• Photonics Research
  • Vol. 8, Issue 10, 1653 (2020)
Kaimin Zheng1,†, Minghao Mi1,†, Ben Wang1, Liang Xu1..., Liyun Hu2, Shengshuai Liu3, Yanbo Lou3, Jietai Jing3,4,5,* and Lijian Zhang1,6,*|Show fewer author(s)
Author Affiliations
  • 1National Laboratory of Solid State Microstructures, Key Laboratory of Intelligent Optical Sensing and Manipulation, College of Engineering and Applied Sciences, and Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
  • 2Center for Quantum Science and Technology, Jiangxi Normal University, Nanchang 330022, China
  • 3State Key Laboratory of Precision Spectroscopy, Joint Institute of Advanced Science and Technology, School of Physics and Electronic Science, East China Normal University, Shanghai 200062, China
  • 4Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China
  • 5e-mail: jtjing@phy.ecnu.edu.cn
  • 6e-mail: lijian.zhang@nju.edu.cn
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    DOI: 10.1364/PRJ.395682 Cite this Article Set citation alerts
    Kaimin Zheng, Minghao Mi, Ben Wang, Liang Xu, Liyun Hu, Shengshuai Liu, Yanbo Lou, Jietai Jing, Lijian Zhang, "Quantum-enhanced stochastic phase estimation with the SU(1,1) interferometer," Photonics Res. 8, 1653 (2020) Copy Citation Text show less

    Abstract

    Quantum stochastic phase estimation has many applications in the precise measurement of various physical parameters. Similar to the estimation of a constant phase, there is a standard quantum limit for stochastic phase estimation, which can be obtained with the Mach–Zehnder interferometer and coherent input state. Recently, it has been shown that the stochastic standard quantum limit can be surpassed with nonclassical resources such as squeezed light. However, practical methods to achieve quantum enhancement in the stochastic phase estimation remain largely unexplored. Here we propose a method utilizing the SU(1,1) interferometer and coherent input states to estimate a stochastic optical phase. As an example, we investigate the Ornstein–Uhlenback stochastic phase. We analyze the performance of this method for three key estimation problems: prediction, tracking, and smoothing. The results show significant reduction of the mean square error compared with the Mach–Zehnder interferometer under the same photon number flux inside the interferometers. In particular, we show that the method with the SU(1,1) interferometer can achieve fundamental quantum scaling, achieve stochastic Heisenberg scaling, and surpass the precision of the canonical measurement.
    c^out=G(Gc^in+gd^in)eiΦ(t)+g(gc^in+Gd^in)eiφ(t),d^out=g(Gc^in+gd^in)eiΦ(t)+G(gc^in+Gd^in)eiφ(t).(1)

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    r(t)2Gg|β|G2+g2φ(t)+2G2g2σf2+1n(t).(2)

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    df(t)=tdτho(t,τ)r(τ),(3)

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    Kdr(tη)=tho(tϵ)Kr(ϵη)dϵ,(4)

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    dφ(t)dt=λφ(t)+κdV(t)dt.(5)

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    Ho(ω)={κPeiωεNλ(1+1+Λ)(λ1+Λ+iω),ε>0,κPeiωεN[λ2(1+Λ)+ω2][1eε(λ1+Λiω)(λ+iω)λ(1+1+Λ)],ε0,(6)

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    ξ=[d(t)df(t)]2=Kd(0)0Kdz2(τ)dτ,(7)

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    Kdz(τ)={PκNλ11+1+Λeλ(τ+ε),τ+ε0,PκNλ11+1+Λeλ1+Λ(τ+ε),τ+ε<0.(8)

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    σf2=(λG2g2κ)+(λG2g2κ)2+4G2g2(|β|2G2+g2+λ)κ4G2g2(|β|2G2+g2+λ).(9)

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    ξNLI={κ2λ[1Λ(1+1+Λ)2e2λε],ε>0,κ2λ[11+Λ+Λe2λ1+Λε(1+1+Λ)21+Λ],ε<0,(10)

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    ξMZI={κ2λ[1Λ1(1+1+Λ1)2e2λε],ε>0,κ2λ[11+Λ1+Λ1e2λ1+Λ1ε(1+1+Λ1)21+Λ1],ε0,(11)

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    σf221/3(κ|β|2)2/3.(12)

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    ξ(κ2|β|2)2/3,(13)

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    X^dout[θ(t)]=d^outeiθ(t)+d^outeiθ(t),(A1)

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    X^dout[θ(t)]=d^outeiθ(t)+d^outeiθ(t)=4GgcosΦ(t)+φ(t)2cos[Φ(t)φ(t)2+θ(t)]|α|4Ggsinφ(t)φf(t)2|α|2Gg[φ(t)φf(t)]|α|,(A2)

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    Δ2X^dout[θ(t)]=X^dout2[θ(t)]X^dout[θ(t)]2=4G2g2{1+cos[Φ(t)+φ(t)]}+18G2g2[sin2φ(t)φf(t)2]+12G2g2[φ(t)φf(t)]2+1.(A3)

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    Xdout(t)=X^dout[θ(t)]+ΔX^dout[θ(t)]n(t)2Gg[φ(t)φf(t)]|α|+2G2g2σf2+1n(t),(A4)

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    r(t)2Gg|α|φ(t)+2G2g2σf2+1n(t)=2Gg|β|G2+g2φ(t)+2G2g2σf2+1n(t),(A5)

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    c^out=12{[eiφ(t)eiΦ(t)]c^in+i[eiφ(t)+eiΦ(t)]d^in},d^out=12{i[eiφ(t)+eiΦ(t)]c^in[eiφ(t)eiΦ(t)]d^in}.(A6)

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    X^dout[θ(t)]=d^outeiθ(t)+d^outeiθ(t)={2cosΦ(t)φ(t)2cos[Φ(t)+φ(t)2θ(t)+π2]}|β|2|β|cos[φf(t)+φ(t)2θ(t)+π2]|β|[φ(t)φf(t)],(A7)

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    Δ2X^dout[θ(t)]=X^dout2[θ(t)]X^dout[θ(t)]21.(A8)

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    Xdout(t)=X^dout[θ(t)]+ΔX^dout[θ(t)]n(t)|β|[φ(t)φf(t)]+n(t).(A9)

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    r(t)|β|φ(t)+n(t).(A10)

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    Kdr(tσ)=tho(tϵ)Kr(ϵσ)dϵ(B1)

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    Kdr(τ)=0ho(υ)Kr(τυ)dυ.(B2)

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    z(τ)=r(t)ω(τt)dt.(B3)

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    |W(ω)|2Sr(ω)=1,(B4)

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    Sr(ω)=4G2g2|α|2κω2+λ2+2G2g2σf2+1.(B5)

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    Kdz(τ)=0fo(υ)Kz(τυ)dυ,τ0.(B6)

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    Kdz(τ)=d(t)ω(υ)r(tτυ)dυ=ω(μ)Kdr(τμ)dμ.(B7)

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    [Sdz(ω)]+=[W*(ω)Sdr(ω)]+=[Sdr(ω)[H+(ω)]*]+,(B8)

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    F(ω)=[Sdr(ω)[H+(ω)]*]+.(B9)

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    Ho(ω)=F(ω)H+(ω)=1H+(ω)[Sdr(ω)[H+(ω)]*]+.(B10)

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    Sdz(ω)=Sdr(ω)[H+(ω)]*=κPeiωεω2+λ2λiωN(λ1+Λiω)=κPeiωελ+iω1N(λ1+Λiω)=κPeiωεNλ(1+1+Λ)(1λ+iω+1λ1+Λiω).(B11)

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    Kdz(τ)=F1[Sdz(ω)]=F1[κPeiωεNλ(1+1+Λ)(1λ+iω+1λ1+Λiω)]=κPeλ(τ+ε)Nλ(1+1+Λ)u(τ+ε)+κPeλ1+Λ(τ+ε)Nλ(1+1+Λ)u(τε),(B12)

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    fo(τ)=Kdz(τ)=κPeλτNλ(1+1+Λ)u(τ)(B13)

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    F(ω)=[Sdz(ω)]+=κPNλ(1+1+Λ)1λ+iω,(B14)

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    Hot(ω)=F(ω)H+(ω)=κPNλ(1+1+Λ)1λ+iωλ+iωN(λ1+Λ+iω)=κPNλ(1+1+Λ)(λ1+Λ+iω).(B15)

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    F(ω)=[Sdz(ω)]+=κPN[eiωε(λ+iω)(λ1+Λiω)eελ1+Λλ(1+1+Λ)(λ1+Λiω)].(B16)

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    Hos(ω)=F(ω)H+(ω)=κPN[eiωε(λ+iω)(λ1+Λiω)eελ1+Λλ(1+1+Λ)(λ1+Λiω)]λ+iωN(λ1+Λ+iω)=κPeiωεN[λ2(1+Λ)+ω2][1eε(λ1+Λiω)(λ+iω)λ(1+1+Λ)].(B17)

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    F(ω)=[Sdz(ω)]+=κPeiωεNλ(1+1+Λ)1λ+iω,(B18)

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    Hop(ω)=F(ω)H+(ω)=κPeiωεNλ(1+1+Λ)1λ+iωλ+iωN(λ1+Λ+iω)=κPeiωεNλ(1+1+Λ)(λ1+Λ+iω).(B19)

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    Ho(ω)={κ|β|eiωελ(1+1+Λ1)(λ1+Λ1+iω),ε>0,κ|β|eiωελ2(1+Λ1)+ω2[1eε(λ1+Λ1iω)(λ+iω)λ(1+1+Λ1)],ε0,(B20)

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    ξ(t)=[d(t)tr(τ)ho(tτ)dτ]2=Kd(0)tho(tτ)Kdr(tτ)dτ=Kd(0)0ho(γ)Kdr(γ)dγ=Kd(0)0Kdz(t)dt[12πejωtdωKdr(τ)ejωτdτH+(ω)].(C1)

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    ξ(t)=Kd(0)0Kdz2(t)dt,(C2)

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    σf2=κ2λ0Pκ2Nλ21(1+1+Λ)2e2λτdτ=κ2λ[1Λ(1+1+Λ)2],(C3)

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    σf2=(λG2g2κ)+(λG2g2κ)2+4G2g2(|β|2G2+g2+λ)κ4G2g2(|β|2G2+g2+λ).(C4)

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    ξp=κ2λ0Pκ2Nλ21(1+1+Λ)2e2λ(τ+ε)dτ=κ2λ[1Λ(1+1+Λ)2e2λε].(C5)

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    ξs=κ2λPκ2Nλ20εe2λ1+Λ(τ+ε)dτ+εe2λ(τ+ε)dτ(1+1+Λ)2=κ2λPκ2Nλ21(1+1+Λ)2(1e2λ1+Λε2λ1+Λ+12λ)=κ2λ[11+Λ+Λe2λ1+Λε(1+1+Λ)21+Λ].(C6)

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    ξMZI={κ2λ[1Λ1(1+1+Λ1)2e2λε],ε>0,κ2λ[11+Λ1+Λ1e2λ1+Λ1ε(1+1+Λ1)21+Λ1],ε0.(C7)

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    SNRNLI=4G2g2|α|22G2g2σf2+1=4G2(G21)|β|2[(2G4G2)σf2+1](2G21).(D1)

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    σf2=4[2Go2(Go21)+1][4Go2(Go21)]212Go4.(D2)

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    12Go4=18(Go4Go2)(|β|22Go21+λ){2[λ(Go4Go2)κ]+4[λ(Go4Go2)κ]2+16(Go4Go2)(|β|22Go21+λ)κ}.(D3)

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    12Go42[(Go4)κ]+4[(Go4)κ]2+8Go2|β|2κ8(Go4)(|β|22Go2).(D4)

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    Go(|β|2κ2)1/322/3κ.(D5)

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    σf221/3(κ|β|2)2/3.(D6)

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    Kaimin Zheng, Minghao Mi, Ben Wang, Liang Xu, Liyun Hu, Shengshuai Liu, Yanbo Lou, Jietai Jing, Lijian Zhang, "Quantum-enhanced stochastic phase estimation with the SU(1,1) interferometer," Photonics Res. 8, 1653 (2020)
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