Quantum-limited directional amplifier based on a triple-cavity optomechanical system

Cheng Jiang; Baowei Ji; Yuanshun Cui; Fen Zuo; Jin Shi; Guibin Chen

doi:10.1364/OE.26.015255

Quantum-limited directional amplifier based on a triple-cavity optomechanical system

Cheng Jiang, Baowei Ji, Yuanshun Cui, Fen Zuo, Jin Shi, and Guibin Chen

Optics Express
Vol. 26,
Issue 12,
pp. 15255-15267
(2018)
•https://doi.org/10.1364/OE.26.015255

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Cheng Jiang, Baowei Ji, Yuanshun Cui, Fen Zuo, Jin Shi, and Guibin Chen, "Quantum-limited directional amplifier based on a triple-cavity optomechanical system," Opt. Express 26, 15255-15267 (2018)

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Table of Contents Category
- Optoelectronics
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Optomechanics (120.4880)
- Parametric oscillators and amplifiers (190.4970)
- Coherent optical effects (270.1670)

About this Article
History
- Original Manuscript: March 22, 2018
- Revised Manuscript: May 20, 2018
- Manuscript Accepted: May 20, 2018
- Published: June 1, 2018

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Abstract

We theoretically propose a scheme for realizing a quantum-limited directional amplifier in a triple-cavity optomechanical system, where one microwave cavity and two optical cavities are, respectively, coupled to a common mechanical resonator. Moreover, the two optical cavities are coupled directly to facilitate the directional amplification between microwave and optical photons. We find that directional amplification between the three cavity modes is achieved with two gain process and one conversion process, and the direction of amplification can be modulated by controlling the phase difference between the field-enhanced optomechanical coupling strengths. Furthermore, with increasing the optomechanical cooperativity, both gain and bandwidth of the directional amplifier can be enhanced, and the noise added to the amplifier can be suppressed to approach the standard quantum limit on the phase-preserving linear amplifier.

1. Introduction

The electromagnetic wave transmission in media is generally invariant under exchange of source and detector [1], but it is one of the basic requirements to realize directional transmission in both classical and quantum information processing. Consequently, nonreciprocal devices, including isolators, circulators and directional amplifiers are of crucial importance, which can protect the signal source from spurious interferences. The conventional solution to achieve nonreciprocity is to bias the propagation channel by the external magnetic field, which requires the magneto-optic materials and therefore prevents the integration and miniaturization [2–4]. In the past decades, a variety of alternative strategies have been developed to break reciprocity without needing magneto-optical effects, such as refractive-index modulation [5,6], optical nonlinearity [7–9], angular momentum biasing in photonic or acoustic systems [10–12], and the quantum Hall effect [13].

Furthermore, cavity optomechanics studies the parametric coupling between an electromagnetic cavity and a mechanical resonator via radiation pressure [14–16]. Based on optomechanical couplings, significant achievements have been made in this field, including ground state cooling of the mechanical resonator [17,18], optomechanically induced transparency (OMIT) [19–24], microwave amplification [25–27], bidirectional state conversion between optical and microwave photons [28–31], phonon laser [32–35], and parity-time-symmetry-breaking chaos [36]. Recently, the radiation-pressure-induced optomechanical coupling has been exploited to break time-reversal symmetry and realize electromagnetic nonreciprocity at optical [37–48] and microwave frequencies [49–54]. Many of these works rely on controlling the relative phases of the field-enhanced optomechanical coupling strengths to achieve nonreciprocity. It is worth noting that directional amplifier can be realized with optomechanical systems by reservoir engineering [45], coherent mechanical drive [46], and blue-detuned optical pumping [47, 51]. Furthermore, directional amplifier has also been demonstrated in the superconducting microwave circuit with Josephson nonlinearity and parametric pumping [55–57].

Motivated by the above achievements, we theoretically investigate how to realize a quantum-limited directional amplifier based on a triple-cavity optomechanical system, where one microwave cavity and two directly coupled optical cavities are, respectively, coupled to a common mechanical resonator [49]. When the microwave cavity is pumped on its red sideband and the two optical cavities are pumped on their respective blue sidebands, directional amplification between microwave and optical photons can be achieved with two gain process and one conversion process. We show that the direction of the amplification depends on the relative phase between the field-enhanced optomechanical strengths, and the gain and bandwidth of the amplifier can be enhanced by increasing the optomechanical cooperativity. Moreover, the mechanical noise can be suppressed in the large cooperativity limit, and this phase-preserving amplifier can approach the quantum limit of a half added quanta.

Compared with the optical nonreciprocity in Ref. [37–48], our scheme can realize the directional amplification between optical and microwave photons without requiring the direct coupling between the optical and microwave cavities. Instead, another optical cavity is introduced to control the direction of amplification. In particular, the proposed directional amplifier is robust against mechanical noise and can reach the quantum-limited value of added noise. Note that quantum-limited directional amplifier for microwave signals has been proposed recently in an electromechanical setup comprising two microwave cavities and two mechanical resonators [51]. Compared with Ref. [51], our work can realize the directional amplification among the three cavities by combing two gain process and one conversion process, which has been demonstrated based on the Josephson Parametric Converter [56]. Furthermore, the phase difference at which isolation occurs depends on the detuning between the pump and cavity field in Ref. [51], but directional amplification can be achieved when the phase difference ϕ = ±π/2 in our work, which is easier to control.

The remainder of the paper is organized as follows. In Sec. 2, we introduce the theoretical model and derive the transmission matrix between the input and output operators. In Sec. 3, we investigate how to realize the quantum-limited directional amplifier based on this optomechanical system. The effects of phase difference and optomechanical cooperativity on the amplifier are discussed. We finally summarize our work in Sec. 4.

2. Model and theory

We consider the optomechanical system schematically shown in Fig. 1, where three cavity modes a₁, a₂, a₃ are independently coupled to a common mechanical mode b via radiation pressure. Furthermore, cavities a₂ and a₃ are coupled directly via hopping interaction to facilitate the nonreciprocal transmission. Cavity a₁ can have vastly different frequency from that of cavities a₂ and a₃, e.g., cavity a₁ is a microwave cavity while cavities a₂ and a₃ are two optical cavities, or vice versa. This model is still valid when all the three cavities are optical cavities, but here we focus on the directional amplification between optical and microwave photons based on this system, which can be realized with current experimental technology [28, 31, 58–60]. The system which consists of two cavity modes coupled to a common mechanical resonator has been realized [28,31]. When a₁ is a microwave cavity and a₂ (a₃) are optical cavities, time-dependent interaction between two optical cavities needs to be introduced, which can be generated by connecting the cavities to other cavity modes or waveguides [49, 60, 61]. In order to realize directional amplification, we apply a red-detuned pump field to drive the cavity mode a₁ and two blue-detuned pump fields to drive the cavity modes a₂ and a₃, respectively. The Hamiltonian of this optomechanical system reads (ħ = 1)

\begin{array}{l} H = \sum_{k = 1}^{3} ω_{k} a_{k}^{†} a_{k} + ω_{m} b^{†} b + \sum_{k = 1}^{3} g_{k} a_{k}^{†} a_{k} (b^{†} + b) + J (a_{2}^{†} a_{3} + a_{3}^{†} a_{2}) \\ + \sum_{k = 1}^{3} ε_{k} (a_{k}^{†} e^{- i ω_{d, k} t} + a_{k} e^{i ω_{d, k} t}), \end{array}

where

a_{k} (a_{k}^{†})

is the annihilation (creation) operator of the cavity mode a_k (k = 1, 2, 3) with resonance frequency ω_k, and b (b^†) is the annihilation (creation) operator of the mechanical mode b with resonance frequency ω_m. The third term represents the intercation between the cavity mode a_k and the mechanical mode b, where g_k is the single-photon coupling strength between the cavity mode a_k and the mechanical mode. The fourth term describes the interaction between the cavity mode a₂ and a₃ with the coupling strength J. The last term shows the interaction between the pump fields and the respective cavity modes, where ε_k and ω_d,_k are the amplitude and frequency of the pump field applied to the cavity mode a_k. The driving terms in Eq. (1) can be made time independent by applying the unitary transformation

U = \exp (i \sum_{k = 1}^{3} ω_{d, k} a_{k}^{†} a_{k} t)

, and the new Hamiltonian H_rot = UHU^† − iU∂U^†/∂t is then given by

H_{rot} = \sum_{k = 1}^{3} Δ_{k} a_{k}^{†} a_{k} + ω_{m} b^{†} b + \sum_{k = 1}^{3} g_{k} a_{k}^{†} a_{k} (b^{†} + b) + J (a_{2}^{†} a_{3} + a_{3}^{†} a_{2}) + \sum_{k = 1}^{3} ε_{k} (a_{k}^{†} + a_{k}),

where we have set ω_d_,2 = ω_d_,3, and ∆_k = ω_k − ω_d,_k is the detuning between the cavity mode a_k and the respective pump field.

Fig. 1 Schematic diagram of the triple-cavity optomechanical system. One microwave cavity a₁ and two optical cavities a₂ and a₃ are respectively coupled to the common mechanical resonator with the field-enhanced optomechanical coupling strength G_k(k = 1, 2, 3), where ϕ is the phase difference between G_k. Moreover, the two optical cavities are coupled directly via hopping interaction J to facilitate the nonreciprocal transmission. Cavity a₁ is driven on its red sideband by a pump field at the frequency ω_d_,1, and cavities a₂ and a₃ are driven on their respective blue sidebands by the pump fields at the frequencies ω_d_,2 and ω_d_,3. The solid arrows represent the transmission elements between different modes.

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According to Eq. (2), we can get the following quantum Langevin equations (QLEs):

{\dot{a}}_{1}^{†} = - (κ_{1} / 2 - i Δ_{1}) a_{1}^{†} + i g_{1} a_{1}^{†} (b^{†} + b) + i ε_{1} + \sqrt{κ_{1}} a_{1, in}^{†},

{\dot{a}}_{2} = - (κ_{2} / 2 + i Δ_{2}) a_{2} - i g_{2} a_{2} (b^{†} + b) - i J a_{3} - i ε_{2} + \sqrt{κ_{2}} a_{2, in},

{\dot{a}}_{3} = - (κ_{3} / 2 + i Δ_{3}) a_{3} - i g_{3} a_{3} (b^{†} + b) - i J a_{2} - i ε_{3} + \sqrt{κ_{3}} a_{3, in},

{\dot{b}}^{†} = - (γ_{m} / 2 - i ω_{m}) b^{†} + i \sum_{k} g_{k} a_{k}^{†} a_{k} + \sqrt{γ_{m}} b_{in}^{†} .

where the decay and noise terms have been added phenomenologically. κ₁, κ₂ and κ₃ are the decay rates of cavity mode a_k (k = 1, 2, 3), and γ_m is the damping rate of the mechanical mode b. Furthermore, a_k_,in (k = 1, 2, 3) and b_in are the noise operators for the cavity modes and mechanical mode with zero mean value and satisfy the following correlation functions

\begin{array}{l} 〈 a_{k, in} (t) a_{k, in}^{†} (t^{'}) = δ (t - t^{'}), 〈 a_{k, in}^{†} (t) a_{k, in} (t^{'}) = 0, \\ 〈 b_{in} (t) b_{in}^{†} (t^{'}) 〉 = (n_{m} + 1) δ (t - t^{'}), 〈 b_{in}^{†} (t) b_{in} (t^{'}) 〉 = n_{m} δ (t - t^{'}), \end{array}

where we have assumed that the thermal photon number equals to zero and the thermal phonon number n_m of the mechanical resonator is given by

n_{m} = 1 / (e^{ℏ ω_{m} / k_{B} T_{m}} - 1)

, with T_m being the environmental temperature and k_B being the Boltzmann constant.

The steady-state solutions for the cavity and mechanical modes can be obtained by setting all the time derivatives in Eqs. (3)–(6) to be zero and neglecting all the noise terms, which obey the following algebraic equations:

α_{1}^{*} = \frac{i ε_{1}}{κ_{1} / 2 - i Δ_{1}^{'}}, α_{2} = - \frac{i J α_{3} + i ε_{2}}{κ_{2} / 2 + i Δ_{2}^{'}}, α_{3} = - \frac{i J α_{2} + i ε_{3}}{κ_{3} / 2 + i Δ_{3}^{'}}, β^{*} = \frac{i \sum_{k} g_{k} {| α_{k} |}^{2}}{γ_{m} / 2 - i ω_{m}},

where α_k (k = 1, 2, 3) and β represent, respectively, the mean values of the cavity mode a_k and mechanical mode b, and

Δ_{k}^{'} = Δ_{k} + g_{k} (β^{*} + β)

is the effective cavity detuning including radiation pressure effect. Subsequently, we can linearize the QLEs (3)–(6) by rewriting each Heisenberg operator as the sum of its steady-state mean value and a small fluctuation, i.e., a_k = α_k + δa_k and b = β + δb. The linearized QLEs are given by:

δ {\dot{a}}_{1}^{†} = - (κ_{1} / 2 - i Δ_{1}^{'}) δ a_{1}^{†} + i g α_{1}^{*} (δ b^{†} + δ b) + \sqrt{κ_{1}} a_{1, in},

δ {\dot{a}}_{2} = - (κ_{2} / 2 + i Δ_{2}^{'}) δ a_{2} - i g_{2} α_{2} (δ b^{†} + δ b) - i J δ a_{3} + \sqrt{κ_{2}} a_{2, in},

δ {\dot{a}}_{3} = - (κ_{3} / 2 + i Δ_{3}^{'}) δ a_{3} - i g_{3} α_{3} (δ b^{†} + δ b) - i J δ a_{2} + \sqrt{κ_{3}} a_{3, in},

δ {\dot{b}}^{†} = - (γ_{m} / 2 - i ω_{m}) δ b^{†} + i \sum_{k} g_{k} (α_{k}^{*} δ a_{k} + α_{k} δ a_{k}^{†}) + \sqrt{γ_{m}} b_{in}^{†} .

To realize the directional amplification, we consider the case that $Δ_{1}^{'} = ω_{m}$ and $Δ_{2}^{'} = Δ_{3}^{'} = - ω_{m}$ . In the resolved sideband limit with ω_m ≫ γ_m, κ_k (k = 1, 2, 3), the rotating wave approximation (RWA) can be applied. For simplicity, we can move into another interaction picture by introducing $δ a_{k} \to δ a_{k} e^{- Δ_{k}^{'} t}$ , $δ b \to δ b^{- i ω_{m} t}$ , $a_{k, in} \to a_{k, in} e^{- Δ_{k}^{'} t}$ , and $b_{in} \to b_{in} e^{- i ω_{m} t}$ [62], and then Eqs. (9)–(12) become

δ {\dot{a}}_{1}^{†} = - \frac{κ_{1}}{2} δ a_{1}^{†} + i G_{1} e^{i ϕ_{1}} δ b^{†} + \sqrt{κ_{1}} a_{1, in}^{†},

δ {\dot{a}}_{2} = - \frac{κ_{2}}{2} δ a_{2} - i G_{2} e^{- i ϕ_{2}} δ b^{†} - i J δ a_{3} + \sqrt{κ_{2}} a_{2, in},

δ {\dot{a}}_{3} = \frac{κ_{3}}{2} δ a_{3} - i G_{3} e^{- i ϕ_{3}} δ b^{†} - i J δ a_{2} + \sqrt{κ_{3}} a_{3, in},

δ {\dot{b}}^{†} = - \frac{γ_{m}}{2} δ b^{†} + i (G_{1} e^{- i ϕ_{1}} δ a_{1}^{†} + G_{2} e^{i ϕ_{2}} δ a_{2} + G_{3} e^{i ϕ_{3}} δ a_{3}) + \sqrt{γ_{m}} b_{i n}^{†},

where we have set

g_{k} α_{k} = g_{k} | α_{k} | e^{- i ϕ_{k}} = G_{k} e^{- i ϕ_{k}}

with G_k being the field-enhanced coupling strength. The effective Hamiltonian associated with Eqs. (13)–(16) is

H_{eff} = G_{1} δ a_{1} δ b^{†} e^{i ϕ_{1}} + G_{2} δ a_{2} δ b e^{i ϕ_{2}} + G_{3} δ a_{3} δ b e^{i ϕ_{3}} + J δ a_{2} δ a_{3}^{†} + H . c ..

The phases ϕ_k (k = 1, 2, 3) can be absorbed by redefining the operators δa_k and δb, and only the phase difference ϕ = ϕ₃ − ϕ₂ in the closed-loop interaction formed by G₂, G₃ and J has physical effects. For convenience, we can set ϕ₁ = ϕ₂ = 0 and ϕ₃ = ϕ, and the symbol δ can be neglected, i.e., δa_k → a_k and δb → b. Subsequently, the QLEs (13)–(16) can be written in the following matrix form:

\dot{μ} = M μ + \sqrt{K} μ_{i n},

where the vector

μ = {(a_{1}^{†}, a_{2}, a_{3}, b^{†})}^{T}

μ_{in} = {(a_{1, in}^{†}, a_{2, in}, a_{3, in}, b_{in}^{†})}^{T}

, the diagonal matrix K = Diag[κ₁, κ₂, κ₃, γ_m], and the coefficient matrix

M = (\begin{matrix} - κ_{1} / 2 & 0 & 0 & i G_{1} \\ 0 & - κ_{2} / 2 & - i J & - i G_{2} \\ 0 & - i J & - κ_{3} / 2 & - i G_{3} e^{- i ϕ} \\ i G_{1} & i G_{2} & i G_{3} e^{i ϕ} & - γ_{m} / 2 \end{matrix}) .

The system is stable only if the real parts of all the eigenvalues of matrix M are negative, The stability condition can be derived analytically by the Routh-Hurwitz criterion [63, 64], whose general form is too cumbersome to give here. However, we will check the stability diagram numerically and choose the parameters that satisfy the stability condition.

By introducing the Fourier transform of the operators

o (ω) = \int_{- \infty}^{+ \infty} o (t) e^{i ω t} d t,

o^{†} (ω) = \int_{- \infty}^{+ \infty} o^{†} (t) e^{i ω t} d t,

the solution to the Eq. (18) in the frequency domain is

μ (ω) = - {(M + i ω I)}^{- 1} \sqrt{K} μ_{in} (ω),

where I represents the unitary matrix. Upon substituting Eq. (22) into the standard input-output relation [65]

μ_{out} (ω) = μ_{in} (ω) - \sqrt{K} μ (ω)

, we can obtain

μ_{out} (ω) = T (ω) μ_{in} (ω),

where the output field vector µ_out(ω) is the Fourier transform of

μ_{out} = {(a_{1, out}^{†}, a_{2, out}, a_{3, out}, b_{out}^{†})}^{T}

, and the transmission matrix is given by

T (ω) = I + \sqrt{K} {(M + i ω I)}^{- 1} \sqrt{K} .

Here the matrix element T_ij(ω) (i, j = 1, 2, 3, 4 correspond to a₁, a₂, a₃, b, respectively) describes the transmission amplitude from mode j to mode i.

3. Quantum-limited directional amplifier

In this section, we discuss in detail how to realize the directional amplifier based on this triple-cavity optomechanical system. We have assumed that cavity a₁ is driven on its red sideband while cavities a₂ and a₃ are driven on their respective blue sidebands. For convenience, we first discuss the condition of directional amplification between cavities a₁ and a₂ when an input field is resonant with the cavity frequency, i.e., ω = 0. According to Eqs. (19) and (24), the transmission matrix elements T₂₁ and T₁₂ can be given by

T_{12} (ω) = - \frac{\sqrt{κ_{1} κ_{2}}}{A (ω)} G_{1} Γ_{3} (G_{2} + i J e^{i ϕ} G_{3} / Γ_{3}),

T_{21} (ω) = - \frac{\sqrt{κ_{1} κ_{2}}}{A (ω)} G_{1} Γ_{3} (G_{2} + i J e^{- i ϕ} G_{3} / Γ_{3}),

where

A (ω) = Γ_{1} (Γ_{2} Γ_{3} Γ_{m} - Γ_{2} G_{3}^{2} - Γ_{3} G_{2}^{2} + Γ_{m} J^{2} - 2 i J \cos ϕ G_{2} G_{3}) + G_{1}^{2} (Γ_{2} Γ_{3} + J^{2})

, Γ₁ = −κ₁/2 +iω, Γ_m = −γ_m/2 + iω, Γ₂ = −κ₂/2 +iω, and Γ₃ = −κ₃/2 +iω. Without loss of generality, we consider the case that the probe field incident on cavity a₁ can be amplified when it is transmitted from cavity a₂, but the probe field cannot be transmitted in the reverse direction, i.e., |T₂₁|² > 1 and |T₁₂|² = 0. According to Eqs. (25) and (26), it can be easily obtained that |T₁₂|² = 0 and |T₂₁|² ≠ 1 when ϕ = −π/2 and G₃ = G₂κ₃/(2J), and we will show that |T₂₁|² can be much larger than 1 due to constructive interference in this hybrid system. Furthermore, to prevent loss of the input field to other modes, it requires that |T_i₁/T₂₁| ≪ 1 (i = 3, 4), which can be realized by choosing

J = \sqrt{κ_{2} κ_{3}} / 3

. Under the above the conditions, the transmission amplitude T₂₁ on resonance can be given by

\begin{array}{l} T_{21} (0) = - \frac{8 \sqrt{κ_{1} κ_{2}} G_{1} G_{2}}{4 κ_{2} G_{1}^{2} - 4 κ_{1} G_{2}^{2} + κ_{1} κ_{2} γ_{m}} \\ = - \frac{2 \sqrt{C_{1} C_{2}}}{C_{1} - C_{2} + 1}, \end{array}

with the optomechanical cooperativity

C_{k} = 4 G_{k}^{2} / (κ_{k} γ_{m})

for k = 1, 2. Furthermore, the full transmission matrix on resonance is

T (0) = (\begin{matrix} \frac{C_{1} + C_{2} - 1}{C_{1} - C_{2} + 1} & 0 & - \frac{2 i \sqrt{C_{1} C_{2}}}{C_{1} - C_{2} + 1} & - \frac{2 i \sqrt{C_{1}}}{C_{1} - C_{2} + 1} \\ \frac{- 2 \sqrt{C_{1} C_{2}}}{C_{1} - C_{2} + 1} & 0 & i \frac{C_{1} + C_{2} + 1}{C_{1} - C_{2} + 1} & \frac{2 i \sqrt{C_{2}}}{C_{1} - C_{2} + 1} \\ 0 & i & 0 & 0 \\ - \frac{2 i \sqrt{C_{1}}}{C_{1} - C_{2} + 1} & 0 & \frac{- 2 \sqrt{C_{2}}}{C_{1} - C_{2} + 1} & \frac{C_{1} - C_{2} - 1}{C_{1} - C_{2} + 1} \end{matrix}),

In the following, we demonstrate numerically the directional amplification between different cavity modes. The parameters are chosen from the recent experiments [17, 18,20]: the decay rates of both optical and microwave cavities can be κ_k =1–10 MHz and the field-enhanced coupling strength G_k can reach a few tens of MHz. We assume that κ₁/2π = 2 MHz, κ₂/2π = κ₃/2π = 3 MHz, and the damping rate of the mechanical resonator γ_m/2π = 30 kHz. Because cavity modes a₂ and a₃ are driven on their blue sidebands, we first plot the stability diagram with respect to the cooperativities C₁ and C₂ in Fig. 2, and we will choose the parameters in the stable regime to realize the directional amplifier. It can be seen that the system is stable when C₂ < C₁ ≤ 60, and we can choose $C_{2} = C_{1} - 0.1 \sqrt{C_{1}}$ for a given C₁ to ensure the system operates in the stable regime.

Fig. 2 Stability diagram with respect to C₁ and C₂. Other parameters are κ₁/2π = 2 MHz, κ₂/2π = 3 MHz, κ₃/2π = 3 MHz, γ_m = κ₂/100, ω = 0, ϕ = −π/2, G₃ = G₂κ₃/(2J), and $J \sqrt{κ_{2} κ_{3}} / 2$ .

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According to Eqs. (25) and (26), we can investigate the directional amplification between cavities a₁ and a₂. Figure 3 plots |T₁₂|² and |T₂₁|² as functions of the probe detuning ω for ϕ = −π/2, 0, π/2, and π, respectively. It can be seen from Fig. 3(a) that |T₂₁(0)|² ≈ 23 dB while |T₁₂(0)|² = 0, which represents that the resonant probe field incident on cavity a₁ can be greatly amplified when it is transmitted from cavity a₂, but the transmission from cavity a₂ to cavity a₁ is totally forbidden. Therefore, directional amplification between microwave and optical photons is realized based on this optomechanical system. This phenomenon can be explained in terms of the interference between two possible paths, where one is along a₁ → b → a₂ with a transmission amplitude proportional to G₂, another is along a₁ → b → a₃ → a₂ with a transmission amplitude proportional to iJe^iϕG₃/Γ₃. When ϕ = −π/2, ω = 0, and G₃ = G₂ κ₃/(2J), constructive interference between the two paths results in the amplification from cavity a₁ to cavity a₂, but the transmission from a₂ to a₁ is suppressed due to destructive interference. If the phase difference ϕ is tuned to be π/2, we can see from Fig. 3(c) that the direction of amplification is reversed compared with that in Fig. 3(a), which can be illustrated by changing the direction of all the arrows in Fig. 1. In both cases, φ ≠ nπ (n is an integer), the time-reversal symmetry is broken and the nonreciprocal transmission appears in this optomechanical system. However, when φ = ±π, the transmission probability |T₂₁|² is equal to |T₁₂|² when the probe detuning ω varies, as shown in Fig. 3(b) and 3(d). The transmission in different direction is reciprocal and there’s no directional amplification.

Fig. 3 Transmission probabilities |T₁₂|² and |T₂₁|² as functions of the probe detuning ω for ϕ = −π/2, 0, π/2, and π, respectively. Other parameters are κ₁/2π = 2 MHz, κ₂/2π = 3 MHz, κ₃/2π = 3 MHz, γ_m = κ₂/100, C₁ = 10, $C_{2} = C_{1} - 0.1 \sqrt{C_{1}}$ , G₃ = G₂κ₃/(2J), and $J \sqrt{κ_{2} κ_{3}} / 2$ .

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Furthermore, directional amplification between the three cavity modes is discussed in Fig. 4 with phase difference ϕ = −π/2. It can be seen from Fig. 4(a) that the signal field incident on cavity cavity a₁ can be directionally amplified when it is transmitted from cavity a₂, and the signal field incident on cavity a₂ can be transmitted to cavity a₃ with unity gain, as shown in Fig. 4(b). Furthermore, Fig. 4(c) shows that the signal field incident on cavity a₃ can be directionally amplified when it is transmitted from cavity a₁, but not vice versa. Therefore, directional amplification in this optomechanical system is realized by combining two gain (G) process and one conversion (C) process, which is similar to the directional amplifier based on the Josephson Parametric Converter (JPC) [56]. Here, the three cavities can be viewed as the three ports in the directional amplifier, which can be labeled as the Signal (S) input, Idler (I) input, and Vacuum (V) input, respectively. The signal field incident on S port (amplifier input) can be directionally amplified from I port (amplifier output), and the signal incident on V port is transmitted with unity gain back to the S port. When ϕ = −π/2, cavity a₃ serves as the S port, cavity a₁ is the I port, and a₂ can be labeled as V port. The directional amplification is along the path $a_{3} \overset{G}{\to} a_{1} \overset{G}{\to} a_{2} \overset{C}{\to} a_{3}$ , as shown in Fig. 4(d). In addition, if we tune the phase difference ϕ to be π/2, the direction of the amplification will be reversed, which is along $a_{2} \overset{G}{\to} a_{1} \overset{G}{\to} a_{3} \overset{C}{\to} a_{2}$ . In this case, cavity a₂ serves as the S port and cavity a₃ is the V port. As discussed in Fig. 4, cavity a₃ serves as the amplifier input port and cavity a₁ can be viewed as the amplifier output port when the phase difference ϕ = −π/2. However, other modes such as the mechanical mode will inevitably add some noise to the directional amplifier. The added number of noise quanta of the amplifier can be obtained by calculating the output spectra of cavity a₁, which is given by [53, 66]

\begin{array}{l} S_{1, out} (ω) = \frac{1}{2} \int \frac{d Ω}{2 π} 〈 a_{1, out} (ω) a_{1, out}^{†} (Ω) + a_{1, out}^{†} (Ω) a_{1, out} (ω) 〉 \\ = \frac{1}{2} {| T_{11} (ω) |}^{2} + \frac{1}{2} {| T_{12} (ω) |}^{2} + \frac{1}{2} {| T_{13} (ω) |}^{2} + (n_{m} + \frac{1}{2}) {| T_{14} (ω) |}^{2}, \end{array}

where we have used the noise correlation function in the frequency domain and the relation o^†(ω) = [o(−ω)]^†. The noise added to the signal is defined as [51,66–68]

N_{1} (ω) = G^{- 1} \sum_{i \neq 3} (n_{i} + 1 / 2) {| T_{1 i} (ω) |}^{2}

, where the power gain of the directional amplifier is given by

G = {| T_{31} (0) |}^{2} = \frac{4 C_{1} C_{2}}{{(C_{1} - C_{2} + 1)}^{2}} .

On resonance, the noise added to the output of cavity a₁ is given by

N_{1} (0) = \frac{1}{2} \frac{{(C_{1} + C_{2} - 1)}^{2}}{4 C_{1} C_{2}} + (n_{m} + \frac{1}{2}) \frac{1}{C_{2}},

where we have assumed that the thermal photon number n₁ = n₂ = n₃ = 0 for the cavity modes and the thermal phonon number n₄ = n_m for the mechanical mode. We can see from Eq. (31) that the thermal noise from the mechanical resonator can be suppressed by increasing the cooperativity C₂. In addition, for large C₁ ≥ C₂, we find at zero frequency

N_{1} (0) \to \frac{1}{2}

, which is the quantum-limited value of the added noise for the phase-preserving linear amplifier [51,66–68].

Fig. 4 Directional amplifier. (a)–(c) Transmission probabilities |T_ij|² (i, j = 1, 2, 3) as functions of the probe detuning ω/2π. (d) Graphical representation of the amplification process, where S, I, and V represent/the “Signal”, “Idler”, and “Vacuum” ports, respectively. G and C correspond to the gain and conversion process. Other parameters used are ϕ = −π/2, κ₁/2π = 2 MHz, κ₂/2π = 3 MHz, κ₃/2π = 3 MHz, γ_m = κ₂/100, C₁ = 3, $C_{2} = C_{1} - 0.1 \sqrt{C_{1}}$ , G₃ = G₂κ₃/(2J), and $J \sqrt{κ_{2} κ_{3}} / 2$ .

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To see more clearly the effect of the cooperativity on the gain of the amplifier and the added noise, in Fig. 5, we plot (a) the transmission probability |T₁₃|² and (b) the added noise $N_{1}$ with respect to the probe detuning ω for different values of cooperativity C₁ with $C_{2} = C_{1} - 0.1 \sqrt{C_{1}}$ . It can be seen from Fig. 5(a) that the peak value of |T₁₃|² and the bandwidth become larger when C₁ increases from 1 to 50. Furthermore, Fig. 5(b) shows that the added number of noise quanta $N_{1}$ decreases with increasing the cooperativity C₁. In particular, when C₁ and C₂ are big enough, $N_{1}$ on resonance can approach 1/2. Therefore, we can realize a quantum-limited directional amplifier based on this triple-cavity optomechanical system.

Fig. 5 (a) Transmission probability |T₁₃|² and (b) added noise $N_{1}$ of the cavity a₁ as a function of the probe detuning ω with C₁ = {1, 5, 10, 50} and $C_{2} = C_{1} - 0.1 \sqrt{C_{1}}$ . Other parameters used are the same as those in Fig. 4 except n_m = 50.

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Finally, Fig. 5 shows that the gain and the bandwidth of the directional amplifier are enhanced by increasing the cooperativity C₁ with $C_{2} = C_{1} - 0.1 \sqrt{C_{1}}$ . The expressions of bandwidth and gain-bandwidth product can be approximated obtained as follows. If we assume the decay rates of the three cavities are the same, i.e., κ₁ = κ₂ = κ₃ = κ, then the denominator in Eq. (12) $A (ω) \approx \frac{1}{8} κ^{3} γ_{m} (C_{1} - C_{2} + 1) - \frac{1}{4} γ_{m} κ^{2} (κ / γ_{m} + C_{1} - 2 C_{2} + 2) i ω$ , where only the terms to the first order of ω is kept [51]. The bandwidth Γ is approximated by the smallest |ω| at which 2|A(0)|² = |A(ω)|² and we obtain the bandwidth Γ = |κ (C₁ − C₂ + 1)/(κ/γ_m + C₁ − 2C₂ + 2)|. For κ/γ_m ≫ {1, C₁, C₂} the bandwidth Γ ≈ γ_m (C₁ − C₂ + 1), which is constrained by the damping rate of the mechanical resonator. Consequently, the gain-bandwidth product $P \equiv Γ \sqrt{G} \approx 2 γ_{m} \sqrt{C_{1} C_{2}}$ . In addition, one should also consider the isolation bandwidth, where sufficient isolation is attained. Close to ω = 0, the reverse transmission amplitude $T_{12} (ω) \approx - i ω \sqrt{G} / κ$ . Thus, the isolation bandwidth is of order $κ / \sqrt{G}$ , which can be broadened by increasing the cavity decay rate κ. In our scheme, the isolation bandwidth is larger than the gain bandwidth, and therefore the signal field can be directionally amplified within the gain bandwidth.

4. Conclusion

To conclude, we have presented a quantum-limited directional amplifier based on a triple-cavity optomechanical system, where one microwave cavity is pumped on its red sideband and two coupled optical cavities are pumped on their respective blue sidebands. Quantum interference between two possible paths in this optomechanical systems results in the directional amplification between microwave and optical photons, where the phase difference between the optomechanical coupling strengths plays a vital role. Furthermore, we have shown that the gain and bandwidth of the amplifier can be improved by increasing the optomechanical cooperativity, and the noise added to this amplifier can approach the quantum limit of half a quanta in the large cooperativity.

Funding

Natural Science Foundation of China (No. 11304110, No. 11604115); Postdoctoral Science Foundation of China (Grant No. 2017M620593); Qing Lan Project of Universities in Jiangsu Province.

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References

View by:
Article Order
|
Year
|
Author
|
Publication

L. Deák and T. Fülöp, “Reciprocity in quantum, electromagnetic and other wave scattering,” Ann. Phys. 327, 1050–1077 (2012).
[Crossref]
L. J. Aplet and J. W. Carson, “A Faraday effect optical isolator,” Appl. Opt. 3, 544–545 (1964).
[Crossref]
R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717 (2004).
[Crossref]
L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photon. 5, 758–762 (2011).
[Crossref]
Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photon. 3, 91–94 (2009).
[Crossref]
H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocity induced by interband photonic transition on a dilicon vhip,” Phys. Rev. Lett. 109, 033901 (2012).
[Crossref]
L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524–529 (2014).
[Crossref]
B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
[Crossref]
X. Guo, C.-L. Zou, H. Jung, and H. X. Tang, “On-chip strong coupling and efficient frequency conversion between telecom and visible optical modes,” Phys. Rev. Lett. 117, 123902 (2016).
[Crossref] [PubMed]
R. Fleury, D. L. Sounas, C. F. Sieck, M. R. Haberman, and A. Alù, “Sound isolation and giant linear nonreciprocity in a compact acoustic circulator,” Science 343, 516–519 (2014).
[Crossref] [PubMed]
N. A. Estep, D. L. Sounas, J. Soric, and A. Alù, “Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops,” Nat. Phys. 10, 923–927 (2014).
[Crossref]
D.-W. Wang, H.-T. Zhou, M.-J. Guo, J.-X. Zhang, J. Evers, and S.-Y. Zhu, “Optical diode made from a moving photonic crystal,” Phys. Rev. Lett. 110, 093901 (2013).
[Crossref] [PubMed]
A. C. Mahoney, J. I. Colless, S. J. Pauka, J. M. Hornibrook, J. D. Watson, G. C. Gardner, M. J. Manfra, A. C. Doherty, and D. J. Reilly, “On-Chip microwave quantum Hall circulator,” Phys. Rev. X 7, 011007 (2017).
M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391 (2014).
[Crossref]
F. Marquardt and S. M. Girvin, “Optomechanics,” Physics 2, 40 (2009).
[Crossref]
H. Xiong, L. G. Si, X. Y. Lv, X. X. Yang, and Y. Wu, “Review of cavity optomechanics in the weak-coupling regime: from linearization to intrinsic nonlinear interactions,” Sci. China Phys. Mech. Astron. 58, 1–13 (2015).
[Crossref]
J. Chan, T. P. M. 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 (London) 478, 89–92 (2011).
[Crossref]
J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature (London) 475, 359–363 (2011).
[Crossref]
G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803 (2010).
[Crossref]
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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 (London) 472, 69–73 (2011).
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X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, and T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nat. Phys. 9, 179–184 (2013).
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H. Jing, Ş. K. Özdemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Sci. Rep. 5, 9663 (2015).
[Crossref] [PubMed]
Y. Jiao, H. Lü, J. Qian, Y. Li, and H. Jing, “Nonlinear optomechanics with gain and loss: amplifying higher-order sideband and group delay,” New J. Phys. 18, 083034 (2016).
[Crossref]
F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. J. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature (London) 480, 351–354 (2011).
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C. F. Ockeloen-Korppi, E. Damskägg, J.-M. Pirkkalainen, T. T. Heikkilä, F. Massel, and M. A. Sillanpää, “Low-noise amplification and frequency conversion with a multiport microwave optomechanical device,” Phys. Rev. X 6, 041024 (2016).
L. D. Tóth, N. R. Bernier, A. Nunnenkamp, A. K. Feofanov, and T. J. Kippenberg, “A dissipative quantum reservoir for microwave light using a mechanical oscillator,” Nat. Phys. 13, 787–793 (2017).
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R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, “Bidirectional and efficient conversion between microwave and optical light,” Nat. Phys. 10, 321–326 (2014).
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L. Tian, “Optoelectromechanical transducer: Reversible conversion between microwave and optical photons,” Ann. Phys. (Berlin) 527, 1 (2015).
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C. Dong, V. Fiore, M. C. Kuzyk, L. Tian, and H. Wang, “Optical wavelength conversion via optomechanical coupling in a silica resonator,” Ann. Phys. (Berlin) 527, 100 (2015).
[Crossref]
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]
I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two-level system,” Phys. Rev. Lett. 104, 083901 (2010).
[Crossref] [PubMed]
J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Gröblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, “Phonon counting and intensity interferometry of a nanomechanical resonator,” Nature (London) 520, 522–525 (2015).
[Crossref]
H. Jing, S. K. Özdemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “P T-symmetric phonon laser,” Phys. Rev. Lett. 113, 053604 (2014).
[Crossref]
H. Lü, S. K. Özdemir, L.-M. Kuang, F. Nori, and H. Jing, “Exceptional points in random-defect phonon lasers,” Phys. Rev. Applied 8, 044020 (2017).
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X.-Y. Lü, H. Jing, J.-Y. Ma, and Y. Wu, “P T-symmetry-breaking chaos in optomechanics,” Phys. Rev. Lett. 114, 253601 (2015).
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M. Hafezi and P. Rabl, “Optomechanically induced non-reciprocity in microring resonators,” Opt. Express 20, 7672–7684 (2012).
[Crossref] [PubMed]
A. Metelmann and A. A. Clerk, “Nonreciprocal photon transmission and amplification via reservoir engineering,” Phys. Rev. X 5, 021025 (2015).
J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Phys. 11, 275 (2015).
[Crossref]
C. H. Dong, Z. Shen, C. L. Zou, Y. L. Zhang, W. Fu, and G. C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
[Crossref] [PubMed]
Z. Shen, Y.-L. Zhang, Y. Chen, C.-L. Zou, Y.-F. Xiao, X.-B. Zou, F.-W. Sun, G.-C. Guo, and C.-H. Dong, “Experimental realization of optomechanically induced non-reciprocity,” Nat. Photon. 10, 657 (2016).
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X.-W. Xu and Y. Li, “Optical nonreciprocity and optomechanical circulator in three-mode optomechanical systems,” Phys. Rev. A 91, 053854 (2015).
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F. Ruesink, M.-A. Miri, A. Alù, and E. Verhagen, “Nonreciprocity and magnetic-free isolation based on optomechanical interactions,” Nat. Commun. 7, 13662 (2016).
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M.-A. Miri, F. Ruesink, E. Verhagen, and A. Alù, “Optical nonreciprocity based on optomechanical coupling,” Phys. Rev. Appl. 7, 064014 (2017).
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K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, “Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13, 465–471 (2017).
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Y. Li, Y. Y. Huang, X. Z. Zhang, and L. Tian, “Optical directional amplification in a three-mode optomechanical system,” Opt. Express 25, 18907–18916 (2017).
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Z. Shen, Y.-L. Zhang, Y. Chen, F.-W. Sun, X.-B. Zou, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Reconfigurable optomechanical circulator and directional amplifier,” Nat. Commun. 9, 1797 (2018).
[Crossref] [PubMed]
D. B. Sohn, S. Kim, and Gaurav Bahl, “Time-reversal symmetry breaking with acoustic pumping of nanophotonic circuits,” Nat. Photon. 12, 91–97 (2018).
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L. Tian and Z. Li, “Nonreciprocal quantum-state conversion between microwave and optical photons,” Phys. Rev. A 96, 013808 (2017).
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X.-W. Xu, Y. Li, A.-X. Chen, and Y.-x. Liu, “Nonreciprocal conversion between microwave and optical photons in electro-optomechanical systems,” Phys. Rev. A 93, 023827 (2016).
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D. Malz, L. D. Tóth, N. R. Bernier, A. K. Feofanov, T. J. Kippenberg, and A. Nunnenkamp, “Quantum-limited directional amplifiers with optomechanics,” Phys. Rev. Lett. 120, 023601 (2018).
[Crossref] [PubMed]
G. A. Peterson, F. Lecocq, K. Cicak, R. W. Simmonds, J. Aumentado, and J. D. Teufel, “Demonstration of efficient nonreciprocity in a microwave optomechanical circuit,” Phys. Rev. X 7, 031001 (2017).
N. R. Bernier, L. D. Tóth, A. Koottandavida, M. A. Ioannou, D. Malz, A. Nunnenkamp, A. K. Feofanov, and T. J. Kippenberg, “Nonreciprocal reconfigurable microwave optomechanical circuit,” Nat. Commun. 8, 604 (2017).
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S. Barzanjeh, M. Wulf, M. Peruzzo, M. Kalaee, P. B. Dieterle, O. Painter, and J. M. Fink, “Mechanical on-chip microwave circulator,” Nat. Commun. 8, 953 (2017).
[Crossref] [PubMed]
B. Abdo, K. Sliwa, L. Frunzio, and M. Devoret, “Directional amplification with a Josephson circuit,” Phys. Rev. X 3, 031001 (2013).
K. M. Sliwa, M. Hatridge, A. Narla, S. Shankar, L. Frunzio, R. J. Schoelkopf, and M. H. Devoret, “Reconfigurable Josephson circulator/directional amplifier,” Phys. Rev. X 5, 041020 (2015).
F. Lecocq, L. Ranzani, G. A. Peterson, K. Cicak, R. W. Simmonds, J. D. Teufel, and J. Aumentado, “Nonreciprocal microwave signal processing with a field-programmable Josephson amplifier,” Phys. Rev. Appl. 7, 024028 (2017).
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J. Bochmann, A. Vainsencher, D. D. Awschalom, and A. N. Cleland, “Nanomechanical coupling between microwave and optical photons,” Nat. Phys. 9, 712–716 (2013).
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T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342, 710–713 (2013).
[Crossref] [PubMed]
Y. Sato, Y. Tanaka, J. Upham, Y. Takahashi, T. Asano, and S. Noda, “Strong coupling between distant photonic nanocavities and its dynamic control,” Nat. Photon. 6, 56–61 (2012).
[Crossref]
K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nat. Photon. 6, 782–787 (2012).
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C. Genes, A. Mari, P. Tombesi, and D. Vitali, “Robust entanglement of a micromechanical resonator with output optical fields,” Phys. Rev. A 78, 032316 (2008).
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E. X. DeJesus and C. Kaufman, “Routh-Hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations,” Phys. Rev. A 35, 5288 (1987).
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I. S. Gradshteyn and I. M. Ryzhik, in Table of Integrals, Series and Products (Academic, 1980), p. 1119.
C. W. Gardiner and M. J. Collett, “Input and output in damped quantum system: quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761 (1985).
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A. Metelmann and A. A. Clerk, “Quantum-limited amplification via reservoir engineering,” Phys. Rev. Lett. 112, 133904 (2014).
[Crossref] [PubMed]
C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26, 1817 (1982).
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A. A. Clerk, M. H. Devoret, S. M. Girvin, Florian Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155 (2010).
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2018 (3)

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

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

D. B. Sohn, S. Kim, and Gaurav Bahl, “Time-reversal symmetry breaking with acoustic pumping of nanophotonic circuits,” Nat. Photon. 12, 91–97 (2018).
[Crossref]

2017 (11)

M.-A. Miri, F. Ruesink, E. Verhagen, and A. Alù, “Optical nonreciprocity based on optomechanical coupling,” Phys. Rev. Appl. 7, 064014 (2017).
[Crossref]

L. D. Tóth, N. R. Bernier, A. Nunnenkamp, A. K. Feofanov, and T. J. Kippenberg, “A dissipative quantum reservoir for microwave light using a mechanical oscillator,” Nat. Phys. 13, 787–793 (2017).
[Crossref]

F. Lecocq, L. Ranzani, G. A. Peterson, K. Cicak, R. W. Simmonds, J. D. Teufel, and J. Aumentado, “Nonreciprocal microwave signal processing with a field-programmable Josephson amplifier,” Phys. Rev. Appl. 7, 024028 (2017).
[Crossref]

N. R. Bernier, L. D. Tóth, A. Koottandavida, M. A. Ioannou, D. Malz, A. Nunnenkamp, A. K. Feofanov, and T. J. Kippenberg, “Nonreciprocal reconfigurable microwave optomechanical circuit,” Nat. Commun. 8, 604 (2017).
[Crossref] [PubMed]

H. Lü, S. K. Özdemir, L.-M. Kuang, F. Nori, and H. Jing, “Exceptional points in random-defect phonon lasers,” Phys. Rev. Applied 8, 044020 (2017).
[Crossref]

L. Tian and Z. Li, “Nonreciprocal quantum-state conversion between microwave and optical photons,” Phys. Rev. A 96, 013808 (2017).
[Crossref]

A. C. Mahoney, J. I. Colless, S. J. Pauka, J. M. Hornibrook, J. D. Watson, G. C. Gardner, M. J. Manfra, A. C. Doherty, and D. J. Reilly, “On-Chip microwave quantum Hall circulator,” Phys. Rev. X 7, 011007 (2017).

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

K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, “Generalized non-reciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering,” Nat. Phys. 13, 465–471 (2017).
[Crossref]

G. A. Peterson, F. Lecocq, K. Cicak, R. W. Simmonds, J. Aumentado, and J. D. Teufel, “Demonstration of efficient nonreciprocity in a microwave optomechanical circuit,” Phys. Rev. X 7, 031001 (2017).

S. Barzanjeh, M. Wulf, M. Peruzzo, M. Kalaee, P. B. Dieterle, O. Painter, and J. M. Fink, “Mechanical on-chip microwave circulator,” Nat. Commun. 8, 953 (2017).
[Crossref] [PubMed]

2016 (6)

X. Guo, C.-L. Zou, H. Jung, and H. X. Tang, “On-chip strong coupling and efficient frequency conversion between telecom and visible optical modes,” Phys. Rev. Lett. 117, 123902 (2016).
[Crossref] [PubMed]

F. Ruesink, M.-A. Miri, A. Alù, and E. Verhagen, “Nonreciprocity and magnetic-free isolation based on optomechanical interactions,” Nat. Commun. 7, 13662 (2016).
[Crossref] [PubMed]

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

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

Y. Jiao, H. Lü, J. Qian, Y. Li, and H. Jing, “Nonlinear optomechanics with gain and loss: amplifying higher-order sideband and group delay,” New J. Phys. 18, 083034 (2016).
[Crossref]

C. F. Ockeloen-Korppi, E. Damskägg, J.-M. Pirkkalainen, T. T. Heikkilä, F. Massel, and M. A. Sillanpää, “Low-noise amplification and frequency conversion with a multiport microwave optomechanical device,” Phys. Rev. X 6, 041024 (2016).

2015 (11)

J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Gröblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, “Phonon counting and intensity interferometry of a nanomechanical resonator,” Nature (London) 520, 522–525 (2015).
[Crossref]

C. H. Dong, Z. Shen, C. L. Zou, Y. L. Zhang, W. Fu, and G. C. Guo, “Brillouin-scattering-induced transparency and non-reciprocal light storage,” Nat. Commun. 6, 6193 (2015).
[Crossref] [PubMed]

C. Dong, V. Fiore, M. C. Kuzyk, L. Tian, and H. Wang, “Optical wavelength conversion via optomechanical coupling in a silica resonator,” Ann. Phys. (Berlin) 527, 100 (2015).
[Crossref]

L. Tian, “Optoelectromechanical transducer: Reversible conversion between microwave and optical photons,” Ann. Phys. (Berlin) 527, 1 (2015).
[Crossref]

A. Metelmann and A. A. Clerk, “Nonreciprocal photon transmission and amplification via reservoir engineering,” Phys. Rev. X 5, 021025 (2015).

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

H. Jing, Ş. K. Özdemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Sci. Rep. 5, 9663 (2015).
[Crossref] [PubMed]

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

H. Xiong, L. G. Si, X. Y. Lv, X. X. Yang, and Y. Wu, “Review of cavity optomechanics in the weak-coupling regime: from linearization to intrinsic nonlinear interactions,” Sci. China Phys. Mech. Astron. 58, 1–13 (2015).
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K. M. Sliwa, M. Hatridge, A. Narla, S. Shankar, L. Frunzio, R. J. Schoelkopf, and M. H. Devoret, “Reconfigurable Josephson circulator/directional amplifier,” Phys. Rev. X 5, 041020 (2015).

J. Kim, M. C. Kuzyk, K. Han, H. Wang, and G. Bahl, “Non-reciprocal Brillouin scattering induced transparency,” Nat. Phys. 11, 275 (2015).
[Crossref]

2014 (8)

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity-time symmetry and variable optical isolation in active-passive-coupled microresonators,” Nat. Photon. 8, 524–529 (2014).
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R. Fleury, D. L. Sounas, C. F. Sieck, M. R. Haberman, and A. Alù, “Sound isolation and giant linear nonreciprocity in a compact acoustic circulator,” Science 343, 516–519 (2014).
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R. W. Andrews, R. W. Peterson, T. P. Purdy, K. Cicak, R. W. Simmonds, C. A. Regal, and K. W. Lehnert, “Bidirectional and efficient conversion between microwave and optical light,” Nat. Phys. 10, 321–326 (2014).
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M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391 (2014).
[Crossref]

A. Metelmann and A. A. Clerk, “Quantum-limited amplification via reservoir engineering,” Phys. Rev. Lett. 112, 133904 (2014).
[Crossref] [PubMed]

H. Jing, S. K. Özdemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “P T-symmetric phonon laser,” Phys. Rev. Lett. 113, 053604 (2014).
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N. A. Estep, D. L. Sounas, J. Soric, and A. Alù, “Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops,” Nat. Phys. 10, 923–927 (2014).
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B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014).
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2013 (5)

J. Bochmann, A. Vainsencher, D. D. Awschalom, and A. N. Cleland, “Nanomechanical coupling between microwave and optical photons,” Nat. Phys. 9, 712–716 (2013).
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D.-W. Wang, H.-T. Zhou, M.-J. Guo, J.-X. Zhang, J. Evers, and S.-Y. Zhu, “Optical diode made from a moving photonic crystal,” Phys. Rev. Lett. 110, 093901 (2013).
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T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342, 710–713 (2013).
[Crossref] [PubMed]

B. Abdo, K. Sliwa, L. Frunzio, and M. Devoret, “Directional amplification with a Josephson circuit,” Phys. Rev. X 3, 031001 (2013).

X. Zhou, F. Hocke, A. Schliesser, A. Marx, H. Huebl, R. Gross, and T. J. Kippenberg, “Slowing, advancing and switching of microwave signals using circuit nanoelectromechanics,” Nat. Phys. 9, 179–184 (2013).
[Crossref]

2012 (6)

M. Hafezi and P. Rabl, “Optomechanically induced non-reciprocity in microring resonators,” Opt. Express 20, 7672–7684 (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. Sato, Y. Tanaka, J. Upham, Y. Takahashi, T. Asano, and S. Noda, “Strong coupling between distant photonic nanocavities and its dynamic control,” Nat. Photon. 6, 56–61 (2012).
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H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocity induced by interband photonic transition on a dilicon vhip,” Phys. Rev. Lett. 109, 033901 (2012).
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K. Fang, Z. Yu, and S. Fan, “Realizing effective magnetic field for photons by controlling the phase of dynamic modulation,” Nat. Photon. 6, 782–787 (2012).
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L. Deák and T. Fülöp, “Reciprocity in quantum, electromagnetic and other wave scattering,” Ann. Phys. 327, 1050–1077 (2012).
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2011 (5)

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 (London) 472, 69–73 (2011).
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L. Bi, J. Hu, P. Jiang, D. H. Kim, G. F. Dionne, L. C. Kimerling, and C. A. Ross, “On-chip optical isolation in monolithically integrated non-reciprocal optical resonators,” Nat. Photon. 5, 758–762 (2011).
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F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. J. Hakonen, and M. A. Sillanpää, “Microwave amplification with nanomechanical resonators,” Nature (London) 480, 351–354 (2011).
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J. Chan, T. P. M. 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 (London) 478, 89–92 (2011).
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J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature (London) 475, 359–363 (2011).
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2010 (4)

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

A. A. Clerk, M. H. Devoret, S. M. Girvin, Florian Marquardt, and R. J. Schoelkopf, “Introduction to quantum noise, measurement, and amplification,” Rev. Mod. Phys. 82, 1155 (2010).
[Crossref]

I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon laser action in a tunable two-level system,” Phys. Rev. Lett. 104, 083901 (2010).
[Crossref] [PubMed]

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

2009 (2)

F. Marquardt and S. M. Girvin, “Optomechanics,” Physics 2, 40 (2009).
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Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photon. 3, 91–94 (2009).
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2008 (1)

C. Genes, A. Mari, P. Tombesi, and D. Vitali, “Robust entanglement of a micromechanical resonator with output optical fields,” Phys. Rev. A 78, 032316 (2008).
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2004 (1)

R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717 (2004).
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1987 (1)

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

1985 (1)

C. W. Gardiner and M. J. Collett, “Input and output in damped quantum system: quantum stochastic differential equations and the master equation,” Phys. Rev. A 31, 3761 (1985).
[Crossref]

1982 (1)

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26, 1817 (1982).
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1964 (1)

L. J. Aplet and J. W. Carson, “A Faraday effect optical isolator,” Appl. Opt. 3, 544–545 (1964).
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Abdo, B.

B. Abdo, K. Sliwa, L. Frunzio, and M. Devoret, “Directional amplification with a Josephson circuit,” Phys. Rev. X 3, 031001 (2013).

Agarwal, G. S.

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

Alegre, T. P. M.

Allman, M. S.

Alù, A.

M.-A. Miri, F. Ruesink, E. Verhagen, and A. Alù, “Optical nonreciprocity based on optomechanical coupling,” Phys. Rev. Appl. 7, 064014 (2017).
[Crossref]

F. Ruesink, M.-A. Miri, A. Alù, and E. Verhagen, “Nonreciprocity and magnetic-free isolation based on optomechanical interactions,” Nat. Commun. 7, 13662 (2016).
[Crossref] [PubMed]

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Fig. 1 Schematic diagram of the triple-cavity optomechanical system. One microwave cavity a₁ and two optical cavities a₂ and a₃ are respectively coupled to the common mechanical resonator with the field-enhanced optomechanical coupling strength G_k(k = 1, 2, 3), where ϕ is the phase difference between G_k. Moreover, the two optical cavities are coupled directly via hopping interaction J to facilitate the nonreciprocal transmission. Cavity a₁ is driven on its red sideband by a pump field at the frequency ω_d_,1, and cavities a₂ and a₃ are driven on their respective blue sidebands by the pump fields at the frequencies ω_d_,2 and ω_d_,3. The solid arrows represent the transmission elements between different modes.

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Fig. 2 Stability diagram with respect to C₁ and C₂. Other parameters are κ₁/2π = 2 MHz, κ₂/2π = 3 MHz, κ₃/2π = 3 MHz, γ_m = κ₂/100, ω = 0, ϕ = −π/2, G₃ = G₂κ₃/(2J), and

J \sqrt{κ_{2} κ_{3}} / 2

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C_{2} = C_{1} - 0.1 \sqrt{C_{1}}

, G₃ = G₂κ₃/(2J), and

J \sqrt{κ_{2} κ_{3}} / 2

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Fig. 4 Directional amplifier. (a)–(c) Transmission probabilities |T_ij|² (i, j = 1, 2, 3) as functions of the probe detuning ω/2π. (d) Graphical representation of the amplification process, where S, I, and V represent/the “Signal”, “Idler”, and “Vacuum” ports, respectively. G and C correspond to the gain and conversion process. Other parameters used are ϕ = −π/2, κ₁/2π = 2 MHz, κ₂/2π = 3 MHz, κ₃/2π = 3 MHz, γ_m = κ₂/100, C₁ = 3,

C_{2} = C_{1} - 0.1 \sqrt{C_{1}}

, G₃ = G₂κ₃/(2J), and

J \sqrt{κ_{2} κ_{3}} / 2

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Fig. 5 (a) Transmission probability |T₁₃|² and (b) added noise

N_{1}

of the cavity a₁ as a function of the probe detuning ω with C₁ = {1, 5, 10, 50} and

C_{2} = C_{1} - 0.1 \sqrt{C_{1}}

. Other parameters used are the same as those in Fig. 4 except n_m = 50.

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Equations (31)

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\begin{array}{l} H = \sum_{k = 1}^{3} ω_{k} a_{k}^{†} a_{k} + ω_{m} b^{†} b + \sum_{k = 1}^{3} g_{k} a_{k}^{†} a_{k} (b^{†} + b) + J (a_{2}^{†} a_{3} + a_{3}^{†} a_{2}) \\ + \sum_{k = 1}^{3} ε_{k} (a_{k}^{†} e^{- i ω_{d, k} t} + a_{k} e^{i ω_{d, k} t}), \end{array}

H_{rot} = \sum_{k = 1}^{3} Δ_{k} a_{k}^{†} a_{k} + ω_{m} b^{†} b + \sum_{k = 1}^{3} g_{k} a_{k}^{†} a_{k} (b^{†} + b) + J (a_{2}^{†} a_{3} + a_{3}^{†} a_{2}) + \sum_{k = 1}^{3} ε_{k} (a_{k}^{†} + a_{k}),

{\dot{a}}_{1}^{†} = - (κ_{1} / 2 - i Δ_{1}) a_{1}^{†} + i g_{1} a_{1}^{†} (b^{†} + b) + i ε_{1} + \sqrt{κ_{1}} a_{1, in}^{†},

{\dot{a}}_{2} = - (κ_{2} / 2 + i Δ_{2}) a_{2} - i g_{2} a_{2} (b^{†} + b) - i J a_{3} - i ε_{2} + \sqrt{κ_{2}} a_{2, in},

{\dot{a}}_{3} = - (κ_{3} / 2 + i Δ_{3}) a_{3} - i g_{3} a_{3} (b^{†} + b) - i J a_{2} - i ε_{3} + \sqrt{κ_{3}} a_{3, in},

{\dot{b}}^{†} = - (γ_{m} / 2 - i ω_{m}) b^{†} + i \sum_{k} g_{k} a_{k}^{†} a_{k} + \sqrt{γ_{m}} b_{in}^{†} .

\begin{array}{l} 〈 a_{k, in} (t) a_{k, in}^{†} (t^{'}) = δ (t - t^{'}), 〈 a_{k, in}^{†} (t) a_{k, in} (t^{'}) = 0, \\ 〈 b_{in} (t) b_{in}^{†} (t^{'}) 〉 = (n_{m} + 1) δ (t - t^{'}), 〈 b_{in}^{†} (t) b_{in} (t^{'}) 〉 = n_{m} δ (t - t^{'}), \end{array}

α_{1}^{*} = \frac{i ε_{1}}{κ_{1} / 2 - i Δ_{1}^{'}}, α_{2} = - \frac{i J α_{3} + i ε_{2}}{κ_{2} / 2 + i Δ_{2}^{'}}, α_{3} = - \frac{i J α_{2} + i ε_{3}}{κ_{3} / 2 + i Δ_{3}^{'}}, β^{*} = \frac{i \sum_{k} g_{k} {| α_{k} |}^{2}}{γ_{m} / 2 - i ω_{m}},

δ {\dot{a}}_{1}^{†} = - (κ_{1} / 2 - i Δ_{1}^{'}) δ a_{1}^{†} + i g α_{1}^{*} (δ b^{†} + δ b) + \sqrt{κ_{1}} a_{1, in},

δ {\dot{a}}_{2} = - (κ_{2} / 2 + i Δ_{2}^{'}) δ a_{2} - i g_{2} α_{2} (δ b^{†} + δ b) - i J δ a_{3} + \sqrt{κ_{2}} a_{2, in},

δ {\dot{a}}_{3} = - (κ_{3} / 2 + i Δ_{3}^{'}) δ a_{3} - i g_{3} α_{3} (δ b^{†} + δ b) - i J δ a_{2} + \sqrt{κ_{3}} a_{3, in},

δ {\dot{b}}^{†} = - (γ_{m} / 2 - i ω_{m}) δ b^{†} + i \sum_{k} g_{k} (α_{k}^{*} δ a_{k} + α_{k} δ a_{k}^{†}) + \sqrt{γ_{m}} b_{in}^{†} .

δ {\dot{a}}_{1}^{†} = - \frac{κ_{1}}{2} δ a_{1}^{†} + i G_{1} e^{i ϕ_{1}} δ b^{†} + \sqrt{κ_{1}} a_{1, in}^{†},

δ {\dot{a}}_{2} = - \frac{κ_{2}}{2} δ a_{2} - i G_{2} e^{- i ϕ_{2}} δ b^{†} - i J δ a_{3} + \sqrt{κ_{2}} a_{2, in},

δ {\dot{a}}_{3} = \frac{κ_{3}}{2} δ a_{3} - i G_{3} e^{- i ϕ_{3}} δ b^{†} - i J δ a_{2} + \sqrt{κ_{3}} a_{3, in},

δ {\dot{b}}^{†} = - \frac{γ_{m}}{2} δ b^{†} + i (G_{1} e^{- i ϕ_{1}} δ a_{1}^{†} + G_{2} e^{i ϕ_{2}} δ a_{2} + G_{3} e^{i ϕ_{3}} δ a_{3}) + \sqrt{γ_{m}} b_{i n}^{†},

H_{eff} = G_{1} δ a_{1} δ b^{†} e^{i ϕ_{1}} + G_{2} δ a_{2} δ b e^{i ϕ_{2}} + G_{3} δ a_{3} δ b e^{i ϕ_{3}} + J δ a_{2} δ a_{3}^{†} + H . c ..

\dot{μ} = M μ + \sqrt{K} μ_{i n},

M = (\begin{matrix} - κ_{1} / 2 & 0 & 0 & i G_{1} \\ 0 & - κ_{2} / 2 & - i J & - i G_{2} \\ 0 & - i J & - κ_{3} / 2 & - i G_{3} e^{- i ϕ} \\ i G_{1} & i G_{2} & i G_{3} e^{i ϕ} & - γ_{m} / 2 \end{matrix}) .

o (ω) = \int_{- \infty}^{+ \infty} o (t) e^{i ω t} d t,

o^{†} (ω) = \int_{- \infty}^{+ \infty} o^{†} (t) e^{i ω t} d t,

μ (ω) = - {(M + i ω I)}^{- 1} \sqrt{K} μ_{in} (ω),

μ_{out} (ω) = T (ω) μ_{in} (ω),

T (ω) = I + \sqrt{K} {(M + i ω I)}^{- 1} \sqrt{K} .

T_{12} (ω) = - \frac{\sqrt{κ_{1} κ_{2}}}{A (ω)} G_{1} Γ_{3} (G_{2} + i J e^{i ϕ} G_{3} / Γ_{3}),

T_{21} (ω) = - \frac{\sqrt{κ_{1} κ_{2}}}{A (ω)} G_{1} Γ_{3} (G_{2} + i J e^{- i ϕ} G_{3} / Γ_{3}),

\begin{array}{l} T_{21} (0) = - \frac{8 \sqrt{κ_{1} κ_{2}} G_{1} G_{2}}{4 κ_{2} G_{1}^{2} - 4 κ_{1} G_{2}^{2} + κ_{1} κ_{2} γ_{m}} \\ = - \frac{2 \sqrt{C_{1} C_{2}}}{C_{1} - C_{2} + 1}, \end{array}

T (0) = (\begin{matrix} \frac{C_{1} + C_{2} - 1}{C_{1} - C_{2} + 1} & 0 & - \frac{2 i \sqrt{C_{1} C_{2}}}{C_{1} - C_{2} + 1} & - \frac{2 i \sqrt{C_{1}}}{C_{1} - C_{2} + 1} \\ \frac{- 2 \sqrt{C_{1} C_{2}}}{C_{1} - C_{2} + 1} & 0 & i \frac{C_{1} + C_{2} + 1}{C_{1} - C_{2} + 1} & \frac{2 i \sqrt{C_{2}}}{C_{1} - C_{2} + 1} \\ 0 & i & 0 & 0 \\ - \frac{2 i \sqrt{C_{1}}}{C_{1} - C_{2} + 1} & 0 & \frac{- 2 \sqrt{C_{2}}}{C_{1} - C_{2} + 1} & \frac{C_{1} - C_{2} - 1}{C_{1} - C_{2} + 1} \end{matrix}),

\begin{array}{l} S_{1, out} (ω) = \frac{1}{2} \int \frac{d Ω}{2 π} 〈 a_{1, out} (ω) a_{1, out}^{†} (Ω) + a_{1, out}^{†} (Ω) a_{1, out} (ω) 〉 \\ = \frac{1}{2} {| T_{11} (ω) |}^{2} + \frac{1}{2} {| T_{12} (ω) |}^{2} + \frac{1}{2} {| T_{13} (ω) |}^{2} + (n_{m} + \frac{1}{2}) {| T_{14} (ω) |}^{2}, \end{array}

G = {| T_{31} (0) |}^{2} = \frac{4 C_{1} C_{2}}{{(C_{1} - C_{2} + 1)}^{2}} .

N_{1} (0) = \frac{1}{2} \frac{{(C_{1} + C_{2} - 1)}^{2}}{4 C_{1} C_{2}} + (n_{m} + \frac{1}{2}) \frac{1}{C_{2}},