Abstract

We study the controllable optical response in a three-mode optomechanical system comprised of two indirectly coupled cavity modes and an intermediate mechanical mode. The two cavity modes are assumed to have different frequencies and driven by two control fields on the red and blue sidebands, respectively. When the system is perturbed by two probe fields satisfying the specific matching condition, a series of intriguing phenomena can be observed by adjusting phases and amplitudes of the control fields, such as absorption-amplification switching, ultra-narrow response windows, frequency-independent perfect reflection, and ultralong optical group delay. We also compare our system with conventional optomechanical systems to highlight its distinct features. Our results may have potential applications in optical communication and quantum information processing.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

In the past decades, due to the rapid development of nanotechnology, optomechanical systems have become a powerful platform for exploring and testing quantum sciences at the macroscopic scale [1, 2]. As an interdiscipline, optomechanics primarily studies the nonlinear interactions between electromagnetic fields and mechanical objects. Such interactions, arising from the radiation pressure of optical modes, have led to a host of important applications, such as ground-state cooling of mechanical oscillators [3,4], quantum synchronization [5–7], ultralow-threshold chaos [8], and tunable optical nonreciprocity [9–15]. Similar to multilevel atomic systems, some intriguing phenomena have been observed in multimode optomechanical systems due to quantum interference between different excitation paths [1,2]. This has sparked an upsurge in the study of multimode optomechanical systems with diverse driving schemes [9–23]. For instance, optomechanically induced transparency (OMIT) [24–26] and absorption (OMIA) [27,28] have been well explored as the optomechanical analogues of electromagnetically induced transparency (EIT) [29,30] and absorption (EIA) [31,32], respectively. We note that OMIT is usually realized in optomechanical systems driven on the red sidebands, whose beam-splitter interactions can result in the required destructive interference. On the other hand, the realization of OMIA depends on constructive interference, which typically arises from mode-squeezing interactions in the case of a blue-sideband driving [1,2,33]. Quite recently, it has been shown that optomechanical systems driven on multiple sidebands can provide a promising platform for many important applications, such as strong entanglement [34–36] and directional amplifier [33,37]. Exploiting optomechanical interference skillfully, it is also viable to realize other coherent phenomena like coherent perfect absorption (CPA), transmission (CPT), and synthesis (CPS) [18,20,38], or attain phase-dependent optical responses as a closed-loop transition structure is formed [21, 39, 40]. In spite of a few seminal works [33–37], combining the techniques of phase modulation and multi-sidebands excitation is still a less cultivated land in optomechanical studies.

In this paper, we consider a three-mode optomechanical system, whose left and right cavity modes interact linearly with an intermediate mechanical mode and are driven by two control fields on the red and blue sidebands, respectively. Our model is different from a similar one [19] where two cavity modes are coupled with a mechanical mode via quadratic interactions and driven by different red-detuned fields. We also employ two probe fields to perturb our optomechanical system by assuming that they are near resonant with corresponding cavities modes. When the probe frequencies satisfy the specific matching condition, our system is found to exhibit distinct features as compared to conventional three-mode optomechanical systems [16–19], including ultra-narrow response windows and significant amplification. It is also viable to realize other intriguing phenomena like absorption-amplification switching, frequency-independent reflection, and ultraslow light propagation in our model.

2. Model and equations

We consider in Fig. 1(a) a three-mode optomechanical system based on Fabry-Pérot cavities with two optical cavities separated by a mechanical membrane. The two cavities are assumed to have different mode frequencies so that they cannot interact directly even if the membrane is not a perfect mirror. Here the left (right) cavity mode is driven by a control field of amplitude εc1 (εc2) and frequency ωc1 (ωc2), and also perturbed by a probe field of amplitude εp1 (εp2) and frequency ωp1 (ωp2). Fig. 1(b) shows a four-level configuration with na1 (na2) and nb being the excitation numbers of the left (right) cavity mode and the mechanical mode, respectively. It is clear that the couplings between different levels form a closed-loop transition so that the optical response should be phase-dependent. In addition, our results do not depend on the quality factors, or more precisely, the frequencies of cavity and mechanical modes as shown below, thus our model can also be realized with other platforms like microring optomechanical systems [41], superconducting circuit systems [42], and optomechanical crystal systems [43]. The Hamiltonian of our optomechanical system can be written as

=ωmbb+j=1,2[ωjajaj+(1)jgjajaj(b+b)+i(εcjajeiϑjeiωcjt+εpajeiωpjtH.c.)],
where aj (aj) is the annihilation (creation) operator of the jth cavity mode with frequency ωj, b (b) is the annihilation (creation) operator of the mechanical mode with frequency ωm, and gj is the single-photon optomechanical coupling strength between cavity mode aj and mechanical mode b. The factor (−1)j stands for the different dependence of the left and right cavity mode frequencies on the mechanical motion. We have taken ϑ1 and ϑ2 as phases of the two control fields while assuming vanishing phases for both probe fields because it is the relative phase of all external fields that determines the optical response in a closed-loop transition. We have also set εp1 = εp2 = εp for simplicity.

 

Fig. 1 (a) Fabry-Pérot implementation of our three-mode optomechanical system. The intermediate membrane oscillator with damping rate γm is linearly coupled with two optical cavities simultaneously. The left (right) cavity mode with decay rate κ is driven by a control field with amplitude εc1 (εc2), frequency ωc1 (ωc2), and phase ϑ1 (ϑ2). We also employ a probe field with amplitude εp1 (εp2) and frequency ωp1 (ωp2) to perturb the left (right) cavity mode. Phases of the two probe fields, however, have been assumed to be vanishing without the loss of generality. (b) Effective energy level configuration exhibiting the closed-loop transition for our three-mode optomechanical system. Relevant states of the left cavity, intermediate mechanical, and right cavity modes are denoted by excitation numbers na1, nb, and na2 in order.

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In the rotating frame with respect to 0=j=1,2ωcjajaj, the Hamiltonian can be rewritten as

=ωmbb+j=1,2[Δjajaj+(1)jgjajaj(b+b)+i(εcjajeiϑj+εpajeiδjtH.c.)],
where Δj = ωjωcj is the detuning between cavity mode aj and control field εcj; δj = ωpjωcj is the detuning between probe field εpj and control field εcj. Taking relevant dissipation and noise terms into account, one can attain with ℋ′ the following Heisenberg-Langevin equations
a˙1=(κ+iΔ1)a1+ig1a1(b+b)+εc1eiϑ1+εpeiδ1t+2κa1in,a˙2=(κ+iΔ2)a2+ig2a2(b+b)+εc2eiϑ2+εpeiδ2t+2κa2in,b˙=(γm+iωm)b+i(g1a1a1g2a2a2)+2γmbin,
where κ represents the identical decay rate of both cavity modes (we consider here the overcoupled regime [44,45] where the total cavity loss almost roots in the external coupling loss) while γm is the damping rate of the mechanical mode. Moreover, ajin and bin denote respectively the input vacuum noise of cavity mode aj and the input thermal noise of mechanical mode b, both of which have zero mean values. For strong pumping, each mode operator can be expressed as a sum of its steady-state value and quantum fluctuation, i.e., aj = αjs + δaj and b = βs + δb.

Neglecting the weak probe fields and all time derivatives, we attain from Eq. (3) the following steady-state values

α1s=εc1eiϑ1κ+iΔ1,α2s=εc2eiϑ2κ+iΔ2,βs=i(g1|α1s|2g2|α2s|2)γm+iωm
with Δ′j = Δj + (−1)j2gjRe(βs) being the effective detuning between cavity mode aj and control field εcj. Meanwhile, the linearized equations for the quantum fluctuations can be written as
δa1.=(κ+iΔ1)δa1+iG1(δb+δb)+εpeiδ1t+2κa1in,δa2.=(κ+iΔ2)δa2iG2(δb+δb)+εpeiδ2t+2κa2in,δb.=(γm+iωm)δb+i(G1*δa1+G1δa1G2*δa2G2δa2)+2γmbin,
where Gj = gjαjs is an enhanced (effective) optomechanical coupling strength. To determine dynamics of the quantum fluctuations, we need to simultaneously solve Eq. (5) and its complex conjugate counterparts
δa1.=(κiΔ1)δa1+iG1*(δb+δb)+εpeiδ1t+2κa1in,δa2.=(κiΔ2)δa2+iG2*(δb+δb)+εpeiδ2t+2κa2in,δb.=(γmiωm)δbi(G1*δa1+G1δa1G2*δa2G2δa2)+2γmbin.

Now we assume that (i.) the two cavity modes are driven on the red and blue sidebands respectively with Δ′1 = −Δ′2 = ωm and (ii.) our optomechanical system works in the resolved sideband regime with ωm ≫ {κ, γm, G1,2}. In this case, we can perform the transformation

δa1δa1eiΔ1t,δa1inδa1ineiΔ1t,δa2δa2eiΔ2t,δa2inδa2ineiΔ2t,δbδbeiωmt,δbinδbineiωmt,
under the rotating-wave approximation so that Eqs. (5) and (6) can be simplified into
δa1.=κδa1+iG1δb+εpeiσ1t+2κa1in,δa2.=κδa2+iG2*δb+εpeiσ2t+2κa2in,δb.=γmδb+i(G1*δa1G2δa2)+2γmbin,
where σj = δj − Δ′jωpjωj is the detuning between cavity mode aj and probe field εpj. Since it is the relative phase of all external fields that determines the optical response, we can further simplify Eq. (8) into
δa1.=κδa1+iGδb+εpeiσ1t+2κa1in,δa2.=κδa2+iηGeiθδb+εpeiσ2t+2κa2in,δb.=γmδb+iG(δa1ηeiθδa2)+2γmbin
by assuming G1 = G and G2 = ηGe with η (θ) being the strength ratio (phase difference) of the two coupling strengths. For convenience, one can express Eq. (9) in the matrix form as
u˙=Mu+f+n,
where u=(δa1,δa2,δb)T, f = (εpe1t, εpe2t, 0)T, and n=(2κa1in,2κa2in,2γmbin)T are the vectors of quantum fluctuations, probe fields, and system noises in order, while the coefficient matrix M can be written as
M=(κ0iG0κiηGeiθiGiηGeiθγm).

Note our system may be unstable due to the introduction of a blue-detuned control field with its stability typically examined via the Routh-Hurwitz criterion [46,47]. Since the analytic formula is too cumbersome, we adopt here an alternative method to judge the stability: the system is stable when the real parts of all eigenvalues of matrix M are negative. As shown in Fig. 2, the system stability is independent of the phase difference θ and the stable regime can be attained only in case of 0 ⩽ η ⩽ 1 (|G2| ≤ |G1|).

 

Fig. 2 Stability diagrams attained with G1 = κ (a) and θ = 0 (b) in the case of Δ′1 = −Δ′2 = ωm and γm = 10−4κ. The yellow and blue regions denote the stable and unstable regimes, respectively.

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Our aim is to realize controllable optical response for the left probe field with the help of other fields. To achieve this aim, we further assume σ1 = −σ2 = σ, which implies in fact ωp1 + ωp2 = ω1 + ω2. Assuming that all steady-state solutions to Eq. (9) have the form 〈δo〉 = O +eiσt + Oeiσt, it is viable to attain the following two terms

A1+=f1f2+ηG2(eiθη)f1[f1f2+G2(1η2)]εp,A2+=0
for δa1 and δa2, respectively, with f1 = κ and f2 = γm. We can find from Eq. (12) that only the left cavity responds to the left probe field while the right cavity just serves as an auxiliary cavity. According to the input-output relation [48], the output amplitude of the left cavity can be written as
εoutL=2κδa1εpeiσt,
where the oscillating term can be removed, if we also assume εoutL = εL+eiσt + εLeiσt, to yield
εL+=2κA1+εp,
which describes the left output component oscillating at frequency ωp1. In this way, we can define the left reflection coefficient rL = 2κA1+/εp − 1 and the left reflectivity RL = |rL|2.

For a conventional three-mode optomechanical system driven by two red-sideband control fields (Δ′1 = Δ′2 = ωm) and perturbed by two identical probe fields (σ1 = σ2 = σ) [18, 20], Eq. (12) turns out to be

A1+=f1f2+ηG2(eiθ+η)f1[f1f2+G2(1+η2)]εp,
from which we can define the left reflectivity R′L = |r′L|2 with r′L = 2κA′1+/εp − 1.

3. Optomechanical switching with ultra-narrow linewidth

We first consider that the two control fields are in phase (θ = 0). Fig. 3(a) shows the influence of coupling strength G and detuning σ on the reflectivity RL with an absorption window found near the resonance position (σ = 0). Clearly, for a fixed strength ratio η = |G2/G1| = 0.9, this absorption window becomes narrower and narrower as G decreases. For comparison, we make a similar calculation on R′L in Fig. 3(b) and find that the absorption window is much wider for the same parameters in a conventional three-mode system. Reflectivities RL and R′L are also plotted against η and σ, respectively, in Figs. 3(c) and 3(d). We find that the absorption window deepens and narrows gradually as η increases in our system. In the conventional three-mode system, however, the window depth (width) exhibits a fast (slow) non-monotonic variation, i.e., first increases and then reduces, with the increase of η. Moreover, RL and R′L can approach zero for η → 1 and η → 0.4, respectively, implying the existence of CPA phenomenon [18, 38] in both three-mode systems. The CPA window in our system is certainly much narrower than that in the conventional three-mode system. As is well known, one major advantage of optomechanical systems is that they can serve as interferometers to allow a direct measurement of mechanical displacements via the phase shifts of transmitted or reflected fields [1,49]. In general, a narrower window enables a more precise measurement, so our three-mode system may have potential applications in, e.g., weak force sensing [50–53] and mechanical Fock-state detection [54–56].

 

Fig. 3 Reflectivity RL against G and σ (a) or η and σ (c). Reflectivity R′L against G and σ (b) or η and σ (d). We assume that η = 0.9 in (a) and (b) while G = κ in (c) and (d). Other parameters are γm = 10−4κ and θ = 0.

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To gain a deeper insight, we further examine in Fig. 4 the profiles of reflectivity RL and relevant effective mechanical damping rates. Figs. 4(a) and 4(b) show that the absorption window becomes much narrower with the decrease of G and/or the increase of η, and a larger η typically corresponds a stronger absorption. In particular, the green dot-dashed line in Fig. 4(b) verifies that our three-mode system supports an ultra-narrow CPA window. For η = 0, the right cavity is decoupled with the mechanical mode so that our system reduces to a two-mode optomechanical system driven on the red sideband. In this case, as shown in Fig. 4(c), the left probe field is almost unabsorbed with a very shallow window in reflectivity. On the other hand, for η = 1, we may observe a frequency-independent perfect reflection (FIPR) phenomenon with RL(σ) ≡ 1, as shown by the yellow dotted line in Fig. 4(b). This is because we have rL = εL+/εp = (κ + )/(κ) when η = 1 and θ = 0 according to Eq. (12). Thus FIPR is a result of perfect destructive interference between the input probe field and the corresponding reflected field. This phenomenon may also attributed to the phononic dark mode [20], an optomechanical analogue of the atomic dark state [57,58], in which the mechanical excitation is completely suppressed (βs = B± = 0).

 

Fig. 4 Profiles of reflectivity RL with η = 0.4 (a), G = κ (b), and η = 0 (c). (d) Effective mechanical damping rates against G with η = 0.9 (red) or against η with G = κ (blue) in the case of σ = 0, where the solid, dashed and dot-dashed lines correspond to Γopt,1, Γopt,2 and Γopt,3, respectively. Other parameters are γm = 10−4κ and θ = 0 if not given in relevant panels.

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Considering that the linewidth of an absorption, transparency, or amplification window in optomechanics is determined by the effective mechanical damping rate [1,59,60], we calculate the steady-state solutions to Eq. (9) to attain

(γmiσ+|G1|2|G2|2κiσ)B+=i(G1*G2)κiσεp
for the mechanical quantum fluctuation δb. It is clear that the damping rate γm and the detuning σ are modified by a complex quantity (|G1|2 − |G2|2)/(κ). In this case, the effective mechanical damping rate of our system reads
Γopt,1=γm+κG2(1η2)κ2+σ2.
Similarly, we can also obtain the effective mechanical damping rates
Γopt,2=γm+κG2κ2+σ2,
for a single-cavity optomechanical system with η = 0 [38] and
Γopt,3=γm+κG2(1+η2)κ2+σ2,
for the conventional three-mode optomechanical system [18,20]. Eqs. (17)(19) not only prove that the increase of G and/or the decrease of η will yield a widened window in our system, but also show that Γopt,3 > Γopt,2 > Γopt,1 always holds as long as η and G are nonvanishing, as shown intuitively in Fig. 4(d). We find in particular from the three red lines that Γopt,1 increases at a much lower rate as compared to Γopt,2 and Γopt,3 as G becomes larger, while from the three blue lines that only Γopt,1 decreases as η becomes larger and can approach zero for η → 1.

Now, we turn to consider that the two control fields are out of phase (θ = π). Fig. 5 is plotted in a way similar to Fig. 3 by examining the dependence of RL and R′L on G and η for comparison. In this case, it is an amplification (instead of absorption) window that appears near the resonant position σ = 0. We also find that (i.) the amplification window in our system is much narrower than that in the conventional three-mode optomechanical system and (ii.) the corresponding linewidths are sensitive to G and η in different ways just as found in the case of θ = 0. Figs. 5(c) and 5(d) show, in particular, that the amplification effect in our system can be exponentially enhanced as η is increased in the stable regime while that in the conventional three-mode system first increases and then reduces in a roughly linear way with the maximum R′L = 2 located at η ≈ 0.42. Therefore, our three-mode system may have potential application in precise optical amplification owing to the tunable ultra-narrow window.

 

Fig. 5 Reflectivity RL against G and σ (a) or η and σ (c). Reflectivity R′L against G and σ (b) or η and σ (d). We assume that η = 0.9 in (a) and (b) while G = κ in (c) and (d). Other parameters are γm = 10−4κ and θ = π.

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Heretofore our calculations have been restricted to the cases of θ = 0 and θ = π. Next, we examine in Fig. 6 the reflectivity profiles of our system for different values of θ. It is clear that we can realize an absorption-amplification transition by adjusting θ with the optimal absorption (amplification) corresponding to θ = 0 (θ = π), and the reflectivity profiles are asymmetric with respect to σ = 0 as long as we have θ (n = 0, ±1, ±2, ...).

 

Fig. 6 Profiles of reflectivity RL for different values of θ. Relevant parameters are γm = 10−4κ, G = κ, and η = 0.9 except θ specified in the figure.

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4. Slow light effect

It is well known that an optical signal with its carrier frequency falling into an OMIT window experience a rapid phase dispersion and thus exhibit a largely reduced group velocity, similar to what happens in an EIT window of atomic systems. Such a slow light effect is essential for optical communications and quantum information process and have been widely studied in various optomechanical systems [24–26]. This urges us to further examine the slow light effect of our system by evaluating the optical group delay

τL=dϕdωp1,
where ϕ =arg[rL(ωp1)] is the phase of the left output field at frequency ωp1. Similarly, we can define
τL=dϕdωp
for the left output field of the conventional three-mode optomechanical system [18, 20] with ϕ′ =arg[r′L(ωp)]. As usual, the slow (fast) light effect will be attained in case that the optical group delay is positive (negative). The group refractive index ng (n′g), another typical figure of merit for evaluating the slow and fast light effects, can be easily attained because it is proportional to the optical group delay τL (τ′L) [61,62].

We plot in Fig. 7 the profiles of optical group delays in the amplification regime (θ = π). Figs. 7(a) and 7(b) show that both τL and τ′L can be largely modified as G is slightly changed and a smaller G always yields a larger optical group delay. It is also clear that τL increases at a larger rate than τ′L as G gradually decreases, indicating that our system is more promising than the conventional three-mode optomechanical system in achieving ultralong optical group delays. Fig. 7(c) further shows that in our system the slow light effect can be greatly enhanced by slightly increasing η, which is especially true when η → 1. This is however impossible in the conventional three-mode system because τ′L seems not sensitive to η as shown in Fig. 7(d), which is more evident when η → 1.

 

Fig. 7 Profiles of optical group delay τL (a, c) and τ′L (b, d). Relevant parameters are the same as in Fig. 5 except G specified for panels (a, b) and η specified for panels (c, d) in the figure.

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Finally, we examine in Fig. 8 how the slow light effect depends on the phase difference of control fields. Fig. 8(a) shows that a deviation of θ from (n = 0, ±1, ±2, ...) can not change the optical group delay near the resonance position (σ = 0), but may lead to an additional slow light window with its position, width, and height being θ-dependent. This is consistent with the results described by output phase ϕ in Fig. 8(b), where the slope of phase dispersion is not sensitive to θ near σ = 0 but exhibits an evident difference near σ = κ for different values of θ. It is worth noting that η should not be too large in order to avoid working near the unstable regime, and G should not be too small in order to ensure the strong-pump assumption. That is, there is a tradeoff between the amplification degree, the slow light effect, and the stability condition, as discussed in [12]. In spite of this tradeoff, it is still viable to realize remarkable optical amplification accompanied by ultralong group delay in our system.

 

Fig. 8 Profiles of optical group delay τL (a) and left output phase ϕ (b). Relevant parameters are γm = 10−4κ, G = κ, and η = 0.98 except θ specified for panels (a, b) in the figure.

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5. Conclusions

In summary, we have proposed a three-mode optomechanical system, comprising two indirectly coupled cavity modes and an intermediate mechanical mode. The two cavity modes are driven on the red and blue sidebands respectively by two control fields, and are also perturbed by two probe fields of identical amplitudes and matching frequencies. Compared to optomechanical systems driven only on the red sidebands, our system exhibits some distinct advantages, such as ultra-narrow response windows and significant amplification effect. In particular, the linewidth of the response window and the amplification effect can be flexibly controlled by adjusting the amplitudes and phase difference of the control fields. It is also viable to realize phase-dependent absorption-amplification transition, frequency-independent perfect reflection, and ultralong optical group delay in our optomechanical system.

Funding

National Natural Science Foundation of China (NSFC) (10534002, 11674049, 11774024, and 11704063); Jilin Scientific and Technological Development Program (20180520205JH); Program of State Key Laboratory of Quantum Optics and Quantum Optics Devices (KF201807).

Acknowledgments

Thanks go to Z.-H. Wang, Y. Zhang, and Y.-M. Liu for helpful discussions.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

References

1. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014). [CrossRef]  

2. T. Kippenberg and K. Vahala, “Cavity opto-mechanics,” Opt. Express 15, 17172–17205 (2007). [CrossRef]   [PubMed]  

3. J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011). [CrossRef]   [PubMed]  

4. Y.-C. Liu, Y.-F. Xiao, X. Luan, and C. W. Wong, “Dynamic dissipative cooling of a mechanical oscillator in strong coupling optomechanics,” Phys. Rev. Lett. 110, 153606 (2013). [CrossRef]  

5. A. Mari, A. Farace, N. Didier, V. Giovannetti, and R. Fazio, “Measures of quantum synchronization in continuous variable systems,” Phys. Rev. Lett. 111, 103605 (2013). [CrossRef]  

6. E. Amitai, N. Lörch, A. Nunnenkamp, S. Walter, and C. Bruder, “Synchronization of an optomechanical system to an external drive,” Phys. Rev. A 95, 053858 (2017). [CrossRef]  

7. L. Du, C.-H. Fan, H.-X. Zhang, and J.-H. Wu, “Synchronization enhancement of indirectly coupled oscillators via periodic modulation in an optomechanical system,” Sci. Rep. 7, 15834 (2017). [CrossRef]   [PubMed]  

8. X.-Y. Lü, H. Jing, J.-Y. Ma, and Y. Wu, “𝒫𝒯-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015). [CrossRef]  

9. M. Hafezi and P. Rabl, “Optomechanically induced non-reciprocity in microring resonators,” Opt. Express 20, 7672–7684 (2012). [CrossRef]   [PubMed]  

10. X.-W. Xu and Y. Li, “Optical nonreciprocity and optomechanical circulator in three-mode optomechanical systems,” Phys. Rev. A 91, 053854 (2015). [CrossRef]  

11. X.-W. Xu, Y. Li, A.-X. Chen, and Y.-X. Liu, “Nonreciprocal conversion between microwave and optical photons in electro-optomechanical systems,” Phys. Rev. A 93, 023827 (2016). [CrossRef]  

12. Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016). [CrossRef]  

13. Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9, 1797 (2018). [CrossRef]   [PubMed]  

14. X. Z. Zhang, L. Tian, and Y. Li, “Optomechanical transistor with mechanical gain,” Phys. Rev. A 97, 043818 (2018). [CrossRef]  

15. C. Jiang, L. N. Song, and Y. Li, “Directional amplifier in an optomechanical system with optical gain,” Phys. Rev. A 97, 053812 (2018). [CrossRef]  

16. M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99, 250401 (2007). [CrossRef]  

17. C.-G. Liao, R.-X. Chen, H. Xie, and X.-M. Lin, “Reservoir-engineered entanglement in a hybrid modulated three-mode optomechanical system,” Phys. Rev. A 97, 042314 (2018). [CrossRef]  

18. X.-B. Yan, C.-L. Cui, K.-H. Gu, X.-D. Tian, C.-B. Fu, and J.-H. Wu, “Coherent perfect absorption, transmission, and synthesis in a double-cavity optomechanical system,” Opt. Express 22, 4886–4895 (2014). [CrossRef]   [PubMed]  

19. C. Bai, B. P. Hou, D. G. Lai, and D. Wu, “Tunable optomechanically induced transparency in double quadratically coupled optomechanical cavities within a common reservoir,” Phys. Rev. A 93, 043804 (2016). [CrossRef]  

20. L. Du, Y.-M. Liu, B. Jiang, and Y. Zhang, “All-optical photon switching, router and amplifier using a passive-active optomechanical system,” Europhys. Lett. 122, 24001 (2018). [CrossRef]  

21. Y. Li, Y. Y. Huang, X. Z. Zhang, and L. Tian, “Optical directional amplification in a three-mode optomechanical system,” Opt. Express 25, 18907–18916 (2017). [CrossRef]   [PubMed]  

22. K. Ullah, H. Jing, and F. Saif, “Multiple electromechanically-induced-transparency windows and fano resonances in hybrid nano-electro-optomechanics,” Phys. Rev. A 97, 033812 (2018). [CrossRef]  

23. L.-L. Zheng, T.-S. Yin, Q. Bin, X.-Y. Lü, and Y. Wu, “Single-photon-induced phonon blockade in a hybrid spin-optomechanical system,” Phys. Rev. A 99, 013804 (2019). [CrossRef]  

24. G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803(R) (2010). [CrossRef]  

25. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010). [CrossRef]   [PubMed]  

26. A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011). [CrossRef]   [PubMed]  

27. F. Hocke, X. Zhou, A. Schliesser, T. J. Kippenberg, H. Huebl, and R. Gross, “Electromechanically induced absorption in a circuit nano-elctromechanical system,” New J. Phys. 14, 123037 (2012). [CrossRef]  

28. K. Qu and G. S. Agarwal, “Photon-mediated electromagnetically induced absorption in hybrid opto-electromechanical systems,” Phys. Rev. A 87, 031802(R) (2013). [CrossRef]  

29. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005). [CrossRef]  

30. Y. Wu and X. Yang, “Electromagnetically induced transparency in v-, λ-, and cascade-type schemes beyond steady-state analysis,” Phys. Rev. A 71, 053806 (2005). [CrossRef]  

31. A. M. Akulshin, S. Barreiro, and A. Lezama, “Electromagnetically induced absorption and transparency due to resonant two-field excitation of quasidegenerate levels in rb vapor,” Phys. Rev. A 57, 2996–3002 (1998). [CrossRef]  

32. A. Lezama, S. Barreiro, and A. M. Akulshin, “Electromagnetically induced absorption,” Phys. Rev. A 59, 4732–4735 (1999). [CrossRef]  

33. D. Malz, L. D. Tóth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, “Quantum-limited directional amplifiers with optomechanics,” Phys. Rev. Lett. 120, 023601 (2018). [CrossRef]   [PubMed]  

34. L. Tian, “Robust photon entanglement via quantum interference in optomechanical interfaces,” Phys. Rev. Lett. 110, 233602 (2013). [CrossRef]   [PubMed]  

35. X.-B. Yan, “Enhanced output entanglement with reservoir engineering,” Phys. Rev. A 96, 053831 (2017). [CrossRef]  

36. Y.-D. Wang and A. A. Clerk, “Reservoir-engineered entanglement in optomechanical systems,” Phys. Rev. Lett. 110, 253601 (2013). [CrossRef]   [PubMed]  

37. C. Jiang, L. N. Song, and Y. Li, “Directional phase-sensitive amplifier between microwave and optical photons,” Phys. Rev. A 99, 023823 (2019). [CrossRef]  

38. G. S. Agarwal and S. Huang, “Nanomechanical inverse electromagnetically induced transparency and confinement of light in normal modes,” New J. Phys. 16, 033023 (2014). [CrossRef]  

39. W. Z. Jia, L. F. Wei, Y. Li, and Y.-X. Liu, “Phase-dependent optical response properties in an optomechanical system by coherently driving the mechanical resonator,” Phys. Rev. A 91, 043843 (2015). [CrossRef]  

40. Q. Bin, X.-Y. Lü, T.-S. Yin, Y. Li, and Y. Wu, “Collective radiance effects in the ultrastrong-coupling regime,” Phys. Rev. A 99, 033809 (2019). [CrossRef]  

41. Y.-L. Zhang, C.-H. Dong, C.-L. Zou, X.-B. Zou, Y.-D. Wang, and G.-C. Guo, “Optomechanical devices based on traveling-wave microresonators,” Phys. Rev. A 95, 043815 (2017). [CrossRef]  

42. S. Barzanjeh, D. Vitali, P. Tombesi, and G. J. Milburn, “Entangling optical and microwave cavity modes by means of a nanomechanical resonator,” Phys. Rev. A 84, 042342 (2011). [CrossRef]  

43. M. Schmidt, S. Kessler, V. Peano, O. Painter, and F. Marquardt, “Optomechanical creation of magnetic fields for photons on a lattice,” Optica 2, 635–641 (2015). [CrossRef]  

44. M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000). [CrossRef]   [PubMed]  

45. S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003). [CrossRef]   [PubMed]  

46. E. X. DeJesus and C. Kaufman, “Routh-hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations,” Phys. Rev. A 35, 5288–5290 (1987). [CrossRef]  

47. E. M.-O. Shahverdiev, “Boundedness of dynamical systems and chaos synchronization,” Phys. Rev. E 60, 3905–3909 (1999). [CrossRef]  

48. D. F. Walls and G. J. Milburn, “Input-output formulation of optical cavities,” in Quantum Optics, (Springer-Verlag, Berlin, 1994).

49. A. A. Clerk, F. Marquardt, and K. Jacobs, “Back-action evasion and squeezing of a mechanical resonator using a cavity detector,” New J. Phys. 10, 095010 (2008). [CrossRef]  

50. A. Arvanitaki and A. A. Geraci, “Detecting high-frequency gravitational waves with optically levitated sensors,” Phys. Rev. Lett. 110, 071105 (2013). [CrossRef]   [PubMed]  

51. X. Xu and J. M. Taylor, “Squeezing in a coupled two-mode optomechanical system for force sensing below the standard quantum limit,” Phys. Rev. A 90, 043848 (2014). [CrossRef]  

52. S. Huang and G. S. Agarwal, “Robust force sensing for a free particle in a dissipative optomechanical system with a parametric amplifier,” Phys. Rev. A 95, 023844 (2017). [CrossRef]  

53. W.-Z. Zhang, Y. Han, B. Xiong, and L. Zhou, “Optomechanical force sensor in a non-markovian regime,” New J. Phys. 19, 083022 (2017). [CrossRef]  

54. A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008). [CrossRef]  

55. J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008). [CrossRef]   [PubMed]  

56. A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010). [CrossRef]  

57. M. O. Scully and M. S. Zubairy, “Coherent trapping–dark states,” in Quantum Optics, (Cambridge University Press, New York, 1997).

58. M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60, 3225–3228 (1999). [CrossRef]  

59. S. Shahidani, M. H. Naderi, and M. Soltanolkotabi, “Control and manipulation of electromagnetically induced transparency in a nonlinear optomechanical system with two movable mirrors,” Phys. Rev. A 88, 053813 (2013). [CrossRef]  

60. B. P. Hou, L. F. Wei, and S. J. Wang, “Optomechanically induced transparency and absorption in hybridized optomechanical systems,” Phys. Rev. A 92, 033829 (2015). [CrossRef]  

61. J. Yu, S. Yuan, J.-Y. Gao, and L. Sun, “Optical pulse propagation in a fabry-perot etalon: analytical discussion,” J. Opt. Soc. Am. A 18, 2153–2160 (2001). [CrossRef]  

62. Y. Zhang, N. Wang, H. Tian, H. Wang, W. Qiu, J. Wang, and P. Yuan, “A high sensitivity optical gyroscope based on slow light in coupled-resonator-induced transparency,” Phys. Lett. A 372, 5848–5852 (2008). [CrossRef]  

References

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  1. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
    [Crossref]
  2. T. Kippenberg and K. Vahala, “Cavity opto-mechanics,” Opt. Express 15, 17172–17205 (2007).
    [Crossref] [PubMed]
  3. J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
    [Crossref] [PubMed]
  4. Y.-C. Liu, Y.-F. Xiao, X. Luan, and C. W. Wong, “Dynamic dissipative cooling of a mechanical oscillator in strong coupling optomechanics,” Phys. Rev. Lett. 110, 153606 (2013).
    [Crossref]
  5. A. Mari, A. Farace, N. Didier, V. Giovannetti, and R. Fazio, “Measures of quantum synchronization in continuous variable systems,” Phys. Rev. Lett. 111, 103605 (2013).
    [Crossref]
  6. E. Amitai, N. Lörch, A. Nunnenkamp, S. Walter, and C. Bruder, “Synchronization of an optomechanical system to an external drive,” Phys. Rev. A 95, 053858 (2017).
    [Crossref]
  7. L. Du, C.-H. Fan, H.-X. Zhang, and J.-H. Wu, “Synchronization enhancement of indirectly coupled oscillators via periodic modulation in an optomechanical system,” Sci. Rep. 7, 15834 (2017).
    [Crossref] [PubMed]
  8. X.-Y. Lü, H. Jing, J.-Y. Ma, and Y. Wu, “𝒫𝒯-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
    [Crossref]
  9. M. Hafezi and P. Rabl, “Optomechanically induced non-reciprocity in microring resonators,” Opt. Express 20, 7672–7684 (2012).
    [Crossref] [PubMed]
  10. X.-W. Xu and Y. Li, “Optical nonreciprocity and optomechanical circulator in three-mode optomechanical systems,” Phys. Rev. A 91, 053854 (2015).
    [Crossref]
  11. X.-W. Xu, Y. Li, A.-X. Chen, and Y.-X. Liu, “Nonreciprocal conversion between microwave and optical photons in electro-optomechanical systems,” Phys. Rev. A 93, 023827 (2016).
    [Crossref]
  12. Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
    [Crossref]
  13. Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9, 1797 (2018).
    [Crossref] [PubMed]
  14. X. Z. Zhang, L. Tian, and Y. Li, “Optomechanical transistor with mechanical gain,” Phys. Rev. A 97, 043818 (2018).
    [Crossref]
  15. C. Jiang, L. N. Song, and Y. Li, “Directional amplifier in an optomechanical system with optical gain,” Phys. Rev. A 97, 053812 (2018).
    [Crossref]
  16. M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99, 250401 (2007).
    [Crossref]
  17. C.-G. Liao, R.-X. Chen, H. Xie, and X.-M. Lin, “Reservoir-engineered entanglement in a hybrid modulated three-mode optomechanical system,” Phys. Rev. A 97, 042314 (2018).
    [Crossref]
  18. X.-B. Yan, C.-L. Cui, K.-H. Gu, X.-D. Tian, C.-B. Fu, and J.-H. Wu, “Coherent perfect absorption, transmission, and synthesis in a double-cavity optomechanical system,” Opt. Express 22, 4886–4895 (2014).
    [Crossref] [PubMed]
  19. C. Bai, B. P. Hou, D. G. Lai, and D. Wu, “Tunable optomechanically induced transparency in double quadratically coupled optomechanical cavities within a common reservoir,” Phys. Rev. A 93, 043804 (2016).
    [Crossref]
  20. L. Du, Y.-M. Liu, B. Jiang, and Y. Zhang, “All-optical photon switching, router and amplifier using a passive-active optomechanical system,” Europhys. Lett. 122, 24001 (2018).
    [Crossref]
  21. Y. Li, Y. Y. Huang, X. Z. Zhang, and L. Tian, “Optical directional amplification in a three-mode optomechanical system,” Opt. Express 25, 18907–18916 (2017).
    [Crossref] [PubMed]
  22. K. Ullah, H. Jing, and F. Saif, “Multiple electromechanically-induced-transparency windows and fano resonances in hybrid nano-electro-optomechanics,” Phys. Rev. A 97, 033812 (2018).
    [Crossref]
  23. L.-L. Zheng, T.-S. Yin, Q. Bin, X.-Y. Lü, and Y. Wu, “Single-photon-induced phonon blockade in a hybrid spin-optomechanical system,” Phys. Rev. A 99, 013804 (2019).
    [Crossref]
  24. G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803(R) (2010).
    [Crossref]
  25. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
    [Crossref] [PubMed]
  26. A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
    [Crossref] [PubMed]
  27. F. Hocke, X. Zhou, A. Schliesser, T. J. Kippenberg, H. Huebl, and R. Gross, “Electromechanically induced absorption in a circuit nano-elctromechanical system,” New J. Phys. 14, 123037 (2012).
    [Crossref]
  28. K. Qu and G. S. Agarwal, “Photon-mediated electromagnetically induced absorption in hybrid opto-electromechanical systems,” Phys. Rev. A 87, 031802(R) (2013).
    [Crossref]
  29. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
    [Crossref]
  30. Y. Wu and X. Yang, “Electromagnetically induced transparency in v-, λ-, and cascade-type schemes beyond steady-state analysis,” Phys. Rev. A 71, 053806 (2005).
    [Crossref]
  31. A. M. Akulshin, S. Barreiro, and A. Lezama, “Electromagnetically induced absorption and transparency due to resonant two-field excitation of quasidegenerate levels in rb vapor,” Phys. Rev. A 57, 2996–3002 (1998).
    [Crossref]
  32. A. Lezama, S. Barreiro, and A. M. Akulshin, “Electromagnetically induced absorption,” Phys. Rev. A 59, 4732–4735 (1999).
    [Crossref]
  33. D. Malz, L. D. Tóth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, “Quantum-limited directional amplifiers with optomechanics,” Phys. Rev. Lett. 120, 023601 (2018).
    [Crossref] [PubMed]
  34. L. Tian, “Robust photon entanglement via quantum interference in optomechanical interfaces,” Phys. Rev. Lett. 110, 233602 (2013).
    [Crossref] [PubMed]
  35. X.-B. Yan, “Enhanced output entanglement with reservoir engineering,” Phys. Rev. A 96, 053831 (2017).
    [Crossref]
  36. Y.-D. Wang and A. A. Clerk, “Reservoir-engineered entanglement in optomechanical systems,” Phys. Rev. Lett. 110, 253601 (2013).
    [Crossref] [PubMed]
  37. C. Jiang, L. N. Song, and Y. Li, “Directional phase-sensitive amplifier between microwave and optical photons,” Phys. Rev. A 99, 023823 (2019).
    [Crossref]
  38. G. S. Agarwal and S. Huang, “Nanomechanical inverse electromagnetically induced transparency and confinement of light in normal modes,” New J. Phys. 16, 033023 (2014).
    [Crossref]
  39. W. Z. Jia, L. F. Wei, Y. Li, and Y.-X. Liu, “Phase-dependent optical response properties in an optomechanical system by coherently driving the mechanical resonator,” Phys. Rev. A 91, 043843 (2015).
    [Crossref]
  40. Q. Bin, X.-Y. Lü, T.-S. Yin, Y. Li, and Y. Wu, “Collective radiance effects in the ultrastrong-coupling regime,” Phys. Rev. A 99, 033809 (2019).
    [Crossref]
  41. Y.-L. Zhang, C.-H. Dong, C.-L. Zou, X.-B. Zou, Y.-D. Wang, and G.-C. Guo, “Optomechanical devices based on traveling-wave microresonators,” Phys. Rev. A 95, 043815 (2017).
    [Crossref]
  42. S. Barzanjeh, D. Vitali, P. Tombesi, and G. J. Milburn, “Entangling optical and microwave cavity modes by means of a nanomechanical resonator,” Phys. Rev. A 84, 042342 (2011).
    [Crossref]
  43. M. Schmidt, S. Kessler, V. Peano, O. Painter, and F. Marquardt, “Optomechanical creation of magnetic fields for photons on a lattice,” Optica 2, 635–641 (2015).
    [Crossref]
  44. M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
    [Crossref] [PubMed]
  45. S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003).
    [Crossref] [PubMed]
  46. E. X. DeJesus and C. Kaufman, “Routh-hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations,” Phys. Rev. A 35, 5288–5290 (1987).
    [Crossref]
  47. E. M.-O. Shahverdiev, “Boundedness of dynamical systems and chaos synchronization,” Phys. Rev. E 60, 3905–3909 (1999).
    [Crossref]
  48. D. F. Walls and G. J. Milburn, “Input-output formulation of optical cavities,” in Quantum Optics, (Springer-Verlag, Berlin, 1994).
  49. A. A. Clerk, F. Marquardt, and K. Jacobs, “Back-action evasion and squeezing of a mechanical resonator using a cavity detector,” New J. Phys. 10, 095010 (2008).
    [Crossref]
  50. A. Arvanitaki and A. A. Geraci, “Detecting high-frequency gravitational waves with optically levitated sensors,” Phys. Rev. Lett. 110, 071105 (2013).
    [Crossref] [PubMed]
  51. X. Xu and J. M. Taylor, “Squeezing in a coupled two-mode optomechanical system for force sensing below the standard quantum limit,” Phys. Rev. A 90, 043848 (2014).
    [Crossref]
  52. S. Huang and G. S. Agarwal, “Robust force sensing for a free particle in a dissipative optomechanical system with a parametric amplifier,” Phys. Rev. A 95, 023844 (2017).
    [Crossref]
  53. W.-Z. Zhang, Y. Han, B. Xiong, and L. Zhou, “Optomechanical force sensor in a non-markovian regime,” New J. Phys. 19, 083022 (2017).
    [Crossref]
  54. A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).
    [Crossref]
  55. J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
    [Crossref] [PubMed]
  56. A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
    [Crossref]
  57. M. O. Scully and M. S. Zubairy, “Coherent trapping–dark states,” in Quantum Optics, (Cambridge University Press, New York, 1997).
  58. M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60, 3225–3228 (1999).
    [Crossref]
  59. S. Shahidani, M. H. Naderi, and M. Soltanolkotabi, “Control and manipulation of electromagnetically induced transparency in a nonlinear optomechanical system with two movable mirrors,” Phys. Rev. A 88, 053813 (2013).
    [Crossref]
  60. B. P. Hou, L. F. Wei, and S. J. Wang, “Optomechanically induced transparency and absorption in hybridized optomechanical systems,” Phys. Rev. A 92, 033829 (2015).
    [Crossref]
  61. J. Yu, S. Yuan, J.-Y. Gao, and L. Sun, “Optical pulse propagation in a fabry-perot etalon: analytical discussion,” J. Opt. Soc. Am. A 18, 2153–2160 (2001).
    [Crossref]
  62. Y. Zhang, N. Wang, H. Tian, H. Wang, W. Qiu, J. Wang, and P. Yuan, “A high sensitivity optical gyroscope based on slow light in coupled-resonator-induced transparency,” Phys. Lett. A 372, 5848–5852 (2008).
    [Crossref]

2019 (3)

L.-L. Zheng, T.-S. Yin, Q. Bin, X.-Y. Lü, and Y. Wu, “Single-photon-induced phonon blockade in a hybrid spin-optomechanical system,” Phys. Rev. A 99, 013804 (2019).
[Crossref]

C. Jiang, L. N. Song, and Y. Li, “Directional phase-sensitive amplifier between microwave and optical photons,” Phys. Rev. A 99, 023823 (2019).
[Crossref]

Q. Bin, X.-Y. Lü, T.-S. Yin, Y. Li, and Y. Wu, “Collective radiance effects in the ultrastrong-coupling regime,” Phys. Rev. A 99, 033809 (2019).
[Crossref]

2018 (7)

K. Ullah, H. Jing, and F. Saif, “Multiple electromechanically-induced-transparency windows and fano resonances in hybrid nano-electro-optomechanics,” Phys. Rev. A 97, 033812 (2018).
[Crossref]

C.-G. Liao, R.-X. Chen, H. Xie, and X.-M. Lin, “Reservoir-engineered entanglement in a hybrid modulated three-mode optomechanical system,” Phys. Rev. A 97, 042314 (2018).
[Crossref]

L. Du, Y.-M. Liu, B. Jiang, and Y. Zhang, “All-optical photon switching, router and amplifier using a passive-active optomechanical system,” Europhys. Lett. 122, 24001 (2018).
[Crossref]

D. Malz, L. D. Tóth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, “Quantum-limited directional amplifiers with optomechanics,” Phys. Rev. Lett. 120, 023601 (2018).
[Crossref] [PubMed]

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9, 1797 (2018).
[Crossref] [PubMed]

X. Z. Zhang, L. Tian, and Y. Li, “Optomechanical transistor with mechanical gain,” Phys. Rev. A 97, 043818 (2018).
[Crossref]

C. Jiang, L. N. Song, and Y. Li, “Directional amplifier in an optomechanical system with optical gain,” Phys. Rev. A 97, 053812 (2018).
[Crossref]

2017 (7)

E. Amitai, N. Lörch, A. Nunnenkamp, S. Walter, and C. Bruder, “Synchronization of an optomechanical system to an external drive,” Phys. Rev. A 95, 053858 (2017).
[Crossref]

L. Du, C.-H. Fan, H.-X. Zhang, and J.-H. Wu, “Synchronization enhancement of indirectly coupled oscillators via periodic modulation in an optomechanical system,” Sci. Rep. 7, 15834 (2017).
[Crossref] [PubMed]

Y. Li, Y. Y. Huang, X. Z. Zhang, and L. Tian, “Optical directional amplification in a three-mode optomechanical system,” Opt. Express 25, 18907–18916 (2017).
[Crossref] [PubMed]

Y.-L. Zhang, C.-H. Dong, C.-L. Zou, X.-B. Zou, Y.-D. Wang, and G.-C. Guo, “Optomechanical devices based on traveling-wave microresonators,” Phys. Rev. A 95, 043815 (2017).
[Crossref]

X.-B. Yan, “Enhanced output entanglement with reservoir engineering,” Phys. Rev. A 96, 053831 (2017).
[Crossref]

S. Huang and G. S. Agarwal, “Robust force sensing for a free particle in a dissipative optomechanical system with a parametric amplifier,” Phys. Rev. A 95, 023844 (2017).
[Crossref]

W.-Z. Zhang, Y. Han, B. Xiong, and L. Zhou, “Optomechanical force sensor in a non-markovian regime,” New J. Phys. 19, 083022 (2017).
[Crossref]

2016 (3)

C. Bai, B. P. Hou, D. G. Lai, and D. Wu, “Tunable optomechanically induced transparency in double quadratically coupled optomechanical cavities within a common reservoir,” Phys. Rev. A 93, 043804 (2016).
[Crossref]

X.-W. Xu, Y. Li, A.-X. Chen, and Y.-X. Liu, “Nonreciprocal conversion between microwave and optical photons in electro-optomechanical systems,” Phys. Rev. A 93, 023827 (2016).
[Crossref]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
[Crossref]

2015 (5)

X.-W. Xu and Y. Li, “Optical nonreciprocity and optomechanical circulator in three-mode optomechanical systems,” Phys. Rev. A 91, 053854 (2015).
[Crossref]

X.-Y. Lü, H. Jing, J.-Y. Ma, and Y. Wu, “𝒫𝒯-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
[Crossref]

B. P. Hou, L. F. Wei, and S. J. Wang, “Optomechanically induced transparency and absorption in hybridized optomechanical systems,” Phys. Rev. A 92, 033829 (2015).
[Crossref]

W. Z. Jia, L. F. Wei, Y. Li, and Y.-X. Liu, “Phase-dependent optical response properties in an optomechanical system by coherently driving the mechanical resonator,” Phys. Rev. A 91, 043843 (2015).
[Crossref]

M. Schmidt, S. Kessler, V. Peano, O. Painter, and F. Marquardt, “Optomechanical creation of magnetic fields for photons on a lattice,” Optica 2, 635–641 (2015).
[Crossref]

2014 (4)

X. Xu and J. M. Taylor, “Squeezing in a coupled two-mode optomechanical system for force sensing below the standard quantum limit,” Phys. Rev. A 90, 043848 (2014).
[Crossref]

G. S. Agarwal and S. Huang, “Nanomechanical inverse electromagnetically induced transparency and confinement of light in normal modes,” New J. Phys. 16, 033023 (2014).
[Crossref]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

X.-B. Yan, C.-L. Cui, K.-H. Gu, X.-D. Tian, C.-B. Fu, and J.-H. Wu, “Coherent perfect absorption, transmission, and synthesis in a double-cavity optomechanical system,” Opt. Express 22, 4886–4895 (2014).
[Crossref] [PubMed]

2013 (7)

L. Tian, “Robust photon entanglement via quantum interference in optomechanical interfaces,” Phys. Rev. Lett. 110, 233602 (2013).
[Crossref] [PubMed]

K. Qu and G. S. Agarwal, “Photon-mediated electromagnetically induced absorption in hybrid opto-electromechanical systems,” Phys. Rev. A 87, 031802(R) (2013).
[Crossref]

Y.-C. Liu, Y.-F. Xiao, X. Luan, and C. W. Wong, “Dynamic dissipative cooling of a mechanical oscillator in strong coupling optomechanics,” Phys. Rev. Lett. 110, 153606 (2013).
[Crossref]

A. Mari, A. Farace, N. Didier, V. Giovannetti, and R. Fazio, “Measures of quantum synchronization in continuous variable systems,” Phys. Rev. Lett. 111, 103605 (2013).
[Crossref]

Y.-D. Wang and A. A. Clerk, “Reservoir-engineered entanglement in optomechanical systems,” Phys. Rev. Lett. 110, 253601 (2013).
[Crossref] [PubMed]

A. Arvanitaki and A. A. Geraci, “Detecting high-frequency gravitational waves with optically levitated sensors,” Phys. Rev. Lett. 110, 071105 (2013).
[Crossref] [PubMed]

S. Shahidani, M. H. Naderi, and M. Soltanolkotabi, “Control and manipulation of electromagnetically induced transparency in a nonlinear optomechanical system with two movable mirrors,” Phys. Rev. A 88, 053813 (2013).
[Crossref]

2012 (2)

M. Hafezi and P. Rabl, “Optomechanically induced non-reciprocity in microring resonators,” Opt. Express 20, 7672–7684 (2012).
[Crossref] [PubMed]

F. Hocke, X. Zhou, A. Schliesser, T. J. Kippenberg, H. Huebl, and R. Gross, “Electromechanically induced absorption in a circuit nano-elctromechanical system,” New J. Phys. 14, 123037 (2012).
[Crossref]

2011 (3)

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref] [PubMed]

S. Barzanjeh, D. Vitali, P. Tombesi, and G. J. Milburn, “Entangling optical and microwave cavity modes by means of a nanomechanical resonator,” Phys. Rev. A 84, 042342 (2011).
[Crossref]

2010 (3)

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[Crossref]

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803(R) (2010).
[Crossref]

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

2008 (4)

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).
[Crossref]

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref] [PubMed]

Y. Zhang, N. Wang, H. Tian, H. Wang, W. Qiu, J. Wang, and P. Yuan, “A high sensitivity optical gyroscope based on slow light in coupled-resonator-induced transparency,” Phys. Lett. A 372, 5848–5852 (2008).
[Crossref]

A. A. Clerk, F. Marquardt, and K. Jacobs, “Back-action evasion and squeezing of a mechanical resonator using a cavity detector,” New J. Phys. 10, 095010 (2008).
[Crossref]

2007 (2)

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99, 250401 (2007).
[Crossref]

T. Kippenberg and K. Vahala, “Cavity opto-mechanics,” Opt. Express 15, 17172–17205 (2007).
[Crossref] [PubMed]

2005 (2)

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

Y. Wu and X. Yang, “Electromagnetically induced transparency in v-, λ-, and cascade-type schemes beyond steady-state analysis,” Phys. Rev. A 71, 053806 (2005).
[Crossref]

2003 (1)

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003).
[Crossref] [PubMed]

2001 (1)

2000 (1)

M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[Crossref] [PubMed]

1999 (3)

E. M.-O. Shahverdiev, “Boundedness of dynamical systems and chaos synchronization,” Phys. Rev. E 60, 3905–3909 (1999).
[Crossref]

A. Lezama, S. Barreiro, and A. M. Akulshin, “Electromagnetically induced absorption,” Phys. Rev. A 59, 4732–4735 (1999).
[Crossref]

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60, 3225–3228 (1999).
[Crossref]

1998 (1)

A. M. Akulshin, S. Barreiro, and A. Lezama, “Electromagnetically induced absorption and transparency due to resonant two-field excitation of quasidegenerate levels in rb vapor,” Phys. Rev. A 57, 2996–3002 (1998).
[Crossref]

1987 (1)

E. X. DeJesus and C. Kaufman, “Routh-hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations,” Phys. Rev. A 35, 5288–5290 (1987).
[Crossref]

Agarwal, G. S.

S. Huang and G. S. Agarwal, “Robust force sensing for a free particle in a dissipative optomechanical system with a parametric amplifier,” Phys. Rev. A 95, 023844 (2017).
[Crossref]

G. S. Agarwal and S. Huang, “Nanomechanical inverse electromagnetically induced transparency and confinement of light in normal modes,” New J. Phys. 16, 033023 (2014).
[Crossref]

K. Qu and G. S. Agarwal, “Photon-mediated electromagnetically induced absorption in hybrid opto-electromechanical systems,” Phys. Rev. A 87, 031802(R) (2013).
[Crossref]

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803(R) (2010).
[Crossref]

Akulshin, A. M.

A. Lezama, S. Barreiro, and A. M. Akulshin, “Electromagnetically induced absorption,” Phys. Rev. A 59, 4732–4735 (1999).
[Crossref]

A. M. Akulshin, S. Barreiro, and A. Lezama, “Electromagnetically induced absorption and transparency due to resonant two-field excitation of quasidegenerate levels in rb vapor,” Phys. Rev. A 57, 2996–3002 (1998).
[Crossref]

Alegre, T. P. M.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

Allman, M. S.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref] [PubMed]

Amitai, E.

E. Amitai, N. Lörch, A. Nunnenkamp, S. Walter, and C. Bruder, “Synchronization of an optomechanical system to an external drive,” Phys. Rev. A 95, 053858 (2017).
[Crossref]

Ansmann, M.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[Crossref]

Arcizet, O.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

Arvanitaki, A.

A. Arvanitaki and A. A. Geraci, “Detecting high-frequency gravitational waves with optically levitated sensors,” Phys. Rev. Lett. 110, 071105 (2013).
[Crossref] [PubMed]

Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99, 250401 (2007).
[Crossref]

Bai, C.

C. Bai, B. P. Hou, D. G. Lai, and D. Wu, “Tunable optomechanically induced transparency in double quadratically coupled optomechanical cavities within a common reservoir,” Phys. Rev. A 93, 043804 (2016).
[Crossref]

Barreiro, S.

A. Lezama, S. Barreiro, and A. M. Akulshin, “Electromagnetically induced absorption,” Phys. Rev. A 59, 4732–4735 (1999).
[Crossref]

A. M. Akulshin, S. Barreiro, and A. Lezama, “Electromagnetically induced absorption and transparency due to resonant two-field excitation of quasidegenerate levels in rb vapor,” Phys. Rev. A 57, 2996–3002 (1998).
[Crossref]

Barzanjeh, S.

S. Barzanjeh, D. Vitali, P. Tombesi, and G. J. Milburn, “Entangling optical and microwave cavity modes by means of a nanomechanical resonator,” Phys. Rev. A 84, 042342 (2011).
[Crossref]

Bernier, N. R.

D. Malz, L. D. Tóth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, “Quantum-limited directional amplifiers with optomechanics,” Phys. Rev. Lett. 120, 023601 (2018).
[Crossref] [PubMed]

Bialczak, R. C.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[Crossref]

Bin, Q.

Q. Bin, X.-Y. Lü, T.-S. Yin, Y. Li, and Y. Wu, “Collective radiance effects in the ultrastrong-coupling regime,” Phys. Rev. A 99, 033809 (2019).
[Crossref]

L.-L. Zheng, T.-S. Yin, Q. Bin, X.-Y. Lü, and Y. Wu, “Single-photon-induced phonon blockade in a hybrid spin-optomechanical system,” Phys. Rev. A 99, 013804 (2019).
[Crossref]

Bruder, C.

E. Amitai, N. Lörch, A. Nunnenkamp, S. Walter, and C. Bruder, “Synchronization of an optomechanical system to an external drive,” Phys. Rev. A 95, 053858 (2017).
[Crossref]

Brukner, C.

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99, 250401 (2007).
[Crossref]

Cai, M.

M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[Crossref] [PubMed]

Chan, J.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

Chang, D. E.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

Chen, A.-X.

X.-W. Xu, Y. Li, A.-X. Chen, and Y.-X. Liu, “Nonreciprocal conversion between microwave and optical photons in electro-optomechanical systems,” Phys. Rev. A 93, 023827 (2016).
[Crossref]

Chen, R.-X.

C.-G. Liao, R.-X. Chen, H. Xie, and X.-M. Lin, “Reservoir-engineered entanglement in a hybrid modulated three-mode optomechanical system,” Phys. Rev. A 97, 042314 (2018).
[Crossref]

Chen, Y.

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9, 1797 (2018).
[Crossref] [PubMed]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
[Crossref]

Cicak, K.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref] [PubMed]

Cleland, A. N.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[Crossref]

Clerk, A. A.

Y.-D. Wang and A. A. Clerk, “Reservoir-engineered entanglement in optomechanical systems,” Phys. Rev. Lett. 110, 253601 (2013).
[Crossref] [PubMed]

A. A. Clerk, F. Marquardt, and K. Jacobs, “Back-action evasion and squeezing of a mechanical resonator using a cavity detector,” New J. Phys. 10, 095010 (2008).
[Crossref]

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).
[Crossref]

Cui, C.-L.

DeJesus, E. X.

E. X. DeJesus and C. Kaufman, “Routh-hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations,” Phys. Rev. A 35, 5288–5290 (1987).
[Crossref]

Deléglise, S.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

Didier, N.

A. Mari, A. Farace, N. Didier, V. Giovannetti, and R. Fazio, “Measures of quantum synchronization in continuous variable systems,” Phys. Rev. Lett. 111, 103605 (2013).
[Crossref]

Dong, C.-H.

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9, 1797 (2018).
[Crossref] [PubMed]

Y.-L. Zhang, C.-H. Dong, C.-L. Zou, X.-B. Zou, Y.-D. Wang, and G.-C. Guo, “Optomechanical devices based on traveling-wave microresonators,” Phys. Rev. A 95, 043815 (2017).
[Crossref]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
[Crossref]

Donner, T.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref] [PubMed]

Du, L.

L. Du, Y.-M. Liu, B. Jiang, and Y. Zhang, “All-optical photon switching, router and amplifier using a passive-active optomechanical system,” Europhys. Lett. 122, 24001 (2018).
[Crossref]

L. Du, C.-H. Fan, H.-X. Zhang, and J.-H. Wu, “Synchronization enhancement of indirectly coupled oscillators via periodic modulation in an optomechanical system,” Sci. Rep. 7, 15834 (2017).
[Crossref] [PubMed]

Eichenfield, M.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

Eisert, J.

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99, 250401 (2007).
[Crossref]

Fan, C.-H.

L. Du, C.-H. Fan, H.-X. Zhang, and J.-H. Wu, “Synchronization enhancement of indirectly coupled oscillators via periodic modulation in an optomechanical system,” Sci. Rep. 7, 15834 (2017).
[Crossref] [PubMed]

Farace, A.

A. Mari, A. Farace, N. Didier, V. Giovannetti, and R. Fazio, “Measures of quantum synchronization in continuous variable systems,” Phys. Rev. Lett. 111, 103605 (2013).
[Crossref]

Fazio, R.

A. Mari, A. Farace, N. Didier, V. Giovannetti, and R. Fazio, “Measures of quantum synchronization in continuous variable systems,” Phys. Rev. Lett. 111, 103605 (2013).
[Crossref]

Feofanov, A. K.

D. Malz, L. D. Tóth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, “Quantum-limited directional amplifiers with optomechanics,” Phys. Rev. Lett. 120, 023601 (2018).
[Crossref] [PubMed]

Fleischhauer, M.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60, 3225–3228 (1999).
[Crossref]

Fu, C.-B.

Gao, J.-Y.

Gavartin, E.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

Geraci, A. A.

A. Arvanitaki and A. A. Geraci, “Detecting high-frequency gravitational waves with optically levitated sensors,” Phys. Rev. Lett. 110, 071105 (2013).
[Crossref] [PubMed]

Gigan, S.

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99, 250401 (2007).
[Crossref]

Giovannetti, V.

A. Mari, A. Farace, N. Didier, V. Giovannetti, and R. Fazio, “Measures of quantum synchronization in continuous variable systems,” Phys. Rev. Lett. 111, 103605 (2013).
[Crossref]

Girvin, S. M.

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref] [PubMed]

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).
[Crossref]

Gross, R.

F. Hocke, X. Zhou, A. Schliesser, T. J. Kippenberg, H. Huebl, and R. Gross, “Electromechanically induced absorption in a circuit nano-elctromechanical system,” New J. Phys. 14, 123037 (2012).
[Crossref]

Gu, K.-H.

Guo, G.-C.

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9, 1797 (2018).
[Crossref] [PubMed]

Y.-L. Zhang, C.-H. Dong, C.-L. Zou, X.-B. Zou, Y.-D. Wang, and G.-C. Guo, “Optomechanical devices based on traveling-wave microresonators,” Phys. Rev. A 95, 043815 (2017).
[Crossref]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
[Crossref]

Hafezi, M.

Han, Y.

W.-Z. Zhang, Y. Han, B. Xiong, and L. Zhou, “Optomechanical force sensor in a non-markovian regime,” New J. Phys. 19, 083022 (2017).
[Crossref]

Harlow, J. W.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref] [PubMed]

Harris, J. G. E.

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).
[Crossref]

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref] [PubMed]

Hill, J. T.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

Hocke, F.

F. Hocke, X. Zhou, A. Schliesser, T. J. Kippenberg, H. Huebl, and R. Gross, “Electromechanically induced absorption in a circuit nano-elctromechanical system,” New J. Phys. 14, 123037 (2012).
[Crossref]

Hofheinz, M.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[Crossref]

Hou, B. P.

C. Bai, B. P. Hou, D. G. Lai, and D. Wu, “Tunable optomechanically induced transparency in double quadratically coupled optomechanical cavities within a common reservoir,” Phys. Rev. A 93, 043804 (2016).
[Crossref]

B. P. Hou, L. F. Wei, and S. J. Wang, “Optomechanically induced transparency and absorption in hybridized optomechanical systems,” Phys. Rev. A 92, 033829 (2015).
[Crossref]

Huang, S.

S. Huang and G. S. Agarwal, “Robust force sensing for a free particle in a dissipative optomechanical system with a parametric amplifier,” Phys. Rev. A 95, 023844 (2017).
[Crossref]

G. S. Agarwal and S. Huang, “Nanomechanical inverse electromagnetically induced transparency and confinement of light in normal modes,” New J. Phys. 16, 033023 (2014).
[Crossref]

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803(R) (2010).
[Crossref]

Huang, Y. Y.

Huebl, H.

F. Hocke, X. Zhou, A. Schliesser, T. J. Kippenberg, H. Huebl, and R. Gross, “Electromechanically induced absorption in a circuit nano-elctromechanical system,” New J. Phys. 14, 123037 (2012).
[Crossref]

Imamoglu, A.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

Jacobs, K.

A. A. Clerk, F. Marquardt, and K. Jacobs, “Back-action evasion and squeezing of a mechanical resonator using a cavity detector,” New J. Phys. 10, 095010 (2008).
[Crossref]

Jayich, A. M.

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).
[Crossref]

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref] [PubMed]

Jia, W. Z.

W. Z. Jia, L. F. Wei, Y. Li, and Y.-X. Liu, “Phase-dependent optical response properties in an optomechanical system by coherently driving the mechanical resonator,” Phys. Rev. A 91, 043843 (2015).
[Crossref]

Jiang, B.

L. Du, Y.-M. Liu, B. Jiang, and Y. Zhang, “All-optical photon switching, router and amplifier using a passive-active optomechanical system,” Europhys. Lett. 122, 24001 (2018).
[Crossref]

Jiang, C.

C. Jiang, L. N. Song, and Y. Li, “Directional phase-sensitive amplifier between microwave and optical photons,” Phys. Rev. A 99, 023823 (2019).
[Crossref]

C. Jiang, L. N. Song, and Y. Li, “Directional amplifier in an optomechanical system with optical gain,” Phys. Rev. A 97, 053812 (2018).
[Crossref]

Jing, H.

K. Ullah, H. Jing, and F. Saif, “Multiple electromechanically-induced-transparency windows and fano resonances in hybrid nano-electro-optomechanics,” Phys. Rev. A 97, 033812 (2018).
[Crossref]

X.-Y. Lü, H. Jing, J.-Y. Ma, and Y. Wu, “𝒫𝒯-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
[Crossref]

Kaufman, C.

E. X. DeJesus and C. Kaufman, “Routh-hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations,” Phys. Rev. A 35, 5288–5290 (1987).
[Crossref]

Kessler, S.

Kim, M. S.

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99, 250401 (2007).
[Crossref]

Kippenberg, T.

Kippenberg, T. J.

D. Malz, L. D. Tóth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, “Quantum-limited directional amplifiers with optomechanics,” Phys. Rev. Lett. 120, 023601 (2018).
[Crossref] [PubMed]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

F. Hocke, X. Zhou, A. Schliesser, T. J. Kippenberg, H. Huebl, and R. Gross, “Electromechanically induced absorption in a circuit nano-elctromechanical system,” New J. Phys. 14, 123037 (2012).
[Crossref]

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003).
[Crossref] [PubMed]

Lai, D. G.

C. Bai, B. P. Hou, D. G. Lai, and D. Wu, “Tunable optomechanically induced transparency in double quadratically coupled optomechanical cavities within a common reservoir,” Phys. Rev. A 93, 043804 (2016).
[Crossref]

Lehnert, K. W.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref] [PubMed]

Lenander, M.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[Crossref]

Lezama, A.

A. Lezama, S. Barreiro, and A. M. Akulshin, “Electromagnetically induced absorption,” Phys. Rev. A 59, 4732–4735 (1999).
[Crossref]

A. M. Akulshin, S. Barreiro, and A. Lezama, “Electromagnetically induced absorption and transparency due to resonant two-field excitation of quasidegenerate levels in rb vapor,” Phys. Rev. A 57, 2996–3002 (1998).
[Crossref]

Li, D.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref] [PubMed]

Li, Y.

C. Jiang, L. N. Song, and Y. Li, “Directional phase-sensitive amplifier between microwave and optical photons,” Phys. Rev. A 99, 023823 (2019).
[Crossref]

Q. Bin, X.-Y. Lü, T.-S. Yin, Y. Li, and Y. Wu, “Collective radiance effects in the ultrastrong-coupling regime,” Phys. Rev. A 99, 033809 (2019).
[Crossref]

C. Jiang, L. N. Song, and Y. Li, “Directional amplifier in an optomechanical system with optical gain,” Phys. Rev. A 97, 053812 (2018).
[Crossref]

X. Z. Zhang, L. Tian, and Y. Li, “Optomechanical transistor with mechanical gain,” Phys. Rev. A 97, 043818 (2018).
[Crossref]

Y. Li, Y. Y. Huang, X. Z. Zhang, and L. Tian, “Optical directional amplification in a three-mode optomechanical system,” Opt. Express 25, 18907–18916 (2017).
[Crossref] [PubMed]

X.-W. Xu, Y. Li, A.-X. Chen, and Y.-X. Liu, “Nonreciprocal conversion between microwave and optical photons in electro-optomechanical systems,” Phys. Rev. A 93, 023827 (2016).
[Crossref]

X.-W. Xu and Y. Li, “Optical nonreciprocity and optomechanical circulator in three-mode optomechanical systems,” Phys. Rev. A 91, 053854 (2015).
[Crossref]

W. Z. Jia, L. F. Wei, Y. Li, and Y.-X. Liu, “Phase-dependent optical response properties in an optomechanical system by coherently driving the mechanical resonator,” Phys. Rev. A 91, 043843 (2015).
[Crossref]

Liao, C.-G.

C.-G. Liao, R.-X. Chen, H. Xie, and X.-M. Lin, “Reservoir-engineered entanglement in a hybrid modulated three-mode optomechanical system,” Phys. Rev. A 97, 042314 (2018).
[Crossref]

Lin, Q.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

Lin, X.-M.

C.-G. Liao, R.-X. Chen, H. Xie, and X.-M. Lin, “Reservoir-engineered entanglement in a hybrid modulated three-mode optomechanical system,” Phys. Rev. A 97, 042314 (2018).
[Crossref]

Liu, Y.-C.

Y.-C. Liu, Y.-F. Xiao, X. Luan, and C. W. Wong, “Dynamic dissipative cooling of a mechanical oscillator in strong coupling optomechanics,” Phys. Rev. Lett. 110, 153606 (2013).
[Crossref]

Liu, Y.-M.

L. Du, Y.-M. Liu, B. Jiang, and Y. Zhang, “All-optical photon switching, router and amplifier using a passive-active optomechanical system,” Europhys. Lett. 122, 24001 (2018).
[Crossref]

Liu, Y.-X.

X.-W. Xu, Y. Li, A.-X. Chen, and Y.-X. Liu, “Nonreciprocal conversion between microwave and optical photons in electro-optomechanical systems,” Phys. Rev. A 93, 023827 (2016).
[Crossref]

W. Z. Jia, L. F. Wei, Y. Li, and Y.-X. Liu, “Phase-dependent optical response properties in an optomechanical system by coherently driving the mechanical resonator,” Phys. Rev. A 91, 043843 (2015).
[Crossref]

Lörch, N.

E. Amitai, N. Lörch, A. Nunnenkamp, S. Walter, and C. Bruder, “Synchronization of an optomechanical system to an external drive,” Phys. Rev. A 95, 053858 (2017).
[Crossref]

Lü, X.-Y.

L.-L. Zheng, T.-S. Yin, Q. Bin, X.-Y. Lü, and Y. Wu, “Single-photon-induced phonon blockade in a hybrid spin-optomechanical system,” Phys. Rev. A 99, 013804 (2019).
[Crossref]

Q. Bin, X.-Y. Lü, T.-S. Yin, Y. Li, and Y. Wu, “Collective radiance effects in the ultrastrong-coupling regime,” Phys. Rev. A 99, 033809 (2019).
[Crossref]

X.-Y. Lü, H. Jing, J.-Y. Ma, and Y. Wu, “𝒫𝒯-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
[Crossref]

Luan, X.

Y.-C. Liu, Y.-F. Xiao, X. Luan, and C. W. Wong, “Dynamic dissipative cooling of a mechanical oscillator in strong coupling optomechanics,” Phys. Rev. Lett. 110, 153606 (2013).
[Crossref]

Lucero, E.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[Crossref]

Lukin, M. D.

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60, 3225–3228 (1999).
[Crossref]

Ma, J.-Y.

X.-Y. Lü, H. Jing, J.-Y. Ma, and Y. Wu, “𝒫𝒯-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
[Crossref]

Malz, D.

D. Malz, L. D. Tóth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, “Quantum-limited directional amplifiers with optomechanics,” Phys. Rev. Lett. 120, 023601 (2018).
[Crossref] [PubMed]

Marangos, J. P.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

Mari, A.

A. Mari, A. Farace, N. Didier, V. Giovannetti, and R. Fazio, “Measures of quantum synchronization in continuous variable systems,” Phys. Rev. Lett. 111, 103605 (2013).
[Crossref]

Marquardt, F.

M. Schmidt, S. Kessler, V. Peano, O. Painter, and F. Marquardt, “Optomechanical creation of magnetic fields for photons on a lattice,” Optica 2, 635–641 (2015).
[Crossref]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

A. A. Clerk, F. Marquardt, and K. Jacobs, “Back-action evasion and squeezing of a mechanical resonator using a cavity detector,” New J. Phys. 10, 095010 (2008).
[Crossref]

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref] [PubMed]

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).
[Crossref]

Martinis, J. M.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[Crossref]

Milburn, G. J.

S. Barzanjeh, D. Vitali, P. Tombesi, and G. J. Milburn, “Entangling optical and microwave cavity modes by means of a nanomechanical resonator,” Phys. Rev. A 84, 042342 (2011).
[Crossref]

D. F. Walls and G. J. Milburn, “Input-output formulation of optical cavities,” in Quantum Optics, (Springer-Verlag, Berlin, 1994).

Naderi, M. H.

S. Shahidani, M. H. Naderi, and M. Soltanolkotabi, “Control and manipulation of electromagnetically induced transparency in a nonlinear optomechanical system with two movable mirrors,” Phys. Rev. A 88, 053813 (2013).
[Crossref]

Neeley, M.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[Crossref]

Nunnenkamp, A.

D. Malz, L. D. Tóth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, “Quantum-limited directional amplifiers with optomechanics,” Phys. Rev. Lett. 120, 023601 (2018).
[Crossref] [PubMed]

E. Amitai, N. Lörch, A. Nunnenkamp, S. Walter, and C. Bruder, “Synchronization of an optomechanical system to an external drive,” Phys. Rev. A 95, 053858 (2017).
[Crossref]

O’Connell, A. D.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[Crossref]

Painter, O.

M. Schmidt, S. Kessler, V. Peano, O. Painter, and F. Marquardt, “Optomechanical creation of magnetic fields for photons on a lattice,” Optica 2, 635–641 (2015).
[Crossref]

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[Crossref] [PubMed]

Painter, O. J.

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003).
[Crossref] [PubMed]

Paternostro, M.

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99, 250401 (2007).
[Crossref]

Peano, V.

Qiu, W.

Y. Zhang, N. Wang, H. Tian, H. Wang, W. Qiu, J. Wang, and P. Yuan, “A high sensitivity optical gyroscope based on slow light in coupled-resonator-induced transparency,” Phys. Lett. A 372, 5848–5852 (2008).
[Crossref]

Qu, K.

K. Qu and G. S. Agarwal, “Photon-mediated electromagnetically induced absorption in hybrid opto-electromechanical systems,” Phys. Rev. A 87, 031802(R) (2013).
[Crossref]

Rabl, P.

Rivière, R.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

Safavi-Naeini, A. H.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

Saif, F.

K. Ullah, H. Jing, and F. Saif, “Multiple electromechanically-induced-transparency windows and fano resonances in hybrid nano-electro-optomechanics,” Phys. Rev. A 97, 033812 (2018).
[Crossref]

Sank, D.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[Crossref]

Sankey, J. C.

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).
[Crossref]

Schliesser, A.

F. Hocke, X. Zhou, A. Schliesser, T. J. Kippenberg, H. Huebl, and R. Gross, “Electromechanically induced absorption in a circuit nano-elctromechanical system,” New J. Phys. 14, 123037 (2012).
[Crossref]

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

Schmidt, M.

Scully, M. O.

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60, 3225–3228 (1999).
[Crossref]

M. O. Scully and M. S. Zubairy, “Coherent trapping–dark states,” in Quantum Optics, (Cambridge University Press, New York, 1997).

Shahidani, S.

S. Shahidani, M. H. Naderi, and M. Soltanolkotabi, “Control and manipulation of electromagnetically induced transparency in a nonlinear optomechanical system with two movable mirrors,” Phys. Rev. A 88, 053813 (2013).
[Crossref]

Shahverdiev, E. M.-O.

E. M.-O. Shahverdiev, “Boundedness of dynamical systems and chaos synchronization,” Phys. Rev. E 60, 3905–3909 (1999).
[Crossref]

Shen, Z.

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9, 1797 (2018).
[Crossref] [PubMed]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
[Crossref]

Simmonds, R. W.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref] [PubMed]

Sirois, A. J.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref] [PubMed]

Soltanolkotabi, M.

S. Shahidani, M. H. Naderi, and M. Soltanolkotabi, “Control and manipulation of electromagnetically induced transparency in a nonlinear optomechanical system with two movable mirrors,” Phys. Rev. A 88, 053813 (2013).
[Crossref]

Song, L. N.

C. Jiang, L. N. Song, and Y. Li, “Directional phase-sensitive amplifier between microwave and optical photons,” Phys. Rev. A 99, 023823 (2019).
[Crossref]

C. Jiang, L. N. Song, and Y. Li, “Directional amplifier in an optomechanical system with optical gain,” Phys. Rev. A 97, 053812 (2018).
[Crossref]

Spillane, S. M.

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003).
[Crossref] [PubMed]

Sun, F.-W.

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9, 1797 (2018).
[Crossref] [PubMed]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
[Crossref]

Sun, L.

Taylor, J. M.

X. Xu and J. M. Taylor, “Squeezing in a coupled two-mode optomechanical system for force sensing below the standard quantum limit,” Phys. Rev. A 90, 043848 (2014).
[Crossref]

Teufel, J. D.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref] [PubMed]

Thompson, J. D.

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).
[Crossref]

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref] [PubMed]

Tian, H.

Y. Zhang, N. Wang, H. Tian, H. Wang, W. Qiu, J. Wang, and P. Yuan, “A high sensitivity optical gyroscope based on slow light in coupled-resonator-induced transparency,” Phys. Lett. A 372, 5848–5852 (2008).
[Crossref]

Tian, L.

X. Z. Zhang, L. Tian, and Y. Li, “Optomechanical transistor with mechanical gain,” Phys. Rev. A 97, 043818 (2018).
[Crossref]

Y. Li, Y. Y. Huang, X. Z. Zhang, and L. Tian, “Optical directional amplification in a three-mode optomechanical system,” Opt. Express 25, 18907–18916 (2017).
[Crossref] [PubMed]

L. Tian, “Robust photon entanglement via quantum interference in optomechanical interfaces,” Phys. Rev. Lett. 110, 233602 (2013).
[Crossref] [PubMed]

Tian, X.-D.

Tombesi, P.

S. Barzanjeh, D. Vitali, P. Tombesi, and G. J. Milburn, “Entangling optical and microwave cavity modes by means of a nanomechanical resonator,” Phys. Rev. A 84, 042342 (2011).
[Crossref]

Tóth, L. D.

D. Malz, L. D. Tóth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, “Quantum-limited directional amplifiers with optomechanics,” Phys. Rev. Lett. 120, 023601 (2018).
[Crossref] [PubMed]

Ullah, K.

K. Ullah, H. Jing, and F. Saif, “Multiple electromechanically-induced-transparency windows and fano resonances in hybrid nano-electro-optomechanics,” Phys. Rev. A 97, 033812 (2018).
[Crossref]

Vahala, K.

Vahala, K. J.

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003).
[Crossref] [PubMed]

M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[Crossref] [PubMed]

Vitali, D.

S. Barzanjeh, D. Vitali, P. Tombesi, and G. J. Milburn, “Entangling optical and microwave cavity modes by means of a nanomechanical resonator,” Phys. Rev. A 84, 042342 (2011).
[Crossref]

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99, 250401 (2007).
[Crossref]

Walls, D. F.

D. F. Walls and G. J. Milburn, “Input-output formulation of optical cavities,” in Quantum Optics, (Springer-Verlag, Berlin, 1994).

Walter, S.

E. Amitai, N. Lörch, A. Nunnenkamp, S. Walter, and C. Bruder, “Synchronization of an optomechanical system to an external drive,” Phys. Rev. A 95, 053858 (2017).
[Crossref]

Wang, H.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[Crossref]

Y. Zhang, N. Wang, H. Tian, H. Wang, W. Qiu, J. Wang, and P. Yuan, “A high sensitivity optical gyroscope based on slow light in coupled-resonator-induced transparency,” Phys. Lett. A 372, 5848–5852 (2008).
[Crossref]

Wang, J.

Y. Zhang, N. Wang, H. Tian, H. Wang, W. Qiu, J. Wang, and P. Yuan, “A high sensitivity optical gyroscope based on slow light in coupled-resonator-induced transparency,” Phys. Lett. A 372, 5848–5852 (2008).
[Crossref]

Wang, N.

Y. Zhang, N. Wang, H. Tian, H. Wang, W. Qiu, J. Wang, and P. Yuan, “A high sensitivity optical gyroscope based on slow light in coupled-resonator-induced transparency,” Phys. Lett. A 372, 5848–5852 (2008).
[Crossref]

Wang, S. J.

B. P. Hou, L. F. Wei, and S. J. Wang, “Optomechanically induced transparency and absorption in hybridized optomechanical systems,” Phys. Rev. A 92, 033829 (2015).
[Crossref]

Wang, Y.-D.

Y.-L. Zhang, C.-H. Dong, C.-L. Zou, X.-B. Zou, Y.-D. Wang, and G.-C. Guo, “Optomechanical devices based on traveling-wave microresonators,” Phys. Rev. A 95, 043815 (2017).
[Crossref]

Y.-D. Wang and A. A. Clerk, “Reservoir-engineered entanglement in optomechanical systems,” Phys. Rev. Lett. 110, 253601 (2013).
[Crossref] [PubMed]

Wei, L. F.

W. Z. Jia, L. F. Wei, Y. Li, and Y.-X. Liu, “Phase-dependent optical response properties in an optomechanical system by coherently driving the mechanical resonator,” Phys. Rev. A 91, 043843 (2015).
[Crossref]

B. P. Hou, L. F. Wei, and S. J. Wang, “Optomechanically induced transparency and absorption in hybridized optomechanical systems,” Phys. Rev. A 92, 033829 (2015).
[Crossref]

Weides, M.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[Crossref]

Weis, S.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

Wenner, J.

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[Crossref]

Whittaker, J. D.

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref] [PubMed]

Winger, M.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

Wong, C. W.

Y.-C. Liu, Y.-F. Xiao, X. Luan, and C. W. Wong, “Dynamic dissipative cooling of a mechanical oscillator in strong coupling optomechanics,” Phys. Rev. Lett. 110, 153606 (2013).
[Crossref]

Wu, D.

C. Bai, B. P. Hou, D. G. Lai, and D. Wu, “Tunable optomechanically induced transparency in double quadratically coupled optomechanical cavities within a common reservoir,” Phys. Rev. A 93, 043804 (2016).
[Crossref]

Wu, J.-H.

L. Du, C.-H. Fan, H.-X. Zhang, and J.-H. Wu, “Synchronization enhancement of indirectly coupled oscillators via periodic modulation in an optomechanical system,” Sci. Rep. 7, 15834 (2017).
[Crossref] [PubMed]

X.-B. Yan, C.-L. Cui, K.-H. Gu, X.-D. Tian, C.-B. Fu, and J.-H. Wu, “Coherent perfect absorption, transmission, and synthesis in a double-cavity optomechanical system,” Opt. Express 22, 4886–4895 (2014).
[Crossref] [PubMed]

Wu, Y.

L.-L. Zheng, T.-S. Yin, Q. Bin, X.-Y. Lü, and Y. Wu, “Single-photon-induced phonon blockade in a hybrid spin-optomechanical system,” Phys. Rev. A 99, 013804 (2019).
[Crossref]

Q. Bin, X.-Y. Lü, T.-S. Yin, Y. Li, and Y. Wu, “Collective radiance effects in the ultrastrong-coupling regime,” Phys. Rev. A 99, 033809 (2019).
[Crossref]

X.-Y. Lü, H. Jing, J.-Y. Ma, and Y. Wu, “𝒫𝒯-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
[Crossref]

Y. Wu and X. Yang, “Electromagnetically induced transparency in v-, λ-, and cascade-type schemes beyond steady-state analysis,” Phys. Rev. A 71, 053806 (2005).
[Crossref]

Xiao, Y.-F.

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
[Crossref]

Y.-C. Liu, Y.-F. Xiao, X. Luan, and C. W. Wong, “Dynamic dissipative cooling of a mechanical oscillator in strong coupling optomechanics,” Phys. Rev. Lett. 110, 153606 (2013).
[Crossref]

Xie, H.

C.-G. Liao, R.-X. Chen, H. Xie, and X.-M. Lin, “Reservoir-engineered entanglement in a hybrid modulated three-mode optomechanical system,” Phys. Rev. A 97, 042314 (2018).
[Crossref]

Xiong, B.

W.-Z. Zhang, Y. Han, B. Xiong, and L. Zhou, “Optomechanical force sensor in a non-markovian regime,” New J. Phys. 19, 083022 (2017).
[Crossref]

Xu, X.

X. Xu and J. M. Taylor, “Squeezing in a coupled two-mode optomechanical system for force sensing below the standard quantum limit,” Phys. Rev. A 90, 043848 (2014).
[Crossref]

Xu, X.-W.

X.-W. Xu, Y. Li, A.-X. Chen, and Y.-X. Liu, “Nonreciprocal conversion between microwave and optical photons in electro-optomechanical systems,” Phys. Rev. A 93, 023827 (2016).
[Crossref]

X.-W. Xu and Y. Li, “Optical nonreciprocity and optomechanical circulator in three-mode optomechanical systems,” Phys. Rev. A 91, 053854 (2015).
[Crossref]

Yan, X.-B.

Yang, C.

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).
[Crossref]

Yang, X.

Y. Wu and X. Yang, “Electromagnetically induced transparency in v-, λ-, and cascade-type schemes beyond steady-state analysis,” Phys. Rev. A 71, 053806 (2005).
[Crossref]

Yelin, S. F.

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60, 3225–3228 (1999).
[Crossref]

Yin, T.-S.

Q. Bin, X.-Y. Lü, T.-S. Yin, Y. Li, and Y. Wu, “Collective radiance effects in the ultrastrong-coupling regime,” Phys. Rev. A 99, 033809 (2019).
[Crossref]

L.-L. Zheng, T.-S. Yin, Q. Bin, X.-Y. Lü, and Y. Wu, “Single-photon-induced phonon blockade in a hybrid spin-optomechanical system,” Phys. Rev. A 99, 013804 (2019).
[Crossref]

Yu, J.

Yuan, P.

Y. Zhang, N. Wang, H. Tian, H. Wang, W. Qiu, J. Wang, and P. Yuan, “A high sensitivity optical gyroscope based on slow light in coupled-resonator-induced transparency,” Phys. Lett. A 372, 5848–5852 (2008).
[Crossref]

Yuan, S.

Zhang, H.-X.

L. Du, C.-H. Fan, H.-X. Zhang, and J.-H. Wu, “Synchronization enhancement of indirectly coupled oscillators via periodic modulation in an optomechanical system,” Sci. Rep. 7, 15834 (2017).
[Crossref] [PubMed]

Zhang, W.-Z.

W.-Z. Zhang, Y. Han, B. Xiong, and L. Zhou, “Optomechanical force sensor in a non-markovian regime,” New J. Phys. 19, 083022 (2017).
[Crossref]

Zhang, X. Z.

Zhang, Y.

L. Du, Y.-M. Liu, B. Jiang, and Y. Zhang, “All-optical photon switching, router and amplifier using a passive-active optomechanical system,” Europhys. Lett. 122, 24001 (2018).
[Crossref]

Y. Zhang, N. Wang, H. Tian, H. Wang, W. Qiu, J. Wang, and P. Yuan, “A high sensitivity optical gyroscope based on slow light in coupled-resonator-induced transparency,” Phys. Lett. A 372, 5848–5852 (2008).
[Crossref]

Zhang, Y.-L.

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9, 1797 (2018).
[Crossref] [PubMed]

Y.-L. Zhang, C.-H. Dong, C.-L. Zou, X.-B. Zou, Y.-D. Wang, and G.-C. Guo, “Optomechanical devices based on traveling-wave microresonators,” Phys. Rev. A 95, 043815 (2017).
[Crossref]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
[Crossref]

Zheng, L.-L.

L.-L. Zheng, T.-S. Yin, Q. Bin, X.-Y. Lü, and Y. Wu, “Single-photon-induced phonon blockade in a hybrid spin-optomechanical system,” Phys. Rev. A 99, 013804 (2019).
[Crossref]

Zhou, L.

W.-Z. Zhang, Y. Han, B. Xiong, and L. Zhou, “Optomechanical force sensor in a non-markovian regime,” New J. Phys. 19, 083022 (2017).
[Crossref]

Zhou, X.

F. Hocke, X. Zhou, A. Schliesser, T. J. Kippenberg, H. Huebl, and R. Gross, “Electromechanically induced absorption in a circuit nano-elctromechanical system,” New J. Phys. 14, 123037 (2012).
[Crossref]

Zou, C.-L.

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9, 1797 (2018).
[Crossref] [PubMed]

Y.-L. Zhang, C.-H. Dong, C.-L. Zou, X.-B. Zou, Y.-D. Wang, and G.-C. Guo, “Optomechanical devices based on traveling-wave microresonators,” Phys. Rev. A 95, 043815 (2017).
[Crossref]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
[Crossref]

Zou, X.-B.

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9, 1797 (2018).
[Crossref] [PubMed]

Y.-L. Zhang, C.-H. Dong, C.-L. Zou, X.-B. Zou, Y.-D. Wang, and G.-C. Guo, “Optomechanical devices based on traveling-wave microresonators,” Phys. Rev. A 95, 043815 (2017).
[Crossref]

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
[Crossref]

Zubairy, M. S.

M. O. Scully and M. S. Zubairy, “Coherent trapping–dark states,” in Quantum Optics, (Cambridge University Press, New York, 1997).

Zwickl, B. M.

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).
[Crossref]

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref] [PubMed]

Europhys. Lett. (1)

L. Du, Y.-M. Liu, B. Jiang, and Y. Zhang, “All-optical photon switching, router and amplifier using a passive-active optomechanical system,” Europhys. Lett. 122, 24001 (2018).
[Crossref]

J. Opt. Soc. Am. A (1)

Nat. Commun. (1)

Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9, 1797 (2018).
[Crossref] [PubMed]

Nat. Photonics (1)

Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photonics 10, 657–661 (2016).
[Crossref]

Nature (4)

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475, 359–363 (2011).
[Crossref] [PubMed]

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, “Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane,” Nature 452, 72–75 (2008).
[Crossref] [PubMed]

A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature 464, 697–703 (2010).
[Crossref]

New J. Phys. (5)

W.-Z. Zhang, Y. Han, B. Xiong, and L. Zhou, “Optomechanical force sensor in a non-markovian regime,” New J. Phys. 19, 083022 (2017).
[Crossref]

A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G. E. Harris, “Dispersive optomechanics: a membrane inside a cavity,” New J. Phys. 10, 095008 (2008).
[Crossref]

A. A. Clerk, F. Marquardt, and K. Jacobs, “Back-action evasion and squeezing of a mechanical resonator using a cavity detector,” New J. Phys. 10, 095010 (2008).
[Crossref]

G. S. Agarwal and S. Huang, “Nanomechanical inverse electromagnetically induced transparency and confinement of light in normal modes,” New J. Phys. 16, 033023 (2014).
[Crossref]

F. Hocke, X. Zhou, A. Schliesser, T. J. Kippenberg, H. Huebl, and R. Gross, “Electromechanically induced absorption in a circuit nano-elctromechanical system,” New J. Phys. 14, 123037 (2012).
[Crossref]

Opt. Express (4)

Optica (1)

Phys. Lett. A (1)

Y. Zhang, N. Wang, H. Tian, H. Wang, W. Qiu, J. Wang, and P. Yuan, “A high sensitivity optical gyroscope based on slow light in coupled-resonator-induced transparency,” Phys. Lett. A 372, 5848–5852 (2008).
[Crossref]

Phys. Rev. A (26)

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60, 3225–3228 (1999).
[Crossref]

S. Shahidani, M. H. Naderi, and M. Soltanolkotabi, “Control and manipulation of electromagnetically induced transparency in a nonlinear optomechanical system with two movable mirrors,” Phys. Rev. A 88, 053813 (2013).
[Crossref]

B. P. Hou, L. F. Wei, and S. J. Wang, “Optomechanically induced transparency and absorption in hybridized optomechanical systems,” Phys. Rev. A 92, 033829 (2015).
[Crossref]

E. X. DeJesus and C. Kaufman, “Routh-hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations,” Phys. Rev. A 35, 5288–5290 (1987).
[Crossref]

W. Z. Jia, L. F. Wei, Y. Li, and Y.-X. Liu, “Phase-dependent optical response properties in an optomechanical system by coherently driving the mechanical resonator,” Phys. Rev. A 91, 043843 (2015).
[Crossref]

Q. Bin, X.-Y. Lü, T.-S. Yin, Y. Li, and Y. Wu, “Collective radiance effects in the ultrastrong-coupling regime,” Phys. Rev. A 99, 033809 (2019).
[Crossref]

Y.-L. Zhang, C.-H. Dong, C.-L. Zou, X.-B. Zou, Y.-D. Wang, and G.-C. Guo, “Optomechanical devices based on traveling-wave microresonators,” Phys. Rev. A 95, 043815 (2017).
[Crossref]

S. Barzanjeh, D. Vitali, P. Tombesi, and G. J. Milburn, “Entangling optical and microwave cavity modes by means of a nanomechanical resonator,” Phys. Rev. A 84, 042342 (2011).
[Crossref]

X. Xu and J. M. Taylor, “Squeezing in a coupled two-mode optomechanical system for force sensing below the standard quantum limit,” Phys. Rev. A 90, 043848 (2014).
[Crossref]

S. Huang and G. S. Agarwal, “Robust force sensing for a free particle in a dissipative optomechanical system with a parametric amplifier,” Phys. Rev. A 95, 023844 (2017).
[Crossref]

C. Bai, B. P. Hou, D. G. Lai, and D. Wu, “Tunable optomechanically induced transparency in double quadratically coupled optomechanical cavities within a common reservoir,” Phys. Rev. A 93, 043804 (2016).
[Crossref]

X. Z. Zhang, L. Tian, and Y. Li, “Optomechanical transistor with mechanical gain,” Phys. Rev. A 97, 043818 (2018).
[Crossref]

C. Jiang, L. N. Song, and Y. Li, “Directional amplifier in an optomechanical system with optical gain,” Phys. Rev. A 97, 053812 (2018).
[Crossref]

X.-W. Xu and Y. Li, “Optical nonreciprocity and optomechanical circulator in three-mode optomechanical systems,” Phys. Rev. A 91, 053854 (2015).
[Crossref]

X.-W. Xu, Y. Li, A.-X. Chen, and Y.-X. Liu, “Nonreciprocal conversion between microwave and optical photons in electro-optomechanical systems,” Phys. Rev. A 93, 023827 (2016).
[Crossref]

E. Amitai, N. Lörch, A. Nunnenkamp, S. Walter, and C. Bruder, “Synchronization of an optomechanical system to an external drive,” Phys. Rev. A 95, 053858 (2017).
[Crossref]

K. Ullah, H. Jing, and F. Saif, “Multiple electromechanically-induced-transparency windows and fano resonances in hybrid nano-electro-optomechanics,” Phys. Rev. A 97, 033812 (2018).
[Crossref]

L.-L. Zheng, T.-S. Yin, Q. Bin, X.-Y. Lü, and Y. Wu, “Single-photon-induced phonon blockade in a hybrid spin-optomechanical system,” Phys. Rev. A 99, 013804 (2019).
[Crossref]

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803(R) (2010).
[Crossref]

K. Qu and G. S. Agarwal, “Photon-mediated electromagnetically induced absorption in hybrid opto-electromechanical systems,” Phys. Rev. A 87, 031802(R) (2013).
[Crossref]

Y. Wu and X. Yang, “Electromagnetically induced transparency in v-, λ-, and cascade-type schemes beyond steady-state analysis,” Phys. Rev. A 71, 053806 (2005).
[Crossref]

A. M. Akulshin, S. Barreiro, and A. Lezama, “Electromagnetically induced absorption and transparency due to resonant two-field excitation of quasidegenerate levels in rb vapor,” Phys. Rev. A 57, 2996–3002 (1998).
[Crossref]

A. Lezama, S. Barreiro, and A. M. Akulshin, “Electromagnetically induced absorption,” Phys. Rev. A 59, 4732–4735 (1999).
[Crossref]

X.-B. Yan, “Enhanced output entanglement with reservoir engineering,” Phys. Rev. A 96, 053831 (2017).
[Crossref]

C.-G. Liao, R.-X. Chen, H. Xie, and X.-M. Lin, “Reservoir-engineered entanglement in a hybrid modulated three-mode optomechanical system,” Phys. Rev. A 97, 042314 (2018).
[Crossref]

C. Jiang, L. N. Song, and Y. Li, “Directional phase-sensitive amplifier between microwave and optical photons,” Phys. Rev. A 99, 023823 (2019).
[Crossref]

Phys. Rev. E (1)

E. M.-O. Shahverdiev, “Boundedness of dynamical systems and chaos synchronization,” Phys. Rev. E 60, 3905–3909 (1999).
[Crossref]

Phys. Rev. Lett. (10)

M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[Crossref] [PubMed]

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett. 91, 043902 (2003).
[Crossref] [PubMed]

A. Arvanitaki and A. A. Geraci, “Detecting high-frequency gravitational waves with optically levitated sensors,” Phys. Rev. Lett. 110, 071105 (2013).
[Crossref] [PubMed]

Y.-D. Wang and A. A. Clerk, “Reservoir-engineered entanglement in optomechanical systems,” Phys. Rev. Lett. 110, 253601 (2013).
[Crossref] [PubMed]

D. Malz, L. D. Tóth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, “Quantum-limited directional amplifiers with optomechanics,” Phys. Rev. Lett. 120, 023601 (2018).
[Crossref] [PubMed]

L. Tian, “Robust photon entanglement via quantum interference in optomechanical interfaces,” Phys. Rev. Lett. 110, 233602 (2013).
[Crossref] [PubMed]

Y.-C. Liu, Y.-F. Xiao, X. Luan, and C. W. Wong, “Dynamic dissipative cooling of a mechanical oscillator in strong coupling optomechanics,” Phys. Rev. Lett. 110, 153606 (2013).
[Crossref]

A. Mari, A. Farace, N. Didier, V. Giovannetti, and R. Fazio, “Measures of quantum synchronization in continuous variable systems,” Phys. Rev. Lett. 111, 103605 (2013).
[Crossref]

M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer, “Creating and probing multipartite macroscopic entanglement with light,” Phys. Rev. Lett. 99, 250401 (2007).
[Crossref]

X.-Y. Lü, H. Jing, J.-Y. Ma, and Y. Wu, “𝒫𝒯-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
[Crossref]

Rev. Mod. Phys. (2)

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

Sci. Rep. (1)

L. Du, C.-H. Fan, H.-X. Zhang, and J.-H. Wu, “Synchronization enhancement of indirectly coupled oscillators via periodic modulation in an optomechanical system,” Sci. Rep. 7, 15834 (2017).
[Crossref] [PubMed]

Science (1)

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010).
[Crossref] [PubMed]

Other (2)

M. O. Scully and M. S. Zubairy, “Coherent trapping–dark states,” in Quantum Optics, (Cambridge University Press, New York, 1997).

D. F. Walls and G. J. Milburn, “Input-output formulation of optical cavities,” in Quantum Optics, (Springer-Verlag, Berlin, 1994).

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Figures (8)

Fig. 1
Fig. 1 (a) Fabry-Pérot implementation of our three-mode optomechanical system. The intermediate membrane oscillator with damping rate γm is linearly coupled with two optical cavities simultaneously. The left (right) cavity mode with decay rate κ is driven by a control field with amplitude εc1 (εc2), frequency ωc1 (ωc2), and phase ϑ1 (ϑ2). We also employ a probe field with amplitude εp1 (εp2) and frequency ωp1 (ωp2) to perturb the left (right) cavity mode. Phases of the two probe fields, however, have been assumed to be vanishing without the loss of generality. (b) Effective energy level configuration exhibiting the closed-loop transition for our three-mode optomechanical system. Relevant states of the left cavity, intermediate mechanical, and right cavity modes are denoted by excitation numbers na1, nb, and na2 in order.
Fig. 2
Fig. 2 Stability diagrams attained with G1 = κ (a) and θ = 0 (b) in the case of Δ′1 = −Δ′2 = ωm and γm = 10−4κ. The yellow and blue regions denote the stable and unstable regimes, respectively.
Fig. 3
Fig. 3 Reflectivity RL against G and σ (a) or η and σ (c). Reflectivity R′L against G and σ (b) or η and σ (d). We assume that η = 0.9 in (a) and (b) while G = κ in (c) and (d). Other parameters are γm = 10−4κ and θ = 0.
Fig. 4
Fig. 4 Profiles of reflectivity RL with η = 0.4 (a), G = κ (b), and η = 0 (c). (d) Effective mechanical damping rates against G with η = 0.9 (red) or against η with G = κ (blue) in the case of σ = 0, where the solid, dashed and dot-dashed lines correspond to Γopt,1, Γopt,2 and Γopt,3, respectively. Other parameters are γm = 10−4κ and θ = 0 if not given in relevant panels.
Fig. 5
Fig. 5 Reflectivity RL against G and σ (a) or η and σ (c). Reflectivity R′L against G and σ (b) or η and σ (d). We assume that η = 0.9 in (a) and (b) while G = κ in (c) and (d). Other parameters are γm = 10−4κ and θ = π.
Fig. 6
Fig. 6 Profiles of reflectivity RL for different values of θ. Relevant parameters are γm = 10−4κ, G = κ, and η = 0.9 except θ specified in the figure.
Fig. 7
Fig. 7 Profiles of optical group delay τL (a, c) and τ′L (b, d). Relevant parameters are the same as in Fig. 5 except G specified for panels (a, b) and η specified for panels (c, d) in the figure.
Fig. 8
Fig. 8 Profiles of optical group delay τL (a) and left output phase ϕ (b). Relevant parameters are γm = 10−4κ, G = κ, and η = 0.98 except θ specified for panels (a, b) in the figure.

Equations (21)

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= ω m b b + j = 1 , 2 [ ω j a j a j + ( 1 ) j g j a j a j ( b + b ) + i ( ε c j a j e i ϑ j e i ω c j t + ε p a j e i ω p j t H . c . ) ] ,
= ω m b b + j = 1 , 2 [ Δ j a j a j + ( 1 ) j g j a j a j ( b + b ) + i ( ε c j a j e i ϑ j + ε p a j e i δ j t H . c . ) ] ,
a ˙ 1 = ( κ + i Δ 1 ) a 1 + i g 1 a 1 ( b + b ) + ε c 1 e i ϑ 1 + ε p e i δ 1 t + 2 κ a 1 in , a ˙ 2 = ( κ + i Δ 2 ) a 2 + i g 2 a 2 ( b + b ) + ε c 2 e i ϑ 2 + ε p e i δ 2 t + 2 κ a 2 in , b ˙ = ( γ m + i ω m ) b + i ( g 1 a 1 a 1 g 2 a 2 a 2 ) + 2 γ m b in ,
α 1 s = ε c 1 e i ϑ 1 κ + i Δ 1 , α 2 s = ε c 2 e i ϑ 2 κ + i Δ 2 , β s = i ( g 1 | α 1 s | 2 g 2 | α 2 s | 2 ) γ m + i ω m
δ a 1 . = ( κ + i Δ 1 ) δ a 1 + i G 1 ( δ b + δ b ) + ε p e i δ 1 t + 2 κ a 1 in , δ a 2 . = ( κ + i Δ 2 ) δ a 2 i G 2 ( δ b + δ b ) + ε p e i δ 2 t + 2 κ a 2 in , δ b . = ( γ m + i ω m ) δ b + i ( G 1 * δ a 1 + G 1 δ a 1 G 2 * δ a 2 G 2 δ a 2 ) + 2 γ m b in ,
δ a 1 . = ( κ i Δ 1 ) δ a 1 + i G 1 * ( δ b + δ b ) + ε p e i δ 1 t + 2 κ a 1 in , δ a 2 . = ( κ i Δ 2 ) δ a 2 + i G 2 * ( δ b + δ b ) + ε p e i δ 2 t + 2 κ a 2 in , δ b . = ( γ m i ω m ) δ b i ( G 1 * δ a 1 + G 1 δ a 1 G 2 * δ a 2 G 2 δ a 2 ) + 2 γ m b in .
δ a 1 δ a 1 e i Δ 1 t , δ a 1 in δ a 1 in e i Δ 1 t , δ a 2 δ a 2 e i Δ 2 t , δ a 2 in δ a 2 in e i Δ 2 t , δ b δ b e i ω m t , δ b in δ b in e i ω m t ,
δ a 1 . = κ δ a 1 + i G 1 δ b + ε p e i σ 1 t + 2 κ a 1 in , δ a 2 . = κ δ a 2 + i G 2 * δ b + ε p e i σ 2 t + 2 κ a 2 in , δ b . = γ m δ b + i ( G 1 * δ a 1 G 2 δ a 2 ) + 2 γ m b in ,
δ a 1 . = κ δ a 1 + i G δ b + ε p e i σ 1 t + 2 κ a 1 in , δ a 2 . = κ δ a 2 + i η G e i θ δ b + ε p e i σ 2 t + 2 κ a 2 in , δ b . = γ m δ b + i G ( δ a 1 η e i θ δ a 2 ) + 2 γ m b in
u ˙ = M u + f + n ,
M = ( κ 0 i G 0 κ i η G e i θ i G i η G e i θ γ m ) .
A 1 + = f 1 f 2 + η G 2 ( e i θ η ) f 1 [ f 1 f 2 + G 2 ( 1 η 2 ) ] ε p , A 2 + = 0
ε outL = 2 κ δ a 1 ε p e i σ t ,
ε L + = 2 κ A 1 + ε p ,
A 1 + = f 1 f 2 + η G 2 ( e i θ + η ) f 1 [ f 1 f 2 + G 2 ( 1 + η 2 ) ] ε p ,
( γ m i σ + | G 1 | 2 | G 2 | 2 κ i σ ) B + = i ( G 1 * G 2 ) κ i σ ε p
Γ opt , 1 = γ m + κ G 2 ( 1 η 2 ) κ 2 + σ 2 .
Γ opt , 2 = γ m + κ G 2 κ 2 + σ 2 ,
Γ opt , 3 = γ m + κ G 2 ( 1 + η 2 ) κ 2 + σ 2 ,
τ L = d ϕ d ω p 1 ,
τ L = d ϕ d ω p

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