Abstract

Homodyne detection is often used for interferometers based on nonlinear optical gain media. For the configuration of a seeded, “truncated SU(1,1)” interferometer Anderson, et al. [ Phys. Rev. A 95, 063843 (2017)] showed how to optimize the homodyne detection scheme and demonstrated theoretically that it can saturate the quantum Cramer-Rao bound for phase estimation. In this work we extend those results by taking into account loss in the truncated SU(1,1) interferometer and determining the optimized homodyne detection scheme for phase measurement. Further, we build a truncated SU(1,1) interferometer and experimentally demonstrate that this optimized scheme achieves a reduction in noise level, corresponding to an enhanced potential phase sensitivity, compared to a typical homodyne detection scheme for a two-mode squeezed state. In doing so, we also demonstrate an improvement in the degree to which we can beat the standard quantum limit with this device.

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

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References

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  1. B. E. Anderson, B. L. Schmittberger, P. Gupta, K. M. Jones, and P. D. Lett, “Optimal phase measurements with bright- and vacuum-seeded SU(1,1) interferometers,” Phys. Rev. A 95, 063843 (2017).
    [Crossref]
  2. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23(8), 1693–1708 (1981).
    [Crossref]
  3. P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed-light-enhanced polarization interferometer,” Phys. Rev. Lett. 59(19), 2153–2156 (1987).
    [Crossref] [PubMed]
  4. M. Xiao, L.-A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59(3), 278–281 (1987).
    [Crossref] [PubMed]
  5. M. J. Holland and K. Burnett, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. 71(9), 1355–1358 (1993).
    [Crossref] [PubMed]
  6. M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161–164 (2004).
    [Crossref] [PubMed]
  7. G. Y. Xiang, B. L. Higgins, D. W. Berry, H. M. Wiseman, and G. J. Pryde, “Entanglement-enhanced measurement of a completely unknown optical phase,” Nat. Photon. 5, 43–47 (2011).
    [Crossref]
  8. B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1,1) interferometers,” Phys. Rev. A 33, 4033–4054 (1986).
    [Crossref]
  9. V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled images from four-wave mixing,” Science 321(5888), 544–547 (2008).
    [Crossref] [PubMed]
  10. C. F. McCormick, V. Boyer, E. Arimondo, and P. D. Lett, “Strong relative intensity squeezing by four-wave mixing in rubidium vapor,” Opt. Lett. 32, 178–180 (2007).
    [Crossref]
  11. A. A. Berni, T. Gehring, B. M. Nielsen, V. Händchen, M. G. A. Paris, and U. L. Andersen, “Ab initio quantum-enhanced optical phase estimation using real-time feedback control,” Nat. Photon. 9, 577–581 (2015).
    [Crossref]
  12. H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwasawa, S. Takeda, H. Arao, K. Ohki, K. Tsumura, D. W. Berry, T. C. Ralph, H. M. Wiseman, E. H. Huntington, and A. Furusawa, “Quantum-enhanced optical-phase tracking,” Science 337, 1514–1517 (2012).
    [Crossref] [PubMed]
  13. F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
    [Crossref] [PubMed]
  14. M. Manceau, G. Leuchs, F. Khalili, and M. Chekhova, “Detection loss tolerant supersensitive phase measurement with an SU(1,1) interferometer,” arXiv:1705.02662v2 (2017).
  15. Z. Y. Ou, “Enhancement of the phase-measurement sensitivity beyond the standard quantum limit by a nonlinear interferometer,” Phys. Rev. A 85, 023815 (2012).
    [Crossref]
  16. 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, 023844 (2012).
    [Crossref]
  17. 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]
  18. 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]
  19. C. W. Helstrom, Quantum detection and estimation theory, (Academic, 1976), Vol. 123.
  20. M. G. A. Paris, “Quantum estimation for quantum technology,” Int. J. Quantum Inf. 07, 125–137 (2009).
    [Crossref]
  21. C. Sparaciari, S. Olivares, and M. G. A. Paris, “Bounds to precision for quantum interferometry with Gaussian states and operations,” J. Opt. Soc. Am. B 32(7), 1354–1359 (2015).
    [Crossref]
  22. C. Sparaciari, S. Olivares, and M. G. A. Paris, “Gaussian-state interferometry with passive and active elements,” Phys. Rev. A 93, 023810 (2016).
    [Crossref]
  23. U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
    [Crossref] [PubMed]
  24. G. Gilbert, M. Hamrick, and Y. S. Weinstein, “Use of maximally entangled n-photon states for practical quantum interferometry,” J. Opt. Soc. Am. B 25(8), 1336–1340 (2008).
    [Crossref]
  25. T. Kim, O. Pfister, M. J. Holland, J. Noh, and J. L. Hall, “Influence of decorrelation on Heisenberg-limited interferometry with quantum correlated photons,” Phys. Rev. A 57, 4004–4013 (1998).
    [Crossref]
  26. T. Ono and H. F. Hofmann, “Effects of photon losses on phase estimation near the Heisenberg limit using coherent light and squeezed vacuum,” Phys. Rev. A 81, 033819 (2010).
    [Crossref]
  27. R. C. Pooser and O. Pfister, “Particle-number scaling of the phase sensitivity in realistic Bayesian twin-mode Heisenberg-limited interferometry,” Phys. Rev. A 69, 043616 (2004).
    [Crossref]
  28. M. A. Rubin and S. Kaushik, “Loss-induced limits to phase measurement precision with maximally entangled states,” Phys. Rev. A 75, 053805 (2007).
    [Crossref]
  29. X.-Y. Chen and L.-Z. Jiang, “The entanglement and phase measurement performance of the damped NOON state,” J. Phys. Pt. B Atom. M. P. 40(14), 2799 (2007).

2017 (2)

B. E. Anderson, B. L. Schmittberger, P. Gupta, K. M. Jones, and P. D. Lett, “Optimal phase measurements with bright- and vacuum-seeded SU(1,1) interferometers,” Phys. Rev. A 95, 063843 (2017).
[Crossref]

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]

2016 (1)

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Gaussian-state interferometry with passive and active elements,” Phys. Rev. A 93, 023810 (2016).
[Crossref]

2015 (2)

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Bounds to precision for quantum interferometry with Gaussian states and operations,” J. Opt. Soc. Am. B 32(7), 1354–1359 (2015).
[Crossref]

A. A. Berni, T. Gehring, B. M. Nielsen, V. Händchen, M. G. A. Paris, and U. L. Andersen, “Ab initio quantum-enhanced optical phase estimation using real-time feedback control,” Nat. Photon. 9, 577–581 (2015).
[Crossref]

2014 (1)

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

2012 (3)

Z. Y. Ou, “Enhancement of the phase-measurement sensitivity beyond the standard quantum limit by a nonlinear interferometer,” Phys. Rev. A 85, 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, 023844 (2012).
[Crossref]

H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwasawa, S. Takeda, H. Arao, K. Ohki, K. Tsumura, D. W. Berry, T. C. Ralph, H. M. Wiseman, E. H. Huntington, and A. Furusawa, “Quantum-enhanced optical-phase tracking,” Science 337, 1514–1517 (2012).
[Crossref] [PubMed]

2011 (1)

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

2010 (2)

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]

T. Ono and H. F. Hofmann, “Effects of photon losses on phase estimation near the Heisenberg limit using coherent light and squeezed vacuum,” Phys. Rev. A 81, 033819 (2010).
[Crossref]

2009 (2)

M. G. A. Paris, “Quantum estimation for quantum technology,” Int. J. Quantum Inf. 07, 125–137 (2009).
[Crossref]

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[Crossref] [PubMed]

2008 (2)

G. Gilbert, M. Hamrick, and Y. S. Weinstein, “Use of maximally entangled n-photon states for practical quantum interferometry,” J. Opt. Soc. Am. B 25(8), 1336–1340 (2008).
[Crossref]

V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled images from four-wave mixing,” Science 321(5888), 544–547 (2008).
[Crossref] [PubMed]

2007 (3)

C. F. McCormick, V. Boyer, E. Arimondo, and P. D. Lett, “Strong relative intensity squeezing by four-wave mixing in rubidium vapor,” Opt. Lett. 32, 178–180 (2007).
[Crossref]

M. A. Rubin and S. Kaushik, “Loss-induced limits to phase measurement precision with maximally entangled states,” Phys. Rev. A 75, 053805 (2007).
[Crossref]

X.-Y. Chen and L.-Z. Jiang, “The entanglement and phase measurement performance of the damped NOON state,” J. Phys. Pt. B Atom. M. P. 40(14), 2799 (2007).

2004 (2)

R. C. Pooser and O. Pfister, “Particle-number scaling of the phase sensitivity in realistic Bayesian twin-mode Heisenberg-limited interferometry,” Phys. Rev. A 69, 043616 (2004).
[Crossref]

M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161–164 (2004).
[Crossref] [PubMed]

1998 (1)

T. Kim, O. Pfister, M. J. Holland, J. Noh, and J. L. Hall, “Influence of decorrelation on Heisenberg-limited interferometry with quantum correlated photons,” Phys. Rev. A 57, 4004–4013 (1998).
[Crossref]

1993 (1)

M. J. Holland and K. Burnett, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. 71(9), 1355–1358 (1993).
[Crossref] [PubMed]

1987 (2)

P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed-light-enhanced polarization interferometer,” Phys. Rev. Lett. 59(19), 2153–2156 (1987).
[Crossref] [PubMed]

M. Xiao, L.-A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59(3), 278–281 (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, 4033–4054 (1986).
[Crossref]

1981 (1)

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

Andersen, U. L.

A. A. Berni, T. Gehring, B. M. Nielsen, V. Händchen, M. G. A. Paris, and U. L. Andersen, “Ab initio quantum-enhanced optical phase estimation using real-time feedback control,” Nat. Photon. 9, 577–581 (2015).
[Crossref]

Anderson, B. E.

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]

B. E. Anderson, B. L. Schmittberger, P. Gupta, K. M. Jones, and P. D. Lett, “Optimal phase measurements with bright- and vacuum-seeded SU(1,1) interferometers,” Phys. Rev. A 95, 063843 (2017).
[Crossref]

Arao, H.

H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwasawa, S. Takeda, H. Arao, K. Ohki, K. Tsumura, D. W. Berry, T. C. Ralph, H. M. Wiseman, E. H. Huntington, and A. Furusawa, “Quantum-enhanced optical-phase tracking,” Science 337, 1514–1517 (2012).
[Crossref] [PubMed]

Arimondo, E.

Banaszek, K.

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[Crossref] [PubMed]

Berni, A. A.

A. A. Berni, T. Gehring, B. M. Nielsen, V. Händchen, M. G. A. Paris, and U. L. Andersen, “Ab initio quantum-enhanced optical phase estimation using real-time feedback control,” Nat. Photon. 9, 577–581 (2015).
[Crossref]

Berry, D. W.

H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwasawa, S. Takeda, H. Arao, K. Ohki, K. Tsumura, D. W. Berry, T. C. Ralph, H. M. Wiseman, E. H. Huntington, and A. Furusawa, “Quantum-enhanced optical-phase tracking,” Science 337, 1514–1517 (2012).
[Crossref] [PubMed]

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

Boyer, V.

V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled images from four-wave mixing,” Science 321(5888), 544–547 (2008).
[Crossref] [PubMed]

C. F. McCormick, V. Boyer, E. Arimondo, and P. D. Lett, “Strong relative intensity squeezing by four-wave mixing in rubidium vapor,” Opt. Lett. 32, 178–180 (2007).
[Crossref]

Burnett, K.

M. J. Holland and K. Burnett, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. 71(9), 1355–1358 (1993).
[Crossref] [PubMed]

Caves, C. M.

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

Chekhova, M.

M. Manceau, G. Leuchs, F. Khalili, and M. Chekhova, “Detection loss tolerant supersensitive phase measurement with an SU(1,1) interferometer,” arXiv:1705.02662v2 (2017).

Chen, X.-Y.

X.-Y. Chen and L.-Z. Jiang, “The entanglement and phase measurement performance of the damped NOON state,” J. Phys. Pt. B Atom. M. P. 40(14), 2799 (2007).

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, 023844 (2012).
[Crossref]

Demkowicz-Dobrzanski, R.

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[Crossref] [PubMed]

Dorner, U.

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[Crossref] [PubMed]

Dowling, J. P

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]

Furusawa, A.

H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwasawa, S. Takeda, H. Arao, K. Ohki, K. Tsumura, D. W. Berry, T. C. Ralph, H. M. Wiseman, E. H. Huntington, and A. Furusawa, “Quantum-enhanced optical-phase tracking,” Science 337, 1514–1517 (2012).
[Crossref] [PubMed]

Gehring, T.

A. A. Berni, T. Gehring, B. M. Nielsen, V. Händchen, M. G. A. Paris, and U. L. Andersen, “Ab initio quantum-enhanced optical phase estimation using real-time feedback control,” Nat. Photon. 9, 577–581 (2015).
[Crossref]

Gilbert, G.

Grangier, P.

P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed-light-enhanced polarization interferometer,” Phys. Rev. Lett. 59(19), 2153–2156 (1987).
[Crossref] [PubMed]

Gupta, P.

B. E. Anderson, B. L. Schmittberger, P. Gupta, K. M. Jones, and P. D. Lett, “Optimal phase measurements with bright- and vacuum-seeded SU(1,1) interferometers,” Phys. Rev. A 95, 063843 (2017).
[Crossref]

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]

Hall, J. L.

T. Kim, O. Pfister, M. J. Holland, J. Noh, and J. L. Hall, “Influence of decorrelation on Heisenberg-limited interferometry with quantum correlated photons,” Phys. Rev. A 57, 4004–4013 (1998).
[Crossref]

Hamrick, M.

Händchen, V.

A. A. Berni, T. Gehring, B. M. Nielsen, V. Händchen, M. G. A. Paris, and U. L. Andersen, “Ab initio quantum-enhanced optical phase estimation using real-time feedback control,” Nat. Photon. 9, 577–581 (2015).
[Crossref]

Helstrom, C. W.

C. W. Helstrom, Quantum detection and estimation theory, (Academic, 1976), Vol. 123.

Hermann-Avigliano, C.

Higgins, B. L.

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

Hofmann, H. F.

T. Ono and H. F. Hofmann, “Effects of photon losses on phase estimation near the Heisenberg limit using coherent light and squeezed vacuum,” Phys. Rev. A 81, 033819 (2010).
[Crossref]

Holland, M. J.

T. Kim, O. Pfister, M. J. Holland, J. Noh, and J. L. Hall, “Influence of decorrelation on Heisenberg-limited interferometry with quantum correlated photons,” Phys. Rev. A 57, 4004–4013 (1998).
[Crossref]

M. J. Holland and K. Burnett, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. 71(9), 1355–1358 (1993).
[Crossref] [PubMed]

Horrom, T.

Hudelist, F.

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

Huntington, E. H.

H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwasawa, S. Takeda, H. Arao, K. Ohki, K. Tsumura, D. W. Berry, T. C. Ralph, H. M. Wiseman, E. H. Huntington, and A. Furusawa, “Quantum-enhanced optical-phase tracking,” Science 337, 1514–1517 (2012).
[Crossref] [PubMed]

Iwasawa, K.

H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwasawa, S. Takeda, H. Arao, K. Ohki, K. Tsumura, D. W. Berry, T. C. Ralph, H. M. Wiseman, E. H. Huntington, and A. Furusawa, “Quantum-enhanced optical-phase tracking,” Science 337, 1514–1517 (2012).
[Crossref] [PubMed]

Jiang, L.-Z.

X.-Y. Chen and L.-Z. Jiang, “The entanglement and phase measurement performance of the damped NOON state,” J. Phys. Pt. B Atom. M. P. 40(14), 2799 (2007).

Jing, J.

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

Jones, K. M.

B. E. Anderson, B. L. Schmittberger, P. Gupta, K. M. Jones, and P. D. Lett, “Optimal phase measurements with bright- and vacuum-seeded SU(1,1) interferometers,” Phys. Rev. A 95, 063843 (2017).
[Crossref]

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]

Kaushik, S.

M. A. Rubin and S. Kaushik, “Loss-induced limits to phase measurement precision with maximally entangled states,” Phys. Rev. A 75, 053805 (2007).
[Crossref]

Khalili, F.

M. Manceau, G. Leuchs, F. Khalili, and M. Chekhova, “Detection loss tolerant supersensitive phase measurement with an SU(1,1) interferometer,” arXiv:1705.02662v2 (2017).

Kim, T.

T. Kim, O. Pfister, M. J. Holland, J. Noh, and J. L. Hall, “Influence of decorrelation on Heisenberg-limited interferometry with quantum correlated photons,” Phys. Rev. A 57, 4004–4013 (1998).
[Crossref]

Kimble, H. J.

M. Xiao, L.-A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59(3), 278–281 (1987).
[Crossref] [PubMed]

Klauder, J. R.

B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1,1) interferometers,” Phys. Rev. A 33, 4033–4054 (1986).
[Crossref]

Kong, J.

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

LaPorta, A.

P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed-light-enhanced polarization interferometer,” Phys. Rev. Lett. 59(19), 2153–2156 (1987).
[Crossref] [PubMed]

Lett, P. D.

B. E. Anderson, B. L. Schmittberger, P. Gupta, K. M. Jones, and P. D. Lett, “Optimal phase measurements with bright- and vacuum-seeded SU(1,1) interferometers,” Phys. Rev. A 95, 063843 (2017).
[Crossref]

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]

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, 023844 (2012).
[Crossref]

V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled images from four-wave mixing,” Science 321(5888), 544–547 (2008).
[Crossref] [PubMed]

C. F. McCormick, V. Boyer, E. Arimondo, and P. D. Lett, “Strong relative intensity squeezing by four-wave mixing in rubidium vapor,” Opt. Lett. 32, 178–180 (2007).
[Crossref]

Leuchs, G.

M. Manceau, G. Leuchs, F. Khalili, and M. Chekhova, “Detection loss tolerant supersensitive phase measurement with an SU(1,1) interferometer,” arXiv:1705.02662v2 (2017).

Liu, C.

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

Lundeen, J. S.

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[Crossref] [PubMed]

M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161–164 (2004).
[Crossref] [PubMed]

Manceau, M.

M. Manceau, G. Leuchs, F. Khalili, and M. Chekhova, “Detection loss tolerant supersensitive phase measurement with an SU(1,1) interferometer,” arXiv:1705.02662v2 (2017).

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, 023844 (2012).
[Crossref]

V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled images from four-wave mixing,” Science 321(5888), 544–547 (2008).
[Crossref] [PubMed]

McCall, S. L.

B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1,1) interferometers,” Phys. Rev. A 33, 4033–4054 (1986).
[Crossref]

McCormick, C. F.

Mitchell, M. W.

M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161–164 (2004).
[Crossref] [PubMed]

Nakane, D.

H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwasawa, S. Takeda, H. Arao, K. Ohki, K. Tsumura, D. W. Berry, T. C. Ralph, H. M. Wiseman, E. H. Huntington, and A. Furusawa, “Quantum-enhanced optical-phase tracking,” Science 337, 1514–1517 (2012).
[Crossref] [PubMed]

Nielsen, B. M.

A. A. Berni, T. Gehring, B. M. Nielsen, V. Händchen, M. G. A. Paris, and U. L. Andersen, “Ab initio quantum-enhanced optical phase estimation using real-time feedback control,” Nat. Photon. 9, 577–581 (2015).
[Crossref]

Noh, J.

T. Kim, O. Pfister, M. J. Holland, J. Noh, and J. L. Hall, “Influence of decorrelation on Heisenberg-limited interferometry with quantum correlated photons,” Phys. Rev. A 57, 4004–4013 (1998).
[Crossref]

Ohki, K.

H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwasawa, S. Takeda, H. Arao, K. Ohki, K. Tsumura, D. W. Berry, T. C. Ralph, H. M. Wiseman, E. H. Huntington, and A. Furusawa, “Quantum-enhanced optical-phase tracking,” Science 337, 1514–1517 (2012).
[Crossref] [PubMed]

Olivares, S.

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Gaussian-state interferometry with passive and active elements,” Phys. Rev. A 93, 023810 (2016).
[Crossref]

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Bounds to precision for quantum interferometry with Gaussian states and operations,” J. Opt. Soc. Am. B 32(7), 1354–1359 (2015).
[Crossref]

Ono, T.

T. Ono and H. F. Hofmann, “Effects of photon losses on phase estimation near the Heisenberg limit using coherent light and squeezed vacuum,” Phys. Rev. A 81, 033819 (2010).
[Crossref]

Ou, Z. Y.

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

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

Paris, M. G. A.

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Gaussian-state interferometry with passive and active elements,” Phys. Rev. A 93, 023810 (2016).
[Crossref]

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Bounds to precision for quantum interferometry with Gaussian states and operations,” J. Opt. Soc. Am. B 32(7), 1354–1359 (2015).
[Crossref]

A. A. Berni, T. Gehring, B. M. Nielsen, V. Händchen, M. G. A. Paris, and U. L. Andersen, “Ab initio quantum-enhanced optical phase estimation using real-time feedback control,” Nat. Photon. 9, 577–581 (2015).
[Crossref]

M. G. A. Paris, “Quantum estimation for quantum technology,” Int. J. Quantum Inf. 07, 125–137 (2009).
[Crossref]

Pfister, O.

R. C. Pooser and O. Pfister, “Particle-number scaling of the phase sensitivity in realistic Bayesian twin-mode Heisenberg-limited interferometry,” Phys. Rev. A 69, 043616 (2004).
[Crossref]

T. Kim, O. Pfister, M. J. Holland, J. Noh, and J. L. Hall, “Influence of decorrelation on Heisenberg-limited interferometry with quantum correlated photons,” Phys. Rev. A 57, 4004–4013 (1998).
[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, 083014 (2010).
[Crossref]

Pooser, R. C.

V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled images from four-wave mixing,” Science 321(5888), 544–547 (2008).
[Crossref] [PubMed]

R. C. Pooser and O. Pfister, “Particle-number scaling of the phase sensitivity in realistic Bayesian twin-mode Heisenberg-limited interferometry,” Phys. Rev. A 69, 043616 (2004).
[Crossref]

Pryde, G. J.

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

Ralph, T. C.

H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwasawa, S. Takeda, H. Arao, K. Ohki, K. Tsumura, D. W. Berry, T. C. Ralph, H. M. Wiseman, E. H. Huntington, and A. Furusawa, “Quantum-enhanced optical-phase tracking,” Science 337, 1514–1517 (2012).
[Crossref] [PubMed]

Rubin, M. A.

M. A. Rubin and S. Kaushik, “Loss-induced limits to phase measurement precision with maximally entangled states,” Phys. Rev. A 75, 053805 (2007).
[Crossref]

Schmittberger, B. L.

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]

B. E. Anderson, B. L. Schmittberger, P. Gupta, K. M. Jones, and P. D. Lett, “Optimal phase measurements with bright- and vacuum-seeded SU(1,1) interferometers,” Phys. Rev. A 95, 063843 (2017).
[Crossref]

Slusher, R. E.

P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed-light-enhanced polarization interferometer,” Phys. Rev. Lett. 59(19), 2153–2156 (1987).
[Crossref] [PubMed]

Smith, B. J.

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[Crossref] [PubMed]

Sparaciari, C.

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Gaussian-state interferometry with passive and active elements,” Phys. Rev. A 93, 023810 (2016).
[Crossref]

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Bounds to precision for quantum interferometry with Gaussian states and operations,” J. Opt. Soc. Am. B 32(7), 1354–1359 (2015).
[Crossref]

Steinberg, A. M.

M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161–164 (2004).
[Crossref] [PubMed]

Takeda, S.

H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwasawa, S. Takeda, H. Arao, K. Ohki, K. Tsumura, D. W. Berry, T. C. Ralph, H. M. Wiseman, E. H. Huntington, and A. Furusawa, “Quantum-enhanced optical-phase tracking,” Science 337, 1514–1517 (2012).
[Crossref] [PubMed]

Tsumura, K.

H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwasawa, S. Takeda, H. Arao, K. Ohki, K. Tsumura, D. W. Berry, T. C. Ralph, H. M. Wiseman, E. H. Huntington, and A. Furusawa, “Quantum-enhanced optical-phase tracking,” Science 337, 1514–1517 (2012).
[Crossref] [PubMed]

Walmsley, I. A.

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[Crossref] [PubMed]

Wasilewski, W.

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[Crossref] [PubMed]

Weinstein, Y. S.

Wheatley, T. A.

H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwasawa, S. Takeda, H. Arao, K. Ohki, K. Tsumura, D. W. Berry, T. C. Ralph, H. M. Wiseman, E. H. Huntington, and A. Furusawa, “Quantum-enhanced optical-phase tracking,” Science 337, 1514–1517 (2012).
[Crossref] [PubMed]

Wiseman, H. M.

H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwasawa, S. Takeda, H. Arao, K. Ohki, K. Tsumura, D. W. Berry, T. C. Ralph, H. M. Wiseman, E. H. Huntington, and A. Furusawa, “Quantum-enhanced optical-phase tracking,” Science 337, 1514–1517 (2012).
[Crossref] [PubMed]

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

Wu, L.-A.

M. Xiao, L.-A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59(3), 278–281 (1987).
[Crossref] [PubMed]

Xiang, G. Y.

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

Xiao, M.

M. Xiao, L.-A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59(3), 278–281 (1987).
[Crossref] [PubMed]

Yonezawa, H.

H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwasawa, S. Takeda, H. Arao, K. Ohki, K. Tsumura, D. W. Berry, T. C. Ralph, H. M. Wiseman, E. H. Huntington, and A. Furusawa, “Quantum-enhanced optical-phase tracking,” Science 337, 1514–1517 (2012).
[Crossref] [PubMed]

Yurke, B.

P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed-light-enhanced polarization interferometer,” Phys. Rev. Lett. 59(19), 2153–2156 (1987).
[Crossref] [PubMed]

B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1,1) interferometers,” Phys. Rev. A 33, 4033–4054 (1986).
[Crossref]

Zhang, W.

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

Int. J. Quantum Inf. (1)

M. G. A. Paris, “Quantum estimation for quantum technology,” Int. J. Quantum Inf. 07, 125–137 (2009).
[Crossref]

J. Opt. Soc. Am. B (2)

J. Phys. Pt. B Atom. M. P. (1)

X.-Y. Chen and L.-Z. Jiang, “The entanglement and phase measurement performance of the damped NOON state,” J. Phys. Pt. B Atom. M. P. 40(14), 2799 (2007).

Nat. Commun. (1)

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

Nat. Photon. (2)

A. A. Berni, T. Gehring, B. M. Nielsen, V. Händchen, M. G. A. Paris, and U. L. Andersen, “Ab initio quantum-enhanced optical phase estimation using real-time feedback control,” Nat. Photon. 9, 577–581 (2015).
[Crossref]

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

Nature (1)

M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161–164 (2004).
[Crossref] [PubMed]

New J. Phys. (1)

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]

Opt. Lett. (1)

Optica (1)

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, 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, 023844 (2012).
[Crossref]

B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1,1) interferometers,” Phys. Rev. A 33, 4033–4054 (1986).
[Crossref]

B. E. Anderson, B. L. Schmittberger, P. Gupta, K. M. Jones, and P. D. Lett, “Optimal phase measurements with bright- and vacuum-seeded SU(1,1) interferometers,” Phys. Rev. A 95, 063843 (2017).
[Crossref]

T. Kim, O. Pfister, M. J. Holland, J. Noh, and J. L. Hall, “Influence of decorrelation on Heisenberg-limited interferometry with quantum correlated photons,” Phys. Rev. A 57, 4004–4013 (1998).
[Crossref]

T. Ono and H. F. Hofmann, “Effects of photon losses on phase estimation near the Heisenberg limit using coherent light and squeezed vacuum,” Phys. Rev. A 81, 033819 (2010).
[Crossref]

R. C. Pooser and O. Pfister, “Particle-number scaling of the phase sensitivity in realistic Bayesian twin-mode Heisenberg-limited interferometry,” Phys. Rev. A 69, 043616 (2004).
[Crossref]

M. A. Rubin and S. Kaushik, “Loss-induced limits to phase measurement precision with maximally entangled states,” Phys. Rev. A 75, 053805 (2007).
[Crossref]

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Gaussian-state interferometry with passive and active elements,” Phys. Rev. A 93, 023810 (2016).
[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. (4)

P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed-light-enhanced polarization interferometer,” Phys. Rev. Lett. 59(19), 2153–2156 (1987).
[Crossref] [PubMed]

M. Xiao, L.-A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59(3), 278–281 (1987).
[Crossref] [PubMed]

M. J. Holland and K. Burnett, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. 71(9), 1355–1358 (1993).
[Crossref] [PubMed]

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102, 040403 (2009).
[Crossref] [PubMed]

Science (2)

V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled images from four-wave mixing,” Science 321(5888), 544–547 (2008).
[Crossref] [PubMed]

H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwasawa, S. Takeda, H. Arao, K. Ohki, K. Tsumura, D. W. Berry, T. C. Ralph, H. M. Wiseman, E. H. Huntington, and A. Furusawa, “Quantum-enhanced optical-phase tracking,” Science 337, 1514–1517 (2012).
[Crossref] [PubMed]

Other (2)

M. Manceau, G. Leuchs, F. Khalili, and M. Chekhova, “Detection loss tolerant supersensitive phase measurement with an SU(1,1) interferometer,” arXiv:1705.02662v2 (2017).

C. W. Helstrom, Quantum detection and estimation theory, (Academic, 1976), Vol. 123.

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Figures (7)

Fig. 1
Fig. 1 Schematics of SU(1,1) type interferometers. A coherent state |αe0〉 with amplitude α and phase ϕ0 and a vacuum state |0〉 are mixed with a strong pump beam in an NLO medium to produce a probe beam and a conjugate beam, which together form a two-mode squeezed state. A phase shift δϕ is applied to the seeded arm, i.e., the probe beam. a) The SU(1,1) interferometer recombines the probe and the conjugate beams in another nonlinear medium. The diagram shows homodyne detectors (HD) after the second cell that measure the output and thus the phase shift δϕ. One could also perform direct intensity detection on the outputs to measure the phase shift δϕ. b) The truncated SU(1,1) interferometer sends the probe and the conjugate beams directly to homodyne detectors [1,18]. The output of the conjugate beam homodyne detector is attenuated by a factor λ using an electronic attenuator before it is combined with the output of the probe homodyne detector to perform the phase measurement. We discuss the electronic attenuation in Sec. 2. c) A schematic of a homodyne detector for phase measurement [11–13, 15] in which a signal beam interferes with a strong local oscillator (LO) of the same spatial mode and frequency as the signal beam on a 50:50 beam splitter. The difference in the photo-currents of the two outputs gives the quadrature of the signal beam amplified by the amplitude of the LO.
Fig. 2
Fig. 2 Measured power on a spectrum analyzer as a function of frequency in a truncated SU(1,1) interferometer using two-mode squeezed light (blue, solid line) and with a coherent beam (red, dashed line) of the same optical power in the phase sensing (probe beam) arm of the interferometer, as shown in Ref. [18]. The coherent beam SNR represents the SQL, and the difference in the noise floor of the traces shows a 4.0(1) dB improvement in SNR with the squeezed light over the SQL. The 4WM gain used for the measurement was 3.3, higher than what we discuss here.
Fig. 3
Fig. 3 Theoretical peak sensitivity, multiplied by the amplitude (|α|) of the coherent seed beam of the interferometer, achieved by an ideal lossless truncated SU(1,1) interferometer as a function of gain in the 4WM process. The solid orange curve shows the phase sensitivity of the observable Q and the solid blue curve represents the phase sensitivity of the observable λoptQ (as defined in the text). The thick dashed green curve indicates the QCRB for the two-mode squeezed state.
Fig. 4
Fig. 4 a) Theoretical noise in the measurement of the operator λQ in an ideal lossless truncated SU(1,1) interferometer as a function of λ. The zero on the vertical axis represents the shot noise of a single homodyne detector without any attenuation on its output, calculated by replacing the probe or the conjugate beam with vacuum. b) The value of λopt as a function of the 4WM gain at different optical transmissions of the probe (ηp) and the conjugate beams (ηc) in the interferometer, assuming ηp = ηc.
Fig. 5
Fig. 5 Theoretical improvement in the SNR as a function of λ with the measurement λQ over the conditions of SQL2 (SNRISQL2), and SQL1 (SNRISQL1) for a lossless truncated SU(1,1) interferometer. The measurement conditions are defined in the text and the different curves are for different values of the 4WM gain. Positive values represent improved SNR (and phase sensitivity) over the SQL as defined for those conditions.
Fig. 6
Fig. 6 a) Noise in the joint homodyne detection measuring λQ with squeezed light as a function of the attenuation parameter λ. b) Improvement in the SNR as a function of the attenuation λ with the measurements λQ over the conditions of SQL1 (SNRISQL1) and SQL2 (SNRISQL2). The left side plots have an estimated 4WM gain of 1.67, a probe transmission of 76% and a conjugate transmission of 79%. The right side plots have an estimated 4WM gain of 1.2, a probe transmission of 73% and a conjugate transmission of 76%. The gain and the loss values were estimated from the theoretical fit of the data. In these fits, we assume that the probe suffers 3% more loss than the conjugate beam, which we measure experimentally and occurs because the probe is closer to the absorption resonance of 85Rb.
Fig. 7
Fig. 7 λopt as a function of the 4WM gain. The points are experimental measurements determined from plots like those in Figs. 2a and 2b. A theoretical curve is generated using a probe beam transmission of 74.5% and a conjugate beam transmission of 77.5%. These values represent the typical losses in our system.

Equations (5)

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

SNR = ( ϕ M ) 2 ( δ ϕ ) 2 Δ 2 M ,
M ^ Q = ( X ^ p + X ^ c ) ,
M ^ λ Q = X ^ p + λ X ^ c ,
λ opt = η p η c sinh ( 2 r ) 1 η c + η c cosh ( 2 r ) ,
SNRI SQLi = SNR tSUI SNR SQLi = 10 Log 10 [ ( Δ ϕ SQLi Δ ϕ tSUI ) 2 ] ,

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