Abstract

We address high-precision measurements by active and passive interferometric schemes based on Gaussian states and operations. In particular, we look for the best states to be injected into their ports according to the quantum Cramér–Rao bound, i.e., maximizing the quantum Fisher information over all the involved parameters, given a constraint on the overall mean number of photons entering into the interferometer. We found that for passive interferometers involving only beam splitters, the optimal input leading to Heisenberg scaling is a pair of identical squeezed-coherent states with at most one-third of the total energy employed in squeezing. For active interferometers involving optical amplifiers, input coherent signals are enough to achieve Heisenberg scaling, given an optimal value of the amplification gain. For passive schemes our results clarify the role of squeezing in improving both the reference phase and the signal phase of an interferometer.

© 2015 Optical Society of America

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  1. R. X. Adhikari, “Gravitational radiation detection with laser interferometry,” Rev. Mod. Phys. 86, 121–151 (2014).
    [Crossref]
  2. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
    [Crossref]
  3. B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1, 1) interferometers,” Phys Rev. A 33, 4033–4054 (1986).
    [Crossref]
  4. M. J. Holland and K. Burnett, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. 71, 1355–1358 (1993).
    [Crossref]
  5. M. G. A. Paris, “Small amount of squeezing in high-sensitive realistic interferometry,” Phys. Lett. A 201, 132–138 (1995).
    [Crossref]
  6. 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]
  7. C. C. Gerry, “Heisenberg-limit interferometry with four-wave mixers operating in a nonlinear regime,” Phys. Rev. A 61, 043811 (2000).
    [Crossref]
  8. R. A. Campos, C. C. Gerry, and A. Benmoussa, “Optical interferometry at the Heisenberg limit with twin Fock states and parity measurements,” Phys. Rev. A 68, 023810 (2003).
    [Crossref]
  9. F. W. Sun, B. H. Liu, Y. X. Gong, Y. F. Huang, Z. Y. Ou, and G. C. Guo, “Experimental demonstration of phase measurement precision beating standard quantum limit by projection measurement,” EPL 82, 24001 (2008).
    [Crossref]
  10. L. Pezzé and A. Smerzi, “Mach-Zehnder interferometry at the Heisenberg limit with coherent and squeezed-vacuum light,” Phys. Rev. Lett. 100, 073601 (2008).
    [Crossref]
  11. M. Kacprowicz, R. Demkowicz-Dobrzanski, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Experimental quantum-enhanced estimation of a lossy phase shift,” Nat. Photonics 4, 357–360 (2010).
    [Crossref]
  12. H. Cable and G. A. Durkin, “Parameter estimation with entangled photons produced by parametric down-conversion,” Phys. Rev. Lett. 105, 013603 (2010).
    [Crossref]
  13. S. Knysh, V. N. Smelyanskiy, and G. A. Durkin, “Scaling laws for precision in quantum interferometry and the bifurcation landscape of the optimal state,” Phys. Rev. A 83, 021804(R) (2011).
    [Crossref]
  14. M. G. Genoni, S. Olivares, and M. G. A. Paris, “Optical phase estimation in the presence of phase diffusion,” Phys. Rev. Lett. 106, 153603 (2011).
    [Crossref]
  15. S. Pirandola, “Quantum reading of a classical digital memory,” Phys. Rev. Lett. 106, 090504 (2011).
    [Crossref]
  16. A. Bisio, M. Dall’Arno, and G. M. D’Ariano, “Tradeoff between energy and error in the discrimination of quantum-optical devices,” Phys. Rev. A 84, 012310 (2011).
    [Crossref]
  17. S. Pirandola, C. Lupo, V. Giovannetti, S. Mancini, and S. L. Braunstein, “Quantum reading capacity,” New J. Phys. 13, 113012 (2011).
    [Crossref]
  18. L. Pezzè, P. Hyllus, and A. Smerzi, “Phase-sensitivity bounds for two-mode interferometers,” Phys. Rev. A 91, 032103 (2015).
    [Crossref]
  19. J. P. Dowling, “Quantum optical metrology at the lowdown on high-N00N states,” Contemp. Phys. 49, 125–143 (2008).
    [Crossref]
  20. S. Olivares, “Quantum optics in the phase space,” EPJ-ST 203, 3–24 (2012).
    [Crossref]
  21. A. Monras, “Phase space formalism for quantum estimation of Gaussian states,” arXiv:1303.3682 (2013).
  22. O. Pinel, P. Jian, N. Treps, C. Fabre, and D. Braun, “Quantum parameter estimation using general single-mode Gaussian states,” Phys. Rev. A 88, 040102(R) (2013).
    [Crossref]
  23. Z. Jiang, “Quantum Fisher information for states in exponential form,” Phys. Rev. A 89, 032128 (2014).
    [Crossref]
  24. D. Šafránek, A. R. Lee, and I. Fuentes, “Quantum parameter estimation using multi-mode Gaussian states,” arXiv:1502.07924 (2015).
  25. G. Adesso, “Gaussian interferometric power,” Phys. Rev. A 90, 022321 (2014).
    [Crossref]
  26. M. D. Lang and C. M. Caves, “Optimal quantum-enhanced interferometry using a laser power source,” Phys. Rev. Lett. 111, 173601 (2013).
    [Crossref]
  27. C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, 1976).
  28. D. C. Brody and L. P. Hughston, “Statistical geometry in quantum mechanics,” Proc. R. Soc. London A 454, 2445–2475 (1998).
    [Crossref]
  29. D. C. Brody and L. P. Hughston, “Geometrization of statistical mechanics,” Proc. R. Soc. London A 455, 1683–1715 (1999).
    [Crossref]
  30. M. G. A. Paris, “Quantum estimation for quantum technology,” Int. J. Quantum Inform. 7, 125–137 (2009).
    [Crossref]
  31. S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439–3443 (1994).
    [Crossref]
  32. S. L. Braunstein, C. M. Caves, and G. J. Milburn, “Generalized uncertainty relations: Theory, examples, and Lorentz invariance,” Ann. Phys. 247, 135–173 (1996).
    [Crossref]
  33. S. Olivares and M. G. A. Paris, “Optimized interferometry with Gaussian states,” Opt. Spectrosc. 103, 231–236 (2007).
    [Crossref]
  34. S. Olivares and M. G. A. Paris, “Fidelity matters: The birth of entanglement in the mixing of Gaussian states,” Phys. Rev. Lett 107, 170505 (2011).
    [Crossref]
  35. A. Monras, “Optimal phase measurements with pure Gaussian states,” Phys. Rev. A 73, 033821 (2006).
    [Crossref]
  36. M. Jarzyna and R. Demkowicz-Dobrzański, “Quantum interferometry with and without an external phase reference,” Phys. Rev. A 85, 011801(R) (2012).
  37. Since the squeezing of the states is the same, after the evolution we still have factorized states, otherwise correlations will arise reducing the QFI, since now the single mode involved in the phase shift is no longer a pure state. It is worth noting that in the single-mode case one has Hsm>Hmax, but it is not possible to fix a phase reference.
  38. J. Abadie, B. P. Abbott, R. Abbott, R. Adhikari, S. B. Anderson, K. Arai, M. C. Araya, J. Betzwieser, G. Billingsley, E. Black, J. K. Blackburn, R. Bork, A. F. Brooks, C. Cepeda, T. Chalermsongsak, A. Corsi, D. C. Coyne, R. Dannenberg, B. Daudert, V. Dergachev, R. DeSalvo, J. C. Driggers, P. Ehrens, R. Engel, T. Etzel, N. Fotopoulos, E. K. Gustafson, C. Hanna, J. Harms, J. Heefner, A. W. Heptonstall, K. A. Hodge, A. Ivanov, M. Jacobson, P. Kalmus, W. Kells, P. J. King, V. Kondrashov, W. Z. Korth, D. Kozak, A. Lazzarini, P. E. Lindquist, M. Mageswaran, K. Mailand, E. Maros, J. N. Marx, G. McIntyre, S. Meshkov, T. Nash, G. H. Ogin, C. Osthelder, P. Ajith, P. Patel, M. Pedraza, M. Phelps, L. R. Price, S. Privitera, N. A. Robertson, V. Sannibale, L. Santamaría, A. C. Searle, F. Seifert, A. S. Sengupta, A. Singer, L. Singer, M. R. Smith, A. Stochino, R. Taylor, C. I. Torrie, S. Vass, A. E. Villar, L. Wallace, R. L. Ward, A. J. Weinstein, S. E. Whitcomb, P. A. Willems, H. Yamamoto, D. Yeaton-Massey, L. Zhang, and J. Zweizig (the LIGO Scientific Collaboration), “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962–965 (2011).
    [Crossref]
  39. K. McKenzie, D. Shaddock, D. McClelland, B. Buchler, and P. Lam, “Experimental demonstration of a squeezing-enhanced power-recycled Michelson interferometer for gravitational wave detection,” Phys. Rev. Lett. 88, 231102 (2002).
    [Crossref]
  40. N. P. William, J. P. Dowling, and G. S. Agarwal, “Coherent-light-boosted, sub-shot noise, quantum interferometry,” New J. Phys. 12, 083014 (2010).
    [Crossref]
  41. A. Ferraro, M. G. A. Paris, A. Allevi, A. Andreoni, M. Bondani, and E. Puddu, “Three-mode entanglement by interlinked nonlinear interactions in optical χ(2) media,” J. Opt. Soc. Am. B 21, 1241 (2004).
    [Crossref]
  42. A. Allevi, A. Andreoni, M. Bondani, E. Puddu, A. Ferraro, and M. G. A. Paris, “Properties of two interlinked χ(2) interactions in non-collinear phase-matching,” Opt. Lett. 29, 180 (2004).
    [Crossref]
  43. A. Ferraro and M. G. A. Paris, “Multimode entanglement and telecloning in a noisy environment,” Phys. Rev. A 72, 032312 (2005).
    [Crossref]
  44. C. Sparaciari, S. Olivares, and M. G. A. Paris, “Bounds to precision in realistic Gaussian interferometry,” in preparation.
  45. M. D. Lang and C. M. Caves, “Optimal quantum-enhanced interferometry,” Phys. Rev. A 90, 025802 (2014).
    [Crossref]
  46. R. Demkowicz-Dobrzanski, K. Banaszek, and R. Schnabel, “Fundamental quantum interferometry bound for the squeezed-light-enhanced gravitational wave detector GEO 600,” Phys. Rev. A 88, 041802(R) (2013).
    [Crossref]
  47. L. Pezzé, A. Smerzi, G. Khoury, J. F. Hodelin, and D. Bouwmeester, “Phase detection at the quantum limit with multiphoton Mach–Zehnder interferometry,” Phys. Rev. Lett. 99, 223602 (2007).
    [Crossref]
  48. S. Olivares and M. G. A. Paris, “Bayesian estimation in homodyne interferometry,” J. Phys. B 42, 055506 (2009).
    [Crossref]
  49. A. S. Lane, S. L. Braunstein, and C. M. Caves, “Maximum-likelihood statistics of multiple quantum phase measurements,” Phys. Rev. A 47, 1667–1696 (1993).
    [Crossref]
  50. Z. Hradil, R. Myška, T. Opatrný, and J. Bajer, “Entropy of phase measurement: Quantum phase via quadrature measurement,” Phys. Rev. A 53, 3738–3742 (1996).
    [Crossref]

2015 (1)

L. Pezzè, P. Hyllus, and A. Smerzi, “Phase-sensitivity bounds for two-mode interferometers,” Phys. Rev. A 91, 032103 (2015).
[Crossref]

2014 (4)

Z. Jiang, “Quantum Fisher information for states in exponential form,” Phys. Rev. A 89, 032128 (2014).
[Crossref]

G. Adesso, “Gaussian interferometric power,” Phys. Rev. A 90, 022321 (2014).
[Crossref]

R. X. Adhikari, “Gravitational radiation detection with laser interferometry,” Rev. Mod. Phys. 86, 121–151 (2014).
[Crossref]

M. D. Lang and C. M. Caves, “Optimal quantum-enhanced interferometry,” Phys. Rev. A 90, 025802 (2014).
[Crossref]

2013 (3)

R. Demkowicz-Dobrzanski, K. Banaszek, and R. Schnabel, “Fundamental quantum interferometry bound for the squeezed-light-enhanced gravitational wave detector GEO 600,” Phys. Rev. A 88, 041802(R) (2013).
[Crossref]

M. D. Lang and C. M. Caves, “Optimal quantum-enhanced interferometry using a laser power source,” Phys. Rev. Lett. 111, 173601 (2013).
[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, 040102(R) (2013).
[Crossref]

2012 (2)

S. Olivares, “Quantum optics in the phase space,” EPJ-ST 203, 3–24 (2012).
[Crossref]

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

2011 (7)

J. Abadie, B. P. Abbott, R. Abbott, R. Adhikari, S. B. Anderson, K. Arai, M. C. Araya, J. Betzwieser, G. Billingsley, E. Black, J. K. Blackburn, R. Bork, A. F. Brooks, C. Cepeda, T. Chalermsongsak, A. Corsi, D. C. Coyne, R. Dannenberg, B. Daudert, V. Dergachev, R. DeSalvo, J. C. Driggers, P. Ehrens, R. Engel, T. Etzel, N. Fotopoulos, E. K. Gustafson, C. Hanna, J. Harms, J. Heefner, A. W. Heptonstall, K. A. Hodge, A. Ivanov, M. Jacobson, P. Kalmus, W. Kells, P. J. King, V. Kondrashov, W. Z. Korth, D. Kozak, A. Lazzarini, P. E. Lindquist, M. Mageswaran, K. Mailand, E. Maros, J. N. Marx, G. McIntyre, S. Meshkov, T. Nash, G. H. Ogin, C. Osthelder, P. Ajith, P. Patel, M. Pedraza, M. Phelps, L. R. Price, S. Privitera, N. A. Robertson, V. Sannibale, L. Santamaría, A. C. Searle, F. Seifert, A. S. Sengupta, A. Singer, L. Singer, M. R. Smith, A. Stochino, R. Taylor, C. I. Torrie, S. Vass, A. E. Villar, L. Wallace, R. L. Ward, A. J. Weinstein, S. E. Whitcomb, P. A. Willems, H. Yamamoto, D. Yeaton-Massey, L. Zhang, and J. Zweizig (the LIGO Scientific Collaboration), “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962–965 (2011).
[Crossref]

S. Olivares and M. G. A. Paris, “Fidelity matters: The birth of entanglement in the mixing of Gaussian states,” Phys. Rev. Lett 107, 170505 (2011).
[Crossref]

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

M. G. Genoni, S. Olivares, and M. G. A. Paris, “Optical phase estimation in the presence of phase diffusion,” Phys. Rev. Lett. 106, 153603 (2011).
[Crossref]

S. Pirandola, “Quantum reading of a classical digital memory,” Phys. Rev. Lett. 106, 090504 (2011).
[Crossref]

A. Bisio, M. Dall’Arno, and G. M. D’Ariano, “Tradeoff between energy and error in the discrimination of quantum-optical devices,” Phys. Rev. A 84, 012310 (2011).
[Crossref]

S. Pirandola, C. Lupo, V. Giovannetti, S. Mancini, and S. L. Braunstein, “Quantum reading capacity,” New J. Phys. 13, 113012 (2011).
[Crossref]

2010 (3)

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

H. Cable and G. A. Durkin, “Parameter estimation with entangled photons produced by parametric down-conversion,” Phys. Rev. Lett. 105, 013603 (2010).
[Crossref]

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

2009 (2)

S. Olivares and M. G. A. Paris, “Bayesian estimation in homodyne interferometry,” J. Phys. B 42, 055506 (2009).
[Crossref]

M. G. A. Paris, “Quantum estimation for quantum technology,” Int. J. Quantum Inform. 7, 125–137 (2009).
[Crossref]

2008 (3)

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

F. W. Sun, B. H. Liu, Y. X. Gong, Y. F. Huang, Z. Y. Ou, and G. C. Guo, “Experimental demonstration of phase measurement precision beating standard quantum limit by projection measurement,” EPL 82, 24001 (2008).
[Crossref]

L. Pezzé and A. Smerzi, “Mach-Zehnder interferometry at the Heisenberg limit with coherent and squeezed-vacuum light,” Phys. Rev. Lett. 100, 073601 (2008).
[Crossref]

2007 (2)

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

S. Olivares and M. G. A. Paris, “Optimized interferometry with Gaussian states,” Opt. Spectrosc. 103, 231–236 (2007).
[Crossref]

2006 (1)

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

2005 (1)

A. Ferraro and M. G. A. Paris, “Multimode entanglement and telecloning in a noisy environment,” Phys. Rev. A 72, 032312 (2005).
[Crossref]

2004 (2)

2003 (1)

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

2002 (1)

K. McKenzie, D. Shaddock, D. McClelland, B. Buchler, and P. Lam, “Experimental demonstration of a squeezing-enhanced power-recycled Michelson interferometer for gravitational wave detection,” Phys. Rev. Lett. 88, 231102 (2002).
[Crossref]

2000 (1)

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

1999 (1)

D. C. Brody and L. P. Hughston, “Geometrization of statistical mechanics,” Proc. R. Soc. London A 455, 1683–1715 (1999).
[Crossref]

1998 (2)

D. C. Brody and L. P. Hughston, “Statistical geometry in quantum mechanics,” Proc. R. Soc. London A 454, 2445–2475 (1998).
[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]

1996 (2)

S. L. Braunstein, C. M. Caves, and G. J. Milburn, “Generalized uncertainty relations: Theory, examples, and Lorentz invariance,” Ann. Phys. 247, 135–173 (1996).
[Crossref]

Z. Hradil, R. Myška, T. Opatrný, and J. Bajer, “Entropy of phase measurement: Quantum phase via quadrature measurement,” Phys. Rev. A 53, 3738–3742 (1996).
[Crossref]

1995 (1)

M. G. A. Paris, “Small amount of squeezing in high-sensitive realistic interferometry,” Phys. Lett. A 201, 132–138 (1995).
[Crossref]

1994 (1)

S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439–3443 (1994).
[Crossref]

1993 (2)

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

A. S. Lane, S. L. Braunstein, and C. M. Caves, “Maximum-likelihood statistics of multiple quantum phase measurements,” Phys. Rev. A 47, 1667–1696 (1993).
[Crossref]

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, 1693–1708 (1981).
[Crossref]

Abadie, J.

J. Abadie, B. P. Abbott, R. Abbott, R. Adhikari, S. B. Anderson, K. Arai, M. C. Araya, J. Betzwieser, G. Billingsley, E. Black, J. K. Blackburn, R. Bork, A. F. Brooks, C. Cepeda, T. Chalermsongsak, A. Corsi, D. C. Coyne, R. Dannenberg, B. Daudert, V. Dergachev, R. DeSalvo, J. C. Driggers, P. Ehrens, R. Engel, T. Etzel, N. Fotopoulos, E. K. Gustafson, C. Hanna, J. Harms, J. Heefner, A. W. Heptonstall, K. A. Hodge, A. Ivanov, M. Jacobson, P. Kalmus, W. Kells, P. J. King, V. Kondrashov, W. Z. Korth, D. Kozak, A. Lazzarini, P. E. Lindquist, M. Mageswaran, K. Mailand, E. Maros, J. N. Marx, G. McIntyre, S. Meshkov, T. Nash, G. H. Ogin, C. Osthelder, P. Ajith, P. Patel, M. Pedraza, M. Phelps, L. R. Price, S. Privitera, N. A. Robertson, V. Sannibale, L. Santamaría, A. C. Searle, F. Seifert, A. S. Sengupta, A. Singer, L. Singer, M. R. Smith, A. Stochino, R. Taylor, C. I. Torrie, S. Vass, A. E. Villar, L. Wallace, R. L. Ward, A. J. Weinstein, S. E. Whitcomb, P. A. Willems, H. Yamamoto, D. Yeaton-Massey, L. Zhang, and J. Zweizig (the LIGO Scientific Collaboration), “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962–965 (2011).
[Crossref]

Abbott, B. P.

J. Abadie, B. P. Abbott, R. Abbott, R. Adhikari, S. B. Anderson, K. Arai, M. C. Araya, J. Betzwieser, G. Billingsley, E. Black, J. K. Blackburn, R. Bork, A. F. Brooks, C. Cepeda, T. Chalermsongsak, A. Corsi, D. C. Coyne, R. Dannenberg, B. Daudert, V. Dergachev, R. DeSalvo, J. C. Driggers, P. Ehrens, R. Engel, T. Etzel, N. Fotopoulos, E. K. Gustafson, C. Hanna, J. Harms, J. Heefner, A. W. Heptonstall, K. A. Hodge, A. Ivanov, M. Jacobson, P. Kalmus, W. Kells, P. J. King, V. Kondrashov, W. Z. Korth, D. Kozak, A. Lazzarini, P. E. Lindquist, M. Mageswaran, K. Mailand, E. Maros, J. N. Marx, G. McIntyre, S. Meshkov, T. Nash, G. H. Ogin, C. Osthelder, P. Ajith, P. Patel, M. Pedraza, M. Phelps, L. R. Price, S. Privitera, N. A. Robertson, V. Sannibale, L. Santamaría, A. C. Searle, F. Seifert, A. S. Sengupta, A. Singer, L. Singer, M. R. Smith, A. Stochino, R. Taylor, C. I. Torrie, S. Vass, A. E. Villar, L. Wallace, R. L. Ward, A. J. Weinstein, S. E. Whitcomb, P. A. Willems, H. Yamamoto, D. Yeaton-Massey, L. Zhang, and J. Zweizig (the LIGO Scientific Collaboration), “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962–965 (2011).
[Crossref]

Abbott, R.

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J. Abadie, B. P. Abbott, R. Abbott, R. Adhikari, S. B. Anderson, K. Arai, M. C. Araya, J. Betzwieser, G. Billingsley, E. Black, J. K. Blackburn, R. Bork, A. F. Brooks, C. Cepeda, T. Chalermsongsak, A. Corsi, D. C. Coyne, R. Dannenberg, B. Daudert, V. Dergachev, R. DeSalvo, J. C. Driggers, P. Ehrens, R. Engel, T. Etzel, N. Fotopoulos, E. K. Gustafson, C. Hanna, J. Harms, J. Heefner, A. W. Heptonstall, K. A. Hodge, A. Ivanov, M. Jacobson, P. Kalmus, W. Kells, P. J. King, V. Kondrashov, W. Z. Korth, D. Kozak, A. Lazzarini, P. E. Lindquist, M. Mageswaran, K. Mailand, E. Maros, J. N. Marx, G. McIntyre, S. Meshkov, T. Nash, G. H. Ogin, C. Osthelder, P. Ajith, P. Patel, M. Pedraza, M. Phelps, L. R. Price, S. Privitera, N. A. Robertson, V. Sannibale, L. Santamaría, A. C. Searle, F. Seifert, A. S. Sengupta, A. Singer, L. Singer, M. R. Smith, A. Stochino, R. Taylor, C. I. Torrie, S. Vass, A. E. Villar, L. Wallace, R. L. Ward, A. J. Weinstein, S. E. Whitcomb, P. A. Willems, H. Yamamoto, D. Yeaton-Massey, L. Zhang, and J. Zweizig (the LIGO Scientific Collaboration), “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962–965 (2011).
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Mancini, S.

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Maros, E.

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McIntyre, G.

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Osthelder, C.

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Patel, P.

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Phelps, M.

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Privitera, S.

J. Abadie, B. P. Abbott, R. Abbott, R. Adhikari, S. B. Anderson, K. Arai, M. C. Araya, J. Betzwieser, G. Billingsley, E. Black, J. K. Blackburn, R. Bork, A. F. Brooks, C. Cepeda, T. Chalermsongsak, A. Corsi, D. C. Coyne, R. Dannenberg, B. Daudert, V. Dergachev, R. DeSalvo, J. C. Driggers, P. Ehrens, R. Engel, T. Etzel, N. Fotopoulos, E. K. Gustafson, C. Hanna, J. Harms, J. Heefner, A. W. Heptonstall, K. A. Hodge, A. Ivanov, M. Jacobson, P. Kalmus, W. Kells, P. J. King, V. Kondrashov, W. Z. Korth, D. Kozak, A. Lazzarini, P. E. Lindquist, M. Mageswaran, K. Mailand, E. Maros, J. N. Marx, G. McIntyre, S. Meshkov, T. Nash, G. H. Ogin, C. Osthelder, P. Ajith, P. Patel, M. Pedraza, M. Phelps, L. R. Price, S. Privitera, N. A. Robertson, V. Sannibale, L. Santamaría, A. C. Searle, F. Seifert, A. S. Sengupta, A. Singer, L. Singer, M. R. Smith, A. Stochino, R. Taylor, C. I. Torrie, S. Vass, A. E. Villar, L. Wallace, R. L. Ward, A. J. Weinstein, S. E. Whitcomb, P. A. Willems, H. Yamamoto, D. Yeaton-Massey, L. Zhang, and J. Zweizig (the LIGO Scientific Collaboration), “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962–965 (2011).
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Puddu, E.

Robertson, N. A.

J. Abadie, B. P. Abbott, R. Abbott, R. Adhikari, S. B. Anderson, K. Arai, M. C. Araya, J. Betzwieser, G. Billingsley, E. Black, J. K. Blackburn, R. Bork, A. F. Brooks, C. Cepeda, T. Chalermsongsak, A. Corsi, D. C. Coyne, R. Dannenberg, B. Daudert, V. Dergachev, R. DeSalvo, J. C. Driggers, P. Ehrens, R. Engel, T. Etzel, N. Fotopoulos, E. K. Gustafson, C. Hanna, J. Harms, J. Heefner, A. W. Heptonstall, K. A. Hodge, A. Ivanov, M. Jacobson, P. Kalmus, W. Kells, P. J. King, V. Kondrashov, W. Z. Korth, D. Kozak, A. Lazzarini, P. E. Lindquist, M. Mageswaran, K. Mailand, E. Maros, J. N. Marx, G. McIntyre, S. Meshkov, T. Nash, G. H. Ogin, C. Osthelder, P. Ajith, P. Patel, M. Pedraza, M. Phelps, L. R. Price, S. Privitera, N. A. Robertson, V. Sannibale, L. Santamaría, A. C. Searle, F. Seifert, A. S. Sengupta, A. Singer, L. Singer, M. R. Smith, A. Stochino, R. Taylor, C. I. Torrie, S. Vass, A. E. Villar, L. Wallace, R. L. Ward, A. J. Weinstein, S. E. Whitcomb, P. A. Willems, H. Yamamoto, D. Yeaton-Massey, L. Zhang, and J. Zweizig (the LIGO Scientific Collaboration), “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962–965 (2011).
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Šafránek, D.

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Sannibale, V.

J. Abadie, B. P. Abbott, R. Abbott, R. Adhikari, S. B. Anderson, K. Arai, M. C. Araya, J. Betzwieser, G. Billingsley, E. Black, J. K. Blackburn, R. Bork, A. F. Brooks, C. Cepeda, T. Chalermsongsak, A. Corsi, D. C. Coyne, R. Dannenberg, B. Daudert, V. Dergachev, R. DeSalvo, J. C. Driggers, P. Ehrens, R. Engel, T. Etzel, N. Fotopoulos, E. K. Gustafson, C. Hanna, J. Harms, J. Heefner, A. W. Heptonstall, K. A. Hodge, A. Ivanov, M. Jacobson, P. Kalmus, W. Kells, P. J. King, V. Kondrashov, W. Z. Korth, D. Kozak, A. Lazzarini, P. E. Lindquist, M. Mageswaran, K. Mailand, E. Maros, J. N. Marx, G. McIntyre, S. Meshkov, T. Nash, G. H. Ogin, C. Osthelder, P. Ajith, P. Patel, M. Pedraza, M. Phelps, L. R. Price, S. Privitera, N. A. Robertson, V. Sannibale, L. Santamaría, A. C. Searle, F. Seifert, A. S. Sengupta, A. Singer, L. Singer, M. R. Smith, A. Stochino, R. Taylor, C. I. Torrie, S. Vass, A. E. Villar, L. Wallace, R. L. Ward, A. J. Weinstein, S. E. Whitcomb, P. A. Willems, H. Yamamoto, D. Yeaton-Massey, L. Zhang, and J. Zweizig (the LIGO Scientific Collaboration), “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962–965 (2011).
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Santamaría, L.

J. Abadie, B. P. Abbott, R. Abbott, R. Adhikari, S. B. Anderson, K. Arai, M. C. Araya, J. Betzwieser, G. Billingsley, E. Black, J. K. Blackburn, R. Bork, A. F. Brooks, C. Cepeda, T. Chalermsongsak, A. Corsi, D. C. Coyne, R. Dannenberg, B. Daudert, V. Dergachev, R. DeSalvo, J. C. Driggers, P. Ehrens, R. Engel, T. Etzel, N. Fotopoulos, E. K. Gustafson, C. Hanna, J. Harms, J. Heefner, A. W. Heptonstall, K. A. Hodge, A. Ivanov, M. Jacobson, P. Kalmus, W. Kells, P. J. King, V. Kondrashov, W. Z. Korth, D. Kozak, A. Lazzarini, P. E. Lindquist, M. Mageswaran, K. Mailand, E. Maros, J. N. Marx, G. McIntyre, S. Meshkov, T. Nash, G. H. Ogin, C. Osthelder, P. Ajith, P. Patel, M. Pedraza, M. Phelps, L. R. Price, S. Privitera, N. A. Robertson, V. Sannibale, L. Santamaría, A. C. Searle, F. Seifert, A. S. Sengupta, A. Singer, L. Singer, M. R. Smith, A. Stochino, R. Taylor, C. I. Torrie, S. Vass, A. E. Villar, L. Wallace, R. L. Ward, A. J. Weinstein, S. E. Whitcomb, P. A. Willems, H. Yamamoto, D. Yeaton-Massey, L. Zhang, and J. Zweizig (the LIGO Scientific Collaboration), “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962–965 (2011).
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Schnabel, R.

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Searle, A. C.

J. Abadie, B. P. Abbott, R. Abbott, R. Adhikari, S. B. Anderson, K. Arai, M. C. Araya, J. Betzwieser, G. Billingsley, E. Black, J. K. Blackburn, R. Bork, A. F. Brooks, C. Cepeda, T. Chalermsongsak, A. Corsi, D. C. Coyne, R. Dannenberg, B. Daudert, V. Dergachev, R. DeSalvo, J. C. Driggers, P. Ehrens, R. Engel, T. Etzel, N. Fotopoulos, E. K. Gustafson, C. Hanna, J. Harms, J. Heefner, A. W. Heptonstall, K. A. Hodge, A. Ivanov, M. Jacobson, P. Kalmus, W. Kells, P. J. King, V. Kondrashov, W. Z. Korth, D. Kozak, A. Lazzarini, P. E. Lindquist, M. Mageswaran, K. Mailand, E. Maros, J. N. Marx, G. McIntyre, S. Meshkov, T. Nash, G. H. Ogin, C. Osthelder, P. Ajith, P. Patel, M. Pedraza, M. Phelps, L. R. Price, S. Privitera, N. A. Robertson, V. Sannibale, L. Santamaría, A. C. Searle, F. Seifert, A. S. Sengupta, A. Singer, L. Singer, M. R. Smith, A. Stochino, R. Taylor, C. I. Torrie, S. Vass, A. E. Villar, L. Wallace, R. L. Ward, A. J. Weinstein, S. E. Whitcomb, P. A. Willems, H. Yamamoto, D. Yeaton-Massey, L. Zhang, and J. Zweizig (the LIGO Scientific Collaboration), “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962–965 (2011).
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Seifert, F.

J. Abadie, B. P. Abbott, R. Abbott, R. Adhikari, S. B. Anderson, K. Arai, M. C. Araya, J. Betzwieser, G. Billingsley, E. Black, J. K. Blackburn, R. Bork, A. F. Brooks, C. Cepeda, T. Chalermsongsak, A. Corsi, D. C. Coyne, R. Dannenberg, B. Daudert, V. Dergachev, R. DeSalvo, J. C. Driggers, P. Ehrens, R. Engel, T. Etzel, N. Fotopoulos, E. K. Gustafson, C. Hanna, J. Harms, J. Heefner, A. W. Heptonstall, K. A. Hodge, A. Ivanov, M. Jacobson, P. Kalmus, W. Kells, P. J. King, V. Kondrashov, W. Z. Korth, D. Kozak, A. Lazzarini, P. E. Lindquist, M. Mageswaran, K. Mailand, E. Maros, J. N. Marx, G. McIntyre, S. Meshkov, T. Nash, G. H. Ogin, C. Osthelder, P. Ajith, P. Patel, M. Pedraza, M. Phelps, L. R. Price, S. Privitera, N. A. Robertson, V. Sannibale, L. Santamaría, A. C. Searle, F. Seifert, A. S. Sengupta, A. Singer, L. Singer, M. R. Smith, A. Stochino, R. Taylor, C. I. Torrie, S. Vass, A. E. Villar, L. Wallace, R. L. Ward, A. J. Weinstein, S. E. Whitcomb, P. A. Willems, H. Yamamoto, D. Yeaton-Massey, L. Zhang, and J. Zweizig (the LIGO Scientific Collaboration), “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962–965 (2011).
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Sengupta, A. S.

J. Abadie, B. P. Abbott, R. Abbott, R. Adhikari, S. B. Anderson, K. Arai, M. C. Araya, J. Betzwieser, G. Billingsley, E. Black, J. K. Blackburn, R. Bork, A. F. Brooks, C. Cepeda, T. Chalermsongsak, A. Corsi, D. C. Coyne, R. Dannenberg, B. Daudert, V. Dergachev, R. DeSalvo, J. C. Driggers, P. Ehrens, R. Engel, T. Etzel, N. Fotopoulos, E. K. Gustafson, C. Hanna, J. Harms, J. Heefner, A. W. Heptonstall, K. A. Hodge, A. Ivanov, M. Jacobson, P. Kalmus, W. Kells, P. J. King, V. Kondrashov, W. Z. Korth, D. Kozak, A. Lazzarini, P. E. Lindquist, M. Mageswaran, K. Mailand, E. Maros, J. N. Marx, G. McIntyre, S. Meshkov, T. Nash, G. H. Ogin, C. Osthelder, P. Ajith, P. Patel, M. Pedraza, M. Phelps, L. R. Price, S. Privitera, N. A. Robertson, V. Sannibale, L. Santamaría, A. C. Searle, F. Seifert, A. S. Sengupta, A. Singer, L. Singer, M. R. Smith, A. Stochino, R. Taylor, C. I. Torrie, S. Vass, A. E. Villar, L. Wallace, R. L. Ward, A. J. Weinstein, S. E. Whitcomb, P. A. Willems, H. Yamamoto, D. Yeaton-Massey, L. Zhang, and J. Zweizig (the LIGO Scientific Collaboration), “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962–965 (2011).
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Shaddock, D.

K. McKenzie, D. Shaddock, D. McClelland, B. Buchler, and P. Lam, “Experimental demonstration of a squeezing-enhanced power-recycled Michelson interferometer for gravitational wave detection,” Phys. Rev. Lett. 88, 231102 (2002).
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Singer, A.

J. Abadie, B. P. Abbott, R. Abbott, R. Adhikari, S. B. Anderson, K. Arai, M. C. Araya, J. Betzwieser, G. Billingsley, E. Black, J. K. Blackburn, R. Bork, A. F. Brooks, C. Cepeda, T. Chalermsongsak, A. Corsi, D. C. Coyne, R. Dannenberg, B. Daudert, V. Dergachev, R. DeSalvo, J. C. Driggers, P. Ehrens, R. Engel, T. Etzel, N. Fotopoulos, E. K. Gustafson, C. Hanna, J. Harms, J. Heefner, A. W. Heptonstall, K. A. Hodge, A. Ivanov, M. Jacobson, P. Kalmus, W. Kells, P. J. King, V. Kondrashov, W. Z. Korth, D. Kozak, A. Lazzarini, P. E. Lindquist, M. Mageswaran, K. Mailand, E. Maros, J. N. Marx, G. McIntyre, S. Meshkov, T. Nash, G. H. Ogin, C. Osthelder, P. Ajith, P. Patel, M. Pedraza, M. Phelps, L. R. Price, S. Privitera, N. A. Robertson, V. Sannibale, L. Santamaría, A. C. Searle, F. Seifert, A. S. Sengupta, A. Singer, L. Singer, M. R. Smith, A. Stochino, R. Taylor, C. I. Torrie, S. Vass, A. E. Villar, L. Wallace, R. L. Ward, A. J. Weinstein, S. E. Whitcomb, P. A. Willems, H. Yamamoto, D. Yeaton-Massey, L. Zhang, and J. Zweizig (the LIGO Scientific Collaboration), “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962–965 (2011).
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Singer, L.

J. Abadie, B. P. Abbott, R. Abbott, R. Adhikari, S. B. Anderson, K. Arai, M. C. Araya, J. Betzwieser, G. Billingsley, E. Black, J. K. Blackburn, R. Bork, A. F. Brooks, C. Cepeda, T. Chalermsongsak, A. Corsi, D. C. Coyne, R. Dannenberg, B. Daudert, V. Dergachev, R. DeSalvo, J. C. Driggers, P. Ehrens, R. Engel, T. Etzel, N. Fotopoulos, E. K. Gustafson, C. Hanna, J. Harms, J. Heefner, A. W. Heptonstall, K. A. Hodge, A. Ivanov, M. Jacobson, P. Kalmus, W. Kells, P. J. King, V. Kondrashov, W. Z. Korth, D. Kozak, A. Lazzarini, P. E. Lindquist, M. Mageswaran, K. Mailand, E. Maros, J. N. Marx, G. McIntyre, S. Meshkov, T. Nash, G. H. Ogin, C. Osthelder, P. Ajith, P. Patel, M. Pedraza, M. Phelps, L. R. Price, S. Privitera, N. A. Robertson, V. Sannibale, L. Santamaría, A. C. Searle, F. Seifert, A. S. Sengupta, A. Singer, L. Singer, M. R. Smith, A. Stochino, R. Taylor, C. I. Torrie, S. Vass, A. E. Villar, L. Wallace, R. L. Ward, A. J. Weinstein, S. E. Whitcomb, P. A. Willems, H. Yamamoto, D. Yeaton-Massey, L. Zhang, and J. Zweizig (the LIGO Scientific Collaboration), “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962–965 (2011).
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Smelyanskiy, V. N.

S. Knysh, V. N. Smelyanskiy, and G. A. Durkin, “Scaling laws for precision in quantum interferometry and the bifurcation landscape of the optimal state,” Phys. Rev. A 83, 021804(R) (2011).
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L. Pezzè, P. Hyllus, and A. Smerzi, “Phase-sensitivity bounds for two-mode interferometers,” Phys. Rev. A 91, 032103 (2015).
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Smith, M. R.

J. Abadie, B. P. Abbott, R. Abbott, R. Adhikari, S. B. Anderson, K. Arai, M. C. Araya, J. Betzwieser, G. Billingsley, E. Black, J. K. Blackburn, R. Bork, A. F. Brooks, C. Cepeda, T. Chalermsongsak, A. Corsi, D. C. Coyne, R. Dannenberg, B. Daudert, V. Dergachev, R. DeSalvo, J. C. Driggers, P. Ehrens, R. Engel, T. Etzel, N. Fotopoulos, E. K. Gustafson, C. Hanna, J. Harms, J. Heefner, A. W. Heptonstall, K. A. Hodge, A. Ivanov, M. Jacobson, P. Kalmus, W. Kells, P. J. King, V. Kondrashov, W. Z. Korth, D. Kozak, A. Lazzarini, P. E. Lindquist, M. Mageswaran, K. Mailand, E. Maros, J. N. Marx, G. McIntyre, S. Meshkov, T. Nash, G. H. Ogin, C. Osthelder, P. Ajith, P. Patel, M. Pedraza, M. Phelps, L. R. Price, S. Privitera, N. A. Robertson, V. Sannibale, L. Santamaría, A. C. Searle, F. Seifert, A. S. Sengupta, A. Singer, L. Singer, M. R. Smith, A. Stochino, R. Taylor, C. I. Torrie, S. Vass, A. E. Villar, L. Wallace, R. L. Ward, A. J. Weinstein, S. E. Whitcomb, P. A. Willems, H. Yamamoto, D. Yeaton-Massey, L. Zhang, and J. Zweizig (the LIGO Scientific Collaboration), “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962–965 (2011).
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Sparaciari, C.

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Bounds to precision in realistic Gaussian interferometry,” in preparation.

Stochino, A.

J. Abadie, B. P. Abbott, R. Abbott, R. Adhikari, S. B. Anderson, K. Arai, M. C. Araya, J. Betzwieser, G. Billingsley, E. Black, J. K. Blackburn, R. Bork, A. F. Brooks, C. Cepeda, T. Chalermsongsak, A. Corsi, D. C. Coyne, R. Dannenberg, B. Daudert, V. Dergachev, R. DeSalvo, J. C. Driggers, P. Ehrens, R. Engel, T. Etzel, N. Fotopoulos, E. K. Gustafson, C. Hanna, J. Harms, J. Heefner, A. W. Heptonstall, K. A. Hodge, A. Ivanov, M. Jacobson, P. Kalmus, W. Kells, P. J. King, V. Kondrashov, W. Z. Korth, D. Kozak, A. Lazzarini, P. E. Lindquist, M. Mageswaran, K. Mailand, E. Maros, J. N. Marx, G. McIntyre, S. Meshkov, T. Nash, G. H. Ogin, C. Osthelder, P. Ajith, P. Patel, M. Pedraza, M. Phelps, L. R. Price, S. Privitera, N. A. Robertson, V. Sannibale, L. Santamaría, A. C. Searle, F. Seifert, A. S. Sengupta, A. Singer, L. Singer, M. R. Smith, A. Stochino, R. Taylor, C. I. Torrie, S. Vass, A. E. Villar, L. Wallace, R. L. Ward, A. J. Weinstein, S. E. Whitcomb, P. A. Willems, H. Yamamoto, D. Yeaton-Massey, L. Zhang, and J. Zweizig (the LIGO Scientific Collaboration), “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962–965 (2011).
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Sun, F. W.

F. W. Sun, B. H. Liu, Y. X. Gong, Y. F. Huang, Z. Y. Ou, and G. C. Guo, “Experimental demonstration of phase measurement precision beating standard quantum limit by projection measurement,” EPL 82, 24001 (2008).
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Taylor, R.

J. Abadie, B. P. Abbott, R. Abbott, R. Adhikari, S. B. Anderson, K. Arai, M. C. Araya, J. Betzwieser, G. Billingsley, E. Black, J. K. Blackburn, R. Bork, A. F. Brooks, C. Cepeda, T. Chalermsongsak, A. Corsi, D. C. Coyne, R. Dannenberg, B. Daudert, V. Dergachev, R. DeSalvo, J. C. Driggers, P. Ehrens, R. Engel, T. Etzel, N. Fotopoulos, E. K. Gustafson, C. Hanna, J. Harms, J. Heefner, A. W. Heptonstall, K. A. Hodge, A. Ivanov, M. Jacobson, P. Kalmus, W. Kells, P. J. King, V. Kondrashov, W. Z. Korth, D. Kozak, A. Lazzarini, P. E. Lindquist, M. Mageswaran, K. Mailand, E. Maros, J. N. Marx, G. McIntyre, S. Meshkov, T. Nash, G. H. Ogin, C. Osthelder, P. Ajith, P. Patel, M. Pedraza, M. Phelps, L. R. Price, S. Privitera, N. A. Robertson, V. Sannibale, L. Santamaría, A. C. Searle, F. Seifert, A. S. Sengupta, A. Singer, L. Singer, M. R. Smith, A. Stochino, R. Taylor, C. I. Torrie, S. Vass, A. E. Villar, L. Wallace, R. L. Ward, A. J. Weinstein, S. E. Whitcomb, P. A. Willems, H. Yamamoto, D. Yeaton-Massey, L. Zhang, and J. Zweizig (the LIGO Scientific Collaboration), “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962–965 (2011).
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Torrie, C. I.

J. Abadie, B. P. Abbott, R. Abbott, R. Adhikari, S. B. Anderson, K. Arai, M. C. Araya, J. Betzwieser, G. Billingsley, E. Black, J. K. Blackburn, R. Bork, A. F. Brooks, C. Cepeda, T. Chalermsongsak, A. Corsi, D. C. Coyne, R. Dannenberg, B. Daudert, V. Dergachev, R. DeSalvo, J. C. Driggers, P. Ehrens, R. Engel, T. Etzel, N. Fotopoulos, E. K. Gustafson, C. Hanna, J. Harms, J. Heefner, A. W. Heptonstall, K. A. Hodge, A. Ivanov, M. Jacobson, P. Kalmus, W. Kells, P. J. King, V. Kondrashov, W. Z. Korth, D. Kozak, A. Lazzarini, P. E. Lindquist, M. Mageswaran, K. Mailand, E. Maros, J. N. Marx, G. McIntyre, S. Meshkov, T. Nash, G. H. Ogin, C. Osthelder, P. Ajith, P. Patel, M. Pedraza, M. Phelps, L. R. Price, S. Privitera, N. A. Robertson, V. Sannibale, L. Santamaría, A. C. Searle, F. Seifert, A. S. Sengupta, A. Singer, L. Singer, M. R. Smith, A. Stochino, R. Taylor, C. I. Torrie, S. Vass, A. E. Villar, L. Wallace, R. L. Ward, A. J. Weinstein, S. E. Whitcomb, P. A. Willems, H. Yamamoto, D. Yeaton-Massey, L. Zhang, and J. Zweizig (the LIGO Scientific Collaboration), “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962–965 (2011).
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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, 040102(R) (2013).
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Vass, S.

J. Abadie, B. P. Abbott, R. Abbott, R. Adhikari, S. B. Anderson, K. Arai, M. C. Araya, J. Betzwieser, G. Billingsley, E. Black, J. K. Blackburn, R. Bork, A. F. Brooks, C. Cepeda, T. Chalermsongsak, A. Corsi, D. C. Coyne, R. Dannenberg, B. Daudert, V. Dergachev, R. DeSalvo, J. C. Driggers, P. Ehrens, R. Engel, T. Etzel, N. Fotopoulos, E. K. Gustafson, C. Hanna, J. Harms, J. Heefner, A. W. Heptonstall, K. A. Hodge, A. Ivanov, M. Jacobson, P. Kalmus, W. Kells, P. J. King, V. Kondrashov, W. Z. Korth, D. Kozak, A. Lazzarini, P. E. Lindquist, M. Mageswaran, K. Mailand, E. Maros, J. N. Marx, G. McIntyre, S. Meshkov, T. Nash, G. H. Ogin, C. Osthelder, P. Ajith, P. Patel, M. Pedraza, M. Phelps, L. R. Price, S. Privitera, N. A. Robertson, V. Sannibale, L. Santamaría, A. C. Searle, F. Seifert, A. S. Sengupta, A. Singer, L. Singer, M. R. Smith, A. Stochino, R. Taylor, C. I. Torrie, S. Vass, A. E. Villar, L. Wallace, R. L. Ward, A. J. Weinstein, S. E. Whitcomb, P. A. Willems, H. Yamamoto, D. Yeaton-Massey, L. Zhang, and J. Zweizig (the LIGO Scientific Collaboration), “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962–965 (2011).
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Villar, A. E.

J. Abadie, B. P. Abbott, R. Abbott, R. Adhikari, S. B. Anderson, K. Arai, M. C. Araya, J. Betzwieser, G. Billingsley, E. Black, J. K. Blackburn, R. Bork, A. F. Brooks, C. Cepeda, T. Chalermsongsak, A. Corsi, D. C. Coyne, R. Dannenberg, B. Daudert, V. Dergachev, R. DeSalvo, J. C. Driggers, P. Ehrens, R. Engel, T. Etzel, N. Fotopoulos, E. K. Gustafson, C. Hanna, J. Harms, J. Heefner, A. W. Heptonstall, K. A. Hodge, A. Ivanov, M. Jacobson, P. Kalmus, W. Kells, P. J. King, V. Kondrashov, W. Z. Korth, D. Kozak, A. Lazzarini, P. E. Lindquist, M. Mageswaran, K. Mailand, E. Maros, J. N. Marx, G. McIntyre, S. Meshkov, T. Nash, G. H. Ogin, C. Osthelder, P. Ajith, P. Patel, M. Pedraza, M. Phelps, L. R. Price, S. Privitera, N. A. Robertson, V. Sannibale, L. Santamaría, A. C. Searle, F. Seifert, A. S. Sengupta, A. Singer, L. Singer, M. R. Smith, A. Stochino, R. Taylor, C. I. Torrie, S. Vass, A. E. Villar, L. Wallace, R. L. Ward, A. J. Weinstein, S. E. Whitcomb, P. A. Willems, H. Yamamoto, D. Yeaton-Massey, L. Zhang, and J. Zweizig (the LIGO Scientific Collaboration), “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962–965 (2011).
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Wallace, L.

J. Abadie, B. P. Abbott, R. Abbott, R. Adhikari, S. B. Anderson, K. Arai, M. C. Araya, J. Betzwieser, G. Billingsley, E. Black, J. K. Blackburn, R. Bork, A. F. Brooks, C. Cepeda, T. Chalermsongsak, A. Corsi, D. C. Coyne, R. Dannenberg, B. Daudert, V. Dergachev, R. DeSalvo, J. C. Driggers, P. Ehrens, R. Engel, T. Etzel, N. Fotopoulos, E. K. Gustafson, C. Hanna, J. Harms, J. Heefner, A. W. Heptonstall, K. A. Hodge, A. Ivanov, M. Jacobson, P. Kalmus, W. Kells, P. J. King, V. Kondrashov, W. Z. Korth, D. Kozak, A. Lazzarini, P. E. Lindquist, M. Mageswaran, K. Mailand, E. Maros, J. N. Marx, G. McIntyre, S. Meshkov, T. Nash, G. H. Ogin, C. Osthelder, P. Ajith, P. Patel, M. Pedraza, M. Phelps, L. R. Price, S. Privitera, N. A. Robertson, V. Sannibale, L. Santamaría, A. C. Searle, F. Seifert, A. S. Sengupta, A. Singer, L. Singer, M. R. Smith, A. Stochino, R. Taylor, C. I. Torrie, S. Vass, A. E. Villar, L. Wallace, R. L. Ward, A. J. Weinstein, S. E. Whitcomb, P. A. Willems, H. Yamamoto, D. Yeaton-Massey, L. Zhang, and J. Zweizig (the LIGO Scientific Collaboration), “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962–965 (2011).
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Walmsley, I. A.

M. Kacprowicz, R. Demkowicz-Dobrzanski, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Experimental quantum-enhanced estimation of a lossy phase shift,” Nat. Photonics 4, 357–360 (2010).
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Ward, R. L.

J. Abadie, B. P. Abbott, R. Abbott, R. Adhikari, S. B. Anderson, K. Arai, M. C. Araya, J. Betzwieser, G. Billingsley, E. Black, J. K. Blackburn, R. Bork, A. F. Brooks, C. Cepeda, T. Chalermsongsak, A. Corsi, D. C. Coyne, R. Dannenberg, B. Daudert, V. Dergachev, R. DeSalvo, J. C. Driggers, P. Ehrens, R. Engel, T. Etzel, N. Fotopoulos, E. K. Gustafson, C. Hanna, J. Harms, J. Heefner, A. W. Heptonstall, K. A. Hodge, A. Ivanov, M. Jacobson, P. Kalmus, W. Kells, P. J. King, V. Kondrashov, W. Z. Korth, D. Kozak, A. Lazzarini, P. E. Lindquist, M. Mageswaran, K. Mailand, E. Maros, J. N. Marx, G. McIntyre, S. Meshkov, T. Nash, G. H. Ogin, C. Osthelder, P. Ajith, P. Patel, M. Pedraza, M. Phelps, L. R. Price, S. Privitera, N. A. Robertson, V. Sannibale, L. Santamaría, A. C. Searle, F. Seifert, A. S. Sengupta, A. Singer, L. Singer, M. R. Smith, A. Stochino, R. Taylor, C. I. Torrie, S. Vass, A. E. Villar, L. Wallace, R. L. Ward, A. J. Weinstein, S. E. Whitcomb, P. A. Willems, H. Yamamoto, D. Yeaton-Massey, L. Zhang, and J. Zweizig (the LIGO Scientific Collaboration), “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962–965 (2011).
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Wasilewski, W.

M. Kacprowicz, R. Demkowicz-Dobrzanski, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Experimental quantum-enhanced estimation of a lossy phase shift,” Nat. Photonics 4, 357–360 (2010).
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Weinstein, A. J.

J. Abadie, B. P. Abbott, R. Abbott, R. Adhikari, S. B. Anderson, K. Arai, M. C. Araya, J. Betzwieser, G. Billingsley, E. Black, J. K. Blackburn, R. Bork, A. F. Brooks, C. Cepeda, T. Chalermsongsak, A. Corsi, D. C. Coyne, R. Dannenberg, B. Daudert, V. Dergachev, R. DeSalvo, J. C. Driggers, P. Ehrens, R. Engel, T. Etzel, N. Fotopoulos, E. K. Gustafson, C. Hanna, J. Harms, J. Heefner, A. W. Heptonstall, K. A. Hodge, A. Ivanov, M. Jacobson, P. Kalmus, W. Kells, P. J. King, V. Kondrashov, W. Z. Korth, D. Kozak, A. Lazzarini, P. E. Lindquist, M. Mageswaran, K. Mailand, E. Maros, J. N. Marx, G. McIntyre, S. Meshkov, T. Nash, G. H. Ogin, C. Osthelder, P. Ajith, P. Patel, M. Pedraza, M. Phelps, L. R. Price, S. Privitera, N. A. Robertson, V. Sannibale, L. Santamaría, A. C. Searle, F. Seifert, A. S. Sengupta, A. Singer, L. Singer, M. R. Smith, A. Stochino, R. Taylor, C. I. Torrie, S. Vass, A. E. Villar, L. Wallace, R. L. Ward, A. J. Weinstein, S. E. Whitcomb, P. A. Willems, H. Yamamoto, D. Yeaton-Massey, L. Zhang, and J. Zweizig (the LIGO Scientific Collaboration), “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962–965 (2011).
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Whitcomb, S. E.

J. Abadie, B. P. Abbott, R. Abbott, R. Adhikari, S. B. Anderson, K. Arai, M. C. Araya, J. Betzwieser, G. Billingsley, E. Black, J. K. Blackburn, R. Bork, A. F. Brooks, C. Cepeda, T. Chalermsongsak, A. Corsi, D. C. Coyne, R. Dannenberg, B. Daudert, V. Dergachev, R. DeSalvo, J. C. Driggers, P. Ehrens, R. Engel, T. Etzel, N. Fotopoulos, E. K. Gustafson, C. Hanna, J. Harms, J. Heefner, A. W. Heptonstall, K. A. Hodge, A. Ivanov, M. Jacobson, P. Kalmus, W. Kells, P. J. King, V. Kondrashov, W. Z. Korth, D. Kozak, A. Lazzarini, P. E. Lindquist, M. Mageswaran, K. Mailand, E. Maros, J. N. Marx, G. McIntyre, S. Meshkov, T. Nash, G. H. Ogin, C. Osthelder, P. Ajith, P. Patel, M. Pedraza, M. Phelps, L. R. Price, S. Privitera, N. A. Robertson, V. Sannibale, L. Santamaría, A. C. Searle, F. Seifert, A. S. Sengupta, A. Singer, L. Singer, M. R. Smith, A. Stochino, R. Taylor, C. I. Torrie, S. Vass, A. E. Villar, L. Wallace, R. L. Ward, A. J. Weinstein, S. E. Whitcomb, P. A. Willems, H. Yamamoto, D. Yeaton-Massey, L. Zhang, and J. Zweizig (the LIGO Scientific Collaboration), “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962–965 (2011).
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Willems, P. A.

J. Abadie, B. P. Abbott, R. Abbott, R. Adhikari, S. B. Anderson, K. Arai, M. C. Araya, J. Betzwieser, G. Billingsley, E. Black, J. K. Blackburn, R. Bork, A. F. Brooks, C. Cepeda, T. Chalermsongsak, A. Corsi, D. C. Coyne, R. Dannenberg, B. Daudert, V. Dergachev, R. DeSalvo, J. C. Driggers, P. Ehrens, R. Engel, T. Etzel, N. Fotopoulos, E. K. Gustafson, C. Hanna, J. Harms, J. Heefner, A. W. Heptonstall, K. A. Hodge, A. Ivanov, M. Jacobson, P. Kalmus, W. Kells, P. J. King, V. Kondrashov, W. Z. Korth, D. Kozak, A. Lazzarini, P. E. Lindquist, M. Mageswaran, K. Mailand, E. Maros, J. N. Marx, G. McIntyre, S. Meshkov, T. Nash, G. H. Ogin, C. Osthelder, P. Ajith, P. Patel, M. Pedraza, M. Phelps, L. R. Price, S. Privitera, N. A. Robertson, V. Sannibale, L. Santamaría, A. C. Searle, F. Seifert, A. S. Sengupta, A. Singer, L. Singer, M. R. Smith, A. Stochino, R. Taylor, C. I. Torrie, S. Vass, A. E. Villar, L. Wallace, R. L. Ward, A. J. Weinstein, S. E. Whitcomb, P. A. Willems, H. Yamamoto, D. Yeaton-Massey, L. Zhang, and J. Zweizig (the LIGO Scientific Collaboration), “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962–965 (2011).
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William, N. P.

N. P. William, 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. Šafránek, A. R. Lee, and I. Fuentes, “Quantum parameter estimation using multi-mode Gaussian states,” arXiv:1502.07924 (2015).

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Since the squeezing of the states is the same, after the evolution we still have factorized states, otherwise correlations will arise reducing the QFI, since now the single mode involved in the phase shift is no longer a pure state. It is worth noting that in the single-mode case one has Hsm>Hmax, but it is not possible to fix a phase reference.

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Bounds to precision in realistic Gaussian interferometry,” in preparation.

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Figures (3)

Fig. 1.
Fig. 1. Schematic diagram of a two-mode interferometric setup. The two modes, prepared in the factorized state | ψ a 〉 and | ψ b 〉 , are coupled by the unitary operator U int ( Θ ) . Then, one of the modes is phase-shifted by an amount ϕ and, finally, the two-mode state undergoes a measurement described by the POVM { Π } .

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Fig. 2.
Fig. 2. Top: optimal value of the total squeezing fraction β tot , maximizing the QFI for a passive interferometer, as a function of N tot . Bottom: corresponding phase sensitivity δ ϕ as a function of N tot (blue solid line). The shot-noise limit N tot − 1 / 2 (red dashed line) and the Heisenberg limit N tot − 1 (green dotted line) are also reported.

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Fig. 3.
Fig. 3. Top: optimal value of the signal fraction Δ max (red dashed line) and of the squeezing fraction β max (blue solid line), maximizing the QFI for the CLBI, as functions of N tot . Bottom: corresponding sensitivity δ ϕ as a function of N tot . The shot-noise limit N tot − 1 / 2 (red dashed line) and the Heisenberg limit N tot − 1 (green dotted line) are also reported.

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Equations (16)

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| Ψ ( Θ , ϕ ) 〉 = U ph ( ϕ ) U int ( Θ ) | ψ a 〉 ⊗ | ψ b 〉 .
δ ϕ 2 ≥ 1 M H ( ϕ ) ,
H ( ϕ ) = Tr [ R ϕ L ϕ 2 ] ,
∂ ϕ R ϕ = 1 2 ( L ϕ R ϕ + R ϕ L ϕ ) .
L ϕ = 2 ( | ∂ ϕ Ψ ( Θ , ϕ ) 〉 〈 Ψ ( Θ , ϕ ) | + | Ψ ( Θ , ϕ ) 〉 〈 ∂ ϕ Ψ ( Θ , ϕ ) | ) ,
H = 1 4 { 4 ( α + γ ) 2 [ e 2 ξ + cosh   2 r ] + cosh   4 ξ + cosh   4 r + 2   cos   θ   sinh   2 r [ 2 ( α + γ ) 2 + sinh   2 ξ ] + cosh   2 ( r − ξ ) + cosh   2 ( ξ + r ) − 4 } .
N tot = α 2 + γ 2 + sinh 2   r + sinh 2   ξ ,
Δ = α 2 / ( α 2 + γ 2 ) ,
β tot = ( sinh 2   r + sinh 2   ξ ) / N tot β = sinh 2   ξ / N tot ,
H max ( N tot , β tot ) = 2 N tot [ 2 + N tot β tot ( 2 − β tot ) + 2 ( 1 − β tot ) N tot β tot ( 2 + N tot β tot ) ] ,
H max ≃ 8 9 N tot 2 [ 2 + ( 1 + 3 N tot − 1 ) 1 / 2 ] + 4 N tot ≃ 8 3 N tot 2 .
U BS | α , r 〉 ⊗ | α , r 〉 = | 2 α , r 〉 ⊗ | 0 , r 〉 ,
H ( gw ) ( N tot , β ) = 2 N tot [ 1 + ( 1 2 + N tot ) β + ( 1 − β ) N tot β ( 1 + N tot β ) ] ,
H max ( gw ) = 2 N tot ( 3 2 + N tot ) ≃ N tot ≫ 1 2 N tot 2 .
H ( CLBI ) = ( α 2 + γ 2 + 1 2 ) ( 1 + cosh   4 r ) − 1 + 2 α γ   cos   θ   sinh   4 r + 2 ( α 2 − γ 2 ) cosh   2 r .
N tot = ( α 2 + γ 2 + 1 ) cosh ( 2 r ) + 2 α γ   cos ( θ ) sinh ( 2 r ) − 1 ,

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