• Photonics Research
  • Vol. 9, Issue 11, 11002152 (2021)
Chang-Long Zhu1, Yu-Long Liu2, Lan Yang3, Yu-Xi Liu4、5、6、*, and Jing Zhang1、5、7、*
Author Affiliations
  • 1Department of Automation, Tsinghua University, Beijing 100084, China
  • 2Beijing Academy of Quantum Information Sciences, Beijing 100193, China
  • 3Department of Electrical and Systems Engineering, Washington University, St. Louis, Missouri 63130, USA
  • 4Institute of Microelectronics, Tsinghua University, Beijing 100084, China
  • 5Center for Quantum Information Science and Technology, BNRist, Beijing 100084, China
  • 6e-mail: yuxiliu@mail.tsinghua.edu.cn
  • 7e-mail: jing-zhang@mail.tsinghua.edu.cn
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    Abstract

    Synchronization has great impacts in various fields such as self-clocking, communication, and neural networks. Here, we present a mechanism of synchronization for two mechanical modes in two coupled optomechanical resonators with a parity-time (PT)-symmetric structure. It is shown that the degree of synchronization between the two far-off-resonant mechanical modes can be increased by decreasing the coupling strength between the two optomechanical resonators due to the large amplification of optomechanical interaction near the exceptional point. Additionally, when we consider the stochastic noises in the optomechanical resonators by working near the exceptional point, we find that more noises can enhance the degree of synchronization of the system under a particular parameter regime. Our results open up a new dimension of research for PT-symmetric systems and synchronization.
    α˙k=Γopkαkiκα3kigomαk(βk+βk*)+2γkexϵk,β˙k=(Γmk+iΩk)βkigom|αk|2,

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    ddt[α1α2]=i[2Δ¯1+iγκκ2Δ¯2iγ][α1α2]+[2γ1exϵ12γ2exϵ2],

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    ωo+=Δ¯1+Δ¯2±i[γ+i(Δ¯2Δ¯1)]2κ2.

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    ωo±=Δ1+Δ22±i(γ+iΔ2)2κ2.

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    gomΔ<Δ1,2γ,κ,

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    Δsplitγ2|Δ+4gom2Ω2Ω1Ω1Ω21Δ2ϵ2|1γ,

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    δΩ1=δΩ2gom2Δ(γ2+κ2)2γϵ2[(κ2γ2)2+γ2Δ2]2.

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    κmech4gom2Δκ2γ3ϵ2[(κ2γ2)2+γ2Δ2]2.

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    Mcc=max0<t<+1ϕ1ϕ20+x1(τt)x2(τ)dτ,ϕi=0+xi2(τ)dτ.

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    α˙k=i(Δk+gomxk)αk+(1)1+kγkαkiκα3k+2γkexϵk+ξk(t),x¨k=2Γmkx˙kΩk2xkgom|αk|2,

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    x¨1=2Γmx˙1Ω˜12x1κmechx2+Γnoise1(t),x¨2=2Γmx˙2Ω˜22x2κmechx1+Γnoise2(t),

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    Γnoisei(t)=0,Γnoisei(t)Γnoisej(t)=4ΓmkTδ(tt),

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    R(τ,t)12Ω˜12τt+q2κmechΩ˜12τt2+q3κmechΩ˜12τt312Ω˜12τt+2ΓmkTκmechΩ˜12τt2+43ΓmkTκmechΩ˜12τt3,

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    0=[(1)1+kγkiΔk]αksiκα3ksigomαks(βks+βks*)+2γkexϵk,0=(Γmk+iΩk)βksigom|αks|2.(A1)

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    β1s=gomΩ1+iΓm1Γm12+Ω12|α1s|2,β2s=gomΩ2+iΓm2Γm22+Ω22|α2s|2,(A2)

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    [(1)1+kγki(Δk+Δks)]αksiκα3ks+2γkexϵk=0,k=1,2,(A3)

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    Δ1s=2Ω1gom2Γm12+Ω12|α1s|2,Δ2s=2Ω2gom2Γm22+Ω22|α2s|2.(A4)

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    α˙k=[(1)1+kγki(Δk+Δks)]αkiκα3k+2γkexϵk.(A5)

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    ωo±=iγ+(Δ¯1+Δ¯2)±i[γ++i(Δ¯2Δ¯1)]2κ2,(A6)

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    ωo±=Δ¯1+Δ¯2±i[γ+i(Δ¯2Δ¯1)]2κ2.(A7)

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    ωo±(Δ¯1+Δ¯2)±iγ2κ2.(A8)

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    Δsplitγ2|Δ+4gom2Ω2Ω1Ω1Ω21Δ2ϵ2|1γ,(A9)

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    gomΔ<Δ1,2γ,(A10)

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    [α˙1α˙2]=M[α1α2]+[igomα1(β1+β1*)igomα2(β2+β2*)]+[2γ1exϵ12γ2exϵ2],(B1)

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    α±ss=μ[ωigom(β2+β2*)]2γ1exϵ1Ξ2±λ[ωigom(β1+β1*)]2γ2exϵ2Ξ2,(B2)

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    Ξ2=ω+ω+igom(ω+λωλ+)(β1+β1*)+igom(ω+λ++ωλ+)(β2+β2*)gom2(λ+λ)2(β1+β1*)(β2+β2*).(B3)

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    α+ssμ2γ1exϵ1λ2γ2exϵ2ω++igomΞ3+λω+ω2γ2exϵ2(ω+ω)2(β1+β1*)+igomΞ3+μω+ω2γ1exϵ1(ω+ω)2(β2+β2*),αssμ2γ1exϵ1λ+2γ2exϵ2ω+igomΞ4++λ+ω+ω2γ2exϵ2(ω+ω)2(β1+β1*)+igomΞ4μω+ω2γ1exϵ1(ω+ω)2(β2+β2*),(B4)

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    α1ss=τ+α+ss+ταss=(σ2e1iκe2)Cigom(σ22e1iκσ2e2)C2(β1+β1*)+igom(κ2e1+iκσ1e2)C2(β2+β2*),α2ss=α+ss+αss=(iκe1+σ1e2)C+igom(iκσ2e1+κ2e2)C2(β1+β1*)igom(iκσ1e1+σ12e2)C2(β2+β2*),(B5)

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    [β˙1β˙2]=[Γm1i(Ω1+δΩ1)κmechκmechΓm2i(Ω2+δΩ2)]×[β1β2][iη1iη2],(B6)

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    δΩk=4gom2[(κ2Δ1Δ2)Δ3kΔkγ3k2]f2×[γ3k2γkexϵk2+(Δ3kγkexϵkκγ3kexϵ3k)2],κmech=4gom2κ{κγ1exϵ12[Δ2(κ2Δ1Δ22γ1γ2)+Δ1γ22]f2+κγ2exϵ22[Δ1(κ2Δ1Δ2)Δ2γ12]f2γ1exγ2exϵ1ϵ2f2[(κ2γ1γ2)2Δ12Δ22(Δ12γ22Δ22γ12)]},ηk=gom[γ3k2ϵk2+(Δ3kϵkκϵ3k)2]f,(B7)

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    δΩ1=δΩ2gom2Δ(γ2+κ2)2γϵ2[(κ2γ2)2+γ2Δ2]2.(C1)

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    κmech4gom2Δκ2γ3ϵ2[(κ2γ2)2+γ2Δ2]2,(C2)

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    λ±=ΓmiΩAve+±iΩAve2κmech2,(D1)

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    λ+=Γmi(2Ω˜2δΩcoup),λ=Γmi(2Ω˜1+δΩcoup),(D2)

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    δΩcoup=Ω˜2Ω˜1(Ω˜2Ω˜1)2κmech2(D3)

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    Ω1,eff=Ω1+δΩ1+δΩcoup,Ω2,eff=Ω2+δΩ2δΩcoup,(D4)

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    Hint=goma1a1(b1+b1)+goma2a2(b2+b2),(E1)

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    geffgom2[γ2(κ2γ2)2+γ2Δ2+1].(E2)

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    α˙k=(γkiΔk)αkiκα3kigomαk(βk+βk*)+2γkexϵk,β˙k=(Γmk+iΩk)βkigom|αk|2.(F1)

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    δΩ1=δΩ2gom2Δϵ2[(κ2+γ1γ2)+Δ+2]2,κmech2δΩ1κ/Δ,(F2)

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    δΩk2gom2Δ(κ2+γ3k2)2ϵ2[(κ2γ1γ2)2+(γ1+γ2)2Δ2/4+Γ2Δ+2]2,κmech2δΩ2κ2γ1γ2κ2+γ12.(F3)

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    ξ1=x1,ξ2=ξ˙1,ξ3=x2,ξ4=ξ˙3;(G1)

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    [ξ˙1(t)ξ˙2(t)ξ˙3(t)ξ˙4(t)]=[0100Ω˜122Γmκmech00001κmech0Ω˜222Γm][ξ1(t)ξ2(t)ξ3(t)ξ4(t)]+[Γ1Γ2Γ3Γ4]=A[ξ1(t)ξ2(t)ξ3(t)ξ4(t)]+[Γ1Γ2Γ3Γ4],(G2)

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    ξi(t)=k=14Gik(t)zk+k=140tGik(t)Γk(tt)dt,(G3)

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    G(t)=eAtIAt[1t00Ω˜12t12Γmtκmecht0001tκmecht0Ω˜22t12Γmt];(G4)

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    ξ1(t)=G11z1+G12z2+G13z3+G14z4+0tG11(t)Γ1(tt)+G12(t)Γ2(tt)+G13(t)Γ3(tt)+G14(t)Γ4(tt)dt=z1+tz2+0ttΓ2(tt)dt.(G5)

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    ξ2(t)=(κmechz3+Ω˜12z1)t+(12Γmt)z2+0t(12Γmt)Γ2(tt)dt,ξ3(t)=z3+z4t+0ttΓ4(tt)dt,ξ4(t)=(κmechz1+Ω˜22z3)t+(12Γmt)z4+0t(12Γmt)Γ4(tt)dt.(G6)

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    Rij(τ,t)=ξi(t+τ)ξj(t),(G7)

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    Rij(τ,t)=k=14Gik(τ)ξk(t)ξj(t),0τ.(G8)

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    R13(τ,t)=(z1+tz2)(z3+z4t)+q3t3+τ[(κmechz3+Ω˜12z1)(z3+z4t)t+(12Γmt)(z3+z4t)z2+q(12t223Γmt3)],R11(0,t)=(z1+z2t)2+q3t3,R33(0,t)=(z3+z4t)2+q3t3.(G9)

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    R(τ,t)=|R13(τ,t)|R11(0,t)R33(0,t)=|12Ω˜12τt+qκmechΩ˜122τt2+qκmechΩ˜123τt3|1+q3κmech2t31+q3Ω˜14t312Ω˜12τt+q2κmechΩ˜12τt2+q3κmechΩ˜12τt312Ω˜12τt+2ΓmkTκmechΩ˜12τt2+43ΓmkTκmechΩ˜12τt3.(G10)

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    Copy Citation Text
    Chang-Long Zhu, Yu-Long Liu, Lan Yang, Yu-Xi Liu, Jing Zhang. Synchronization in PT-symmetric optomechanical resonators[J]. Photonics Research, 2021, 9(11): 11002152
    Download Citation
    Category: Nonlinear Optics
    Received: Feb. 23, 2021
    Accepted: Aug. 30, 2021
    Published Online: Oct. 9, 2021
    The Author Email: Yu-Xi Liu (yuxiliu@mail.tsinghua.edu.cn), Jing Zhang (jing-zhang@mail.tsinghua.edu.cn)