Local modulation of double optomechanically induced transparency and amplification

Q. Yang; B. P. Hou; D. G. Lai

doi:10.1364/OE.25.009697

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 E_L = (2P_Lκ/ħω_L)^1/2 with frequency ω_L = 2πc/λ (λ is the wavelength) and a weak probe field E_P = (2P_Pκ/ħω_P)^1/2 with frequency ω_p, in which P_i (i = L, p) denotes its power. The two NRs are driven by the external time-dependent forces D₁ (D₂) with frequency ω_D1 (ω_D2) and initial phase ϕ₁ (ϕ₂), respectively. The resonance frequency and decay rate of the optomechanical cavity field with optical mode $\hat{a}$ are denoted by ω_c and κ, respectively. And ${\hat{b}}_{i} ({\hat{b}}_{i}^{+})$ (i = 1, 2) is the annihilation (creation) operator for the vibration mode of the charged NRi (i = 1, 2) with frequency ω_mi and effective mass m_i. The Hamiltonian of the two-mechanical-mode optomechanical system in the rotating frame at the frequency ω_L reads [38,39,44,45]

\begin{array}{l} H = ℏ Δ_{c} {\hat{a}}^{+} \hat{a} + ℏ ω_{m 1} {\hat{b}}_{1}^{+} {\hat{b}}_{1} + ℏ ω_{m 2} {\hat{b}}_{2}^{+} {\hat{b}}_{2} + ℏ g {\hat{a}}^{+} \hat{a} ({\hat{b}}_{1}^{+} + {\hat{b}}_{1}) + ℏ V ({\hat{b}}_{1}^{+} {\hat{b}}_{2} + {\hat{b}}_{1} {\hat{b}}_{2}^{+}) \\ + i ℏ D_{1} ({\hat{b}}_{1}^{+} e^{- i ϕ_{1}} e^{- i ω_{D_{1}} t} - {\hat{b}}_{1}^{} e^{i ϕ_{1}} e^{i ω_{D_{1}} t}) + i ℏ D_{2} ({\hat{b}}_{2}^{+} e^{- i ϕ_{2}} e^{- i ω_{D_{2}} t} - {\hat{b}}_{2}^{} e^{i ϕ_{2}} e^{i ω_{D_{2}} t}) \\ + i ℏ E_{L} ({\hat{a}}^{+} - \hat{a}) + i ℏ E_{P} ({\hat{a}}^{+} e^{- i δ t} - H . c .) . \end{array}

Here Δ_c = ω_c − ω_L is the detuning of the coupling field from the cavity, and δ = ω_p − ω_L 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/2m₁ω_m1)^1/2ω_c /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 ω_m1 + ω_m2 [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.

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:

\frac{d \hat{a}}{d t} = - {κ + i [Δ_{c} + g ({\hat{b}}_{1}^{+} + {\hat{b}}_{1})]} \hat{a} + E_{P} e^{- i δ t} + E_{L} + \sqrt{2 κ} {\hat{a}}_{in},

\frac{d {\hat{b}}_{1}}{d t} = - (γ_{1} + i ω_{m 1}) {\hat{b}}_{1} - i g {\hat{a}}^{+} \hat{a} - i V {\hat{b}}_{2} + D_{1} e^{- i ϕ_{1}} e^{- i ω_{D_{1}} t} + \sqrt{2 γ_{1}} {\hat{b}}_{1, i n},

\frac{d {\hat{b}}_{2}}{d t} = - (γ_{2} + i ω_{m 2}) {\hat{b}}_{2} - i V {\hat{b}}_{1} + D_{2} e^{- i ϕ_{2}} e^{- i ω_{D_{2}} t} + \sqrt{2 γ_{2}} {\hat{b}}_{2, i n} .

Here,

{\hat{a}}_{i n}

is the input vacuum in the cavity with zero mean value. And, γ_i (i = 1, 2) and

{\hat{b}}_{i, i n}

are the damping rate and the Langevin force coming from the interaction of the ith NR with its environment. Using

\hat{a} = a_{s} + δ \hat{a}

and

{\hat{b}}_{i} = b_{i s} + δ {\hat{b}}_{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

a_{s} = \frac{E_{L}}{κ + i Δ},

b_{1 s} = \frac{- i g E_{L}^{2}}{(γ_{1} + i ω_{m 1} + \frac{V^{2}}{γ_{2} + i ω_{m 2}}) (κ^{2} + Δ^{2})},

b_{2 s} = \frac{- V g E_{L}^{2}}{[(γ_{1} + i ω_{m 1}) (γ_{2} + i ω_{m 2}) + V^{2}] (κ^{2} + Δ^{2})},

where Δ = Δ_c + g(

b_{1 s}^{*}

+

b_{1 s}

) 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 (

δ a

,

δ b_{1}

,

δ b_{2}

) of the fluctuations (

δ \hat{a}

,

δ {\hat{b}}_{1}

,

δ {\hat{b}}_{2}

) can be given by

δ \dot{a} = - (κ + i Δ) δ a - i g (δ b_{1}^{+} + δ b_{1}) a_{s} + E_{P} e^{- i δ t},

δ {\dot{b}}_{1} = - (γ_{1} + i ω_{m 1}) δ b_{1} - i g (a_{s}^{*} δ a + δ a^{*} a_{s}) - i V δ b_{2} + D_{1} e^{- i ϕ_{1}} e^{- i ω_{D_{1}} t},

δ {\dot{b}}_{2} = - (γ_{2} + i ω_{m 2}) δ b_{2} - i V δ b_{1} + D_{2} e^{- i ϕ_{2}} e^{- i ω_{D_{2}} t} .

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 ω_L as follows [42]:

δ a = a_{1 +} e^{- i δ t} + a_{1 -} e^{i δ t} + 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^{- i δ t} + b_{1 -} e^{i δ t} + 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^{- i δ t} + b_{2 -} e^{i δ t} + 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} .

To investigate the optical property of the optomechanical system, the experimentally accessible transmitted field amplitude a_out(t) can be calculated by using the input–output relation a_out(t) + a_in(t) = (2κ)^1/2 a(t), in which the mean value of a_in(t) equals zero. In the present paper, we assume that the frequencies ω_Di (i = 1, 2) of the external forces are detuned by zero from probe field, i.e. ω_Di = δ, then the corresponding components of the output field are obtained as

a_{1 +} = \frac{E_{P}}{(1 - M_{2} - \frac{M_{1} M_{3}^{*}}{1 - M_{4}^{*}}) Γ_{1 -}},

a_{11 +} = \frac{- {igD}_{1} e^{- i ϕ_{1}} [\frac{a_{s}^{*} M_{5}}{(1 - M_{8}^{*}) G_{3} Γ_{21 +}^{*}} + \frac{a_{s}}{G_{3} Γ_{21 -}}]}{1 - M_{6} - \frac{M_{5} M_{7}^{*}}{1 - M_{8}^{*}}},

a_{12 +} = \frac{- \frac{g V D_{2} e^{- i ϕ_{2}}}{Γ_{62 -}} [\frac{a_{s}^{*} M_{9}}{(1 - M_{12}^{*}) G_{5} Γ_{22 +}^{*}} + \frac{a_{s}}{G_{5} Γ_{22 -}}]}{1 - M_{10} - \frac{M_{9} M_{11}^{*}}{1 - M_{12}^{*}}} .

Here,

\begin{array}{l} M_{1} = \frac{- g^{2} a_{s}^{2}}{Γ_{1 -}} (\frac{1}{G_{1 +}^{*}} - \frac{1}{G_{1 -}}), M_{2} = \frac{g^{2} {| a_{s} |}^{2}}{Γ_{1 -}} (\frac{1}{G_{1 +}^{*}} - \frac{1}{G_{1 -}}), \\ M_{3} = \frac{- g^{2} a_{s}^{2}}{Γ_{1 +}} (\frac{1}{G_{1 -}^{*}} - \frac{1}{G_{1 +}}), M_{4} = \frac{g^{2} {| a_{s} |}^{2}}{Γ_{1 +}} (\frac{1}{G_{1 -}^{*}} - \frac{1}{G_{1 +}}), \\ M_{5} = \frac{- g^{2} a_{s}^{2}}{Γ_{21 -}} (\frac{1}{G_{21 +}^{*}} - \frac{1}{G_{21 -}}), M_{6} = \frac{g^{2} {| a_{s} |}^{2}}{Γ_{21 -}} (\frac{1}{G_{21 +}^{*}} - \frac{1}{G_{21 -}}), \\ M_{7} = \frac{- g^{2} a_{s}^{2}}{Γ_{21 +}} (\frac{1}{G_{21 -}^{*}} - \frac{1}{G_{21 +}}), M_{8} = \frac{g^{2} {| a_{s} |}^{2}}{Γ_{21 +}} (\frac{1}{G_{21 -}^{*}} - \frac{1}{G_{21 +}}), \\ M_{9} = \frac{- g^{2} a_{s}^{2}}{Γ_{22 -}} (\frac{1}{G_{22 +}^{*}} - \frac{1}{G_{22 -}}), M_{10} = \frac{g^{2} {| a_{s} |}^{2}}{Γ_{22 -}} (\frac{1}{G_{22 +}^{*}} - \frac{1}{G_{22 -}}), \\ M_{11} = \frac{- g^{2} a_{s}^{2}}{Γ_{22 +}} (\frac{1}{G_{22 -}^{*}} - \frac{1}{G_{22 +}}), M_{12} = \frac{g^{2} {| a_{s} |}^{2}}{Γ_{22 +}} (\frac{1}{G_{22 -}^{*}} - \frac{1}{G_{22 +}}), \end{array}

where

G_{1 \pm} = Γ_{3 \pm} + \frac{V^{2}}{Γ_{5 \pm}}, G_{2 j \pm} = Γ_{4 j \pm} + \frac{V^{2}}{Γ_{6 j \pm}}, (j = 1, 2)

And

\begin{array}{l} Γ_{1 \pm} = κ \pm i (δ \pm Δ), Γ_{2 j \pm} = κ \pm i (ω_{D_{j}} \pm Δ), \\ Γ_{3 \pm} = γ_{1} + i (ω_{m_{1}} \pm δ), Γ_{4 j \pm} = γ_{1} + i (ω_{m_{1}} \pm ω_{D_{j}}), \\ Γ_{5 \pm} = γ_{2} + i (ω_{m_{2}} \pm Δ), Γ_{6 j \pm} = γ_{2} + i (ω_{m_{2}} \pm ω_{D_{j}}) . \end{array}

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]

ε_{T} = \frac{2 κ (a_{1 +} + a_{11 +} + a_{12 +})}{E_{p}} .

The real part Re[ε_T] 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, ω_m1 = ω_m2 = 2π × 947 × 10³Hz, ω_c = 1.77 × 10¹⁵Hz, λ = 1064nm, E_p = E_L/10 and L = 25mm. Under the resonance condition of Δ = ω_m1, the absorption Re[ε_T] of the output field are plotted as a function of σ = δ−ω_m1, which is redefined to display the main optical properties around the resonance δ = ω_m1.

In Fig. 2, we plot the probe absorption Re[ε_T] 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 = ω_m1). 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.

Fig. 2 The absorption Re[ε_T] as a function of σ/ω_m1 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: P_L = 2μW, m₁ = m₂ = 145ng, κ = 2π × 215 × 10³Hz, ω_m1 = ω_m2 = 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/ω_m1 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 ${\hat{b}}_{+} = ({\hat{b}}_{1} - {\hat{b}}_{2}) / \sqrt{2}$ and ${\hat{b}}_{-} = ({\hat{b}}_{1} + {\hat{b}}_{2}) / 2$ 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 ω_m+−ω_m− = 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 ω_m1 = ω_m2 = ω_m can be rewritten as

\begin{array}{l} H = ℏ Δ_{c} {\hat{a}}^{+} \hat{a} + ℏ ω_{m -} {\hat{b}}_{-}^{+} {\hat{b}}_{-} + ℏ ω_{m +} {\hat{b}}_{+}^{+} {\hat{b}}_{+} + \frac{1}{\sqrt{2}} ℏ g {\hat{a}}^{+} \hat{a} ({\hat{b}}_{+}^{+} + {\hat{b}}_{+}) + \frac{1}{\sqrt{2}} ℏ g {\hat{a}}^{+} \hat{a} ({\hat{b}}_{-}^{+} + {\hat{b}}_{-}) \\ + i ℏ E_{L} ({\hat{a}}^{+} - \hat{a}) + i ℏ E_{P} ({\hat{a}}^{+} e^{- i δ t} - H . c .), \end{array}

where ω_{m ±} = ω_m

\mp

V. The Hamiltonian in Eq. (11) is composed by two parts which describe the optomechanical interactions with coupling constants G₁₊ = g/2^1/2 and G₂₊ = g/2^1/2 between the cavity and the dressed mechanical modes

{\hat{b}}_{+}

and

{\hat{b}}_{-}

, respectively. It is well known that the OMIT occurs only when the two-photon resonance condition δ = ω_mi (with δ = ω_p−ω_L) is met [10–12]. By using the redefined tuning σ = δ −ω_mi, 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.

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[ε_T] as a function of σ /ω_m1 with V = 1.5 × 8 × 10⁵ Hz/m² under the conditions of ω_D1 = 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[ε_T] as a function of D₁ at σ = −0.2ω_m1 in Fig. 5. It is found that the OMIA amplitude monotonously increases with the force amplitude D₁.

Fig. 4 The absorption Re[ε_T ] as a function of σ/ω_m1 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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Fig. 5 The absorption Re[ε_T] as a function of D₁ at σ = −0.2ω_m1. 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.

Fig. 6 The absorption Re[ε_T] as a function of σ /ω_m1 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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Fig. 7 The absorptions Re[ε_T ] as a function of D₂ at σ = −0.2ω_m1 (black, solid curve) and σ = 0.2ω_m1 (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 2π, which is different from the variation of the absorption with the force amplitude D₁ in a monotonous decrease pattern shown in Fig. 5.

Fig. 8 The absorption Re[ε_T ] as a function of σ /ω_m1 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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Fig. 9 The absorption Re[ε_T] as a function of ϕ₁ at σ = −0.2ω_m1. 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 = −0.2ω_m1 increases with the given ϕ₂ and turns into amplification, while the maximal transparency around σ = 0.2ω_m1 decreases with the ϕ₂. The explicit dependences of the probe absorptions on the phase ϕ₂ at σ = −0.2ω_m1 and σ = 0.2ω_m1 are displayed by solid and dashed curves in Fig. 11, respectively. They both oscillate with a period of 2π but they are antiphase with each other. Additionally, it is found that the variation of the absorption at σ = 0.2ω_m1 with ϕ₂ is similar that with ϕ₁ shown in Fig. 9, but the amplification amplitude is larger.

Fig. 10 The absorption Re[ε_T] as a function of σ /ω_m1 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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Fig. 11 The absorption Re[ε_T] as a function of ϕ₂: σ = −0.2ω_m1 (black, solid curve); σ = 0.2ω_m1 (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 ${\hat{b}}_{\pm}$ . Now, we define the displaced dressed mechanical mode as ${\hat{B}}_{\pm} = {\hat{b}}_{\pm} + i (D_{1} \exp [- i (ϕ_{1} + ω_{D_{1}} t)] \mp D_{2} \exp [- i (ϕ_{2} + ω_{D_{2}} t)]) / \sqrt{2} ω_{m \pm}$ in terms of the mechanical dressed mode ${\hat{b}}_{\pm}$ . When including the external forces D₁ and D₂ applied on the system, the Hamiltonian in Eq. (11) becomes

\begin{array}{l} H = ℏ [Δ_{c} - \frac{g}{ω_{m +}} (D_{1} \sin (ϕ_{1} + ω_{D_{1}} t) - D_{2} \sin (ϕ_{2} + ω_{D_{2}} t)) \\ \begin{matrix}  \end{matrix} - \frac{g}{ω_{m -}} (D_{1} \sin (ϕ_{1} + ω_{D_{1}} t) + D_{2} \sin (ϕ_{2} + ω_{D_{2}} t))] {\hat{a}}^{+} \hat{a} \\ \begin{matrix}  \end{matrix} + ℏ \sum_{j = +, -} ω_{m j} {\hat{B}}_{j}^{+} {\hat{B}}_{j} + \frac{1}{\sqrt{2}} ℏ g {\hat{a}}^{+} \hat{a} \sum_{j = +, -} ({\hat{B}}_{j}^{+} + {\hat{B}}_{j}) \\ \begin{matrix}  \end{matrix} + i ℏ (E_{L} {\hat{a}}^{+} + E_{P} {\hat{a}}^{+} e^{- i δ t} - H . c .), \end{array}

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 ω_m- (σ = V) and ω_m+ (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 ω_m- (σ = V), which is similar to the amplitude D₁, while it suppresses the transparency at ω_m+ (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 2π 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 2π.

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).

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Local modulation of double optomechanically induced transparency and amplification

Abstract

1. Introduction

2. Model and dynamical equation

3. Double-OMIT induced by mechanical interaction

4. Double OMIA induced by time-dependent forces

5. Phase-dependent double OMIA

6. Conclusions

Funding

References and links.

Cited By

Figures (11)

Equations (22)

Optics Express