonzero diffusion coefficients can be calculated based on equation (29). For simplicity, the cumbersome expression of the diffusion matrix is not presented here.

In figure 9, we plot the logarithmic negativity $E_\mathrm{N}$ versus the input power and OAM for adiabatical and nonadiabatical elimination cases. The parameters are chosen as $\omega_{\phi_{1}} = 2\pi\times10$ MHz, $\omega_{\phi_{2}} = 2\pi\times6$ MHz, $d = -0.014$, $\gamma = 2\pi\times20$ MHz, $m_{1} = m_{2} = 50$ ng, $Q_{1} = Q_{2} = 2\times10^{7}$, $\kappa_{1} = \kappa_{2} = 2\pi\times0.5$ MHz and the other parameters are the same as those in table 1. It is seen that the results of adiabatical elimination cases are in well agreement with the nonadiabatical elimination cases when the condition of $\gamma_{\phi}\ll\kappa_{a}$ is satisfied, indicating that both methods are valid for calculating the quantum entanglement.

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3.4. Experimental implementations

In summary, the macroscopic entanglement between two rotating mirrors are theoretically investigated based on the atomic reservoir effects in a DLGC system. It is found out that the microscopic photon–photon entanglement prepared by a single pathway of Bogoliubov dissipation can be transferred to two macroscopic rotating mirrors at proper frequency conditions. We explore that the optimal entanglement is obtained when the driving field is nearly resonant with the atomic transition and the stable entanglement is possible to obtain when the atomic decay rates are larger than the cavity losses. Finally, it turns out that two different methods with or without adiabatically elimination of cavities to calculate the mirror–mirror entanglement are equivalent. The present scheme provides a way to establish macroscopic entanglement between the mirrors without direct interaction, which may be useful for the long-distance quantum communications and quantum sensing technology.

This work is supported by the National Natural Science Foundation of China (Grant No. 11574179) and is funded by National '111 Research Center' Microelectronics Circuits.

No new data were created or analysed in this study.