1. Introduction

Quantum entanglement, implying that the existence of global states of a composite system which cannot describe the states of individual subsystems [1]. It has become the physical resource for the applications including quantum metrology, computation and communication [2–4]. Cavity optomechanical system, as an ideal candidate for implementing quantum information carriers, has been used in the generation of different types of entanglements, for example two optical fields [5], the mechanical and optical mode [6–12], and the mechanical and magnon mode [13,14]. Moreover, macroscopic entanglement between two mechanical oscillators has been observed in experiments [15–18]. To improve the bosonic modes entanglement in cavity optomechanical systems, various schemes have been proposed, for example, by introducing an ancillary mechanical mode [9], employing lesser pulsed interactions [19] and increasing an auxiliary cavity [20].

In order to realize large-scale light-vibration entanglement, it is necessary to eliminate the suppression influence induced by optical dark-mode (DM) effect [8,21–29]. This effect comes from the coupling of an optical mode to multiple degenerate or near-degenerate vibrational modes [25–29]. Recently, the method of breaking the dark-mode effect has become a new research hotspot, which can be used to enhance the photon-phonon [30] and phonon-phonon entanglement [8].

However, the reason for the realization of DM effect and how to make it continuously adjustable have not yet been fully revealed. Here, we propose a new opinion to understand the dark-mode effect of the optomechanical system. By taking the exceptional points (EPs) of the system, the DM effect can be explained as the quantum states coalesce.

The remainder of this paper is organized as follows. In Sec. 2, we present the Hamiltonian of the system and demonstrate that the dark-mode effect will be broken at EPs, these critical points are induced by asymmetric coupling of counterpropagating optical modes. In Sec. 3, we prove that quantum entanglement will be destroyed at EPs and give the physical explanation of it. Meanwhile, we propose light-vibration entanglement influenced by chirality of the system. Finally, we conclude this paper in Sec. 4.

2. Model and system

In this paper, we consider a system consists of a silica microtoroid whispering-gallery-mode (WGM) resonator that allows for both sides of light through the tapered optical fiber waveguides as shown in Fig. 1(a). This optomechanical device supports two counterpropagating optical modes, i.e., the clockwise (CW) and counterclockwise (CCW) modes. These two modes can be asymmetric coupled by controlling the relative size and position of two nano scatters (i.e., relative phase angle $\beta $) placed within the mode volume of the resonator [31,32]. Besides, the resonator can hold a mechanical breathing mode with frequency ${\omega _b}$. In our system, both of the two optical modes are coupled to the common mechanical mode. In a rotating frame with respect to the transformation operator $U = \exp [ - i({\omega _{d,CW}}a_{CW}^\dagger {a_{CW}} + {\omega _{d,CCW}}a_{CCW}^\dagger {a_{CCW}})t]$ under ${\omega _{d,CW}} = {\omega _{d,CCW}}$, the Hamiltonian of the physical system considered in the text reads as (with $\hbar = 1$):

 

Fig. 1. (a) The WGM resonator with mechanical mode driven by two classical fields propagating in the opposite direction with driving amplitude ${\Omega _1}$ and ${\Omega _2}$, respectively. (b) ${J_{12(21)}}$ is the non-Hermitian interaction between CW and CCW mode and the optomechanical interaction between jth cavity-field mode and mechanical mode is ${g_j}$. The real and imagery parts of the eigenvalues for the system are shown in (c) and (d), respectively. The exceptional points emerge at $\beta \in \{ 0.5\pi ,1.5\pi \} $. (e) ${J_{12(21)}}$ versus the relative phase angle $\beta $. In this letter, we choose ${\varepsilon _1} \approx {\varepsilon _2} = 2\pi \times (1.5 - 0.1i)\; \text{MHz}$.

Download Full Size | PDF

Quantum Langevin equations of this optomechanical system can read

For simplicity, we choose the same decay rates of optical modes as ${\kappa _{CW}} = {\kappa _{CCW}} = \kappa $ and the same optomechanical coupling strength as ${g_{CW}} = {g_{CCW}} = g$ in the following work. The linearized equations of motion for these quantum fluctuations can be written as:

The linearized optomechanical can be given according to Eq. (6). For studying the optomechanical entanglement of the system, the beam-splitting-type interaction between these bosonic modes are expected to exist in the linearized Hamiltonian, and hence neglecting the two-mode-squeezing interaction terms by making the rotating-wave approximation (RWA). The linearized optomechanical Hamiltonian in the RWA can be taken in the following form as:

In order to observe the exceptional point (EP) in the optomechanical system, we take the Hamiltonian in Eq. (7) with dissipative terms in a matrix form as

The eigenvalues of this Hamiltonian can be obtained by solving $Det({H_r} - \lambda I) = 0$, where I is an identity matrix. Due to the analytical solution of the eigenvalues are too tedious, hence we numerically solve the eigenvalues and the results have been shown in Fig. 1(c) and 1(d). It is obvious that the eigenvalues become coalescence at the points $\beta = 0.5\pi $ and $1.5\pi $, indicating the emergence of EPs in the system. Interestingly, we give the asymmetric coupling strength ${J_{12(21)}}$ as the functions of relative phase angle $\beta $ in Fig. 1(e). We find that ${J_{12(21)}} = 0$ at EPs. Different from the general coherent coupling, this type of asymmetric coupling is non-Hermitian, that is, ${J_{12}} \ne J_{21}^ \ast $.

To find the dark-mode area in this three bosonic modes optomechanical system, we first tune the relative phase angle $\beta $ at EPs, which means that the non-Hermitian coupling between CW and CCW modes is zero. In this case, the system possesses two hybrid cavity modes: bright (${a_B}$) and dark (${a_D}$) modes, which defined as [23]

We can see that the effective coupling strength of the dark-mode is absent (i.e., ${G_D} = 0$) when $\tilde{\Delta}_{C W}=\tilde{\Delta}_{C C W}$. Consequently, the mode ${a_D}$ is decoupled from both the bright mode (BM) and mechanical mode, while ${a_B}$ is a bright mode and it always couples with the mechanical mode due to ${G_B} > 0$. We will demonstrate that the existence of dark-mode can be controlled by utilizing the EPs in the following section.

By defining the optical and mechanical quadratures $\delta {X_o} = (\delta {o^\dagger } + \delta o)/\sqrt 2 $ and $\delta {Y_o} = i(\delta {o^\dagger } - \delta o)/\sqrt 2 $, and the corresponding input-noise operators $X_o^{in} = (o_{in}^\dagger + {o_{in}})/\sqrt 2 $ and $Y_o^{in} = i(o_{in}^\dagger - {o_{in}})/\sqrt 2 $, we can get the linearized Langevin equations as $\mathop u\limits^\cdot (t) = Au(t) + N(t)$, where $u(t) = {[\delta {X_{aCW}},\delta {Y_{aCW}},\delta {X_{aCCW}},\delta {Y_{aCCW}},\delta {X_b},\delta {Y_b}]^T}$ is the fluctuation operator vector. The noise operator vector reads as $N(t) = \sqrt 2 {[\sqrt \kappa X_{aCW}^{in},\sqrt \kappa Y_{aCW}^{in},\sqrt \kappa X_{aCCW}^{in},\sqrt \kappa Y_{aCCW}^{in},\sqrt {{\gamma _b}} X_b^{in},\sqrt {{\gamma _b}} Y_b^{in}]^T}$, and the coefficient matrix is

The formal solution of the Langevin equations is given by $u(t) = M(t)u(0) + \int_0^t {M(t - s)N(s)ds} $, where $M(t) = \exp (At)$. Especially, the parameters used in our simulations satisfy the stability conditions according to Routh-Hurwitz criterion [33]. For studying the tripartite entanglement, we calculate the steady-state solution of the covariance matrix V, which defined as ${V_{kl}} = \frac{1}{2}[\left\langle {{u_k}(\infty ){u_l}(\infty )} \right\rangle + \left\langle {{u_l}(\infty ){u_k}(\infty )} \right\rangle ]$, for $k,l = 1, \ldots ,6$. The covariance matrix V fulfills the Lyapunov equation $AV + V{A^T} ={-} Q$, where $Q = diag\{ \kappa ,\kappa ,\kappa ,\kappa ,{\gamma _b}(2{n_b} + 1),{\gamma _b}(2{n_b} + 1)\} $. We can quantify the bipartite and tripartite entanglement by using the logarithmic negativity ${E_{N,j}}$ and the minimum residual contangle $E_\tau ^{r|s|t}$, respectively, which defined as

Here, ${\xi _j} \equiv {2^{ - 1/2}}{\left\{ {\sum {({V_j^{\prime}} )- {{\left[ {\sum {{{({V_j^{\prime}} )}^2} - 4\det V_j^{\prime}} } \right]}^{1/2}}} } \right\}^{1/2}}$, with $\sum {({V_j^{\prime}} )} \equiv \det {\mathrm{{\cal A}}_j} + \det \mathrm{{\cal B}} - 2\det {\mathrm{{\cal C}}_j}$, is the smallest eigenvalue of the partial transpose of the reduced correlation matrix $V_j^{\prime} = \left( {\begin{array}{{cc}} {{\mathrm{{\cal A}}_j}}&{{\mathrm{{\cal C}}_j}}\\ {\mathrm{{\cal C}}_j^T}&\mathrm{{\cal B}} \end{array}} \right)$, which can be obtained by removing the rows and columns of the modes in V. The $E_\tau ^{r|s|t}$ in Eq. (13) is the minimum residual contangle, where $(r,s,t) \in ({a_{CW}},{a_{CCW}},b)$ denote all the permutations of the three mode indices. $E_\tau ^{r|(st)}$($E_\tau ^{r|s}$ or $E_\tau ^{r|t}$) means the contangle of subsystems of r and st (s or t), which is a proper entanglement monotone defined as the squared logarithmic negativity. The minimum value of the residual contangle satisfies $E_\tau ^{r|s|t} > 0$.

3. Result and analysis

We display the bipartite and tripartite optomechanical entanglement ${E_{N,j}}$ versus the effective detuning $\tilde{\Delta}_j$ in both BM and DM regimes in Fig. 2(a) and 2(b), respectively. We show that in BM regime, i.e., relative phase angle is tuned as $\beta = n\pi (n = 0,1,2)$, a large bipartite entanglement is achieved around the detuning at $\tilde{\Delta}_j \approx 0.7{\omega _b}$ while the optimal tripartite entanglement is corresponding to the detuning at $\tilde{\Delta}_j \approx 0.5{\omega _b}$. However, in the DM regime, i.e., relative phase angle is tuned as $\beta = 0.5\pi ,1.5\pi $, the thermal phonons concealed in the dark mode will make the quantum entanglement destroyed [8,30].

 

Fig. 2. (a) Bipartite entanglement ${E_{N,j}}$ and (b) tripartite entanglement $E_\tau ^{r|s|t}$ versus $\tilde{\Delta} _{j}/{\omega _b}$ in the BM regime ($\beta = n\pi $, dashed lines) and in the DM regime ($\beta = 0.5\pi ,1.5\pi $, horizontal solid lines). Here, we choose $\tilde{\Delta} _{CW} = \tilde{\Delta} _{CCW}$. (c) ${E_{N,j}}$ and (d) $E_\tau ^{r|s|t}$ versus $\beta $ under the optimal detunings $\tilde{\Delta} _{j} = 0.7{\omega _b}$ for ${E_{N,j}}$ and $\tilde{\Delta} _{j} = 0.5{\omega _b}$ for $E_\tau ^{r|s|t}$. (e) ${E_{N,j}}$ and (f) $E_\tau ^{r|s|t}$ versus $\beta $ and $\tilde{\Delta} _{CCW}/\tilde{\Delta} _{CW}$. Other parameters are ${\gamma _b}/2\pi = 100Hz$, $\kappa /2\pi = 1 \text{MHz}$, $T = 0.01K$, ${\omega _b}/2\pi = 10\text{MHz}$ and ${G_j}/{\omega _b} = 0.02$ for ${E_{N,j}}$ and ${G_j}/{\omega _b} = 0.03$ for $E_\tau ^{r|s|t}$.

Download Full Size | PDF

In Fig. 2(c) and 2(d), we display ${E_{N,j}}$ and $E_\tau ^{r|s|t}$ versus $\beta $. We emphasize the regime of DM effect, as the green areas show. We find that the entanglement of the optical and mechanical mode is absent in DM regime, where the relative phase angle is tuned at EPs ($\beta = 0.5\pi ,1.5\pi $). However, when the relative phase angle is tuned away from EPs, that is, the whole system is in the BM regime, where the optomechanical entanglement is recovery. These findings indicate that one can control multimode quantum entanglement flexibly via the relative phase angle away from EPs.

Moreover, we can also see the DM effect on bipartite and tripartite entanglement in Fig. 2(e) and 2(f), respectively. We plot ${E_{N,j}}$ and $E_\tau ^{r|s|t}$ as functions of $\tilde{\Delta} _{CCW}/\tilde{\Delta} _{CW}$ and $\beta $. We can analyze the results in two cases: (i) when ${J_{12(21)}} = 0$, i.e., the phase angle is tuned at EPs (as the red stars show), the dark-mode emerge under $\tilde{\Delta} _{CCW} = \tilde{\Delta} _{CW}$, where only the bright-mode coupled with the mechanical mode. The dashed lines in Fig. 2(e) and 2(f) are degeneracy lines. (ii) When ${J_{12(21)}} \ne 0$, especially for $\beta = n\pi $, the dark-mode will be broken, hence the constructive and destructive interference caused by the optical-fields coupling will lead to the optomechanical entanglement.

The underlying physics can be understood as follows. The asymmetric coupling ${J_{12(21)}}$ provides the origin for breaking the dark-mode effect and builds the channel to exchange the excitation energy between the two optical modes. Once ${J_{12(21)}} = 0$, this channel is closed, resulting in the entanglement is almost unfeasible. By choosing proper ratio of the normalized driving detuning, the entanglement can be partially recovery, and the degree of it is really small. However, when ${J_{12(21)}} \ne 0$, the entanglement will completely feasible even if $\tilde{\Delta} _{CCW} = \tilde{\Delta} _{CW}$.

It is well known that the thermal noise has a great influence on quantum entanglement, therefore, we should study a new mechanism to realize a noise-tolerant entanglement. From Fig. 3(a), we can see that the ground-state cooling of the mechanical mode (i.e., ${n_f} < 1$) can be achieved in the regime of BM. However, at $\beta = 0.5\pi ,1.5\pi $ (corresponding to EPs), the ground-state of mechanical mode cannot be cooled (${n_f} > 1$). Interestingly, the populations of two optical modes (i.e., CW and CCW modes) are decreased rapidly at EPs (see Fig. 3(b)), which means that few numbers of optical modes can entangle with the mechanical mode. This point can also be used to explain why the light-vibration entanglement decreased in the regimes of dark mode. However, the populations of the two optical modes increased dramatical in the regimes of BM regimes, where the entanglement will be recovered. In addition, we can see that the optomechanical entanglement in the BM regime is more robust against thermal noise than that in the DM regime, as the pink area shows in Fig. 3(c).

 

Fig. 3. (a) The average phonon number ${n_f}$ in this optomechanical system versus the relative phase angle $\beta $ under $\tilde{\Delta} _{j} = 0.5{\omega _b}$. (b) The populations of the CW and CCW modes versus $\beta $. (c) ${E_{N,CW}}$ versus phonon number ${n_b}$ in the BM (blue curves) and DM (red curves) regimes, respectively. The other parameters are the same as those in Fig. 2.

Download Full Size | PDF

In the following, we analysis the general case for ${G_{CW}} \ne {G_{CCW}}$. Therefore, we define the chirality of the system as:

The effective coupling strength can be adjusted by controlling the driving amplitude, and $k \in [0,1]$. For the balanced bidirectional coupling in Eq. (14), the chirality is $k = 0$, which is the nonchiral case, while for the chirality $k = 1$, represents the pure chiral case.

In Fig. 4(a) and 4(b), we plot ${E_{N,CW}}$ and $E_\tau ^{r|s|t}$ as the functions of the relative phase angle and the chirality, respectively. Similarly, regardless of the changing of chirality, we can also demonstrate that the optomechanical entanglement is fully suppressed in the DM regime (i.e., the relative phase angles at EPs). And the entanglement is recovery in the BM regime. In addition, we also note that the value of entanglement can be influenced by the chirality of the system: with the increase of the chirality, the entanglement will decrease until it fully disappears. In Fig. 4(c), we can see that the different values of ${E_{N,j}}$, for $k \in (0.57,0.64)$, only entanglement ${E_{N,CW}}$ exists, while the other entanglement ${E_{N,CCW}}$ disappears, as the blue area shows. In the fully chiral case ($k = 1$), the entanglement will be completely destroyed even in the BM regime, as shown in the green area. Furthermore, we find that in the DM regime, thermal phonons concealed in the dark-mode will be excited with the increase of chirality (see Fig. 4(d)), which can also demonstrate that the optomechanical entanglement will completely be suppressed by the thermal noise.

 

Fig. 4. (a) ${E_{N,CW}}$ and (b) $E_\tau ^{r|s|t}$ as functions of chirality and the relative phase angles. (c) Different values of entanglement ${E_{N,CW}}$ (red lines) and ${E_{N,CCW}}$ (blue lines) versus chirality with $\beta = \pi $. (d) The mechanical mode number ${n_b}$ versus chirality and the relative phase angles under ${G_j}/{\omega _b} = 0.01$. Other parameters are the same as those in Fig. 2.

Download Full Size | PDF

4. Conclusion

We have shown that how to explain the dark-mode effect via the exceptional points of the system. And the optomechanical entanglement can be flexible controlled by adjusting the relative phase angle of the non-Hermitian coupling. The entanglement is destroyed at EPs while recovery when the relative phase angle is tuned away from EPs. In particular, the ground-state cooling of the mechanical mode is achievable in DM regime, and the thermal noise robustness of the bipartite entanglement is large than that in the BM regime. Moreover, the chiral coupling of the system also influence the degree of the quantum entanglement, which provides another method to switch between with or without entanglement.