Researching | Tunable mechanical-mode coupling based on nanobeam-double optomechanical cavities

Journals >Photonics Research >Volume 10 >Issue 8 >Page 1819 > Article

Photonics Research
Vol. 10, Issue 8, 1819 (2022)

Tunable mechanical-mode coupling based on nanobeam-double optomechanical cavities | Editors' Pick

Qiancheng Xu^1,2, Kaiyu Cui^1,2,*, Ning Wu^1,2, Xue Feng^1,2..., Fang Liu^1,2, Wei Zhang^1,2,3 and Yidong Huang^1,2,3|Show fewer author(s)

Author Affiliations

¹Department of Electronic Engineering, Tsinghua University, Beijing 100084, China

²Beijing National Research Center for Information Science and Technology (BNRist), Tsinghua University, Beijing 100084, China

³Beijing Academy of Quantum Information Sciences, Beijing, China

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DOI: 10.1364/PRJ.447711 Cite this Article Set citation alerts

Qiancheng Xu, Kaiyu Cui, Ning Wu, Xue Feng, Fang Liu, Wei Zhang, Yidong Huang, "Tunable mechanical-mode coupling based on nanobeam-double optomechanical cavities," Photonics Res. 10, 1819 (2022) Copy Citation Text

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Abstract

Tunable coupled mechanical resonators with nonequilibrium dynamic phenomena have attracted considerable attention in quantum simulations, quantum computations, and non-Hermitian systems. In this study, we propose tunable mechanical-mode coupling based on nanobeam-double optomechanical cavities. The excited optical mode interacts with both symmetric and antisymmetric mechanical supermodes and mediates coupling at a frequency of approximately 4.96 GHz. The mechanical-mode coupling is tuned through both optical spring and gain effects, and the reduced coupled frequency difference in non-Hermitian parameter space is observed. These results benefit research on the microscopic mechanical parity–time symmetry for topology and on-chip high-sensitivity sensors.

{\ddot{x}}_{1} + (Γ_{m} + Γ_{opt}) {\dot{x}}_{1} + {(Ω_{m} + δ Ω_{m})}^{2} x_{1} + Ω_{m} k x_{2} = F_{1} / m_{eff}, {\ddot{x}}_{2} + Γ_{m} {\dot{x}}_{2} + Ω_{m}^{2} x_{2} + Ω_{m} k x_{1} = F_{2} / m_{eff},

(1)

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\dot{a} = (i Δ - \frac{γ_{o}}{2}) a + i G x_{1} a + \sqrt{γ_{oc}} s_{in}, {\ddot{x}}_{1} = - Ω_{m}^{2} x_{1} - Γ_{m} {\dot{x}}_{1} - Ω_{m} κ x_{2} + ℏ G {| a |}^{2} / m_{eff} + F_{1} / m_{eff}, {\ddot{x}}_{2} = - Ω_{m}^{2} x_{2} - Γ_{m} {\dot{x}}_{2} - Ω_{m} κ x_{1} + F_{2} / m_{eff},

(B1)

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\dot{a} = (i \bar{Δ} - \frac{γ_{o}}{2}) a + i G \bar{a} x_{1}, {\ddot{x}}_{1} = - Ω_{m}^{2} x_{1} - Γ_{m} {\dot{x}}_{1} - Ω_{m} κ x_{2} + ℏ G ({\bar{a}}^{*} a + \bar{a} a^{*}) / m_{eff} + F_{1} / m_{eff}, {\ddot{x}}_{2} = - Ω_{m}^{2} x_{2} - Γ_{m} {\dot{x}}_{2} - Ω_{m} κ x_{1} + F_{2} / m_{eff},

(B2)

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\bar{a} = \frac{\sqrt{γ_{o c}} s_{in}}{- i \bar{Δ} + γ_{o} / 2}, {\bar{x}}_{1} = \frac{ℏ G {\bar{a}}^{2}}{m_{eff} (Ω_{m}^{2} - κ^{2})}, {\bar{x}}_{2} = - \frac{κ}{Ω_{m}} \frac{ℏ G {\bar{a}}^{2}}{m_{eff} (Ω_{m}^{2} - κ^{2})},

(B3)

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{\ddot{x}}_{+} + (Γ_{m} + \frac{1}{2} Γ_{opt}) {\dot{x}}_{+} + (Ω_{m}^{2} + Ω_{m} κ + Ω_{m} δ Ω_{m}) x_{+} = - Ω_{m} δ Ω_{m} x_{-} - \frac{1}{2} Γ_{opt} {\dot{x}}_{-} + F_{+} / m_{eff}, {\ddot{x}}_{-} + (Γ_{m} + \frac{1}{2} Γ_{opt}) {\dot{x}}_{-} + (Ω_{m}^{2} - Ω_{m} κ + Ω_{m} δ Ω_{m}) x_{-} = - Ω_{m} δ Ω_{m} x_{+} - \frac{1}{2} Γ_{opt} {\dot{x}}_{+} + F_{-} / m_{eff},

(B4)

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δ Ω_{m} = \frac{8 g^{2} Ω_{m}}{γ_{o}^{2} + 16 Ω_{m}^{2}}, Γ_{opt} = - \frac{4 g^{2}}{γ_{o}} \frac{16 Ω_{m}^{2}}{γ_{o}^{2} + 16 Ω_{m}^{2}}

(B5)

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ϵ^{- 1} (ω) a (ω) = i G \bar{a} x_{1} (ω), χ_{1}^{- 1} (ω) x_{1} (ω) = - Ω_{m} κ x_{2} (ω) + F_{1} (ω) / m_{eff}, χ_{2}^{- 1} (ω) x_{2} (ω) = - Ω_{m} κ x_{1} (ω) + F_{2} (ω) / m_{eff},

(B6)

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ϵ^{- 1} (ω) = - i (ω + \bar{Δ}) + \frac{γ_{o}}{2}, χ_{1}^{- 1} (ω) = Ω_{m}^{2} - ω^{2} - i ω Γ_{m} - i Σ (ω), χ_{2}^{- 1} (ω) = Ω_{m}^{2} - ω^{2} - i ω Γ_{m}, Σ (ω) = \frac{ℏ G^{2} {| \bar{a} |}^{2}}{m_{eff}} [ϵ (ω) - ϵ^{*} (- ω)],

(B7)

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x_{1} (ω) = \frac{F_{1} / m_{eff} - Ω_{m} κ F_{2} / m_{eff} χ_{2}}{χ_{1}^{- 1} - Ω_{m}^{2} κ^{2} χ_{2}} .

(B8)

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I_{o} = {| s_{in} - \sqrt{γ_{o c}} (\bar{a} + a) |}^{2} \approx {\bar{I}}_{o} - \sqrt{γ_{o c}} (d_{in}^{*} a + d_{in} a^{*}),

(B9)

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d_{in} = (1 - \frac{γ_{o c}}{- i Δ + γ_{o} / 2}) s_{in} .

(B10)

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S_{II} (ω) = {| H (ω) |}^{2} S_{x x} (ω),

(B11)

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H (ω) = - i \sqrt{γ_{o c}} G \bar{a} [d_{in}^{*} ϵ (ω) - d_{in} ϵ^{*} (- ω)],

(B12)

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S_{x x} (ω) = \frac{1 + Ω_{m}^{2} κ^{2} {| χ_{2} (ω) |}^{2}}{{| χ_{1}^{- 1} (ω) - Ω_{m}^{2} κ^{2} χ_{2} (ω) |}^{2}} \frac{S_{FF} (ω)}{m_{eff}^{2}} .

(B13)

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\dot{a} (t) = [i Δ (t) - \frac{γ_{o}}{2}] a (t) + \sqrt{γ_{c}} a_{in} (t), Δ \dot{T} (t) = - γ_{th} Δ T (t) + \frac{η_{s}}{c_{p, m}} (γ_{lin} + γ_{FCA}) {| a (t) |}^{2}, \dot{N} (t) = - γ_{fc} N (t) + β_{fc} {| a (t) |}^{4},

(C1)

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Δ (t) = ω_{l} - ω_{o} + ω_{o} [\frac{1}{n} \frac{d n}{d T} Δ T (t) + \frac{1}{n} \frac{d n}{d N} N (t)] .

(C2)

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Qiancheng Xu, Kaiyu Cui, Ning Wu, Xue Feng, Fang Liu, Wei Zhang, Yidong Huang, "Tunable mechanical-mode coupling based on nanobeam-double optomechanical cavities," Photonics Res. 10, 1819 (2022)

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