• Photonics Research
  • Vol. 9, Issue 11, 2152 (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
  • show less
    DOI: 10.1364/PRJ.423506 Cite this Article Set citation alerts
    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): 2152 Copy Citation Text show less

    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,

    View in Article

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

    View in Article

    ωo+=Δ¯1+Δ¯2±i[γ+i(Δ¯2Δ¯1)]2κ2.

    View in Article

    ωo±=Δ1+Δ22±i(γ+iΔ2)2κ2.

    View in Article

    gomΔ<Δ1,2γ,κ,

    View in Article

    Δsplitγ2|Δ+4gom2Ω2Ω1Ω1Ω21Δ2ϵ2|1γ,

    View in Article

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

    View in Article

    κmech4gom2Δκ2γ3ϵ2[(κ2γ2)2+γ2Δ2]2.

    View in Article

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

    View in Article

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

    View in Article

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

    View in Article

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

    View in Article

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

    View in Article

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

    View in Article

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

    View in Article

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

    View in Article

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

    View in Article

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

    View in Article

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

    View in Article

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

    View in Article

    ωo±(Δ¯1+Δ¯2)±iγ2κ2.(A8)

    View in Article

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

    View in Article

    gomΔ<Δ1,2γ,(A10)

    View in Article

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

    View in Article

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

    View in Article

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

    View in Article

    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)

    View in Article

    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)

    View in Article

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

    View in Article

    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)

    View in Article

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

    View in Article

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

    View in Article

    λ±=ΓmiΩAve+±iΩAve2κmech2,(D1)

    View in Article

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

    View in Article

    δΩcoup=Ω˜2Ω˜1(Ω˜2Ω˜1)2κmech2(D3)

    View in Article

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

    View in Article

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

    View in Article

    geffgom2[γ2(κ2γ2)2+γ2Δ2+1].(E2)

    View in Article

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

    View in Article

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

    View in Article

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

    View in Article

    ξ1=x1,ξ2=ξ˙1,ξ3=x2,ξ4=ξ˙3;(G1)

    View in Article

    [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)

    View in Article

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

    View in Article

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

    View in Article

    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)

    View in Article

    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)

    View in Article

    Rij(τ,t)=ξi(t+τ)ξj(t),(G7)

    View in Article

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

    View in Article

    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)

    View in Article

    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)

    View in Article

    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): 2152
    Download Citation