Abstract

We consider the probe absorption properties in a mechanically coupled optomechanical system in which the two coupled nanomechanical oscillators are driven by the time-dependent forces, respectively. It is found that the mechanical interaction splits the transparency window for a usual single-mode optomechanical system into two parts and then leads to appearance of the double optomechanically induced transparency. The distance between the two transparency positions (the frequency for the maximal transparency) is determined by the mechanical interaction amplitude. This can be explained by using optomechanical dressed-mode picture which is analogue to the interacting dark resonances in coherent atoms. When the mechanical resonators are driven by the external forces, the transparencies in the double-transparency spectrum can be increased into amplifications or be suppressed by tuning the amplitude of the forces. Additionally, it is shown that the double transparencies or the amplifications oscillate with the initial phases of the forces with a period of 2π. These investigations will be useful for more flexible controllability of multi-channel optical communication based on the optomechanical systems.

© 2017 Optical Society of America

1. Introduction

A cavity optomechanical system (OMS) consists of a highly reflective fixed mirror and a movable mirror which is harmonically coupled to the cavity optical field by radiation pressure [1–3]. The optomechanical system has currently attracted much attention due to its potential applications in different topics of physics, such as the cooling of the nanomechanical oscillators to the quantum ground state [4–6], the entanglement between the macroscopic oscillator and the cavity field [7–9], the discovery of the optomechanically induced transparency (OMIT) phenomenon [10–12], the demonstration of the quantum nonlinearities [13–17], the proposal of the coherent wavelength conversion [18–20], the generation of the macroscopic quantum superposition [21], spectroscopy application [22] and other various applications [23].

It is well known that the OMIT phenomenon is one of the important absorption properties in the optomechanical system. The OMIT has been demonstrated theoretically [10] and experimentally [11,12] in the optomechanical system, respectively. Recently, the study on the OMIT has been generalized from the single-mode optomechanical system to the two-mode ones, which include the two-optical OMS constituted by two cavities, and the two-mechanical OMS composed by two mechanical oscillators. For example, the OMIT in a two-cavity-mode optomechanical system, in which two cavity modes are coupled to a common mechanical oscillator, was investigated [24]. In such system, the mechanical-mode splitting of the movable mirror as well as the OMIT in the splitting region were identified [25]. Also, the optomechanically induced absorption in the double-cavity configurations of the hybrid opto-electromechanical systems was predicted [26]. Additionally, the OMITs in the two-coupled-cavity optomechanical system were theoretically investigated [27,28]. Also, three-pathway OMIT in the coupled-cavity optomechanical system was investigated [29]. Recently, we have considered the OMIT and optomechanically induced absorption accompanied by normal-mode splitting (NMS) in strongly tunnel-coupled optomechanical cavities [30]. Subsequently, the OMIT identified by the polarization of optical fields in the vector cavity optomechanics was studied [31]. The optomechanical interactions in these systems are of linear coupling between the mechanical oscillator and the cavity field. Meanwhile, the quadratically coupled optomechanical systems have been used to investigate the OMIT [32]. Recently, we have considered the OMIT in a double quadratically coupled optomechanical cavities within a common reservoir [33]. Additionally, the OMIT in the nonlinear quantum regime [34–36] and with higher-order sidebands [37] have been investigated.

On the other hand, the OMIT in the two-mechanical-mode optomechanical system, which is realized by the mechanical interaction between the two oscillators, was demonstrated [38,39]. For example, the OMIT in the optomechanical cavity coupled to a charged nanomechanical resonator via Coulomb interaction was investigated [38], and this scheme can be used to measure the environmental temperature by injecting a squeezed field into the cavity [39]. Additionally, the OMIT properties in an optomechanical cavity formed by two moving mirrors with a Kerr-down-conversion nonlinear crystal inside has been demonstrated [40]. The OMIT spectra in the two-mechanical-mode optomechanical systems take on a feature of the double transparency windows. Another mechanical manipulation of the OMIT is to drive the mechanical oscillator by using external coherent forces [41–43]. The mechanical modulation of the OMIT can be realized by using Coulomb force to drive the charged mechanical oscillator in the optomechanical system, which can be used to precisely measure the charge number of small charged objects [41]. The spectrum modification of the cavity field and output field in the OMIT by driving an external time-dependent force on the mechanical resonator was studied [42]. Meanwhile, the phase-dependent OMIT in an optomechanical system by coherently driving the mechanical resonator was theoretically investigated [43].

The double OMIT can be created by mechanical interaction in the optomechanical system [38, 39], in which the transparency frequencies can be tuned by adjusting the mechanical interaction amplitude. It is interesting to simultaneously tune the absorption properties (the transparency or the amplification) at different frequencies in double OMIT, which can be useful in multi-channel optical communication. In the present paper, we propose a scheme of a two-mode optomechanical system with both mechanical interaction between two oscillators and the drivings of the external forces on the oscillators. In such scheme, we cannot only tune the transparency frequencies in double OMIT by the mechanical interaction, but also can modulate the absorption properties at different frequencies by the external forces. To our knowledge, the two-coupled-oscillator optomechanical system driven by the external forces has not been investigated. We find that the transparency properties in the double OMIT are remarkably dependent on the mechanical interaction, the forces and their initial phases.

This paper is organized as follows: In Sec. 2 we describe the model and solve its dynamical equation to the system. The double OMIT induced by the mechanical interaction between the two nanomechanical oscillators is displayed in Sec. 3. The manipulations of the double OMIT and optomechanically induced amplification (OMIA) by using the external forces are investigated in Sec. 4. The phase-dependent OMIT and OMIA are discussed in Sec. 5. Finally, we summarize our main results in Sec. 6.

2. Model and dynamical equation

The optomechanical cavity under consideration, which is shown in Fig. 1, consists of one fixed partially transmitting mirror and a charged nanomechanical resonator (NR1), which is coupled to another charged resonator (NR2) by Coulomb interaction. The left fixed mirror is simultaneously driven by a strong coupling field EL = (2PLκ/ħωL)1/2 with frequency ωL = 2πc/λ (λ is the wavelength) and a weak probe field EP = (2PPκ/ħωP)1/2 with frequency ωp, in which Pi (i = L, p) denotes its power. The two NRs are driven by the external time-dependent forces D1 (D2) with frequency ωD1 (ωD2) and initial phase ϕ1 (ϕ2), respectively. The resonance frequency and decay rate of the optomechanical cavity field with optical mode a^ are denoted by ωc and κ, respectively. And b^i(b^i+)(i = 1, 2) is the annihilation (creation) operator for the vibration mode of the charged NRi (i = 1, 2) with frequency ωmi and effective mass mi. The Hamiltonian of the two-mechanical-mode optomechanical system in the rotating frame at the frequency ωL reads [38,39,44,45]

H=Δca^+a^+ωm1b^1+b^1+ωm2b^2+b^2+ga^+a^(b^1++b^1)+V(b^1+b^2+b^1b^2+)+iD1(b^1+eiϕ1eiωD1tb^1eiϕ1eiωD1t)+iD2(b^2+eiϕ2eiωD2tb^2eiϕ2eiωD2t)+iEL(a^+a^)+iEP(a^+eiδtH.c.).
Here Δc = ωc − ωL is the detuning of the coupling field from the cavity, and δ = ωp − ωL is the detuning of the coupling from the probe fields. The first three terms describe the free energies of the cavity and the two NRs, respectively. The optomechanical coupling between the cavity field and the NR1 is given by the fourth term, in which the optomechanical coupling coefficient is denoted by g = (ħ/2m1ωm1)1/2ωc /L with L being the length of the cavity. The fifth term denotes the mechanical coupling between the two charged NRs with coupling amplitude V. The mechanical interaction can be realized by the Coulomb interaction between the two charged NRs after adiabatically eliminating the fast (micro-) motion at the mechanical frequency ωm1 + ωm2 [38,39], or by using waveguide mediated mechanical coupling [44,45]. The coupling amplitude V of the Coulomb interaction is determined by the capacitance and the voltage of the bias gate, and the distance between the equilibrium positions of the NR centers [38,39]. Alternatively, this mechanical interaction can be achieved by using piezoelectric effect [46] or the photothermal effect induced by the irradiation of the laser [47] to stress the coupling overhang, which connects the two one-end-vibrating cantilevers. The mechanical coupling amplitude can be modulated by tuning the effective spring constant of the coupling overhang via the stress generated by the piezoelectric effect or the photothermal effect.

 figure: Fig. 1

Fig. 1 Schematic diagram of a mechanically coupled optomechanical system, in which the two coupled nanomechanical resonators are driven by the time-dependent forces, respectively.

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The next two terms describe the mechanical drivings of the time-dependent forces on the NR1 and NR2, respectively [38,48–50]. The mechanical driving of the nanomechanical oscillators can be experimentally realized by the Coulomb force due to the interaction of the charged NR with the charged body nearby [41], or by the interaction between the two parallel conductors with a time-harmonic current [42]. If the mechanical driving of the nanomechanical oscillators is realized by the Coulomb force, the mechanical coupling between the two oscillators can be better realized by using piezoelectric effect or the photothermal effect. This is due to the fact that the Coulomb interaction in the mechanical driving will interfere that in the mechanical coupling between the two oscillators. The last two terms show the interactions of the cavity field with the coupling and probe fields, respectively.

The Heisenberg equations for the resonators and the cavity variables, including the corresponding noise and damping terms, can be written as follows:

da^dt={κ+i[Δc+g(b^1++b^1)]}a^+EPe-iδt+EL+2κa^in,
db^1dt=(γ1+iωm1)b^1iga^+a^iVb^2+D1eiϕ1eiωD1t+2γ1b^1,in,
db^2dt=(γ2+iωm2)b^2iVb^1+D2eiϕ2eiωD2t+2γ2b^2,in.
Here, a^inis the input vacuum in the cavity with zero mean value. And, γi (i = 1, 2) and b^i,in are the damping rate and the Langevin force coming from the interaction of the ith NR with its environment. Using a^=as+δa^ and b^i=bis+δb^i(i = 1, 2), Eqs. (2a)-(2c) can be divided into the steady parts and the fluctuation ones. Substituting the division forms into Eqs. (2a)-(2c) and setting all the time derivations at the steady parts to be zero, we obtain the steady-state solutions of the variables, which are given by
as=ELκ+iΔ,
b1s=igEL2(γ1+iωm1+V2γ2+iωm2)(κ2+Δ2),
b2s=VgEL2[(γ1+iωm1)(γ2+iωm2)+V2](κ2+Δ2),
where Δ = Δc + g(b1s + b1s) is the effective detuning. As all the control fields are assumed to be sufficiently strong, all the operators can be identified with their expectation values. After being linearized by neglecting nonlinear terms in the fluctuations, the Langevin equations for the expectation values (δa,δb1,δb2) of the fluctuations (δa^,δb^1,δb^2) can be given by
δa˙=(κ+iΔ)δaig(δb1++δb1)as+EPeiδt,
δb˙1=(γ1+iωm1)δb1ig(asδa+δaas)iVδb2+D1eiϕ1eiωD1t,
δb˙2=(γ2+iωm2)δb2iVδb1+D2eiϕ2eiωD2t.
To discuss the response of the probe absorption to the optomechanical interaction and the external time dependent forces, we should expand the expectation values of the intracavity field fluctuations in a frame rotating at the coupling frequency ωL as follows [42]:
δa=a1+eiδt+a1eiδt+a11+eiωD1t+a11eiωD1t+a12+eiωD2t+a12eiωD2t,
δb1=b1+eiδt+b1eiδt+b11+eiωD1t+b11eiωD1t+b12+eiωD2t+b12eiωD2t,
δb2=b2+eiδt+b2eiδt+b21+eiωD1t+b21eiωD1t+b22+eiωD2t+b22eiωD2t.
To investigate the optical property of the optomechanical system, the experimentally accessible transmitted field amplitude aout(t) can be calculated by using the input–output relation aout(t) + ain(t) = (2κ)1/2 a(t), in which the mean value of ain(t) equals zero. In the present paper, we assume that the frequencies ωDi (i = 1, 2) of the external forces are detuned by zero from probe field, i.e. ωDi = δ, then the corresponding components of the output field are obtained as
a1+=EP(1M2M1M31M4)Γ1,
a11+=igD1eiϕ1[asM5(1M8)G3Γ21++asG3Γ21]1M6M5M71M8,
a12+=gVD2eiϕ2Γ62[asM9(1M12)G5Γ22++asG5Γ22]1M10M9M111M12.
Here,
M1=g2as2Γ1(1G1+1G1),M2=g2|as|2Γ1(1G1+1G1),M3=g2as2Γ1+(1G11G1+),M4=g2|as|2Γ1+(1G11G1+),M5=g2as2Γ21(1G21+1G21),M6=g2|as|2Γ21(1G21+1G21),M7=g2as2Γ21+(1G211G21+),M8=g2|as|2Γ21+(1G211G21+),M9=g2as2Γ22(1G22+1G22),M10=g2|as|2Γ22(1G22+1G22),M11=g2as2Γ22+(1G221G22+),M12=g2|as|2Γ22+(1G221G22+),
where
G1±=Γ3±+V2Γ5±,G2j±=Γ4j±+V2Γ6j±,(j=1,2)
And
Γ1±=κ±i(δ±Δ),Γ2j±=κ±i(ωDj±Δ),Γ3±=γ1+i(ωm1±δ),Γ4j±=γ1+i(ωm1±ωDj),Γ5±=γ2+i(ωm2±Δ),Γ6j±=γ2+i(ωm2±ωDj).
The component of the output field a1+ describes the response of the probe to the optomechanical interaction induced by the coupling field. Meanwhile, the component a11+ indicates the effect of the external force D1 on the probe response through the optomechanical interaction, and a12+ demonstrates how the force D2 affect the probe absorption via the mechanical and optomechanical interactions. The output field can be expressed by using Eqs. (6a)-(6c) [10,11,32]
εT=2κ(a1++a11++a12+)Ep.
The real part Re[εT] of the output field exhibits the absorption properties, which is used to describe the OMIT [10–12].

3. Double-OMIT induced by mechanical interaction

We investigate the OMIT in the optomechanical cavity in which the nanomechanical resonator (NR1) interacts with another resonator. Here we focus on the effects of the mechanical interaction on the optical properties. To make the following result within experimental realizations, we use the parameters in the experiment for the observation of the normal-mode splitting [51]: m1 = m2 = 145ng, κ = 2π × 215 × 103Hz, γ1 = γ2 = 2π × 140Hz, ωm1 = ωm2 = 2π × 947 × 103Hz, ωc = 1.77 × 1015Hz, λ = 1064nm, Ep = EL/10 and L = 25mm. Under the resonance condition of Δ = ωm1, the absorption Re[εT] of the output field are plotted as a function of σ = δ−ωm1, which is redefined to display the main optical properties around the resonance δ = ωm1.

In Fig. 2, we plot the probe absorption Re[εT] for different mechanical couplings without considering the drivings of the external forces on the NRs. It is shown from the dashed curve in Fig. 2, with the mechanical coupling amplitude given by V = 1.5 × 8 × 105Hz/m2, that the transparency window in the usual optomechanical system (without mechanical interaction), shown by solid curve, is split into two parts due to the absorption peak around the resonance σ = 0 (δ = ωm1). This is termed as double-OMIT due to the spectrum structure of two transparency windows. When the mechanical coupling is set as V = 3 × 8 × 10 Hz/m2 , shown by the dotted curve, the central absorption peak becomes broader and the distance between the two transparency dips turns longer by comparing with the dashed curve.

 figure: Fig. 2

Fig. 2 The absorption Re[εT] as a function of σ/ωm1 for different values of the mechanical interaction: V = 0 (black, solid curve); V = 1.5 × 8 × 105 Hz/m2 (red, dashed curve); V = 3 × 8 × 105Hz/m2 (blue, dotted curve). The other values of the parameters are given by: PL = 2μW, m1 = m2 = 145ng, κ = 2π × 215 × 103Hz, ωm1 = ωm2 = 2π × 947 × 103Hz, γ1 = γ2 = 2π × 140Hz, L = 25mm, D1 = 0, D2 = 0, ϕ1 = 0 and ϕ2 = 0.

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In fact, the distance between the two transparency dips is determined by the mechanical interaction. This can be confirmed by the quantitative findings that the normalized mechanical coupling amplitudes 2V/ωm1 are given by 0.404 in dashed curve and 0.808 in dotted curve, which well coincide with the corresponding distances between the two transparency dips. In other words, we can use the positions of the transparency dips or the distance between the two transparency dips to accurately measure the mechanical coupling strength V.

How to explain the double OMIT features which are displayed by the absorption spectra in Fig. 2. The OMIT can be explained by the destructive interference between the probe field and the anti-Stokes field scattering from the strong coupling field, which suppresses the build-up of an intracavity probe field [11]. Alternatively, the dressed mode picture withΛ configuration constructed by the optical and mechanical modes is another intuitive explanation of the OMIT, in which the coherent cancellation of the loss channels in the dressed optical and mechanical modes leads to the transparency [12]. This picture is analogy to the coherent three-level atoms which is used to discuss the electromagnetically induced transparency (EIT) [52].

Here the physical process in the mechanically interacting optomechanical system can be described by the level-diagram picture shown in Fig. 3(a), which is analogy to atomic structure of interacting dark resonances [53]. Correspondingly, the level-diagram picture with the dressed mechanical modes is shown by Fig. 3(b). The dark resonances corresponding to the two Λ-configuration transitions with mechanical dressed modes b^+=(b^1b^2)/2and b^=(b^1+b^2)/2 indicate two transparency dips shown by the dashed or dotted curves in Fig. 2, and the distance between the two transparency dips is determined by ωm+−ωm− = 2V . The central absorption peaks come from the interference induced by the coherent interaction between the two dark resonances. In detail, the Hamiltonian in Eq. (1) without the drivings of the external forces and under the condition of ωm1 = ωm2 = ωm can be rewritten as

H=Δca^+a^+ωmb^+b^+ωm+b^++b^++12ga^+a^(b^+++b^+)+12ga^+a^(b^++b^)+iEL(a^+a^)+iEP(a^+eiδtH.c.),
where ωm ± = ωmV. The Hamiltonian in Eq. (11) is composed by two parts which describe the optomechanical interactions with coupling constants G1+ = g/21/2 and G2+ = g/21/2 between the cavity and the dressed mechanical modes b^+ and b^, respectively. It is well known that the OMIT occurs only when the two-photon resonance condition δ = ωmi (with δ = ωp−ωL) is met [10–12]. By using the redefined tuning σ = δ −ωmi, the two-photon resonance conditions for the OMIT in the dressed subsystem are given by σ = V, or σ = -V. This is why the two transparency windows in the double OMIT spectra, shown in Fig. 2, are located at σ = V and σ = -V, respectively.

 figure: Fig. 3

Fig. 3 The level-diagram picture constructed by the optical and mechanical modes (a), and the dressed mode picture with double Λ configuration (b).

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4. Double OMIA induced by time-dependent forces

We shall proceed to discuss the transparency properties in the mechanically interacting optomechanical cavity when the two nanomechanical resonators are driven by the external time-dependent forces. Firstly, we only consider the effect of the external time-dependent force on the NR1, which is directly coupled to the cavity field, and switch off the force driving on the NR2 mechanically interacting with NR1. In Fig. 4, we plot the absorption Re[εT] as a function of σ /ωm1 with V = 1.5 × 8 × 105 Hz/m2 under the conditions of ωD1 = δ and ϕ1 = 0 for different values of the external force D1: D1 = 0 (solid curve); D1 = 1 × 109N (dashed curve); D1 = 3 × 109N (dotted curve). It is shown from the dashed curve in Fig. 4 that the probe absorptions at the two maximal transparency frequencies become negative, i.e. the optomechanically induced amplification (OMIA) is created by the external time-dependent force. When the force is increased up to D1 = 3 × 109N shown by dotted curve, the two dips symmetrically become deeper and their amplification amplitudes turn larger. To show explicit variation of the OMIA with the external force driving on the NR1, we plot the absorption Re[εT] as a function of D1 at σ = −0.2ωm1 in Fig. 5. It is found that the OMIA amplitude monotonously increases with the force amplitude D1.

 figure: Fig. 4

Fig. 4 The absorption Re[εT ] as a function of σ/ωm1 with V = 1.5 × 8 × 105 Hz/m2 for different values of the external force D1: D1 = 0 (black, solid curve); D1 = 1 × 109N (red, dashed curve); D1 = 3 × 109N (blue, dotted curve). The other values of the parameters are set with the same values as in Fig. 2.

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 figure: Fig. 5

Fig. 5 The absorption Re[εT] as a function of D1 at σ = −0.2ωm1. The other values of the parameters are set with the same values as in Fig. 4.

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Next, we shall consider the probe absorption properties in the mechanical interacting optomechanical system when the NR2 is driven by the time-dependent force D2. The same settings of the parameters as Fig. 4 are used in Fig. 6 except for considering the effects of the external force amplitude D2 on the probe OMIT. At this time, the left transparency dip in the double OMIT induced by the mechanical interaction becomes shallower but the right transparency dip becomes deeper with increase of the external time-dependent force D2. In other words, the external force D2 suppresses the transparency in the left window and improves the transparency in the right window into the amplification. The asymmetric variation of the absorption spectrum with the external force D2 is different from symmetric variation with the force D1. This can be well illustrated by the explicit variations of the double-OMIT at the two transparency positions with the external force D2, which is shown by Fig. 7. The difference between the variations of the absorption spectra with the forces D1 and D2 comes from their different interaction patterns of the nanomechanical resonators with the optomechanical cavity: the force D1 is used to drive the NR1 which is optomechanically coupled to the cavity field, while the force D2 is applied on the NR2 which is only coupled to the NR1 and not directly interacted with the opomechanical cavity.

 figure: Fig. 6

Fig. 6 The absorption Re[εT] as a function of σ /ωm1 with V = 1.5 × 8 × 105 Hz/m2 for different values of the external force D2: D2 = 0 (black, solid curve); D2 = 1 × 109N (red, dashed curve); D2 = 3 × 109N (blue, dotted curve). The other values of the parameters are set with the same values as in Fig. 2.

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 figure: Fig. 7

Fig. 7 The absorptions Re[εT ] as a function of D2 at σ = −0.2ωm1 (black, solid curve) and σ = 0.2ωm1 (red, dashed curve). The other values of the parameters are set with the same values as in Fig. 6.

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5. Phase-dependent double OMIA

The optical properties of the optomechanical system can be affected by the initial phases of the external fields driving on the nanomechanical resonators. The variation of the probe absorption spectrum with the initial phase ϕ1 of the external field D1 is shown by Fig. 8. It is shown from the dashed curve (ϕ1 = π/2) that the minimal absorptions in the two windows simultaneously become larger by comparing with the solid curve (ϕ1 = 0). When the phase ϕ1 increases up to π, shown by the dotted curve in Fig. 8, the two transparency dips become further shallower. In other words, the double OMIT is suppressed by the initial phase ϕ1 in the given region. This can be demonstrated by the explicit variation of the maximal transparency in the left window with the initial phase ϕ1, which is shown by Fig. 9. It is shown that the probe absorption oscillates with a period of , which is different from the variation of the absorption with the force amplitude D1 in a monotonous decrease pattern shown in Fig. 5.

 figure: Fig. 8

Fig. 8 The absorption Re[εT ] as a function of σ /ωm1 with D1 = 1 × 109N for different values of the phase ϕ1: ϕ1 = 0 (black, solid curve); ϕ1 = π/2 (red, dashed curve); ϕ1 = π (blue, dotted curve). The other values of the parameters are set with the same values as in Fig. 3.

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 figure: Fig. 9

Fig. 9 The absorption Re[εT] as a function of ϕ1 at σ = −0.2ωm1. The other values of the parameters are set with the same values as in Fig. 8. The dotted line denotes the X-axis.

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Following the pattern in investigating the variation behaviors of the probe absorption with the initial phase ϕ1, we shall consider the dependence of the absorption spectrum on the phase ϕ2. In Fig. 10, the similar setting of the parameters is used but for the initial phase ϕ2 of the force D2 driving the NR2. It is shown that the maximal transparency around σ = −0.2ωm1 increases with the given ϕ2 and turns into amplification, while the maximal transparency around σ = 0.2ωm1 decreases with the ϕ2. The explicit dependences of the probe absorptions on the phase ϕ2 at σ = −0.2ωm1 and σ = 0.2ωm1 are displayed by solid and dashed curves in Fig. 11, respectively. They both oscillate with a period of but they are antiphase with each other. Additionally, it is found that the variation of the absorption at σ = 0.2ωm1 with ϕ2 is similar that with ϕ1 shown in Fig. 9, but the amplification amplitude is larger.

 figure: Fig. 10

Fig. 10 The absorption Re[εT] as a function of σ /ωm1 for different values of the phase ϕ2 with D2 = 1 × 109N: ϕ2 = 0 (black, solid curve); ϕ2 = π/2 (red, dashed curve); ϕ2 = π (blue, dotted curve). The other values of the parameters are set with the same values as in Fig. 6.

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 figure: Fig. 11

Fig. 11 The absorption Re[εT] as a function of ϕ2: σ = −0.2ωm1 (black, solid curve); σ = 0.2ωm1 (red, dashed curve). The other values of the parameters are set with the same values as in Fig. 10. The dotted line denotes the X-axis.

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Finally, we explain the variations of the double optomechanically induced transparency and amplification with the amplitude and initial phases of the external forces presented in above discussions by using the mechanical dressed mode b^±. Now, we define the displaced dressed mechanical mode as B^±=b^±+i(D1exp[i(ϕ1+ωD1t)]D2exp[i(ϕ2+ωD2t)])/2ωm± in terms of the mechanical dressed mode b^±. When including the external forces D1 and D2 applied on the system, the Hamiltonian in Eq. (11) becomes

H=[Δcgωm+(D1sin(ϕ1+ωD1t)D2sin(ϕ2+ωD2t))gωm(D1sin(ϕ1+ωD1t)+D2sin(ϕ2+ωD2t))]a^+a^+j=+,ωmjB^j+B^j+12ga^+a^j=+,(B^j++B^j)+i(ELa^++EPa^+eiδtH.c.),
where the constant term is omitted.

The behaviors of the double OMIT and OMIA varying with the amplitudes and the initial phases of the external forces can be well explained by the effective Hamiltonian in Eq. (12), which is constructed by the displaced dressed mechanical variables. The terms in the last line in Eq. (12) lead to the double OMIT with the two transparency windows located at ωm- (σ = V) and ωm+ (σ = -V), which is addressed in above discussions. The second and third term in square bracket in Eq. (12) come from the effects of the external forces on the two mechanical oscillators. It is clearly seen that the phases and the amplitudes of the external forces can contribute to the intracavity field through the optomechanical interactions, and then induce the interferences of the intracavity fields. For example, the amplitude D1 can simultaneously lead to the amplifications at σ = V and σ = -V due to its negative contribution to the intracavity field. And, the amplitude D2 provides a negative contribution to the intracavity field at ωm- (σ = V), which is similar to the amplitude D1, while it suppresses the transparency at ωm+ (σ = -V) due to its positive contribution to the intracavity field. This well explain why the amplitude D1 can induce the amplifications both at σ = -V and σ = V shown in Figs. 4 and 5, and why the amplitude D2 causes the suppression of the OMIT at σ = -V while it leads to the amplification at σ = V, which are shown in Figs. 6 and 7.

Additionally, we can see that the intracavity fields induced by the external forces vary with the initial phases in a sine pattern. This can explain why the absorption spectra, which are shown by Figs. 8-11, synchronously oscillate with the initial phase ϕ1 for a period of at σ = -V and σ = V, but they oscillating with ϕ2 are antiphase at the two transparency frequencies because of their different signs of the expressions. From the viewpoint of energy transfer, the field induced by the external forces can interfere with the intracavity field and lead to the probe OMIT or the OMIA. In the interference process, the mechanical driving can transfer the energy from the intracavity field to the probe field.

6. Conclusions

In summary, we have investigated the probe absorption properties in a mechanically coupled optomechanical system, in which the two nanomechanical resonators are driven by the time-dependent forces. It is found that the mechanical interaction splits the single transparency window in a usual single-mode optomechanical system into the double optomechanically induced transparency (OMIT), in which the distance between the two transparency positions is determined by the mechanical interaction amplitude. When the nanomechanical resonators are driven by the time-dependent forces, the maximal transparencies in the double OMIT spectra can be improved into the amplification or be suppressed. If only the time-dependent force D1 is used to drive the NR1, the two maximal transparencies are simultaneously improved and turn into the amplifications with increase of D1. When switching on the force D2 driving the NR2, the left maximal transparency is improved and changed into the amplification, while the right maximal transparency is suppressed. Additionally, the transparency and amplification amplitudes at two dips are markedly affected by the initial phases of the external fields and oscillate with a period of .

This study will be useful for more flexible controllability of multi-channel optical communication based on the optomechanical systems. For example, in the double-channel optical communication the transparency frequencies for the probe field can be stabilized by tuning the amplitude of the mechanical interaction between the oscillators, and the simultaneous propagation or amplification at two different frequencies for an optical field can be realized by increasing the amplitude D1 or by tuning the initial phase ϕ1 of the external force applied on the NR1. When the probe field is transparent or amplified at one frequency while it is suppressed at another frequency, we can tune the amplitude D2 or the initial phase ϕ2 of the force driving the NR2. The double-channel optical communication can be generalized to the multi-channel one only by adding more nanomechanical oscillators to mechanically interact with the optomechanical oscillator.

Funding

National Natural Science Foundation of China (10647007); the Young Foundation of Sichuan Province (09ZQ026-008).

References and links.

1. P. Meystre, “A short walk through quantum optomechanics,” Ann. Phys. 525(3), 215–233 (2013). [CrossRef]  

2. T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15(25), 17172–17205 (2007). [CrossRef]   [PubMed]  

3. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014). [CrossRef]  

4. J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011). [CrossRef]   [PubMed]  

5. A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108(3), 033602 (2012). [CrossRef]   [PubMed]  

6. Y. C. Liu, Y. F. Xiao, X. Luan, and C. W. Wong, “Optomechanically-induced-transparency cooling of massive mechanical resonators to the quantum ground state,” Sci. China Phys. Mech. Astron. 58, 1–6 (2015).

7. D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007). [CrossRef]   [PubMed]  

8. M. Gao, F. C. Lei, C. G. Du, and G. L. Long, “Dynamics and entanglement of a membrane-in-the-middle optomechanical system in the extremely-large-amplitude regime,” Sci. China Phys. Mech. Astron. 59(1), 610301 (2016). [CrossRef]  

9. J. Q. Liao, Q. Q. Wu, and F. Nori, “Entangling two macroscopic mechanical mirrors in a two-cavity optomechanical system,” Phys. Rev. A 89(1), 014302 (2014). [CrossRef]  

10. G. S. Agarwal and S. Huang, “The electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010). [CrossRef]  

11. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010). [CrossRef]   [PubMed]  

12. A. H. Safavi-Naeini, T. P. Mayer 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(7341), 69–73 (2011). [CrossRef]   [PubMed]  

13. P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107(6), 063601 (2011). [CrossRef]   [PubMed]  

14. A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107(6), 063602 (2011). [CrossRef]   [PubMed]  

15. M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Rev. Lett. 109(6), 063601 (2012). [CrossRef]   [PubMed]  

16. X. Y. Lü, W. M. Zhang, S. Ashhab, Y. Wu, and F. Nori, “Quantum-criticality-induced strong Kerr nonlinearities in optomechanical systems,” Sci. Rep. 3, 2943 (2013). [CrossRef]   [PubMed]  

17. K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109(1), 013603 (2012). [CrossRef]   [PubMed]  

18. J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3, 1196 (2012). [CrossRef]   [PubMed]  

19. Y. D. Wang and A. A. Clerk, “Using interference for high fidelity quantum state transfer in optomechanics,” Phys. Rev. Lett. 108(15), 153603 (2012). [CrossRef]   [PubMed]  

20. L. Tian and H. Wang, “Optical wavelength conversion of quantum states with optomechanics,” Phys. Rev. A 82(5), 053806 (2010). [CrossRef]  

21. J. Q. Liao and L. Tian, “Macroscopic quantum superposition in cavity optomechanics,” Phys. Rev. Lett. 116(16), 163602 (2016). [CrossRef]   [PubMed]  

22. X. G. Zhuang, L. L. Wang, Q. Chen, X. Y. Wu, and J. X. Fang, “Identification of green tea origins by near-infrared (NIR) spectroscopy and different regression tools,” Sci. China Technol. Sci. 60(1), 84–90 (2017). [CrossRef]  

23. Z. H. Zhou, F. J. Shu, Z. Shen, C. H. Dong, and G. C. Guo, “High-Q whispering gallery modes in a polymer microresonator with broad strain tuning,” Sci. China Phys. Mech. Astron. 58(11), 114208 (2015). [CrossRef]  

24. C. Jiang, H. Liu, Y. Cui, X. Li, G. Chen, and B. Chen, “Electromagnetically induced transparency and slow light in two-mode optomechanics,” Opt. Express 21(10), 12165–12173 (2013). [CrossRef]   [PubMed]  

25. J. Ma, C. You, L. G. Si, H. Xiong, X. Yang, and Y. Wu, “Optomechanically induced transparency in the mechanical-mode splitting regime,” Opt. Lett. 39(14), 4180–4183 (2014). [CrossRef]   [PubMed]  

26. K. Qu and G. S. Agarwal, “Phonon mediated electromagnetically induced absorption in hybrid opto-electro mechanical systems,” Phys. Rev. A 87(3), 031802 (2013). [CrossRef]  

27. Y. J. Guo, K. Li, W. J. Nie, and Y. Li, “Electromagnetically-induced-transparency-like ground-state cooling in a double-cavity optomechanical system,” Phys. Rev. A 90(5), 053841 (2014). [CrossRef]  

28. Y. He, “Optomechanically induced transparency associated with steady-state entanglement,” Phys. Rev. A 91(1), 013827 (2015). [CrossRef]  

29. F. C. Lei, M. Gao, C. Du, Q. L. Jing, and G. L. Long, “Three-pathway electromagnetically induced transparency in coupled-cavity optomechanical system,” Opt. Express 23(9), 11508–11517 (2015). [CrossRef]   [PubMed]  

30. B. P. Hou, L. F. Wei, and S. J. Wang, “Optomechanically induced transparency and absorption in hybridized optomechanical systems,” Phys. Rev. A 92(3), 033829 (2015). [CrossRef]  

31. H. Xiong, Y. M. Huang, L. L. Wan, and Y. Wu, “Vector cavity optomechanics in the parameter configuration of optomechanically induced transparency,” Phys. Rev. A 94(1), 013816 (2016). [CrossRef]  

32. S. Huang and G. S. Agarwal, “Electromagnetically induced transparency from two-phonon processes in quadratically coupled membranes,” Phys. Rev. A 83(2), 023823 (2011). [CrossRef]  

33. 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(4), 043804 (2016). [CrossRef]  

34. M. A. Lemonde, N. Didier, and A. A. Clerk, “Nonlinear interaction effects in a strongly driven optomechanical cavity,” Phys. Rev. Lett. 111(5), 053602 (2013). [CrossRef]   [PubMed]  

35. K. Børkje, A. Nunnenkamp, J. D. Teufel, and S. M. Girvin, “Signatures of nonlinear cavity optomechanics in the weak coupling regime,” Phys. Rev. Lett. 111(5), 053603 (2013). [CrossRef]   [PubMed]  

36. A. Kronwald and F. Marquardt, “Optomechanically induced transparency in the nonlinear quantum regime,” Phys. Rev. Lett. 111(13), 133601 (2013). [CrossRef]   [PubMed]  

37. H. Xiong, L. G. Si, A. S. Zheng, X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86(1), 013815 (2012). [CrossRef]  

38. P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90(4), 043825 (2014). [CrossRef]  

39. Q. Wang, J. Q. Zhang, P. C. Ma, C. M. Yao, and M. Feng, “Precision measurement of the environmental temperature by tunable double optomechanically induced transparency with a squeezed field,” Phys. Rev. A 91(6), 063827 (2015). [CrossRef]  

40. S. Shahidani, M. H. Naderi, and M. Soltanolkotabi, “Control and manipulation of electromagentically induced transparency in a nonlinear optomechanical system with two movable mirrors,” Phys. Rev. A 88(5), 053813 (2013). [CrossRef]  

41. J. Q. Zhang, Y. Li, M. Feng, and Y. Xu, “Precision measurement of electrical charge with optomechanically induced transparency,” Phys. Rev. A 86(5), 053806 (2012). [CrossRef]  

42. J. Ma, C. You, L. G. Si, H. Xiong, J. Li, X. Yang, and Y. Wu, “Optomechanically induced transparency in the presence of an external time-harmonic-driving force,” Sci. Rep. 5, 11278 (2015). [CrossRef]   [PubMed]  

43. 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(4), 043843 (2015). [CrossRef]  

44. Y. Sato, Y. Tanaka, J. Upham, Y. Takahashi, T. Asano, and S. Noda, “Strong coupling between distant photonic nanocavities and its dynamic control,” Nat. Photonics 6(1), 56–61 (2011). [CrossRef]  

45. K. J. Fang, M. H. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photonics 10(7), 489–496 (2016). [CrossRef]  

46. H. Okamoto, A. Gourgout, C. Y. Chang, K. Onomitsu, I. Mahboob, E. Y. Chang, and H. Yamaguchi, “Coherent phonon manipulation in coupled mechanical resonators,” Nat. Phys. 9(8), 480–484 (2013). [CrossRef]  

47. H. Okamoto, T. Kamada, K. Onomitsu, I. Mahboob, and H. Yamaguchi, “Optical tuning of coupled micromechanical resonators,” Appl. Phys. Express 2, 062202 (2009). [CrossRef]  

48. X. W. Xu and Y. Li, “Controllable optical output fields from an optomechanical system with a mechanical driving,” Phys. Rev. A 92(2), 023855 (2015). [CrossRef]  

49. H. Suzuki, E. Brown, and R. Sterling, “Nonlinear dynamics of an optomechanical system with a coherent mechanical pump: Second-order sideband generation,” Phys. Rev. A 92(3), 033823 (2015). [CrossRef]  

50. C. Jiang, Y. S. Cui, X. T. Bian, F. Zuo, H. L. Yu, and G. B. Chen, “Phase-dependent multiple optomechanically induced absorption in multimode optomechanical systems with mechanical driving,” Phys. Rev. A 94(2), 023837 (2016). [CrossRef]  

51. S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460(7256), 724–727 (2009). [CrossRef]   [PubMed]  

52. S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997). [CrossRef]  

53. M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60(4), 3225–3228 (1999). [CrossRef]  

References

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  1. P. Meystre, “A short walk through quantum optomechanics,” Ann. Phys. 525(3), 215–233 (2013).
    [Crossref]
  2. T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15(25), 17172–17205 (2007).
    [Crossref] [PubMed]
  3. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
    [Crossref]
  4. J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
    [Crossref] [PubMed]
  5. A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108(3), 033602 (2012).
    [Crossref] [PubMed]
  6. Y. C. Liu, Y. F. Xiao, X. Luan, and C. W. Wong, “Optomechanically-induced-transparency cooling of massive mechanical resonators to the quantum ground state,” Sci. China Phys. Mech. Astron. 58, 1–6 (2015).
  7. D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
    [Crossref] [PubMed]
  8. M. Gao, F. C. Lei, C. G. Du, and G. L. Long, “Dynamics and entanglement of a membrane-in-the-middle optomechanical system in the extremely-large-amplitude regime,” Sci. China Phys. Mech. Astron. 59(1), 610301 (2016).
    [Crossref]
  9. J. Q. Liao, Q. Q. Wu, and F. Nori, “Entangling two macroscopic mechanical mirrors in a two-cavity optomechanical system,” Phys. Rev. A 89(1), 014302 (2014).
    [Crossref]
  10. G. S. Agarwal and S. Huang, “The electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010).
    [Crossref]
  11. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
    [Crossref] [PubMed]
  12. A. H. Safavi-Naeini, T. P. Mayer 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(7341), 69–73 (2011).
    [Crossref] [PubMed]
  13. P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107(6), 063601 (2011).
    [Crossref] [PubMed]
  14. A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107(6), 063602 (2011).
    [Crossref] [PubMed]
  15. M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Rev. Lett. 109(6), 063601 (2012).
    [Crossref] [PubMed]
  16. X. Y. Lü, W. M. Zhang, S. Ashhab, Y. Wu, and F. Nori, “Quantum-criticality-induced strong Kerr nonlinearities in optomechanical systems,” Sci. Rep. 3, 2943 (2013).
    [Crossref] [PubMed]
  17. K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109(1), 013603 (2012).
    [Crossref] [PubMed]
  18. J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3, 1196 (2012).
    [Crossref] [PubMed]
  19. Y. D. Wang and A. A. Clerk, “Using interference for high fidelity quantum state transfer in optomechanics,” Phys. Rev. Lett. 108(15), 153603 (2012).
    [Crossref] [PubMed]
  20. L. Tian and H. Wang, “Optical wavelength conversion of quantum states with optomechanics,” Phys. Rev. A 82(5), 053806 (2010).
    [Crossref]
  21. J. Q. Liao and L. Tian, “Macroscopic quantum superposition in cavity optomechanics,” Phys. Rev. Lett. 116(16), 163602 (2016).
    [Crossref] [PubMed]
  22. X. G. Zhuang, L. L. Wang, Q. Chen, X. Y. Wu, and J. X. Fang, “Identification of green tea origins by near-infrared (NIR) spectroscopy and different regression tools,” Sci. China Technol. Sci. 60(1), 84–90 (2017).
    [Crossref]
  23. Z. H. Zhou, F. J. Shu, Z. Shen, C. H. Dong, and G. C. Guo, “High-Q whispering gallery modes in a polymer microresonator with broad strain tuning,” Sci. China Phys. Mech. Astron. 58(11), 114208 (2015).
    [Crossref]
  24. C. Jiang, H. Liu, Y. Cui, X. Li, G. Chen, and B. Chen, “Electromagnetically induced transparency and slow light in two-mode optomechanics,” Opt. Express 21(10), 12165–12173 (2013).
    [Crossref] [PubMed]
  25. J. Ma, C. You, L. G. Si, H. Xiong, X. Yang, and Y. Wu, “Optomechanically induced transparency in the mechanical-mode splitting regime,” Opt. Lett. 39(14), 4180–4183 (2014).
    [Crossref] [PubMed]
  26. K. Qu and G. S. Agarwal, “Phonon mediated electromagnetically induced absorption in hybrid opto-electro mechanical systems,” Phys. Rev. A 87(3), 031802 (2013).
    [Crossref]
  27. Y. J. Guo, K. Li, W. J. Nie, and Y. Li, “Electromagnetically-induced-transparency-like ground-state cooling in a double-cavity optomechanical system,” Phys. Rev. A 90(5), 053841 (2014).
    [Crossref]
  28. Y. He, “Optomechanically induced transparency associated with steady-state entanglement,” Phys. Rev. A 91(1), 013827 (2015).
    [Crossref]
  29. F. C. Lei, M. Gao, C. Du, Q. L. Jing, and G. L. Long, “Three-pathway electromagnetically induced transparency in coupled-cavity optomechanical system,” Opt. Express 23(9), 11508–11517 (2015).
    [Crossref] [PubMed]
  30. B. P. Hou, L. F. Wei, and S. J. Wang, “Optomechanically induced transparency and absorption in hybridized optomechanical systems,” Phys. Rev. A 92(3), 033829 (2015).
    [Crossref]
  31. H. Xiong, Y. M. Huang, L. L. Wan, and Y. Wu, “Vector cavity optomechanics in the parameter configuration of optomechanically induced transparency,” Phys. Rev. A 94(1), 013816 (2016).
    [Crossref]
  32. S. Huang and G. S. Agarwal, “Electromagnetically induced transparency from two-phonon processes in quadratically coupled membranes,” Phys. Rev. A 83(2), 023823 (2011).
    [Crossref]
  33. 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(4), 043804 (2016).
    [Crossref]
  34. M. A. Lemonde, N. Didier, and A. A. Clerk, “Nonlinear interaction effects in a strongly driven optomechanical cavity,” Phys. Rev. Lett. 111(5), 053602 (2013).
    [Crossref] [PubMed]
  35. K. Børkje, A. Nunnenkamp, J. D. Teufel, and S. M. Girvin, “Signatures of nonlinear cavity optomechanics in the weak coupling regime,” Phys. Rev. Lett. 111(5), 053603 (2013).
    [Crossref] [PubMed]
  36. A. Kronwald and F. Marquardt, “Optomechanically induced transparency in the nonlinear quantum regime,” Phys. Rev. Lett. 111(13), 133601 (2013).
    [Crossref] [PubMed]
  37. H. Xiong, L. G. Si, A. S. Zheng, X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86(1), 013815 (2012).
    [Crossref]
  38. P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90(4), 043825 (2014).
    [Crossref]
  39. Q. Wang, J. Q. Zhang, P. C. Ma, C. M. Yao, and M. Feng, “Precision measurement of the environmental temperature by tunable double optomechanically induced transparency with a squeezed field,” Phys. Rev. A 91(6), 063827 (2015).
    [Crossref]
  40. S. Shahidani, M. H. Naderi, and M. Soltanolkotabi, “Control and manipulation of electromagentically induced transparency in a nonlinear optomechanical system with two movable mirrors,” Phys. Rev. A 88(5), 053813 (2013).
    [Crossref]
  41. J. Q. Zhang, Y. Li, M. Feng, and Y. Xu, “Precision measurement of electrical charge with optomechanically induced transparency,” Phys. Rev. A 86(5), 053806 (2012).
    [Crossref]
  42. J. Ma, C. You, L. G. Si, H. Xiong, J. Li, X. Yang, and Y. Wu, “Optomechanically induced transparency in the presence of an external time-harmonic-driving force,” Sci. Rep. 5, 11278 (2015).
    [Crossref] [PubMed]
  43. 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(4), 043843 (2015).
    [Crossref]
  44. Y. Sato, Y. Tanaka, J. Upham, Y. Takahashi, T. Asano, and S. Noda, “Strong coupling between distant photonic nanocavities and its dynamic control,” Nat. Photonics 6(1), 56–61 (2011).
    [Crossref]
  45. K. J. Fang, M. H. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photonics 10(7), 489–496 (2016).
    [Crossref]
  46. H. Okamoto, A. Gourgout, C. Y. Chang, K. Onomitsu, I. Mahboob, E. Y. Chang, and H. Yamaguchi, “Coherent phonon manipulation in coupled mechanical resonators,” Nat. Phys. 9(8), 480–484 (2013).
    [Crossref]
  47. H. Okamoto, T. Kamada, K. Onomitsu, I. Mahboob, and H. Yamaguchi, “Optical tuning of coupled micromechanical resonators,” Appl. Phys. Express 2, 062202 (2009).
    [Crossref]
  48. X. W. Xu and Y. Li, “Controllable optical output fields from an optomechanical system with a mechanical driving,” Phys. Rev. A 92(2), 023855 (2015).
    [Crossref]
  49. H. Suzuki, E. Brown, and R. Sterling, “Nonlinear dynamics of an optomechanical system with a coherent mechanical pump: Second-order sideband generation,” Phys. Rev. A 92(3), 033823 (2015).
    [Crossref]
  50. C. Jiang, Y. S. Cui, X. T. Bian, F. Zuo, H. L. Yu, and G. B. Chen, “Phase-dependent multiple optomechanically induced absorption in multimode optomechanical systems with mechanical driving,” Phys. Rev. A 94(2), 023837 (2016).
    [Crossref]
  51. S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460(7256), 724–727 (2009).
    [Crossref] [PubMed]
  52. S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
    [Crossref]
  53. M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60(4), 3225–3228 (1999).
    [Crossref]

2017 (1)

X. G. Zhuang, L. L. Wang, Q. Chen, X. Y. Wu, and J. X. Fang, “Identification of green tea origins by near-infrared (NIR) spectroscopy and different regression tools,” Sci. China Technol. Sci. 60(1), 84–90 (2017).
[Crossref]

2016 (6)

J. Q. Liao and L. Tian, “Macroscopic quantum superposition in cavity optomechanics,” Phys. Rev. Lett. 116(16), 163602 (2016).
[Crossref] [PubMed]

H. Xiong, Y. M. Huang, L. L. Wan, and Y. Wu, “Vector cavity optomechanics in the parameter configuration of optomechanically induced transparency,” Phys. Rev. A 94(1), 013816 (2016).
[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(4), 043804 (2016).
[Crossref]

M. Gao, F. C. Lei, C. G. Du, and G. L. Long, “Dynamics and entanglement of a membrane-in-the-middle optomechanical system in the extremely-large-amplitude regime,” Sci. China Phys. Mech. Astron. 59(1), 610301 (2016).
[Crossref]

K. J. Fang, M. H. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photonics 10(7), 489–496 (2016).
[Crossref]

C. Jiang, Y. S. Cui, X. T. Bian, F. Zuo, H. L. Yu, and G. B. Chen, “Phase-dependent multiple optomechanically induced absorption in multimode optomechanical systems with mechanical driving,” Phys. Rev. A 94(2), 023837 (2016).
[Crossref]

2015 (10)

X. W. Xu and Y. Li, “Controllable optical output fields from an optomechanical system with a mechanical driving,” Phys. Rev. A 92(2), 023855 (2015).
[Crossref]

H. Suzuki, E. Brown, and R. Sterling, “Nonlinear dynamics of an optomechanical system with a coherent mechanical pump: Second-order sideband generation,” Phys. Rev. A 92(3), 033823 (2015).
[Crossref]

Q. Wang, J. Q. Zhang, P. C. Ma, C. M. Yao, and M. Feng, “Precision measurement of the environmental temperature by tunable double optomechanically induced transparency with a squeezed field,” Phys. Rev. A 91(6), 063827 (2015).
[Crossref]

J. Ma, C. You, L. G. Si, H. Xiong, J. Li, X. Yang, and Y. Wu, “Optomechanically induced transparency in the presence of an external time-harmonic-driving force,” Sci. Rep. 5, 11278 (2015).
[Crossref] [PubMed]

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(4), 043843 (2015).
[Crossref]

Y. C. Liu, Y. F. Xiao, X. Luan, and C. W. Wong, “Optomechanically-induced-transparency cooling of massive mechanical resonators to the quantum ground state,” Sci. China Phys. Mech. Astron. 58, 1–6 (2015).

Y. He, “Optomechanically induced transparency associated with steady-state entanglement,” Phys. Rev. A 91(1), 013827 (2015).
[Crossref]

F. C. Lei, M. Gao, C. Du, Q. L. Jing, and G. L. Long, “Three-pathway electromagnetically induced transparency in coupled-cavity optomechanical system,” Opt. Express 23(9), 11508–11517 (2015).
[Crossref] [PubMed]

B. P. Hou, L. F. Wei, and S. J. Wang, “Optomechanically induced transparency and absorption in hybridized optomechanical systems,” Phys. Rev. A 92(3), 033829 (2015).
[Crossref]

Z. H. Zhou, F. J. Shu, Z. Shen, C. H. Dong, and G. C. Guo, “High-Q whispering gallery modes in a polymer microresonator with broad strain tuning,” Sci. China Phys. Mech. Astron. 58(11), 114208 (2015).
[Crossref]

2014 (5)

Y. J. Guo, K. Li, W. J. Nie, and Y. Li, “Electromagnetically-induced-transparency-like ground-state cooling in a double-cavity optomechanical system,” Phys. Rev. A 90(5), 053841 (2014).
[Crossref]

J. Ma, C. You, L. G. Si, H. Xiong, X. Yang, and Y. Wu, “Optomechanically induced transparency in the mechanical-mode splitting regime,” Opt. Lett. 39(14), 4180–4183 (2014).
[Crossref] [PubMed]

J. Q. Liao, Q. Q. Wu, and F. Nori, “Entangling two macroscopic mechanical mirrors in a two-cavity optomechanical system,” Phys. Rev. A 89(1), 014302 (2014).
[Crossref]

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
[Crossref]

P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90(4), 043825 (2014).
[Crossref]

2013 (9)

S. Shahidani, M. H. Naderi, and M. Soltanolkotabi, “Control and manipulation of electromagentically induced transparency in a nonlinear optomechanical system with two movable mirrors,” Phys. Rev. A 88(5), 053813 (2013).
[Crossref]

H. Okamoto, A. Gourgout, C. Y. Chang, K. Onomitsu, I. Mahboob, E. Y. Chang, and H. Yamaguchi, “Coherent phonon manipulation in coupled mechanical resonators,” Nat. Phys. 9(8), 480–484 (2013).
[Crossref]

P. Meystre, “A short walk through quantum optomechanics,” Ann. Phys. 525(3), 215–233 (2013).
[Crossref]

X. Y. Lü, W. M. Zhang, S. Ashhab, Y. Wu, and F. Nori, “Quantum-criticality-induced strong Kerr nonlinearities in optomechanical systems,” Sci. Rep. 3, 2943 (2013).
[Crossref] [PubMed]

K. Qu and G. S. Agarwal, “Phonon mediated electromagnetically induced absorption in hybrid opto-electro mechanical systems,” Phys. Rev. A 87(3), 031802 (2013).
[Crossref]

C. Jiang, H. Liu, Y. Cui, X. Li, G. Chen, and B. Chen, “Electromagnetically induced transparency and slow light in two-mode optomechanics,” Opt. Express 21(10), 12165–12173 (2013).
[Crossref] [PubMed]

M. A. Lemonde, N. Didier, and A. A. Clerk, “Nonlinear interaction effects in a strongly driven optomechanical cavity,” Phys. Rev. Lett. 111(5), 053602 (2013).
[Crossref] [PubMed]

K. Børkje, A. Nunnenkamp, J. D. Teufel, and S. M. Girvin, “Signatures of nonlinear cavity optomechanics in the weak coupling regime,” Phys. Rev. Lett. 111(5), 053603 (2013).
[Crossref] [PubMed]

A. Kronwald and F. Marquardt, “Optomechanically induced transparency in the nonlinear quantum regime,” Phys. Rev. Lett. 111(13), 133601 (2013).
[Crossref] [PubMed]

2012 (7)

H. Xiong, L. G. Si, A. S. Zheng, X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86(1), 013815 (2012).
[Crossref]

A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108(3), 033602 (2012).
[Crossref] [PubMed]

M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Rev. Lett. 109(6), 063601 (2012).
[Crossref] [PubMed]

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109(1), 013603 (2012).
[Crossref] [PubMed]

J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3, 1196 (2012).
[Crossref] [PubMed]

Y. D. Wang and A. A. Clerk, “Using interference for high fidelity quantum state transfer in optomechanics,” Phys. Rev. Lett. 108(15), 153603 (2012).
[Crossref] [PubMed]

J. Q. Zhang, Y. Li, M. Feng, and Y. Xu, “Precision measurement of electrical charge with optomechanically induced transparency,” Phys. Rev. A 86(5), 053806 (2012).
[Crossref]

2011 (6)

Y. Sato, Y. Tanaka, J. Upham, Y. Takahashi, T. Asano, and S. Noda, “Strong coupling between distant photonic nanocavities and its dynamic control,” Nat. Photonics 6(1), 56–61 (2011).
[Crossref]

A. H. Safavi-Naeini, T. P. Mayer 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(7341), 69–73 (2011).
[Crossref] [PubMed]

P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107(6), 063601 (2011).
[Crossref] [PubMed]

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107(6), 063602 (2011).
[Crossref] [PubMed]

J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

S. Huang and G. S. Agarwal, “Electromagnetically induced transparency from two-phonon processes in quadratically coupled membranes,” Phys. Rev. A 83(2), 023823 (2011).
[Crossref]

2010 (3)

G. S. Agarwal and S. Huang, “The electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010).
[Crossref]

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref] [PubMed]

L. Tian and H. Wang, “Optical wavelength conversion of quantum states with optomechanics,” Phys. Rev. A 82(5), 053806 (2010).
[Crossref]

2009 (2)

H. Okamoto, T. Kamada, K. Onomitsu, I. Mahboob, and H. Yamaguchi, “Optical tuning of coupled micromechanical resonators,” Appl. Phys. Express 2, 062202 (2009).
[Crossref]

S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460(7256), 724–727 (2009).
[Crossref] [PubMed]

2007 (2)

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref] [PubMed]

T. J. Kippenberg and K. J. Vahala, “Cavity opto-mechanics,” Opt. Express 15(25), 17172–17205 (2007).
[Crossref] [PubMed]

1999 (1)

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60(4), 3225–3228 (1999).
[Crossref]

1997 (1)

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
[Crossref]

Agarwal, G. S.

K. Qu and G. S. Agarwal, “Phonon mediated electromagnetically induced absorption in hybrid opto-electro mechanical systems,” Phys. Rev. A 87(3), 031802 (2013).
[Crossref]

S. Huang and G. S. Agarwal, “Electromagnetically induced transparency from two-phonon processes in quadratically coupled membranes,” Phys. Rev. A 83(2), 023823 (2011).
[Crossref]

G. S. Agarwal and S. Huang, “The electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010).
[Crossref]

Alegre, T. P.

A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108(3), 033602 (2012).
[Crossref] [PubMed]

J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

Arcizet, O.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref] [PubMed]

Asano, T.

Y. Sato, Y. Tanaka, J. Upham, Y. Takahashi, T. Asano, and S. Noda, “Strong coupling between distant photonic nanocavities and its dynamic control,” Nat. Photonics 6(1), 56–61 (2011).
[Crossref]

Ashhab, S.

X. Y. Lü, W. M. Zhang, S. Ashhab, Y. Wu, and F. Nori, “Quantum-criticality-induced strong Kerr nonlinearities in optomechanical systems,” Sci. Rep. 3, 2943 (2013).
[Crossref] [PubMed]

Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
[Crossref]

J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460(7256), 724–727 (2009).
[Crossref] [PubMed]

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref] [PubMed]

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(4), 043804 (2016).
[Crossref]

Bennett, S. D.

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109(1), 013603 (2012).
[Crossref] [PubMed]

Bian, X. T.

C. Jiang, Y. S. Cui, X. T. Bian, F. Zuo, H. L. Yu, and G. B. Chen, “Phase-dependent multiple optomechanically induced absorption in multimode optomechanical systems with mechanical driving,” Phys. Rev. A 94(2), 023837 (2016).
[Crossref]

Böhm, H. R.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref] [PubMed]

Børkje, K.

K. Børkje, A. Nunnenkamp, J. D. Teufel, and S. M. Girvin, “Signatures of nonlinear cavity optomechanics in the weak coupling regime,” Phys. Rev. Lett. 111(5), 053603 (2013).
[Crossref] [PubMed]

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107(6), 063602 (2011).
[Crossref] [PubMed]

Brown, E.

H. Suzuki, E. Brown, and R. Sterling, “Nonlinear dynamics of an optomechanical system with a coherent mechanical pump: Second-order sideband generation,” Phys. Rev. A 92(3), 033823 (2015).
[Crossref]

Chan, J.

A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108(3), 033602 (2012).
[Crossref] [PubMed]

J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3, 1196 (2012).
[Crossref] [PubMed]

J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

A. H. Safavi-Naeini, T. P. Mayer 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(7341), 69–73 (2011).
[Crossref] [PubMed]

Chang, C. Y.

H. Okamoto, A. Gourgout, C. Y. Chang, K. Onomitsu, I. Mahboob, E. Y. Chang, and H. Yamaguchi, “Coherent phonon manipulation in coupled mechanical resonators,” Nat. Phys. 9(8), 480–484 (2013).
[Crossref]

Chang, D. E.

A. H. Safavi-Naeini, T. P. Mayer 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(7341), 69–73 (2011).
[Crossref] [PubMed]

Chang, E. Y.

H. Okamoto, A. Gourgout, C. Y. Chang, K. Onomitsu, I. Mahboob, E. Y. Chang, and H. Yamaguchi, “Coherent phonon manipulation in coupled mechanical resonators,” Nat. Phys. 9(8), 480–484 (2013).
[Crossref]

Chen, B.

Chen, G.

Chen, G. B.

C. Jiang, Y. S. Cui, X. T. Bian, F. Zuo, H. L. Yu, and G. B. Chen, “Phase-dependent multiple optomechanically induced absorption in multimode optomechanical systems with mechanical driving,” Phys. Rev. A 94(2), 023837 (2016).
[Crossref]

Chen, Q.

X. G. Zhuang, L. L. Wang, Q. Chen, X. Y. Wu, and J. X. Fang, “Identification of green tea origins by near-infrared (NIR) spectroscopy and different regression tools,” Sci. China Technol. Sci. 60(1), 84–90 (2017).
[Crossref]

Clerk, A. A.

M. A. Lemonde, N. Didier, and A. A. Clerk, “Nonlinear interaction effects in a strongly driven optomechanical cavity,” Phys. Rev. Lett. 111(5), 053602 (2013).
[Crossref] [PubMed]

Y. D. Wang and A. A. Clerk, “Using interference for high fidelity quantum state transfer in optomechanics,” Phys. Rev. Lett. 108(15), 153603 (2012).
[Crossref] [PubMed]

Cui, Y.

Cui, Y. S.

C. Jiang, Y. S. Cui, X. T. Bian, F. Zuo, H. L. Yu, and G. B. Chen, “Phase-dependent multiple optomechanically induced absorption in multimode optomechanical systems with mechanical driving,” Phys. Rev. A 94(2), 023837 (2016).
[Crossref]

Deléglise, S.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref] [PubMed]

Didier, N.

M. A. Lemonde, N. Didier, and A. A. Clerk, “Nonlinear interaction effects in a strongly driven optomechanical cavity,” Phys. Rev. Lett. 111(5), 053602 (2013).
[Crossref] [PubMed]

Dong, C. H.

Z. H. Zhou, F. J. Shu, Z. Shen, C. H. Dong, and G. C. Guo, “High-Q whispering gallery modes in a polymer microresonator with broad strain tuning,” Sci. China Phys. Mech. Astron. 58(11), 114208 (2015).
[Crossref]

Du, C.

Du, C. G.

M. Gao, F. C. Lei, C. G. Du, and G. L. Long, “Dynamics and entanglement of a membrane-in-the-middle optomechanical system in the extremely-large-amplitude regime,” Sci. China Phys. Mech. Astron. 59(1), 610301 (2016).
[Crossref]

Eichenfield, M.

A. H. Safavi-Naeini, T. P. Mayer 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(7341), 69–73 (2011).
[Crossref] [PubMed]

Fang, J. X.

X. G. Zhuang, L. L. Wang, Q. Chen, X. Y. Wu, and J. X. Fang, “Identification of green tea origins by near-infrared (NIR) spectroscopy and different regression tools,” Sci. China Technol. Sci. 60(1), 84–90 (2017).
[Crossref]

Fang, K. J.

K. J. Fang, M. H. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photonics 10(7), 489–496 (2016).
[Crossref]

Feng, M.

Q. Wang, J. Q. Zhang, P. C. Ma, C. M. Yao, and M. Feng, “Precision measurement of the environmental temperature by tunable double optomechanically induced transparency with a squeezed field,” Phys. Rev. A 91(6), 063827 (2015).
[Crossref]

P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90(4), 043825 (2014).
[Crossref]

J. Q. Zhang, Y. Li, M. Feng, and Y. Xu, “Precision measurement of electrical charge with optomechanically induced transparency,” Phys. Rev. A 86(5), 053806 (2012).
[Crossref]

Ferreira, A.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref] [PubMed]

Fleischhauer, M.

M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully, “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60(4), 3225–3228 (1999).
[Crossref]

Gao, M.

M. Gao, F. C. Lei, C. G. Du, and G. L. Long, “Dynamics and entanglement of a membrane-in-the-middle optomechanical system in the extremely-large-amplitude regime,” Sci. China Phys. Mech. Astron. 59(1), 610301 (2016).
[Crossref]

F. C. Lei, M. Gao, C. Du, Q. L. Jing, and G. L. Long, “Three-pathway electromagnetically induced transparency in coupled-cavity optomechanical system,” Opt. Express 23(9), 11508–11517 (2015).
[Crossref] [PubMed]

Gavartin, E.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref] [PubMed]

Gigan, S.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref] [PubMed]

Girvin, S. M.

K. Børkje, A. Nunnenkamp, J. D. Teufel, and S. M. Girvin, “Signatures of nonlinear cavity optomechanics in the weak coupling regime,” Phys. Rev. Lett. 111(5), 053603 (2013).
[Crossref] [PubMed]

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107(6), 063602 (2011).
[Crossref] [PubMed]

Gourgout, A.

H. Okamoto, A. Gourgout, C. Y. Chang, K. Onomitsu, I. Mahboob, E. Y. Chang, and H. Yamaguchi, “Coherent phonon manipulation in coupled mechanical resonators,” Nat. Phys. 9(8), 480–484 (2013).
[Crossref]

Gröblacher, S.

J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
[Crossref] [PubMed]

S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460(7256), 724–727 (2009).
[Crossref] [PubMed]

Guerreiro, A.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref] [PubMed]

Guo, G. C.

Z. H. Zhou, F. J. Shu, Z. Shen, C. H. Dong, and G. C. Guo, “High-Q whispering gallery modes in a polymer microresonator with broad strain tuning,” Sci. China Phys. Mech. Astron. 58(11), 114208 (2015).
[Crossref]

Guo, Y. J.

Y. J. Guo, K. Li, W. J. Nie, and Y. Li, “Electromagnetically-induced-transparency-like ground-state cooling in a double-cavity optomechanical system,” Phys. Rev. A 90(5), 053841 (2014).
[Crossref]

Habraken, S. J. M.

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109(1), 013603 (2012).
[Crossref] [PubMed]

Hammerer, K.

S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460(7256), 724–727 (2009).
[Crossref] [PubMed]

Harris, S. E.

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
[Crossref]

He, Y.

Y. He, “Optomechanically induced transparency associated with steady-state entanglement,” Phys. Rev. A 91(1), 013827 (2015).
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Wang, H.

L. Tian and H. Wang, “Optical wavelength conversion of quantum states with optomechanics,” Phys. Rev. A 82(5), 053806 (2010).
[Crossref]

Wang, L. L.

X. G. Zhuang, L. L. Wang, Q. Chen, X. Y. Wu, and J. X. Fang, “Identification of green tea origins by near-infrared (NIR) spectroscopy and different regression tools,” Sci. China Technol. Sci. 60(1), 84–90 (2017).
[Crossref]

Wang, Q.

Q. Wang, J. Q. Zhang, P. C. Ma, C. M. Yao, and M. Feng, “Precision measurement of the environmental temperature by tunable double optomechanically induced transparency with a squeezed field,” Phys. Rev. A 91(6), 063827 (2015).
[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(3), 033829 (2015).
[Crossref]

Wang, Y. D.

Y. D. Wang and A. A. Clerk, “Using interference for high fidelity quantum state transfer in optomechanics,” Phys. Rev. Lett. 108(15), 153603 (2012).
[Crossref] [PubMed]

Wei, L. F.

B. P. Hou, L. F. Wei, and S. J. Wang, “Optomechanically induced transparency and absorption in hybridized optomechanical systems,” Phys. Rev. A 92(3), 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(4), 043843 (2015).
[Crossref]

Weis, S.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330(6010), 1520–1523 (2010).
[Crossref] [PubMed]

Winger, M.

A. H. Safavi-Naeini, T. P. Mayer 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(7341), 69–73 (2011).
[Crossref] [PubMed]

Wong, C. W.

Y. C. Liu, Y. F. Xiao, X. Luan, and C. W. Wong, “Optomechanically-induced-transparency cooling of massive mechanical resonators to the quantum ground state,” Sci. China Phys. Mech. Astron. 58, 1–6 (2015).

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(4), 043804 (2016).
[Crossref]

Wu, Q. Q.

J. Q. Liao, Q. Q. Wu, and F. Nori, “Entangling two macroscopic mechanical mirrors in a two-cavity optomechanical system,” Phys. Rev. A 89(1), 014302 (2014).
[Crossref]

Wu, X. Y.

X. G. Zhuang, L. L. Wang, Q. Chen, X. Y. Wu, and J. X. Fang, “Identification of green tea origins by near-infrared (NIR) spectroscopy and different regression tools,” Sci. China Technol. Sci. 60(1), 84–90 (2017).
[Crossref]

Wu, Y.

H. Xiong, Y. M. Huang, L. L. Wan, and Y. Wu, “Vector cavity optomechanics in the parameter configuration of optomechanically induced transparency,” Phys. Rev. A 94(1), 013816 (2016).
[Crossref]

J. Ma, C. You, L. G. Si, H. Xiong, J. Li, X. Yang, and Y. Wu, “Optomechanically induced transparency in the presence of an external time-harmonic-driving force,” Sci. Rep. 5, 11278 (2015).
[Crossref] [PubMed]

J. Ma, C. You, L. G. Si, H. Xiong, X. Yang, and Y. Wu, “Optomechanically induced transparency in the mechanical-mode splitting regime,” Opt. Lett. 39(14), 4180–4183 (2014).
[Crossref] [PubMed]

X. Y. Lü, W. M. Zhang, S. Ashhab, Y. Wu, and F. Nori, “Quantum-criticality-induced strong Kerr nonlinearities in optomechanical systems,” Sci. Rep. 3, 2943 (2013).
[Crossref] [PubMed]

H. Xiong, L. G. Si, A. S. Zheng, X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86(1), 013815 (2012).
[Crossref]

Xiao, Y.

P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90(4), 043825 (2014).
[Crossref]

Xiao, Y. F.

Y. C. Liu, Y. F. Xiao, X. Luan, and C. W. Wong, “Optomechanically-induced-transparency cooling of massive mechanical resonators to the quantum ground state,” Sci. China Phys. Mech. Astron. 58, 1–6 (2015).

Xiong, H.

H. Xiong, Y. M. Huang, L. L. Wan, and Y. Wu, “Vector cavity optomechanics in the parameter configuration of optomechanically induced transparency,” Phys. Rev. A 94(1), 013816 (2016).
[Crossref]

J. Ma, C. You, L. G. Si, H. Xiong, J. Li, X. Yang, and Y. Wu, “Optomechanically induced transparency in the presence of an external time-harmonic-driving force,” Sci. Rep. 5, 11278 (2015).
[Crossref] [PubMed]

J. Ma, C. You, L. G. Si, H. Xiong, X. Yang, and Y. Wu, “Optomechanically induced transparency in the mechanical-mode splitting regime,” Opt. Lett. 39(14), 4180–4183 (2014).
[Crossref] [PubMed]

H. Xiong, L. G. Si, A. S. Zheng, X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86(1), 013815 (2012).
[Crossref]

Xu, X. W.

X. W. Xu and Y. Li, “Controllable optical output fields from an optomechanical system with a mechanical driving,” Phys. Rev. A 92(2), 023855 (2015).
[Crossref]

Xu, Y.

J. Q. Zhang, Y. Li, M. Feng, and Y. Xu, “Precision measurement of electrical charge with optomechanically induced transparency,” Phys. Rev. A 86(5), 053806 (2012).
[Crossref]

Yamaguchi, H.

H. Okamoto, A. Gourgout, C. Y. Chang, K. Onomitsu, I. Mahboob, E. Y. Chang, and H. Yamaguchi, “Coherent phonon manipulation in coupled mechanical resonators,” Nat. Phys. 9(8), 480–484 (2013).
[Crossref]

H. Okamoto, T. Kamada, K. Onomitsu, I. Mahboob, and H. Yamaguchi, “Optical tuning of coupled micromechanical resonators,” Appl. Phys. Express 2, 062202 (2009).
[Crossref]

Yang, X.

J. Ma, C. You, L. G. Si, H. Xiong, J. Li, X. Yang, and Y. Wu, “Optomechanically induced transparency in the presence of an external time-harmonic-driving force,” Sci. Rep. 5, 11278 (2015).
[Crossref] [PubMed]

J. Ma, C. You, L. G. Si, H. Xiong, X. Yang, and Y. Wu, “Optomechanically induced transparency in the mechanical-mode splitting regime,” Opt. Lett. 39(14), 4180–4183 (2014).
[Crossref] [PubMed]

H. Xiong, L. G. Si, A. S. Zheng, X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86(1), 013815 (2012).
[Crossref]

Yao, C. M.

Q. Wang, J. Q. Zhang, P. C. Ma, C. M. Yao, and M. Feng, “Precision measurement of the environmental temperature by tunable double optomechanically induced transparency with a squeezed field,” Phys. Rev. A 91(6), 063827 (2015).
[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(4), 3225–3228 (1999).
[Crossref]

You, C.

J. Ma, C. You, L. G. Si, H. Xiong, J. Li, X. Yang, and Y. Wu, “Optomechanically induced transparency in the presence of an external time-harmonic-driving force,” Sci. Rep. 5, 11278 (2015).
[Crossref] [PubMed]

J. Ma, C. You, L. G. Si, H. Xiong, X. Yang, and Y. Wu, “Optomechanically induced transparency in the mechanical-mode splitting regime,” Opt. Lett. 39(14), 4180–4183 (2014).
[Crossref] [PubMed]

Yu, H. L.

C. Jiang, Y. S. Cui, X. T. Bian, F. Zuo, H. L. Yu, and G. B. Chen, “Phase-dependent multiple optomechanically induced absorption in multimode optomechanical systems with mechanical driving,” Phys. Rev. A 94(2), 023837 (2016).
[Crossref]

Zeilinger, A.

D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
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Zhang, J. Q.

Q. Wang, J. Q. Zhang, P. C. Ma, C. M. Yao, and M. Feng, “Precision measurement of the environmental temperature by tunable double optomechanically induced transparency with a squeezed field,” Phys. Rev. A 91(6), 063827 (2015).
[Crossref]

P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90(4), 043825 (2014).
[Crossref]

J. Q. Zhang, Y. Li, M. Feng, and Y. Xu, “Precision measurement of electrical charge with optomechanically induced transparency,” Phys. Rev. A 86(5), 053806 (2012).
[Crossref]

Zhang, W. M.

X. Y. Lü, W. M. Zhang, S. Ashhab, Y. Wu, and F. Nori, “Quantum-criticality-induced strong Kerr nonlinearities in optomechanical systems,” Sci. Rep. 3, 2943 (2013).
[Crossref] [PubMed]

Zhang, Z. M.

P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90(4), 043825 (2014).
[Crossref]

Zheng, A. S.

H. Xiong, L. G. Si, A. S. Zheng, X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86(1), 013815 (2012).
[Crossref]

Zhou, Z. H.

Z. H. Zhou, F. J. Shu, Z. Shen, C. H. Dong, and G. C. Guo, “High-Q whispering gallery modes in a polymer microresonator with broad strain tuning,” Sci. China Phys. Mech. Astron. 58(11), 114208 (2015).
[Crossref]

Zhuang, X. G.

X. G. Zhuang, L. L. Wang, Q. Chen, X. Y. Wu, and J. X. Fang, “Identification of green tea origins by near-infrared (NIR) spectroscopy and different regression tools,” Sci. China Technol. Sci. 60(1), 84–90 (2017).
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Zoller, P.

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109(1), 013603 (2012).
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Zuo, F.

C. Jiang, Y. S. Cui, X. T. Bian, F. Zuo, H. L. Yu, and G. B. Chen, “Phase-dependent multiple optomechanically induced absorption in multimode optomechanical systems with mechanical driving,” Phys. Rev. A 94(2), 023837 (2016).
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Ann. Phys. (1)

P. Meystre, “A short walk through quantum optomechanics,” Ann. Phys. 525(3), 215–233 (2013).
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Appl. Phys. Express (1)

H. Okamoto, T. Kamada, K. Onomitsu, I. Mahboob, and H. Yamaguchi, “Optical tuning of coupled micromechanical resonators,” Appl. Phys. Express 2, 062202 (2009).
[Crossref]

Nat. Commun. (1)

J. T. Hill, A. H. Safavi-Naeini, J. Chan, and O. Painter, “Coherent optical wavelength conversion via cavity optomechanics,” Nat. Commun. 3, 1196 (2012).
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Nat. Photonics (2)

Y. Sato, Y. Tanaka, J. Upham, Y. Takahashi, T. Asano, and S. Noda, “Strong coupling between distant photonic nanocavities and its dynamic control,” Nat. Photonics 6(1), 56–61 (2011).
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K. J. Fang, M. H. Matheny, X. Luan, and O. Painter, “Optical transduction and routing of microwave phonons in cavity-optomechanical circuits,” Nat. Photonics 10(7), 489–496 (2016).
[Crossref]

Nat. Phys. (1)

H. Okamoto, A. Gourgout, C. Y. Chang, K. Onomitsu, I. Mahboob, E. Y. Chang, and H. Yamaguchi, “Coherent phonon manipulation in coupled mechanical resonators,” Nat. Phys. 9(8), 480–484 (2013).
[Crossref]

Nature (3)

S. Gröblacher, K. Hammerer, M. R. Vanner, and M. Aspelmeyer, “Observation of strong coupling between a micromechanical resonator and an optical cavity field,” Nature 460(7256), 724–727 (2009).
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A. H. Safavi-Naeini, T. P. Mayer 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(7341), 69–73 (2011).
[Crossref] [PubMed]

J. Chan, T. P. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
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Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. A (20)

K. Qu and G. S. Agarwal, “Phonon mediated electromagnetically induced absorption in hybrid opto-electro mechanical systems,” Phys. Rev. A 87(3), 031802 (2013).
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Y. J. Guo, K. Li, W. J. Nie, and Y. Li, “Electromagnetically-induced-transparency-like ground-state cooling in a double-cavity optomechanical system,” Phys. Rev. A 90(5), 053841 (2014).
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Y. He, “Optomechanically induced transparency associated with steady-state entanglement,” Phys. Rev. A 91(1), 013827 (2015).
[Crossref]

H. Xiong, L. G. Si, A. S. Zheng, X. Yang, and Y. Wu, “Higher-order sidebands in optomechanically induced transparency,” Phys. Rev. A 86(1), 013815 (2012).
[Crossref]

P. C. Ma, J. Q. Zhang, Y. Xiao, M. Feng, and Z. M. Zhang, “Tunable double optomechanically induced transparency in an optomechanical system,” Phys. Rev. A 90(4), 043825 (2014).
[Crossref]

Q. Wang, J. Q. Zhang, P. C. Ma, C. M. Yao, and M. Feng, “Precision measurement of the environmental temperature by tunable double optomechanically induced transparency with a squeezed field,” Phys. Rev. A 91(6), 063827 (2015).
[Crossref]

S. Shahidani, M. H. Naderi, and M. Soltanolkotabi, “Control and manipulation of electromagentically induced transparency in a nonlinear optomechanical system with two movable mirrors,” Phys. Rev. A 88(5), 053813 (2013).
[Crossref]

J. Q. Zhang, Y. Li, M. Feng, and Y. Xu, “Precision measurement of electrical charge with optomechanically induced transparency,” Phys. Rev. A 86(5), 053806 (2012).
[Crossref]

B. P. Hou, L. F. Wei, and S. J. Wang, “Optomechanically induced transparency and absorption in hybridized optomechanical systems,” Phys. Rev. A 92(3), 033829 (2015).
[Crossref]

H. Xiong, Y. M. Huang, L. L. Wan, and Y. Wu, “Vector cavity optomechanics in the parameter configuration of optomechanically induced transparency,” Phys. Rev. A 94(1), 013816 (2016).
[Crossref]

S. Huang and G. S. Agarwal, “Electromagnetically induced transparency from two-phonon processes in quadratically coupled membranes,” Phys. Rev. A 83(2), 023823 (2011).
[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(4), 043804 (2016).
[Crossref]

J. Q. Liao, Q. Q. Wu, and F. Nori, “Entangling two macroscopic mechanical mirrors in a two-cavity optomechanical system,” Phys. Rev. A 89(1), 014302 (2014).
[Crossref]

G. S. Agarwal and S. Huang, “The electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81(4), 041803 (2010).
[Crossref]

L. Tian and H. Wang, “Optical wavelength conversion of quantum states with optomechanics,” Phys. Rev. A 82(5), 053806 (2010).
[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(4), 043843 (2015).
[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(4), 3225–3228 (1999).
[Crossref]

X. W. Xu and Y. Li, “Controllable optical output fields from an optomechanical system with a mechanical driving,” Phys. Rev. A 92(2), 023855 (2015).
[Crossref]

H. Suzuki, E. Brown, and R. Sterling, “Nonlinear dynamics of an optomechanical system with a coherent mechanical pump: Second-order sideband generation,” Phys. Rev. A 92(3), 033823 (2015).
[Crossref]

C. Jiang, Y. S. Cui, X. T. Bian, F. Zuo, H. L. Yu, and G. B. Chen, “Phase-dependent multiple optomechanically induced absorption in multimode optomechanical systems with mechanical driving,” Phys. Rev. A 94(2), 023837 (2016).
[Crossref]

Phys. Rev. Lett. (11)

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109(1), 013603 (2012).
[Crossref] [PubMed]

J. Q. Liao and L. Tian, “Macroscopic quantum superposition in cavity optomechanics,” Phys. Rev. Lett. 116(16), 163602 (2016).
[Crossref] [PubMed]

Y. D. Wang and A. A. Clerk, “Using interference for high fidelity quantum state transfer in optomechanics,” Phys. Rev. Lett. 108(15), 153603 (2012).
[Crossref] [PubMed]

P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107(6), 063601 (2011).
[Crossref] [PubMed]

A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-photon optomechanics,” Phys. Rev. Lett. 107(6), 063602 (2011).
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M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Rev. Lett. 109(6), 063601 (2012).
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D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, “Optomechanical entanglement between a movable mirror and a cavity field,” Phys. Rev. Lett. 98(3), 030405 (2007).
[Crossref] [PubMed]

A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. Alegre, A. Krause, and O. Painter, “Observation of quantum motion of a nanomechanical resonator,” Phys. Rev. Lett. 108(3), 033602 (2012).
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M. A. Lemonde, N. Didier, and A. A. Clerk, “Nonlinear interaction effects in a strongly driven optomechanical cavity,” Phys. Rev. Lett. 111(5), 053602 (2013).
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K. Børkje, A. Nunnenkamp, J. D. Teufel, and S. M. Girvin, “Signatures of nonlinear cavity optomechanics in the weak coupling regime,” Phys. Rev. Lett. 111(5), 053603 (2013).
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A. Kronwald and F. Marquardt, “Optomechanically induced transparency in the nonlinear quantum regime,” Phys. Rev. Lett. 111(13), 133601 (2013).
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Phys. Today (1)

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
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Rev. Mod. Phys. (1)

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
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Sci. China Phys. Mech. Astron. (3)

Y. C. Liu, Y. F. Xiao, X. Luan, and C. W. Wong, “Optomechanically-induced-transparency cooling of massive mechanical resonators to the quantum ground state,” Sci. China Phys. Mech. Astron. 58, 1–6 (2015).

M. Gao, F. C. Lei, C. G. Du, and G. L. Long, “Dynamics and entanglement of a membrane-in-the-middle optomechanical system in the extremely-large-amplitude regime,” Sci. China Phys. Mech. Astron. 59(1), 610301 (2016).
[Crossref]

Z. H. Zhou, F. J. Shu, Z. Shen, C. H. Dong, and G. C. Guo, “High-Q whispering gallery modes in a polymer microresonator with broad strain tuning,” Sci. China Phys. Mech. Astron. 58(11), 114208 (2015).
[Crossref]

Sci. China Technol. Sci. (1)

X. G. Zhuang, L. L. Wang, Q. Chen, X. Y. Wu, and J. X. Fang, “Identification of green tea origins by near-infrared (NIR) spectroscopy and different regression tools,” Sci. China Technol. Sci. 60(1), 84–90 (2017).
[Crossref]

Sci. Rep. (2)

X. Y. Lü, W. M. Zhang, S. Ashhab, Y. Wu, and F. Nori, “Quantum-criticality-induced strong Kerr nonlinearities in optomechanical systems,” Sci. Rep. 3, 2943 (2013).
[Crossref] [PubMed]

J. Ma, C. You, L. G. Si, H. Xiong, J. Li, X. Yang, and Y. Wu, “Optomechanically induced transparency in the presence of an external time-harmonic-driving force,” Sci. Rep. 5, 11278 (2015).
[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(6010), 1520–1523 (2010).
[Crossref] [PubMed]

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Figures (11)

Fig. 1
Fig. 1 Schematic diagram of a mechanically coupled optomechanical system, in which the two coupled nanomechanical resonators are driven by the time-dependent forces, respectively.
Fig. 2
Fig. 2 The absorption Re[εT] as a function of σ/ωm1 for different values of the mechanical interaction: V = 0 (black, solid curve); V = 1.5 × 8 × 105 Hz/m2 (red, dashed curve); V = 3 × 8 × 105Hz/m2 (blue, dotted curve). The other values of the parameters are given by: PL = 2μW, m1 = m2 = 145ng, κ = 2π × 215 × 103Hz, ωm1 = ωm2 = 2π × 947 × 103Hz, γ1 = γ2 = 2π × 140Hz, L = 25mm, D1 = 0, D2 = 0, ϕ1 = 0 and ϕ2 = 0.
Fig. 3
Fig. 3 The level-diagram picture constructed by the optical and mechanical modes (a), and the dressed mode picture with double Λ configuration (b).
Fig. 4
Fig. 4 The absorption Re[εT ] as a function of σ/ωm1 with V = 1.5 × 8 × 105 Hz/m2 for different values of the external force D1: D1 = 0 (black, solid curve); D1 = 1 × 109N (red, dashed curve); D1 = 3 × 109N (blue, dotted curve). The other values of the parameters are set with the same values as in Fig. 2.
Fig. 5
Fig. 5 The absorption Re[εT] as a function of D1 at σ = −0.2ωm1. The other values of the parameters are set with the same values as in Fig. 4.
Fig. 6
Fig. 6 The absorption Re[εT] as a function of σ /ωm1 with V = 1.5 × 8 × 105 Hz/m2 for different values of the external force D2: D2 = 0 (black, solid curve); D2 = 1 × 109N (red, dashed curve); D2 = 3 × 109N (blue, dotted curve). The other values of the parameters are set with the same values as in Fig. 2.
Fig. 7
Fig. 7 The absorptions Re[εT ] as a function of D2 at σ = −0.2ωm1 (black, solid curve) and σ = 0.2ωm1 (red, dashed curve). The other values of the parameters are set with the same values as in Fig. 6.
Fig. 8
Fig. 8 The absorption Re[εT ] as a function of σ /ωm1 with D1 = 1 × 109N for different values of the phase ϕ1: ϕ1 = 0 (black, solid curve); ϕ1 = π/2 (red, dashed curve); ϕ1 = π (blue, dotted curve). The other values of the parameters are set with the same values as in Fig. 3.
Fig. 9
Fig. 9 The absorption Re[εT] as a function of ϕ1 at σ = −0.2ωm1. The other values of the parameters are set with the same values as in Fig. 8. The dotted line denotes the X-axis.
Fig. 10
Fig. 10 The absorption Re[εT] as a function of σ /ωm1 for different values of the phase ϕ2 with D2 = 1 × 109N: ϕ2 = 0 (black, solid curve); ϕ2 = π/2 (red, dashed curve); ϕ2 = π (blue, dotted curve). The other values of the parameters are set with the same values as in Fig. 6.
Fig. 11
Fig. 11 The absorption Re[εT] as a function of ϕ2: σ = −0.2ωm1 (black, solid curve); σ = 0.2ωm1 (red, dashed curve). The other values of the parameters are set with the same values as in Fig. 10. The dotted line denotes the X-axis.

Equations (22)

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H= Δ c a ^ + a ^ + ω m1 b ^ 1 + b ^ 1 + ω m2 b ^ 2 + b ^ 2 +g a ^ + a ^ ( b ^ 1 + + b ^ 1 )+V( b ^ 1 + b ^ 2 + b ^ 1 b ^ 2 + ) + i D 1 ( b ^ 1 + e i ϕ 1 e i ω D 1 t b ^ 1 e i ϕ 1 e i ω D 1 t )+i D 2 ( b ^ 2 + e i ϕ 2 e i ω D 2 t b ^ 2 e i ϕ 2 e i ω D 2 t ) + i E L ( a ^ + a ^ )+i E P ( a ^ + e iδt H.c.).
d a ^ dt ={ κ+i[ Δ c +g( b ^ 1 + + b ^ 1 ) ] } a ^ + E P e -iδt + E L + 2κ a ^ in ,
d b ^ 1 dt =( γ 1 +i ω m1 ) b ^ 1 ig a ^ + a ^ iV b ^ 2 + D 1 e i ϕ 1 e i ω D 1 t + 2 γ 1 b ^ 1,in ,
d b ^ 2 dt =( γ 2 +i ω m2 ) b ^ 2 iV b ^ 1 + D 2 e i ϕ 2 e i ω D 2 t + 2 γ 2 b ^ 2,in .
a s = E L κ+iΔ ,
b 1s = ig E L 2 ( γ 1 +i ω m1 + V 2 γ 2 +i ω m2 )( κ 2 + Δ 2 ) ,
b 2s = Vg E L 2 [ ( γ 1 +i ω m1 )( γ 2 +i ω m2 )+ V 2 ]( κ 2 + Δ 2 ) ,
δ a ˙ =( κ+iΔ )δaig( δ b 1 + +δ b 1 ) a s + E P e iδt ,
δ b ˙ 1 =( γ 1 +i ω m1 )δ b 1 ig( a s δa+δ a a s )iVδ b 2 + D 1 e i ϕ 1 e i ω D 1 t ,
δ b ˙ 2 =( γ 2 +i ω m2 )δ b 2 iVδ b 1 + D 2 e i ϕ 2 e i ω D 2 t .
δa= a 1+ e iδt + a 1 e iδt + a 11+ e i ω D 1 t + a 11 e i ω D 1 t + a 12+ e i ω D 2 t + a 12 e i ω D 2 t ,
δ b 1 = b 1+ e iδt + b 1 e iδt + b 11+ e i ω D 1 t + b 11 e i ω D 1 t + b 12+ e i ω D 2 t + b 12 e i ω D 2 t ,
δ b 2 = b 2+ e iδt + b 2 e iδt + b 21+ e i ω D 1 t + b 21 e i ω D 1 t + b 22+ e i ω D 2 t + b 22 e i ω D 2 t .
a 1+ = E P ( 1 M 2 M 1 M 3 1 M 4 ) Γ 1 ,
a 11+ = igD 1 e i ϕ 1 [ a s M 5 ( 1 M 8 ) G 3 Γ 21+ + a s G 3 Γ 21 ] 1 M 6 M 5 M 7 1 M 8 ,
a 12+ = gV D 2 e i ϕ 2 Γ 62 [ a s M 9 ( 1 M 12 ) G 5 Γ 22+ + a s G 5 Γ 22 ] 1 M 10 M 9 M 11 1 M 12 .
M 1 = g 2 a s 2 Γ 1 ( 1 G 1+ 1 G 1 ), M 2 = g 2 | a s | 2 Γ 1 ( 1 G 1+ 1 G 1 ), M 3 = g 2 a s 2 Γ 1+ ( 1 G 1 1 G 1+ ), M 4 = g 2 | a s | 2 Γ 1+ ( 1 G 1 1 G 1+ ), M 5 = g 2 a s 2 Γ 21 ( 1 G 21+ 1 G 21 ), M 6 = g 2 | a s | 2 Γ 21 ( 1 G 21+ 1 G 21 ), M 7 = g 2 a s 2 Γ 21+ ( 1 G 21 1 G 21+ ), M 8 = g 2 | a s | 2 Γ 21+ ( 1 G 21 1 G 21+ ), M 9 = g 2 a s 2 Γ 22 ( 1 G 22+ 1 G 22 ), M 10 = g 2 | a s | 2 Γ 22 ( 1 G 22+ 1 G 22 ), M 11 = g 2 a s 2 Γ 22+ ( 1 G 22 1 G 22+ ), M 12 = g 2 | a s | 2 Γ 22+ ( 1 G 22 1 G 22+ ),
G 1± = Γ 3± + V 2 Γ 5± , G 2j± = Γ 4j± + V 2 Γ 6j± , (j=1,2)
Γ 1± =κ±i( δ±Δ ), Γ 2j± =κ±i( ω D j ±Δ ), Γ 3± = γ 1 +i( ω m 1 ±δ ), Γ 4j± = γ 1 +i( ω m 1 ± ω D j ), Γ 5± = γ 2 +i( ω m 2 ±Δ ), Γ 6j± = γ 2 +i( ω m 2 ± ω D j ).
ε T = 2κ( a 1+ + a 11+ + a 12+ ) E p .
H= Δ c a ^ + a ^ + ω m b ^ + b ^ + ω m+ b ^ + + b ^ + + 1 2 g a ^ + a ^ ( b ^ + + + b ^ + )+ 1 2 g a ^ + a ^ ( b ^ + + b ^ ) + i E L ( a ^ + a ^ )+i E P ( a ^ + e iδt H.c.),
H= [ Δ c g ω m+ ( D 1 sin( ϕ 1 + ω D 1 t ) D 2 sin( ϕ 2 + ω D 2 t ) ) g ω m ( D 1 sin( ϕ 1 + ω D 1 t )+ D 2 sin( ϕ 2 + ω D 2 t ) ) ] a ^ + a ^ + j=+, ω mj B ^ j + B ^ j + 1 2 g a ^ + a ^ j=+, ( B ^ j + + B ^ j ) + i( E L a ^ + + E P a ^ + e iδt H.c.),

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