Abstract

In this paper, we study the phase sensitivity of an SU(1,1) interferometer with coherent and displaced-squeezed-vacuum (DSV) states as inputs, and parity and on-off as detection strategies. Our scheme with parity is sub-shotnoise limited and approaches the Heisenberg limit with increasing squeezing strength of the optical parametric amplifier (OPA). Also, for the on-off detection scheme, we show that sub-shotnoise sensitivity is possible by increasing the squeezing strength of the OPA.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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  1. B. C. Barish and R. Weiss, “LIGO and the detection of gravitational waves,” Phys. Today 52(10), 44 (1999).
    [Crossref]
  2. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23(8), 1693–1708 (1981).
    [Crossref]
  3. J. P. Dowling, “Quantum optical metrology—the lowdown on high-N00N states,” Contemp. Phys. 49(2), 125–143 (2008).
    [Crossref]
  4. H. Lee, P. Kok, and J. P. Dowling, “A quantum Rosetta stone for interferometry,” J. Mod. Opt. 49(14–15), 2325–2338 (2002).
    [Crossref]
  5. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306(5700), 1330–1336 (2004).
    [Crossref] [PubMed]
  6. K. R. Motes, J. P. Olson, E. J. Rabeaux, J. P. Dowling, S. J. Olson, and P. P. Rohde, “Linear optical quantum metrology with single photons: exploiting spontaneously generated entanglement to beat the shot-noise limit,” Phys. Lett. Rev. 114(17), 170802 (2015).
    [Crossref]
  7. 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]
  8. R. Demkowicz-Dobrzański, M. Jarzyna, and J. Kołodyński, “Quantum limits in optical interferometry,” Prog. Opt. 60, 345 (2015).
    [Crossref]
  9. B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1, 1) interferometers,” Phys. Rev. A 33(6), 4033 (1986).
    [Crossref]
  10. W. N. Plick, J. P. Dowling, and G. S. Agarwal, “Coherent-light-boosted, sub-shot noise, quantum interferometry,” New J. Phys. 12(8), 083014 (2010).
    [Crossref]
  11. Z. Y. Ou, “Enhancement of the phase-measurement sensitivity beyond the standard quantum limit by a nonlinear interferometer,” Phys. Rev. A 85(2), 023815 (2012).
    [Crossref]
  12. D. Li, C. H. Yuan, Z. Y. Ou, and W. P. Zhang, “The phase sensitivity of an SU(1, 1) interferometer with coherent and squeezed-vacuum light,” New J. Phys. 16(7), 073020 (2014).
    [Crossref]
  13. D. Li, B. T. Gard, Y. Gao, C. H. Yuan, W. P. Zhang, H. Lee, and J. P. Dowling, “Phase sensitivity at the Heisenberg limit in an SU (1, 1) interferometer via parity detection,” Phys. Rev. A 94(6), 063840 (2016).
    [Crossref]
  14. X.Y. Hu, C.P. Wei, Y.F. Yu, and Z.M. Zhang, “Enhanced phase sensitivity of an SU(1,1) interferometer with displaced squeezed vacuum light,” Front. Phys. 11(3), 114203 (2016).
    [Crossref]
  15. A. Monras, “Optimal phase measurements with pure Gaussian states,” Phys. Rev. A 73(3), 033821 (2006).
    [Crossref]
  16. O. Pinel, P. Jian, N. Treps, C. Fabre, and D. Braun, “Quantum parameter estimation using general single-mode Gaussian states,” Phys. Rev. A 88(4), 040102 (2013).
    [Crossref]
  17. D. Li, C. H. Yuan, Y. Yao, W. Jiang, M. Li, and W. Zhang,“Effects of loss on the phase sensitivity with parity detection in an SU(1,1) interferometer,” J. Opt. B 35(5), 1080 (2018).
    [Crossref]
  18. A. M. Marino, N. V. Corzo Trejo, and P. D. Lett, “Effect of losses on the performance of an SU(1,1) interferometer,” Phys. Rev. A 86(2), 023844 (2012).
    [Crossref]
  19. X. L. Hu, D. Li, L. Q. Chen, K. Zhang, W. P. Zhang, and C. H. Yuan, “Phase estimation for an SU(1,1) interferometer in the presence of phase diffusion and photon losses,” Phys. Rev. A 98(2), 023803 (2018).
    [Crossref]
  20. X. Ma, C. You, S. Adhikari, E. S. Matekole, and J. P. Dowling, “Sub-shot-noise-limited phase estimation via SU(1,1) interferometer with thermal states,” Opt. Express 26(14), 18492–18504 (2018).
    [Crossref] [PubMed]
  21. S. S. Szigeti, R. J. Lewis-Swan, and S. A. Haine, “Pumped-up SU(1, 1) interferometry,” Phys. Rev. Lett. 118(15), 150401 (2017).
    [Crossref] [PubMed]
  22. B. E. Anderson, P. Gupta, B. L. Schmittberger, T. Horrom, C. Hermann-Avigliano, K. M. Jones, and P. D. Lett, “Phase sensing beyond the standard quantum limit with a variation on the SU(1,1) interferometer,” Optica 4(7), 752–756 (2017).
    [Crossref]
  23. C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84(2), 621–669 (2012).
    [Crossref]
  24. 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]
  25. C. C. Gerry, “Heisenberg-limit interferometry with four-wave mixers operating in a nonlinear regime,” Phys. Rev. A 61 (4), 043811 (2000).
    [Crossref]
  26. K. P. Seshadreesan, P. M. Anisimov, H. Lee, and J. P. Dowling, “Parity detection achieves the Heisenberg limit in interferometry with coherent mixed with squeezed vacuum light,” New J. Phys. 13(8), 083026 (2011).
    [Crossref]
  27. M. Takeoka, R. B. Jin, and M. Sasaki, “Full analysis of multi-photon pair effects in spontaneous parametric down conversion based photonic quantum information processing,” New J. Phys. 17(4), 043030 (2015).
    [Crossref]
  28. S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, 1993).
  29. C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, 1976).
  30. E. L. Lehmann and G. Casella, Theory of Point Estimation (Springer, 1998).
  31. 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]

2018 (3)

D. Li, C. H. Yuan, Y. Yao, W. Jiang, M. Li, and W. Zhang,“Effects of loss on the phase sensitivity with parity detection in an SU(1,1) interferometer,” J. Opt. B 35(5), 1080 (2018).
[Crossref]

X. L. Hu, D. Li, L. Q. Chen, K. Zhang, W. P. Zhang, and C. H. Yuan, “Phase estimation for an SU(1,1) interferometer in the presence of phase diffusion and photon losses,” Phys. Rev. A 98(2), 023803 (2018).
[Crossref]

X. Ma, C. You, S. Adhikari, E. S. Matekole, and J. P. Dowling, “Sub-shot-noise-limited phase estimation via SU(1,1) interferometer with thermal states,” Opt. Express 26(14), 18492–18504 (2018).
[Crossref] [PubMed]

2017 (2)

2016 (2)

D. Li, B. T. Gard, Y. Gao, C. H. Yuan, W. P. Zhang, H. Lee, and J. P. Dowling, “Phase sensitivity at the Heisenberg limit in an SU (1, 1) interferometer via parity detection,” Phys. Rev. A 94(6), 063840 (2016).
[Crossref]

X.Y. Hu, C.P. Wei, Y.F. Yu, and Z.M. Zhang, “Enhanced phase sensitivity of an SU(1,1) interferometer with displaced squeezed vacuum light,” Front. Phys. 11(3), 114203 (2016).
[Crossref]

2015 (3)

K. R. Motes, J. P. Olson, E. J. Rabeaux, J. P. Dowling, S. J. Olson, and P. P. Rohde, “Linear optical quantum metrology with single photons: exploiting spontaneously generated entanglement to beat the shot-noise limit,” Phys. Lett. Rev. 114(17), 170802 (2015).
[Crossref]

R. Demkowicz-Dobrzański, M. Jarzyna, and J. Kołodyński, “Quantum limits in optical interferometry,” Prog. Opt. 60, 345 (2015).
[Crossref]

M. Takeoka, R. B. Jin, and M. Sasaki, “Full analysis of multi-photon pair effects in spontaneous parametric down conversion based photonic quantum information processing,” New J. Phys. 17(4), 043030 (2015).
[Crossref]

2014 (1)

D. Li, C. H. Yuan, Z. Y. Ou, and W. P. Zhang, “The phase sensitivity of an SU(1, 1) interferometer with coherent and squeezed-vacuum light,” New J. Phys. 16(7), 073020 (2014).
[Crossref]

2013 (1)

O. Pinel, P. Jian, N. Treps, C. Fabre, and D. Braun, “Quantum parameter estimation using general single-mode Gaussian states,” Phys. Rev. A 88(4), 040102 (2013).
[Crossref]

2012 (3)

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84(2), 621–669 (2012).
[Crossref]

Z. Y. Ou, “Enhancement of the phase-measurement sensitivity beyond the standard quantum limit by a nonlinear interferometer,” Phys. Rev. A 85(2), 023815 (2012).
[Crossref]

A. M. Marino, N. V. Corzo Trejo, and P. D. Lett, “Effect of losses on the performance of an SU(1,1) interferometer,” Phys. Rev. A 86(2), 023844 (2012).
[Crossref]

2011 (1)

K. P. Seshadreesan, P. M. Anisimov, H. Lee, and J. P. Dowling, “Parity detection achieves the Heisenberg limit in interferometry with coherent mixed with squeezed vacuum light,” New J. Phys. 13(8), 083026 (2011).
[Crossref]

2010 (3)

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]

W. N. Plick, J. P. Dowling, and G. S. Agarwal, “Coherent-light-boosted, sub-shot noise, quantum interferometry,” New J. Phys. 12(8), 083014 (2010).
[Crossref]

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]

2008 (1)

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

2006 (1)

A. Monras, “Optimal phase measurements with pure Gaussian states,” Phys. Rev. A 73(3), 033821 (2006).
[Crossref]

2004 (1)

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306(5700), 1330–1336 (2004).
[Crossref] [PubMed]

2002 (1)

H. Lee, P. Kok, and J. P. Dowling, “A quantum Rosetta stone for interferometry,” J. Mod. Opt. 49(14–15), 2325–2338 (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)

B. C. Barish and R. Weiss, “LIGO and the detection of gravitational waves,” Phys. Today 52(10), 44 (1999).
[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), 1693–1708 (1981).
[Crossref]

Adhikari, S.

Agarwal, G. S.

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]

W. N. Plick, J. P. Dowling, and G. S. Agarwal, “Coherent-light-boosted, sub-shot noise, quantum interferometry,” New J. Phys. 12(8), 083014 (2010).
[Crossref]

Anderson, B. E.

Anisimov, P. M.

K. P. Seshadreesan, P. M. Anisimov, H. Lee, and J. P. Dowling, “Parity detection achieves the Heisenberg limit in interferometry with coherent mixed with squeezed vacuum light,” New J. Phys. 13(8), 083026 (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]

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]

Barish, B. C.

B. C. Barish and R. Weiss, “LIGO and the detection of gravitational waves,” Phys. Today 52(10), 44 (1999).
[Crossref]

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]

Braun, D.

O. Pinel, P. Jian, N. Treps, C. Fabre, and D. Braun, “Quantum parameter estimation using general single-mode Gaussian states,” Phys. Rev. A 88(4), 040102 (2013).
[Crossref]

Casella, G.

E. L. Lehmann and G. Casella, Theory of Point Estimation (Springer, 1998).

Caves, C. M.

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23(8), 1693–1708 (1981).
[Crossref]

Cerf, N. J.

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84(2), 621–669 (2012).
[Crossref]

Chen, L. Q.

X. L. Hu, D. Li, L. Q. Chen, K. Zhang, W. P. Zhang, and C. H. Yuan, “Phase estimation for an SU(1,1) interferometer in the presence of phase diffusion and photon losses,” Phys. Rev. A 98(2), 023803 (2018).
[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]

Corzo Trejo, N. V.

A. M. Marino, N. V. Corzo Trejo, and P. D. Lett, “Effect of losses on the performance of an SU(1,1) interferometer,” Phys. Rev. A 86(2), 023844 (2012).
[Crossref]

Demkowicz-Dobrzanski, R.

R. Demkowicz-Dobrzański, M. Jarzyna, and J. Kołodyński, “Quantum limits in optical interferometry,” Prog. Opt. 60, 345 (2015).
[Crossref]

Dowling, J. P.

X. Ma, C. You, S. Adhikari, E. S. Matekole, and J. P. Dowling, “Sub-shot-noise-limited phase estimation via SU(1,1) interferometer with thermal states,” Opt. Express 26(14), 18492–18504 (2018).
[Crossref] [PubMed]

D. Li, B. T. Gard, Y. Gao, C. H. Yuan, W. P. Zhang, H. Lee, and J. P. Dowling, “Phase sensitivity at the Heisenberg limit in an SU (1, 1) interferometer via parity detection,” Phys. Rev. A 94(6), 063840 (2016).
[Crossref]

K. R. Motes, J. P. Olson, E. J. Rabeaux, J. P. Dowling, S. J. Olson, and P. P. Rohde, “Linear optical quantum metrology with single photons: exploiting spontaneously generated entanglement to beat the shot-noise limit,” Phys. Lett. Rev. 114(17), 170802 (2015).
[Crossref]

K. P. Seshadreesan, P. M. Anisimov, H. Lee, and J. P. Dowling, “Parity detection achieves the Heisenberg limit in interferometry with coherent mixed with squeezed vacuum light,” New J. Phys. 13(8), 083026 (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]

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. N. Plick, J. P. Dowling, and G. S. Agarwal, “Coherent-light-boosted, sub-shot noise, quantum interferometry,” New J. Phys. 12(8), 083014 (2010).
[Crossref]

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

H. Lee, P. Kok, and J. P. Dowling, “A quantum Rosetta stone for interferometry,” J. Mod. Opt. 49(14–15), 2325–2338 (2002).
[Crossref]

Fabre, C.

O. Pinel, P. Jian, N. Treps, C. Fabre, and D. Braun, “Quantum parameter estimation using general single-mode Gaussian states,” Phys. Rev. A 88(4), 040102 (2013).
[Crossref]

Gao, Y.

D. Li, B. T. Gard, Y. Gao, C. H. Yuan, W. P. Zhang, H. Lee, and J. P. Dowling, “Phase sensitivity at the Heisenberg limit in an SU (1, 1) interferometer via parity detection,” Phys. Rev. A 94(6), 063840 (2016).
[Crossref]

García-Patrón, R.

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84(2), 621–669 (2012).
[Crossref]

Gard, B. T.

D. Li, B. T. Gard, Y. Gao, C. H. Yuan, W. P. Zhang, H. Lee, and J. P. Dowling, “Phase sensitivity at the Heisenberg limit in an SU (1, 1) interferometer via parity detection,” Phys. Rev. A 94(6), 063840 (2016).
[Crossref]

Gerry, C. C.

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, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306(5700), 1330–1336 (2004).
[Crossref] [PubMed]

Gupta, P.

Haine, S. A.

S. S. Szigeti, R. J. Lewis-Swan, and S. A. Haine, “Pumped-up SU(1, 1) interferometry,” Phys. Rev. Lett. 118(15), 150401 (2017).
[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]

Helstrom, C. W.

C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, 1976).

Hermann-Avigliano, C.

Horrom, T.

Hu, X. L.

X. L. Hu, D. Li, L. Q. Chen, K. Zhang, W. P. Zhang, and C. H. Yuan, “Phase estimation for an SU(1,1) interferometer in the presence of phase diffusion and photon losses,” Phys. Rev. A 98(2), 023803 (2018).
[Crossref]

Hu, X.Y.

X.Y. Hu, C.P. Wei, Y.F. Yu, and Z.M. Zhang, “Enhanced phase sensitivity of an SU(1,1) interferometer with displaced squeezed vacuum light,” Front. Phys. 11(3), 114203 (2016).
[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]

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]

Jarzyna, M.

R. Demkowicz-Dobrzański, M. Jarzyna, and J. Kołodyński, “Quantum limits in optical interferometry,” Prog. Opt. 60, 345 (2015).
[Crossref]

Jian, P.

O. Pinel, P. Jian, N. Treps, C. Fabre, and D. Braun, “Quantum parameter estimation using general single-mode Gaussian states,” Phys. Rev. A 88(4), 040102 (2013).
[Crossref]

Jiang, W.

D. Li, C. H. Yuan, Y. Yao, W. Jiang, M. Li, and W. Zhang,“Effects of loss on the phase sensitivity with parity detection in an SU(1,1) interferometer,” J. Opt. B 35(5), 1080 (2018).
[Crossref]

Jin, R. B.

M. Takeoka, R. B. Jin, and M. Sasaki, “Full analysis of multi-photon pair effects in spontaneous parametric down conversion based photonic quantum information processing,” New J. Phys. 17(4), 043030 (2015).
[Crossref]

Jones, K. M.

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, 1993).

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]

Kok, P.

H. Lee, P. Kok, and J. P. Dowling, “A quantum Rosetta stone for interferometry,” J. Mod. Opt. 49(14–15), 2325–2338 (2002).
[Crossref]

Kolodynski, J.

R. Demkowicz-Dobrzański, M. Jarzyna, and J. Kołodyński, “Quantum limits in optical interferometry,” Prog. Opt. 60, 345 (2015).
[Crossref]

Lee, H.

D. Li, B. T. Gard, Y. Gao, C. H. Yuan, W. P. Zhang, H. Lee, and J. P. Dowling, “Phase sensitivity at the Heisenberg limit in an SU (1, 1) interferometer via parity detection,” Phys. Rev. A 94(6), 063840 (2016).
[Crossref]

K. P. Seshadreesan, P. M. Anisimov, H. Lee, and J. P. Dowling, “Parity detection achieves the Heisenberg limit in interferometry with coherent mixed with squeezed vacuum light,” New J. Phys. 13(8), 083026 (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]

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]

H. Lee, P. Kok, and J. P. Dowling, “A quantum Rosetta stone for interferometry,” J. Mod. Opt. 49(14–15), 2325–2338 (2002).
[Crossref]

Lehmann, E. L.

E. L. Lehmann and G. Casella, Theory of Point Estimation (Springer, 1998).

Lett, P. D.

Lewis-Swan, R. J.

S. S. Szigeti, R. J. Lewis-Swan, and S. A. Haine, “Pumped-up SU(1, 1) interferometry,” Phys. Rev. Lett. 118(15), 150401 (2017).
[Crossref] [PubMed]

Li, D.

X. L. Hu, D. Li, L. Q. Chen, K. Zhang, W. P. Zhang, and C. H. Yuan, “Phase estimation for an SU(1,1) interferometer in the presence of phase diffusion and photon losses,” Phys. Rev. A 98(2), 023803 (2018).
[Crossref]

D. Li, C. H. Yuan, Y. Yao, W. Jiang, M. Li, and W. Zhang,“Effects of loss on the phase sensitivity with parity detection in an SU(1,1) interferometer,” J. Opt. B 35(5), 1080 (2018).
[Crossref]

D. Li, B. T. Gard, Y. Gao, C. H. Yuan, W. P. Zhang, H. Lee, and J. P. Dowling, “Phase sensitivity at the Heisenberg limit in an SU (1, 1) interferometer via parity detection,” Phys. Rev. A 94(6), 063840 (2016).
[Crossref]

D. Li, C. H. Yuan, Z. Y. Ou, and W. P. Zhang, “The phase sensitivity of an SU(1, 1) interferometer with coherent and squeezed-vacuum light,” New J. Phys. 16(7), 073020 (2014).
[Crossref]

Li, M.

D. Li, C. H. Yuan, Y. Yao, W. Jiang, M. Li, and W. Zhang,“Effects of loss on the phase sensitivity with parity detection in an SU(1,1) interferometer,” J. Opt. B 35(5), 1080 (2018).
[Crossref]

Lloyd, S.

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84(2), 621–669 (2012).
[Crossref]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306(5700), 1330–1336 (2004).
[Crossref] [PubMed]

Ma, X.

Maccone, L.

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306(5700), 1330–1336 (2004).
[Crossref] [PubMed]

Marino, A. M.

A. M. Marino, N. V. Corzo Trejo, and P. D. Lett, “Effect of losses on the performance of an SU(1,1) interferometer,” Phys. Rev. A 86(2), 023844 (2012).
[Crossref]

Matekole, E. S.

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]

Monras, A.

A. Monras, “Optimal phase measurements with pure Gaussian states,” Phys. Rev. A 73(3), 033821 (2006).
[Crossref]

Motes, K. R.

K. R. Motes, J. P. Olson, E. J. Rabeaux, J. P. Dowling, S. J. Olson, and P. P. Rohde, “Linear optical quantum metrology with single photons: exploiting spontaneously generated entanglement to beat the shot-noise limit,” Phys. Lett. Rev. 114(17), 170802 (2015).
[Crossref]

Olson, J. P.

K. R. Motes, J. P. Olson, E. J. Rabeaux, J. P. Dowling, S. J. Olson, and P. P. Rohde, “Linear optical quantum metrology with single photons: exploiting spontaneously generated entanglement to beat the shot-noise limit,” Phys. Lett. Rev. 114(17), 170802 (2015).
[Crossref]

Olson, S. J.

K. R. Motes, J. P. Olson, E. J. Rabeaux, J. P. Dowling, S. J. Olson, and P. P. Rohde, “Linear optical quantum metrology with single photons: exploiting spontaneously generated entanglement to beat the shot-noise limit,” Phys. Lett. Rev. 114(17), 170802 (2015).
[Crossref]

Ou, Z. Y.

D. Li, C. H. Yuan, Z. Y. Ou, and W. P. Zhang, “The phase sensitivity of an SU(1, 1) interferometer with coherent and squeezed-vacuum light,” New J. Phys. 16(7), 073020 (2014).
[Crossref]

Z. Y. Ou, “Enhancement of the phase-measurement sensitivity beyond the standard quantum limit by a nonlinear interferometer,” Phys. Rev. A 85(2), 023815 (2012).
[Crossref]

Pinel, O.

O. Pinel, P. Jian, N. Treps, C. Fabre, and D. Braun, “Quantum parameter estimation using general single-mode Gaussian states,” Phys. Rev. A 88(4), 040102 (2013).
[Crossref]

Pirandola, S.

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84(2), 621–669 (2012).
[Crossref]

Plick, W. N.

W. N. Plick, J. P. Dowling, and G. S. Agarwal, “Coherent-light-boosted, sub-shot noise, quantum interferometry,” New J. Phys. 12(8), 083014 (2010).
[Crossref]

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. 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]

Rabeaux, E. J.

K. R. Motes, J. P. Olson, E. J. Rabeaux, J. P. Dowling, S. J. Olson, and P. P. Rohde, “Linear optical quantum metrology with single photons: exploiting spontaneously generated entanglement to beat the shot-noise limit,” Phys. Lett. Rev. 114(17), 170802 (2015).
[Crossref]

Ralph, T. C.

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84(2), 621–669 (2012).
[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]

Rohde, P. P.

K. R. Motes, J. P. Olson, E. J. Rabeaux, J. P. Dowling, S. J. Olson, and P. P. Rohde, “Linear optical quantum metrology with single photons: exploiting spontaneously generated entanglement to beat the shot-noise limit,” Phys. Lett. Rev. 114(17), 170802 (2015).
[Crossref]

Sasaki, M.

M. Takeoka, R. B. Jin, and M. Sasaki, “Full analysis of multi-photon pair effects in spontaneous parametric down conversion based photonic quantum information processing,” New J. Phys. 17(4), 043030 (2015).
[Crossref]

Schmittberger, B. L.

Seshadreesan, K. P.

K. P. Seshadreesan, P. M. Anisimov, H. Lee, and J. P. Dowling, “Parity detection achieves the Heisenberg limit in interferometry with coherent mixed with squeezed vacuum light,” New J. Phys. 13(8), 083026 (2011).
[Crossref]

Shapiro, J. H.

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84(2), 621–669 (2012).
[Crossref]

Szigeti, S. S.

S. S. Szigeti, R. J. Lewis-Swan, and S. A. Haine, “Pumped-up SU(1, 1) interferometry,” Phys. Rev. Lett. 118(15), 150401 (2017).
[Crossref] [PubMed]

Takeoka, M.

M. Takeoka, R. B. Jin, and M. Sasaki, “Full analysis of multi-photon pair effects in spontaneous parametric down conversion based photonic quantum information processing,” New J. Phys. 17(4), 043030 (2015).
[Crossref]

Treps, N.

O. Pinel, P. Jian, N. Treps, C. Fabre, and D. Braun, “Quantum parameter estimation using general single-mode Gaussian states,” Phys. Rev. A 88(4), 040102 (2013).
[Crossref]

Weedbrook, C.

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84(2), 621–669 (2012).
[Crossref]

Wei, C.P.

X.Y. Hu, C.P. Wei, Y.F. Yu, and Z.M. Zhang, “Enhanced phase sensitivity of an SU(1,1) interferometer with displaced squeezed vacuum light,” Front. Phys. 11(3), 114203 (2016).
[Crossref]

Weiss, R.

B. C. Barish and R. Weiss, “LIGO and the detection of gravitational waves,” Phys. Today 52(10), 44 (1999).
[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]

Yao, Y.

D. Li, C. H. Yuan, Y. Yao, W. Jiang, M. Li, and W. Zhang,“Effects of loss on the phase sensitivity with parity detection in an SU(1,1) interferometer,” J. Opt. B 35(5), 1080 (2018).
[Crossref]

You, C.

Yu, Y.F.

X.Y. Hu, C.P. Wei, Y.F. Yu, and Z.M. Zhang, “Enhanced phase sensitivity of an SU(1,1) interferometer with displaced squeezed vacuum light,” Front. Phys. 11(3), 114203 (2016).
[Crossref]

Yuan, C. H.

D. Li, C. H. Yuan, Y. Yao, W. Jiang, M. Li, and W. Zhang,“Effects of loss on the phase sensitivity with parity detection in an SU(1,1) interferometer,” J. Opt. B 35(5), 1080 (2018).
[Crossref]

X. L. Hu, D. Li, L. Q. Chen, K. Zhang, W. P. Zhang, and C. H. Yuan, “Phase estimation for an SU(1,1) interferometer in the presence of phase diffusion and photon losses,” Phys. Rev. A 98(2), 023803 (2018).
[Crossref]

D. Li, B. T. Gard, Y. Gao, C. H. Yuan, W. P. Zhang, H. Lee, and J. P. Dowling, “Phase sensitivity at the Heisenberg limit in an SU (1, 1) interferometer via parity detection,” Phys. Rev. A 94(6), 063840 (2016).
[Crossref]

D. Li, C. H. Yuan, Z. Y. Ou, and W. P. Zhang, “The phase sensitivity of an SU(1, 1) interferometer with coherent and squeezed-vacuum light,” New J. Phys. 16(7), 073020 (2014).
[Crossref]

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]

Zhang, K.

X. L. Hu, D. Li, L. Q. Chen, K. Zhang, W. P. Zhang, and C. H. Yuan, “Phase estimation for an SU(1,1) interferometer in the presence of phase diffusion and photon losses,” Phys. Rev. A 98(2), 023803 (2018).
[Crossref]

Zhang, W.

D. Li, C. H. Yuan, Y. Yao, W. Jiang, M. Li, and W. Zhang,“Effects of loss on the phase sensitivity with parity detection in an SU(1,1) interferometer,” J. Opt. B 35(5), 1080 (2018).
[Crossref]

Zhang, W. P.

X. L. Hu, D. Li, L. Q. Chen, K. Zhang, W. P. Zhang, and C. H. Yuan, “Phase estimation for an SU(1,1) interferometer in the presence of phase diffusion and photon losses,” Phys. Rev. A 98(2), 023803 (2018).
[Crossref]

D. Li, B. T. Gard, Y. Gao, C. H. Yuan, W. P. Zhang, H. Lee, and J. P. Dowling, “Phase sensitivity at the Heisenberg limit in an SU (1, 1) interferometer via parity detection,” Phys. Rev. A 94(6), 063840 (2016).
[Crossref]

D. Li, C. H. Yuan, Z. Y. Ou, and W. P. Zhang, “The phase sensitivity of an SU(1, 1) interferometer with coherent and squeezed-vacuum light,” New J. Phys. 16(7), 073020 (2014).
[Crossref]

Zhang, Z.M.

X.Y. Hu, C.P. Wei, Y.F. Yu, and Z.M. Zhang, “Enhanced phase sensitivity of an SU(1,1) interferometer with displaced squeezed vacuum light,” Front. Phys. 11(3), 114203 (2016).
[Crossref]

Contemp. Phys. (1)

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

Front. Phys. (1)

X.Y. Hu, C.P. Wei, Y.F. Yu, and Z.M. Zhang, “Enhanced phase sensitivity of an SU(1,1) interferometer with displaced squeezed vacuum light,” Front. Phys. 11(3), 114203 (2016).
[Crossref]

J. Mod. Opt. (1)

H. Lee, P. Kok, and J. P. Dowling, “A quantum Rosetta stone for interferometry,” J. Mod. Opt. 49(14–15), 2325–2338 (2002).
[Crossref]

J. Opt. B (1)

D. Li, C. H. Yuan, Y. Yao, W. Jiang, M. Li, and W. Zhang,“Effects of loss on the phase sensitivity with parity detection in an SU(1,1) interferometer,” J. Opt. B 35(5), 1080 (2018).
[Crossref]

New J. Phys. (5)

D. Li, C. H. Yuan, Z. Y. Ou, and W. P. Zhang, “The phase sensitivity of an SU(1, 1) interferometer with coherent and squeezed-vacuum light,” New J. Phys. 16(7), 073020 (2014).
[Crossref]

W. N. Plick, J. P. Dowling, and G. S. Agarwal, “Coherent-light-boosted, sub-shot noise, quantum interferometry,” New J. Phys. 12(8), 083014 (2010).
[Crossref]

K. P. Seshadreesan, P. M. Anisimov, H. Lee, and J. P. Dowling, “Parity detection achieves the Heisenberg limit in interferometry with coherent mixed with squeezed vacuum light,” New J. Phys. 13(8), 083026 (2011).
[Crossref]

M. Takeoka, R. B. Jin, and M. Sasaki, “Full analysis of multi-photon pair effects in spontaneous parametric down conversion based photonic quantum information processing,” New J. Phys. 17(4), 043030 (2015).
[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]

Opt. Express (1)

Optica (1)

Phys. Lett. Rev. (1)

K. R. Motes, J. P. Olson, E. J. Rabeaux, J. P. Dowling, S. J. Olson, and P. P. Rohde, “Linear optical quantum metrology with single photons: exploiting spontaneously generated entanglement to beat the shot-noise limit,” Phys. Lett. Rev. 114(17), 170802 (2015).
[Crossref]

Phys. Rev. A (9)

Z. Y. Ou, “Enhancement of the phase-measurement sensitivity beyond the standard quantum limit by a nonlinear interferometer,” Phys. Rev. A 85(2), 023815 (2012).
[Crossref]

D. Li, B. T. Gard, Y. Gao, C. H. Yuan, W. P. Zhang, H. Lee, and J. P. Dowling, “Phase sensitivity at the Heisenberg limit in an SU (1, 1) interferometer via parity detection,” Phys. Rev. A 94(6), 063840 (2016).
[Crossref]

A. Monras, “Optimal phase measurements with pure Gaussian states,” Phys. Rev. A 73(3), 033821 (2006).
[Crossref]

O. Pinel, P. Jian, N. Treps, C. Fabre, and D. Braun, “Quantum parameter estimation using general single-mode Gaussian states,” Phys. Rev. A 88(4), 040102 (2013).
[Crossref]

A. M. Marino, N. V. Corzo Trejo, and P. D. Lett, “Effect of losses on the performance of an SU(1,1) interferometer,” Phys. Rev. A 86(2), 023844 (2012).
[Crossref]

X. L. Hu, D. Li, L. Q. Chen, K. Zhang, W. P. Zhang, and C. H. Yuan, “Phase estimation for an SU(1,1) interferometer in the presence of phase diffusion and photon losses,” Phys. Rev. A 98(2), 023803 (2018).
[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]

Phys. Rev. D (1)

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23(8), 1693–1708 (1981).
[Crossref]

Phys. Rev. Lett. (2)

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]

S. S. Szigeti, R. J. Lewis-Swan, and S. A. Haine, “Pumped-up SU(1, 1) interferometry,” Phys. Rev. Lett. 118(15), 150401 (2017).
[Crossref] [PubMed]

Phys. Today (1)

B. C. Barish and R. Weiss, “LIGO and the detection of gravitational waves,” Phys. Today 52(10), 44 (1999).
[Crossref]

Prog. Opt. (1)

R. Demkowicz-Dobrzański, M. Jarzyna, and J. Kołodyński, “Quantum limits in optical interferometry,” Prog. Opt. 60, 345 (2015).
[Crossref]

Rev. Mod. Phys. (1)

C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, “Gaussian quantum information,” Rev. Mod. Phys. 84(2), 621–669 (2012).
[Crossref]

Science (1)

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306(5700), 1330–1336 (2004).
[Crossref] [PubMed]

Other (3)

S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, 1993).

C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, 1976).

E. L. Lehmann and G. Casella, Theory of Point Estimation (Springer, 1998).

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Figures (4)

Fig. 1
Fig. 1 A schematic of a SU(1,1) interferometer. Two OPAs with the same squeezing parameter g is used. The pump field between the two OPAs has a π phase difference. Parity measurement is performed in mode b, and the on-off detection is done in both modes a and b.
Fig. 2
Fig. 2 The effect on the phase sensitivity with the increase in the squeezing parameter r. The HL (blue) is given by Eq. 18. Plotted with 1 = 16, 2 = 4, g = 2.
Fig. 3
Fig. 3 Phase sensitivity as a function of the squeezing strength g of the OPA. The sensitivity with coherent and DSV (pink) is obtained by numerically optimizing ϕ. The HL (blue) and SNL (red) are given by Eqs. (18) and (19). Plotted with 1 = 16, 2 = 4 and r = 2.
Fig. 4
Fig. 4 Phase sensitivity with coherent and DSV state with on-off detector. The sensitivity (green) is obtained by numerically optimizing ϕ from the classical CRB. The SNL (red), parity (pink) and HL (blue) are also shown for comparison. Plotted with 1 = 16, 2 = 4 and r = 2.

Equations (61)

Equations on this page are rendered with MathJax. Learn more.

x ^ a k = a ^ k + a ^ k , p ^ a k = i ( a ^ k a ^ k ) ,
x ^ b k = b ^ k + b ^ k , p ^ a k = i ( b ^ k b ^ k ) .
X k = ( X ^ k , 1 , X ^ k , 2 , X ^ k , 3 , X ^ k , 4 ) T = ( x ^ a k , p ^ a k , x ^ b k , p ^ b k ) T .
X ¯ k = ( X ¯ k , 1 , X ¯ k , 2 , X ¯ k , 3 , X ¯ k , 4 ) T ,
Γ k m n = Tr [ ( Δ X ^ k , m Δ X ^ k , n + Δ X ^ k , n Δ X ^ k , m ) ρ ] ,
W ( X 0 ) = exp [ ( X 0 X ¯ 0 ) T ( Γ 0 ) 1 ( X 0 X ¯ 0 ) ] | Γ 0 | .
S OPA 1 = ( cosh ( g ) 0 sinh ( g ) 0 0 cosh ( g ) 0 sinh ( g ) sinh ( g ) 0 cosh ( g ) 0 0 sinh ( g ) 0 cosh ( g ) ) ,
S ϕ = ( cos ( ϕ ) sin ( ϕ ) 0 0 sin ( ϕ ) cos ( ϕ ) 0 0 0 0 1 0 0 0 0 1 ) ,
S OPA 2 = ( cosh ( g ) 0 sinh ( g ) 0 0 cosh ( g ) 0 sinh ( g ) sinh ( g ) 0 cosh ( g ) 0 0 sinh ( g ) 0 cosh ( g ) ) ,
X ¯ 2 = S X ¯ 0 ,
Γ 2 = S Γ 0 S T .
Π ^ b = ( 1 ) b ^ 2 b ^ 2 .
Δ ϕ = Δ Π ^ b | Π ^ b ϕ |
Π ^ off = | 0 0 | , Π ^ on = I ^ | 0 0 |
P on = 1 2 det ( Γ + I )
Δ ϕ 1 / F ,
F = 1 P on ( d P on d ϕ ) 2 .
Δ ϕ HL = 1 n ¯ total = 1 n ¯ 1 + n ¯ 2 + n ¯ ξ + n ¯ opa ( 1 + n ¯ 1 + n ¯ 2 + n ¯ ξ ) + 2 n ¯ 1 n ¯ 2 n ¯ opa ( n ¯ opa + 2 )
Δ ϕ SNL = 1 n ¯ total = 1 n ¯ 1 + n ¯ 2 + n ¯ ξ + n ¯ opa ( 1 + n ¯ 1 + n ¯ 2 + n ¯ ξ ) + 2 n ¯ 1 n ¯ 2 n ¯ opa ( n ¯ opa + 2 )
Π ^ b = exp ( X ¯ 22 T Γ 22 1 X ¯ 22 ) | Γ 22 | ,
X coh = ( 2 α 1 cos ( θ 1 ) 2 α 1 sin ( θ 1 ) ) ,
Γ coh = ( 1 0 0 1 ) .
X DSV = ( 2 α 2 cos ( θ 2 ) 2 α 2 sin ( θ 2 ) ) ,
Γ DSV = ( ( cosh ( r ) + sinh ( r ) ) 2 0 0 ( cosh ( r ) sinh ( r ) ) 2 ) .
X 0 = X coh X DSV = ( 2 α 1 cos ( θ 1 ) 2 α 1 sin ( θ 1 ) 2 α 2 cos ( θ 2 ) 2 α 2 sin ( θ 2 ) ) ,
Γ 0 = ( 1 0 0 0 0 1 0 0 0 0 ( cosh ( r ) + sinh ( r ) ) 2 0 0 0 0 ( cosh ( r ) sinh ( r ) ) 2 ) .
X 2 = ( 2 α 1 ( cos 2 ( g ) cos ( θ 1 + ϕ ) sin 2 ( g ) cos ( θ 1 ) ) 2 α 2 sinh ( 2 g ) sin ( ϕ 2 ) sin ( ϕ 2 θ 2 ) 2 α 1 ( cosh 2 ( g ) sin ( θ 1 + ϕ ) sin 2 ( g ) sin ( θ 1 ) ) + 2 α 2 sinh ( 2 g ) sin ( ϕ 2 ) cos ( ϕ 2 θ 2 ) 2 α 1 sinh ( 2 g ) sin ( ϕ 2 ) sin ( θ 1 + ϕ 2 ) + 2 α 2 ( cosh 2 ( g ) cos ( θ 2 ) sinh 2 ( g ) cos ( ϕ θ 2 ) ) 2 α 1 sinh ( 2 g ) sin ( ϕ 2 ) cos ( θ 1 + ϕ 2 ) + 2 α 2 ( sinh 2 ( g ) sin ( ϕ θ 2 ) + cosh 2 ( g ) sin ( θ 2 ) ) ) .
Γ 2 = ( Γ 2 11 Γ 2 12 Γ 2 13 Γ 2 14 Γ 2 21 Γ 2 22 Γ 2 23 Γ 2 24 Γ 2 31 Γ 2 32 Γ 2 33 Γ 2 34 Γ 2 41 Γ 2 42 Γ 2 43 Γ 2 44 ) .
Γ 2 11 = e 2 r sin 2 ( 2 g ) sin 4 ( ϕ 2 ) + sinh 4 ( g ) + cosh 4 ( g ) + sinh 2 ( g ) cosh 2 ( g ) ( e 2 r sin 2 ( ϕ ) 2 cos ( ϕ ) ) ,
Γ 2 12 = 4 sinh 2 ( g ) cosh 2 ( g ) sinh ( r ) cosh ( r ) sin ( ϕ ) ( cos ( ϕ ) 1 ) ,
Γ 2 12 = 1 4 e 2 r ( e 2 r 1 ) ( e 2 r + 1 ) sinh g cosh ( g ) ( 2 sinh 2 ( g ) cos ( 2 ϕ + 1 ) ) 1 4 e 2 r ( e 2 r + 1 ) sinh ( g ) cosh ( g ) cosh ( 2 g ) ( 4 e 2 r cos ( ϕ ) + 3 e 2 r + 1 ) ,
Γ 2 14 = sinh g cosh ( g ) sin ( ϕ ) ( sinh 2 ( r ) + cosh 2 ( r ) + 1 ) sinh ( g ) cosh ( g ) sinh ( 2 r ) sin ( ϕ ) ( cosh ( 2 g ) 2 sinh 2 ( g ) cos ( ϕ ) ) ,
Γ 2 12 = Γ 2 12 = 4 sinh 2 ( g ) cosh 2 ( g ) sinh ( r ) cosh ( r ) sin ( ϕ ) ( cos ( ϕ ) 1 ) ,
Γ 2 22 = e 2 r sinh 2 ( 2 g ) sin 4 ( ϕ 2 ) sinh 2 ( g ) cosh 2 ( g ) ( 2 cos ( ϕ ) e 2 r sin 2 ( ϕ ) ) + sinh 4 ( g ) + cosh 4 ( g ) ,
Γ 2 23 = sinh ( g ) cosh ( g ) sin ( ϕ ) sinh ( 2 r ) ( cosh ( 2 g ) 2 sinh 2 ( g ) cos ( ϕ ) ) + sinh ( g ) cosh ( g ) sin ( ϕ ) ( sinh 2 ( r ) + cosh 2 ( r ) + 1 ) ,
Γ 2 24 = 1 2 e 2 r ( e 2 r + 1 ) sinh ( g ) sinh 2 ( g ) cosh ( g ) ( 2 e 2 r sin 2 ( ϕ ) + cos ( 2 ϕ ) ) + 1 4 e 2 r ( e 2 r + 1 ) sinh ( g ) cosh ( g ) cosh ( 2 g ) ( 3 4 cos ( ϕ ) ) + 1 ) ,
Γ 2 31 = Γ 2 13 = 1 4 e 2 r ( e 2 r 1 ) ( e 2 r + 1 ) sinh ( g ) cosh ( g ) ( 2 sinh 2 ( g ) cos ( 2 ϕ + 1 ) ) 1 4 e 2 r ( e 2 r + 1 ) sinh ( g ) cosh ( g ) cosh ( 2 g ) ( 4 e 2 r cos ( ϕ ) + 3 e 2 r + 1 ) ,
Γ 2 32 = Γ 2 23 = sinh ( g ) cosh ( g ) sin ( ϕ ) sinh ( 2 r ) ( cosh ( 2 g ) 2 sinh 2 ( g ) cos ( ϕ ) ) + sinh ( g ) cosh ( g ) sin ( ϕ ) ( sinh 2 ( r ) + cosh 2 ( r ) + 1 ) ,
Γ 2 33 = e 2 r sinh 4 ( g ) sin 2 ( ϕ ) + e 2 r ( cosh 2 ( g ) sinh 2 ( g ) cos ( ϕ ) ) 2 + sinh 2 ( 2 g ) sin 2 ( ϕ 2 ) ,
Γ 2 34 = 4 sinh 2 ( g ) sinh ( r ) cosh ( r ) sin ( ϕ ) ( cosh 2 ( g ) sinh 2 ( g ) cos ( ϕ ) ) ,
Γ 2 41 = sinh ( g ) cosh ( g ) sin ( ϕ ) ( sinh 2 ( r ) + cosh 2 ( r ) + 1 ) sinh ( g ) cosh ( g ) sinh ( 2 r ) sin ( ϕ ) ( cosh ( 2 g ) 2 sinh 2 ( g ) cos ( ϕ ) ) ,
Γ 2 42 = Γ 2 24 = 1 2 e 2 r ( e 2 r + 1 ) sinh ( g ) sinh 2 ( g ) cosh ( g ) ( 2 e 2 r sin 2 ( ϕ ) + cos ( 2 ϕ ) ) + 1 4 e 2 r ( e 2 r + 1 ) sinh ( g ) cosh ( g ) cosh ( 2 g ) ( 3 4 cos ( ϕ ) ) + 1 ) ,
Γ 2 43 = Γ 2 34 = 4 sinh 2 ( g ) sinh ( r ) cosh ( r ) sin ( ϕ ) ( cosh 2 ( g ) sinh 2 ( g ) cos ( ϕ ) ) ,
Γ 2 44 = e 2 r sinh 4 ( g ) sin 2 ( ϕ ) + e 2 r ( cosh 2 ( g ) sinh 2 ( g ) cos ( ϕ ) ) 2 + sinh 2 ( 2 g ) sin 2 ( ϕ 2 ) .
Π ^ = 8 v 6 exp [ 128 ( v 1 × v 2 v 3 × v 4 ) v 5 ]
v 1 = α 1 sinh ( 2 g ) sin ( ϕ 2 ) cos ( θ 1 + ϕ 2 ) + α 2 ( sinh 2 ( g ) sin ( ϕ θ 2 ) + cosh 2 ( g ) sin ( θ 2 ) )
v 2 = α 1 sinh ( 2 g ) sin ( ϕ 2 ) sin ( θ 1 + ϕ 2 ) + α 2 ( cosh 2 ( g ) cos ( θ 2 ) sinh 2 ( g ) cos ( ϕ θ 2 ) ) × 4 sinh 2 ( g ) sinh ( r ) cosh ( r ) sin ( ϕ ) ( cosh 2 ( g ) sinh 2 ( g ) cos ( ϕ ) ) ( α 1 sinh ( 2 g ) sin ( ϕ 2 ) cos ( θ 1 + ϕ 2 ) + α 2 ( sinh 2 ( g ) sin ( ϕ θ 2 ) + cosh 2 ( g ) sin ( θ 2 ) ) ) × ( e 2 r sinh 4 ( g ) sin 2 ( ϕ ) + e 2 r ( cosh 2 ( g ) sinh 2 ( g ) cos ( ϕ ) ) 2 + sinh 2 ( 2 g ) sin 2 ( ϕ 2 ) )
v 3 = α 1 sinh ( 2 g ) sin ( ϕ 2 ) sin ( θ 1 + ϕ 2 ) + α 2 ( cosh 2 ( g ) cos ( θ 2 ) sinh 2 ( g ) cos ( ϕ θ 2 ) )
v 4 = ( e 2 r sin 4 ( g ) sin 2 ( ϕ ) + e 2 r ( cosh 2 ( g ) sinh 2 ( g ) cos ( ϕ ) ) 2 + sinh 2 ( 2 g ) sin 2 ( ϕ 2 ) ) × ( α 1 sinh ( 2 g ) sin ( ϕ 2 ) sin ( θ 1 + ϕ 2 ) + α 2 ( cosh 2 ( g ) cos ( θ 2 ) sinh 2 ( g ) cos ( ϕ θ 2 ) ) ) ( 4 sinh 2 ( g ) sinh ( r ) cosh ( r ) sin ( ϕ ) ) ( cosh 2 ( g ) sinh 2 ( g ) cos ( ϕ ) ) × ( α 1 sinh ( 2 g ) sin ( ϕ 2 ) cos ( θ 1 + ϕ 2 ) + α 2 ( sinh 2 ( g ) sin ( ϕ θ 2 ) + cosh 2 ( g ) sin ( θ 2 ) ) )
v 5 = 32 cosh 2 ( r ) ( sinh 4 ( 2 g ) cos ( 2 ϕ ) sinh 2 ( 4 g ) cos ( ϕ ) ) + 4 cosh ( 4 g 2 r ) + 3 cosh ( 8 g 2 r ) + 8 cosh ( 4 g ) + 6 cosh ( 8 g ) + 50 + 4 cosh ( 2 ( 2 g + r ) ) + 3 cosh ( 2 ( 4 g + r ) ) 14 cosh ( 2 r )
v 6 = 8 sinh 4 ( 2 g ) cos ( 2 ϕ ) 8 sinh 2 ( 4 g ) cos ( ϕ ) + 4 cosh ( 4 g ) + 3 cosh ( 8 g ) × 4 cosh ( r ) 14 cosh ( 2 r ) + 50
P on a = 1 16 q 1 + q 2
q 1 = 8 cosh ( 4 g ) ( 18 cosh 2 ( r ) + 8 cos ( ϕ ) + cos ( 2 ϕ ) ) 78 cosh ( 2 r ) + 72 cos ( ϕ ) + 6 cos ( 2 ϕ ) + 242 + 2 cosh ( 8 g ) ( 6 cosh 2 ( r ) 4 cos ( ϕ ) + cos ( 2 ϕ ) ) ,
q 2 = 16 sinh 2 ( 2 g ) ( cosh ( 2 r ) ( sinh 2 ( 2 g ) cos ( 2 ϕ ) 2 ( cosh ( 4 g ) + 5 ) cos ( ϕ ) ) )
16 sinh 2 ( 2 g ) ( 16 sinh ( 2 r ) sin 2 ( ϕ 2 ) sin ( Γ ϕ ) ) .
P on b = 1 16 t 1 + t 2 + t 3 + t 4 + t 5 .
t 1 = 8 cosh ( 4 g ) ( 18 cosh 2 ( r ) + 8 cos ( ϕ ) + cos ( 2 ϕ ) ) 32 cos ( ϕ ) cosh ( 4 g 2 r ) 4 cos ( ϕ ) cosh ( 8 g 2 r ) 4 cos ( 2 ϕ ) cosh ( 4 g 2 r ) + cos ( 2 ϕ ) cosh ( 8 g 2 r ) + 72 cos ( ϕ ) + 6 cos ( 2 ϕ ) 14 ,
t 2 = 2 cosh ( 8 g ) ( 6 cosh 2 ( r ) 4 cos ( ϕ ) + cos ( 2 ϕ ) ) 32 cos ( ϕ ) cosh ( 4 g + 2 r ) 4 cos ( 2 ϕ ) cosh ( 4 g + 2 r ) 4 cos ( ϕ ) cosh ( 8 g + 2 r ) + cos ( 2 ϕ ) cosh ( 8 g + 2 r ) + 72 cosh ( 2 r ) cos ( ϕ ) + 6 cosh ( 2 r ) cos ( 2 ϕ ) + 178 cosh ( 2 r ) ,
t 3 = 64 sin ( Γ ) sinh ( 2 g 2 r ) + 64 cos ( Γ ) sin ( 2 ϕ ) sinh ( 2 g 2 r ) + 16 sin ( Γ ) sinh ( 4 g 2 r ) 32 sin ( Γ ) cos ( ϕ ) sinh ( 4 g 2 r ) 64 sin ( Γ ) cos ( 2 ϕ ) sinh ( 2 g 2 r ) ,
t 4 = 16 sin ( Γ ) cos ( 2 ϕ ) sinh ( 4 g 2 r ) + 32 cos ( Γ ) sin ( ϕ ) sinh ( 4 g 2 r ) 96 sin ( Γ ) sinh ( 2 r ) + 64 cos ( Γ ) sinh ( 2 r ) 64 sin ( Γ ) sinh ( 2 r ) cos ( 2 ϕ ) 96 sin ( Γ ) sinh ( 2 r ) cos ( 2 ϕ ) 16 cos ( Γ ) sin ( 2 ϕ ) sinh ( 4 g 2 r ) ,
t 5 = 64 sin 2 ( ϕ 2 ) sin ( Γ ϕ ) sinh ( 4 g + 2 r ) 128 sin ( ϕ ) cos ( Γ ϕ ) sinh ( 2 ( g + r ) ) + 96 cos ( Γ ) sinh ( 2 r ) sin ( 2 ϕ ) ,

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