• Photonics Research
  • Vol. 9, Issue 8, 1522 (2021)
Ben Wang, Liang Xu, Jun-chi Li, and Lijian Zhang*
Author Affiliations
  • National Laboratory of Solid State Microstructures and College of Engineering and Applied Sciences, Nanjing University, Nanjing 210093, China
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    DOI: 10.1364/PRJ.417613 Cite this Article Set citation alerts
    Ben Wang, Liang Xu, Jun-chi Li, Lijian Zhang. Quantum-limited localization and resolution in three dimensions[J]. Photonics Research, 2021, 9(8): 1522 Copy Citation Text show less

    Abstract

    As a method to extract information from optical systems, imaging can be viewed as a parameter estimation problem. The fundamental precision in locating one emitter or estimating the separation between two incoherent emitters is bounded below by the multiparameter quantum Cramér-Rao bound (QCRB). Multiparameter QCRB gives an intrinsic bound in parameter estimation. We determine the ultimate potential of quantum-limited imaging for improving the resolution of a far-field, diffraction-limited optical field within the paraxial approximation. We show that the quantum Fisher information matrix (QFIm) in about one emitter’s position is independent on its true value. We calculate the QFIm of two unequal-brightness emitters’ relative positions and intensities; the results show that only when the relative intensity and centroids of two-point sources, including longitudinal and transverse directions, are known exactly, the separation in different directions can be estimated simultaneously with finite precision. Our results give the upper bounds on certain far-field imaging technology and will find wide use in applications from microscopy to astrometry.
    T2Ψ+2k2Ψ+i2kzΨ=0,

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    |Ψ˜=exp(iG^zeip^xxeip^yye)|Ψ,

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    ρ=q|Ψ1Ψ1|+(1q)|Ψ2Ψ2|,

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    ρ(x)=q|Ψ(xx1,z1)|2+(1q)|Ψ(xx2,z2)|2.

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    Cov[θ]jp(j|θ)[θθ̌(j)]T[θθ̌(j)],

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    Cov[θ]1M[F(ρθ,Πj)]1,

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    [F(ρθ,Πj)]μν=j1p(j|θ)p(j|θ)θμp(j|θ)θν,

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    Fμνdirect=1p(x,y|θ)p(x,y|θ)θμp(x,y|θ)θνdxdy,

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    Cov[θ]1M[F(ρθ,Πj)]11M[Q(ρθ)]1,

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    [Q(ρθ)]μν=12Tr[ρθ{Lμ,Lν}],

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    κρθ=Lκρθ+ρθLκ2.

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    Tr[ρθ{Lμ,Lν}]=0.

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    Ψ(x,y,z)=Ψ(x,y,z)=Ψ(x,y,z).

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    Lκ=2(|Ψ˜κΨ˜|+|κΨ˜Ψ˜|),

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    [Qloc(θ)]jk=4Re(jΨ˜|kΨ˜jΨ˜|Ψ˜Ψ˜|kΨ˜),

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    |xeΨ˜=ip^x|Ψ˜,|yeΨ˜=ip^y|Ψ˜,|zeΨ˜=iG^|Ψ˜,

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    [Γloc(θ)]jk=4Im(jΨ˜|kΨ˜jΨ˜|Ψ˜Ψ˜|kΨ˜),

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    Qloc=4[px2000py2000gz2Gz2],

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    Γloc=[000000000].

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    |Ψ=x,ydxdy2πw02exp(x2+y2w02)|x,y,

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    |Ψ˜=x,ydxdy2πw(ze)2exp[(xxe)2+(yye)2w(ze)2]exp[ikzeik(xxe)2+(yye)22R(ze)+iζ(ze)]|x,y,

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    4[1w020001w0200014zr2].

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    Fμν=x,ydxdy1I(x,y)I(x,y)θμI(x,y)θν,

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    Fxexe=4zr2w02(z2+zr2),Fyeye=4zr2w02(z2+zr2),Fzeze=4z2(z2+zr2)2.

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    {|Ψ1,|Ψ2,|Ψ3,|Ψ4,|Ψ5,|Ψ6},

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    |Ψ1=exp(iG^z1ip^x1)|Ψ,|Ψ2=exp(iG^z2ip^x2)|Ψ,|Ψ3=ip^exp(iG^z1ip^x1)|Ψp,|Ψ4=iG^exp(iG^z1ip^x1)|Ψg,|Ψ5=ip^exp(iG^z2ip^x2)|Ψp,|Ψ6=iG^exp(iG^z2ip^x2)|Ψg,

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    Q=[Qx0x02p2(12q)Qx0z004wsw2p2(12q)p2000Qx0z00Qz0z02(g2G2)(1+2q)4wtw002(g2G2)(1+2q)g2G204wsw04wtw01+w2(1+q)q],

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    Γ=[0Γx0sΓx0z0Γx0t4sϕ(1+2q)w2Γx0s0Γsz002sϕw2Γx0z0Γsz00Γz0t4(G+tϕ)(1+2q)w2Γx0t0Γz0t02(G+tϕ)w24sϕ(1+2q)w22sϕw24(G+tϕ)(1+2q)w22(G+tϕ)w20],

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    weiϕ=Ψ1|Ψ2,G=Ψ|G^|Ψ,Qx0x0=4[p24(sw)2(1q)q4(sϕ)2(1q)qw21w2],Qx0z0=16swtw(1+q)q16sϕ(G+tϕ)(1+q)qw21+w2,Qz0z0=4{G24(tw)2(1+q)q[G24(Gtw+tϕ)(G+tw+tϕ)q(1q)]w2}1+w2+4g2,Γx0s=8swsϕ(1+q)qw31+w2,Γx0z0=16[sϕtw+sw(G+tϕ)](1+q)q(1+2q)w,Γx0t=8(1+q)qw[sϕtw+sw(G+tϕ)(1+w2)]1+w2,Γsz0=8(1+q)qw[sw(G+tϕ)+sϕtw(1+w2)]1+w2,Γz0t=8tw(G+tϕ)(1+q)qw31+w2.

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    G+tϕ=0,

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    [p200g2G2],

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    0DTr(Q1),

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    lims,t0QG=[2kzrk(12q)zr000k(12q)zrk2zr000001zr21+2q2zr20001+2q2zr214zr2000000],

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    lims,t0ΓG=[0000000000000000000000000],

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    ρ|Ψj=qΠ1j|Ψ1+(1q)Π2j|Ψ2,(A1)

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    R=[qΠ11qΠ12qΠ13qΠ14qΠ15Π15(1q)Π21(1q)Π22(1q)Π23(1q)Π24(1q)Π25(1q)Π26000000000000000000000000].(A2)

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    x1ρ=qp(|Ψ3Ψ1|+|Ψ1Ψ3|),x2ρ=(1q)p(|Ψ5Ψ2|+|Ψ2Ψ5|),z1ρ=qg(|Ψ4Ψ1|+|Ψ1Ψ4|),z2ρ=(1q)g(|Ψ6Ψ2|+|Ψ2Ψ6|),qρ=|Ψ1Ψ1||Ψ2Ψ2|.(A3)

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    Ξθi=RLθi+LθiR2,(A4)

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    Ξx1=qp[Π31Π32Π33Π34Π35Π36000000Π11Π12Π13Π14Π15Π16000000000000000000],(A5)

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    Ξx2=(1q)p[000000Π51Π52Π53Π54Π55Π56000000000000Π21Π22Π23Π24Π25Π26000000],(A6)

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    Ξz1=qg[Π41Π42Π43Π44Π45Π46000000000000Π11Π12Π13Π14Π15Π16000000000000],(A7)

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    Ξz2=(1q)g[000000Π61Π62Π63Π64Π65Π66000000000000Π21Π22Π23Π24Π25Π26000000],(A8)

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    Ξq=[Π11Π12Π13Π14Π15Π16Π21Π22Π23Π24Π25Π26000000000000000000000000].(A9)

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    θ1=x0=x2+x12,θ2=s=x2x1,θ3=z0=z2+z12,θ4=t=z2z1,θ5=q.(A10)

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    (L^x0L^sL^z0L^tL^q)=(11000121200000110001212000001)(L^x1L^x2L^z1L^z2L^q).(A11)

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    |Ψ1=exp(iG^z1ip^x1)|Ψ=exp[iG^z1ip^(x0s2)]|Ψ,|Ψ2=exp(iG^z2ip^x2)|Ψ=exp[iG^z2ip^(x0+s2)]|Ψ.(A12)

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    x0ρ=i[ρ,p^]=x1ρ+x2ρ.(A13)

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    L^x0=L^x1+L^x2.(A14)

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    [Q(ρ)]μν+i[Γ(ρ)]μν=Tr[ρLμLν],(A15)

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    Tr[ρLμLν]=Tr[TRT1TLμT1TLνT1]=Tr[RLμLν].(A16)

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    Ben Wang, Liang Xu, Jun-chi Li, Lijian Zhang. Quantum-limited localization and resolution in three dimensions[J]. Photonics Research, 2021, 9(8): 1522
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