Reservoir-engineered entanglement in an unresolved-sideband optomechanical system
本站小编 Free考研考试/2022-01-02
Yang-Yang Wang1,2, Rong Zhang1,2, Stefano Chesi3,4, Ying-Dan Wang,1,2,51Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China 2School of Physical Sciences, University of Chinese Academy of Sciences, No. 19A Yuquan Road, Beijing 100049, China 3Beijing Computational Science Research Center, Beijing 100193, China 4Department of Physics, Beijing Normal University, Beijing 100875, China 5Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University, Changsha 410081, China
National Natural Science Foundation of China.No. 11434011 National Natural Science Foundation of China.No. 11574330 National Natural Science Foundation of China.No. 1171101295 National Natural Science Foundation of China.No.11974040
Abstract We study theoretically the generation of strong entanglement of two mechanical oscillators in an unresolved-sideband optomechanical cavity, using a reservoir engineering approach. In our proposal, the effect of unwanted counter-rotating terms is suppressed via destructive quantum interference by the optical field of two auxiliary cavities. For arbitrary values of the optomechanical interaction, the entanglement is obtained numerically. In the weak-coupling regime, we derive an analytical expression for the entanglement of the two mechanical oscillators based on an effective master equation, and obtain the optimal parameters to achieve strong entanglement. Our analytical results are in accord with numerical simulations. Keywords：quantum entanglement;optomechanical cavity;the unresolved-sideband regime;reservoir engineering
PDF (756KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Yang-Yang Wang, Rong Zhang, Stefano Chesi, Ying-Dan Wang. Reservoir-engineered entanglement in an unresolved-sideband optomechanical system. Communications in Theoretical Physics, 2021, 73(5): 055105- doi:10.1088/1572-9494/abe2f8
1. Introduction
Quantum entanglement is a key concept in fundamental quantum physics and a crucial resource for quantum information and quantum communication [1]. Significant efforts have been devoted to generating quantum entanglement in several physical systems such as atoms [2, 3], trapped ions [4, 5] and defects in solid-state systems [6, 7]. Particularly, cavity optomechanical systems [8], which allow convenient manipulation and precise measurement of interacting mechanical and optical degrees of freedom, are promising candidates to study quantum mechanical features from mesoscopic to macroscopic scales, including quantum ground state cooling of mechanical modes [9–12], nonlinear quantum effects [13–16], nonreciprocity [17, 18], quantum entanglement and state transfer [19–25].
Of particular interest is the generation of entangled states with optomechanical systems, which is a difficult task due to unavoidable decoherence and dissipation induced by the environment. Reservoir engineering is a unique approach utilizing decoherence to achieve a certain steady state. Recently, several steady-state entanglement proposals have been put forward based on reservior engineering in superconducting qubits [26, 27] and superconducting resonators [28], Rydberg atoms [29], and trapped ions [30, 31]. Inspired by these schemes, the idea of generating entanglement in optomechanical systems was discussed as well. For example, steady-state two-mode mechanical squeezed states based on reservoir engineering have been proposed theoretically, with the aid of dissipation induced by two cavities [32]. Subsequently, entanglement of two cavities coupled to a mechanical oscillator, or two mechanical oscillators coupled to the same cavity, were discussed [33, 34]. Furthermore, Woolley and Clerk have studied two-mode mechanical squeezed states via a single reservoir which can cool both Bogoliubov modes [35]. Recently, the stabilized entanglement of two mechanical oscillators has been demonstrated experimentally [36] on the basis of previous theoretical proposals [32, 33, 35].
Note that all the above proposals strictly require the good-cavity limit, where the frequency of the mechanical oscillator is much larger than the dissipation of the cavity (i.e. ωm ≫ κc), and the counter-rotating interaction can be dropped. However, on one hand, it is difficult to achieve a high-finesse cavity; on the other hand, from a practical experimental perspective, being able to relax the good-cavity limit has benefits associated with a large κc, which allows one to use a small drive and still achieve a large effective cavity-mechanical coupling [37]. Luckily, to alleviate this limit, some approaches were proposed based on novel coupling mechanisms [37–39], parameter modulation [40, 41] and hybrid systems [42]. Recently, several schemes to achieve ground-state cooling and strong mechanical squeezing in the unresolved sideband regime were proposed, utilizing an auxiliary cavity [12, 43–45]. Nevertheless, entanglement generation via reservoir engineering is difficult to achieve in the unresolved sideband regime.
In this paper, we propose a scheme to generate steady-state entanglement of the two mechanical oscillators via reservoir engineering in the unresolved sideband regime, using the setup shown in figure 1. We add two auxiliary high-Q cavities coupled with a typical optomechanical cavity, which is driven on the red and blue side bands with different laser amplitudes. From numerical simulations, we find that strong entanglement can be generated in the unresolved sideband regime. Considering a weak-coupling condition, we regard the three cavities as a reservoir and hence obtain an effective master equation for the two mechanical oscillators. From this effective master equation, an analytical solution for the variance of the two mechanical oscillators can be derived. We find that the effect of counter-rotating terms can be suppressed by choosing optimal parameters, such that the frequencies of counter-rotating terms correspond to minima of the optical spectrum. This phenomenon is due to destructive quantum interference. Additionally, we find that having two mechanical modes with a sufficiency large mismatch in frequency allows for much stronger entanglement, because the two Bogoliubov operators which result from a canonical transformation of the two mechanical modes are both cooled efficiently at different frequencies of the cavity spectrum. However, our scheme also generates entanglement when the mechanical modes are degenerate in frequency. Finally, our theoretical results are compared with numerical simulations and good agreement is found.
Figure 1.
New window|Download| PPT slide Figure 1.System schematics. ${\hat{b}}_{i}$ and ${\hat{a}}_{i}$ denote the mechanical and cavity modes, respectively. κc, κ1 and κ2 are the dissipative rates of three cavity. γ1 and γ2 are the dissipative rates of the two mechanical modes. ωd1 and ωd2 are the red and blue detuning laser drives.
Our paper is organized as follows. In section 2, we describe in detail our system, review the principle of entanglement generation, and discuss the direct numerical simulations of our system. In section 3, we derive the effective master equation and obtain the expression of the steady-state mechanical variance in the weak-coupling regime with a nonzero detuning between the mechanical oscillators. In section 4, we analyze the optical spectrum of the three cavities. Furthermore, in section 5, we discuss in detail the optimal parameters to achieve maximum entanglement. Finally, additional discussions and our conclusion are given in sections 6 and 7.
2. System and Hamiltonian
As shown in figure 1, we consider a cavity optomechanical system involving two mechanical oscillators and a main cavity which is coupled to two auxiliary cavities. The Hamiltonian is$\begin{eqnarray}\begin{array}{rcl}\hat{H} & = & \displaystyle \sum _{i=1,2}[{{\rm{\Omega }}}_{i}{\hat{b}}_{i}^{\dagger }{\hat{b}}_{i}+{g}_{i}({\hat{b}}_{i}^{\dagger }+{\hat{b}}_{i}){\hat{a}}_{{c}}^{\dagger }{\hat{a}}_{{c}}+{J}_{i}({\hat{a}}_{i}^{\dagger }{\hat{a}}_{{c}}+{\hat{a}}_{{c}}^{\dagger }{\hat{a}}_{i})]\\ & & +\ \displaystyle \sum _{j={c},1,2}{\omega }_{j}{\hat{a}}_{j}^{\dagger }{\hat{a}}_{j}+{\hat{H}}_{\mathrm{dr}}+{\hat{H}}_{\mathrm{env}},\end{array}\end{eqnarray}$$\begin{eqnarray}{\hat{H}}_{\mathrm{dr}}=({\alpha }_{1}{{\rm{e}}}^{-{\rm{i}}{\omega }_{{\rm{d}}1}t}+{\alpha }_{2}{{\rm{e}}}^{-{\rm{i}}{\omega }_{{\rm{d}}2}t}){\hat{a}}_{{c}}^{\dagger }+{\rm{h}}.{\rm{c}}.,\end{eqnarray}$where ${\hat{b}}_{i}$$({\hat{b}}_{i}^{\dagger })$ are the annihilation (creation) operators of the mechanical modes, with frequencies ωi. ${\hat{a}}_{j}$$({\hat{a}}_{j}^{\dagger })$ are the annihilation (creation) operators of the cavity modes, with cavities frequencies ωj. gi are the single-photon optomechanical coupling strengths between the main cavity and the i mechanical oscillator. Ji are the coupling strengths between the main cavity and the i auxiliary cavity. ${\hat{H}}_{\mathrm{dr}}$ denotes the driving of the electromagnetic mode. To generate entanglement, red and blue lasers are applied to simultaneously drive the main cavity with the amplitudes α1, α2 and frequencies ωd1 = ωc + ωm, ωd2 = ωc − ωm, with ${\omega }_{{\rm{m}}}=\tfrac{{{\rm{\Omega }}}_{1}+{{\rm{\Omega }}}_{2}}{2}$. The last term ${\hat{H}}_{\mathrm{env}}$ accounts for dissipation of the modes subject to damping at rates γ1, γ2, κc, κ1 and κ2.
In the following, we assume that the single-photon optomechanical coupling rates are equal, g1 = g2 = g. After applying the linearized transformation ${\hat{a}}_{{c}}\to \hat{d}+{\bar{\alpha }}_{1}{{\rm{e}}}^{-{\rm{i}}{\omega }_{{\rm{d}}1}t}$ + ${\bar{\alpha }}_{2}{{\rm{e}}}^{-{\rm{i}}{\omega }_{{\rm{d}}2}t}$, ${\hat{a}}_{1}\to {\hat{d}}_{1}+{\bar{\alpha }}_{1}^{{\prime} }{{\rm{e}}}^{-{\rm{i}}{\omega }_{{\rm{d}}1}t}$+${\bar{\alpha }}_{2}^{{\prime} }{{\rm{e}}}^{-{\rm{i}}{\omega }_{{\rm{d}}2}t}$, ${\hat{a}}_{2}\to {\hat{d}}_{2}\,+{\bar{\alpha }}_{1}^{{\prime\prime} }{{\rm{e}}}^{-{\rm{i}}{\omega }_{{\rm{d}}1}t}$ + ${\bar{\alpha }}_{2}^{{\prime\prime} }{{\rm{e}}}^{-{\rm{i}}{\omega }_{{\rm{d}}2}t}$ and going into the interaction picture, the linearized Hamiltonian reads$\begin{eqnarray}\begin{array}{rcl}{\hat{H}}_{I} & = & \delta ({\hat{b}}_{1}^{\dagger }{\hat{b}}_{1}-{\hat{b}}_{2}^{\dagger }{\hat{b}}_{2})+[{\hat{d}}^{\dagger }({G}_{1}{\hat{b}}_{1}^{\dagger }+{G}_{2}{\hat{b}}_{1})+{\rm{h}}.{\rm{c}}.]\\ & & +\ [{\hat{d}}^{\dagger }({G}_{1}{\hat{b}}_{2}^{\dagger }+{G}_{2}{\hat{b}}_{2})+{\rm{h}}.{\rm{c}}.]\\ & & +\ [{\hat{d}}^{\dagger }({G}_{1}{\hat{b}}_{1}{{\rm{e}}}^{-2{\rm{i}}{\omega }_{m}t}+{G}_{2}{\hat{b}}_{1}^{\dagger }{{\rm{e}}}^{2{\rm{i}}{\omega }_{{\rm{m}}}t})+{\rm{h}}.{\rm{c}}.]\\ & & +\ [{\hat{d}}^{\dagger }({G}_{1}{\hat{b}}_{2}{{\rm{e}}}^{-2{\rm{i}}{\omega }_{{\rm{m}}}t}+{G}_{2}{\hat{b}}_{2}^{\dagger }{{\rm{e}}}^{2{\rm{i}}{\omega }_{{\rm{m}}}t})+{\rm{h}}.{\rm{c}}.]\\ & & +\ {{\rm{\Delta }}}_{1}{\hat{d}}_{1}^{\dagger }{\hat{d}}_{1}+{{\rm{\Delta }}}_{2}{\hat{d}}_{2}^{\dagger }{\hat{d}}_{2}\\ & & +\ {J}_{1}({\hat{d}}_{1}^{\dagger }\hat{d}+{\hat{d}}^{\dagger }{\hat{d}}_{1})+{J}_{2}({\hat{d}}_{2}^{\dagger }\hat{d}+{\hat{d}}^{\dagger }{\hat{d}}_{2})+{\hat{H}}_{\mathrm{env}},\end{array}\end{eqnarray}$where ${G}_{\mathrm{1,2}}=g{\bar{\alpha }}_{\mathrm{1,2}}$, Δ1 = ω1 − ωc, Δ2 = ω2 − ωc and $\delta =\tfrac{{{\rm{\Omega }}}_{1}-{{\rm{\Omega }}}_{2}}{2}$, which denotes the detuning between the two mechanical oscillators. Here, ${\bar{\alpha }}_{\mathrm{1,2}}$ is the coherent light field amplitudes, with the explicit values given in appendix A. We focus on the regime G1 < G2, such that the dynamics is stable.
The first two lines of the Hamiltonian can generate a two-mode squeezed state based on reservoir engineering approach [35] in the good-cavity limit κc ≪ ωm. This can be explained with the following argument, based on the two-mode Bogoliubov operators$\begin{eqnarray}{\hat{\beta }}_{1}={\hat{b}}_{1}\cosh \xi +{\hat{b}}_{2}^{\dagger }\sinh \xi =\hat{S}(\xi ){\hat{b}}_{1}{\hat{S}}^{\dagger }(\xi ),\end{eqnarray}$$\begin{eqnarray}{\hat{\beta }}_{2}={\hat{b}}_{2}\cosh \xi +{\hat{b}}_{1}^{\dagger }\sinh \xi =\hat{S}(\xi ){\hat{b}}_{2}{\hat{S}}^{\dagger }(\xi ).\end{eqnarray}$Here $\hat{S}(\xi )\equiv \exp [\xi ({\hat{b}}_{1}{\hat{b}}_{2}-{\hat{b}}_{1}^{\dagger }{\hat{b}}_{2}^{\dagger })]$ is the two-mode squeezing operator with squeezing parameter $\xi =\mathrm{arctanh}({G}_{1}/{G}_{2})$. Thus, the ground state of ${\hat{\beta }}_{1}$ and ${\hat{\beta }}_{2}$ is a two-mode squeezed state $| \xi \rangle =\hat{S}(\xi )| 0,0\rangle $, where ∣0, 0⟩ is the vacuum state of ${\hat{b}}_{1}$, ${\hat{b}}_{2}$. To realize entanglement of two mechanical oscillators, it is sufficient to cool the modes ${\hat{\beta }}_{1}$ and ${\hat{\beta }}_{2}$ to their ground state. With equations (4) and (5), the first two lines of the Hamiltonian can be written as$\begin{eqnarray*}{\hat{H}}_{1}=\delta ({\hat{\beta }}_{1}^{\dagger }{\hat{\beta }}_{1}-{\hat{\beta }}_{2}^{\dagger }{\hat{\beta }}_{2})+g[({\hat{\beta }}_{1}^{\dagger }+{\hat{\beta }}_{2}^{\dagger })\hat{d}+{\rm{h}}.{\rm{c}}.].\end{eqnarray*}$Furthermore, we define ${\hat{\beta }}_{{\rm{s}},{\rm{d}}}=\tfrac{1}{\sqrt{2}}({\hat{\beta }}_{1}\pm {\hat{\beta }}_{2})$. The above Hamiltonian is rewritten as$\begin{eqnarray*}{\hat{H}}_{1}^{{\prime} }=\delta ({\hat{\beta }}_{{\rm{s}}}^{\dagger }{\hat{\beta }}_{{\rm{d}}}-{\hat{\beta }}_{{\rm{d}}}^{\dagger }{\hat{\beta }}_{{\rm{s}}})+\sqrt{2}g[{\hat{\beta }}_{{\rm{s}}}^{\dagger }\hat{d}+{\rm{h}}.{\rm{c}}.].\end{eqnarray*}$When δ = 0, the cavity can only cool mode ${\hat{\beta }}_{{\rm{s}}}$. Instead, with δ ≠ 0, the cavity can simultaneously and efficiently cool both ${\hat{\beta }}_{{\rm{s}}}$ and ${\hat{\beta }}_{{\rm{d}}}$ to the ground states, implying that the mechanical oscillators are maximally entangled.
However, the effect of counter-rotating terms in the third and fourth lines of equation (3) is not negligible in an unresolved sideband regime κc ≫ ωm, and the ideal cooling of ${\hat{\beta }}_{1}$ and ${\hat{\beta }}_{2}$ becomes impossible. Here we circumvent this problem by adding two auxiliary cavities, to reduce the effect of counter-rotating terms through destructive quantum interference.
To show the feasibility of our proposal, we firstly compute the entanglement of two mechanical modes with and without auxiliary cavities in figure 2. Generally, the entanglement can be quantified by the logarithmic negativity EN (see appendix B). For a parametric-amplifier interaction Hamiltonian, the maximum stationary intracavity entanglement is less than ${E}_{N}=\mathrm{ln}2$ (the green-dashed line in figure 2), due to a constraint from the stability condition [33]. We find that the logarithmic negativity EN decreases with increasing cavity damping, and larger entanglement is realized in the resolved-sideband regime. Furthermore, whether δ is zero or nonzero, the logarithmic negativity EN does not beat $\mathrm{ln}2$ with κc > ωm without two auxiliary cavities. When the two auxiliary cavities are included, the entanglement is nonzero also in the unresolved-sideband regime. This phenomenon comes from reservior engineering, with the two auxiliary cavities tailoring the optical density of states and alleviating the effect of the counter-rotating terms Furthermore, when δ = 0, entanglement beats $\mathrm{ln}2$ until κc ≃ 5ωm. With a nonzero detuning, δ ≠ 0, the entanglement surpasses $\mathrm{ln}2$ even for a bad cavity with κc ≃ 25ωm under suitable parameters. This result is consistent with the previous analysis, because the detuning between two mechanical frequencies ensures that both ${\hat{\beta }}_{1}$ and ${\hat{\beta }}_{2}$ can be efficiently cooled.
Figure 2.
New window|Download| PPT slide Figure 2.The logarithmic negativity EN versus κc/ωm in four representative cases. All the lines are obtained with the optimal ropt = G1/G2 at fixed G2. Sold blue and red lines refer to δ = 0 and δ = 0.01ωm, respectively, with cavity coupling strength J1 = J2 = J = 5ωm. The dotted lines are without two auxiliary cavities. The green-dashed line is ${E}_{N}=\mathrm{ln}2$. Other parameters: G2 = 0.1ωm, γ1 = γ2 = 10−5ωm, Δ1 = −Δ2 = 2ωm, κ1 = κ2 = 0.3ωm, nth = 0.
Note that all above results are obtained by numerically simulating the Langevin equations [46] with the full Hamiltonian equation (3). In the following, we will present a theoretical approach to optimize the driving and optical coupling coefficients to achieve the maximum entanglement of two mechanical oscillators based on an effective master equation in the weak coupling regime.
3. Mechanical entanglement in the weak-coupling regime
In the weak coupling regime G1,2 ≪ κc, κ1,2, three cavities act as a reservoir for the mechanical system. One can trace out the cavity degrees of freedom and keep the degrees of freedom of two mechanical oscillators. In the derivation, since the term eiδt affects the form of the effective master equation, we first consider δ ≠ 0. The case of δ = 0 will be considered in section 6. Several time-dependent terms can be omitted when γ1,2 ≪ δ, thus we can obtain an effective master equation of the form$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{\rm{d}}\hat{\rho }(t)}{{\rm{d}}t} & = & {{\rm{\Gamma }}}_{\uparrow ,1}{ \mathcal D }[{\hat{b}}_{1}^{\dagger }]\hat{\rho }(t)+{{\rm{\Gamma }}}_{\downarrow ,1}{ \mathcal D }[{\hat{b}}_{1}]\hat{\rho }(t)+{{\rm{\Gamma }}}_{\uparrow ,2}{ \mathcal D }[{\hat{b}}_{2}^{\dagger }]\hat{\rho }(t)\\ & & +\ {{\rm{\Gamma }}}_{\downarrow ,2}{ \mathcal D }[{\hat{b}}_{2}]\hat{\rho }(t)+{{\rm{\Gamma }}}_{{\rm{s}},12}{{ \mathcal D }}_{{\rm{s}}}[{\hat{b}}_{1}^{\dagger },{\hat{b}}_{2}^{\dagger }]\hat{\rho }(t)\\ & & +\ {{\rm{\Gamma }}}_{{\rm{s}},21}{{ \mathcal D }}_{{\rm{s}}}[{\hat{b}}_{1},{\hat{b}}_{2}]\hat{\rho }(t)\\ & & +\ {{\rm{\Gamma }}}_{{\rm{s}},21}{{ \mathcal D }}_{{\rm{s}}}[{\hat{b}}_{2}^{\dagger },{\hat{b}}_{1}^{\dagger }]\hat{\rho }(t)+{{\rm{\Gamma }}}_{{\rm{s}},12}{{ \mathcal D }}_{{\rm{s}}}[{\hat{b}}_{2},{\hat{b}}_{1}]\hat{\rho }(t),\end{array}\end{eqnarray}$with the dissipative superoperators ${ \mathcal D }[{\hat{A}}_{i}]\hat{\rho }(t)={\hat{A}}_{i}\hat{\rho }(t){\hat{A}}_{i}^{\dagger }\,-\tfrac{1}{2}{\hat{A}}_{i}^{\dagger }{\hat{A}}_{i}\hat{\rho }(t)-\tfrac{1}{2}\hat{\rho }(t){\hat{A}}_{i}^{\dagger }{\hat{A}}_{i}$ and ${{ \mathcal D }}_{{\rm{s}}}[{\hat{A}}_{i},{\hat{A}}_{j}]\hat{\rho }(t)={\hat{A}}_{i}\hat{\rho }(t){\hat{A}}_{j}\,-\tfrac{1}{2}{\hat{A}}_{i}{\hat{A}}_{j}\hat{\rho }(t)-\tfrac{1}{2}\hat{\rho }(t){\hat{A}}_{i}{\hat{A}}_{j}$ with i, j = 1, 2. ${ \mathcal D }[{\hat{A}}_{i}]\hat{\rho }(t)$ and ${ \mathcal D }[{\hat{A}}_{i}^{\dagger }]\hat{\rho }(t)$ denote cooling and heating effects caused by the thermal environment and optical fields, while the terms ${{ \mathcal D }}_{{\rm{s}}}[{\hat{A}}_{i},{\hat{A}}_{j}]\hat{\rho }(t)$ can lead the system to a steady state where a two-mode squeezed state is achieved. The corresponding dissipative rates are $\begin{eqnarray*}\begin{array}{rcl}{{\rm{\Gamma }}}_{\uparrow ,1} & = & {G}_{1}^{2}{S}_{\mathrm{op}}(-\delta )+{G}_{2}^{2}{S}_{\mathrm{op}}[-(2{\omega }_{{\rm{m}}}+\delta )]+{\gamma }_{1}{n}_{\mathrm{th},1},\\ {{\rm{\Gamma }}}_{\downarrow ,1} & = & {G}_{2}^{2}{S}_{\mathrm{op}}(\delta )+{G}_{1}^{2}{S}_{\mathrm{op}}(2{\omega }_{{\rm{m}}}+\delta )+{\gamma }_{1}(1+{n}_{\mathrm{th},1}),\\ {{\rm{\Gamma }}}_{\uparrow ,2} & = & {G}_{1}^{2}{S}_{\mathrm{op}}(\delta )+{G}_{2}^{2}{S}_{\mathrm{op}}[-(2{\omega }_{{\rm{m}}}-\delta )]+{\gamma }_{2}{n}_{\mathrm{th},2},\\ {{\rm{\Gamma }}}_{\downarrow ,2} & = & {G}_{2}^{2}{S}_{\mathrm{op}}(-\delta )+{G}_{1}^{2}{S}_{\mathrm{op}}(2{\omega }_{{\rm{m}}}-\delta )+{\gamma }_{2}(1+{n}_{\mathrm{th},2}),\\ {{\rm{\Gamma }}}_{{\rm{s}},12} & = & {G}_{1}{G}_{2}{S}_{\mathrm{op}}(-\delta ),\\ {{\rm{\Gamma }}}_{{\rm{s}},21} & = & {G}_{1}{G}_{2}{S}_{\mathrm{op}}(\delta ),\end{array}\end{eqnarray*}$ with the optical spectral function$\begin{eqnarray*}{S}_{\mathrm{op}}(\omega )={\int }_{-\infty }^{\infty }{\rm{d}}{t}{{\rm{e}}}^{{\rm{i}}\omega t}\langle \hat{d}(t){\hat{d}}^{\dagger }(0)\rangle ,\end{eqnarray*}$which can be computed in a standard way (see [12] and section 4). The dissipative terms of the effective master equation are related to the enhanced coupling coefficients G1,2, the damping of mechanical oscillators γ1,2, thermal occupation nth,1 and nth,2, and optical spectrum. Notably, ${G}_{1,2}^{2}{S}_{\mathrm{op}}(\pm \delta )$ come from the rotating-wave terms, while ${G}_{1}^{2}{S}_{\mathrm{op}}(2{\omega }_{{\rm{m}}}\pm \delta )$ and ${G}_{2}^{2}{S}_{\mathrm{op}}(-2{\omega }_{{\rm{m}}}\pm \delta )$ originate from the counter-rotating terms in equation (3). In the resolved-sideband regime, ${G}_{1,2}^{2}{S}_{\mathrm{op}}(\pm \delta )$ play a dominant role in generating entanglement. However, in the unresolved sideband regime, the effects of counter-rotating terms is comparable to the rotating-wave terms.
We now assume a symmetric mechanical damping (γ1 = γ2 = γ, nth,1 = nth,2 = nth) and symmetric optical spectrum [Sop(δ) = Sop(−δ), Sop(2ωm + δ) = Sop(−2ωm + δ), Sop(2ωm − δ) = Sop(−2ωm + δ)], which allows us to obtain simple analytical results for the steady-state second moments. Generally, collective quadrature operators can be defined by$\begin{eqnarray*}\begin{array}{rcl}{\hat{X}}_{\pm } & = & ({\hat{X}}_{1}\pm {\hat{X}}_{2})/\sqrt{2},\\ {\hat{P}}_{\pm } & = & ({\hat{P}}_{1}\pm {\hat{P}}_{2})/\sqrt{2},\end{array}\end{eqnarray*}$where the quadratures for each mechanical mode are ${\hat{X}}_{i}=({\hat{b}}_{i}+{\hat{b}}_{i}^{\dagger })/\sqrt{2}$ and ${\hat{P}}_{i}=-{\rm{i}}({\hat{b}}_{i}-{\hat{b}}_{i}^{\dagger })/\sqrt{2}$ with i = 1, 2. The steady-state second moments from equation (6) are$\begin{eqnarray}\begin{array}{l}\langle {\hat{X}}_{\pm }^{2}\rangle =\langle {\hat{P}}_{\mp }^{2}\rangle =\displaystyle \frac{1}{2}+{n}_{\mathrm{th}}+{C}_{{\rm{e}}}\\ \ \pm \ \displaystyle \frac{{C}_{{\rm{e}}}\sinh 2\xi }{{C}_{{\rm{e}}}({\epsilon }_{1}+{\epsilon }_{2}-2)-\cosh 2\xi -1}\\ \ +\ \displaystyle \frac{{C}_{{\rm{e}}}}{2}\displaystyle \sum _{i=1,2}\\ \ \times \ \displaystyle \frac{2+2{C}_{{\rm{e}}}+2{n}_{\mathrm{th}}-{\epsilon }_{i}(1+2{C}_{{\rm{e}}}+2{n}_{\mathrm{th}}+\cosh 2\xi )}{-1-\cosh 2\xi -2{C}_{{\rm{e}}}+2{C}_{{\rm{e}}}{\epsilon }_{i}},\end{array}\end{eqnarray}$with$\begin{eqnarray}{\epsilon }_{1}=\displaystyle \frac{{S}_{\mathrm{op}}(2{\omega }_{{\rm{m}}}+\delta )}{{S}_{\mathrm{op}}(\delta )},\end{eqnarray}$$\begin{eqnarray}{\epsilon }_{2}=\displaystyle \frac{{S}_{\mathrm{op}}(2{\omega }_{{\rm{m}}}-\delta )}{{S}_{\mathrm{op}}(\delta )},\end{eqnarray}$$\begin{eqnarray}{C}_{{\rm{e}}}=\displaystyle \frac{{G}_{2}^{2}{S}_{\mathrm{op}}(\delta )}{\gamma }.\end{eqnarray}$Without the counter-rotating terms, ε1 = ε2 = 0, the steady-state second moments are simplified to$\begin{eqnarray}\langle {\hat{X}}_{\pm }^{2}\rangle =\langle {\hat{P}}_{\mp }^{2}\rangle =\displaystyle \frac{1}{2}\displaystyle \frac{{{\rm{e}}}^{\mp 2\xi }+\tfrac{1+2{n}_{\mathrm{th}}}{2{C}_{{\rm{e}}}}\cosh 2\xi +\tfrac{1+2{n}_{\mathrm{th}}}{2{C}_{{\rm{e}}}}}{\tfrac{1+\cosh 2\xi }{2{C}_{{\rm{e}}}}+1},\end{eqnarray}$and ${S}_{\mathrm{op}}(\delta )=\tfrac{4{\kappa }_{{\rm{c}}}}{{\kappa }_{{\rm{c}}}^{2}+4{\delta }^{2}}$. Considering κc ≫ δ, the optical spectrum is ${S}_{\mathrm{op}}(\delta )=\tfrac{4}{{\kappa }_{{\rm{c}}}}$. Thus, we obtain$\begin{eqnarray}\begin{array}{rcl}\langle {\hat{X}}_{\pm }^{2}\rangle & = & \langle {\hat{P}}_{\mp }^{2}\rangle =\displaystyle \frac{\gamma {\kappa }_{{\rm{c}}}}{\gamma {\kappa }_{{\rm{c}}}+4({G}_{2}^{2}-{G}_{1}^{2})}\left({n}_{\mathrm{th}}+\displaystyle \frac{1}{2}\right)\\ & & +\ \displaystyle \frac{2{\left({G}_{2}\mp {G}_{1}\right)}^{2}}{\gamma {\kappa }_{{\rm{c}}}+4({G}_{2}^{2}-{G}_{1}^{2})},\end{array}\end{eqnarray}$in agreement with [35].
Here we are interested in the effect of the counter-rotating terms In equation (7), because ε1 and ε2 have equivalent properties, we set ε1 = ε2 = ε. This can be achieved by choosing a small δ which ensures that the spectral density entering ε1,2 has a very small shift to the left and right sides of 2ωm. Therefore,$\begin{eqnarray*}\langle {\hat{X}}_{\pm }^{2}\rangle =\langle {\hat{P}}_{\mp }^{2}\rangle =\displaystyle \frac{1}{2}\displaystyle \frac{{{\rm{e}}}^{\mp 2\xi }+\left(\tfrac{1+2{n}_{\mathrm{th}}}{2{C}_{{\rm{e}}}}+\epsilon \right)\cosh 2\xi +\tfrac{1+2{n}_{\mathrm{th}}}{2{C}_{{\rm{e}}}}}{\tfrac{1+\cosh 2\xi }{2{C}_{{\rm{e}}}}+(1-\epsilon )}.\end{eqnarray*}$
According to Duan inequality [47], an inseparable state should satisfy$\begin{eqnarray}V\equiv \langle {\hat{X}}_{+}^{2}\rangle +\langle {\hat{P}}_{-}^{2}\rangle \lt 1.\end{eqnarray}$For our system, the variance is$\begin{eqnarray}V=\displaystyle \frac{{{\rm{e}}}^{-2\xi }+\left[\tfrac{1+2{n}_{\mathrm{th}}}{2{C}_{{\rm{e}}}}+\epsilon \right]\cosh 2\xi +\tfrac{1+2{n}_{\mathrm{th}}}{2{C}_{{\rm{e}}}}}{\tfrac{1+\cosh 2\xi }{2{C}_{{\rm{e}}}}+(1-\epsilon )}\end{eqnarray}$$\begin{eqnarray}=\ \displaystyle \frac{{G}_{2}^{2}(1+2{n}_{\mathrm{th}})+{C}_{{\rm{e}}}[{\left({G}_{1}-{G}_{2}\right)}^{2}+({G}_{2}^{2}+{G}_{1}^{2})\epsilon ]}{{G}_{2}^{2}+{C}_{{\rm{e}}}({G}_{1}^{2}-{G}_{2}^{2})(\epsilon -1)}.\end{eqnarray}$From above expression of variance in equation (14), small ε can make a minimum value of V.
In figure 3, the logarithmic negativity EN and V from equation (15) are shown as a function of $\tfrac{{G}_{1}}{{G}_{2}}$. We find that the logarithmic negativity EN surpasses $\mathrm{ln}2\approx 0.69$ and the value of V is below 1 even with a nonzero thermal phonon occupation. More importantly, we see that the maximum logarithmic negativity EN also corresponds to the minimum value of V, which indicates that the two mechanical oscillators are entangled in a steady state in the unresolved-sideband regime. Interestingly, both the logarithmic negativity EN and V are non-monotone functions of $\tfrac{{G}_{1}}{{G}_{2}}$, and here is an optimal value of $\tfrac{{G}_{1}}{{G}_{2}}$ to achieve strong entanglement. This optimal point is a trade-off between two factors. A strong entanglement requires a large squeezing, which needs $\tfrac{{G}_{1}}{{G}_{2}}\to 1$. On the other hand, the system is effectively cooled to the steady state when $\tfrac{{G}_{1}}{{G}_{2}}\to 0$.
Figure 3.
New window|Download| PPT slide Figure 3.(a) The logarithmic negativity EN. (b) The quantity $V\equiv \langle {\hat{X}}_{+}^{2}\rangle +\langle {\hat{P}}_{-}^{2}\rangle $ entering the Duan inequality with equation (13). The red , blue, orange lines are with G2 = 0.2ωm, 0.1ωm, 0.05ωm, respectively. Other parameters: δ = 0.01ωm, γ = 10−5ωm, κc = 5ωm, Δ1 = −Δ2 = 2ωm, κ1 = κ2 = 0.3ωm, J1 = J2 = 5ωm , nth = 5.
We compare analytical and numerical results in figure 4(a), showing that with a small coupling G2 the analytical and numerical results are in a good agreement. However, increasing G2 close to κ1 and κ2, a visible deviation appears due to the violation of weak-coupling approximation. This deviation can be seen more clearly in the figure 4(b) which shows the minimum optimal variance V as function of G2. In the weak-coupling regime G2 < κ, the analytical results are in agreement with numerical simulations. While in the strong coupling regime, G2 > κ, the numerical results deviate from the analytical ones, showing a nonmonotonic behavior with increasing G2. Because the mechanical modes are mixed with the optical cavity mode in the strong coupling regime, the strong hybridization between the cavity field and two mechanical modes cannot be neglected. We also find that the variance has more visible deviations with a large thermal phonon number. Therefore, our treatment based on the effective master equation is valid in the weak coupling regime and for sufficiently small thermal phonon number.
Figure 4.
New window|Download| PPT slide Figure 4.Comparison of the numerical and analytical results. (a) V versus $\tfrac{{G}_{1}}{{G}_{2}}$. The blue, green, red lines are for different G2 = 0.05ωm, 0.8ωm, 0.3ωm. The thermal phonon number is nth = 5. (b) V versus G2. All the sold lines are the analytical results, while the dots and dashed curves are numerical results. The blue, green, red lines denote nth = 0, nth = 10, nth = 30, respectively. Other parameters: δ = 0.01ωm, γ1 = γ2 = 10−5ωm, κc = 5ωm, Δ1 = −Δ2 = 2ωm, κ1 = κ2 = κ = 0.2ωm, J1 = J2 = 5ωm.
In figure 3, strong entanglement corresponds to a minimum value of V, which is related to the driving coupling rate, cooperativity Ce and counter-rotating factor ε. Ce and ε have a direct connection with the optical spectrum. Therefore, in the next section, we will analyze in detail the optical field allowing to achieve a minimum value of V.
4. Theoretical analysis of optical spectrum
From the above discussion, the values of the optical spectrum Sop(ω) at ω = ±δ, ±(2ωm + δ), ±(2ωm − δ) are crucial to achieve a strong entanglement. In this section, we will discuss the dependence of the optical spectrum on system parameters and show how to choose suitable parameters to generate strong entanglement. In the weak coupling regime, the back-action of mechanics to the cavities can be neglected, thus the Hamiltonian of the optical fields is$\begin{eqnarray}\begin{array}{l}{\hat{H}}_{\mathrm{op}}={{\rm{\Delta }}}_{1}{\hat{d}}_{1}^{\dagger }\hat{{d}_{1}}+{{\rm{\Delta }}}_{2}{\hat{d}}_{2}^{\dagger }\hat{{d}_{2}}+{J}_{1}({\hat{d}}_{1}^{\dagger }\hat{d}+{\hat{d}}^{\dagger }\hat{{d}_{1}})\\ \qquad \ +\ {J}_{2}({\hat{d}}_{2}^{\dagger }\hat{d}+{\hat{d}}^{\dagger }\hat{{d}_{2}}),\end{array}\end{eqnarray}$with the corresponding quantum Langevin equations$\begin{eqnarray}\dot{\hat{d}}=-{\rm{i}}{J}_{1}{\hat{d}}_{1}-{\rm{i}}{J}_{2}{\hat{d}}_{2}-\displaystyle \frac{{\kappa }_{{\rm{c}}}}{2}\hat{d}+\sqrt{{\kappa }_{{\rm{c}}}}{\hat{d}}_{\mathrm{in}}\end{eqnarray}$$\begin{eqnarray}\dot{\hat{{d}_{1}}}=-{\rm{i}}{J}_{1}\hat{d}-{\rm{i}}{{\rm{\Delta }}}_{1}{\hat{d}}_{1}-\displaystyle \frac{{\kappa }_{1}}{2}{\hat{d}}_{1}+\sqrt{{\kappa }_{1}}{\hat{d}}_{1,\mathrm{in}}\end{eqnarray}$$\begin{eqnarray}\dot{\hat{{d}_{2}}}=-{\rm{i}}{J}_{2}\hat{d}-{\rm{i}}{{\rm{\Delta }}}_{2}{\hat{d}}_{2}-\displaystyle \frac{{\kappa }_{2}}{2}{\hat{d}}_{2}+\sqrt{{\kappa }_{2}}{\hat{d}}_{2,\mathrm{in}},\end{eqnarray}$where ${\hat{d}}_{i,{\rm{in}}}$ are noise operators with nonzero correlation functions satisfying $\left\langle {\hat{d}}_{i,{\rm{in}}}(t){\hat{d}}_{j,{\rm{in}}}^{\dagger }({t}^{{\prime} })\right\rangle ={\delta }_{{ij}}\delta (t-{t}^{{\prime} })$. With the above equations, we can obtain the optical spectrum$\begin{eqnarray}{S}_{\mathrm{op}}(\omega )=\displaystyle \frac{1}{A(\omega )}+\displaystyle \frac{1}{{A}^{* }(\omega )},\end{eqnarray}$with$\begin{eqnarray}A(\omega )=\displaystyle \frac{{\kappa }_{{\rm{c}}}}{2}-{\rm{i}}\omega +\displaystyle \frac{{J}_{1}^{2}}{{\rm{i}}({{\rm{\Delta }}}_{1}-\omega )+\tfrac{{\kappa }_{1}}{2}}+\displaystyle \frac{{J}_{2}^{2}}{{\rm{i}}({{\rm{\Delta }}}_{2}-\omega )+\tfrac{{\kappa }_{2}}{2}}.\end{eqnarray}$Normally, the optical spectrum without the two auxiliary cavities has a standard Lorentzian shape with a single peak at ω = 0 of half-width κc, as shown in the red line of figure 5, but in the unresolved-sideband regime κc ≫ ωm, the effect of counter-rotating terms is comparable to the rotating-wave terms, which suppresses entanglement of the two mechanical oscillators. Due to the two auxiliary cavities, the single peak splits into three peaks when J1,2 are large. This splitting can be understood by diagonalizing the optical Hamiltonian. When Δ1 = −Δ2 = Δ, J1 = J2 = J and κ1 = κ2 = κ, the peaks are symmetrically placed at ω = 0, $\pm \sqrt{2{J}^{2}+{{\rm{\Delta }}}^{2}}$ and the position of the central peak is unchanged. At a small J1,2, there are two dips at ω = Δ1 and Δ2, a result of two-photon resonance in the EIT phenomenon of a three-level atomic system due to destructive quantum interference [48].
For our system, we need large Sop(± δ), small Sop(± (2ωm + δ)) and Sop(± (2ωm − δ)) to make ε smaller. Intuitively, we set Δ1 = −Δ2 = Δ = 2ωm, and a small δ. The optical spectrum changes with different coupling J1 = J2 = J as shown in figure 5(a). With a small J ≪ κc, the optical spectrum has a lineshape similar to EIT with two narrow dips at ω = ±2ωm. These two dips allow to achieve a smaller value of ε1,2 = Sop(± 2ωm)/Sop(0) than without auxiliary cavities. Increasing J leads to a further decrease of ε1,2, due to the vanishing value of Sop(±2ωm ± δ). Furthermore, the blue line of figure 5(b) assumes different couplings J1 = ωm and J2 = 3ωm. We find this case cannot simultaneously suppress Sop(2ωm) and Sop(−2ωm) to very small values. Hence, it is desirable to consider a symmetrical optical spectrum.
In the limit of J ≫ κ and a small δ, the optical spectrum gives$\begin{eqnarray}{S}_{\mathrm{op}}(\delta )\simeq {S}_{\mathrm{op}}(0)=\displaystyle \frac{4(16{\omega }_{{\rm{m}}}^{2}+{\kappa }^{2})}{8{J}^{2}\kappa +(16{\omega }_{{\rm{m}}}^{2}+{\kappa }^{2}){\kappa }_{{\rm{c}}}},\end{eqnarray}$$\begin{eqnarray}{S}_{\mathrm{op}}(2{\omega }_{{\rm{m}}}\pm \delta )\simeq {S}_{\mathrm{op}}(\pm 2{\omega }_{{\rm{m}}})=\displaystyle \frac{\kappa }{{J}^{2}}.\end{eqnarray}$Thus, with κ ≪ ωm,$\begin{eqnarray}{\epsilon }_{1}={\epsilon }_{2}=\epsilon \simeq \displaystyle \frac{{\kappa }_{{\rm{c}}}\kappa }{4{J}^{2}}+\displaystyle \frac{{\kappa }^{2}}{8{\omega }_{{\rm{m}}}^{2}}.\end{eqnarray}$From equations (10) and (22) and considering the limit ωm ≫ κ, the effective cooperativity Ce can be expressed as$\begin{eqnarray}{C}_{{\rm{e}}}\simeq C{\left(1+\displaystyle \frac{{J}^{2}\kappa }{2{\omega }_{{\rm{m}}}^{2}{\kappa }_{{\rm{c}}}}\right)}^{-1}\end{eqnarray}$with the standard cooperativity $C=\tfrac{4{G}_{2}^{2}}{\gamma {\kappa }_{{\rm{c}}}}$. From these simplified expressions, we see that ε and Ce decrease with increasing J, as also shown in figure 6. A strong entanglement needs small ε and large Ce. Thus, the entanglement depends on J in a non-monotonic way: the beneficial decrease of ε saturates at large J, see equation (24), and the entanglement gets corrupted by the decrease of Ce. According to the above analysis, the best choice of detuning of the two auxiliary cavities with a main cavity is Δ1 = −Δ2 = Δ = 2ωm, and the detuning δ between the two mechanical frequencies needs to be small. Meanwhile, the coupling J needs to be optimized, as detailed in the next section.
Figure 6.
New window|Download| PPT slide Figure 6.V versus J at the optimal r. The parameters are δ = 0.01ωm, γ1 = γ2 = 10−5ωm, κc = 5ωm, Δ1 = −Δ2 = 2ωm, κ1 = κ2 = κ = 0.2ωm, G2 = 0.1ωm, nth = 5.
5. Optimal parameters to achieve entanglement
In this section, we will elaborate on the optimal parameters to generate a strong entanglement, complementing the spectrum analysis in the previous section. Like in the resolved sideband regime (i.e. without two auxiliary cavities), there is an optimal coupling ratio r = G1/G2 to achieve a minimum variance. The optimal value can be derived from equation (15):$\begin{eqnarray}{r}_{\mathrm{opt}}=\displaystyle \frac{{G}_{1}}{{G}_{2}}=\displaystyle \frac{D-\sqrt{{D}^{2}-{C}_{{\rm{e}}}[{C}_{{\rm{e}}}(1-\epsilon )+1](1-\epsilon )}}{{C}_{{\rm{e}}}(1-\epsilon )},\end{eqnarray}$with D = 1 + (1 − ε)nth − Ce(ε2 − 1). The corresponding optimal value of the variance is$\begin{eqnarray}{V}_{\mathrm{opt}}=\displaystyle \frac{2{n}_{\mathrm{th}}+1+{C}_{{\rm{e}}}[({r}_{\mathrm{opt}}^{2}+1)\epsilon +{\left({r}_{\mathrm{opt}}-1\right)}^{2}]}{1+{C}_{{\rm{e}}}(\epsilon -1)({r}_{\mathrm{opt}}^{2}-1)}.\end{eqnarray}$
Considering the limit Ce ≫ 1 and a small counter-rotating effect ε, the above optimal parameter can be simplified to$\begin{eqnarray}\begin{array}{l}{r}_{\mathrm{opt}}\approx 1+\displaystyle \frac{1+{C}_{{\rm{e}}}\epsilon +{n}_{\mathrm{th}}}{{C}_{{\rm{e}}}}\\ \quad \ -\ \sqrt{\displaystyle \frac{1+2{C}_{{\rm{e}}}\epsilon +2{n}_{\mathrm{th}}+(2{n}_{\mathrm{th}}+1)\epsilon }{{C}_{{\rm{e}}}}},\end{array}\end{eqnarray}$and the optimal variance is$\begin{eqnarray}{V}_{\mathrm{opt}}\approx \displaystyle \frac{{n}_{\mathrm{th}}}{{C}_{{\rm{e}}}}+\sqrt{\displaystyle \frac{1+2{n}_{\mathrm{th}}+2{C}_{{\rm{e}}}\epsilon }{{C}_{{\rm{e}}}}}.\end{eqnarray}$Equation (29) shows a lower bound ${V}_{\mathrm{opt}}=\sqrt{2\epsilon }$, which is clearly present in figure 4(b). If the state is entangled, Vopt < 1, which implies κ < 2ωm.
From the above equation, Vopt is a decreasing function of Ce and a increasing function of ε. Since Ce, $\epsilon \propto \tfrac{1}{J}$, there exists an optimal value of J to make Vopt smaller. In figure 6, the optimal variance equation (27) is plotted, which indeed is a nonmonotonic function of J. From our analysis of the spectrum, we have seen that a moderate J can overcome the unresolved sideband limit and decreases the value of ε. With a strong J, ε will saturate to a small value when $J\gg \sqrt{\kappa {\kappa }_{{\rm{c}}}}$, as shown by the blue line in figure 6. Meanwhile, Ce decreases strongly at a large J. A large J will suppress the spectral density at ω = 0, decreasing the value of Sop(0) and results in a very small effective cooperativity. Therefore, despite a tiny ε, a very large J cannot realize a much stronger entanglement.
To realize a strong entanglement, we further optimize the coupling coefficient J under optimal laser-driving. According to equations (24) and (25), the minimum variance in equation (29) can be approximated as$\begin{eqnarray}{V}_{\mathrm{opt}}\simeq \sqrt{\displaystyle \frac{1}{{C}_{\mathrm{th}}}\left(1+\displaystyle \frac{{J}^{2}\kappa }{2{\omega }_{{\rm{m}}}^{2}{\kappa }_{{\rm{c}}}}\right)+\displaystyle \frac{{\kappa }_{{\rm{c}}}\kappa }{2{J}^{2}}+\displaystyle \frac{{\kappa }^{2}}{4{\omega }_{{\rm{m}}}^{2}}},\end{eqnarray}$where we have defined a thermal coefficient$\begin{eqnarray}{C}_{\mathrm{th}}=\displaystyle \frac{C}{2{n}_{\mathrm{th}}+1}.\end{eqnarray}$
In the above derivation, we have considered a small thermal phonon occupation and a large cooperativity Ce ≫ 1, to make $\tfrac{{n}_{\mathrm{th}}}{{C}_{{\rm{e}}}}\ll 1$. Then, the second term of equation (29) is larger than $\sqrt{\tfrac{{n}_{\mathrm{th}}}{{C}_{{\rm{e}}}}}$, and the dominant contribution to the variance is the second term in this regime.
By minimizing the approximated variance, we can obtain the optimal coupling$\begin{eqnarray}{J}_{\mathrm{opt}}={\left({C}_{\mathrm{th}}\right)}^{\tfrac{1}{4}}\sqrt{{\omega }_{{\rm{m}}}{\kappa }_{{\rm{c}}}}.\end{eqnarray}$The corresponding optimal variance with optimal r and J is$\begin{eqnarray}{V}_{\min }\simeq \displaystyle \frac{1}{\sqrt{{C}_{\mathrm{th}}}}+\displaystyle \frac{\kappa }{2{\omega }_{{\rm{m}}}}.\end{eqnarray}$Figure 7 shows that the above approximations (i.e. large Ce and small nth) are compatible with the previous more accurate equation (27). The factor ${({C}_{\mathrm{th}})}^{\tfrac{1}{4}}\sqrt{{\omega }_{{\rm{m}}}{\kappa }_{{\rm{c}}}}$ is of oder unity, hence the optimal J is of the same order of κc. Furthermore, equation (33) shows that the entanglement is influenced by two main factors: the first term is the thermal cooperativity and the second is related to the linewidth of the auxiliary cavities. If κ < 2ωm and Cth ≪ 1, strong entanglement can be achieved.
Figure 7.
New window|Download| PPT slide Figure 7.(a) Jopt versus κc, at the optimal ropt. (b) Vmin versus κc, after optimizing both r and J. Both blue (nth = 0) and red lines (nth = 5) are obtained from miniming equation (27). The dots denote the approximate results of equations (32) and (33). Other parameters: δ = 0.01ωm, G2 = 0.1ωm, γ1 = γ2 = 10−5ωm, Δ1 = −Δ2 = 2ωm, κ1 = κ2 = κ = 0.2ωm.
6. Additional discussions
We complement here the analysis of the previous section, by addressing the special case of two mechanical modes with degenerate frequency. We also discuss in more detail the effect on entanglement of the auxiliary cavities damping.
6.1. Entanglement of two same-frequency mechanical oscillators
We assume here that the two mechanical frequencies are identical (i.e. ω1 = ω2), such that $\delta =\tfrac{{{\rm{\Omega }}}_{1}-{{\rm{\Omega }}}_{2}}{2}=0$. The effective master equation is found in this case as:$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{\rm{d}}\hat{\rho }(t)}{{\rm{d}}t} & = & {{\rm{\Gamma }}}_{s}^{{\prime} }{{ \mathcal D }}_{{\rm{s}}}[{\hat{b}}_{1}^{\dagger },{\hat{b}}_{1}^{\dagger }]\hat{\rho }(t)+{{\rm{\Gamma }}}_{\uparrow ,1}^{{\prime} }{ \mathcal D }[{\hat{b}}_{1}^{\dagger }]\hat{\rho }(t)\\ & & +\ {{\rm{\Gamma }}}_{\downarrow ,1}^{{\prime} }{ \mathcal D }[{\hat{b}}_{1}]\hat{\rho }(t)\\ & & +\ {{\rm{\Gamma }}}_{{\rm{s}}}^{{\prime} }{{ \mathcal D }}_{{\rm{s}}}[{\hat{b}}_{1},{\hat{b}}_{1}]\hat{\rho }(t)+{{\rm{\Gamma }}}_{{\rm{s}}}^{{\prime} }{{ \mathcal D }}_{{\rm{s}}}[{\hat{b}}_{2}^{\dagger },{\hat{b}}_{2}^{\dagger }]\hat{\rho }(t)\\ & & +\ {{\rm{\Gamma }}}_{\uparrow ,2}^{{\prime} }{ \mathcal D }[{\hat{b}}_{2}^{\dagger }]\hat{\rho }(t)\\ & & +\ {{\rm{\Gamma }}}_{\downarrow ,2}^{{\prime} }{ \mathcal D }[{\hat{b}}_{2}]\hat{\rho }(t)+{{\rm{\Gamma }}}_{{\rm{s}}}^{{\prime} }{{ \mathcal D }}_{{\rm{s}}}[{\hat{b}}_{2},{\hat{b}}_{2}]\hat{\rho }(t)\\ & & +\ {{\rm{\Gamma }}}_{{\rm{s}}}^{{\prime} }{{ \mathcal D }}_{{\rm{s}}}[{\hat{b}}_{1}^{\dagger },{\hat{b}}_{2}^{\dagger }]\hat{\rho }(t)\\ & & +\ {{\rm{\Gamma }}}_{\uparrow ,12}^{{\prime} }{ \mathcal D }[{\hat{b}}_{1}^{\dagger },{\hat{b}}_{2}]\hat{\rho }(t)+{{\rm{\Gamma }}}_{\downarrow ,12}^{{\prime} }{{ \mathcal D }}_{{\rm{s}}}[{\hat{b}}_{1},{\hat{b}}_{2}^{\dagger }]\hat{\rho }(t)\\ & & +\ {{\rm{\Gamma }}}_{{\rm{s}}}^{{\prime} }{{ \mathcal D }}_{{\rm{s}}}[{\hat{b}}_{1},{\hat{b}}_{2}]\hat{\rho }(t)\\ & & +\ {{\rm{\Gamma }}}_{{\rm{s}}}^{{\prime} }{{ \mathcal D }}_{{\rm{s}}}[{\hat{b}}_{2}^{\dagger },{\hat{b}}_{1}^{\dagger }]\hat{\rho }(t)+{{\rm{\Gamma }}}_{\uparrow ,12}^{{\prime} }{{ \mathcal D }}_{{\rm{s}}}[{\hat{b}}_{2}^{\dagger },{\hat{b}}_{1}]\hat{\rho }(t)\\ & & +\ {{\rm{\Gamma }}}_{\downarrow ,12}^{{\prime} }{{ \mathcal D }}_{{\rm{s}}}[{\hat{b}}_{2},{\hat{b}}_{1}^{\dagger }]\hat{\rho }(t)\\ & & +\ {{\rm{\Gamma }}}_{{\rm{s}}}^{{\prime} }{{ \mathcal D }}_{{\rm{s}}}[{\hat{b}}_{2},{\hat{b}}_{1}]\hat{\rho }(t).\end{array}\end{eqnarray}$This form is considerably more complex than equation (6). In fact, the previous treatment assumed δ ≫ γ1,2, and several terms could be neglected, due to the presence of fast-oscillating factors of type e±iδt. Here, however, such contributions should be retained. We note that, when δ is large, fast oscillating factors can also appear in dissipative terms where only an individual mechanical mode is involved. For example, the first term of equation (34) does not appear in equation (6). This is due to our choice of drives with frequency ωc ± ωm, where ωm is detuned from both mechanical frequencies ω1,2. The rates of equation (34) read as follows: $\begin{eqnarray*}\begin{array}{rcl}{{\rm{\Gamma }}}_{{\rm{s}}}^{{\prime} } & = & {G}_{1}{G}_{2}{S}_{\mathrm{op}}(0),\\ {{\rm{\Gamma }}}_{\uparrow ,1}^{{\prime} } & = & {G}_{1}^{2}{S}_{\mathrm{op}}(0)+{G}_{2}^{2}{S}_{\mathrm{op}}(-2{\omega }_{{\rm{m}}})+{\gamma }_{1}{n}_{\mathrm{th},1},\\ {{\rm{\Gamma }}}_{\downarrow ,1}^{{\prime} } & = & {G}_{2}^{2}{S}_{\mathrm{op}}(0)+{G}_{1}^{2}{S}_{\mathrm{op}}(2{\omega }_{{\rm{m}}})+{\gamma }_{1}(1+{n}_{\mathrm{th},1}),\\ {{\rm{\Gamma }}}_{\uparrow ,2}^{{\prime} } & = & {G}_{1}^{2}{S}_{\mathrm{op}}(0)+{G}_{2}^{2}{S}_{\mathrm{op}}(-2{\omega }_{{\rm{m}}})+{\gamma }_{2}{n}_{\mathrm{th},2},\\ {{\rm{\Gamma }}}_{\downarrow ,2}^{{\prime} } & = & {G}_{2}^{2}{S}_{\mathrm{op}}(0)+{G}_{1}^{2}{S}_{\mathrm{op}}(2{\omega }_{{\rm{m}}})+{\gamma }_{2}(1+{n}_{\mathrm{th},2}),\\ {{\rm{\Gamma }}}_{\uparrow ,12}^{{\prime} } & = & {G}_{1}^{2}{S}_{\mathrm{op}}(0)+{G}_{2}^{2}{S}_{\mathrm{op}}(-2{\omega }_{{\rm{m}}}),\\ {{\rm{\Gamma }}}_{\downarrow ,12}^{{\prime} } & = & {G}_{2}^{2}{S}_{\mathrm{op}}(0)+{G}_{1}^{2}{S}_{\mathrm{op}}(2{\omega }_{{\rm{m}}}).\end{array}\end{eqnarray*}$ If, as before, we consider symmetric mechanical damping (γ1 = γ2 = γ, nth,1 = nth,2 = nth), the steady-state second moments are$\begin{eqnarray}\begin{array}{l}\langle {\hat{X}}_{+}^{2}\rangle =\displaystyle \frac{{{\rm{e}}}^{-2r}+\left(\tfrac{1+2{n}_{\mathrm{th}}}{2{C}_{{\rm{e}}}}+{\epsilon }_{2}\right){\cosh }^{2}\xi +{\epsilon }_{1}{\sinh }^{2}\xi }{2[1/2{C}_{{\rm{e}}}+({\epsilon }_{1}-{\epsilon }_{2})]{\cosh }^{2}\xi -{\epsilon }_{1}+1}\\ \langle {\hat{X}}_{-}^{2}\rangle =1/2+{n}_{\mathrm{th}},\\ \langle {\hat{P}}_{+}^{2}\rangle \\ \ =\ \displaystyle \frac{1+2{n}_{\mathrm{th}}+2{C}_{{\rm{e}}}(1+{\epsilon }_{2})+2{C}_{{\rm{e}}}\tanh \xi [\tanh \xi (1+{\epsilon }_{1})+2]}{2-4{C}_{{\rm{e}}}({\epsilon }_{2}-1)+4{C}_{{\rm{e}}}({\epsilon }_{1}-1){\tanh }^{2}\xi },\\ \langle {\hat{P}}_{-}^{2}\rangle =1/2+{n}_{\mathrm{th}},\end{array}\end{eqnarray}$with$\begin{eqnarray}{\epsilon }_{1}=\displaystyle \frac{{S}_{\mathrm{op}}(2{\omega }_{{\rm{m}}})}{{S}_{\mathrm{op}}(0)},\end{eqnarray}$$\begin{eqnarray}{\epsilon }_{2}=\displaystyle \frac{{S}_{\mathrm{op}}(-2{\omega }_{{\rm{m}}})}{{S}_{\mathrm{op}}(0)},\end{eqnarray}$$\begin{eqnarray}{C}_{{\rm{e}}}=\displaystyle \frac{{G}_{2}^{2}{S}_{\mathrm{op}}(0)}{\gamma }.\end{eqnarray}$$\langle {\hat{X}}_{-}^{2}\rangle =\langle {\hat{P}}_{-}^{2}\rangle =1/2+{n}_{\mathrm{th}}$ is only related to the temperature of the thermal bath since X− and P− are evading the back-action from the optical field. Nonzero entanglement needs:$\begin{eqnarray}V=\langle {\hat{X}}_{+}^{2}\rangle +\langle {\hat{P}}_{-}^{2}\rangle =\displaystyle \frac{1}{2}+{n}_{\mathrm{th}}+\langle {\hat{X}}_{+}^{2}\rangle \lt 1,\end{eqnarray}$which implies that the thermal phonon occupation nth should be very small. Note that when nth = 0, entanglement generation is equivalent to squeezing. By considering a symmetric optical spectrum (i.e. ε1 = ε2 = ε),$\begin{eqnarray}2\langle {\hat{X}}_{+}^{2}\rangle =\displaystyle \frac{2{n}_{\mathrm{th}}+1+2{C}_{{\rm{e}}}[({r}^{2}+1)\epsilon +{\left(r-1\right)}^{2}]}{1+2{C}_{{\rm{e}}}(\epsilon -1)({r}^{2}-1)}.\end{eqnarray}$When nth = 0 and the variance satisfies $2\langle {\hat{X}}_{+}^{2}\rangle \lt 1$, entanglement can be generated. Since throughout the paper we assumed δ ≪ ω1,2, the counter-rotating term ε and the cooperativity Ce are numerically very close to the case of different mechanical oscillators. Note that equation (40) takes the same form as equation (27) except for the replacement of Ce with 2Ce, thus the analysis of optimal entanglement with respect to ropt and Jopt is the same of the previous section. Although when δ = 0, we need a smaller G2 to realize an entangled state, compared to the case δ ≠ 0, the value of the logarithmic negativity suffers the decoupling of X−, P− from the optical fields. In figure 8, we show analytical and numerical results for the variance V and logarithmic negativity EN, showing that mechanical entanglement is also feasible when δ = 0, but the logarithmic negativity EN is less than the case of δ ≠ 0 due to inefficient cooling of both ${\hat{\beta }}_{1}$ and ${\hat{\beta }}_{2}$.
Figure 8.
New window|Download| PPT slide Figure 8.(a) The logarithmic negativity EN. (b) The variance V. Red , blue, orange lines are G2 = 0.1ωm, 0.04ωm, 0.03ωm. The parameters: δ = 0, γ = 10−5ωm, κc = 3ωm, Δ1 = −Δ2 = 2ωm, κ1 = κ2 = 0.3ωm, J1 = J2 = 5ωm , nth = 0.
6.2. The dependence of entanglement on auxiliary cavity dissipation κ
In the previous section, we considered the impact of J on the optical spectrum. Note that small κ1 = κ2 = κ can strongly reduce ε. This can be seen from equation (24), and is also clearly illustrated by figure 9(a). With a small κ, the central peak becomes much higher and the dips get deeper, which makes Sop(±δ) ≫ Sop(±2ωm ± δ). Due to the weak-coupling condition, the value of κ cannot be too small for the effective master equation to be valid, unless the couplings G1,2 are also decreased. Thus, we examine the dependence of entanglement on κ through numerical simulations in figure 9(b), which allow us to reach very small values of κ. As expected, we find that a smaller κ leads to stronger entanglement.
In conclusion, we theoretically propose a scheme to generate strong entanglement of two mechanical oscillators in an optomechanical cavity in the unresolved-sideband regime. To suppress unwanted counter-rotating terms, we add two optical cavities and cancel their effect via destructive quantum interference. In the weak coupling regime, based on an effective master equation approach, we derive an analytical variance of the two mechanical modes and obtain the optimal coupling parameters to achieve strong entanglement.
Acknowledgments
YDW acknowledges support from NSFC (Grant No. 11 574 330 and No. 11 434 011), MOST (Grant No. 2017FA0304500) and the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB23000000). S Chesi acknowledges support from the National Key Research and Development Program of China (Grant No. 2016YFA0301200), NSAF (Grant No. U1930402), and NSFC (Grants No.11974040 and No. 1 171 101 295). We also thank R Fazio and G C La Rocca for helpful discussions.
,1,2,51Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, 
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
