Abstract

Super-resolved angular displacement estimation is of crucial significance to the field of quantum information processing. Here we report an estimation protocol based on a Sagnac interferometer fed by a coherent state carrying orbital angular momentum. In a lossless scenario, through the use of parity measurement, our protocol can achieve a 4ℓ-fold super-resolved output with quantum number ℓ; meanwhile, a shot-noise-limited sensitivity saturating the quantum Cramér-Rao bound is reachable. We also consider the effects of several realistic factors, including nonideal state preparation, photon loss, and inefficient measurement. Finally, with mean photon number $\bar N=2.297$ and ℓ = 1 taken, we experimentally demonstrate a super-resolved effect of angular displacement with a factor of 7.88.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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  5. Y. Yan, G. Xie, M. P. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5(1), 4876 (2014).
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  6. A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8(3), 234–238 (2014).
    [Crossref]
  7. G. Puentes, N. Hermosa, and J. Torres, “Weak measurements with orbital-angular-momentum pointer states,” Phys. Rev. Lett. 109(4), 040401 (2012).
    [Crossref]
  8. O. S. Magaña-Loaiza and R. W. Boyd, “Quantum imaging and information,” Rep. Prog. Phys. 82(12), 124401 (2019).
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  9. V. D’ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4(1), 2432 (2013).
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  12. G. Thekkadath, L. Giner, Y. Chalich, M. Horton, J. Banker, and J. Lundeen, “Direct measurement of the density matrix of a quantum system,” Phys. Rev. Lett. 117(12), 120401 (2016).
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    [Crossref]
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    [Crossref]
  17. J.-D. Zhang, Z.-J. Zhang, L.-Z. Cen, J.-Y. Hu, and Y. Zhao, “Heisenberg-scaling angular displacement estimation with tunable squeezed bell states,” J. Opt. 21(3), 035201 (2019).
    [Crossref]
  18. J. Liu, W. Liu, S. Li, D. Wei, H. Gao, and F. Li, “Enhancement of the angular rotation measurement sensitivity based on SU(2) and SU(1,1) interferometers,” Photonics Res. 5(6), 617–622 (2017).
    [Crossref]
  19. J.-D. Zhang, C.-F. Jin, Z.-J. Zhang, L.-Z. Cen, J.-Y. Hu, and Y. Zhao, “Super-sensitive angular displacement estimation via an SU(1,1)-SU(2) hybrid interferometer,” Opt. Express 26(25), 33080–33090 (2018).
    [Crossref]
  20. J. Liu, S. Li, D. Wei, H. Gao, and F. Li, “Super-resolution and ultra-sensitivity of angular rotation measurement based on SU(1,1) interferometers using homodyne detection,” J. Opt. 20(2), 025201 (2018).
    [Crossref]
  21. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
    [Crossref]
  22. Z. Zhang, T. Qiao, K. Ma, L. Cen, J. Zhang, F. Wang, and Y. Zhao, “Ultra-sensitive and super-resolving angular rotation measurement based on photon orbital angular momentum using parity measurement,” Opt. Lett. 41(16), 3856–3859 (2016).
    [Crossref]
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    [Crossref]
  24. C. C. Gerry, “Heisenberg-limit interferometry with four-wave mixers operating in a nonlinear regime,” Phys. Rev. A 61(4), 043811 (2000).
    [Crossref]
  25. C. C. Gerry, A. Benmoussa, and R. Campos, “Quantum nondemolition measurement of parity and generation of parity eigenstates in optical fields,” Phys. Rev. A 72(5), 053818 (2005).
    [Crossref]
  26. D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, I. A. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing,” J. Mod. Opt. 51(9-10), 1499–1515 (2004).
    [Crossref]
  27. L. Cohen, D. Istrati, L. Dovrat, and H. Eisenberg, “Super-resolved phase measurements at the shot noise limit by parity measurement,” Opt. Express 22(10), 11945–11953 (2014).
    [Crossref]
  28. O. S. Magaña-Loaiza, R. d. J. León-Montiel, A. Perez-Leija, A. B. U’Ren, C. You, K. Busch, A. E. Lita, S. W. Nam, R. P. Mirin, and T. Gerrits, “Multiphoton quantum-state engineering using conditional measurements,” npj Quantum Inf. 5(1), 80 (2019).
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  29. C. You, M. A. Quiroz-Juárez, A. Lambert, N. Bhusal, C. Dong, A. Perez-Leija, A. Javaid, R. de J. León-Montiel, and O. S. Maga na-Loaiza, “Identification of light sources using machine learning,” (2011).
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  31. T. B. Bahder, “Phase estimation with nonunitary interferometers: Information as a metric,” Phys. Rev. A 83(5), 053601 (2011).
    [Crossref]
  32. J. Zhang, Z. Zhang, L. Cen, M. Yu, S. Li, F. Wang, and Y. Zhao, “Effects of imperfect elements on resolution and sensitivity of quantum metrology using two-mode squeezed vacuum state,” Opt. Express 25(21), 24907–24916 (2017).
    [Crossref]
  33. X. Feng, G. Jin, and W. Yang, “Quantum interferometry with binary-outcome measurements in the presence of phase diffusion,” Phys. Rev. A 90(1), 013807 (2014).
    [Crossref]
  34. M. Kacprowicz, R. Demkowicz-Dobrzański, W. Wasilewski, K. Banaszek, and I. Walmsley, “Experimental quantum-enhanced estimation of a lossy phase shift,” Nat. Photonics 4(6), 357–360 (2010).
    [Crossref]
  35. N. Spagnolo, C. Vitelli, V. G. Lucivero, V. Giovannetti, L. Maccone, and F. Sciarrino, “Phase estimation via quantum interferometry for noisy detectors,” Phys. Rev. Lett. 108(23), 233602 (2012).
    [Crossref]
  36. A. Datta, L. Zhang, N. Thomas-Peter, U. Dorner, B. J. Smith, and I. A. Walmsley, “Quantum metrology with imperfect states and detectors,” Phys. Rev. A 83(6), 063836 (2011).
    [Crossref]
  37. L. Pezzé, A. Smerzi, G. Khoury, J. F. Hodelin, and D. Bouwmeester, “Phase detection at the quantum limit with multiphoton Mach-Zehnder interferometry,” Phys. Rev. Lett. 99(22), 223602 (2007).
    [Crossref]
  38. Z. Huang, K. R. Motes, P. M. Anisimov, J. P. Dowling, and D. W. Berry, “Adaptive phase estimation with two-mode squeezed vacuum and parity measurement,” Phys. Rev. A 95(5), 053837 (2017).
    [Crossref]
  39. Q.-S. Tan, J.-Q. Liao, X. Wang, and F. Nori, “Enhanced interferometry using squeezed thermal states and even or odd states,” Phys. Rev. A 89(5), 053822 (2014).
    [Crossref]
  40. P. Gibilisco, D. Imparato, and T. Isola, “Uncertainty principle and quantum Fisher information. II,” J. Math. Phys. 48(7), 072109 (2007).
    [Crossref]
  41. S. Knysh, V. N. Smelyanskiy, and G. A. Durkin, “Scaling laws for precision in quantum interferometry and the bifurcation landscape of the optimal state,” Phys. Rev. A 83(2), 021804 (2011).
    [Crossref]
  42. J. Liu, X. Jing, and X. Wang, “Phase-matching condition for enhancement of phase sensitivity in quantum metrology,” Phys. Rev. A 88(4), 042316 (2013).
    [Crossref]
  43. M. Jarzyna and R. Demkowicz-Dobrzański, “Quantum interferometry with and without an external phase reference,” Phys. Rev. A 85(1), 011801 (2012).
    [Crossref]
  44. M. Takeoka, K. P. Seshadreesan, C. You, S. Izumi, and J. P. Dowling, “Fundamental precision limit of a Mach-Zehnder interferometric sensor when one of the inputs is the vacuum,” Phys. Rev. A 96(5), 052118 (2017).
    [Crossref]
  45. C. You, S. Adhikari, X. Ma, M. Sasaki, M. Takeoka, and J. P. Dowling, “Conclusive precision bounds for SU(1,1) interferometers,” Phys. Rev. A 99(4), 042122 (2019).
    [Crossref]
  46. M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number resolution using time-multiplexed single-photon detectors,” Phys. Rev. A 68(4), 043814 (2003).
    [Crossref]
  47. W. N. Plick, P. M. Anisimov, J. P. Dowling, H. Lee, and G. S. Agarwal, “Parity detection in quantum optical metrology without number-resolving detectors,” New J. Phys. 12(11), 113025 (2010).
    [Crossref]

2019 (4)

O. S. Magaña-Loaiza and R. W. Boyd, “Quantum imaging and information,” Rep. Prog. Phys. 82(12), 124401 (2019).
[Crossref]

J.-D. Zhang, Z.-J. Zhang, L.-Z. Cen, J.-Y. Hu, and Y. Zhao, “Heisenberg-scaling angular displacement estimation with tunable squeezed bell states,” J. Opt. 21(3), 035201 (2019).
[Crossref]

O. S. Magaña-Loaiza, R. d. J. León-Montiel, A. Perez-Leija, A. B. U’Ren, C. You, K. Busch, A. E. Lita, S. W. Nam, R. P. Mirin, and T. Gerrits, “Multiphoton quantum-state engineering using conditional measurements,” npj Quantum Inf. 5(1), 80 (2019).
[Crossref]

C. You, S. Adhikari, X. Ma, M. Sasaki, M. Takeoka, and J. P. Dowling, “Conclusive precision bounds for SU(1,1) interferometers,” Phys. Rev. A 99(4), 042122 (2019).
[Crossref]

2018 (3)

2017 (5)

Z. Zhang, T. Qiao, J. Song, L. Cen, J. Zhang, S. Li, L. Yan, F. Wang, and Y. Zhao, “Improved resolution and sensitivity of angular rotation measurement using entangled coherent states,” Opt. Commun. 403, 92–96 (2017).
[Crossref]

J. Liu, W. Liu, S. Li, D. Wei, H. Gao, and F. Li, “Enhancement of the angular rotation measurement sensitivity based on SU(2) and SU(1,1) interferometers,” Photonics Res. 5(6), 617–622 (2017).
[Crossref]

J. Zhang, Z. Zhang, L. Cen, M. Yu, S. Li, F. Wang, and Y. Zhao, “Effects of imperfect elements on resolution and sensitivity of quantum metrology using two-mode squeezed vacuum state,” Opt. Express 25(21), 24907–24916 (2017).
[Crossref]

Z. Huang, K. R. Motes, P. M. Anisimov, J. P. Dowling, and D. W. Berry, “Adaptive phase estimation with two-mode squeezed vacuum and parity measurement,” Phys. Rev. A 95(5), 053837 (2017).
[Crossref]

M. Takeoka, K. P. Seshadreesan, C. You, S. Izumi, and J. P. Dowling, “Fundamental precision limit of a Mach-Zehnder interferometric sensor when one of the inputs is the vacuum,” Phys. Rev. A 96(5), 052118 (2017).
[Crossref]

2016 (2)

Z. Zhang, T. Qiao, K. Ma, L. Cen, J. Zhang, F. Wang, and Y. Zhao, “Ultra-sensitive and super-resolving angular rotation measurement based on photon orbital angular momentum using parity measurement,” Opt. Lett. 41(16), 3856–3859 (2016).
[Crossref]

G. Thekkadath, L. Giner, Y. Chalich, M. Horton, J. Banker, and J. Lundeen, “Direct measurement of the density matrix of a quantum system,” Phys. Rev. Lett. 117(12), 120401 (2016).
[Crossref]

2015 (2)

L. Zhang, A. Datta, and I. A. Walmsley, “Precision metrology using weak measurements,” Phys. Rev. Lett. 114(21), 210801 (2015).
[Crossref]

X.-Q. Zhou, H. Cable, R. Whittaker, P. Shadbolt, J. L. O’Brien, and J. C. Matthews, “Quantum-enhanced tomography of unitary processes,” Optica 2(6), 510–516 (2015).
[Crossref]

2014 (6)

O. S. Magaña-Loaiza, M. Mirhosseini, B. Rodenburg, and R. W. Boyd, “Amplification of angular rotations using weak measurements,” Phys. Rev. Lett. 112(20), 200401 (2014).
[Crossref]

Y. Yan, G. Xie, M. P. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5(1), 4876 (2014).
[Crossref]

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8(3), 234–238 (2014).
[Crossref]

L. Cohen, D. Istrati, L. Dovrat, and H. Eisenberg, “Super-resolved phase measurements at the shot noise limit by parity measurement,” Opt. Express 22(10), 11945–11953 (2014).
[Crossref]

Q.-S. Tan, J.-Q. Liao, X. Wang, and F. Nori, “Enhanced interferometry using squeezed thermal states and even or odd states,” Phys. Rev. A 89(5), 053822 (2014).
[Crossref]

X. Feng, G. Jin, and W. Yang, “Quantum interferometry with binary-outcome measurements in the presence of phase diffusion,” Phys. Rev. A 90(1), 013807 (2014).
[Crossref]

2013 (2)

V. D’ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4(1), 2432 (2013).
[Crossref]

J. Liu, X. Jing, and X. Wang, “Phase-matching condition for enhancement of phase sensitivity in quantum metrology,” Phys. Rev. A 88(4), 042316 (2013).
[Crossref]

2012 (3)

M. Jarzyna and R. Demkowicz-Dobrzański, “Quantum interferometry with and without an external phase reference,” Phys. Rev. A 85(1), 011801 (2012).
[Crossref]

G. Puentes, N. Hermosa, and J. Torres, “Weak measurements with orbital-angular-momentum pointer states,” Phys. Rev. Lett. 109(4), 040401 (2012).
[Crossref]

N. Spagnolo, C. Vitelli, V. G. Lucivero, V. Giovannetti, L. Maccone, and F. Sciarrino, “Phase estimation via quantum interferometry for noisy detectors,” Phys. Rev. Lett. 108(23), 233602 (2012).
[Crossref]

2011 (4)

A. Datta, L. Zhang, N. Thomas-Peter, U. Dorner, B. J. Smith, and I. A. Walmsley, “Quantum metrology with imperfect states and detectors,” Phys. Rev. A 83(6), 063836 (2011).
[Crossref]

A. K. Jha, G. S. Agarwal, and R. W. Boyd, “Supersensitive measurement of angular displacements using entangled photons,” Phys. Rev. A 83(5), 053829 (2011).
[Crossref]

T. B. Bahder, “Phase estimation with nonunitary interferometers: Information as a metric,” Phys. Rev. A 83(5), 053601 (2011).
[Crossref]

S. Knysh, V. N. Smelyanskiy, and G. A. Durkin, “Scaling laws for precision in quantum interferometry and the bifurcation landscape of the optimal state,” Phys. Rev. A 83(2), 021804 (2011).
[Crossref]

2010 (2)

W. N. Plick, P. M. Anisimov, J. P. Dowling, H. Lee, and G. S. Agarwal, “Parity detection in quantum optical metrology without number-resolving detectors,” New J. Phys. 12(11), 113025 (2010).
[Crossref]

M. Kacprowicz, R. Demkowicz-Dobrzański, W. Wasilewski, K. Banaszek, and I. Walmsley, “Experimental quantum-enhanced estimation of a lossy phase shift,” Nat. Photonics 4(6), 357–360 (2010).
[Crossref]

2008 (1)

J. P. Dowling, “Quantum optical metrology-the lowdown on high-N00N states,” Contemp. Phys. 49(2), 125–143 (2008).
[Crossref]

2007 (2)

L. Pezzé, A. Smerzi, G. Khoury, J. F. Hodelin, and D. Bouwmeester, “Phase detection at the quantum limit with multiphoton Mach-Zehnder interferometry,” Phys. Rev. Lett. 99(22), 223602 (2007).
[Crossref]

P. Gibilisco, D. Imparato, and T. Isola, “Uncertainty principle and quantum Fisher information. II,” J. Math. Phys. 48(7), 072109 (2007).
[Crossref]

2005 (1)

C. C. Gerry, A. Benmoussa, and R. Campos, “Quantum nondemolition measurement of parity and generation of parity eigenstates in optical fields,” Phys. Rev. A 72(5), 053818 (2005).
[Crossref]

2004 (1)

D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, I. A. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing,” J. Mod. Opt. 51(9-10), 1499–1515 (2004).
[Crossref]

2003 (1)

M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number resolution using time-multiplexed single-photon detectors,” Phys. Rev. A 68(4), 043814 (2003).
[Crossref]

2002 (2)

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
[Crossref]

M. Padgett and L. Allen, “Orbital angular momentum exchange in cylindrical-lens mode converters,” J. Opt. B: Quantum Semiclassical Opt. 4(2), S17–S19 (2002).
[Crossref]

2000 (1)

C. C. Gerry, “Heisenberg-limit interferometry with four-wave mixers operating in a nonlinear regime,” Phys. Rev. A 61(4), 043811 (2000).
[Crossref]

1999 (1)

J. Tabosa and D. Petrov, “Optical pumping of orbital angular momentum of light in cold cesium atoms,” Phys. Rev. Lett. 83(24), 4967–4970 (1999).
[Crossref]

1997 (1)

1996 (1)

J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. Heinzen, “Optimal frequency measurements with maximally correlated states,” Phys. Rev. A 54(6), R4649–R4652 (1996).
[Crossref]

1992 (1)

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Achilles, D.

D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, I. A. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing,” J. Mod. Opt. 51(9-10), 1499–1515 (2004).
[Crossref]

Adhikari, S.

Agarwal, G. S.

A. K. Jha, G. S. Agarwal, and R. W. Boyd, “Supersensitive measurement of angular displacements using entangled photons,” Phys. Rev. A 83(5), 053829 (2011).
[Crossref]

W. N. Plick, P. M. Anisimov, J. P. Dowling, H. Lee, and G. S. Agarwal, “Parity detection in quantum optical metrology without number-resolving detectors,” New J. Phys. 12(11), 113025 (2010).
[Crossref]

Ahmed, N.

Y. Yan, G. Xie, M. P. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5(1), 4876 (2014).
[Crossref]

Allen, L.

M. Padgett and L. Allen, “Orbital angular momentum exchange in cylindrical-lens mode converters,” J. Opt. B: Quantum Semiclassical Opt. 4(2), S17–S19 (2002).
[Crossref]

N. Simpson, K. Dholakia, L. Allen, and M. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22(1), 52–54 (1997).
[Crossref]

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Anisimov, P. M.

Z. Huang, K. R. Motes, P. M. Anisimov, J. P. Dowling, and D. W. Berry, “Adaptive phase estimation with two-mode squeezed vacuum and parity measurement,” Phys. Rev. A 95(5), 053837 (2017).
[Crossref]

W. N. Plick, P. M. Anisimov, J. P. Dowling, H. Lee, and G. S. Agarwal, “Parity detection in quantum optical metrology without number-resolving detectors,” New J. Phys. 12(11), 113025 (2010).
[Crossref]

Aolita, L.

V. D’ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4(1), 2432 (2013).
[Crossref]

Bahder, T. B.

T. B. Bahder, “Phase estimation with nonunitary interferometers: Information as a metric,” Phys. Rev. A 83(5), 053601 (2011).
[Crossref]

Banaszek, K.

M. Kacprowicz, R. Demkowicz-Dobrzański, W. Wasilewski, K. Banaszek, and I. Walmsley, “Experimental quantum-enhanced estimation of a lossy phase shift,” Nat. Photonics 4(6), 357–360 (2010).
[Crossref]

D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, I. A. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing,” J. Mod. Opt. 51(9-10), 1499–1515 (2004).
[Crossref]

Banker, J.

G. Thekkadath, L. Giner, Y. Chalich, M. Horton, J. Banker, and J. Lundeen, “Direct measurement of the density matrix of a quantum system,” Phys. Rev. Lett. 117(12), 120401 (2016).
[Crossref]

Bao, C.

Y. Yan, G. Xie, M. P. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5(1), 4876 (2014).
[Crossref]

Barnett, S. M.

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Z. Huang, K. R. Motes, P. M. Anisimov, J. P. Dowling, and D. W. Berry, “Adaptive phase estimation with two-mode squeezed vacuum and parity measurement,” Phys. Rev. A 95(5), 053837 (2017).
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Itano, W. M.

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M. Takeoka, K. P. Seshadreesan, C. You, S. Izumi, and J. P. Dowling, “Fundamental precision limit of a Mach-Zehnder interferometric sensor when one of the inputs is the vacuum,” Phys. Rev. A 96(5), 052118 (2017).
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D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, I. A. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing,” J. Mod. Opt. 51(9-10), 1499–1515 (2004).
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M. Jarzyna and R. Demkowicz-Dobrzański, “Quantum interferometry with and without an external phase reference,” Phys. Rev. A 85(1), 011801 (2012).
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Jha, A. K.

A. K. Jha, G. S. Agarwal, and R. W. Boyd, “Supersensitive measurement of angular displacements using entangled photons,” Phys. Rev. A 83(5), 053829 (2011).
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Jin, G.

X. Feng, G. Jin, and W. Yang, “Quantum interferometry with binary-outcome measurements in the presence of phase diffusion,” Phys. Rev. A 90(1), 013807 (2014).
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J. Liu, X. Jing, and X. Wang, “Phase-matching condition for enhancement of phase sensitivity in quantum metrology,” Phys. Rev. A 88(4), 042316 (2013).
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M. Kacprowicz, R. Demkowicz-Dobrzański, W. Wasilewski, K. Banaszek, and I. Walmsley, “Experimental quantum-enhanced estimation of a lossy phase shift,” Nat. Photonics 4(6), 357–360 (2010).
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L. Pezzé, A. Smerzi, G. Khoury, J. F. Hodelin, and D. Bouwmeester, “Phase detection at the quantum limit with multiphoton Mach-Zehnder interferometry,” Phys. Rev. Lett. 99(22), 223602 (2007).
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S. Knysh, V. N. Smelyanskiy, and G. A. Durkin, “Scaling laws for precision in quantum interferometry and the bifurcation landscape of the optimal state,” Phys. Rev. A 83(2), 021804 (2011).
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Laurat, J.

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8(3), 234–238 (2014).
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Y. Yan, G. Xie, M. P. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5(1), 4876 (2014).
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J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88(25), 257901 (2002).
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W. N. Plick, P. M. Anisimov, J. P. Dowling, H. Lee, and G. S. Agarwal, “Parity detection in quantum optical metrology without number-resolving detectors,” New J. Phys. 12(11), 113025 (2010).
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J. Liu, S. Li, D. Wei, H. Gao, and F. Li, “Super-resolution and ultra-sensitivity of angular rotation measurement based on SU(1,1) interferometers using homodyne detection,” J. Opt. 20(2), 025201 (2018).
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J. Liu, W. Liu, S. Li, D. Wei, H. Gao, and F. Li, “Enhancement of the angular rotation measurement sensitivity based on SU(2) and SU(1,1) interferometers,” Photonics Res. 5(6), 617–622 (2017).
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Y. Yan, G. Xie, M. P. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5(1), 4876 (2014).
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J. Liu, S. Li, D. Wei, H. Gao, and F. Li, “Super-resolution and ultra-sensitivity of angular rotation measurement based on SU(1,1) interferometers using homodyne detection,” J. Opt. 20(2), 025201 (2018).
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J. Liu, W. Liu, S. Li, D. Wei, H. Gao, and F. Li, “Enhancement of the angular rotation measurement sensitivity based on SU(2) and SU(1,1) interferometers,” Photonics Res. 5(6), 617–622 (2017).
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V. D’ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4(1), 2432 (2013).
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Liu, J.

J. Liu, S. Li, D. Wei, H. Gao, and F. Li, “Super-resolution and ultra-sensitivity of angular rotation measurement based on SU(1,1) interferometers using homodyne detection,” J. Opt. 20(2), 025201 (2018).
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J. Liu, W. Liu, S. Li, D. Wei, H. Gao, and F. Li, “Enhancement of the angular rotation measurement sensitivity based on SU(2) and SU(1,1) interferometers,” Photonics Res. 5(6), 617–622 (2017).
[Crossref]

J. Liu, X. Jing, and X. Wang, “Phase-matching condition for enhancement of phase sensitivity in quantum metrology,” Phys. Rev. A 88(4), 042316 (2013).
[Crossref]

Liu, W.

J. Liu, W. Liu, S. Li, D. Wei, H. Gao, and F. Li, “Enhancement of the angular rotation measurement sensitivity based on SU(2) and SU(1,1) interferometers,” Photonics Res. 5(6), 617–622 (2017).
[Crossref]

Lucivero, V. G.

N. Spagnolo, C. Vitelli, V. G. Lucivero, V. Giovannetti, L. Maccone, and F. Sciarrino, “Phase estimation via quantum interferometry for noisy detectors,” Phys. Rev. Lett. 108(23), 233602 (2012).
[Crossref]

Lundeen, J.

G. Thekkadath, L. Giner, Y. Chalich, M. Horton, J. Banker, and J. Lundeen, “Direct measurement of the density matrix of a quantum system,” Phys. Rev. Lett. 117(12), 120401 (2016).
[Crossref]

Ma, K.

Ma, X.

C. You, S. Adhikari, X. Ma, M. Sasaki, M. Takeoka, and J. P. Dowling, “Conclusive precision bounds for SU(1,1) interferometers,” Phys. Rev. A 99(4), 042122 (2019).
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N. Spagnolo, C. Vitelli, V. G. Lucivero, V. Giovannetti, L. Maccone, and F. Sciarrino, “Phase estimation via quantum interferometry for noisy detectors,” Phys. Rev. Lett. 108(23), 233602 (2012).
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C. You, M. A. Quiroz-Juárez, A. Lambert, N. Bhusal, C. Dong, A. Perez-Leija, A. Javaid, R. de J. León-Montiel, and O. S. Maga na-Loaiza, “Identification of light sources using machine learning,” (2011).

Magaña-Loaiza, O. S.

O. S. Magaña-Loaiza, R. d. J. León-Montiel, A. Perez-Leija, A. B. U’Ren, C. You, K. Busch, A. E. Lita, S. W. Nam, R. P. Mirin, and T. Gerrits, “Multiphoton quantum-state engineering using conditional measurements,” npj Quantum Inf. 5(1), 80 (2019).
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O. S. Magaña-Loaiza and R. W. Boyd, “Quantum imaging and information,” Rep. Prog. Phys. 82(12), 124401 (2019).
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O. S. Magaña-Loaiza, M. Mirhosseini, B. Rodenburg, and R. W. Boyd, “Amplification of angular rotations using weak measurements,” Phys. Rev. Lett. 112(20), 200401 (2014).
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Marrucci, L.

V. D’ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4(1), 2432 (2013).
[Crossref]

Matthews, J. C.

Maxein, D.

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8(3), 234–238 (2014).
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N. Spagnolo, C. Vitelli, V. G. Lucivero, V. Giovannetti, L. Maccone, and F. Sciarrino, “Phase estimation via quantum interferometry for noisy detectors,” Phys. Rev. Lett. 108(23), 233602 (2012).
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C. You, S. Adhikari, X. Ma, M. Sasaki, M. Takeoka, and J. P. Dowling, “Conclusive precision bounds for SU(1,1) interferometers,” Phys. Rev. A 99(4), 042122 (2019).
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M. Takeoka, K. P. Seshadreesan, C. You, S. Izumi, and J. P. Dowling, “Fundamental precision limit of a Mach-Zehnder interferometric sensor when one of the inputs is the vacuum,” Phys. Rev. A 96(5), 052118 (2017).
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G. Puentes, N. Hermosa, and J. Torres, “Weak measurements with orbital-angular-momentum pointer states,” Phys. Rev. Lett. 109(4), 040401 (2012).
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Y. Yan, G. Xie, M. P. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5(1), 4876 (2014).
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O. S. Magaña-Loaiza, R. d. J. León-Montiel, A. Perez-Leija, A. B. U’Ren, C. You, K. Busch, A. E. Lita, S. W. Nam, R. P. Mirin, and T. Gerrits, “Multiphoton quantum-state engineering using conditional measurements,” npj Quantum Inf. 5(1), 80 (2019).
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A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8(3), 234–238 (2014).
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N. Spagnolo, C. Vitelli, V. G. Lucivero, V. Giovannetti, L. Maccone, and F. Sciarrino, “Phase estimation via quantum interferometry for noisy detectors,” Phys. Rev. Lett. 108(23), 233602 (2012).
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V. D’ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4(1), 2432 (2013).
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L. Zhang, A. Datta, and I. A. Walmsley, “Precision metrology using weak measurements,” Phys. Rev. Lett. 114(21), 210801 (2015).
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A. Datta, L. Zhang, N. Thomas-Peter, U. Dorner, B. J. Smith, and I. A. Walmsley, “Quantum metrology with imperfect states and detectors,” Phys. Rev. A 83(6), 063836 (2011).
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D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, I. A. Walmsley, M. J. Fitch, B. C. Jacobs, T. B. Pittman, and J. D. Franson, “Photon-number-resolving detection using time-multiplexing,” J. Mod. Opt. 51(9-10), 1499–1515 (2004).
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Wang, X.

Q.-S. Tan, J.-Q. Liao, X. Wang, and F. Nori, “Enhanced interferometry using squeezed thermal states and even or odd states,” Phys. Rev. A 89(5), 053822 (2014).
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J. Liu, X. Jing, and X. Wang, “Phase-matching condition for enhancement of phase sensitivity in quantum metrology,” Phys. Rev. A 88(4), 042316 (2013).
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M. Kacprowicz, R. Demkowicz-Dobrzański, W. Wasilewski, K. Banaszek, and I. Walmsley, “Experimental quantum-enhanced estimation of a lossy phase shift,” Nat. Photonics 4(6), 357–360 (2010).
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J. Liu, S. Li, D. Wei, H. Gao, and F. Li, “Super-resolution and ultra-sensitivity of angular rotation measurement based on SU(1,1) interferometers using homodyne detection,” J. Opt. 20(2), 025201 (2018).
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J. Liu, W. Liu, S. Li, D. Wei, H. Gao, and F. Li, “Enhancement of the angular rotation measurement sensitivity based on SU(2) and SU(1,1) interferometers,” Photonics Res. 5(6), 617–622 (2017).
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Z. Zhang, T. Qiao, J. Song, L. Cen, J. Zhang, S. Li, L. Yan, F. Wang, and Y. Zhao, “Improved resolution and sensitivity of angular rotation measurement using entangled coherent states,” Opt. Commun. 403, 92–96 (2017).
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X. Feng, G. Jin, and W. Yang, “Quantum interferometry with binary-outcome measurements in the presence of phase diffusion,” Phys. Rev. A 90(1), 013807 (2014).
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C. You, S. Adhikari, X. Ma, M. Sasaki, M. Takeoka, and J. P. Dowling, “Conclusive precision bounds for SU(1,1) interferometers,” Phys. Rev. A 99(4), 042122 (2019).
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O. S. Magaña-Loaiza, R. d. J. León-Montiel, A. Perez-Leija, A. B. U’Ren, C. You, K. Busch, A. E. Lita, S. W. Nam, R. P. Mirin, and T. Gerrits, “Multiphoton quantum-state engineering using conditional measurements,” npj Quantum Inf. 5(1), 80 (2019).
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Yu, M.

Zhang, J.

Zhang, J.-D.

Zhang, L.

L. Zhang, A. Datta, and I. A. Walmsley, “Precision metrology using weak measurements,” Phys. Rev. Lett. 114(21), 210801 (2015).
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A. Datta, L. Zhang, N. Thomas-Peter, U. Dorner, B. J. Smith, and I. A. Walmsley, “Quantum metrology with imperfect states and detectors,” Phys. Rev. A 83(6), 063836 (2011).
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Zhang, Z.

Zhang, Z.-J.

Zhao, Y.

Zhao, Z.

Y. Yan, G. Xie, M. P. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5(1), 4876 (2014).
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J. Mod. Opt. (1)

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J. Opt. (2)

J. Liu, S. Li, D. Wei, H. Gao, and F. Li, “Super-resolution and ultra-sensitivity of angular rotation measurement based on SU(1,1) interferometers using homodyne detection,” J. Opt. 20(2), 025201 (2018).
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J. Opt. B: Quantum Semiclassical Opt. (1)

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Nat. Commun. (2)

Y. Yan, G. Xie, M. P. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5(1), 4876 (2014).
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V. D’ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4(1), 2432 (2013).
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Nat. Photonics (2)

A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, “A quantum memory for orbital angular momentum photonic qubits,” Nat. Photonics 8(3), 234–238 (2014).
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New J. Phys. (1)

W. N. Plick, P. M. Anisimov, J. P. Dowling, H. Lee, and G. S. Agarwal, “Parity detection in quantum optical metrology without number-resolving detectors,” New J. Phys. 12(11), 113025 (2010).
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npj Quantum Inf. (1)

O. S. Magaña-Loaiza, R. d. J. León-Montiel, A. Perez-Leija, A. B. U’Ren, C. You, K. Busch, A. E. Lita, S. W. Nam, R. P. Mirin, and T. Gerrits, “Multiphoton quantum-state engineering using conditional measurements,” npj Quantum Inf. 5(1), 80 (2019).
[Crossref]

Opt. Commun. (1)

Z. Zhang, T. Qiao, J. Song, L. Cen, J. Zhang, S. Li, L. Yan, F. Wang, and Y. Zhao, “Improved resolution and sensitivity of angular rotation measurement using entangled coherent states,” Opt. Commun. 403, 92–96 (2017).
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Opt. Express (4)

Opt. Lett. (2)

Optica (1)

Photonics Res. (1)

J. Liu, W. Liu, S. Li, D. Wei, H. Gao, and F. Li, “Enhancement of the angular rotation measurement sensitivity based on SU(2) and SU(1,1) interferometers,” Photonics Res. 5(6), 617–622 (2017).
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Phys. Rev. A (16)

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
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A. K. Jha, G. S. Agarwal, and R. W. Boyd, “Supersensitive measurement of angular displacements using entangled photons,” Phys. Rev. A 83(5), 053829 (2011).
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Figures (8)

Fig. 1.
Fig. 1. Schematic of the angular displacement estimation protocol. The full names of the abbreviations in the figure: L, laser; P, polarizer; SLM, spatial light modulator; I, iris; BS, beam splitter; DP, Dove prism; M, mirror; PNRD, photon-number-resolving detector.
Fig. 2.
Fig. 2. (a) The output of parity measurement against angular displacement with different quantum numbers, where $N = 10$. (b) The output of parity measurement against angular displacement with different mean photon numbers, where $\ell = 3$.
Fig. 3.
Fig. 3. The FWHM of parity measurement against the mean photon number.
Fig. 4.
Fig. 4. (a) The output of parity measurement against angular displacement, where $N = 10$ and $\ell = 3$. (b) The optimal sensitivity of parity measurement against two path losses, where $N = 10$ and $\ell = 3$.
Fig. 5.
Fig. 5. The sensitivity of parity measurement against angular displacement with different rates. Where the curve of $r = 0$ is the ideal curve, the curves of $r = 10^{-3}$ and $r = 10^{-2}$ respectively correspond to the scenarios: dark counts, and dark counts along with response-time delay.
Fig. 6.
Fig. 6. Diagram of two angular displacement estimation protocols: a Sagnac interferometer and a Mach-Zehnder interferometer. The full names of the abbreviations in the figure: DP, Dove prism; D, detector; BS, beam splitter; SI, Sagnac interferometer; MZI, Mach-Zehnder interferometer.
Fig. 7.
Fig. 7. Experimental data against angular displacement with $\ell = 1$. (a) The blue line is a fit to the output. Error bars are one standard deviation due to propagated Poissonian statistics. (b) The red line is the sensitivity deduced from the fit of output, blue dots are the sensitivities calculated from the experimental data, and the black dashed line is the shot-noise limit defined in accordance with $\bar N$.
Fig. 8.
Fig. 8. (a) The analog voltage signals displayed by oscillograph, and each signal is converted from single statistical trigger counts. (b) The probability distribution of output photon state and a fit based on Poissonian distribution.

Equations (28)

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| ψ out = e N 2 n = 0 [ i α cos ( 2 φ ) ] n n ! m = 0 [ i α sin ( 2 φ ) ] m m ! | n , m .
P ( n , m ) = e N n ! m ! [ N cos 2 ( 2 φ ) ] n [ N sin 2 ( 2 φ ) ] m .
P even = 1 2 { 1 + exp [ 2 N sin 2 ( 2 φ ) ] } ,
P odd = 1 2 { 1 exp [ 2 N sin 2 ( 2 φ ) ] } .
Π ^ = exp [ 2 N sin 2 ( 2 φ ) ] .
F = 1 P even ( P even φ ) 2 + 1 P odd ( P odd φ ) 2 .
Δ φ = Π ^ 2 Π ^ 2 | Π ^ / Π ^ φ φ | ,
Δ φ = exp [ 4 N sin 2 ( 2 φ ) ] 1 | 4 N sin ( 4 φ ) | .
Δ φ min = 1 + 4 N sin 2 ( 2 φ ) 1 | 4 N sin ( 4 φ ) | | φ 0 = 1 4 N .
V = Π ^ max Π ^ min Π ^ max + Π ^ min .
ρ in = [ η | α α | + ( 1 η ) | α 0 α 0 | ] | 0 0 | .
Π ^ 1 = η exp [ 2 N sin 2 ( 2 φ ) ] + 1 η .
Δ φ 1 = sin ( 2 φ ) 1 η N sin 2 ( 2 φ ) | 2 η N sin ( 4 φ ) | | φ 0 = 1 η 1 4 N .
Π ^ 2 = exp [ N T A T B cos ( 4 φ ) N 2 ( T A + T B ) ] ,
Δ φ 2 = exp { N [ T A + T B 2 T A T B cos ( 4 φ ) ] } 1 | 4 ( T A + T B ) N sin ( 4 φ ) | .
Π ^ 3 = exp [ 2 κ N sin 2 ( 2 φ ) ] .
Π ^ 4 = e 2 r Π ^
J ^ x = 1 2 ( a ^ b ^ + a ^ b ^ ) ,
J ^ y = i 2 ( a ^ b ^ a ^ b ^ ) ,
J ^ z = 1 2 ( a ^ a ^ b ^ b ^ )
F S = 4 [ ψ | ( 4 J ^ z ) 2 | ψ ψ | 4 J ^ z | ψ 2 ] = 16 2 N ,
F M = 4 [ ψ | ( 2 n ^ a ) 2 | ψ ψ | 2 n ^ a | ψ 2 ] = 8 2 N .
Π ^ = 0.9507 exp { 4.594 sin 2 [ 2 ( φ 0.7022 ) ] } .
Π ^ = exp [ 4.594 sin 2 ( 2 φ ) 0.0506 ]
ρ ¯ 1 = 1 2 π 0 2 π exp ( i δ n ^ a ) ρ a ρ b exp ( i δ n ^ a ) d δ = n = 0 p n | n n | | 0 0 | .
ρ ¯ 2 = U ^ BS ρ ¯ 1 U ^ BS = n = 0 p n m = 0 n C n m | n m n m | | m m | ,
F Fock = 4 [ U ^ BS ( 2 n ^ a ) 2 U ^ BS U ^ BS ( 2 n ^ a ) U ^ BS 2 ] = 4 2 n .
F ρ ¯ = n = 0 p n 4 2 n = 4 2 N .

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