Tunable double optomechanically induced transparency in photonically and phononically coupled optomechanical systems

Zhuo Qian; Miao-Miao Zhao; Bang-Pin Hou; Yong-Hong Zhao

doi:10.1364/OE.25.033097

1. Introduction

A canonical cavity optomechanical system (OMS) is composed of two mirrors, with one fixed and another movable, coupled to the cavity field by the radiation pressure [1–3]. There are many novel phenomena in the OMS, such as laser cooling of mechanical resonators into the ground state [4–7], optomechanically induced transparency and absorption (OMIT and OMIA) [8–11], and the generation of the macroscopic quantum superposition [12]. It has also been found that the quantum-coherent coupling, where the coherent coupling rate exceeds both the optical and the mechanical decoherence rate, can be used to transfer quantum states between the optical field and the mechanical oscillator [13]. The basis-dependent maximal coherence method, posted by Yao et al, should be useful to deal with this type of problem [14]. Experimently, OMS can be realized using silicon and aluminum nitride, which offer low optical and mechanical loss [15–17].

Although most attention has been paid to the single-cavity OMS, it is of much importance to generalize OMS by integrating more optical or mechanical modes for practical applications, such as quantum information processing, quantum computing [18–20], all-optical bit storage [21], and the high precision measurement [16,22]. Specifically, the optical properties of the normal-mode splitting (NMS), tunable OMIT and OMIA phenomenona in a hybridized strongly tunnel-coupled optomechanical cavity were shown [23]. Subsequently, the OMIT in the double quadratically coupled optomechanical cavities within a common reservoir was considered [24]. Recently, a mechanical resonator is used to couple two cavity modes via radiation pressure, where a correlation can be set up and a photon-phonon entanglement can be transferred to a photon-photon one [25]. When the probe field goes beyond the linear and perturbative regime, a photonic molecule OMS composed of double cavities can generate tunable high-order sideband spectra [26–28]. In other situation where the optomechanical motion is torsional, photons can be shuttled in a multicavity optomechanical device [29]. Also, the mechanical-mode splitting of the movable mirror as well as OMIT in the splitting region in the two-mode optomechanical system were investigated [30].

Besides the two-optical-mode coupled optomechanical system, the two optomechanical cavities can also be coupled via the phononic interaction between two mechanical resonators, which generates the double OMIT phenomenon [31–33]. The distance between two OMIT tips is determined by the mechanical interaction amplitude, which can be explained by the dressed-mode picture similar to the interacting dark resonances in coherent atoms [34]. Holmes et al post a synchronization scheme of many nanomechanical resonators coupled via a cavity field, which should be used to encode and process infomation [35]. The mechanical motion can be either cooled down or amplified by the optical forces in an OMS, which is useful for the non-volatile mechanical memory [36].

Recently, the optomechanical circuits have been fabricated using two optomechanical cavities wired by silicon phononic waveguides, which mediate the couplings between both cavities and resonators [37]. The synthetic magnetism has also been realized in double-cavity and double-resonator optomechanical systems by radiation-pressure-induced parametric photonic and phononic couplings [38]. Based on the phononically and photonically coupled optomechanical system, it is natural to know whether the probe absorption spectrum is simply overlapped by the optical properties contributed from the photonic paths and those by the phononic paths, or resulted from the interference between the two coupling paths, when the photonic and phononic paths are simultaneously connected. It is found that there do exist the explicit competition and interference between the photonic and phononic paths. The competition is demonstrated by the fact that the main optical properties is induced by the path with stronger coupling. However, when the two interactions are comparable, the interference dominates the absorption spectrum explicitly.

This paper is organized as follows: In Sec. II, we give the model and solve the corresponding dynamical equations. OMIT modulated by the photonic or the phononic coupling is presented in Sec. III. Combined effects of the photonic and phononic interactions on the transparency spectra via both photonic and phononic interactions are investigated in Sec. IV. At last, our main results are summarized in Sec. V.

2. Model and dynamical equations

The optomechanical system under consideration is shown in Fig. 1. Two cavities with resonant frequency ω_ci(i = 1, 2) are coupled with tunnel strength J, and two mechanical resonators with mass m_i and frequency ω_mi are interacted with strength V. The optomechanical interactions can be mediated by the radiation pressure, while the direct couplings between the two cavities or between the two resonators can be mediated by phononic crystal waveguides in a optomechanical circuit, such as silicon nanowires, as suggested by Fang et al [37]. The phononic coupling can also be realized by the Coulomb interaction, which makes the two paths independently tunable. Cavities are driven by the control fields with amplitude being $E_{Li} = \sqrt{2 P_{Li} κ_{i} / ℏ ω_{Li}}$ , in which ω_Li and P_Li are their frequency and power. L_i and κ_i are the length and decay rate of the ith cavity. In order to detect the absorption properties of the OMS, a weak field denoted by the amplitude $E_{P} = \sqrt{2 P_{P} κ_{1} / ℏ ω_{P}}$ with power P_P and frequency ω_P, is used to probe the first cavity.

Fig. 1 Schematic diagram of the optomechanical systems composed of two cavities (â₁, â₂) and two mechanical resonators (b̂₁, b̂₂). The coupling between both cavities (J) are due to the quantum tunnelling, and the interaction between two resonators (V) is mediated by the phononic crystal waveguides. Each cavity, driven by a strong control field (E_L1, E_L2) separately, is optomechanically coupled to the corresponding resonator by g₁ or g₂. γ_m1 and γ_m2 represent the damping rates of resonators, while κ₁ and κ₂ denote the noise imposed on cavities. The first cavity is probed by a weak field E_P, and ε_out is the output field.

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The Hamiltonian of the OMS, shown in Fig. 1, in the rotating frame at the frequency ω_Li can be given as follows [37,38]

\begin{array}{l} \hat{H} & = & \sum_{i = 1, 2} ℏ Δ_{ci} {\hat{a}}_{i}^{+} {\hat{a}}_{i} + ℏ J ({\hat{a}}_{1}^{+} {\hat{a}}_{2} + {\hat{a}}_{1} {\hat{a}}_{2}^{+}) + \sum_{i = 1, 2} ℏ ω_{mi} {\hat{b}}_{i}^{+} {\hat{b}}_{i} + ℏ V ({\hat{b}}_{1}^{+} {\hat{b}}_{2} + {\hat{b}}_{1} {\hat{b}}_{2}^{+}) \\ + \sum_{j = 1, 2} ℏ g_{j} ({\hat{b}}_{j}^{+} + {\hat{b}}_{j}) {\hat{a}}_{j}^{+} {\hat{a}}_{j} + i ℏ E_{L 1} ({\hat{a}}_{1}^{+} - {\hat{a}}_{1}) + i ℏ E_{L 2} ({\hat{a}}_{2}^{+} - {\hat{a}}_{2}) \\ + i ℏ E_{P} ({\hat{a}}_{1}^{+} e^{- i δ t} - {\hat{a}}_{1} e^{i δ t}), \end{array}

where δ = ω_P − ω_L1 denotes the detuning between the first coupling field and the probe field and Δ_ci = ω_ci − ω_Li, (i = 1, 2) the detuning of the control field from the corresponding cavity. â_i(

{\hat{a}}_{i}^{†}

) is the annihilation (creation) operator of the ith cavity field, while b̂_i(

{\hat{b}}_{i}^{†}

) is the annihilation (creation) operator of the mechanical resonator located at each cavity. In Eq. (1), the first term denotes the free energies of the two optical cavities, and the second term indicates the tunnel coupling between two cavities with the strength J. The free energies of the mechanical oscillators are given by the third term, and the fourth term describes the interaction between two oscillators with coupling amplitude V. The optomechanical couplings between the cavity fields and the corresponding mechanical oscillators are displayed by the fifth term, in which the optomechanical coupling coefficient is given by

g_{i} = (ω_{ci} / L_{i}) \sqrt{\frac{ℏ}{2 m_{i} ω_{mi}}}

, (i = 1, 2) [39]. The applications of the first and second control fields on the two optomechanical cavities are given by the next two terms, and the last term describes the driving of the probe field on the first cavity.

When including the noises of the cavities and the dampings of the mechanical oscillators, the Langevin equations for the cavity and oscillator variables are given by

{\dot{\hat{a}}}_{1} = - i Δ_{c 1} {\hat{a}}_{1} - i J {\hat{a}}_{2} - i g_{1} ({\hat{b}}_{1}^{+} + {\hat{b}}_{1}) {\hat{a}}_{1} + E_{P} e^{- i δ t} + E_{L 1} - κ_{1} {\hat{a}}_{1} + \sqrt{2 κ_{1}} {\hat{a}}_{in, 1},

{\dot{\hat{a}}}_{2} = - i Δ_{c 2} {\hat{a}}_{2} - i J {\hat{a}}_{1} - i g_{2} ({\hat{b}}_{2}^{+} + {\hat{b}}_{2}) {\hat{a}}_{2} + E_{L 2} - κ_{2} {\hat{a}}_{2} + \sqrt{2 κ_{2}} {\hat{a}}_{in, 2},

{\dot{\hat{b}}}_{1} = - i ω_{m 1} {\hat{b}}_{1} - i V {\hat{b}}_{2} - i g_{1} {\hat{a}}_{1}^{+} {\hat{a}}_{1} - γ_{m 1} {\hat{b}}_{1} + ξ_{1},

{\dot{\hat{b}}}_{2} = - i ω_{m 2} {\hat{b}}_{2} - i V {\hat{b}}_{1} - i g_{2} {\hat{a}}_{2}^{+} {\hat{a}}_{2} - γ_{m 2} {\hat{b}}_{2} + ξ_{2},

where γ_mi and ξ_i are the damping rate and Langevin force arising from the interaction between the ith mechanical resonator and its environment. The input vacuum in the two cavities are denoted by â_in,i with zero mean value. Because the probe field is assumed much weaker than the control fields, the usual linearization approach in quantum optics can be adopted to get an analytical understanding of the topic addressed in the present paper. The oscillator and cavity variables can be then divided into the steady parts and the fluctuation ones as Q̂ = Q_s + δQ̂, with Q̂ = â_α, b̂_α. The steady parts a_s_α, b_s_α are obtained as

a_{s α} = \frac{E_{L α} (i Δ β + κ_{β}) - i J E_{L β}}{(i Δ_{α} + κ_{α}) (i Δ_{β} + κ_{β}) + J^{2}},

\begin{array}{l} b_{s α} & = & \frac{- V g_{β} a_{s β}^{*} a_{s β} - i g_{α} a_{s α}^{*} a_{s α} (i ω_{m β} + γ_{m β})}{(i ω_{m α} + γ_{m α}) (i ω_{m β} + γ_{m β}) + V^{2}}, \\ α, β & = & 1, 2, α \neq β, \end{array}

where the effective detunings are

Δ_{α} = Δ_{c α} + g_{α} b_{s α}^{*} + g_{α} b_{s α}

, α = 1, 2.

By using the expectation value expression 〈Q〉 = Tr(ρQ) and the factorization assumption 〈AB〉 = 〈A〉〈B〉, the dynamical behaviours of the OMS can be obtained by solving the equation of motion for the fluctuation parts around their steady ones. The linearized Langevin equations for the expectation values (δa_i, δb_i) of their fluctuations are given by

\begin{array}{l} δ {\dot{a}}_{1} & = & - (i Δ_{c 1} + κ_{1}) δ a_{1} - i J δ a_{2} + E_{P} e^{- i δ t} \\ - i g_{1} (b_{s 1}^{*} δ a_{1} + b_{s 1} δ a_{1} + δ b_{1}^{+} a_{s 1} + δ b_{1} a_{s 1}), \end{array}

δ {\dot{a}}_{2} = - (i Δ_{c 2} + κ_{2}) δ a_{2} - i J δ a_{1} - i g_{2} (b_{s 2}^{*} δ a_{2} + b_{s 2} δ a_{2} + δ b_{2}^{+} a_{s 2} + δ b_{2} a_{s 2}),

δ {\dot{b}}_{1} = - i ω_{m 1} δ b_{1} - i V δ b_{2} - i g_{1} (a_{s 1}^{*} δ a_{1} + δ a_{1}^{+} a_{s 1}) - γ_{m 1} δ b_{1},

δ {\dot{b}}_{2} = - i ω_{m 2} δ b_{2} - i V δ b_{1} - i g_{2} (a_{s 2}^{*} δ a_{2} + δ a_{2}^{+} a_{s 2}) - γ_{m 2} δ b_{2},

Using ansatz as follows,

\begin{array}{l} δ a_{i} & = & a_{i +} e^{- i δ t} + a_{i -} e^{i δ t}, \\ δ b_{i} & = & b_{i +} e^{- i δ t} + b_{i -} e^{i δ t} . \end{array}

in the sideband regime ω_mi >> κ_i, (i = 1, 2) and under the resonant condition of Δ_i = ω_mi, (i = 1, 2) [8–10], the output field can be given by

ε_{out} = \frac{2 κ_{1} a_{1 +}}{E_{P}},

a_{1 +} = \frac{Λ_{1} + Λ_{5} + Λ_{6} + η_{1}}{1 - Λ_{2} - Λ_{3} - Λ_{4}},

where,

\begin{array}{l} Λ_{1} & = & \frac{Ω_{2} η_{2} (Ω_{8} + Ω_{9} Ω_{11})}{Φ}, Λ_{2} = \frac{Ω_{2} Ψ}{Φ}, \\ Λ_{3} & = & \frac{Ω_{1} [Φ (Ω_{4} + Ω_{6} Ω_{10}) + Ψ (Ω_{5} + Ω_{6} Ω_{12})]}{Φ (1 - Ω_{6} Ω_{11})}, \\ Λ_{4} & = & \frac{Ω_{3} [Φ (Ω_{10} + Ω_{4} Ω_{11}) + Ψ (Ω_{12} + Ω_{5} Ω_{11})]}{Φ (1 - Ω_{6} Ω_{11})}, \\ Λ_{5} & = & \frac{Ω_{1} η_{2} [(Ω_{8} + Ω_{9} Ω_{11}) (Ω_{5} + Ω_{6} Ω_{12}) + Φ]}{Φ (1 - Ω_{6} Ω_{11})}, \\ Λ_{6} & = & \frac{Ω_{3} η_{2} [(Ω_{8} + Ω_{9} Ω_{11}) (Ω_{12} + Ω_{5} Ω_{11}) + Ω_{11} Φ]}{Φ (1 - Ω_{6} Ω_{11})}, \end{array}

with,

\begin{array}{l} Ψ = & (1 - Ω_{6} Ω_{11}) (Ω_{7} + Ω_{9} Ω_{10}) + (Ω_{6} Ω_{10} + Ω_{4}) (Ω_{8} + Ω_{9} Ω_{11}), \\ Φ = & (1 - Ω_{6} Ω_{11}) (1 - Ω_{9} Ω_{11}) - (Ω_{8} + Ω_{9} Ω_{10}) (Ω_{5} + Ω_{6} Ω_{12}) . \end{array}

\begin{array}{l} Ω_{1} & = & \frac{Γ_{2} a_{s 2}^{*}}{O_{1} Y_{1} Y_{2} - Γ_{1} a_{s 1}^{*}}, Ω_{2} = \frac{Γ_{2} a_{s 1}}{O_{1} Y_{1} Y_{2} - Γ_{1} a_{s 1}^{*}}, \\ Ω_{3} & = & \frac{Γ_{2} a_{s 2}}{O_{1} Y_{1} Y_{2} - Γ_{1} a_{s 1}^{*}}, Ω_{4} = \frac{Γ_{3} a_{s 1}^{*}}{O_{1} Y_{1} Y_{2} - Γ_{4} a_{s 2}^{*}}, \\ Ω_{5} & = & \frac{Γ_{3} a_{s 1}}{O_{1} Y_{1} Y_{2} - Γ_{4} a_{s 2}^{*}}, Ω_{6} = \frac{Γ_{4} a_{s 2}}{O_{1} Y_{1} Y_{2} - Γ_{4} a_{s 2}^{*}}, \\ Ω_{7} & = & \frac{Γ_{5} a_{s 1}^{*}}{O_{2} Y_{1} Y_{2} - Γ_{5} a_{s 1}}, Ω_{8} = \frac{Γ_{6} a_{s 2}^{*}}{O_{2} Y_{1} Y_{2} - Γ_{5} a_{s 1}}, \\ Ω_{9} & = & \frac{Γ_{6} a_{s 2}}{O_{2} Y_{1} Y_{2} - Γ_{5} a_{s 1}}, Ω_{10} = \frac{Γ_{7} a_{s 1}^{*}}{O_{2} Y_{1} Y_{2} - Γ_{8} a_{s 2}}, \\ Ω_{11} & = & \frac{Γ_{8} a_{s 2}^{*}}{O_{2} Y_{1} Y_{2} - Γ_{8} a_{s 2}}, Ω_{12} = \frac{Γ_{7} a_{s 1}}{O_{2} Y_{1} Y_{2} - Γ_{8} a_{s 2}} . \end{array}

η_{1} = \frac{Y_{1} Y_{2} L_{2} E_{P}}{(O_{1} Y_{1} Y_{2} - Γ_{1} a_{s 1}^{*})}, η_{2} = \frac{- i J E_{p} Y_{1} Y_{2}}{(O_{1} Y_{1} Y_{2} - Γ_{4} a_{s 2}^{*})} .

and

\begin{array}{l} Γ_{1} & = & g_{1}^{2} a_{s 1} (- Y_{2} L_{2} R_{2} + Y_{1} L_{2} R_{4}) + J V g_{1} g_{2} a_{s 2} (Y_{2} + Y_{1}), \\ Γ_{2} & = & i V g_{1} g_{2} a_{s 1} (Y_{2} L_{2} + Y_{1} L_{2}) - i J g_{2}^{2} a_{s 2} (- Y_{2} R_{1} + Y_{1} R_{3}), \\ Γ_{3} & = & - i J g_{1}^{2} a_{s 1} (- Y_{2} R_{2} + Y_{1} R_{4}) + i V g_{1} g_{2} a_{s 2} (Y_{2} L_{1} + Y_{1} L_{1}), \\ Γ_{4} & = & J V g_{1} g_{2} a_{s 1} (Y_{2} + Y_{1}) + g_{2}^{2} a_{s 2} (- Y_{2} L_{1} R_{1} + Y_{1} L_{1} R_{3}), \\ Γ_{5} & = & - g_{1}^{2} a_{s 1}^{*} (- Y_{2} L_{4} R_{2} + Y_{1} R_{4} L_{4}) + J V g_{1} g_{2} a_{s 2}^{*} (Y_{2} + Y_{1}), \\ Γ_{6} & = & - i V g_{1} g_{2} a_{s 1}^{*} (Y_{2} L_{4} + Y_{1} L_{4}) - i J g_{2}^{2} a_{s 2}^{*} (- Y_{2} R_{1} + Y_{1} R_{3}), \\ Γ_{7} & = & - i J g_{1}^{2} a_{s 1}^{*} (- Y_{2} R_{2} + Y_{1} R_{4}) - i V g_{1} g_{2} a_{s 2}^{*} (Y_{2} L_{3} + Y_{1} L_{3}), \\ Γ_{8} & = & J V g_{1} g_{2} a_{s 1}^{*} (Y_{2} + Y_{1}) - g_{2}^{2} a_{s 2}^{*} (- Y_{2} L_{3} R_{1} + Y_{1} L_{3} R_{3}) . \end{array}

Y_{1} = R_{1} R_{2} + V^{2}, Y_{2} = R_{3} R_{4} + V^{2} .

O_{1} = L_{1} + L_{2} + J^{2}, O_{2} = L_{3} L_{4} + J^{2} .

\begin{array}{l} L_{1} = i Δ_{1} + κ_{1} - i δ, L_{2} = i Δ_{2} + κ_{2} - i δ, \\ L_{3} = κ_{1} - i δ - i Δ_{1}, L_{4} = κ_{2} - i δ - i Δ_{2} . \end{array}

\begin{array}{l} R_{1} = i ω_{m 1} - i δ + γ_{m 1}, R_{2} = i ω_{m 2} - i δ + γ_{m 2}, \\ R_{3} = γ_{m 1} - i δ + i ω_{m 1}, R_{4} = γ_{m 2} - i δ - i ω_{m 2} . \end{array}

The absorption property of the OMS is described by the real part of ε_out, denoted as ε_R = Re [ε_out].

3. Optomechanically induced transparency modulated by the direct interactions between two cavities or two oscillators

Firstly, we consider the absorption properties of the probing (first) optomechanical cavity which photonically (or phononically) coupled to another bare cavity (or mechanical oscillator). In other words, we investigate the effects of the photonic and phononic interaction on the single OMIT which is induced by the optomechanical coupling in the first cavity. The absorption properties in the optomechanical system can be described by the variation of ε_R with the normalized frequency σ/ω_m1 with displaced detuning σ = δ−ω_m1, which makes the main phenomena appearing around σ = 0. In order to make the following results more realizable, we use the experimental parameters as [8,40], λ₁ = λ₂ = 2πc/ω_L = 1064 × 10⁻⁹ m with ω_L1 = ω_L2 = ω_L, L₁ = L₂ = 25 × 10⁻³ m, κ₁ = κ₂ = 2π × 215 × 10³ Hz, ω_m1 = ω_m2 = ω_m = 2π × 947 × 10³ Hz, m₁ = m₂ = 145 × 10⁻¹² kg, ω_c1 = ω_c2 = 1.77 × 10¹⁵ Hz, and γ_m1 = γ_m2 = 2π × 140 Hz.

As is well known, the optomechanical interaction can cause a single transparency window in a canonical OMS, in which one mirror is fixed and the other is movable. When switching on both the first control field and the phononic channel connecting two mechanical oscillators, whereas keeping the second control field and the photonic channel coupling the two cavities switched off, the usual OMIT with a single transparency window is changed into the double one composed by two transparency windows as shown in Fig. 2(a). Additionally, the frequency for the transparency position corresponds to the phononic coupling strength V, or the distance between the two transparency tips is 2V. For example the frequencies for the right (or left) transparency position in the solid, dashed-dotted, and dashed curves in Fig. 2(a) are 0.5V₀, V₀, and 2.0V₀, which coincide with the given values of the phononic coupling strength exactly. This can be used to determine the phononic coupling strength by using the double transparency spectrum. The double transparency spectrum induced by the phononic interaction can be explained by the double optomechanical interactions between the cavity field and the two dressed phononic modes excited by the phononic interaction, which be specified later in this section. Next, we shall investigate the effects of the photonic coupling on the OMIT induced by the first optomechanical interaction. When the photonic coupling lies in the weak regime, i.e. J < κ, where the cavity field decays faster than the photon exchange through the photonic coupling path, the single OMIT features are not destroyed. If the two cavities are strongly coupled (J > κ), where the photons can exchange through the photonic coupling path before the decays of the cavity field, the two absorption peaks of the transparency window, shown in Fig. 2(b), are distorted and move outside with the increase of the photonic coupling. This is due to the fact that the absorption spectrum in this case is superposed by both the single OMIT and the NMS induced by the interaction between the cavities. The deviation of the sidebands from the resonant point is determined by the width of the transparency window and the splitting of the normal mode [23]. It can also be seen from Eq. (17) that the value of a₁₊ on the resonant point becomes smaller along with increase of the first control field.

Fig. 2 Absorption of the output field ε_R as a function of the normalized frequency σ/ω_m1. The first optomechanical cavity is driven by a field of 3 mW, while the optomechanical interaction of the coupled (second) cavity is switched off. In part (a), the photonic tunnel coupling J between two cavities is set to 0, and the mechanical interactions are chosen to be V = 0.5V₀, V₀ and 2.0V₀ where the parameter V₀ is equal to 2π × 10⁵ Hz, while in part (b), the mechanical interaction keeps at zero, and the photonic couplings are chosen to be J = 0.5κ, κ, and 2.0κ, as shown in the red solid, blue dashed dot, and green dashed curves, respectively.

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Secondly, we investigate how the photonic or phononic interactions influence the probe absorption properties when both the optomechanical interactions are considered. First of all, we should know the role in the probe absorption spectrum played by the second optomechanical interaction induced by the second control field E_L2. To highlight the contribution from the second optomechanical interaction, we should turn on the photonic channel and tune it in the weak region (J < κ). Then, we compare the absorption spectra induced by: i) only the first optomechanical interaction arising from the first control field E_L1, ii) only the second optomechanical interaction by turning on the second control field E_L2 together with weaker photonic coupling, and iii) both optomechanical interactions, corresponding to the solid, dashed-dotted and dashed curves in Fig. 3, respectively. The usual OMIT induced by the first optomechanical interaction is shown by the solid curve, while the second optomechanical interaction can lead to the absorption peak on the resonant point σ = 0 shown in the dashed-dotted curve.

Fig. 3 Absorption of the output field ε_R as a function of the normalized frequency σ/ω_m1, where the mechanical interaction is switched off. The red solid curve shows that both cavities are driven by a control field of 3 mW and decoupled with each other. The blue dash-dotted curve shows that the first cavity is bare optical cavity and the second cavity is driven by a control field of 3 mW, where two cavities are coupled by J = 0.5κ. The green dashed curve shows that two cavities are coupled by J = 0.5κ and driven by a control field of 3 mW.

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Now, we shall investigate how the photonic or the phononic interactions modulate the probe absorption when both optomechanical interactions are aroused by the control fields. In Fig. 4(a), the phononic coupling amplitudes are set as different values: 0.5V₀ (solid curve), V₀ (dashed-dotted curve), 2V₀ (dashed curve). It can be seen that the phononic interaction will lead to two transparency dips symmetrically located around the resonant point, which is similar to that including only the first optomechanical interaction shown in Fig. 2(a). However, the double transparency dips in Fig. 4(a) become deeper and the central peaks turn lower when the phononic interaction strength V decreases. These are different from those shown in Fig. 2(a) in which the depth in the double transparency dips and the height of the central absorption peaks are kept unchanged for the same setting of the phononic coupling amplitudes. This difference mainly comes from the second optomechanical interactions. When the phononic interaction tunes weaker, the absorption induced by the second optomechanical interaction is dominate and suppresses the transparency.

Fig. 4 Absorption of the output field ε_R as a function of the normalized frequency σ/ω_m1 with driving field of both cavities taking the value of 3 mW. In part (a), the tunnel coupling is switched off and the mechanical interaction is chosen as V = 0.5V₀, V₀, and 2.0V₀, and in part (b), the mechanical interaction is switched off and the photonic coupling is chosen as J = 0.5κ, κ, and 2.0κ, as shown in the red solid, blue dashed dot, and green dash curves, respectively.

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The effects of the photonic interaction J on the probe absorption are shown in Fig. 4(b) by setting different values of the photonic coupling: J = 0.5κ, κ, and 2.0κ. As discussed in Fig. 3, the optomechanical interaction in the coupled cavity causes the resonance absorption around the resonant point and that in the probing cavity leads to the transparency which splits the absorption peak. This results in the deep transparency dips around the resonant point shown in Fig. 4(b). It is shown from Fig. 4(b) that the two absorption peaks of the central transparency window become smooth when the photonic coupling J is weak. If J is increased up to the strong region (J > κ), as shown by dashed-dotted and dashed curves in Fig. 4(b), there appear two shallow windows on both shoulders of the absorption spectrum. And, the windows become deeper and the peaks turn broader with the photonic interaction. These variations are mainly resulted from the NMS induced by the photonic interaction. The strong photonic interaction splits into the spectrum structure composed by the two outer peaks, which correspond to two normal photonic modes. The distance between the two peaks is determined by the photonic coupling and increases with the coupling strength.

The results presented in above discussions can be explained by the analytical findings and the dressed-mode picture. Firstly, we demonstrate the variations of the absorption spectrum with the photonic interaction via the analytical findings. In the region around the resonance δ ≈ ω_m1 in which the main results are presented, we can get $ω_{mi}^{2} - δ^{2} - i δ γ_{mi} \approx ω_{mi} (2 σ - i γ_{mi})$ . When considering Δ₁ = ω_m1 and Δ₂ = ω_m2, which is the essential condition of OMIT [8, 9], the formula (7) can be reduced as the following compact form in the resolved sideband regime (ω_mi ≫ κ_i, (i = 1, 2)),

a_{1 +} = \frac{E_{p}}{κ_{1} - i σ + \frac{G_{1}^{2}}{\frac{γ_{m 1}}{2} - i σ} + \frac{J^{2}}{κ_{2} - i σ + \frac{G_{2}^{2}}{\frac{γ_{m 2}}{2} - i σ}}},

where the optomechanical coupling reads G_i = g_i|a_si |(i = 1, 2). Without loss of generality, the frequencies of the both two oscillators are assumed to be the same value as ω_m1 = ω_m2 = ω_m. (i) We focus on the NMS induced by the strong coupling between the two cavities, J ≫ κ₁, κ₂. When the two control fields are turned off (P_L1 = P_L2 = 0), Eq. (17) reduces to

a_{1 +} = \frac{E_{p}}{κ_{1} + i σ + \frac{J^{2}}{κ_{2} - i σ}}

. In the strong photonic coupling region (J ≫ κ_i, (i = 1, 2)), its denominator in the absorption coefficient Re[a₁₊] has two complex roots instead of imaginary ones which leads to the appearance of NMS with two peaks located at σ ≈ ±J. These two resonance peaks behave the sideband peaks in the dashed curves of Fig. 3 and those in the dashed-dotted and dashed curves of Fig. 4(b).

(ii) We shall demonstrate the transparency properties at the resonant point δ = ω_m(σ = 0). When the direct coupling between the two cavities and that between the two oscillators are turned off (J = 0 and V = 0), the formula (17) can be reduced to $a_{1 +} = \frac{E_{p}}{κ_{1} - i σ + \frac{G_{1}^{2}}{\frac{γ_{m 1}}{2} - i σ}}$ , which is consistent with the results of Weis, et al [9]. Now, the denominator has two imaginary roots because of smaller driving field than the critical field $P_{LC} = \frac{ℏ ω_{L}}{2 κ g^{2}} {(\frac{2 κ - γ_{m}}{4})}^{2} (κ^{2} + ω_{m}^{2})$ , which characterizes the transparency windows to be the OMIT other than Autler-Townes effects. Taking parameters here, this critical value for the driven field should be P_LC = 3.83 mW, which is exactly larger than 3 mW used here [8]. This transparency dips can be shown in Fig. 2(b), solid and dashed curves in Fig. 3 and all curves in Fig. 4(b).

(iii) Now we turn to the probe absorption properties when the optomechanical interaction in the coupled cavity is switched on by the second control field (E_L2). When the first control field driving the probing cavity is switched off (P_L1 = 0), and the second control field applied on the coupled cavity is turned on (P_L2 ≠ 0), Eq. (17) is simplified as $a_{1 +} = \frac{E_{p}}{κ_{1} - i σ + \frac{J^{2}}{κ_{2} - i σ + \frac{G_{2}^{2}}{\frac{γ_{m 2}}{2} - i σ}}}$ . Its denominator is a cubic form of the frequency detuning σ and then has three roots. Two of them determine the sideband absorption, The third one is responsible for the OMIA on the resonant point σ = 0. This can be further demonstrated by the fact that the absorption coefficient is monotonously increased with the second opto-mechanical coupling $G_{2}^{2}$ . These optical properties can be shown in Fig. 3.

Therefore, the optomechanical interaction induced by the second control field leads to the OMIA on resonance, which can be split by the OMIT resulted from the first control field. And, the direct strong coupling between two cavities causes the NMS which is characterized by the two sideband absorption peaks.

Next, the OMIT influenced by the mechanical coupling can be explained by the dressed-mode picture. The mechanical dressed-mode basis is defined by

{\hat{b}}_{\pm} = \frac{1}{\sqrt{2}} ({\hat{b}}_{1} \mp {\hat{b}}_{2}),

While taking no account of the photonic coupling J between cavities and adopting the dressed-mode basis, the Hamiltonian can be rewritten as

\begin{array}{l} \hat{H} & = & ℏ Δ_{c 1} {\hat{a}}_{1}^{+} {\hat{a}}_{1} + ℏ Δ_{c 2} {\hat{a}}_{2}^{+} {\hat{a}}_{2} + (ℏ ω_{m} + V) {\hat{b}}_{-}^{+} {\hat{b}}_{-} + ℏ (ω_{m} - V) {\hat{b}}_{+}^{+} {\hat{b}}_{+} \\ + \frac{ℏ}{\sqrt{2}} ({\hat{b}}_{-} + {\hat{b}}_{-}^{+}) (g_{1} {\hat{a}}_{1}^{+} {\hat{a}}_{1} + g_{2} {\hat{a}}_{2}^{+} {\hat{a}}_{2}) + \frac{ℏ}{\sqrt{2}} ({\hat{b}}_{+} + {\hat{b}}_{+}^{+}) (g_{1} {\hat{a}}_{1}^{+} {\hat{a}}_{1} - g_{2} {\hat{a}}_{2}^{+} {\hat{a}}_{2}) \\ + i ℏ E_{L 1} ({\hat{a}}_{1}^{+} - {\hat{a}}_{1}) + i ℏ E_{L 2} ({\hat{a}}_{2}^{+} - {\hat{a}}_{2}), \end{array}

As given by Safavi-Naeini et al [10], OMIT from a single OMS is resulted from the destructive interference between the lossy dressed optical and mechanical transitions, which leads to the transparency. When taking account of the phononic coupling, the level diagram according to Eq. (22), which describes the two optomechanical cavities coupled by the phononical interaction, is given by Fig. 5. If the optomechanical interaction in the coupled cavity is switched off, the level diagram is simplified as that in the dashed square, which is a well-known double Λ-type diagram [41]. Then, one OMIT dip corresponding each Λ diagram is created and a half of the distance between the two dips is determined by the mechanical coupling strength, which is also the same as the splitting between two dressed mode 2V as shown in Fig. 2(a). If the coupled cavity is driven by the right control field, the level diagram will change to a double N configuration, as shown in Fig. 5, where the absorption at the resonant point can be suppressed [42]. This is why the absorption on resonance in Fig. 4(a) is smaller than that shown in Fig. 2(a). Moreover, after turning on the tunnel coupling J to be in the weak region as shown in Fig. 6(a), the absorption is similar with that in Fig. 4(a).

Fig. 5 Level diagram of a double N-type for a mechanically coupled optomechanical cavities, where a double Λ-type is shown in the dashed square.

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4. Competition and interference between photonic and phononic interactions

The effects of the photonic or phononic interactions on the probe absorption spectrum have been investigated in above discussion. Now we wonder to know how the absorption properties behave in the presence of both interactions simultaneously. In Fig. 6(a), the probe absorption is plotted versus the detuning with J = 0.5κ and V = 0.5V₀, 1.5V₀, and 2.0V₀, which correspond to solid, dashed-dotted and dashed curves. It is shown that the distances between the two transparency dips become larger and their depths become deeper, which is similar to the Fig. 4(a) in the absence of the photonic interaction. On the other hand, the central peaks of the absorption curves become slightly lower with the phononic interaction, which is different from that shown in Fig. 4(a). Following the same setting as Fig. 6(a) except for J = 2κ, it can be seen from Fig. 6(b) that the central peaks are changed into dips, which become deeper and wider, and lead to the appearance of the triple optomechanically induced transparency for V ≠ 0.

Fig. 6 Absorption of the output field ε_R as a function of the normalized frequency σ/ω_m1 with the control fields driving the cavities set by the value of 3 mW, where the mechanical interactions are taken as V = 0.5V₀, 1.5V₀, and 2.0V₀, as shown in the red solid, blue dashed-dotted and green dashed curves, respectively. The photonic coupling J is taken to be 0.5κ in (a), and 2.0κ in (b). The parameter V₀ takes the same value as Fig. 2 (a).

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Why do there exist distinct difference between the Figs. 6(a) and 6(b) just because of different amplitudes of the photonic interaction? By comparing these two figures, two main results can be addressed in the present work: the competition and interference between the photonic and phononic interactions displayed in the transparency spectrum. It is shown from Fig. 6(a) that the sideband absorption peaks move outward with the increase of the phononic interaction, which is due to the drag of the larger distance between the two transparency dips because of the increase of the phononic coupling. In this case with the smaller value of the photonic interaction, the probe absorption properties is mainly determined by the phononic interaction. When the photonic interaction becomes stronger, for example J = 2κ shown in Fig. 6(b), the sideband dips move outward with the phononic interaction, but the out-most absorption peaks are kept unchanged due to the fixed photonic interaction J. Therefore, we can see that the sidebands in the transparency spectra are inclined to the features of the interaction with stronger strength, when the strength of the two interactions differs substantially. The main features in above discussion can be recured using parameters from [37], except for the ultra sharp transparency windows comparing with those in Fig. 6(a). This result is not displayed here due to the length of the paper.

On the other hand, the effects of the photonic interaction on the absorption spectrum in the presence of the phononic coupling are considered in the following. In Fig. 7, we fix the phononic interaction strength at V = V₀ and set the photonic interaction strength as different values: J = 0.5κ, κ, 2κ. It is shown that positions of the transparency dips located symmetrically around the resonant point are kept at σ = ±V₀ due to the fixed strength V, and the sideband absorption peaks move outward due to the NMS induced by increased strengths of the photonic interaction. It is demonstrated once again that the sideband transparency dips characterize the effects of the phononic interaction, and the sideband absorption peaks exhibit the influences of the photonic interaction on the transparency spectra in this system.

Fig. 7 Absorption of the output field ε_R as a function of the normalized frequency σ/ω_m1, where both optomechanical cavities are driven by a control field of 3 mW, and the mechanical interaction is taken as a constant of V₀. The red solid, blue dashed dot, and green dashed curves show those with the photonic interaction taken as J = 0.5κ, 1.0κ and 2.0κ.

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Meanwhile, we can see the appearance of the interference between the photonic and phononic interactions in Fig. 6(b). In Fig. 4(a), it is seen that the absorption at the resonant point is increased with the phononic interaction strength in the absence of the photonic interaction. After switching on the photonic path to the strong interaction region as shown in Fig. 6(b), the absorption peaks at the resonant point in Fig. 4(a) turn into the transparency windows and their depths become deeper with the phononic interaction strength. The transparency around the resonance characterizes the destructive interference between the photonic and phononic paths. Also, it can be seen from Fig. 7 that the absorption around the resonant point becomes smaller with the increase of the photonic interaction strength and eventually the absorption peak becomes transparency window. This recurs the interference between the photonic and phononic paths, which is similar to investigation in Fig. 6(b).

To demonstrate the transparency on the resonant point resulted from the destructive interference between the photonic and phononic paths, we compare with the absorptions on resonance under the second optomechanical interaction in three cases: only the photonic or the phononic paths are switched on and both of them on. To highlight the interference between the photonic and phononic paths coupling the cavities via the probing absorption spectrum, we turn on the control field applied on the coupled cavity and switch off that of the probing cavity to eliminate its OMIT which will attenuate the interference induced by the two paths. In Fig. 8, the solid red curve shows the absorption with the mechanical coupling V = 0 and the tunnel coupling J = 2κ, and the blue dashed-dotted curve gives that with V = 2V₀ and J = 0, while the green dashed curve gives that with V = 2V₀ and J = 2κ. It is shown that the absorption takes a value of about 2.0 when only one path is turned on. However, the absorption is suppressed to about 0.4 and is not the addition of them when both paths are turned on. It is the destructive interference between the two paths that leads to the reduction of the absorption on the resonant point when the strengths of the photonic and phononic couplings are comparable.

Fig. 8 Absorption of the output field ε_R as a function of the normalized frequency σ/ω_m1 with the control field of the probing cavity switched off and that of the coupled cavity taking the value of 3 mW. The solid red curve shows the absorption with the mechanical coupling V = 0 and the tunnel coupling J = 2κ. The blue dashed-dotted curve gives that with V = 2V₀ and J = 0, while the green dashed curve gives that with V = 2V₀ and J = 2κ. The other parameters are the same as Fig. 2(a).

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The current scheme is originated from the experimental paper [37,38], in which two optomechanical cavities are coupled together via a photon-phonon waveguide, resulting in optical and mechanical inter-cavity couplings. Additionally, we can realize the current scheme in another system, where two cavities are coupled directly through photon waveguide, and direct interaction between two mechanical modes oscillators can be realized by the Coulomb interaction between two charged mechanical oscillators [33, 43]. The phononic coupling V can be adjusted by the capacitance and the voltage of the bias gate. In such scheme, the photonic and phononic interaction can be tuned independently.

In this scheme, the transparency spectrum can be modulated flexibly by the four interactions: the optomechanical interactions in the probing and coupled cavities, the photonic interaction, and the phononic interaction. Specifically, the two optomechanical interactions in the probing and coupled cavities can be tuned by adjusting the left and right control fields. The photonic interaction amplitude can be adjusted by the tunnelling or using the optical waveguide [44,45]. And, the phononic interaction can be realized by the Coulomb interaction which is determined by the capacitance and the voltage of the bias gate [33,43]. These results suggest various applications. The double transparency in the present scheme can be used to realize the multi-channel optical communication. The transition from the absorption peak induced by the optomechanical interaction in the coupled cavity to a deep transparency dip, resulted from the optomechanical interaction in the probing cavity, can be used as optical switching, which can be realized only by adjusting the left or right control fields. Additionally, this type of hybridized OMS can be a part of optomechanical circuits for coherent signal processing [37], and the quantum computation can be realized in multipath interference by direct photonic entanglement [46].

5. Summary

We presented a highly tunable double optomechanically induced transparency scheme based on a recent experiment, where the crystal waveguides are used to mediate direct phononic or photonic interactions between two resonators or two cavities [37]. This optomechanical system is composed of double optical cavities and double mechanical resonators, which are coupled through both the photonic and the phononic paths, respectively. The calculated results show that OMIT with a single transparency window is changed into double OMIT with two transparency windows when switching on the phononic path, and the distance between two transparency dips is linearly dependent with the strength of the mechanical interaction. The double OMIT is suppressed by the second optomechanical interaction as a whole, while enhanced by the mechanical interaction in the presence of the second optomechanical interaction. On the other hand, the sidebands in the absorption spectra characterizes the NMS induced by the strong photonic coupling between the two cavities. Specifically, the widths and the locations of the sideband absorption peaks are determined by the photonically coupling amplitude. These phenomena have also been demonstrated by analytical findings or the dressed-mode picture. Eventually, the dominate features of the stronger interaction over the other with a weaker one displays the competition between photonic and phononic interactions. The transition, from the absorption peak around the resonant point in the presence of the phononic path to the transparency window when both quantum paths are applied simultaneously, identifies the destructive interference between the phononic and photonic paths, which differs the OMIT from Autler-Townes splitting [47]. Our proposed scheme and results have much universality and can be realized by various methods. For example, the mechanical oscillators can be coupled by both phononic waveguide or the Coulumb interaction, which indicate potential applications in ultrahigh precision measurement or manipulation.

Funding

National Natural Science Foundation of China (No. 11104191, 10647007); the Young Foundation of Sichuan Province (09ZQ026-008).

Acknowledgments

The authors thanks the National Natural Science Foundation of China (No. 11104191, 10647007) and the Young Foundation of Sichuan Province (09ZQ026-008) for financial supports.

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Tunable double optomechanically induced transparency in photonically and phononically coupled optomechanical systems

Abstract

1. Introduction

2. Model and dynamical equations

3. Optomechanically induced transparency modulated by the direct interactions between two cavities or two oscillators

4. Competition and interference between photonic and phononic interactions

5. Summary

Funding

Acknowledgments

References and links

Cited By

Figures (8)

Equations (26)

Optics Express