• Photonics Research
  • Vol. 10, Issue 12, 2816 (2022)
Binke Xia1, Jingzheng Huang1、2、*, Hongjing Li1, Miaomiao Liu1, Tailong Xiao1, Chen Fang1, and Guihua Zeng1、3、*
Author Affiliations
  • 1State Key Laboratory of Advanced Optical Communication Systems and Networks, Institute for Quantum Sensing and Information Processing, Shanghai Jiao Tong University, Shanghai 200240, China
  • 2e-mail:
  • 3e-mail:
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    DOI: 10.1364/PRJ.473699 Cite this Article Set citation alerts
    Binke Xia, Jingzheng Huang, Hongjing Li, Miaomiao Liu, Tailong Xiao, Chen Fang, Guihua Zeng. Ultrasensitive measurement of angular rotations via a Hermite–Gaussian pointer[J]. Photonics Research, 2022, 10(12): 2816 Copy Citation Text show less

    Abstract

    Exploring high sensitivity on the measurement of angular rotations is an outstanding challenge in optics and metrology. In this work, we employ the mn-order Hermite–Gaussian (HG) beam in the weak measurement scheme with an angular rotation interaction, where the rotation information is taken by another HG mode state completely after the post-selection. By taking a projective measurement on the final light beam, the precision of angular rotation is improved by a factor of 2mn+m+n. For verification, we perform an optical experiment where the minimum detectable angular rotation improves 15-fold with HG55 mode over that of HG11 mode, and achieves a sub-microradian scale of the measurement precision. Our theoretical framework and experimental results not only provide a more practical and convenient scheme for ultrasensitive measurement of angular rotations but also contribute to a wide range of applications in quantum metrology.
    δα^2ΔΩ^2i14N|Aw|2.

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    ΔL^z2mn=2  mn+m+n,

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    δα^214N|Aw|2(2  mn+m+n),

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    umn(x,y,z)=exp[ik(x2+y2)2q(z)i(m+n+1)χ(z)]σ0σ(z)ψmn[σ0σ(z)x,σ0σ(z)y],

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    12σ2(z)ikq(z)=kb+iz,tanχ(z)=zb,

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    ψmn(x,y)=Hm(x2σ0)Hn(y2σ0)2m+n+1πσ02m!n!exp(x2+y24σ02),

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    iψt=[σ02(p^x2+p^y2)+14σ02(x^2+y^2)]ψ.

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    |m,n=dxdyψmn(x,y)|x,y.

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    |umn(z)=dxdyumn(x,y)|x,y.

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    a^x|m,n=m+1|m+1,n,a^x|m,n=m|m1,n,

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    a^y|m,n=n+1|m,n+1,a^y|m,n=n|m,n1.

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    2ikzu(x,y,z)=(2x2+2y2)u(x,y,z),

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    ddz|un(z)=i2k(p^x2+p^y2)|un(z).

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    U^(z)=exp[i2k(p^x2+p^y2)z].

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    a^x(z)=U^(z)a^xU^(z),a^x(z)=U^(z)a^xU^(z),

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    a^y(z)=U^(z)a^yU^(z),a^y(z)=U^(z)a^yU^(z).

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    {p^x=i2σ0[a^x(z)a^x(z)]x^=σ0[a^x(z)+a^x(z)]+zσ0ib[a^x(z)a^x(z)]p^y=i2σ0[a^y(z)a^y(z)]y^=σ0[a^y(z)+a^y(z)]+zσ0ib[a^y(z)a^y(z)].

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    L^z=x^p^yy^p^x=i[a^x(z)a^y(z)a^x(z)a^y(z)].

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    |ψf|m,n+Awα[m(n+1)|m1,n+1(m+1)n|m+1,n1],

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    |ψL^=12  mn+m+n[m(n+1)|m1,n+1(m+1)n|m+1,n1],

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    F(α)=λ1ψf|Π^λ|ψf(αψf|Π^λ|ψf)2,

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    αminQCR=12  mn+m+n12|cotε|N,

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    Nα=|ψL^|Π^L^|ψL^|2N=(2mn+m+n)(cotε)2αtot2N(2mn+m+n)(cotε)2α02N+2(2mn+m+n)(cotε)2α0αcos(2πft)N.

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    Iα(2mn+m+n)(cotε)2α02I0+2(2mn+m+n)(cotε)2α0αcos(2πft)I0.

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    Iα1  kHz=2(2mn+m+n)(cotε)2α0αI0.

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    δIα=γδNα/τγ2mn+m+n|cotε|α0N/τ.

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    SNR=Iα1  kHzδIα=22mn+m+n|cotε|Nα.

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    αmin(m,n)=12mn+m+n12|cotε|N.

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    SNR(m,n)=Vα(m,n)Vsn(m,n)=Vp(m,n)Vnoise(m,n)Vnoise(m,n)Ven.

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    δg^2ΔΩ^2i14Nα2|gAw|2,

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    δg^2σ02Nα2|gAw|2.

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    δg^2σ02(2m+1)Nα2|gAw|2

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    δg^214(2  mn+m+n)Nα2|gAw|2,

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    |qubit=cosθ2|0+eiϕsinθ2|1,

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    H^I=δ(tt0)ασ^zΩ^,

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    |Ψf=cosθ2|0|ψ++eiϕsinθ2|1|ψ,

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    ρ^f=cos2θ2|ψ+ψ+|+sin2θ2|ψψ|,

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    δθ^2Ω^2i14α2N,

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    δθ^214α2(2mn+m+n)N,

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    U^=exp(iH^Idt)=exp(iαA^Ω^)=12(I^+n·σ)exp(iαΩ^)+12(I^n·σ)exp(iαΩ^)1iαn·σΩ^.(A1)

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    |Ψ˜f=|ff|U^|Ψi|ff|(1iαn·σΩ^)|Ψi=[f|i(1iMwΩ^)|ψi]|f.(A2)

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    |ψf=N(1iMwΩ^)|ψi,(A3)

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    N=11+2Im(Mw)Ω^i+|Mw|2Ω^2i(A4)

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    Q(g)=4[ψ(g)|g|ψ(g)gψ(g)|g|ψ(g)ψ(g)||ψ(g)g].(A5)

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    Q(g)=4N2[|Mwg|2Ω^2iN2(|Mwg|2Ω^i2+ImMw|Mwg|2Ω^iΩ^2i+|Mw|2|Mwg|2Ω^2i2)]4|Mwg|2ΔΩ^2i,(A6)

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    δg^2ΔΩ^2i14N·1|gMw|2.(A7)

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    {Mwα=σxwsinθcosϕ+σywsinθsinϕ+σzwcosθMwθ=α(σxwcosθcosϕ+σywcosθsinϕσzwsinθ)Mwϕ=α(σywsinθcosϕσxwsinθsinϕ),(A8)

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    |Ψf=cosθ2|0|ψ++eiϕsinθ2|1|ψ,(B1)

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    ρ^f=Trqubit(|ΨfΨf|)=cos2θ2|ψ+ψ+|+sin2θ2|ψψ.(B2)

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    Q(θ)=Tr(ρ^fL^θ2),(B3)

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    ρ^fL^θ+L^θρ^f=2θρ^f.(B4)

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    |e1=12(1+δ)(|ψ++|ψ),|e2=12(1δ)(|ψ+|ψ),(B5)

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    ρ^f=1+δ2|e1e1|+1δ2|e2e2|+1δ22cosθ(|e1e2|+|e2e1|),(B6)

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    θρ^f=1δ22sinθ(|e1e2|+|e2e1|).(B7)

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    {2e1|θρ^f|e1=(1+δ)e1|L^θ|e1+1δ22cosθ(e2|L^θ|e1+e1|L^θ|e2)=02e2|θρ^f|e2=(1δ)e2|L^θ|e2+1δ22cosθ(e1|L^θ|e2+e2|L^θ|e1)=02e1|θρ^f|e2=e1|L^θ|e2+1δ22cosθ(e2|L^θ|e2+e1|L^θ|e1)=1δ2sinθ2e2|θρ^f|e1=e2|L^θ|e1+1δ22cosθ(e1|L^θ|e1+e2|L^θ|e2)=1δ2sinθ,(B8)

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    e1|L^θ|e1=(1δ)cotθ,e2|L^θ|e2=(1+δ)cotθ,e1|L^θ|e2=e2|L^θ|e1=1δ2cscθ.(B9)

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    Q(θ)=Tr(ρ^fL^θ2)=Tr(L^θθρ^f)=|exSex|L^θθρ^f|ex=1δ22sinθ(e2|L^θ|e1+e1|L^θ|e2).(B10)

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    Q(θ)=1δ2=1(Reψ+|ψ)24α2Ω^2i,(B11)

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    S(x,y)=Ain(x,y)exp[iϕin(x,y)+iH(x,y)].(C1)

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    H(x,y)=f(Ar)sin(ϕr).(C2)

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    exp[if(a)sin(ϕ)]=Jq[f(a)]exp(iqϕ),(C3)

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    f(Ar)=J11(Ar),(C4)

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    f(u,v)=F[g(x,y)h(x,y)]=+g(x,y)h(x,y)exp[i2π(xu+yv)]dxdy,(D1)

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    η|+f(u,v)exp(u2+v2wf2)dudv|2,(D2)

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    η|f(0,0)|2=|+g(x,y)h(x,y)dxdy|2=|h*|g|2.(D3)

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    P=|ψL^|ψf|2(2mn+m+n)(cotε)2α2.(D4)

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    F(α)=1P(Pα)2+11P[(1P)α]24(2mn+m+n)(cotε)2(D5)

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    αminCCR=12  mn+m+n12|cotε|N,(D6)

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    Binke Xia, Jingzheng Huang, Hongjing Li, Miaomiao Liu, Tailong Xiao, Chen Fang, Guihua Zeng. Ultrasensitive measurement of angular rotations via a Hermite–Gaussian pointer[J]. Photonics Research, 2022, 10(12): 2816
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