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Journals >Photonics Research >Volume 5 >Issue 6 >Page 617 > Article

Photonics Research
Vol. 5, Issue 6, 617 (2017)

Enhancement of the angular rotation measurement sensitivity based on SU(2) and SU(1,1) interferometers

Jun Liu, Wenxiao Liu, Shitao Li, Dong Wei, Hong Gao^*, and Fuli Li

Author Affiliations

Key Laboratory of Quantum Information and Quantum Optoelectronic Devices of Shaanxi Province, School of Science, Xi’an Jiaotong University, Xi’an 710049, China

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DOI: 10.1364/PRJ.5.000617 Cite this Article Set citation alerts

Jun Liu, Wenxiao Liu, Shitao Li, Dong Wei, Hong Gao, Fuli Li. Enhancement of the angular rotation measurement sensitivity based on SU(2) and SU(1,1) interferometers[J]. Photonics Research, 2017, 5(6): 617 Copy Citation Text

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Abstract

We investigate the sensitivity of the angular rotation measurement with the method of homodyne detection in SU(2) and SU(1,1) interferometers by employing orbital angular momentum (OAM). By combining a coherent beam with a vacuum beam in an SU(2) interferometer, we get the sensitivity of the angular rotation measurement as 12Nl. We can surpass the limit of the angular rotation measurement in an SU(1,1) interferometer by combining a coherent beam with a vacuum beam or a squeezed vacuum beam when the probe beam has OAM. Without injection, the sensitivity can reach 12Nl. In addition, by employing another construction of an SU(1,1) interferometer where the pump beam has OAM, with the same injection of an SU(1,1) interferometer, the sensitivity of the angular rotation measurement can be improved by a factor of 2, reaching 14Nl. The results confirm the potential of this technology for precision measurements in angular rotation measurements.

Keywords

(120.3180) Interferometry.(350.5730) Resolution (040.5570) Quantum detectors

{\hat{a}}_{1} = \sqrt{T} {\hat{a}}_{0, + l} + i \sqrt{R} {\hat{b}}_{0}, {\hat{b}}_{1} = i \sqrt{R} {\hat{a}}_{0, + l} + \sqrt{T} {\hat{b}}_{0} .

(1)

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{\hat{a}}_{out} = \frac{1}{2} {\hat{a}}_{0, + l} (e^{2 i l θ_{2}} - 1) + \frac{1}{2} i {\hat{b}}_{0} (e^{2 i l θ_{2}} + 1), {\hat{b}}_{out} = i \frac{1}{2} {\hat{a}}_{0, + l} (e^{2 i l θ_{2}} + 1) + \frac{1}{2} {\hat{b}}_{0} (e^{2 i l θ_{2}} - 1) .

(2)

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I_{ps} = ⟨ {\hat{b}}_{1}^{+} {\hat{b}}_{1} ⟩ = \frac{{| α |}^{2}}{2} .

(3)

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⟨ {\hat{b}}_{out}^{+} {\hat{b}}_{out} ⟩ = \frac{⟨ {\hat{b}}_{0}^{+} {\hat{b}}_{0} ⟩ [1 + \cos (2 l θ_{2})]}{2} = I_{ps} [1 + \cos (2 l θ_{2})] .

(4)

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{\hat{X}}_{out} = {\hat{a}}_{out}^{+} + {\hat{a}}_{out} .

(5)

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⟨ {\hat{X}}_{out}^{2} ⟩ = | α | | α | \sin^{2} (2 l θ_{2}) + 1 = {⟨ {\hat{X}}_{out} ⟩}^{2} + 1 = 2 I_{ps} δ^{2} + 1 .

(6)

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R_{limit} = {| α |}^{2} {(δ)}^{2} / 1 = 2 I_{ps} {(δ)}^{2} = 2 I_{ps} {(2 l θ_{2})}^{2} .

(7)

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δ = \frac{1}{\sqrt{N}} = δ_{SNL} .

(8)

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θ_{2} = \frac{1}{2 l \sqrt{N}} = θ_{SNL} .

(9)

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{\hat{a}}_{1} = \sqrt{G} {\hat{a}}_{0, + l} + \sqrt{G - 1} {\hat{b}}_{0}^{+}, {\hat{b}}_{1} = \sqrt{G - 1} {\hat{a}}_{0, + l}^{+} + \sqrt{G} {\hat{b}}_{0} .

(10)

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{\hat{a}}_{out} = [G + (G - 1) e^{2 i l θ_{2}}] {\hat{a}}_{0, + l} + [\sqrt{G (G - 1)} + \sqrt{G (G - 1)} e^{2 i l θ_{2}}] {\hat{b}}_{0}^{+}, {\hat{b}}_{out} = [\sqrt{G (G - 1)} + \sqrt{G (G - 1)} e^{- 2 i l θ_{2}}] {\hat{a}}_{0, + l}^{+} + [(G - 1) + G e^{- 2 i l θ_{2}}] {\hat{b}}_{0} .

(11)

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{\hat{I}}_{ps} = ⟨ {\hat{b}}_{1}^{+} {\hat{b}}_{1} ⟩ = (G - 1) ({| α |}^{2} + 1) \approx (G - 1) {| α |}^{2}, | α | ≫ 1 .

(12)

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⟨ {\hat{b}}_{out}^{+} {\hat{b}}_{out} ⟩ = [2 G (G - 1) + 2 G (G - 1) \cos (2 l θ_{2})] ({| α |}^{2} + 1) \approx [2 G (G - 1) + 2 G (G - 1) \cos (2 l θ_{2})] ({| α |}^{2}) = 2 G [1 + \cos (2 l θ_{2})] I_{ps} .

(13)

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⟨ {\hat{X}}_{b out}^{2} ⟩ = 2 G (G - 1) {| α |}^{2} [1 - \cos (4 l θ_{2})] + 4 G^{2} - 4 G + 1 + 4 G (G - 1) \cos (2 l θ_{2}) .

(14)

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⟨ {\hat{X}}_{b out}^{2} ⟩ \approx G (G - 1) δ^{2} (4 {| α |}^{2} + 2) + 1 .

(15)

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R_{limit} = G (G - 1) δ^{2} (4 {| α |}^{2} + 2) \approx 4 G I_{ps} δ^{2} .

(16)

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δ = \frac{1}{2 \sqrt{G I_{p s}}} = \frac{δ_{SNL}}{\sqrt{2 G}} .

(17)

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Δ θ = \frac{1}{4 l \sqrt{G I_{ps}}} = \frac{θ_{SNL}}{\sqrt{2 G}} .

(18)

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ξ^{+} {\hat{b}}_{0} ξ = {\hat{b}}_{0} \cosh r - {\hat{b}}_{0}^{+} e^{2 i ϕ} \sinh r, ξ^{+} {\hat{b}}_{0}^{+} ξ = {\hat{b}}_{0}^{+} \cosh r - {\hat{b}}_{0} e^{- 2 i ϕ} \sinh r,

(19)

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⟨ {\hat{X}}_{b out}^{2} ⟩ \approx 4 G (G - 1) δ^{2} {| α |}^{2} + e^{- r} | α | ≫ 1 .

(20)

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R_{limit} = 4 G (G - 1) δ^{2} {| α |}^{2} e^{r} .

(21)

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δ = \frac{1}{2 \sqrt{G I_{ps} e^{r}}} = \frac{δ_{SNL}}{\sqrt{2 G e^{r}}} .

(22)

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Δ θ = \frac{1}{4 l \sqrt{G I_{ps} e^{r}}} = \frac{θ_{SNL}}{\sqrt{2 G e^{r}}} .

(23)

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I_{ps} = G - 1 .

(24)

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⟨ {\hat{b}}_{out}^{+} {\hat{b}}_{out} ⟩ = [2 G (G - 1) + 2 G (G - 1) \cos (2 l θ_{2})] .

(25)

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⟨ {\hat{X}}_{b out}^{2} ⟩ = 1 + 4 G (G - 1) \cos (2 l θ_{2}) \approx 1 + 2 G (G - 1) δ^{2} .

(26)

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R_{limit} = 2 G (G - 1) δ^{2} .

(27)

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δ = \frac{1}{\sqrt{2 G (G - 1)}} = \frac{1}{\sqrt{2 I_{ps} (I_{ps} + 1)}} \approx \frac{1}{N} = δ_{HL} .

(28)

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Δ θ \approx \frac{1}{2 l N} .

(29)

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{\hat{a}}_{1} = \sqrt{G} {\hat{a}}_{0} + \sqrt{G - 1} {\hat{b}}_{0}^{+}, {\hat{b}}_{1} = \sqrt{G - 1} {\hat{a}}_{0}^{+} + \sqrt{G} {\hat{b}}_{0} .

(30)

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{\hat{a}}_{out} = [G + (G - 1) e^{- 4 i l θ_{2}}] {\hat{a}}_{0} + [\sqrt{G (G - 1)} + \sqrt{G (G - 1)} e^{- 4 i l θ_{2}}] {\hat{b}}_{0}^{+}, {\hat{b}}_{out} = [\sqrt{G (G - 1)} + \sqrt{G (G - 1)} e^{4 i l θ_{2}}] {\hat{a}}_{0}^{+} + [(G - 1) + G e^{4 i l θ_{2}}] {\hat{b}}_{0} .

(31)

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I_{ps} = ⟨ {\hat{b}}_{1}^{+} {\hat{b}}_{1} ⟩ = (G - 1) ({| α |}^{2} + 1) \approx (G - 1) {| α |}^{2}, | α | ≫ 1 .

(32)

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⟨ {\hat{b}}_{out}^{+} {\hat{b}}_{out} ⟩ = [2 G (G - 1) + 2 G (G - 1) \cos (4 l θ_{2})] ({| α |}^{2} + 1) \approx [2 G (G - 1) + 2 G (G - 1) \cos (4 l θ_{2})] {| α |}^{2} = 2 G [1 + \cos (4 l θ_{2})] I_{ps} .

(33)

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⟨ {\hat{X}}_{b out}^{2} ⟩ = 2 G (G - 1) {| α |}^{2} [1 - \cos (8 l θ_{2})] + 4 G^{2} - 4 G + 1 + 4 G (G - 1) \cos (4 l θ_{2}) .

(34)

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⟨ {\hat{X}}_{b out}^{2} ⟩ \approx G (G - 1) δ^{2} (4 {| α |}^{2} + 2) + 1 .

(35)

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R_{limit} = G (G - 1) δ^{2} (4 {| α |}^{2} + 2) \approx 4 G I_{ps} δ^{2} .

(36)

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δ = \frac{1}{2 \sqrt{G I_{ps}}} = \frac{δ_{SNL}}{\sqrt{2 G}} .

(37)

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Δ θ = \frac{1}{8 l \sqrt{G I_{ps}}} = \frac{θ_{SNL}}{2 \sqrt{2 G}} .

(38)

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⟨ {\hat{X}}_{b out}^{2} ⟩ \approx 4 G (G - 1) δ^{2} {| α |}^{2} + e^{- r},

(39)

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R_{limit} = 4 G (G - 1) δ^{2} {| α |}^{2} e^{r} .

(40)

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δ = \frac{1}{2 \sqrt{G I_{ps} e^{r}}} = \frac{1}{\sqrt{2 G N e^{r}}} .

(41)

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Δ θ = \frac{1}{8 l \sqrt{G I_{ps} e^{r}}} = \frac{1}{4 l \sqrt{2 G N e^{r}}} .

(42)

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δ \approx \frac{1}{N} = δ_{HL} .

(43)

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Δ θ \approx \frac{1}{4 l N} .

(44)

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References (32)

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Jun Liu, Wenxiao Liu, Shitao Li, Dong Wei, Hong Gao, Fuli Li. Enhancement of the angular rotation measurement sensitivity based on SU(2) and SU(1,1) interferometers[J]. Photonics Research, 2017, 5(6): 617

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