Abstract

It has been demonstrated that using two-mode squeezed vacuum state for phase estimation can break the Heisenberg limit. Our results reveal that the two-mode squeezed vacuum state is also applied to the optical rotation angle measurement. In our scheme, the resolution and sensitivity of the optical rotation angle signal are the same as the case of phase estimation. For the parameter estimation, phase or rotation angle, we discuss the influences of several imperfect factors on the resolution and sensitivity. First, the effect that the upper limit of photon-number resolving has on the maximum amount of available quantum Fisher information has been analyzed. Then, we have also studied the impacts of both the transmission efficiency in the transmission process and the detection efficiency on the detection results. Finally, conditions where all of the above imperfect elements are taken into account at the same time have also been explored. Additionally, other imperfect factors such as squeezing efficiency and dark counts are briefly discussed.

© 2017 Optical Society of America

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References

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  4. J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777 (2012).
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  7. M. D. Lang and C. M. Caves, “Optimal quantum-enhanced interferometry using a laser power source,” Phys. Rev. Lett. 111(17), 173601 (2013).
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  9. M. Lebugle, M. Grafe, R. Heilmann, A. Perez-Leija, S. Nolte, and A. Szameit, “Experimental observation of N00N state Bloch oscillations,” Nat. Commun. 6(1), 8273 (2015).
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    [Crossref] [PubMed]
  13. J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, “Optimal frequency measurements with maximally correlated states,” Phys. Rev. A 54(6), R4649 (1996).
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  14. C. C. Gerry, “Heisenberg-limit interferometry with four-wave mixers operating in a nonlinear regime,” Phys. Rev. A 61(4), 043811 (2000).
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  15. C. C. Gerry and R. A. Campos, “Generation of maximally entangled photonic states with a quantum-optical Fredkin gate,” Phys. Rev. A 64(6), 063814 (2001).
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    [Crossref]
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    [Crossref]
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    [Crossref]
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  22. 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]
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    [Crossref]
  28. 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]
  29. D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, and I. A. Walmsley, “Fiber-assisted detection with photon number resolution,” Opt. Lett. 28(23), 2387–2389 (2003).
    [Crossref] [PubMed]
  30. J. S. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. L. Pregnel, C. Silberhorn, T. C. Ralph, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Tomography of quantum detectors,” Nat. Phys. 5(1), 27–30 (2009).
    [Crossref]
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    [Crossref]

2017 (1)

P. Liu, P. Wang, W. Yang, G. R. Jin, and C. P. Sun, “Fisher information of a squeezed-state interferometer with a finite photon-number resolution,” Phys. Rev. A 95(2), 023824 (2017).
[Crossref]

2015 (1)

M. Lebugle, M. Grafe, R. Heilmann, A. Perez-Leija, S. Nolte, and A. Szameit, “Experimental observation of N00N state Bloch oscillations,” Nat. Commun. 6(1), 8273 (2015).
[Crossref] [PubMed]

2014 (4)

W. Zhang, Q. Qi, J. Zhou, and L. Chen, “Mimicking Faraday rotation to sort the orbital angular momentum of light,” Phys. Rev. Lett. 112(15), 153601 (2014).
[Crossref] [PubMed]

C. E. Vollmer, C. Baune, A. Samblowski, T. Eberle, V. Handchen, J. Fiurasek, and R. Schnabel, “Quantum up-conversion of squeezed vacuum states from 1550 to 532 nm,” Phys. Rev. Lett. 112(7), 073602 (2014).
[Crossref] [PubMed]

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]

Y. Israel, S. Rosen, and Y. Silberberg, “Supersensitive polarization microscopy using NOON states of light,” Phys. Rev. Lett. 112(10), 103604 (2014).
[Crossref] [PubMed]

2013 (2)

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]

M. D. Lang and C. M. Caves, “Optimal quantum-enhanced interferometry using a laser power source,” Phys. Rev. Lett. 111(17), 173601 (2013).
[Crossref] [PubMed]

2012 (1)

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777 (2012).
[Crossref]

2011 (4)

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5(4), 222–229 (2011).
[Crossref]

J. Joo, W. J. Munro, and T. P. Spiller, “Quantum metrology with entangled coherent states,” Phys. Rev. Lett. 107(8) 083601 (2011).
[Crossref] [PubMed]

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]

G. Y. Xiang, B. L. Higgins, D. W. Berry, H. M. Wiseman, and G. J. Pryde, “Entanglement-enhanced measurement of a completely unknown optical phase,” Nat. Photonics 5(1), 43–47 (2011).
[Crossref]

2010 (2)

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science,  328(5980), 879–881 (2010).
[Crossref] [PubMed]

P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum metrology with two-mode squeezed vacuum: parity detection beats the Heisenberg limit,” Phys. Rev. Lett. 104(10), 103602 (2010).
[Crossref] [PubMed]

2009 (2)

J. S. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. L. Pregnel, C. Silberhorn, T. C. Ralph, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Tomography of quantum detectors,” Nat. Phys. 5(1), 27–30 (2009).
[Crossref]

R. H. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photonics 3, 696–705 (2009).
[Crossref]

2008 (1)

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

2006 (2)

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96(1), 010401 (2006).
[Crossref] [PubMed]

C. F. Roos, M. Chwalla, K. Kim, M. Riebe, and R. Blatt, “‘Designer atoms’ for quantum metrology,” Nature 443(7109), 316–319 (2006).
[Crossref] [PubMed]

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 (2)

D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, and I. A. Walmsley, “Fiber-assisted detection with photon number resolution,” Opt. Lett. 28(23), 2387–2389 (2003).
[Crossref] [PubMed]

R. A. Campos, C. C. Gerry, and A. Benmoussa, “Optical interferometry at the Heisenberg limit with twin Fock states and parity measurements,” Phys. Rev. A 68(2), 023810 (2003).
[Crossref]

2002 (1)

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] [PubMed]

2001 (1)

C. C. Gerry and R. A. Campos, “Generation of maximally entangled photonic states with a quantum-optical Fredkin gate,” Phys. Rev. A 64(6), 063814 (2001).
[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]

1996 (1)

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

1986 (1)

B. Yurke, S. L. McCall, and J. R. Klauder, “SU (2) and SU (1, 1) interferometers,” Phys. Rev. A 33(6) 4033 (1986).
[Crossref]

1981 (1)

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23(8), 1693 (1981).
[Crossref]

Abramowitz, M.

M. Abramowitz and I. Stegun, Handbook of Mathematical functions (Dover, 1965).

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]

D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, and I. A. Walmsley, “Fiber-assisted detection with photon number resolution,” Opt. Lett. 28(23), 2387–2389 (2003).
[Crossref] [PubMed]

Afek, I.

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science,  328(5980), 879–881 (2010).
[Crossref] [PubMed]

Ambar, O.

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science,  328(5980), 879–881 (2010).
[Crossref] [PubMed]

Anisimov, P. M.

P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum metrology with two-mode squeezed vacuum: parity detection beats the Heisenberg limit,” Phys. Rev. Lett. 104(10), 103602 (2010).
[Crossref] [PubMed]

Banaszek, K.

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]

D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, and I. A. Walmsley, “Fiber-assisted detection with photon number resolution,” Opt. Lett. 28(23), 2387–2389 (2003).
[Crossref] [PubMed]

Barnett, S. M.

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] [PubMed]

Baune, C.

C. E. Vollmer, C. Baune, A. Samblowski, T. Eberle, V. Handchen, J. Fiurasek, and R. Schnabel, “Quantum up-conversion of squeezed vacuum states from 1550 to 532 nm,” Phys. Rev. Lett. 112(7), 073602 (2014).
[Crossref] [PubMed]

Benmoussa, A.

R. A. Campos, C. C. Gerry, and A. Benmoussa, “Optical interferometry at the Heisenberg limit with twin Fock states and parity measurements,” Phys. Rev. A 68(2), 023810 (2003).
[Crossref]

Berry, D. W.

G. Y. Xiang, B. L. Higgins, D. W. Berry, H. M. Wiseman, and G. J. Pryde, “Entanglement-enhanced measurement of a completely unknown optical phase,” Nat. Photonics 5(1), 43–47 (2011).
[Crossref]

Blatt, R.

C. F. Roos, M. Chwalla, K. Kim, M. Riebe, and R. Blatt, “‘Designer atoms’ for quantum metrology,” Nature 443(7109), 316–319 (2006).
[Crossref] [PubMed]

Bollinger, J. J.

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

Campos, R. A.

R. A. Campos, C. C. Gerry, and A. Benmoussa, “Optical interferometry at the Heisenberg limit with twin Fock states and parity measurements,” Phys. Rev. A 68(2), 023810 (2003).
[Crossref]

C. C. Gerry and R. A. Campos, “Generation of maximally entangled photonic states with a quantum-optical Fredkin gate,” Phys. Rev. A 64(6), 063814 (2001).
[Crossref]

Caves, C. M.

M. D. Lang and C. M. Caves, “Optimal quantum-enhanced interferometry using a laser power source,” Phys. Rev. Lett. 111(17), 173601 (2013).
[Crossref] [PubMed]

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23(8), 1693 (1981).
[Crossref]

Chen, L.

W. Zhang, Q. Qi, J. Zhou, and L. Chen, “Mimicking Faraday rotation to sort the orbital angular momentum of light,” Phys. Rev. Lett. 112(15), 153601 (2014).
[Crossref] [PubMed]

Chen, Z. B.

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777 (2012).
[Crossref]

Chiruvelli, A.

P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum metrology with two-mode squeezed vacuum: parity detection beats the Heisenberg limit,” Phys. Rev. Lett. 104(10), 103602 (2010).
[Crossref] [PubMed]

Chwalla, M.

C. F. Roos, M. Chwalla, K. Kim, M. Riebe, and R. Blatt, “‘Designer atoms’ for quantum metrology,” Nature 443(7109), 316–319 (2006).
[Crossref] [PubMed]

Coldenstrodt-Ronge, H.

J. S. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. L. Pregnel, C. Silberhorn, T. C. Ralph, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Tomography of quantum detectors,” Nat. Phys. 5(1), 27–30 (2009).
[Crossref]

Courtial, J.

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] [PubMed]

Dowling, J. P.

P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum metrology with two-mode squeezed vacuum: parity detection beats the Heisenberg limit,” Phys. Rev. Lett. 104(10), 103602 (2010).
[Crossref] [PubMed]

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

M. Takeoka, K. P. Seshadreesan, C. L. You, S. Izumi, and J. P. Dowling, “Fundamental precision limit of a Mach-Zehnder interferometric sensor when one of the inputs is the vacuum,” arXiv:1705.09506, (2017).

Durkin, G. A.

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]

Eberle, T.

C. E. Vollmer, C. Baune, A. Samblowski, T. Eberle, V. Handchen, J. Fiurasek, and R. Schnabel, “Quantum up-conversion of squeezed vacuum states from 1550 to 532 nm,” Phys. Rev. Lett. 112(7), 073602 (2014).
[Crossref] [PubMed]

Eisert, J.

J. S. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. L. Pregnel, C. Silberhorn, T. C. Ralph, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Tomography of quantum detectors,” Nat. Phys. 5(1), 27–30 (2009).
[Crossref]

Feito, A.

J. S. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. L. Pregnel, C. Silberhorn, T. C. Ralph, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Tomography of quantum detectors,” Nat. Phys. 5(1), 27–30 (2009).
[Crossref]

Fitch, M. J.

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]

Fiurasek, J.

C. E. Vollmer, C. Baune, A. Samblowski, T. Eberle, V. Handchen, J. Fiurasek, and R. Schnabel, “Quantum up-conversion of squeezed vacuum states from 1550 to 532 nm,” Phys. Rev. Lett. 112(7), 073602 (2014).
[Crossref] [PubMed]

Franke-Arnold, S.

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] [PubMed]

Franson, J. 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]

Gerry, C. C.

R. A. Campos, C. C. Gerry, and A. Benmoussa, “Optical interferometry at the Heisenberg limit with twin Fock states and parity measurements,” Phys. Rev. A 68(2), 023810 (2003).
[Crossref]

C. C. Gerry and R. A. Campos, “Generation of maximally entangled photonic states with a quantum-optical Fredkin gate,” Phys. Rev. A 64(6), 063814 (2001).
[Crossref]

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

Giovannetti, V.

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5(4), 222–229 (2011).
[Crossref]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96(1), 010401 (2006).
[Crossref] [PubMed]

Grafe, M.

M. Lebugle, M. Grafe, R. Heilmann, A. Perez-Leija, S. Nolte, and A. Szameit, “Experimental observation of N00N state Bloch oscillations,” Nat. Commun. 6(1), 8273 (2015).
[Crossref] [PubMed]

Hadfield, R. H.

R. H. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photonics 3, 696–705 (2009).
[Crossref]

Handchen, V.

C. E. Vollmer, C. Baune, A. Samblowski, T. Eberle, V. Handchen, J. Fiurasek, and R. Schnabel, “Quantum up-conversion of squeezed vacuum states from 1550 to 532 nm,” Phys. Rev. Lett. 112(7), 073602 (2014).
[Crossref] [PubMed]

Heilmann, R.

M. Lebugle, M. Grafe, R. Heilmann, A. Perez-Leija, S. Nolte, and A. Szameit, “Experimental observation of N00N state Bloch oscillations,” Nat. Commun. 6(1), 8273 (2015).
[Crossref] [PubMed]

Heinzen, D. J.

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

Higgins, B. L.

G. Y. Xiang, B. L. Higgins, D. W. Berry, H. M. Wiseman, and G. J. Pryde, “Entanglement-enhanced measurement of a completely unknown optical phase,” Nat. Photonics 5(1), 43–47 (2011).
[Crossref]

Huver, S. D.

P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum metrology with two-mode squeezed vacuum: parity detection beats the Heisenberg limit,” Phys. Rev. Lett. 104(10), 103602 (2010).
[Crossref] [PubMed]

Israel, Y.

Y. Israel, S. Rosen, and Y. Silberberg, “Supersensitive polarization microscopy using NOON states of light,” Phys. Rev. Lett. 112(10), 103604 (2014).
[Crossref] [PubMed]

Itano, W. M.

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

Izumi, S.

M. Takeoka, K. P. Seshadreesan, C. L. You, S. Izumi, and J. P. Dowling, “Fundamental precision limit of a Mach-Zehnder interferometric sensor when one of the inputs is the vacuum,” arXiv:1705.09506, (2017).

Jacobs, B. C.

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]

Jin, G. R.

P. Liu, P. Wang, W. Yang, G. R. Jin, and C. P. Sun, “Fisher information of a squeezed-state interferometer with a finite photon-number resolution,” Phys. Rev. A 95(2), 023824 (2017).
[Crossref]

Jing, X.

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]

Joo, J.

J. Joo, W. J. Munro, and T. P. Spiller, “Quantum metrology with entangled coherent states,” Phys. Rev. Lett. 107(8) 083601 (2011).
[Crossref] [PubMed]

Kim, K.

C. F. Roos, M. Chwalla, K. Kim, M. Riebe, and R. Blatt, “‘Designer atoms’ for quantum metrology,” Nature 443(7109), 316–319 (2006).
[Crossref] [PubMed]

Klauder, J. R.

B. Yurke, S. L. McCall, and J. R. Klauder, “SU (2) and SU (1, 1) interferometers,” Phys. Rev. A 33(6) 4033 (1986).
[Crossref]

Knysh, S.

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]

Lang, M. D.

M. D. Lang and C. M. Caves, “Optimal quantum-enhanced interferometry using a laser power source,” Phys. Rev. Lett. 111(17), 173601 (2013).
[Crossref] [PubMed]

Leach, J.

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] [PubMed]

Lebugle, M.

M. Lebugle, M. Grafe, R. Heilmann, A. Perez-Leija, S. Nolte, and A. Szameit, “Experimental observation of N00N state Bloch oscillations,” Nat. Commun. 6(1), 8273 (2015).
[Crossref] [PubMed]

Lee, H.

P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum metrology with two-mode squeezed vacuum: parity detection beats the Heisenberg limit,” Phys. Rev. Lett. 104(10), 103602 (2010).
[Crossref] [PubMed]

Liao, J. Q.

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]

Liu, J.

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, P.

P. Liu, P. Wang, W. Yang, G. R. Jin, and C. P. Sun, “Fisher information of a squeezed-state interferometer with a finite photon-number resolution,” Phys. Rev. A 95(2), 023824 (2017).
[Crossref]

Lloyd, S.

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5(4), 222–229 (2011).
[Crossref]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96(1), 010401 (2006).
[Crossref] [PubMed]

Lu, C. Y.

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777 (2012).
[Crossref]

Lundeen, J. S.

J. S. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. L. Pregnel, C. Silberhorn, T. C. Ralph, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Tomography of quantum detectors,” Nat. Phys. 5(1), 27–30 (2009).
[Crossref]

Maccone, L.

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5(4), 222–229 (2011).
[Crossref]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96(1), 010401 (2006).
[Crossref] [PubMed]

McCall, S. L.

B. Yurke, S. L. McCall, and J. R. Klauder, “SU (2) and SU (1, 1) interferometers,” Phys. Rev. A 33(6) 4033 (1986).
[Crossref]

Munro, W. J.

J. Joo, W. J. Munro, and T. P. Spiller, “Quantum metrology with entangled coherent states,” Phys. Rev. Lett. 107(8) 083601 (2011).
[Crossref] [PubMed]

Nolte, S.

M. Lebugle, M. Grafe, R. Heilmann, A. Perez-Leija, S. Nolte, and A. Szameit, “Experimental observation of N00N state Bloch oscillations,” Nat. Commun. 6(1), 8273 (2015).
[Crossref] [PubMed]

Nori, F.

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]

Padgett, M. J.

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] [PubMed]

Pan, J. W.

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777 (2012).
[Crossref]

Perez-Leija, A.

M. Lebugle, M. Grafe, R. Heilmann, A. Perez-Leija, S. Nolte, and A. Szameit, “Experimental observation of N00N state Bloch oscillations,” Nat. Commun. 6(1), 8273 (2015).
[Crossref] [PubMed]

Pittman, T. B.

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]

Plenio, M. B.

J. S. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. L. Pregnel, C. Silberhorn, T. C. Ralph, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Tomography of quantum detectors,” Nat. Phys. 5(1), 27–30 (2009).
[Crossref]

Plick, W. N.

P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum metrology with two-mode squeezed vacuum: parity detection beats the Heisenberg limit,” Phys. Rev. Lett. 104(10), 103602 (2010).
[Crossref] [PubMed]

Pregnel, K. L.

J. S. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. L. Pregnel, C. Silberhorn, T. C. Ralph, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Tomography of quantum detectors,” Nat. Phys. 5(1), 27–30 (2009).
[Crossref]

Pryde, G. J.

G. Y. Xiang, B. L. Higgins, D. W. Berry, H. M. Wiseman, and G. J. Pryde, “Entanglement-enhanced measurement of a completely unknown optical phase,” Nat. Photonics 5(1), 43–47 (2011).
[Crossref]

Qi, Q.

W. Zhang, Q. Qi, J. Zhou, and L. Chen, “Mimicking Faraday rotation to sort the orbital angular momentum of light,” Phys. Rev. Lett. 112(15), 153601 (2014).
[Crossref] [PubMed]

Ralph, T. C.

J. S. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. L. Pregnel, C. Silberhorn, T. C. Ralph, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Tomography of quantum detectors,” Nat. Phys. 5(1), 27–30 (2009).
[Crossref]

Raterman, G. M.

P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum metrology with two-mode squeezed vacuum: parity detection beats the Heisenberg limit,” Phys. Rev. Lett. 104(10), 103602 (2010).
[Crossref] [PubMed]

Riebe, M.

C. F. Roos, M. Chwalla, K. Kim, M. Riebe, and R. Blatt, “‘Designer atoms’ for quantum metrology,” Nature 443(7109), 316–319 (2006).
[Crossref] [PubMed]

Roos, C. F.

C. F. Roos, M. Chwalla, K. Kim, M. Riebe, and R. Blatt, “‘Designer atoms’ for quantum metrology,” Nature 443(7109), 316–319 (2006).
[Crossref] [PubMed]

Rosen, S.

Y. Israel, S. Rosen, and Y. Silberberg, “Supersensitive polarization microscopy using NOON states of light,” Phys. Rev. Lett. 112(10), 103604 (2014).
[Crossref] [PubMed]

Samblowski, A.

C. E. Vollmer, C. Baune, A. Samblowski, T. Eberle, V. Handchen, J. Fiurasek, and R. Schnabel, “Quantum up-conversion of squeezed vacuum states from 1550 to 532 nm,” Phys. Rev. Lett. 112(7), 073602 (2014).
[Crossref] [PubMed]

Schnabel, R.

C. E. Vollmer, C. Baune, A. Samblowski, T. Eberle, V. Handchen, J. Fiurasek, and R. Schnabel, “Quantum up-conversion of squeezed vacuum states from 1550 to 532 nm,” Phys. Rev. Lett. 112(7), 073602 (2014).
[Crossref] [PubMed]

Seshadreesan, K. P.

M. Takeoka, K. P. Seshadreesan, C. L. You, S. Izumi, and J. P. Dowling, “Fundamental precision limit of a Mach-Zehnder interferometric sensor when one of the inputs is the vacuum,” arXiv:1705.09506, (2017).

Silberberg, Y.

Y. Israel, S. Rosen, and Y. Silberberg, “Supersensitive polarization microscopy using NOON states of light,” Phys. Rev. Lett. 112(10), 103604 (2014).
[Crossref] [PubMed]

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science,  328(5980), 879–881 (2010).
[Crossref] [PubMed]

Silberhorn, C.

J. S. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. L. Pregnel, C. Silberhorn, T. C. Ralph, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Tomography of quantum detectors,” Nat. Phys. 5(1), 27–30 (2009).
[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]

D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, and I. A. Walmsley, “Fiber-assisted detection with photon number resolution,” Opt. Lett. 28(23), 2387–2389 (2003).
[Crossref] [PubMed]

Sliwa, C.

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]

D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, and I. A. Walmsley, “Fiber-assisted detection with photon number resolution,” Opt. Lett. 28(23), 2387–2389 (2003).
[Crossref] [PubMed]

Smelyanskiy, V. N.

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]

Spiller, T. P.

J. Joo, W. J. Munro, and T. P. Spiller, “Quantum metrology with entangled coherent states,” Phys. Rev. Lett. 107(8) 083601 (2011).
[Crossref] [PubMed]

Stegun, I.

M. Abramowitz and I. Stegun, Handbook of Mathematical functions (Dover, 1965).

Sun, C. P.

P. Liu, P. Wang, W. Yang, G. R. Jin, and C. P. Sun, “Fisher information of a squeezed-state interferometer with a finite photon-number resolution,” Phys. Rev. A 95(2), 023824 (2017).
[Crossref]

Szameit, A.

M. Lebugle, M. Grafe, R. Heilmann, A. Perez-Leija, S. Nolte, and A. Szameit, “Experimental observation of N00N state Bloch oscillations,” Nat. Commun. 6(1), 8273 (2015).
[Crossref] [PubMed]

Takeoka, M.

M. Takeoka, K. P. Seshadreesan, C. L. You, S. Izumi, and J. P. Dowling, “Fundamental precision limit of a Mach-Zehnder interferometric sensor when one of the inputs is the vacuum,” arXiv:1705.09506, (2017).

Tan, Q. S.

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]

Vollmer, C. E.

C. E. Vollmer, C. Baune, A. Samblowski, T. Eberle, V. Handchen, J. Fiurasek, and R. Schnabel, “Quantum up-conversion of squeezed vacuum states from 1550 to 532 nm,” Phys. Rev. Lett. 112(7), 073602 (2014).
[Crossref] [PubMed]

Walmsley, I. A.

J. S. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. L. Pregnel, C. Silberhorn, T. C. Ralph, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Tomography of quantum detectors,” Nat. Phys. 5(1), 27–30 (2009).
[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]

D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, and I. A. Walmsley, “Fiber-assisted detection with photon number resolution,” Opt. Lett. 28(23), 2387–2389 (2003).
[Crossref] [PubMed]

Wang, P.

P. Liu, P. Wang, W. Yang, G. R. Jin, and C. P. Sun, “Fisher information of a squeezed-state interferometer with a finite photon-number resolution,” Phys. Rev. A 95(2), 023824 (2017).
[Crossref]

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).
[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]

Weinfurter, H.

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777 (2012).
[Crossref]

Wineland, D. J.

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

Wiseman, H. M.

G. Y. Xiang, B. L. Higgins, D. W. Berry, H. M. Wiseman, and G. J. Pryde, “Entanglement-enhanced measurement of a completely unknown optical phase,” Nat. Photonics 5(1), 43–47 (2011).
[Crossref]

Xiang, G. Y.

G. Y. Xiang, B. L. Higgins, D. W. Berry, H. M. Wiseman, and G. J. Pryde, “Entanglement-enhanced measurement of a completely unknown optical phase,” Nat. Photonics 5(1), 43–47 (2011).
[Crossref]

Yang, W.

P. Liu, P. Wang, W. Yang, G. R. Jin, and C. P. Sun, “Fisher information of a squeezed-state interferometer with a finite photon-number resolution,” Phys. Rev. A 95(2), 023824 (2017).
[Crossref]

You, C. L.

M. Takeoka, K. P. Seshadreesan, C. L. You, S. Izumi, and J. P. Dowling, “Fundamental precision limit of a Mach-Zehnder interferometric sensor when one of the inputs is the vacuum,” arXiv:1705.09506, (2017).

Yurke, B.

B. Yurke, S. L. McCall, and J. R. Klauder, “SU (2) and SU (1, 1) interferometers,” Phys. Rev. A 33(6) 4033 (1986).
[Crossref]

Zeilinger, A.

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777 (2012).
[Crossref]

Zhang, W.

W. Zhang, Q. Qi, J. Zhou, and L. Chen, “Mimicking Faraday rotation to sort the orbital angular momentum of light,” Phys. Rev. Lett. 112(15), 153601 (2014).
[Crossref] [PubMed]

Zhou, J.

W. Zhang, Q. Qi, J. Zhou, and L. Chen, “Mimicking Faraday rotation to sort the orbital angular momentum of light,” Phys. Rev. Lett. 112(15), 153601 (2014).
[Crossref] [PubMed]

Zukowski, M.

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777 (2012).
[Crossref]

Contemp. Phys. (1)

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

J. Mod. Opt. (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]

Nat. Commun. (1)

M. Lebugle, M. Grafe, R. Heilmann, A. Perez-Leija, S. Nolte, and A. Szameit, “Experimental observation of N00N state Bloch oscillations,” Nat. Commun. 6(1), 8273 (2015).
[Crossref] [PubMed]

Nat. Photonics (3)

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5(4), 222–229 (2011).
[Crossref]

G. Y. Xiang, B. L. Higgins, D. W. Berry, H. M. Wiseman, and G. J. Pryde, “Entanglement-enhanced measurement of a completely unknown optical phase,” Nat. Photonics 5(1), 43–47 (2011).
[Crossref]

R. H. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photonics 3, 696–705 (2009).
[Crossref]

Nat. Phys. (1)

J. S. Lundeen, A. Feito, H. Coldenstrodt-Ronge, K. L. Pregnel, C. Silberhorn, T. C. Ralph, J. Eisert, M. B. Plenio, and I. A. Walmsley, “Tomography of quantum detectors,” Nat. Phys. 5(1), 27–30 (2009).
[Crossref]

Nature (1)

C. F. Roos, M. Chwalla, K. Kim, M. Riebe, and R. Blatt, “‘Designer atoms’ for quantum metrology,” Nature 443(7109), 316–319 (2006).
[Crossref] [PubMed]

Opt. Lett. (1)

Phys. Rev. A (9)

P. Liu, P. Wang, W. Yang, G. R. Jin, and C. P. Sun, “Fisher information of a squeezed-state interferometer with a finite photon-number resolution,” Phys. Rev. A 95(2), 023824 (2017).
[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]

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]

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]

B. Yurke, S. L. McCall, and J. R. Klauder, “SU (2) and SU (1, 1) interferometers,” Phys. Rev. A 33(6) 4033 (1986).
[Crossref]

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

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

C. C. Gerry and R. A. Campos, “Generation of maximally entangled photonic states with a quantum-optical Fredkin gate,” Phys. Rev. A 64(6), 063814 (2001).
[Crossref]

R. A. Campos, C. C. Gerry, and A. Benmoussa, “Optical interferometry at the Heisenberg limit with twin Fock states and parity measurements,” Phys. Rev. A 68(2), 023810 (2003).
[Crossref]

Phys. Rev. D (1)

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23(8), 1693 (1981).
[Crossref]

Phys. Rev. Lett. (8)

J. Joo, W. J. Munro, and T. P. Spiller, “Quantum metrology with entangled coherent states,” Phys. Rev. Lett. 107(8) 083601 (2011).
[Crossref] [PubMed]

M. D. Lang and C. M. Caves, “Optimal quantum-enhanced interferometry using a laser power source,” Phys. Rev. Lett. 111(17), 173601 (2013).
[Crossref] [PubMed]

C. E. Vollmer, C. Baune, A. Samblowski, T. Eberle, V. Handchen, J. Fiurasek, and R. Schnabel, “Quantum up-conversion of squeezed vacuum states from 1550 to 532 nm,” Phys. Rev. Lett. 112(7), 073602 (2014).
[Crossref] [PubMed]

P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum metrology with two-mode squeezed vacuum: parity detection beats the Heisenberg limit,” Phys. Rev. Lett. 104(10), 103602 (2010).
[Crossref] [PubMed]

W. Zhang, Q. Qi, J. Zhou, and L. Chen, “Mimicking Faraday rotation to sort the orbital angular momentum of light,” Phys. Rev. Lett. 112(15), 153601 (2014).
[Crossref] [PubMed]

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] [PubMed]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96(1), 010401 (2006).
[Crossref] [PubMed]

Y. Israel, S. Rosen, and Y. Silberberg, “Supersensitive polarization microscopy using NOON states of light,” Phys. Rev. Lett. 112(10), 103604 (2014).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84(2), 777 (2012).
[Crossref]

Science (1)

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science,  328(5980), 879–881 (2010).
[Crossref] [PubMed]

Other (2)

M. Takeoka, K. P. Seshadreesan, C. L. You, S. Izumi, and J. P. Dowling, “Fundamental precision limit of a Mach-Zehnder interferometric sensor when one of the inputs is the vacuum,” arXiv:1705.09506, (2017).

M. Abramowitz and I. Stegun, Handbook of Mathematical functions (Dover, 1965).

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Figures (7)

Fig. 1
Fig. 1 The schematic of metrology scheme using TMSV state. (NPBS=non-polarizing beam splitter, QWP=quarter wave plate, PS=phase shifter, RM=rotation medium, PNRD=photon number resolving detector)
Fig. 2
Fig. 2 The surface of normalized signal with phase φ and rotation angle θ as functions when the average photon number N = 5.
Fig. 3
Fig. 3 Normalized Fisher information curve with distinct average photon number and diverse photon-number resolving. N is average photon number in input state, Nup is the upper limit of photon-number resolving, FQ is ideal quantum Fisher information, Fobt is quantum Fisher information which can be obtained when the upper limit of photon-number resolving is Nup.
Fig. 4
Fig. 4 (a)Sensitivity curve with T = 0.8, 0.9, 1 and N = 5, SNL represents the shot noise limit. (b)The minimum transmission efficiency allowed under the premise of realizing super-sensitivity from N = 1 to N = 20.
Fig. 5
Fig. 5 (a)The change curves of normalized signal with detection efficiencies from 0.9 to 1. (b) The change curves of sensitivity with detection efficiencies from 0.9 to 1. Where DE is the abbreviation of detection efficiency in Figs. 5(a) and 5(b).
Fig. 6
Fig. 6 (a)The normalized signal visibility varies with transmission efficiency and detection efficiency. (b) The sensitivity varies with transmission efficiency and detection efficiency. Where the variation range of transmission efficiency and that of detection efficiency are 0.9 to 1 in Figs. 6(a) and 6(b).
Fig. 7
Fig. 7 The effect of response time delay and dark counts on system sensitivity. Where HL=Heisenberg limit, SNL=shot noise limit, curve of r = 0 is ideal curve.

Equations (11)

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| ψ = k = 0 n ( 1 ) n k e i ( 2 n 2 k ) ( φ + θ ) [ ( 2 k k ) ( 2 n 2 k n k ) ( 1 2 ) 2 n ] 1 / 2 | 2 n 2 k , 2 k R ,
ˆ = ψ o u t | ˆ | ψ o u t = ψ | U ˆ B S ˆ U ˆ B S | ψ = 1 1 + N ( N + 2 ) cos 2 ( φ + θ ) .
Δ φ = 1 + N ( N + 2 ) cos 2 ( φ + θ ) N ( N + 2 ) sin φ Δ θ = 1 + N ( N + 2 ) cos 2 ( φ + θ ) N ( N + 2 ) sin θ
U ˆ M Z I = exp ( i J ˆ y δ ) .
W i n ( α , β ) = 4 π 2 exp ( 2 | α cosh r e i κ β * sinh r | 2 2 | e i κ α * sinh r + β cosh r | 2 ) .
α ˜ = 1 T ( α sin δ 2 + β cos δ 2 ) β ˜ = 1 T ( α sin δ 2 + β cos δ 2 )
ˆ = π 2 0 W o u t ( 0 , β ) d 2 β = T 1 + N ( N + 2 ) cos 2 δ .
Δ δ = 1 + N ( N + 2 ) cos 2 δ 1 + N ( N + 2 ) cos 2 δ T 2 T N ( N + 2 ) cos δ sin δ .
μ ˆ n = m = 0 λ n , m | m m |
P e = P e [ D A ( e ) D B ( e ) ] + P o [ D A ( o ) D B ( o ) ] = 1 4 ( 1 + e 4 r ) + 1 2 ˆ e 2 r .
ˆ = P e P o = ˆ e 2 r .

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