Abstract

We theoretically study the phase sensitivity of an SU(1,1) interferometer with a thermal state and a squeezed vacuum state as inputs and parity detection as the measurement. We find that the phase sensitivity can beat the shot-noise limit and approaches the Heisenberg limit, with increasing input photon number, in an ideal situation. We also consider the effect of various noises, including photon loss, dark counts, and thermal photon noise. Our results show that the phase sensitivity is below the shot-noise limit with photon loss and dark counts, but cannot beat the shot-noise limit in the presence of thermal noise.

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

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References

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    [Crossref] [PubMed]
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    [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. A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85(13), 2733 (2000).
    [Crossref] [PubMed]
  6. B. M. Escher, R. L. de Matos Filho, and L. Davidovich, “General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,” Nat. Phys. 7, 406–411 (2011).
    [Crossref]
  7. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306(5700), 1330–1336 (2004).
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  8. V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photon. 5, 222–229 (2011).
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  9. R. Schnabel, N. Mavalvala, D. E. McClelland, and P. K. Lam, “Quantum metrology for gravitational wave astronomy,” Nat. Comm. 1, 121 (2010).
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  10. 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).
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    [Crossref]
  12. B. T. Gard, C. You, D. K. Mishra, R. Singh, H. Lee, T. R. Corbitt, and J. P. Dowling, “Nearly optimal measurement schemes in a noisy Mach-Zehnder interferometer with coherent and squeezed vacuum,” EPJ Quantum Technol. 4(1), 4 (2017).
  13. J. Kong, F. Hudelist, Z. Y. Ou, and W. P. Zhang, “Cancellation of Internal Quantum Noise of an Amplifier by Quantum Correlation,” Phys. Rev. Lett. 111(3), 033608 (2013).
    [Crossref] [PubMed]
  14. B. Chen, C. Qiu, S. Y. Chen, J. X. Guo, L. Q. Chen, Z. Y. Ou, and W. P. Zhang, “Atom-Light Hybrid Interferometer,” Phys. Rev. Lett. 115(4), 043602 (2015).
    [Crossref] [PubMed]
  15. F. Hudelist, J. Kong, C. J. Liu, J. T. Jing, Z.Y. Ou, and W. P. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Comm. 5, 3049 (2014).
    [Crossref]
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  17. 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–R4652 (1996).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  21. 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, 073020 (2014).
    [Crossref]
  22. 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]
  23. 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, 113025 (2010).
    [Crossref]
  24. S. S. Szigeti, R. J. Lewis-Swan, and S. A. Haine, “Pumped-up SU (1, 1) interferometry,” Phys. Rev. Lett. 118(15), 150401 (2017).
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    [Crossref]
  26. M. Manceau, G. Leuchs, F. Khalili, and M. Chekhova, “Detection Loss Tolerant Supersensitive Phase Measurement with an SU(1,1) Interferometer,” Phys. Rev. Lett. 119(22), 223604 (2017).
    [Crossref] [PubMed]
  27. W. Du, J. Jia, J. F. Chen, Z. Y. Ou, and W. P. Zhang, “Absolute sensitivity of phase measurement in an SU(1,1) type interferometer,” Opt. Lett. 43(5), 1051–1054 (2018).
    [Crossref] [PubMed]
  28. E. T. Jaynes, “Information Theory and Statistical Mechanics,” Phys. Rev. 106, 620 (1957).
    [Crossref]
  29. N. Y. Halpern, P. Faist, J. Oppenheim, and A. Winter, “Microcanonical and resource-theoretic derivations of the thermal state of a quantum system with noncommuting charges,” Nat. Comm. 7, 12051 (2016).
    [Crossref]
  30. 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]
  31. H. Shibata, K. Shimizu, H. Takesue, and Y. Tokura, “Ultimate low system dark-count rate for superconducting nanowire single-photon detector,” Opt. Lett. 40(14), 3428–3431 (2015).
    [Crossref] [PubMed]
  32. Z. X. 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]
  33. S. L. Braunstein and P. V. Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77(2), 513–577 (2005).
    [Crossref]
  34. 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]

2018 (1)

2017 (6)

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]

M. Manceau, G. Leuchs, F. Khalili, and M. Chekhova, “Detection Loss Tolerant Supersensitive Phase Measurement with an SU(1,1) Interferometer,” Phys. Rev. Lett. 119(22), 223604 (2017).
[Crossref] [PubMed]

Z. X. 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]

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]

C. You, S. Adhikari, Y. Chi, M. LaBorde, C. Matyas, C. Zhang, Z. Su, T. Byrnes, C. Lu, J. P. Dowling, and J. P. Olson,, “Multiparameter estimation with single photons-linearly-optically generated quantum entanglement beats the shot noise limit,” J. Opt. 19(12), 124002 (2017).
[Crossref]

B. T. Gard, C. You, D. K. Mishra, R. Singh, H. Lee, T. R. Corbitt, and J. P. Dowling, “Nearly optimal measurement schemes in a noisy Mach-Zehnder interferometer with coherent and squeezed vacuum,” EPJ Quantum Technol. 4(1), 4 (2017).

2016 (2)

N. Y. Halpern, P. Faist, J. Oppenheim, and A. Winter, “Microcanonical and resource-theoretic derivations of the thermal state of a quantum system with noncommuting charges,” Nat. Comm. 7, 12051 (2016).
[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]

2015 (2)

H. Shibata, K. Shimizu, H. Takesue, and Y. Tokura, “Ultimate low system dark-count rate for superconducting nanowire single-photon detector,” Opt. Lett. 40(14), 3428–3431 (2015).
[Crossref] [PubMed]

B. Chen, C. Qiu, S. Y. Chen, J. X. Guo, L. Q. Chen, Z. Y. Ou, and W. P. Zhang, “Atom-Light Hybrid Interferometer,” Phys. Rev. Lett. 115(4), 043602 (2015).
[Crossref] [PubMed]

2014 (2)

F. Hudelist, J. Kong, C. J. Liu, J. T. Jing, Z.Y. Ou, and W. P. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Comm. 5, 3049 (2014).
[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, 073020 (2014).
[Crossref]

2013 (1)

J. Kong, F. Hudelist, Z. Y. Ou, and W. P. Zhang, “Cancellation of Internal Quantum Noise of an Amplifier by Quantum Correlation,” Phys. Rev. Lett. 111(3), 033608 (2013).
[Crossref] [PubMed]

2012 (2)

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]

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]

2011 (2)

B. M. Escher, R. L. de Matos Filho, and L. Davidovich, “General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,” Nat. Phys. 7, 406–411 (2011).
[Crossref]

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

2010 (4)

R. Schnabel, N. Mavalvala, D. E. McClelland, and P. K. Lam, “Quantum metrology for gravitational wave astronomy,” Nat. Comm. 1, 121 (2010).
[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, 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, 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]

2005 (1)

S. L. Braunstein and P. V. Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77(2), 513–577 (2005).
[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)

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85(13), 2733 (2000).
[Crossref] [PubMed]

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–R4652 (1996).
[Crossref] [PubMed]

1987 (1)

M. Xiao, L. A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59(3), 278 (1987).
[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]

1957 (1)

E. T. Jaynes, “Information Theory and Statistical Mechanics,” Phys. Rev. 106, 620 (1957).
[Crossref]

1887 (1)

A. A. Michelson and E. W. Morley, “On the Relative Motion of the Earth and of the Luminiferous Ether,” Sidereal Messenger 6, 306–310 (1887).

Abrams, D. S.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85(13), 2733 (2000).
[Crossref] [PubMed]

Adhikari, S.

C. You, S. Adhikari, Y. Chi, M. LaBorde, C. Matyas, C. Zhang, Z. Su, T. Byrnes, C. Lu, J. P. Dowling, and J. P. Olson,, “Multiparameter estimation with single photons-linearly-optically generated quantum entanglement beats the shot noise limit,” J. Opt. 19(12), 124002 (2017).
[Crossref]

Agarwal, G. S.

W. N. Plick, J. P. Dowling, and G. S. Agarwal, “Coherent-light-boosted, sub-shot noise, quantum interferometry,” New J. Phys. 12, 083014 (2010).
[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, 113025 (2010).
[Crossref]

Anderson, B. E.

Anisimov, P. M.

Z. X. 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, 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]

Berry, D. W.

Z. X. 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]

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–R4652 (1996).
[Crossref] [PubMed]

Boto, A. N.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85(13), 2733 (2000).
[Crossref] [PubMed]

Braunstein, S. L.

S. L. Braunstein and P. V. Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77(2), 513–577 (2005).
[Crossref]

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85(13), 2733 (2000).
[Crossref] [PubMed]

Byrnes, T.

C. You, S. Adhikari, Y. Chi, M. LaBorde, C. Matyas, C. Zhang, Z. Su, T. Byrnes, C. Lu, J. P. Dowling, and J. P. Olson,, “Multiparameter estimation with single photons-linearly-optically generated quantum entanglement beats the shot noise limit,” J. Opt. 19(12), 124002 (2017).
[Crossref]

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]

Chekhova, M.

M. Manceau, G. Leuchs, F. Khalili, and M. Chekhova, “Detection Loss Tolerant Supersensitive Phase Measurement with an SU(1,1) Interferometer,” Phys. Rev. Lett. 119(22), 223604 (2017).
[Crossref] [PubMed]

Chen, B.

B. Chen, C. Qiu, S. Y. Chen, J. X. Guo, L. Q. Chen, Z. Y. Ou, and W. P. Zhang, “Atom-Light Hybrid Interferometer,” Phys. Rev. Lett. 115(4), 043602 (2015).
[Crossref] [PubMed]

Chen, J. F.

Chen, L. Q.

B. Chen, C. Qiu, S. Y. Chen, J. X. Guo, L. Q. Chen, Z. Y. Ou, and W. P. Zhang, “Atom-Light Hybrid Interferometer,” Phys. Rev. Lett. 115(4), 043602 (2015).
[Crossref] [PubMed]

Chen, S. Y.

B. Chen, C. Qiu, S. Y. Chen, J. X. Guo, L. Q. Chen, Z. Y. Ou, and W. P. Zhang, “Atom-Light Hybrid Interferometer,” Phys. Rev. Lett. 115(4), 043602 (2015).
[Crossref] [PubMed]

Chi, Y.

C. You, S. Adhikari, Y. Chi, M. LaBorde, C. Matyas, C. Zhang, Z. Su, T. Byrnes, C. Lu, J. P. Dowling, and J. P. Olson,, “Multiparameter estimation with single photons-linearly-optically generated quantum entanglement beats the shot noise limit,” J. Opt. 19(12), 124002 (2017).
[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]

Corbitt, T. R.

B. T. Gard, C. You, D. K. Mishra, R. Singh, H. Lee, T. R. Corbitt, and J. P. Dowling, “Nearly optimal measurement schemes in a noisy Mach-Zehnder interferometer with coherent and squeezed vacuum,” EPJ Quantum Technol. 4(1), 4 (2017).

Davidovich, L.

B. M. Escher, R. L. de Matos Filho, and L. Davidovich, “General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,” Nat. Phys. 7, 406–411 (2011).
[Crossref]

de Matos Filho, R. L.

B. M. Escher, R. L. de Matos Filho, and L. Davidovich, “General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,” Nat. Phys. 7, 406–411 (2011).
[Crossref]

Dowling, J. P.

B. T. Gard, C. You, D. K. Mishra, R. Singh, H. Lee, T. R. Corbitt, and J. P. Dowling, “Nearly optimal measurement schemes in a noisy Mach-Zehnder interferometer with coherent and squeezed vacuum,” EPJ Quantum Technol. 4(1), 4 (2017).

C. You, S. Adhikari, Y. Chi, M. LaBorde, C. Matyas, C. Zhang, Z. Su, T. Byrnes, C. Lu, J. P. Dowling, and J. P. Olson,, “Multiparameter estimation with single photons-linearly-optically generated quantum entanglement beats the shot noise limit,” J. Opt. 19(12), 124002 (2017).
[Crossref]

Z. X. 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]

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]

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, 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, 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]

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]

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85(13), 2733 (2000).
[Crossref] [PubMed]

Du, W.

Escher, B. M.

B. M. Escher, R. L. de Matos Filho, and L. Davidovich, “General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,” Nat. Phys. 7, 406–411 (2011).
[Crossref]

Faist, P.

N. Y. Halpern, P. Faist, J. Oppenheim, and A. Winter, “Microcanonical and resource-theoretic derivations of the thermal state of a quantum system with noncommuting charges,” Nat. Comm. 7, 12051 (2016).
[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).
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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).
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B. T. Gard, C. You, D. K. Mishra, R. Singh, H. Lee, T. R. Corbitt, and J. P. Dowling, “Nearly optimal measurement schemes in a noisy Mach-Zehnder interferometer with coherent and squeezed vacuum,” EPJ Quantum Technol. 4(1), 4 (2017).

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).
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V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photon. 5, 222–229 (2011).
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V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306(5700), 1330–1336 (2004).
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B. Chen, C. Qiu, S. Y. Chen, J. X. Guo, L. Q. Chen, Z. Y. Ou, and W. P. Zhang, “Atom-Light Hybrid Interferometer,” Phys. Rev. Lett. 115(4), 043602 (2015).
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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).
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N. Y. Halpern, P. Faist, J. Oppenheim, and A. Winter, “Microcanonical and resource-theoretic derivations of the thermal state of a quantum system with noncommuting charges,” Nat. Comm. 7, 12051 (2016).
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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–R4652 (1996).
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Horrom, T.

Huang, Z. X.

Z. X. 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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F. Hudelist, J. Kong, C. J. Liu, J. T. Jing, Z.Y. Ou, and W. P. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Comm. 5, 3049 (2014).
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J. Kong, F. Hudelist, Z. Y. Ou, and W. P. Zhang, “Cancellation of Internal Quantum Noise of an Amplifier by Quantum Correlation,” Phys. Rev. Lett. 111(3), 033608 (2013).
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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).
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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–R4652 (1996).
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E. T. Jaynes, “Information Theory and Statistical Mechanics,” Phys. Rev. 106, 620 (1957).
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Jing, J. T.

F. Hudelist, J. Kong, C. J. Liu, J. T. Jing, Z.Y. Ou, and W. P. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Comm. 5, 3049 (2014).
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Jones, K. M.

Khalili, F.

M. Manceau, G. Leuchs, F. Khalili, and M. Chekhova, “Detection Loss Tolerant Supersensitive Phase Measurement with an SU(1,1) Interferometer,” Phys. Rev. Lett. 119(22), 223604 (2017).
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M. Xiao, L. A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59(3), 278 (1987).
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B. Yurke, S. L. McCall, and J. R. Klauder, “SU (2) and SU (1, 1) interferometers,” Phys. Rev. A 33(6), 4033 (1986).
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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).
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A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85(13), 2733 (2000).
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F. Hudelist, J. Kong, C. J. Liu, J. T. Jing, Z.Y. Ou, and W. P. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Comm. 5, 3049 (2014).
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J. Kong, F. Hudelist, Z. Y. Ou, and W. P. Zhang, “Cancellation of Internal Quantum Noise of an Amplifier by Quantum Correlation,” Phys. Rev. Lett. 111(3), 033608 (2013).
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C. You, S. Adhikari, Y. Chi, M. LaBorde, C. Matyas, C. Zhang, Z. Su, T. Byrnes, C. Lu, J. P. Dowling, and J. P. Olson,, “Multiparameter estimation with single photons-linearly-optically generated quantum entanglement beats the shot noise limit,” J. Opt. 19(12), 124002 (2017).
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R. Schnabel, N. Mavalvala, D. E. McClelland, and P. K. Lam, “Quantum metrology for gravitational wave astronomy,” Nat. Comm. 1, 121 (2010).
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B. T. Gard, C. You, D. K. Mishra, R. Singh, H. Lee, T. R. Corbitt, and J. P. Dowling, “Nearly optimal measurement schemes in a noisy Mach-Zehnder interferometer with coherent and squeezed vacuum,” EPJ Quantum Technol. 4(1), 4 (2017).

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]

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, 113025 (2010).
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H. Lee, P. Kok, and J. P. Dowling, “A quantum Rosetta stone for interferometry,” J. Mod. Opt. 49(14–15), 2325–2338 (2002).
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Leuchs, G.

M. Manceau, G. Leuchs, F. Khalili, and M. Chekhova, “Detection Loss Tolerant Supersensitive Phase Measurement with an SU(1,1) Interferometer,” Phys. Rev. Lett. 119(22), 223604 (2017).
[Crossref] [PubMed]

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.

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, 073020 (2014).
[Crossref]

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F. Hudelist, J. Kong, C. J. Liu, J. T. Jing, Z.Y. Ou, and W. P. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Comm. 5, 3049 (2014).
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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, “Advances in quantum metrology,” Nat. Photon. 5, 222–229 (2011).
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V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306(5700), 1330–1336 (2004).
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C. You, S. Adhikari, Y. Chi, M. LaBorde, C. Matyas, C. Zhang, Z. Su, T. Byrnes, C. Lu, J. P. Dowling, and J. P. Olson,, “Multiparameter estimation with single photons-linearly-optically generated quantum entanglement beats the shot noise limit,” J. Opt. 19(12), 124002 (2017).
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Maccone, L.

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photon. 5, 222–229 (2011).
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V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306(5700), 1330–1336 (2004).
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Manceau, M.

M. Manceau, G. Leuchs, F. Khalili, and M. Chekhova, “Detection Loss Tolerant Supersensitive Phase Measurement with an SU(1,1) Interferometer,” Phys. Rev. Lett. 119(22), 223604 (2017).
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Matyas, C.

C. You, S. Adhikari, Y. Chi, M. LaBorde, C. Matyas, C. Zhang, Z. Su, T. Byrnes, C. Lu, J. P. Dowling, and J. P. Olson,, “Multiparameter estimation with single photons-linearly-optically generated quantum entanglement beats the shot noise limit,” J. Opt. 19(12), 124002 (2017).
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R. Schnabel, N. Mavalvala, D. E. McClelland, and P. K. Lam, “Quantum metrology for gravitational wave astronomy,” Nat. Comm. 1, 121 (2010).
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B. Yurke, S. L. McCall, and J. R. Klauder, “SU (2) and SU (1, 1) interferometers,” Phys. Rev. A 33(6), 4033 (1986).
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R. Schnabel, N. Mavalvala, D. E. McClelland, and P. K. Lam, “Quantum metrology for gravitational wave astronomy,” Nat. Comm. 1, 121 (2010).
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A. A. Michelson and E. W. Morley, “On the Relative Motion of the Earth and of the Luminiferous Ether,” Sidereal Messenger 6, 306–310 (1887).

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B. T. Gard, C. You, D. K. Mishra, R. Singh, H. Lee, T. R. Corbitt, and J. P. Dowling, “Nearly optimal measurement schemes in a noisy Mach-Zehnder interferometer with coherent and squeezed vacuum,” EPJ Quantum Technol. 4(1), 4 (2017).

Morley, E. W.

A. A. Michelson and E. W. Morley, “On the Relative Motion of the Earth and of the Luminiferous Ether,” Sidereal Messenger 6, 306–310 (1887).

Motes, K. R.

Z. X. 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]

Olson, J. P.

C. You, S. Adhikari, Y. Chi, M. LaBorde, C. Matyas, C. Zhang, Z. Su, T. Byrnes, C. Lu, J. P. Dowling, and J. P. Olson,, “Multiparameter estimation with single photons-linearly-optically generated quantum entanglement beats the shot noise limit,” J. Opt. 19(12), 124002 (2017).
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Oppenheim, J.

N. Y. Halpern, P. Faist, J. Oppenheim, and A. Winter, “Microcanonical and resource-theoretic derivations of the thermal state of a quantum system with noncommuting charges,” Nat. Comm. 7, 12051 (2016).
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Ou, Z. Y.

W. Du, J. Jia, J. F. Chen, Z. Y. Ou, and W. P. Zhang, “Absolute sensitivity of phase measurement in an SU(1,1) type interferometer,” Opt. Lett. 43(5), 1051–1054 (2018).
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B. Chen, C. Qiu, S. Y. Chen, J. X. Guo, L. Q. Chen, Z. Y. Ou, and W. P. Zhang, “Atom-Light Hybrid Interferometer,” Phys. Rev. Lett. 115(4), 043602 (2015).
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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, 073020 (2014).
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J. Kong, F. Hudelist, Z. Y. Ou, and W. P. Zhang, “Cancellation of Internal Quantum Noise of an Amplifier by Quantum Correlation,” Phys. Rev. Lett. 111(3), 033608 (2013).
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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).
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Ou, Z.Y.

F. Hudelist, J. Kong, C. J. Liu, J. T. Jing, Z.Y. Ou, and W. P. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Comm. 5, 3049 (2014).
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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).
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Plick, W. N.

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, 113025 (2010).
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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).
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W. N. Plick, J. P. Dowling, and G. S. Agarwal, “Coherent-light-boosted, sub-shot noise, quantum interferometry,” New J. Phys. 12, 083014 (2010).
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Qiu, C.

B. Chen, C. Qiu, S. Y. Chen, J. X. Guo, L. Q. Chen, Z. Y. Ou, and W. P. Zhang, “Atom-Light Hybrid Interferometer,” Phys. Rev. Lett. 115(4), 043602 (2015).
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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).
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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).
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Schmittberger, B. L.

Schnabel, R.

R. Schnabel, N. Mavalvala, D. E. McClelland, and P. K. Lam, “Quantum metrology for gravitational wave astronomy,” Nat. Comm. 1, 121 (2010).
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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).
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Shimizu, K.

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B. T. Gard, C. You, D. K. Mishra, R. Singh, H. Lee, T. R. Corbitt, and J. P. Dowling, “Nearly optimal measurement schemes in a noisy Mach-Zehnder interferometer with coherent and squeezed vacuum,” EPJ Quantum Technol. 4(1), 4 (2017).

Su, Z.

C. You, S. Adhikari, Y. Chi, M. LaBorde, C. Matyas, C. Zhang, Z. Su, T. Byrnes, C. Lu, J. P. Dowling, and J. P. Olson,, “Multiparameter estimation with single photons-linearly-optically generated quantum entanglement beats the shot noise limit,” J. Opt. 19(12), 124002 (2017).
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S. S. Szigeti, R. J. Lewis-Swan, and S. A. Haine, “Pumped-up SU (1, 1) interferometry,” Phys. Rev. Lett. 118(15), 150401 (2017).
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Tokura, Y.

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).
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A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85(13), 2733 (2000).
[Crossref] [PubMed]

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–R4652 (1996).
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Winter, A.

N. Y. Halpern, P. Faist, J. Oppenheim, and A. Winter, “Microcanonical and resource-theoretic derivations of the thermal state of a quantum system with noncommuting charges,” Nat. Comm. 7, 12051 (2016).
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M. Xiao, L. A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59(3), 278 (1987).
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M. Xiao, L. A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59(3), 278 (1987).
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C. You, S. Adhikari, Y. Chi, M. LaBorde, C. Matyas, C. Zhang, Z. Su, T. Byrnes, C. Lu, J. P. Dowling, and J. P. Olson,, “Multiparameter estimation with single photons-linearly-optically generated quantum entanglement beats the shot noise limit,” J. Opt. 19(12), 124002 (2017).
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B. T. Gard, C. You, D. K. Mishra, R. Singh, H. Lee, T. R. Corbitt, and J. P. Dowling, “Nearly optimal measurement schemes in a noisy Mach-Zehnder interferometer with coherent and squeezed vacuum,” EPJ Quantum Technol. 4(1), 4 (2017).

Yuan, C. 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]

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, 073020 (2014).
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B. Yurke, S. L. McCall, and J. R. Klauder, “SU (2) and SU (1, 1) interferometers,” Phys. Rev. A 33(6), 4033 (1986).
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C. You, S. Adhikari, Y. Chi, M. LaBorde, C. Matyas, C. Zhang, Z. Su, T. Byrnes, C. Lu, J. P. Dowling, and J. P. Olson,, “Multiparameter estimation with single photons-linearly-optically generated quantum entanglement beats the shot noise limit,” J. Opt. 19(12), 124002 (2017).
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W. Du, J. Jia, J. F. Chen, Z. Y. Ou, and W. P. Zhang, “Absolute sensitivity of phase measurement in an SU(1,1) type interferometer,” Opt. Lett. 43(5), 1051–1054 (2018).
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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]

B. Chen, C. Qiu, S. Y. Chen, J. X. Guo, L. Q. Chen, Z. Y. Ou, and W. P. Zhang, “Atom-Light Hybrid Interferometer,” Phys. Rev. Lett. 115(4), 043602 (2015).
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F. Hudelist, J. Kong, C. J. Liu, J. T. Jing, Z.Y. Ou, and W. P. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Comm. 5, 3049 (2014).
[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, 073020 (2014).
[Crossref]

J. Kong, F. Hudelist, Z. Y. Ou, and W. P. Zhang, “Cancellation of Internal Quantum Noise of an Amplifier by Quantum Correlation,” Phys. Rev. Lett. 111(3), 033608 (2013).
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B. T. Gard, C. You, D. K. Mishra, R. Singh, H. Lee, T. R. Corbitt, and J. P. Dowling, “Nearly optimal measurement schemes in a noisy Mach-Zehnder interferometer with coherent and squeezed vacuum,” EPJ Quantum Technol. 4(1), 4 (2017).

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).
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C. You, S. Adhikari, Y. Chi, M. LaBorde, C. Matyas, C. Zhang, Z. Su, T. Byrnes, C. Lu, J. P. Dowling, and J. P. Olson,, “Multiparameter estimation with single photons-linearly-optically generated quantum entanglement beats the shot noise limit,” J. Opt. 19(12), 124002 (2017).
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Nat. Comm. (3)

F. Hudelist, J. Kong, C. J. Liu, J. T. Jing, Z.Y. Ou, and W. P. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Comm. 5, 3049 (2014).
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R. Schnabel, N. Mavalvala, D. E. McClelland, and P. K. Lam, “Quantum metrology for gravitational wave astronomy,” Nat. Comm. 1, 121 (2010).
[Crossref]

N. Y. Halpern, P. Faist, J. Oppenheim, and A. Winter, “Microcanonical and resource-theoretic derivations of the thermal state of a quantum system with noncommuting charges,” Nat. Comm. 7, 12051 (2016).
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V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photon. 5, 222–229 (2011).
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W. N. Plick, J. P. Dowling, and G. S. Agarwal, “Coherent-light-boosted, sub-shot noise, quantum interferometry,” New J. Phys. 12, 083014 (2010).
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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, 073020 (2014).
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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, 113025 (2010).
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Opt. Lett. (2)

Optica (1)

Phys. Rev. (1)

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Figures (10)

Fig. 1
Fig. 1 Schematic of an ideal SU(1,1) interferometer. An SU(1,1) interferometer is similar to a MZI except the two beam splitters are replaced by two OPAs. |Ψth〉 and |Ψξ〉 are thermal state and squeezed vacuum state respectively. ai and bi(i = 0, 1, 2) denote two light beams in the different processes. ϕ1 and ϕ2 describe the phase shift in the two paths.
Fig. 2
Fig. 2 Phase sensitivity with parity detection as a function of g with constraint r = 0, nth = 0 on the input.
Fig. 3
Fig. 3 Phase sensitivity with parity detection as a function of nth with r = 0, g = 2. The phase sensitivity at nth = 0 is equal to the sensitivity at g = 2 in Fig. 2 and becomes greater with increasing nth.
Fig. 4
Fig. 4 Phase sensitivity with parity detection as a function of nth with constraints r = 2, g = 2. Comparing the values at nth = 20 in Fig. 3, both the phase sensitivity and HL are lower due to replacing vacuum state with squeezed vacuum state. Moreover, the phase sensitivity is much higher.
Fig. 5
Fig. 5 Phase sensitivity with parity detection as a function of nth2 under the condition g = 2 and nth = 20, where nth and nth2 are the mean photon number of the upper thermal state and the lower thermal state respectively.
Fig. 6
Fig. 6 Phase sensitivity with parity detection as a function of nα under the condition g = 2 and nth = 20, where nth and nα are the mean photon number of the upper thermal state and the lower coherent state respectively.
Fig. 7
Fig. 7 Phase sensitivity with parity detection and intensity detection as a function of nth with constraints r = 2, g = 2. Comparing the values, the phase sensitivity with intensity detection is much worse than that using parity detection.
Fig. 8
Fig. 8 Phase sensitivity with parity detection as a function of nth under the condition r = 2, g = 2 in the cases of photon loss L = 1% or photon loss L = 10%.
Fig. 9
Fig. 9 Phase sensitivity with parity detection as a function of nth under the condition r = 2, g = 2 and dark counts d = 0.01.
Fig. 10
Fig. 10 Phase sensitivity with parity detection as a function of nth under the condition r = 2, g = 2 and thermal photon noise is 10−20 or 1.

Equations (52)

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( a ^ 1 b ^ 1 ) = T ^ OPA 1 ( a ^ 0 b ^ 0 ) ,
( a ^ 2 b ^ 2 ) = T ^ ( a ^ 0 b ^ 0 ) .
T ^ OPA 1 = ( u 1 v 1 v 1 * u 1 ) ,
T ^ ϕ = ( e i ϕ 1 0 0 e i ϕ 2 ) ,
T ^ OPA 2 = ( u 2 v 2 v 2 * u 2 ) ,
n ¯ = Ψ in | ( a ^ 1 a ^ 1 + b ^ 1 b ^ 1 ) | Ψ in ,
n ¯ = ( n OPA + 1 ) ( n th + n s ) + n OPA ,
Δ ϕ SNL = 1 n ¯ = 1 ( n OPA + 1 ) ( n th + n s ) + n OPA ,
Δ ϕ HL = 1 n ¯ = 1 ( n OPA + 1 ) ( n th + n s ) + n OPA .
Π ^ b ( 1 ) b ^ 2 b ^ 2 .
Δ ϕ = Δ Π ^ b | Π ^ b / ϕ | ,
Δ ϕ = 2 n OPA ( n OPA + 2 ) [ 1 + ( 1 + 2 n s ) ( 1 + 2 n th ) ] ,
n th = sinh 2 g n s cosh 2 ( 2 g ) .
Π ^ b D C = e 2 d Π ^ b .
x ^ a j = a ^ j + a ^ j , p ^ a j = i ( a ^ j a ^ j ) ,
x ^ b j = b ^ j + b ^ j , p ^ a j = i ( b ^ j b ^ j ) ,
X j = ( X ^ j , 1 , X ^ j , 2 , X ^ j , 3 , X ^ j , 4 ) T = ( x ^ a j , p ^ a j , x ^ b j , p ^ b j ) T .
X ¯ j = ( X ^ j , 1 , X ^ j , 2 , X ^ j , 3 , X ^ j , 4 ) T ,
Γ j k l = 1 2 Tr [ ( X ˜ j , k X ˜ j , l + X ˜ j , l X ˜ j , k ) ρ ] ,
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 2 ) sin ( ϕ 2 ) 0 0 sin ( ϕ 2 ) cos ( ϕ 2 ) 0 0 0 0 cos ( ϕ 2 ) sin ( ϕ 2 ) 0 0 sin ( ϕ 2 ) cos ( ϕ 2 ) ) ,
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 = exp ( X ¯ 22 T Γ 22 1 X ¯ 22 ) | Γ 22 | ,
X ¯ 0 = 0 , Γ 0 = σ th σ sqz ,
σ th = ( 2 n t h + 1 ) ( 1 0 0 1 ) ,
σ sqz = ( e 2 r 0 0 e 2 r ) .
X ¯ 2 = 0 ,
Γ 2 = ( γ 11 γ 12 γ 13 γ 14 γ 21 γ 22 γ 23 γ 24 γ 31 γ 32 γ 33 γ 34 γ 41 γ 42 γ 43 γ 44 ) ,
γ 11 = e 2 r sin 2 ( ϕ 2 ) sinh 2 ( 2 g ) + 1 4 [ 3 + cosh ( 4 g ) 2 cos ϕ sinh 2 ( 2 g ) ] ( 1 + 2 n th )
γ 13 = γ 31 = sin 2 ( ϕ 2 ) sinh ( 4 g ) ( cosh r + sinh r ) ( cosh r + e r n th ) ,
γ 14 = γ 41 = e 2 r cosh g sin ϕ sinh g [ 1 + e 2 r ( 1 + 2 n th ) ] ,
γ 22 = e 2 r sin 2 ( ϕ 2 ) sinh 2 ( 2 g ) + 1 4 [ 3 + cosh ( 4 g ) 2 cos ϕ sinh 2 ( 2 g ) ] ( 1 + 2 n th ) ,
γ 23 = γ 32 = cosh g sin ϕ sinh g ( 1 + e 2 r + 2 n th ) ,
γ 24 = γ 42 = 1 2 sin 2 ( ϕ 2 ) sinh ( 4 g ) ( 1 + e 2 r + 2 n th ) ,
γ 33 = 1 2 e 2 r ( 1 + cos ϕ ) + e 2 r cosh 2 ( 2 g ) sin 2 ( ϕ 2 ) + sin 2 ( ϕ 2 ) sinh 2 ( 2 g ) ( 1 + 2 n th ) ,
γ 34 = γ 43 = cosh ( 2 g ) sin ϕ sinh ( 2 r ) ,
γ 44 = 1 2 e 2 r ( 1 + cos ϕ ) + e 2 r cosh 2 ( 2 g ) sin 2 ( ϕ 2 ) + sin 2 ( ϕ 2 ) sinh 2 ( 2 g ) ( 1 + 2 n th ) .
X ¯ 22 = ( 0 0 ) , Γ 22 = ( γ 33 γ 34 γ 43 γ 44 ) .
Π b = 8 T ,
T = e 2 r { 7 + 50 e 2 r 7 e 4 r + ( 1 + e 2 r ) 2 [ 4 cosh ( 4 g ) + 3 cosh ( 8 g ) + 8 cos ( 2 ϕ ) sinh 4 ( 2 g ) 8 cos ϕ sinh 2 ( 4 g ) ] } + 32 e 2 r sin 2 ( ϕ 2 ) sinh 2 ( 2 g ) n th { ( 1 + e 4 r ) [ 3 + cosh ( 4 g ) 2 cos ϕ sinh 2 ( 2 g ) ] + 8 e 2 r sin 2 ( ϕ 2 ) sinh 2 ( 2 g ) ( 1 + n th ) } .
Δ ϕ = Δ Π ^ b | Π ^ b / ϕ | ,
Δ Π ^ b = { 1 64 / { e 2 r { 7 + 50 e 2 r 7 e 4 r + ( 1 + e 2 r ) 2 [ 4 cosh ( 4 g ) + 3 cosh ( 8 g ) + 8 cos ( 2 ϕ ) sinh 4 ( 2 g ) 8 cos ϕ sinh 2 ( 4 g ) ] } + 32 e 2 r sin 2 ( ϕ 2 ) sinh 2 ( 2 g ) n th { ( 1 + e 4 r ) [ 3 + cosh ( 4 g ) 2 cos ϕ sinh 2 ( 2 g ) ] + 8 e 2 r sin 2 ( ϕ 2 ) sinh 2 ( 2 g ) ( 1 + n th ) } } } 1 / 2 ,
| Π ^ b / ϕ | = { 128 sinh 2 ( 2 g ) { 2 sin ( 2 ϕ ) sinh 2 ( 2 g ) [ cosh 2 r + n th ( 1 + cosh ( 2 r ) + n th ) ] + sin ( ϕ ) { 4 cosh 2 ( 2 g ) cosh 2 r + 4 n th [ cosh 2 ( 2 g ) cosh ( 2 r ) + sinh 2 ( 2 g ) ( 1 + n th ) ] } } } / { e 2 r { 7 + ( 1 + e 2 r ) 2 [ 4 cosh ( 4 g ) + 3 cosh ( 8 g ) + 8 cos ( 2 ϕ ) sinh 4 ( 2 g ) 8 cos ϕ sinh 2 ( 4 g ) ] 7 e 4 r + 50 e 2 r + 32 sin 2 ( ϕ 2 ) sinh 2 ( 2 g ) n th [ ( 1 + e 4 r ) ( 3 + cosh ( 4 g ) 2 cos ϕ sinh 2 ( 2 g ) ) + 8 e 2 r sin 2 ( ϕ 2 ) sinh 2 ( 2 g ) ( 1 + n th ) ] } } 3 / 2 .
Γ 0 * = ( Γ 0 0 4 0 4 σ th σ th ) 8 × 8 ,
S OPAi * = ( S OPAi 0 4 0 4 I 4 ) 8 × 8 ,
S ϕ * = ( S ϕ 0 4 0 4 I 4 ) 8 × 8 ,
S VBS * = ( T I 4 1 T I 4 1 T I 4 T I 4 ) 8 × 8 ,
X ¯ 2 * = S X ¯ 0 * ,
Γ 2 * = S * Γ 0 * ( S * ) T ,
Π ^ b * = exp ( X ¯ 22 * T Γ 22 * 1 X ¯ 22 * ) | Γ 22 * | ,

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