Abstract

When atoms are illuminated by an off-resonant field, the AC Stark effect will lead to phase shifts in atomic states. The phase shifts are proportional to the photon number of the off-resonant illuminating field. By measuring the atomic phase with newly developed atom-light hybrid interferometers, we can achieve quantum non-demolition measurement of the photon number of the optical field. In this paper, we analyze theoretically the performance of this QND measurement scheme by using the QND measurement criteria established by Holland et al [Phys. Rev. A 42, 2995 (1990)]. We find the quality of the QND measurement depends on the phase resolution of the atom-light hybrid interferometers. We apply this QND measurement scheme to a twin-photon state from parametric amplifier to verify the photon correlation in the twin beams. Furthermore, a sequential QND measurement procedure is analyzed for verifying the projection property of quantum measurement and for the quantum information tapping. Finally, we discuss the possibility for single-photon-number-resolving detection via QND measurement.

© 2017 Optical Society of America

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References

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  5. W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71(3), 033819 (2005).
    [Crossref]
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    [Crossref]
  7. M. D. Levenson, R. M. Shelby, M. Reid, and D. F. Walls, “Quantum nondemolition detection of optical quadrature amplitudes,” Phys. Rev. Lett. 57(20), 2473 (1986).
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  8. S. R. Friberg, S. Machida, and Y. Yamamoto, “Quantum-nondemolition measurement of the photon number of an optical soliton,” Phys. Rev. Lett. 69(22), 3165 (1992).
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  9. P. Kok, H. Lee, and J. P. Dowling, “Single-photon quantum-nondemolition detectors constructed with linear optics and projective measurements,” Phys. Rev. A 66(6), 063814(2002).
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  10. G. J. Pryde, J. L. O’Brien, A. G. White, S. D. Bartlett, and T. C. Ralph, “Measuring a photonic qubit without destroying it,” Phys. Rev. Lett. 92(19), 190402 (2004).
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  18. S. H. Autler and C. H. Townes, “Stark effect in rapidly varying fields,” Phys. Rev. 100(2), 703 (1955).
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    [Crossref]
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    [Crossref]
  27. G. Campbell, M. Hosseini, B. M. Sparkes, P. K. Lam, and B. C. Buchler, “Time-and frequency-domain polariton interference,” New J. Phys. 14(3), 033022 (2012).
    [Crossref]
  28. B. Chen, C. Qiu, S. Chen, J. Guo, L. Q. Chen, Z. Y. Ou, and W. Zhang, “Atom-light hybrid interferometer,” Phys. Rev. Lett. 115(4), 043602 (2015).
    [Crossref] [PubMed]
  29. C. Qiu, S. Chen, L. Q. Chen, B. Chen, J. Guo, Z. Y. Ou, and W. Zhang, “Atom–light superposition oscillation and Ramsey-like atom–light interferometer,” Optica 3(7), 775–780 (2016).
    [Crossref]
  30. M. J. Holland, M. J. Collett, D. F. Walls, and M. D. Levenson, “Nonideal quantum nondemolition measurements,” Phys. Rev. A 42(5), 2995 (1990).
    [Crossref] [PubMed]
  31. S. S. Szigeti, R. J. Lewis-Swan, and S. A. Haine, “Pumped-up SU (1, 1) interferometry,” Phys. Rev. Lett. 118(15), 150401 (2017).
    [Crossref] [PubMed]
  32. H. Ma, D. Li, C. H. Yuan, L. Q. Chen, Z. Y. Ou, and W. Zhang, “SU (1, 1)-type light-atom-correlated interferometer,” Phys. Rev. A 92(2), 023847 (2015).
    [Crossref]
  33. Z. D. Chen, C. H. Yuan, H. M. Ma, D. Li, L. Q. Chen, Z. Y. Ou, and W. Zhang, “Effects of losses in the atom-light hybrid SU (1, 1) interferometer,” Opt. Express 24(16), 17766–17778(2016).
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    [Crossref]
  35. Z. Y. Ou, “Enhancement of the phase-measurement sensitivity beyond the standard quantum limit by a nonlinear interferometer,” Phys. Rev. A 85(2), 023815 (2012).
    [Crossref]
  36. 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]
  37. Z. Y. Ou, “Efficient conversion between photons and between photon and atom by stimulated emission,” Phys. Rev. A 78(2), 023819 (2008).
    [Crossref]
  38. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23(8), 1693 (1981).
    [Crossref]
  39. M. D. Reid, “Demonstration of the Einstein-Podolsky-Rosen paradox using nondegenerate parametric amplification,” Phys. Rev. A 40(2), 913 (1989).
    [Crossref]
  40. A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, “Observation of quantum noise reduction on twin laser beams,” Phys. Rev. Lett. 59(22), 2555 (1987).
    [Crossref] [PubMed]
  41. O. Aytür and P. Kumar, “Pulsed twin beams of light,” Phys. Rev. Lett. 65(13), 1551 (1990).
    [Crossref] [PubMed]
  42. L. Q. Chen, C. Bian, G. W. Zhang, Z. Y. Ou, and W. Zhang, “Observation of temporal beating in first-and second-order intensity measurement between independent Raman Stokes fields in atomic vapor,” Phys. Rev. A 82(3), 033832 (2010).
    [Crossref]
  43. J. Geng, G. T. Campbell, J. Bernu, D. B. Higginbottom, B. M. Sparkes, S. M. Assad, W. P. Zhang, N.P. Robins, P. K. Lam, and B. C. Buchler, “Electromagnetically induced transparency and four-wave mixing in a cold atomic ensemble with large optical depth,” New J. Phys. 16(11), 113053 (2014).
    [Crossref]
  44. J. F. Roch, K. Vigneron, P. Grelu, A. Sinatra, J. P. Poizat, and P. Grangier, “Quantum Nondemolition Measurements using Cold Trapped Atoms,” Phys. Rev. Lett. 78(78), 634–637 (1997).
    [Crossref]
  45. D. Kruse, M. Ruder, J. Benhelm, C. V. Cube, C. Zimmermann, P. W. Courteille, T. Elsässer, B. Nagorny, and A. Hemmerich, “Cold atoms in a high-Q ring cavity,” Phys. Rev. A 67(5), 051802 (2003).
    [Crossref]

2017 (1)

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

2016 (3)

2015 (3)

M. Gabbrielli, L. Pezzè, and A. Smerzi, “Spin-mixing interferometry with Bose-Einstein condensates,” Phys. Rev. Lett. 115(16), 163002(2015).
[Crossref] [PubMed]

H. Ma, D. Li, C. H. Yuan, L. Q. Chen, Z. Y. Ou, and W. Zhang, “SU (1, 1)-type light-atom-correlated interferometer,” Phys. Rev. A 92(2), 023847 (2015).
[Crossref]

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

2014 (2)

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]

J. Geng, G. T. Campbell, J. Bernu, D. B. Higginbottom, B. M. Sparkes, S. M. Assad, W. P. Zhang, N.P. Robins, P. K. Lam, and B. C. Buchler, “Electromagnetically induced transparency and four-wave mixing in a cold atomic ensemble with large optical depth,” New J. Phys. 16(11), 113053 (2014).
[Crossref]

2013 (2)

B. Fan, A. F. Kockum, J. Combes, G. Johansson, I.C. Hoi, C.M. Wilson, P. Delsing, G.J. Milburn, and T.M. Stace, “Breakdown of the cross-Kerr scheme for photon counting,” Phys. Rev. Lett. 110(5), 053601 (2013).
[Crossref] [PubMed]

S. Haroche, “Nobel Lecture: Controlling photons in a box and exploring the quantum to classical boundary,” Rev. Mod. Phys. 85(3), 1083 (2013).
[Crossref]

2012 (2)

G. Campbell, M. Hosseini, B. M. Sparkes, P. K. Lam, and B. C. Buchler, “Time-and frequency-domain polariton interference,” New J. Phys. 14(3), 033022 (2012).
[Crossref]

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

2011 (1)

J. Jing, C. Liu, Z. Zhou, Z.Y. Ou, and W. Zhang, “Realization of a nonlinear interferometer with parametric amplifiers,” Appl. Phys. Lett. 99(1), 011110 (2011).
[Crossref]

2010 (1)

L. Q. Chen, C. Bian, G. W. Zhang, Z. Y. Ou, and W. Zhang, “Observation of temporal beating in first-and second-order intensity measurement between independent Raman Stokes fields in atomic vapor,” Phys. Rev. A 82(3), 033832 (2010).
[Crossref]

2008 (1)

Z. Y. Ou, “Efficient conversion between photons and between photon and atom by stimulated emission,” Phys. Rev. A 78(2), 023819 (2008).
[Crossref]

2007 (1)

C. Guerlin, J. Bernu, S. Deleglise, C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune, J.M. Raimond, and S. Haroche, “Progressive field-state collapse and quantum non-demolition photon counting,” Nature 448(7156), 889–893 (2007).
[Crossref] [PubMed]

2005 (2)

W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71(3), 033819 (2005).
[Crossref]

W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71(3), 033819 (2005).
[Crossref]

2004 (1)

G. J. Pryde, J. L. O’Brien, A. G. White, S. D. Bartlett, and T. C. Ralph, “Measuring a photonic qubit without destroying it,” Phys. Rev. Lett. 92(19), 190402 (2004).
[Crossref] [PubMed]

2003 (1)

D. Kruse, M. Ruder, J. Benhelm, C. V. Cube, C. Zimmermann, P. W. Courteille, T. Elsässer, B. Nagorny, and A. Hemmerich, “Cold atoms in a high-Q ring cavity,” Phys. Rev. A 67(5), 051802 (2003).
[Crossref]

2002 (1)

P. Kok, H. Lee, and J. P. Dowling, “Single-photon quantum-nondemolition detectors constructed with linear optics and projective measurements,” Phys. Rev. A 66(6), 063814(2002).
[Crossref]

1999 (1)

A. Kuzmich, L. Mandel, J. Janis, Y. E. Young, R. Ejnisman, and N.P. Bigelow,“Quantum nondemolition measurements of collective atomic spin,” Phys. Rev. A 60(3), 2346 (1999).
[Crossref]

1998 (1)

P. Grangier, J. A. Levenson, and J. P. Poizat, “Quantum non-demolition measurements in optics,” Nature 396(6711), 537–542 (1998).
[Crossref]

1997 (1)

J. F. Roch, K. Vigneron, P. Grelu, A. Sinatra, J. P. Poizat, and P. Grangier, “Quantum Nondemolition Measurements using Cold Trapped Atoms,” Phys. Rev. Lett. 78(78), 634–637 (1997).
[Crossref]

1996 (1)

V. B. Braginsky and F. Y. Khalili, “Quantum nondemolition measurements: the route from toys to tools,” Rev. Mod. Phys. 68(1), 1 (1996).
[Crossref]

1995 (1)

J. Jacobson, G. Björk, and Y. Yamamoto, “Quantum limit for the atom-light interferometer,” Appl. Phys. B: Lasers and Optics,  60(2), 187–191 (1995).
[Crossref]

1994 (1)

S. F. Pereira, Z. Y. Ou, and H. J. Kimble, “Backaction evading measurements for quantum nondemolition detection and quantum optical tapping,” Phys. Rev. Lett. 72(2), 214 (1994).
[Crossref] [PubMed]

1993 (1)

J. P. Poizat and P. Grangier, “Experimental realization of a quantum optical tap,” Phys. Rev. Lett. 70(3), 271 (1993).
[Crossref] [PubMed]

1992 (1)

S. R. Friberg, S. Machida, and Y. Yamamoto, “Quantum-nondemolition measurement of the photon number of an optical soliton,” Phys. Rev. Lett. 69(22), 3165 (1992).
[Crossref] [PubMed]

1990 (2)

M. J. Holland, M. J. Collett, D. F. Walls, and M. D. Levenson, “Nonideal quantum nondemolition measurements,” Phys. Rev. A 42(5), 2995 (1990).
[Crossref] [PubMed]

O. Aytür and P. Kumar, “Pulsed twin beams of light,” Phys. Rev. Lett. 65(13), 1551 (1990).
[Crossref] [PubMed]

1989 (3)

M. D. Reid, “Demonstration of the Einstein-Podolsky-Rosen paradox using nondegenerate parametric amplification,” Phys. Rev. A 40(2), 913 (1989).
[Crossref]

G. J. Milburn, “Quantum optical Fredkin gate,” Phys. Rev. Lett. 62(18), 2124 (1989).
[Crossref] [PubMed]

A. L. Porta, R. E. Slusher, and B. Yurke, “Back-action evading measurements of an optical field using parametric down conversion,” Phys. Rev. Lett. 62(1), 28 (1989).
[Crossref] [PubMed]

1987 (1)

A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, “Observation of quantum noise reduction on twin laser beams,” Phys. Rev. Lett. 59(22), 2555 (1987).
[Crossref] [PubMed]

1986 (2)

M. D. Levenson, R. M. Shelby, M. Reid, and D. F. Walls, “Quantum nondemolition detection of optical quadrature amplitudes,” Phys. Rev. Lett. 57(20), 2473 (1986).
[Crossref] [PubMed]

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

1985 (1)

N. Imoto, H. A. Haus, and Y. Yamamoto, “Quantum nondemolition measurement of the photon number via the optical Kerr effect,” Phys. Rev. A 32(4), 2287 (1985).
[Crossref]

1981 (1)

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

1980 (1)

B. V. Braginsky, Y. I. Vorontsov, and K. S. Thorne, “Quantum nondemolition measurements,” Science 209(4456), 547–557 (1980).
[Crossref] [PubMed]

1955 (2)

S. H. Autler and C. H. Townes, “Stark effect in rapidly varying fields,” Phys. Rev. 100(2), 703 (1955).
[Crossref]

I. Kruse, K. Lange, J. Peise, B. Lücke, L. Pezzè, J. Arlt, W. Ertmer, C. Lisdat, L. Santos, A. Smerzi, and C. Klempt, “Stark effect in rapidly varying fields,” Phys. Rev. 100(2), 703 (1955).
[Crossref]

1950 (2)

D. Linnemann, H. Strobel, W. Muessel, J. Schulz, R. J. Lewis-Swan, K. V. Kheruntsyan, and M. K. Oberthaler, “Quantum-enhanced sensing based on time reversal of nonlinear dynamics,” Phys. Rev. Lett. 117(1), 013001 (1950).
[Crossref]

N. F. Ramsey, “A molecular beam resonance method with separated oscillating fields,” Phys. Rev. 78(6), 695 (1950).
[Crossref]

Arlt, J.

I. Kruse, K. Lange, J. Peise, B. Lücke, L. Pezzè, J. Arlt, W. Ertmer, C. Lisdat, L. Santos, A. Smerzi, and C. Klempt, “Stark effect in rapidly varying fields,” Phys. Rev. 100(2), 703 (1955).
[Crossref]

Assad, S. M.

J. Geng, G. T. Campbell, J. Bernu, D. B. Higginbottom, B. M. Sparkes, S. M. Assad, W. P. Zhang, N.P. Robins, P. K. Lam, and B. C. Buchler, “Electromagnetically induced transparency and four-wave mixing in a cold atomic ensemble with large optical depth,” New J. Phys. 16(11), 113053 (2014).
[Crossref]

Autler, S. H.

S. H. Autler and C. H. Townes, “Stark effect in rapidly varying fields,” Phys. Rev. 100(2), 703 (1955).
[Crossref]

Aytür, O.

O. Aytür and P. Kumar, “Pulsed twin beams of light,” Phys. Rev. Lett. 65(13), 1551 (1990).
[Crossref] [PubMed]

Bartlett, S. D.

G. J. Pryde, J. L. O’Brien, A. G. White, S. D. Bartlett, and T. C. Ralph, “Measuring a photonic qubit without destroying it,” Phys. Rev. Lett. 92(19), 190402 (2004).
[Crossref] [PubMed]

Beausoleil, R. G.

W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71(3), 033819 (2005).
[Crossref]

W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71(3), 033819 (2005).
[Crossref]

Benhelm, J.

D. Kruse, M. Ruder, J. Benhelm, C. V. Cube, C. Zimmermann, P. W. Courteille, T. Elsässer, B. Nagorny, and A. Hemmerich, “Cold atoms in a high-Q ring cavity,” Phys. Rev. A 67(5), 051802 (2003).
[Crossref]

Bernu, J.

J. Geng, G. T. Campbell, J. Bernu, D. B. Higginbottom, B. M. Sparkes, S. M. Assad, W. P. Zhang, N.P. Robins, P. K. Lam, and B. C. Buchler, “Electromagnetically induced transparency and four-wave mixing in a cold atomic ensemble with large optical depth,” New J. Phys. 16(11), 113053 (2014).
[Crossref]

C. Guerlin, J. Bernu, S. Deleglise, C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune, J.M. Raimond, and S. Haroche, “Progressive field-state collapse and quantum non-demolition photon counting,” Nature 448(7156), 889–893 (2007).
[Crossref] [PubMed]

Bian, C.

L. Q. Chen, C. Bian, G. W. Zhang, Z. Y. Ou, and W. Zhang, “Observation of temporal beating in first-and second-order intensity measurement between independent Raman Stokes fields in atomic vapor,” Phys. Rev. A 82(3), 033832 (2010).
[Crossref]

Bigelow, N.P.

A. Kuzmich, L. Mandel, J. Janis, Y. E. Young, R. Ejnisman, and N.P. Bigelow,“Quantum nondemolition measurements of collective atomic spin,” Phys. Rev. A 60(3), 2346 (1999).
[Crossref]

Björk, G.

J. Jacobson, G. Björk, and Y. Yamamoto, “Quantum limit for the atom-light interferometer,” Appl. Phys. B: Lasers and Optics,  60(2), 187–191 (1995).
[Crossref]

Braginsky, B. V.

B. V. Braginsky, Y. I. Vorontsov, and K. S. Thorne, “Quantum nondemolition measurements,” Science 209(4456), 547–557 (1980).
[Crossref] [PubMed]

Braginsky, V. B.

V. B. Braginsky and F. Y. Khalili, “Quantum nondemolition measurements: the route from toys to tools,” Rev. Mod. Phys. 68(1), 1 (1996).
[Crossref]

Brune, M.

C. Guerlin, J. Bernu, S. Deleglise, C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune, J.M. Raimond, and S. Haroche, “Progressive field-state collapse and quantum non-demolition photon counting,” Nature 448(7156), 889–893 (2007).
[Crossref] [PubMed]

Buchler, B. C.

J. Geng, G. T. Campbell, J. Bernu, D. B. Higginbottom, B. M. Sparkes, S. M. Assad, W. P. Zhang, N.P. Robins, P. K. Lam, and B. C. Buchler, “Electromagnetically induced transparency and four-wave mixing in a cold atomic ensemble with large optical depth,” New J. Phys. 16(11), 113053 (2014).
[Crossref]

G. Campbell, M. Hosseini, B. M. Sparkes, P. K. Lam, and B. C. Buchler, “Time-and frequency-domain polariton interference,” New J. Phys. 14(3), 033022 (2012).
[Crossref]

Campbell, G.

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J. Geng, G. T. Campbell, J. Bernu, D. B. Higginbottom, B. M. Sparkes, S. M. Assad, W. P. Zhang, N.P. Robins, P. K. Lam, and B. C. Buchler, “Electromagnetically induced transparency and four-wave mixing in a cold atomic ensemble with large optical depth,” New J. Phys. 16(11), 113053 (2014).
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C. Qiu, S. Chen, L. Q. Chen, B. Chen, J. Guo, Z. Y. Ou, and W. Zhang, “Atom–light superposition oscillation and Ramsey-like atom–light interferometer,” Optica 3(7), 775–780 (2016).
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B. Chen, C. Qiu, S. Chen, J. Guo, L. Q. Chen, Z. Y. Ou, and W. Zhang, “Atom-light hybrid interferometer,” Phys. Rev. Lett. 115(4), 043602 (2015).
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C. Qiu, S. Chen, L. Q. Chen, B. Chen, J. Guo, Z. Y. Ou, and W. Zhang, “Atom–light superposition oscillation and Ramsey-like atom–light interferometer,” Optica 3(7), 775–780 (2016).
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B. Chen, C. Qiu, S. Chen, J. Guo, L. Q. Chen, Z. Y. Ou, and W. Zhang, “Atom-light hybrid interferometer,” Phys. Rev. Lett. 115(4), 043602 (2015).
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M. J. Holland, M. J. Collett, D. F. Walls, and M. D. Levenson, “Nonideal quantum nondemolition measurements,” Phys. Rev. A 42(5), 2995 (1990).
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B. Fan, A. F. Kockum, J. Combes, G. Johansson, I.C. Hoi, C.M. Wilson, P. Delsing, G.J. Milburn, and T.M. Stace, “Breakdown of the cross-Kerr scheme for photon counting,” Phys. Rev. Lett. 110(5), 053601 (2013).
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D. Kruse, M. Ruder, J. Benhelm, C. V. Cube, C. Zimmermann, P. W. Courteille, T. Elsässer, B. Nagorny, and A. Hemmerich, “Cold atoms in a high-Q ring cavity,” Phys. Rev. A 67(5), 051802 (2003).
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D. Kruse, M. Ruder, J. Benhelm, C. V. Cube, C. Zimmermann, P. W. Courteille, T. Elsässer, B. Nagorny, and A. Hemmerich, “Cold atoms in a high-Q ring cavity,” Phys. Rev. A 67(5), 051802 (2003).
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C. Guerlin, J. Bernu, S. Deleglise, C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune, J.M. Raimond, and S. Haroche, “Progressive field-state collapse and quantum non-demolition photon counting,” Nature 448(7156), 889–893 (2007).
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B. Fan, A. F. Kockum, J. Combes, G. Johansson, I.C. Hoi, C.M. Wilson, P. Delsing, G.J. Milburn, and T.M. Stace, “Breakdown of the cross-Kerr scheme for photon counting,” Phys. Rev. Lett. 110(5), 053601 (2013).
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P. Kok, H. Lee, and J. P. Dowling, “Single-photon quantum-nondemolition detectors constructed with linear optics and projective measurements,” Phys. Rev. A 66(6), 063814(2002).
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A. Kuzmich, L. Mandel, J. Janis, Y. E. Young, R. Ejnisman, and N.P. Bigelow,“Quantum nondemolition measurements of collective atomic spin,” Phys. Rev. A 60(3), 2346 (1999).
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Elsässer, T.

D. Kruse, M. Ruder, J. Benhelm, C. V. Cube, C. Zimmermann, P. W. Courteille, T. Elsässer, B. Nagorny, and A. Hemmerich, “Cold atoms in a high-Q ring cavity,” Phys. Rev. A 67(5), 051802 (2003).
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I. Kruse, K. Lange, J. Peise, B. Lücke, L. Pezzè, J. Arlt, W. Ertmer, C. Lisdat, L. Santos, A. Smerzi, and C. Klempt, “Stark effect in rapidly varying fields,” Phys. Rev. 100(2), 703 (1955).
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A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, “Observation of quantum noise reduction on twin laser beams,” Phys. Rev. Lett. 59(22), 2555 (1987).
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B. Fan, A. F. Kockum, J. Combes, G. Johansson, I.C. Hoi, C.M. Wilson, P. Delsing, G.J. Milburn, and T.M. Stace, “Breakdown of the cross-Kerr scheme for photon counting,” Phys. Rev. Lett. 110(5), 053601 (2013).
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S. R. Friberg, S. Machida, and Y. Yamamoto, “Quantum-nondemolition measurement of the photon number of an optical soliton,” Phys. Rev. Lett. 69(22), 3165 (1992).
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M. Gabbrielli, L. Pezzè, and A. Smerzi, “Spin-mixing interferometry with Bose-Einstein condensates,” Phys. Rev. Lett. 115(16), 163002(2015).
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Giacobino, E.

A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, “Observation of quantum noise reduction on twin laser beams,” Phys. Rev. Lett. 59(22), 2555 (1987).
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C. Guerlin, J. Bernu, S. Deleglise, C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune, J.M. Raimond, and S. Haroche, “Progressive field-state collapse and quantum non-demolition photon counting,” Nature 448(7156), 889–893 (2007).
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P. Grangier, J. A. Levenson, and J. P. Poizat, “Quantum non-demolition measurements in optics,” Nature 396(6711), 537–542 (1998).
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C. Guerlin, J. Bernu, S. Deleglise, C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune, J.M. Raimond, and S. Haroche, “Progressive field-state collapse and quantum non-demolition photon counting,” Nature 448(7156), 889–893 (2007).
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Guo, J.

C. Qiu, S. Chen, L. Q. Chen, B. Chen, J. Guo, Z. Y. Ou, and W. Zhang, “Atom–light superposition oscillation and Ramsey-like atom–light interferometer,” Optica 3(7), 775–780 (2016).
[Crossref]

B. Chen, C. Qiu, S. Chen, J. Guo, L. Q. Chen, Z. Y. Ou, and W. Zhang, “Atom-light hybrid interferometer,” Phys. Rev. Lett. 115(4), 043602 (2015).
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S. S. Szigeti, R. J. Lewis-Swan, and S. A. Haine, “Pumped-up SU (1, 1) interferometry,” Phys. Rev. Lett. 118(15), 150401 (2017).
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S. A. Haine and W. Y. S. Lau, “Generation of atom-light entanglement in an optical cavity for quantum enhanced atom interferometry,” Phys. Rev. A 93(2), 023607(2016).
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C. Guerlin, J. Bernu, S. Deleglise, C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune, J.M. Raimond, and S. Haroche, “Progressive field-state collapse and quantum non-demolition photon counting,” Nature 448(7156), 889–893 (2007).
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N. Imoto, H. A. Haus, and Y. Yamamoto, “Quantum nondemolition measurement of the photon number via the optical Kerr effect,” Phys. Rev. A 32(4), 2287 (1985).
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A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, “Observation of quantum noise reduction on twin laser beams,” Phys. Rev. Lett. 59(22), 2555 (1987).
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Hemmerich, A.

D. Kruse, M. Ruder, J. Benhelm, C. V. Cube, C. Zimmermann, P. W. Courteille, T. Elsässer, B. Nagorny, and A. Hemmerich, “Cold atoms in a high-Q ring cavity,” Phys. Rev. A 67(5), 051802 (2003).
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J. Geng, G. T. Campbell, J. Bernu, D. B. Higginbottom, B. M. Sparkes, S. M. Assad, W. P. Zhang, N.P. Robins, P. K. Lam, and B. C. Buchler, “Electromagnetically induced transparency and four-wave mixing in a cold atomic ensemble with large optical depth,” New J. Phys. 16(11), 113053 (2014).
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Hoi, I.C.

B. Fan, A. F. Kockum, J. Combes, G. Johansson, I.C. Hoi, C.M. Wilson, P. Delsing, G.J. Milburn, and T.M. Stace, “Breakdown of the cross-Kerr scheme for photon counting,” Phys. Rev. Lett. 110(5), 053601 (2013).
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Holland, M. J.

M. J. Holland, M. J. Collett, D. F. Walls, and M. D. Levenson, “Nonideal quantum nondemolition measurements,” Phys. Rev. A 42(5), 2995 (1990).
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Horowicz, R. J.

A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, “Observation of quantum noise reduction on twin laser beams,” Phys. Rev. Lett. 59(22), 2555 (1987).
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Hosseini, M.

G. Campbell, M. Hosseini, B. M. Sparkes, P. K. Lam, and B. C. Buchler, “Time-and frequency-domain polariton interference,” New J. Phys. 14(3), 033022 (2012).
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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).
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N. Imoto, H. A. Haus, and Y. Yamamoto, “Quantum nondemolition measurement of the photon number via the optical Kerr effect,” Phys. Rev. A 32(4), 2287 (1985).
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A. Kuzmich, L. Mandel, J. Janis, Y. E. Young, R. Ejnisman, and N.P. Bigelow,“Quantum nondemolition measurements of collective atomic spin,” Phys. Rev. A 60(3), 2346 (1999).
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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).
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J. Jing, C. Liu, Z. Zhou, Z.Y. Ou, and W. Zhang, “Realization of a nonlinear interferometer with parametric amplifiers,” Appl. Phys. Lett. 99(1), 011110 (2011).
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Johansson, G.

B. Fan, A. F. Kockum, J. Combes, G. Johansson, I.C. Hoi, C.M. Wilson, P. Delsing, G.J. Milburn, and T.M. Stace, “Breakdown of the cross-Kerr scheme for photon counting,” Phys. Rev. Lett. 110(5), 053601 (2013).
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V. B. Braginsky and F. Y. Khalili, “Quantum nondemolition measurements: the route from toys to tools,” Rev. Mod. Phys. 68(1), 1 (1996).
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D. Linnemann, H. Strobel, W. Muessel, J. Schulz, R. J. Lewis-Swan, K. V. Kheruntsyan, and M. K. Oberthaler, “Quantum-enhanced sensing based on time reversal of nonlinear dynamics,” Phys. Rev. Lett. 117(1), 013001 (1950).
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S. F. Pereira, Z. Y. Ou, and H. J. Kimble, “Backaction evading measurements for quantum nondemolition detection and quantum optical tapping,” Phys. Rev. Lett. 72(2), 214 (1994).
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B. Yurke, S. L. McCall, and J. R. Klauder, “SU (2) and SU (1, 1) interferometers,” Phys. Rev. A 33(6), 4033 (1986).
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Klempt, C.

I. Kruse, K. Lange, J. Peise, B. Lücke, L. Pezzè, J. Arlt, W. Ertmer, C. Lisdat, L. Santos, A. Smerzi, and C. Klempt, “Stark effect in rapidly varying fields,” Phys. Rev. 100(2), 703 (1955).
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Kockum, A. F.

B. Fan, A. F. Kockum, J. Combes, G. Johansson, I.C. Hoi, C.M. Wilson, P. Delsing, G.J. Milburn, and T.M. Stace, “Breakdown of the cross-Kerr scheme for photon counting,” Phys. Rev. Lett. 110(5), 053601 (2013).
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Kok, P.

P. Kok, H. Lee, and J. P. Dowling, “Single-photon quantum-nondemolition detectors constructed with linear optics and projective measurements,” Phys. Rev. A 66(6), 063814(2002).
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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).
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Kruse, D.

D. Kruse, M. Ruder, J. Benhelm, C. V. Cube, C. Zimmermann, P. W. Courteille, T. Elsässer, B. Nagorny, and A. Hemmerich, “Cold atoms in a high-Q ring cavity,” Phys. Rev. A 67(5), 051802 (2003).
[Crossref]

Kruse, I.

I. Kruse, K. Lange, J. Peise, B. Lücke, L. Pezzè, J. Arlt, W. Ertmer, C. Lisdat, L. Santos, A. Smerzi, and C. Klempt, “Stark effect in rapidly varying fields,” Phys. Rev. 100(2), 703 (1955).
[Crossref]

Kuhr, S.

C. Guerlin, J. Bernu, S. Deleglise, C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune, J.M. Raimond, and S. Haroche, “Progressive field-state collapse and quantum non-demolition photon counting,” Nature 448(7156), 889–893 (2007).
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A. Kuzmich, L. Mandel, J. Janis, Y. E. Young, R. Ejnisman, and N.P. Bigelow,“Quantum nondemolition measurements of collective atomic spin,” Phys. Rev. A 60(3), 2346 (1999).
[Crossref]

Lam, P. K.

J. Geng, G. T. Campbell, J. Bernu, D. B. Higginbottom, B. M. Sparkes, S. M. Assad, W. P. Zhang, N.P. Robins, P. K. Lam, and B. C. Buchler, “Electromagnetically induced transparency and four-wave mixing in a cold atomic ensemble with large optical depth,” New J. Phys. 16(11), 113053 (2014).
[Crossref]

G. Campbell, M. Hosseini, B. M. Sparkes, P. K. Lam, and B. C. Buchler, “Time-and frequency-domain polariton interference,” New J. Phys. 14(3), 033022 (2012).
[Crossref]

Lange, K.

I. Kruse, K. Lange, J. Peise, B. Lücke, L. Pezzè, J. Arlt, W. Ertmer, C. Lisdat, L. Santos, A. Smerzi, and C. Klempt, “Stark effect in rapidly varying fields,” Phys. Rev. 100(2), 703 (1955).
[Crossref]

Lau, W. Y. S.

S. A. Haine and W. Y. S. Lau, “Generation of atom-light entanglement in an optical cavity for quantum enhanced atom interferometry,” Phys. Rev. A 93(2), 023607(2016).
[Crossref]

Lee, H.

P. Kok, H. Lee, and J. P. Dowling, “Single-photon quantum-nondemolition detectors constructed with linear optics and projective measurements,” Phys. Rev. A 66(6), 063814(2002).
[Crossref]

Levenson, J. A.

P. Grangier, J. A. Levenson, and J. P. Poizat, “Quantum non-demolition measurements in optics,” Nature 396(6711), 537–542 (1998).
[Crossref]

Levenson, M. D.

M. J. Holland, M. J. Collett, D. F. Walls, and M. D. Levenson, “Nonideal quantum nondemolition measurements,” Phys. Rev. A 42(5), 2995 (1990).
[Crossref] [PubMed]

M. D. Levenson, R. M. Shelby, M. Reid, and D. F. Walls, “Quantum nondemolition detection of optical quadrature amplitudes,” Phys. Rev. Lett. 57(20), 2473 (1986).
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Lewis-Swan, R. J.

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

D. Linnemann, H. Strobel, W. Muessel, J. Schulz, R. J. Lewis-Swan, K. V. Kheruntsyan, and M. K. Oberthaler, “Quantum-enhanced sensing based on time reversal of nonlinear dynamics,” Phys. Rev. Lett. 117(1), 013001 (1950).
[Crossref]

Li, D.

Z. D. Chen, C. H. Yuan, H. M. Ma, D. Li, L. Q. Chen, Z. Y. Ou, and W. Zhang, “Effects of losses in the atom-light hybrid SU (1, 1) interferometer,” Opt. Express 24(16), 17766–17778(2016).
[Crossref] [PubMed]

H. Ma, D. Li, C. H. Yuan, L. Q. Chen, Z. Y. Ou, and W. Zhang, “SU (1, 1)-type light-atom-correlated interferometer,” Phys. Rev. A 92(2), 023847 (2015).
[Crossref]

Linnemann, D.

D. Linnemann, H. Strobel, W. Muessel, J. Schulz, R. J. Lewis-Swan, K. V. Kheruntsyan, and M. K. Oberthaler, “Quantum-enhanced sensing based on time reversal of nonlinear dynamics,” Phys. Rev. Lett. 117(1), 013001 (1950).
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Lisdat, C.

I. Kruse, K. Lange, J. Peise, B. Lücke, L. Pezzè, J. Arlt, W. Ertmer, C. Lisdat, L. Santos, A. Smerzi, and C. Klempt, “Stark effect in rapidly varying fields,” Phys. Rev. 100(2), 703 (1955).
[Crossref]

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]

J. Jing, C. Liu, Z. Zhou, Z.Y. Ou, and W. Zhang, “Realization of a nonlinear interferometer with parametric amplifiers,” Appl. Phys. Lett. 99(1), 011110 (2011).
[Crossref]

Lücke, B.

I. Kruse, K. Lange, J. Peise, B. Lücke, L. Pezzè, J. Arlt, W. Ertmer, C. Lisdat, L. Santos, A. Smerzi, and C. Klempt, “Stark effect in rapidly varying fields,” Phys. Rev. 100(2), 703 (1955).
[Crossref]

Ma, H.

H. Ma, D. Li, C. H. Yuan, L. Q. Chen, Z. Y. Ou, and W. Zhang, “SU (1, 1)-type light-atom-correlated interferometer,” Phys. Rev. A 92(2), 023847 (2015).
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Ma, H. M.

Machida, S.

S. R. Friberg, S. Machida, and Y. Yamamoto, “Quantum-nondemolition measurement of the photon number of an optical soliton,” Phys. Rev. Lett. 69(22), 3165 (1992).
[Crossref] [PubMed]

Mandel, L.

A. Kuzmich, L. Mandel, J. Janis, Y. E. Young, R. Ejnisman, and N.P. Bigelow,“Quantum nondemolition measurements of collective atomic spin,” Phys. Rev. A 60(3), 2346 (1999).
[Crossref]

McCall, S. L.

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

G. J. Milburn, “Quantum optical Fredkin gate,” Phys. Rev. Lett. 62(18), 2124 (1989).
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Milburn, G.J.

B. Fan, A. F. Kockum, J. Combes, G. Johansson, I.C. Hoi, C.M. Wilson, P. Delsing, G.J. Milburn, and T.M. Stace, “Breakdown of the cross-Kerr scheme for photon counting,” Phys. Rev. Lett. 110(5), 053601 (2013).
[Crossref] [PubMed]

Muessel, W.

D. Linnemann, H. Strobel, W. Muessel, J. Schulz, R. J. Lewis-Swan, K. V. Kheruntsyan, and M. K. Oberthaler, “Quantum-enhanced sensing based on time reversal of nonlinear dynamics,” Phys. Rev. Lett. 117(1), 013001 (1950).
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Munro, W. J.

W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71(3), 033819 (2005).
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W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71(3), 033819 (2005).
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Nagorny, B.

D. Kruse, M. Ruder, J. Benhelm, C. V. Cube, C. Zimmermann, P. W. Courteille, T. Elsässer, B. Nagorny, and A. Hemmerich, “Cold atoms in a high-Q ring cavity,” Phys. Rev. A 67(5), 051802 (2003).
[Crossref]

Nemoto, K.

W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71(3), 033819 (2005).
[Crossref]

W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71(3), 033819 (2005).
[Crossref]

O’Brien, J. L.

G. J. Pryde, J. L. O’Brien, A. G. White, S. D. Bartlett, and T. C. Ralph, “Measuring a photonic qubit without destroying it,” Phys. Rev. Lett. 92(19), 190402 (2004).
[Crossref] [PubMed]

Oberthaler, M. K.

D. Linnemann, H. Strobel, W. Muessel, J. Schulz, R. J. Lewis-Swan, K. V. Kheruntsyan, and M. K. Oberthaler, “Quantum-enhanced sensing based on time reversal of nonlinear dynamics,” Phys. Rev. Lett. 117(1), 013001 (1950).
[Crossref]

Ou, Z. Y.

C. Qiu, S. Chen, L. Q. Chen, B. Chen, J. Guo, Z. Y. Ou, and W. Zhang, “Atom–light superposition oscillation and Ramsey-like atom–light interferometer,” Optica 3(7), 775–780 (2016).
[Crossref]

Z. D. Chen, C. H. Yuan, H. M. Ma, D. Li, L. Q. Chen, Z. Y. Ou, and W. Zhang, “Effects of losses in the atom-light hybrid SU (1, 1) interferometer,” Opt. Express 24(16), 17766–17778(2016).
[Crossref] [PubMed]

H. Ma, D. Li, C. H. Yuan, L. Q. Chen, Z. Y. Ou, and W. Zhang, “SU (1, 1)-type light-atom-correlated interferometer,” Phys. Rev. A 92(2), 023847 (2015).
[Crossref]

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

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(2), 023815 (2012).
[Crossref]

L. Q. Chen, C. Bian, G. W. Zhang, Z. Y. Ou, and W. Zhang, “Observation of temporal beating in first-and second-order intensity measurement between independent Raman Stokes fields in atomic vapor,” Phys. Rev. A 82(3), 033832 (2010).
[Crossref]

Z. Y. Ou, “Efficient conversion between photons and between photon and atom by stimulated emission,” Phys. Rev. A 78(2), 023819 (2008).
[Crossref]

S. F. Pereira, Z. Y. Ou, and H. J. Kimble, “Backaction evading measurements for quantum nondemolition detection and quantum optical tapping,” Phys. Rev. Lett. 72(2), 214 (1994).
[Crossref] [PubMed]

Ou, Z.Y.

J. Jing, C. Liu, Z. Zhou, Z.Y. Ou, and W. Zhang, “Realization of a nonlinear interferometer with parametric amplifiers,” Appl. Phys. Lett. 99(1), 011110 (2011).
[Crossref]

Peise, J.

I. Kruse, K. Lange, J. Peise, B. Lücke, L. Pezzè, J. Arlt, W. Ertmer, C. Lisdat, L. Santos, A. Smerzi, and C. Klempt, “Stark effect in rapidly varying fields,” Phys. Rev. 100(2), 703 (1955).
[Crossref]

Pereira, S. F.

S. F. Pereira, Z. Y. Ou, and H. J. Kimble, “Backaction evading measurements for quantum nondemolition detection and quantum optical tapping,” Phys. Rev. Lett. 72(2), 214 (1994).
[Crossref] [PubMed]

Pezzè, L.

M. Gabbrielli, L. Pezzè, and A. Smerzi, “Spin-mixing interferometry with Bose-Einstein condensates,” Phys. Rev. Lett. 115(16), 163002(2015).
[Crossref] [PubMed]

I. Kruse, K. Lange, J. Peise, B. Lücke, L. Pezzè, J. Arlt, W. Ertmer, C. Lisdat, L. Santos, A. Smerzi, and C. Klempt, “Stark effect in rapidly varying fields,” Phys. Rev. 100(2), 703 (1955).
[Crossref]

Poizat, J. P.

P. Grangier, J. A. Levenson, and J. P. Poizat, “Quantum non-demolition measurements in optics,” Nature 396(6711), 537–542 (1998).
[Crossref]

J. F. Roch, K. Vigneron, P. Grelu, A. Sinatra, J. P. Poizat, and P. Grangier, “Quantum Nondemolition Measurements using Cold Trapped Atoms,” Phys. Rev. Lett. 78(78), 634–637 (1997).
[Crossref]

J. P. Poizat and P. Grangier, “Experimental realization of a quantum optical tap,” Phys. Rev. Lett. 70(3), 271 (1993).
[Crossref] [PubMed]

Porta, A. L.

A. L. Porta, R. E. Slusher, and B. Yurke, “Back-action evading measurements of an optical field using parametric down conversion,” Phys. Rev. Lett. 62(1), 28 (1989).
[Crossref] [PubMed]

Pryde, G. J.

G. J. Pryde, J. L. O’Brien, A. G. White, S. D. Bartlett, and T. C. Ralph, “Measuring a photonic qubit without destroying it,” Phys. Rev. Lett. 92(19), 190402 (2004).
[Crossref] [PubMed]

Qiu, C.

C. Qiu, S. Chen, L. Q. Chen, B. Chen, J. Guo, Z. Y. Ou, and W. Zhang, “Atom–light superposition oscillation and Ramsey-like atom–light interferometer,” Optica 3(7), 775–780 (2016).
[Crossref]

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

Raimond, J.M.

C. Guerlin, J. Bernu, S. Deleglise, C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune, J.M. Raimond, and S. Haroche, “Progressive field-state collapse and quantum non-demolition photon counting,” Nature 448(7156), 889–893 (2007).
[Crossref] [PubMed]

Ralph, T. C.

G. J. Pryde, J. L. O’Brien, A. G. White, S. D. Bartlett, and T. C. Ralph, “Measuring a photonic qubit without destroying it,” Phys. Rev. Lett. 92(19), 190402 (2004).
[Crossref] [PubMed]

Ramsey, N. F.

N. F. Ramsey, “A molecular beam resonance method with separated oscillating fields,” Phys. Rev. 78(6), 695 (1950).
[Crossref]

Reid, M.

M. D. Levenson, R. M. Shelby, M. Reid, and D. F. Walls, “Quantum nondemolition detection of optical quadrature amplitudes,” Phys. Rev. Lett. 57(20), 2473 (1986).
[Crossref] [PubMed]

Reid, M. D.

M. D. Reid, “Demonstration of the Einstein-Podolsky-Rosen paradox using nondegenerate parametric amplification,” Phys. Rev. A 40(2), 913 (1989).
[Crossref]

Reynaud, S.

A. Heidmann, R. J. Horowicz, S. Reynaud, E. Giacobino, C. Fabre, and G. Camy, “Observation of quantum noise reduction on twin laser beams,” Phys. Rev. Lett. 59(22), 2555 (1987).
[Crossref] [PubMed]

Robins, N.P.

J. Geng, G. T. Campbell, J. Bernu, D. B. Higginbottom, B. M. Sparkes, S. M. Assad, W. P. Zhang, N.P. Robins, P. K. Lam, and B. C. Buchler, “Electromagnetically induced transparency and four-wave mixing in a cold atomic ensemble with large optical depth,” New J. Phys. 16(11), 113053 (2014).
[Crossref]

Roch, J. F.

J. F. Roch, K. Vigneron, P. Grelu, A. Sinatra, J. P. Poizat, and P. Grangier, “Quantum Nondemolition Measurements using Cold Trapped Atoms,” Phys. Rev. Lett. 78(78), 634–637 (1997).
[Crossref]

Ruder, M.

D. Kruse, M. Ruder, J. Benhelm, C. V. Cube, C. Zimmermann, P. W. Courteille, T. Elsässer, B. Nagorny, and A. Hemmerich, “Cold atoms in a high-Q ring cavity,” Phys. Rev. A 67(5), 051802 (2003).
[Crossref]

Santos, L.

I. Kruse, K. Lange, J. Peise, B. Lücke, L. Pezzè, J. Arlt, W. Ertmer, C. Lisdat, L. Santos, A. Smerzi, and C. Klempt, “Stark effect in rapidly varying fields,” Phys. Rev. 100(2), 703 (1955).
[Crossref]

Sayrin, C.

C. Guerlin, J. Bernu, S. Deleglise, C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune, J.M. Raimond, and S. Haroche, “Progressive field-state collapse and quantum non-demolition photon counting,” Nature 448(7156), 889–893 (2007).
[Crossref] [PubMed]

Schulz, J.

D. Linnemann, H. Strobel, W. Muessel, J. Schulz, R. J. Lewis-Swan, K. V. Kheruntsyan, and M. K. Oberthaler, “Quantum-enhanced sensing based on time reversal of nonlinear dynamics,” Phys. Rev. Lett. 117(1), 013001 (1950).
[Crossref]

Shelby, R. M.

M. D. Levenson, R. M. Shelby, M. Reid, and D. F. Walls, “Quantum nondemolition detection of optical quadrature amplitudes,” Phys. Rev. Lett. 57(20), 2473 (1986).
[Crossref] [PubMed]

Sinatra, A.

J. F. Roch, K. Vigneron, P. Grelu, A. Sinatra, J. P. Poizat, and P. Grangier, “Quantum Nondemolition Measurements using Cold Trapped Atoms,” Phys. Rev. Lett. 78(78), 634–637 (1997).
[Crossref]

Slusher, R. E.

A. L. Porta, R. E. Slusher, and B. Yurke, “Back-action evading measurements of an optical field using parametric down conversion,” Phys. Rev. Lett. 62(1), 28 (1989).
[Crossref] [PubMed]

Smerzi, A.

M. Gabbrielli, L. Pezzè, and A. Smerzi, “Spin-mixing interferometry with Bose-Einstein condensates,” Phys. Rev. Lett. 115(16), 163002(2015).
[Crossref] [PubMed]

I. Kruse, K. Lange, J. Peise, B. Lücke, L. Pezzè, J. Arlt, W. Ertmer, C. Lisdat, L. Santos, A. Smerzi, and C. Klempt, “Stark effect in rapidly varying fields,” Phys. Rev. 100(2), 703 (1955).
[Crossref]

Sparkes, B. M.

J. Geng, G. T. Campbell, J. Bernu, D. B. Higginbottom, B. M. Sparkes, S. M. Assad, W. P. Zhang, N.P. Robins, P. K. Lam, and B. C. Buchler, “Electromagnetically induced transparency and four-wave mixing in a cold atomic ensemble with large optical depth,” New J. Phys. 16(11), 113053 (2014).
[Crossref]

G. Campbell, M. Hosseini, B. M. Sparkes, P. K. Lam, and B. C. Buchler, “Time-and frequency-domain polariton interference,” New J. Phys. 14(3), 033022 (2012).
[Crossref]

Spiller, T. P.

W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71(3), 033819 (2005).
[Crossref]

W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71(3), 033819 (2005).
[Crossref]

Stace, T.M.

B. Fan, A. F. Kockum, J. Combes, G. Johansson, I.C. Hoi, C.M. Wilson, P. Delsing, G.J. Milburn, and T.M. Stace, “Breakdown of the cross-Kerr scheme for photon counting,” Phys. Rev. Lett. 110(5), 053601 (2013).
[Crossref] [PubMed]

Strobel, H.

D. Linnemann, H. Strobel, W. Muessel, J. Schulz, R. J. Lewis-Swan, K. V. Kheruntsyan, and M. K. Oberthaler, “Quantum-enhanced sensing based on time reversal of nonlinear dynamics,” Phys. Rev. Lett. 117(1), 013001 (1950).
[Crossref]

Szigeti, S. S.

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

Thorne, K. S.

B. V. Braginsky, Y. I. Vorontsov, and K. S. Thorne, “Quantum nondemolition measurements,” Science 209(4456), 547–557 (1980).
[Crossref] [PubMed]

Townes, C. H.

S. H. Autler and C. H. Townes, “Stark effect in rapidly varying fields,” Phys. Rev. 100(2), 703 (1955).
[Crossref]

Vigneron, K.

J. F. Roch, K. Vigneron, P. Grelu, A. Sinatra, J. P. Poizat, and P. Grangier, “Quantum Nondemolition Measurements using Cold Trapped Atoms,” Phys. Rev. Lett. 78(78), 634–637 (1997).
[Crossref]

Vorontsov, Y. I.

B. V. Braginsky, Y. I. Vorontsov, and K. S. Thorne, “Quantum nondemolition measurements,” Science 209(4456), 547–557 (1980).
[Crossref] [PubMed]

Walls, D. F.

M. J. Holland, M. J. Collett, D. F. Walls, and M. D. Levenson, “Nonideal quantum nondemolition measurements,” Phys. Rev. A 42(5), 2995 (1990).
[Crossref] [PubMed]

M. D. Levenson, R. M. Shelby, M. Reid, and D. F. Walls, “Quantum nondemolition detection of optical quadrature amplitudes,” Phys. Rev. Lett. 57(20), 2473 (1986).
[Crossref] [PubMed]

White, A. G.

G. J. Pryde, J. L. O’Brien, A. G. White, S. D. Bartlett, and T. C. Ralph, “Measuring a photonic qubit without destroying it,” Phys. Rev. Lett. 92(19), 190402 (2004).
[Crossref] [PubMed]

Wilson, C.M.

B. Fan, A. F. Kockum, J. Combes, G. Johansson, I.C. Hoi, C.M. Wilson, P. Delsing, G.J. Milburn, and T.M. Stace, “Breakdown of the cross-Kerr scheme for photon counting,” Phys. Rev. Lett. 110(5), 053601 (2013).
[Crossref] [PubMed]

Yamamoto, Y.

J. Jacobson, G. Björk, and Y. Yamamoto, “Quantum limit for the atom-light interferometer,” Appl. Phys. B: Lasers and Optics,  60(2), 187–191 (1995).
[Crossref]

S. R. Friberg, S. Machida, and Y. Yamamoto, “Quantum-nondemolition measurement of the photon number of an optical soliton,” Phys. Rev. Lett. 69(22), 3165 (1992).
[Crossref] [PubMed]

N. Imoto, H. A. Haus, and Y. Yamamoto, “Quantum nondemolition measurement of the photon number via the optical Kerr effect,” Phys. Rev. A 32(4), 2287 (1985).
[Crossref]

Young, Y. E.

A. Kuzmich, L. Mandel, J. Janis, Y. E. Young, R. Ejnisman, and N.P. Bigelow,“Quantum nondemolition measurements of collective atomic spin,” Phys. Rev. A 60(3), 2346 (1999).
[Crossref]

Yuan, C. H.

Z. D. Chen, C. H. Yuan, H. M. Ma, D. Li, L. Q. Chen, Z. Y. Ou, and W. Zhang, “Effects of losses in the atom-light hybrid SU (1, 1) interferometer,” Opt. Express 24(16), 17766–17778(2016).
[Crossref] [PubMed]

H. Ma, D. Li, C. H. Yuan, L. Q. Chen, Z. Y. Ou, and W. Zhang, “SU (1, 1)-type light-atom-correlated interferometer,” Phys. Rev. A 92(2), 023847 (2015).
[Crossref]

Yurke, B.

A. L. Porta, R. E. Slusher, and B. Yurke, “Back-action evading measurements of an optical field using parametric down conversion,” Phys. Rev. Lett. 62(1), 28 (1989).
[Crossref] [PubMed]

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

Zhang, G. W.

L. Q. Chen, C. Bian, G. W. Zhang, Z. Y. Ou, and W. Zhang, “Observation of temporal beating in first-and second-order intensity measurement between independent Raman Stokes fields in atomic vapor,” Phys. Rev. A 82(3), 033832 (2010).
[Crossref]

Zhang, W.

C. Qiu, S. Chen, L. Q. Chen, B. Chen, J. Guo, Z. Y. Ou, and W. Zhang, “Atom–light superposition oscillation and Ramsey-like atom–light interferometer,” Optica 3(7), 775–780 (2016).
[Crossref]

Z. D. Chen, C. H. Yuan, H. M. Ma, D. Li, L. Q. Chen, Z. Y. Ou, and W. Zhang, “Effects of losses in the atom-light hybrid SU (1, 1) interferometer,” Opt. Express 24(16), 17766–17778(2016).
[Crossref] [PubMed]

H. Ma, D. Li, C. H. Yuan, L. Q. Chen, Z. Y. Ou, and W. Zhang, “SU (1, 1)-type light-atom-correlated interferometer,” Phys. Rev. A 92(2), 023847 (2015).
[Crossref]

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

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]

J. Jing, C. Liu, Z. Zhou, Z.Y. Ou, and W. Zhang, “Realization of a nonlinear interferometer with parametric amplifiers,” Appl. Phys. Lett. 99(1), 011110 (2011).
[Crossref]

L. Q. Chen, C. Bian, G. W. Zhang, Z. Y. Ou, and W. Zhang, “Observation of temporal beating in first-and second-order intensity measurement between independent Raman Stokes fields in atomic vapor,” Phys. Rev. A 82(3), 033832 (2010).
[Crossref]

Zhang, W. P.

J. Geng, G. T. Campbell, J. Bernu, D. B. Higginbottom, B. M. Sparkes, S. M. Assad, W. P. Zhang, N.P. Robins, P. K. Lam, and B. C. Buchler, “Electromagnetically induced transparency and four-wave mixing in a cold atomic ensemble with large optical depth,” New J. Phys. 16(11), 113053 (2014).
[Crossref]

Zhou, Z.

J. Jing, C. Liu, Z. Zhou, Z.Y. Ou, and W. Zhang, “Realization of a nonlinear interferometer with parametric amplifiers,” Appl. Phys. Lett. 99(1), 011110 (2011).
[Crossref]

Zimmermann, C.

D. Kruse, M. Ruder, J. Benhelm, C. V. Cube, C. Zimmermann, P. W. Courteille, T. Elsässer, B. Nagorny, and A. Hemmerich, “Cold atoms in a high-Q ring cavity,” Phys. Rev. A 67(5), 051802 (2003).
[Crossref]

Appl. Phys. B: Lasers and Optics (1)

J. Jacobson, G. Björk, and Y. Yamamoto, “Quantum limit for the atom-light interferometer,” Appl. Phys. B: Lasers and Optics,  60(2), 187–191 (1995).
[Crossref]

Appl. Phys. Lett. (1)

J. Jing, C. Liu, Z. Zhou, Z.Y. Ou, and W. Zhang, “Realization of a nonlinear interferometer with parametric amplifiers,” Appl. Phys. Lett. 99(1), 011110 (2011).
[Crossref]

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]

Nature (2)

P. Grangier, J. A. Levenson, and J. P. Poizat, “Quantum non-demolition measurements in optics,” Nature 396(6711), 537–542 (1998).
[Crossref]

C. Guerlin, J. Bernu, S. Deleglise, C. Sayrin, S. Gleyzes, S. Kuhr, M. Brune, J.M. Raimond, and S. Haroche, “Progressive field-state collapse and quantum non-demolition photon counting,” Nature 448(7156), 889–893 (2007).
[Crossref] [PubMed]

New J. Phys. (2)

G. Campbell, M. Hosseini, B. M. Sparkes, P. K. Lam, and B. C. Buchler, “Time-and frequency-domain polariton interference,” New J. Phys. 14(3), 033022 (2012).
[Crossref]

J. Geng, G. T. Campbell, J. Bernu, D. B. Higginbottom, B. M. Sparkes, S. M. Assad, W. P. Zhang, N.P. Robins, P. K. Lam, and B. C. Buchler, “Electromagnetically induced transparency and four-wave mixing in a cold atomic ensemble with large optical depth,” New J. Phys. 16(11), 113053 (2014).
[Crossref]

Opt. Express (1)

Optica (1)

Phys. Rev. (3)

I. Kruse, K. Lange, J. Peise, B. Lücke, L. Pezzè, J. Arlt, W. Ertmer, C. Lisdat, L. Santos, A. Smerzi, and C. Klempt, “Stark effect in rapidly varying fields,” Phys. Rev. 100(2), 703 (1955).
[Crossref]

S. H. Autler and C. H. Townes, “Stark effect in rapidly varying fields,” Phys. Rev. 100(2), 703 (1955).
[Crossref]

N. F. Ramsey, “A molecular beam resonance method with separated oscillating fields,” Phys. Rev. 78(6), 695 (1950).
[Crossref]

Phys. Rev. A (14)

S. A. Haine and W. Y. S. Lau, “Generation of atom-light entanglement in an optical cavity for quantum enhanced atom interferometry,” Phys. Rev. A 93(2), 023607(2016).
[Crossref]

A. Kuzmich, L. Mandel, J. Janis, Y. E. Young, R. Ejnisman, and N.P. Bigelow,“Quantum nondemolition measurements of collective atomic spin,” Phys. Rev. A 60(3), 2346 (1999).
[Crossref]

W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71(3), 033819 (2005).
[Crossref]

N. Imoto, H. A. Haus, and Y. Yamamoto, “Quantum nondemolition measurement of the photon number via the optical Kerr effect,” Phys. Rev. A 32(4), 2287 (1985).
[Crossref]

P. Kok, H. Lee, and J. P. Dowling, “Single-photon quantum-nondemolition detectors constructed with linear optics and projective measurements,” Phys. Rev. A 66(6), 063814(2002).
[Crossref]

W. J. Munro, K. Nemoto, R. G. Beausoleil, and T. P. Spiller, “High-efficiency quantum-nondemolition single-photon-number-resolving detector,” Phys. Rev. A 71(3), 033819 (2005).
[Crossref]

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

M. J. Holland, M. J. Collett, D. F. Walls, and M. D. Levenson, “Nonideal quantum nondemolition measurements,” Phys. Rev. A 42(5), 2995 (1990).
[Crossref] [PubMed]

H. Ma, D. Li, C. H. Yuan, L. Q. Chen, Z. Y. Ou, and W. Zhang, “SU (1, 1)-type light-atom-correlated interferometer,” Phys. Rev. A 92(2), 023847 (2015).
[Crossref]

Z. Y. Ou, “Efficient conversion between photons and between photon and atom by stimulated emission,” Phys. Rev. A 78(2), 023819 (2008).
[Crossref]

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

L. Q. Chen, C. Bian, G. W. Zhang, Z. Y. Ou, and W. Zhang, “Observation of temporal beating in first-and second-order intensity measurement between independent Raman Stokes fields in atomic vapor,” Phys. Rev. A 82(3), 033832 (2010).
[Crossref]

M. D. Reid, “Demonstration of the Einstein-Podolsky-Rosen paradox using nondegenerate parametric amplification,” Phys. Rev. A 40(2), 913 (1989).
[Crossref]

D. Kruse, M. Ruder, J. Benhelm, C. V. Cube, C. Zimmermann, P. W. Courteille, T. Elsässer, B. Nagorny, and A. Hemmerich, “Cold atoms in a high-Q ring cavity,” Phys. Rev. A 67(5), 051802 (2003).
[Crossref]

Phys. Rev. D (1)

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

Phys. Rev. Lett. (15)

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

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

M. Gabbrielli, L. Pezzè, and A. Smerzi, “Spin-mixing interferometry with Bose-Einstein condensates,” Phys. Rev. Lett. 115(16), 163002(2015).
[Crossref] [PubMed]

G. J. Pryde, J. L. O’Brien, A. G. White, S. D. Bartlett, and T. C. Ralph, “Measuring a photonic qubit without destroying it,” Phys. Rev. Lett. 92(19), 190402 (2004).
[Crossref] [PubMed]

G. J. Milburn, “Quantum optical Fredkin gate,” Phys. Rev. Lett. 62(18), 2124 (1989).
[Crossref] [PubMed]

B. Fan, A. F. Kockum, J. Combes, G. Johansson, I.C. Hoi, C.M. Wilson, P. Delsing, G.J. Milburn, and T.M. Stace, “Breakdown of the cross-Kerr scheme for photon counting,” Phys. Rev. Lett. 110(5), 053601 (2013).
[Crossref] [PubMed]

A. L. Porta, R. E. Slusher, and B. Yurke, “Back-action evading measurements of an optical field using parametric down conversion,” Phys. Rev. Lett. 62(1), 28 (1989).
[Crossref] [PubMed]

J. P. Poizat and P. Grangier, “Experimental realization of a quantum optical tap,” Phys. Rev. Lett. 70(3), 271 (1993).
[Crossref] [PubMed]

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Figures (4)

Fig. 1
Fig. 1 (a) Schematic of QND measurement with atom-light interferometers. |g〉 and |m〉 are two ground state levels; |e1〉 and |e2〉 are two excited state levels. Atoms are initially prepared in the |g〉 ground state. The laser frequency of each optical field and corresponding signatures of pump, optical and atomic fields are given in (b) and (c). (b) In the linear interferometer, the strong pump field labeled as S, couples the state |m〉 and |e1〉, and generates an optical signal field a ^ W 1 and atomic spin wave S ^ a 1 , as two beams of the interferometer. The RS is shorted for Raman system. (c) In the nonlinear interferometer, the strong pump field labeled as W, couples |g〉 and |e1〉 and generates an optical signal a ^ S 1 and atomic spin wave S ^ a 1 . Between the two Raman processes in the atom-light interferometers, the atomic spin wave experiences a phase modulatiion ΔφAC via the AC Stark effect by probe field N ^ p .
Fig. 2
Fig. 2 QND measurement on the twin beams with two atom-light interferometers (ALI). The subscriptions s and i mean signal and idler. Xs and Xi are the output fields of the atom-light interferometers for signal and idler beams.
Fig. 3
Fig. 3 Sequential QND measurement with atom-light interferometers(ALI). XS and X s final quadrature outputs of ALI and ALI’.
Fig. 4
Fig. 4 The quadrature amplitude X ^ S o u t as a function of the probe photon number np, showing that the meter signal is sensitive to probe photon number. The errorbar is the fluctuation of meter signal quadrature amplitude Δ 2 X ^ S o u t . The data used in this graph is calculated at κ = 5.30 × 10−4rad, G 2 = 10, and Ips = 8.91 × 105.

Equations (35)

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Δ ω A C = Ω R 2 / Δ ,
φ A C = Δ ω A C Δ T = | μ g e | 2 I Δ T c n ϵ 0 Δ = | μ g e | 2 c n ϵ 0 Δ P Δ T A = | μ g e | 2 c n ϵ 0 Δ N p A = κ N p ,
a ^ W o u t = a ^ W i n cos φ / 2 + S ^ a i n sin φ / 2 S ^ W o u t = S ^ a i n cos φ / 2 a ^ W i n sin φ / 2 ,
N ^ W = a ^ W o u t a ^ W o u t = a ^ W i n a ^ W i n ( 1 + cos φ ) / 2 + S ^ a i n S ^ a i n ( 1 cos φ ) / 2 + sin φ ( a ^ W i n S ^ a i n + h . c . ) / 2 .
N ^ W S ^ a i n S ^ a i n ( 1 + Δ φ A C ) / 2 + ( a ^ W i n S ^ a i n + h . c . ) / 2 = S ^ a i n S ^ a i n ( 1 + κ N ^ p ) / 2 + ( a ^ W i n S ^ a i n + h . c . ) / 2 .
a ^ S o u t = ( G 2 e i φ + g 2 ) a ^ S i n + G g ( 1 + e i φ ) S ^ a i n = G T ( φ ) a ^ S i n + g T ( φ ) S ^ a i n , S ^ a o u t = G g ( 1 + e i φ ) a ^ S i n + ( G 2 e i φ + g 2 ) S ^ a i n = G T ( φ ) a ^ S i n + g T ( φ ) S ^ a i n ,
a ^ S o u t = ( 1 + i G 2 Δ φ A C ) a ^ S i n i G g Δ φ A C S ^ a i n .
X ^ S o u t = a ^ S o u t + a ^ S o u t = X S i n + G 2 Δ φ A C Y ^ S i n G g Δ φ A C Y ^ a i n = X S i n + κ G N ^ p ( G Y ^ S i n g Y ^ a i n ) ,
N ^ W = | α | 2 ( 1 + κ n p ) / 2 .
Δ 2 N ^ W = | α | 2 ( 1 + κ n p ) 2 / 4 + | α | 2 / 4 | α | 2 / 2 .
R L = N ^ W s 2 / Δ 2 N ^ W = | α | 2 κ 2 n p 2 / 2 = κ 2 n p 2 I p s ,
X ^ S o u t = G g κ n p Y ^ a i n ,
Δ 2 X ^ S o u t = 1 + G 2 ( G 2 + g 2 ) κ 2 n p 2 .
R = | X ^ S o u t | 2 Δ 2 X ^ S o u t = 4 G 2 g 2 κ 2 n p 2 | α | 2 1 + G 2 ( G 2 + g 2 ) κ 2 n p 2 4 G 2 g 2 κ 2 n p 2 | α | 2 = 4 G 2 κ 2 n p 2 I p s .
C S i n , M | Δ S i n Δ M | Δ 2 S i n Δ 2 M .
V a r ( S o u t | M ) = V a r ( S o u t ) ( 1 C S o u t , M 2 ) .
V a r ( S o u t ) ( 1 C S o u t , M 2 ) ( Δ S o u t λ Δ M ) 2 m ,
C N W N p | Δ N ^ W Δ N ^ p | Δ 2 N ^ W Δ 2 N ^ p .
Δ N ^ W Δ N ^ p = κ | α | 2 | α p | 2 / 2 .
Δ 2 N ^ W = | α | 2 2 ( 1 + κ | α p | 2 ) + κ 2 | α | 2 | α p | 2 4 ( | α | 2 + | α p | 2 + 1 ) .
Δ 2 N ^ p | α p | 2 .
C N W , N p = 1 1 + 1 + | α p | 2 2 I p s + 1 + κ | α p | 2 κ 2 I p s | α p | 2
C N p X S | Δ N ^ p Δ X ^ S | Δ 2 N ^ p Δ 2 X ^ S .
Δ X ^ S Δ N ^ p = 2 G g | α | κ | α p | 2 .
Δ 2 X ^ S = 1 + G 2 g 2 κ 2 | α p | 2 ( 4 | α | 2 + | α p | 2 + 1 ) + G 4 κ 2 | α p | 2 ( | α p | 2 + 1 ) .
C N p X S = 1 1 + ( 2 g 2 + 1 ) ( 1 + | α p | 2 ) 4 I p s + 1 4 G 2 I p s κ 2 | α p | 2 .
| TwBm = n = 0 n ¯ n ( n ¯ + 1 ) n + 1 | n s | n i ,
C X S ( s ) X S ( i ) = 1 1 + 2 n ¯ + 1 n ¯ + 1 G 2 + g 2 4 I p s + 1 4 G 2 κ 2 I p s n ¯ ( n ¯ + 1 ) .
( Δ X ^ S ( s ) λ Δ X ^ S ( i ) ) 2 Δ 2 X ^ S ( s ) ( 1 C X S ( s ) X S ( i ) 2 ) .
C X S X S | Δ X ^ S Δ X ^ S | Δ 2 X ^ S Δ 2 X ^ S .
( Δ N p λ Δ X S λ Δ X S ) 2 = V a r ( N p | X S , X S ) ,
V a r ( N p | X s , X S ) = Δ 2 N p ( 1 C N p X s 2 + C N p X S 2 2 C N p X S C N p X S C X S X S 1 C X S X S 2 ) = Δ 2 N p 1 C N p X s 2 1 + C N p X s 2 = V a r ( N p | X S ) 1 + C N p X s 2 .
T N p = R N p o u t R N p i n , T X S = R X S o u t R N p i n , T X S = R X S o u t R N p i n ,
T X S = R X S o u t R N p i n = C N p X S 2 = T X S .
T N p + T X S = T N p + T X S = 1 + C N p X S 2 > 1 .

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