Abstract

We present a complete analysis of the amplification process occurring in a nonlinear fiber, either driven with one or two pumps. After determining the solution for the signal and idler fields resulting from this amplification, we analyze the physical transformations undergone by these fields. To this aim, we use a Bloch–Messiah decomposition for the symplectic transformation governing the fields’ evolution. This analysis is shown to be particularly simple and permits us not only to gather results already present in other approaches [Opt. Express 21, 1374 (2013)], but also to predict original features of parametric amplification in fibers. In particular, we present a study of the correlations of the signal-idler fields at the amplifier output, and we clarify the impact these correlations have on the amplifier noise figure. Finally, we address the effect of losses. We determine whether it is advantageous to consider a link consisting in an amplifying nonlinear fiber, followed by a standard fiber-based lossy transmission line or whether the two elements should be reversed by comparing the respective noise figures.

© 2014 Optical Society of America

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    [CrossRef]
  7. Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430 (2011).
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    [CrossRef]
  11. C. Lundström, B. Corcoran, M. Karlsson, and P. A. Andrekson, “Phase and amplitude characteristics of a phase-sensitive amplifier operating in gain saturation,” Opt. Express 20, 21400–21412 (2012).
    [CrossRef]
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    [CrossRef]
  20. M. E. Marhic, “Polarization independence and phase-sensitive parametric amplification,” J. Opt. Soc. Am. B 28, 2685–2689 (2011).
    [CrossRef]
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    [CrossRef]
  22. M. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71, 1539–1589 (1999).
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  23. S. L. Braunstein, “Squeezing as an irreducible resource,” Phys. Rev. A 71, 055801 (2005).
    [CrossRef]
  24. A. B. Dutta, N. Mukunda, and R. Simon, “The real symplectic groups in quantum mechanics and optics,” Pramana 45, 471–497 (1995).
    [CrossRef]
  25. R. Loudon, The Quantum Theory of Light (Oxford University, 2000).
  26. G. Grynberg, A. Aspect, and C. Fabre, Introduction to Quantum Optics (Cambridge University, 2010).
  27. J. Hansryd, P. A. Andrekson, M. Westlund, L. Jie, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
    [CrossRef]
  28. Z. Chen, L. Yan, W. Pan, B. Luo, A. Yi, J. Ye, H. Jiang, and Y. Guo, “Phase sensitive amplifier for PSK signals based on non-degenerate four-wave-mixing in the optical fiber,” Opt. Commun. 285, 2445–2450 (2012).
    [CrossRef]
  29. R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 3028–3038 (1988).
    [CrossRef]
  30. See http://www.specialtyphotonics.com .
  31. Note that since the expression of ϕ in terms of the fiber parameters reported in Eq. (A4) also involves the pump phases, the signal phase is in this way unambiguously defined.
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    [CrossRef]
  33. O. Pinel, J. Fade, D. Braun, P. Jian, N. Treps, and C. Fabre, “Ultimate sensitivity of precision measurements with intense Gaussian quantum light: a multimodal approach,” Phys. Rev. A 85, 010101(R) (2012).
    [CrossRef]
  34. S. Haroche and J. M. Raymond, Exploring the Quantum (Oxford Graduate Texts, 2008).
  35. A. Furusawa and P. Van Loock, Quantum Teleportation and Entanglement (Wiley-VCH, 2011).
  36. L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
    [CrossRef]
  37. E. Desurvire, Erbium-Doped Fiber Amplifiers (Wiley, 2002).
  38. N. V. Corzo, Q. Glorieux, A. M. Marino, J. B. Clark, and P. D. Lett, “Rotation of the noise ellipse for squeezed vacuum light generated via four-wave-mixing,” Phys. Rev. A 88, 043836 (2013).
    [CrossRef]

2013 (5)

2012 (5)

C. Lundström, B. Corcoran, M. Karlsson, and P. A. Andrekson, “Phase and amplitude characteristics of a phase-sensitive amplifier operating in gain saturation,” Opt. Express 20, 21400–21412 (2012).
[CrossRef]

M. E. Marhic, “Noise figure of hybrid optical parametric amplifiers,” Opt. Express 20, 28752–28757 (2012).
[CrossRef]

O. Pinel, J. Fade, D. Braun, P. Jian, N. Treps, and C. Fabre, “Ultimate sensitivity of precision measurements with intense Gaussian quantum light: a multimodal approach,” Phys. Rev. A 85, 010101(R) (2012).
[CrossRef]

Z. Chen, L. Yan, W. Pan, B. Luo, A. Yi, J. Ye, H. Jiang, and Y. Guo, “Phase sensitive amplifier for PSK signals based on non-degenerate four-wave-mixing in the optical fiber,” Opt. Commun. 285, 2445–2450 (2012).
[CrossRef]

Z. Tong, C. Lundström, P. A. Andrekson, M. Karlsson, and A. Bogris, “Ultralow noise, broadband phase-sensitive optical amplifiers, and their applications,” IEEE J. Sel. Top. Quantum Electron. 18, 1016–1032 (2012).
[CrossRef]

2011 (2)

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430 (2011).
[CrossRef]

M. E. Marhic, “Polarization independence and phase-sensitive parametric amplification,” J. Opt. Soc. Am. B 28, 2685–2689 (2011).
[CrossRef]

2010 (3)

Z. Tong, C. J. McKinstrie, C. Lundström, M. Karlsson, and P. Andrekson, “Noise performance of optical fiber transmission links that use non-degenerate cascaded phase-sensitive amplifiers,” Opt. Express 18, 15426–15439 (2010).
[CrossRef]

C. J. McKinstrie, M. Karlsson, and Z. Tong, “Field-quadrature and photon-number correlations produced by parametric processes,” Opt. Express 18, 19792–19823 (2010).
[CrossRef]

R. Slav’k, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrom, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

2006 (1)

C. J. McKinstrie, M. Raymer, S. Radic, and M. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” Opt. Commun. 257, 146–163 (2006).
[CrossRef]

2005 (3)

2004 (2)

2002 (1)

J. Hansryd, P. A. Andrekson, M. Westlund, L. Jie, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[CrossRef]

2001 (1)

2000 (1)

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
[CrossRef]

1999 (1)

M. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71, 1539–1589 (1999).
[CrossRef]

1995 (1)

A. B. Dutta, N. Mukunda, and R. Simon, “The real symplectic groups in quantum mechanics and optics,” Pramana 45, 471–497 (1995).
[CrossRef]

1988 (1)

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 3028–3038 (1988).
[CrossRef]

1985 (1)

C. L. Caves and B. M. Schumaker, “New formalism for two-photon quantum optics. I. Quadrature phases and squeezed states,” Phys. Rev. A 31, 3068–3092 (1985).
[CrossRef]

1982 (1)

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26, 1817–1839 (1982).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2013).

Andrekson, P.

Andrekson, P. A.

C. Lundström, B. Corcoran, M. Karlsson, and P. A. Andrekson, “Phase and amplitude characteristics of a phase-sensitive amplifier operating in gain saturation,” Opt. Express 20, 21400–21412 (2012).
[CrossRef]

Z. Tong, C. Lundström, P. A. Andrekson, M. Karlsson, and A. Bogris, “Ultralow noise, broadband phase-sensitive optical amplifiers, and their applications,” IEEE J. Sel. Top. Quantum Electron. 18, 1016–1032 (2012).
[CrossRef]

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430 (2011).
[CrossRef]

R. Slav’k, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrom, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

J. Hansryd, P. A. Andrekson, M. Westlund, L. Jie, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[CrossRef]

Aspect, A.

G. Grynberg, A. Aspect, and C. Fabre, Introduction to Quantum Optics (Cambridge University, 2010).

Blessing, D. J.

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430 (2011).
[CrossRef]

Bogris, A.

Z. Tong, C. Lundström, P. A. Andrekson, M. Karlsson, and A. Bogris, “Ultralow noise, broadband phase-sensitive optical amplifiers, and their applications,” IEEE J. Sel. Top. Quantum Electron. 18, 1016–1032 (2012).
[CrossRef]

R. Slav’k, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrom, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Braun, D.

O. Pinel, J. Fade, D. Braun, P. Jian, N. Treps, and C. Fabre, “Ultimate sensitivity of precision measurements with intense Gaussian quantum light: a multimodal approach,” Phys. Rev. A 85, 010101(R) (2012).
[CrossRef]

Braunstein, S. L.

S. L. Braunstein, “Squeezing as an irreducible resource,” Phys. Rev. A 71, 055801 (2005).
[CrossRef]

Caves, C. L.

C. L. Caves and B. M. Schumaker, “New formalism for two-photon quantum optics. I. Quadrature phases and squeezed states,” Phys. Rev. A 31, 3068–3092 (1985).
[CrossRef]

Caves, C. M.

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26, 1817–1839 (1982).
[CrossRef]

Chen, Z.

Z. Chen, L. Yan, W. Pan, B. Luo, A. Yi, J. Ye, H. Jiang, and Y. Guo, “Phase sensitive amplifier for PSK signals based on non-degenerate four-wave-mixing in the optical fiber,” Opt. Commun. 285, 2445–2450 (2012).
[CrossRef]

Cirac, J. I.

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
[CrossRef]

Clark, J. B.

N. V. Corzo, Q. Glorieux, A. M. Marino, J. B. Clark, and P. D. Lett, “Rotation of the noise ellipse for squeezed vacuum light generated via four-wave-mixing,” Phys. Rev. A 88, 043836 (2013).
[CrossRef]

Corcoran, B.

Corzo, N. V.

N. V. Corzo, Q. Glorieux, A. M. Marino, J. B. Clark, and P. D. Lett, “Rotation of the noise ellipse for squeezed vacuum light generated via four-wave-mixing,” Phys. Rev. A 88, 043836 (2013).
[CrossRef]

Dasgupta, S.

R. Slav’k, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrom, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Desurvire, E.

E. Desurvire, Erbium-Doped Fiber Amplifiers (Wiley, 2002).

Duan, L.-M.

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
[CrossRef]

Dutta, A. B.

A. B. Dutta, N. Mukunda, and R. Simon, “The real symplectic groups in quantum mechanics and optics,” Pramana 45, 471–497 (1995).
[CrossRef]

Ellis, A. D.

R. Slav’k, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrom, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Fabre, C.

O. Pinel, J. Fade, D. Braun, P. Jian, N. Treps, and C. Fabre, “Ultimate sensitivity of precision measurements with intense Gaussian quantum light: a multimodal approach,” Phys. Rev. A 85, 010101(R) (2012).
[CrossRef]

G. Grynberg, A. Aspect, and C. Fabre, Introduction to Quantum Optics (Cambridge University, 2010).

Fade, J.

O. Pinel, J. Fade, D. Braun, P. Jian, N. Treps, and C. Fabre, “Ultimate sensitivity of precision measurements with intense Gaussian quantum light: a multimodal approach,” Phys. Rev. A 85, 010101(R) (2012).
[CrossRef]

Fiorentino, M.

Furusawa, A.

A. Furusawa and P. Van Loock, Quantum Teleportation and Entanglement (Wiley-VCH, 2011).

Giedke, G.

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
[CrossRef]

Glorieux, Q.

N. V. Corzo, Q. Glorieux, A. M. Marino, J. B. Clark, and P. D. Lett, “Rotation of the noise ellipse for squeezed vacuum light generated via four-wave-mixing,” Phys. Rev. A 88, 043836 (2013).
[CrossRef]

Grüner-Nielsen, L.

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430 (2011).
[CrossRef]

R. Slav’k, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrom, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Grynberg, G.

G. Grynberg, A. Aspect, and C. Fabre, Introduction to Quantum Optics (Cambridge University, 2010).

Guo, Y.

Z. Chen, L. Yan, W. Pan, B. Luo, A. Yi, J. Ye, H. Jiang, and Y. Guo, “Phase sensitive amplifier for PSK signals based on non-degenerate four-wave-mixing in the optical fiber,” Opt. Commun. 285, 2445–2450 (2012).
[CrossRef]

Hansryd, J.

J. Hansryd, P. A. Andrekson, M. Westlund, L. Jie, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[CrossRef]

Haroche, S.

S. Haroche and J. M. Raymond, Exploring the Quantum (Oxford Graduate Texts, 2008).

Hedekvist, P.

J. Hansryd, P. A. Andrekson, M. Westlund, L. Jie, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[CrossRef]

Herstrom, S.

R. Slav’k, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrom, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Jakobsen, D.

R. Slav’k, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrom, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Jian, P.

O. Pinel, J. Fade, D. Braun, P. Jian, N. Treps, and C. Fabre, “Ultimate sensitivity of precision measurements with intense Gaussian quantum light: a multimodal approach,” Phys. Rev. A 85, 010101(R) (2012).
[CrossRef]

Jiang, H.

Z. Chen, L. Yan, W. Pan, B. Luo, A. Yi, J. Ye, H. Jiang, and Y. Guo, “Phase sensitive amplifier for PSK signals based on non-degenerate four-wave-mixing in the optical fiber,” Opt. Commun. 285, 2445–2450 (2012).
[CrossRef]

Jie, L.

J. Hansryd, P. A. Andrekson, M. Westlund, L. Jie, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[CrossRef]

Kakande, J.

R. Slav’k, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrom, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Karlsson, M.

C. J. McKinstrie, J. R. Ott, and M. Karlsson, “Schmidt decompositions of parametric processes II: vector four-wave mixing,” Opt. Express 21, 11009–11020 (2013).
[CrossRef]

C. J. McKinstrie and M. Karlsson, “Schmidt decompositions of parametric processes I: basic theory and simple examples,” Opt. Express 21, 1374–1394 (2013).
[CrossRef]

C. Lundström, B. Corcoran, M. Karlsson, and P. A. Andrekson, “Phase and amplitude characteristics of a phase-sensitive amplifier operating in gain saturation,” Opt. Express 20, 21400–21412 (2012).
[CrossRef]

Z. Tong, C. Lundström, P. A. Andrekson, M. Karlsson, and A. Bogris, “Ultralow noise, broadband phase-sensitive optical amplifiers, and their applications,” IEEE J. Sel. Top. Quantum Electron. 18, 1016–1032 (2012).
[CrossRef]

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430 (2011).
[CrossRef]

Z. Tong, C. J. McKinstrie, C. Lundström, M. Karlsson, and P. Andrekson, “Noise performance of optical fiber transmission links that use non-degenerate cascaded phase-sensitive amplifiers,” Opt. Express 18, 15426–15439 (2010).
[CrossRef]

C. J. McKinstrie, M. Karlsson, and Z. Tong, “Field-quadrature and photon-number correlations produced by parametric processes,” Opt. Express 18, 19792–19823 (2010).
[CrossRef]

Kolobov, M.

M. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71, 1539–1589 (1999).
[CrossRef]

Kumar, P.

Lett, P. D.

N. V. Corzo, Q. Glorieux, A. M. Marino, J. B. Clark, and P. D. Lett, “Rotation of the noise ellipse for squeezed vacuum light generated via four-wave-mixing,” Phys. Rev. A 88, 043836 (2013).
[CrossRef]

Loudon, R.

R. Loudon, The Quantum Theory of Light (Oxford University, 2000).

Lundström, C.

C. Lundström, B. Corcoran, M. Karlsson, and P. A. Andrekson, “Phase and amplitude characteristics of a phase-sensitive amplifier operating in gain saturation,” Opt. Express 20, 21400–21412 (2012).
[CrossRef]

Z. Tong, C. Lundström, P. A. Andrekson, M. Karlsson, and A. Bogris, “Ultralow noise, broadband phase-sensitive optical amplifiers, and their applications,” IEEE J. Sel. Top. Quantum Electron. 18, 1016–1032 (2012).
[CrossRef]

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430 (2011).
[CrossRef]

Z. Tong, C. J. McKinstrie, C. Lundström, M. Karlsson, and P. Andrekson, “Noise performance of optical fiber transmission links that use non-degenerate cascaded phase-sensitive amplifiers,” Opt. Express 18, 15426–15439 (2010).
[CrossRef]

R. Slav’k, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrom, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Luo, B.

Z. Chen, L. Yan, W. Pan, B. Luo, A. Yi, J. Ye, H. Jiang, and Y. Guo, “Phase sensitive amplifier for PSK signals based on non-degenerate four-wave-mixing in the optical fiber,” Opt. Commun. 285, 2445–2450 (2012).
[CrossRef]

Marhic, M. E.

Marino, A. M.

N. V. Corzo, Q. Glorieux, A. M. Marino, J. B. Clark, and P. D. Lett, “Rotation of the noise ellipse for squeezed vacuum light generated via four-wave-mixing,” Phys. Rev. A 88, 043836 (2013).
[CrossRef]

McKinstrie, C. J.

C. J. McKinstrie, J. R. Ott, and M. Karlsson, “Schmidt decompositions of parametric processes II: vector four-wave mixing,” Opt. Express 21, 11009–11020 (2013).
[CrossRef]

C. J. McKinstrie and M. Karlsson, “Schmidt decompositions of parametric processes I: basic theory and simple examples,” Opt. Express 21, 1374–1394 (2013).
[CrossRef]

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430 (2011).
[CrossRef]

Z. Tong, C. J. McKinstrie, C. Lundström, M. Karlsson, and P. Andrekson, “Noise performance of optical fiber transmission links that use non-degenerate cascaded phase-sensitive amplifiers,” Opt. Express 18, 15426–15439 (2010).
[CrossRef]

C. J. McKinstrie, M. Karlsson, and Z. Tong, “Field-quadrature and photon-number correlations produced by parametric processes,” Opt. Express 18, 19792–19823 (2010).
[CrossRef]

C. J. McKinstrie, M. Raymer, S. Radic, and M. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” Opt. Commun. 257, 146–163 (2006).
[CrossRef]

C. J. McKinstrie, M. Yu, M. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13, 4986–5012 (2005).
[CrossRef]

C. J. McKinstrie and S. Radic, “Quantum noise properties of parametric amplifiers driven by two pump waves,” Opt. Express 12, 5037–5066 (2004).
[CrossRef]

C. J. McKinstrie and S. Radic, “Phase-sensitive amplification in a fiber,” Opt. Express 12, 4973–4979 (2004).
[CrossRef]

Mukunda, N.

A. B. Dutta, N. Mukunda, and R. Simon, “The real symplectic groups in quantum mechanics and optics,” Pramana 45, 471–497 (1995).
[CrossRef]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 3028–3038 (1988).
[CrossRef]

O’Gorman, J.

R. Slav’k, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrom, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Ott, J. R.

Pan, W.

Z. Chen, L. Yan, W. Pan, B. Luo, A. Yi, J. Ye, H. Jiang, and Y. Guo, “Phase sensitive amplifier for PSK signals based on non-degenerate four-wave-mixing in the optical fiber,” Opt. Commun. 285, 2445–2450 (2012).
[CrossRef]

Parmigiani, F.

R. Slav’k, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrom, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Petropoulos, P.

R. Slav’k, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrom, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Phelan, R.

R. Slav’k, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrom, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Pinel, O.

O. Pinel, J. Fade, D. Braun, P. Jian, N. Treps, and C. Fabre, “Ultimate sensitivity of precision measurements with intense Gaussian quantum light: a multimodal approach,” Phys. Rev. A 85, 010101(R) (2012).
[CrossRef]

Puttnam, B. J.

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430 (2011).
[CrossRef]

Radic, S.

Raymer, M.

C. J. McKinstrie, M. Raymer, S. Radic, and M. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” Opt. Commun. 257, 146–163 (2006).
[CrossRef]

C. J. McKinstrie, M. Yu, M. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13, 4986–5012 (2005).
[CrossRef]

Raymond, J. M.

S. Haroche and J. M. Raymond, Exploring the Quantum (Oxford Graduate Texts, 2008).

Richardson, D. J.

R. Slav’k, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrom, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Schumaker, B. M.

C. L. Caves and B. M. Schumaker, “New formalism for two-photon quantum optics. I. Quadrature phases and squeezed states,” Phys. Rev. A 31, 3068–3092 (1985).
[CrossRef]

Sharping, J. E.

Simon, R.

A. B. Dutta, N. Mukunda, and R. Simon, “The real symplectic groups in quantum mechanics and optics,” Pramana 45, 471–497 (1995).
[CrossRef]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 3028–3038 (1988).
[CrossRef]

Sjödin, M.

R. Slav’k, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrom, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Slav’k, R.

R. Slav’k, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrom, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Sudarshan, E. C. G.

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 3028–3038 (1988).
[CrossRef]

Sygletos, S.

R. Slav’k, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrom, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Syvridis, D.

R. Slav’k, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrom, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Tipsuwannakul, E.

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430 (2011).
[CrossRef]

Toda, H.

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430 (2011).
[CrossRef]

Tong, Z.

Z. Tong and S. Radic, “Low-noise optical amplification and signal processing in parametric devices,” Adv. Opt. Photon. 5, 318–384 (2013).
[CrossRef]

Z. Tong, C. Lundström, P. A. Andrekson, M. Karlsson, and A. Bogris, “Ultralow noise, broadband phase-sensitive optical amplifiers, and their applications,” IEEE J. Sel. Top. Quantum Electron. 18, 1016–1032 (2012).
[CrossRef]

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430 (2011).
[CrossRef]

Z. Tong, C. J. McKinstrie, C. Lundström, M. Karlsson, and P. Andrekson, “Noise performance of optical fiber transmission links that use non-degenerate cascaded phase-sensitive amplifiers,” Opt. Express 18, 15426–15439 (2010).
[CrossRef]

C. J. McKinstrie, M. Karlsson, and Z. Tong, “Field-quadrature and photon-number correlations produced by parametric processes,” Opt. Express 18, 19792–19823 (2010).
[CrossRef]

Treps, N.

O. Pinel, J. Fade, D. Braun, P. Jian, N. Treps, and C. Fabre, “Ultimate sensitivity of precision measurements with intense Gaussian quantum light: a multimodal approach,” Phys. Rev. A 85, 010101(R) (2012).
[CrossRef]

Van Loock, P.

A. Furusawa and P. Van Loock, Quantum Teleportation and Entanglement (Wiley-VCH, 2011).

Vasilyev, M.

C. J. McKinstrie, M. Raymer, S. Radic, and M. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” Opt. Commun. 257, 146–163 (2006).
[CrossRef]

M. Vasilyev, “Distributed phase-sensitive amplification,” Opt. Express 13, 7563–7571 (2005).
[CrossRef]

Weerasuriya, R.

R. Slav’k, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrom, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Westlund, M.

J. Hansryd, P. A. Andrekson, M. Westlund, L. Jie, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[CrossRef]

Windeler, R. S.

Yan, L.

Z. Chen, L. Yan, W. Pan, B. Luo, A. Yi, J. Ye, H. Jiang, and Y. Guo, “Phase sensitive amplifier for PSK signals based on non-degenerate four-wave-mixing in the optical fiber,” Opt. Commun. 285, 2445–2450 (2012).
[CrossRef]

Ye, J.

Z. Chen, L. Yan, W. Pan, B. Luo, A. Yi, J. Ye, H. Jiang, and Y. Guo, “Phase sensitive amplifier for PSK signals based on non-degenerate four-wave-mixing in the optical fiber,” Opt. Commun. 285, 2445–2450 (2012).
[CrossRef]

Yi, A.

Z. Chen, L. Yan, W. Pan, B. Luo, A. Yi, J. Ye, H. Jiang, and Y. Guo, “Phase sensitive amplifier for PSK signals based on non-degenerate four-wave-mixing in the optical fiber,” Opt. Commun. 285, 2445–2450 (2012).
[CrossRef]

Yu, M.

Zoller, P.

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
[CrossRef]

Adv. Opt. Photon. (1)

IEEE J. Sel. Top. Quantum Electron. (2)

Z. Tong, C. Lundström, P. A. Andrekson, M. Karlsson, and A. Bogris, “Ultralow noise, broadband phase-sensitive optical amplifiers, and their applications,” IEEE J. Sel. Top. Quantum Electron. 18, 1016–1032 (2012).
[CrossRef]

J. Hansryd, P. A. Andrekson, M. Westlund, L. Jie, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[CrossRef]

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

Nat. Photonics (2)

R. Slav’k, F. Parmigiani, J. Kakande, C. Lundström, M. Sjödin, P. A. Andrekson, R. Weerasuriya, S. Sygletos, A. D. Ellis, L. Grüner-Nielsen, D. Jakobsen, S. Herstrom, R. Phelan, J. O’Gorman, A. Bogris, D. Syvridis, S. Dasgupta, P. Petropoulos, and D. J. Richardson, “All-optical phase and amplitude regenerator for next-generation telecommunications systems,” Nat. Photonics 4, 690–695 (2010).
[CrossRef]

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics 5, 430 (2011).
[CrossRef]

Opt. Commun. (2)

C. J. McKinstrie, M. Raymer, S. Radic, and M. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” Opt. Commun. 257, 146–163 (2006).
[CrossRef]

Z. Chen, L. Yan, W. Pan, B. Luo, A. Yi, J. Ye, H. Jiang, and Y. Guo, “Phase sensitive amplifier for PSK signals based on non-degenerate four-wave-mixing in the optical fiber,” Opt. Commun. 285, 2445–2450 (2012).
[CrossRef]

Opt. Express (10)

C. Lundström, B. Corcoran, M. Karlsson, and P. A. Andrekson, “Phase and amplitude characteristics of a phase-sensitive amplifier operating in gain saturation,” Opt. Express 20, 21400–21412 (2012).
[CrossRef]

M. E. Marhic, “Noise figure of hybrid optical parametric amplifiers,” Opt. Express 20, 28752–28757 (2012).
[CrossRef]

C. J. McKinstrie and M. Karlsson, “Schmidt decompositions of parametric processes I: basic theory and simple examples,” Opt. Express 21, 1374–1394 (2013).
[CrossRef]

C. J. McKinstrie, J. R. Ott, and M. Karlsson, “Schmidt decompositions of parametric processes II: vector four-wave mixing,” Opt. Express 21, 11009–11020 (2013).
[CrossRef]

C. J. McKinstrie and S. Radic, “Phase-sensitive amplification in a fiber,” Opt. Express 12, 4973–4979 (2004).
[CrossRef]

C. J. McKinstrie and S. Radic, “Quantum noise properties of parametric amplifiers driven by two pump waves,” Opt. Express 12, 5037–5066 (2004).
[CrossRef]

C. J. McKinstrie, M. Yu, M. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13, 4986–5012 (2005).
[CrossRef]

M. Vasilyev, “Distributed phase-sensitive amplification,” Opt. Express 13, 7563–7571 (2005).
[CrossRef]

Z. Tong, C. J. McKinstrie, C. Lundström, M. Karlsson, and P. Andrekson, “Noise performance of optical fiber transmission links that use non-degenerate cascaded phase-sensitive amplifiers,” Opt. Express 18, 15426–15439 (2010).
[CrossRef]

C. J. McKinstrie, M. Karlsson, and Z. Tong, “Field-quadrature and photon-number correlations produced by parametric processes,” Opt. Express 18, 19792–19823 (2010).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (5)

N. V. Corzo, Q. Glorieux, A. M. Marino, J. B. Clark, and P. D. Lett, “Rotation of the noise ellipse for squeezed vacuum light generated via four-wave-mixing,” Phys. Rev. A 88, 043836 (2013).
[CrossRef]

O. Pinel, J. Fade, D. Braun, P. Jian, N. Treps, and C. Fabre, “Ultimate sensitivity of precision measurements with intense Gaussian quantum light: a multimodal approach,” Phys. Rev. A 85, 010101(R) (2012).
[CrossRef]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian pure states in quantum mechanics and the symplectic group,” Phys. Rev. A 37, 3028–3038 (1988).
[CrossRef]

S. L. Braunstein, “Squeezing as an irreducible resource,” Phys. Rev. A 71, 055801 (2005).
[CrossRef]

C. L. Caves and B. M. Schumaker, “New formalism for two-photon quantum optics. I. Quadrature phases and squeezed states,” Phys. Rev. A 31, 3068–3092 (1985).
[CrossRef]

Phys. Rev. D (1)

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D 26, 1817–1839 (1982).
[CrossRef]

Phys. Rev. Lett. (1)

L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, “Inseparability criterion for continuous variable systems,” Phys. Rev. Lett. 84, 2722–2725 (2000).
[CrossRef]

Pramana (1)

A. B. Dutta, N. Mukunda, and R. Simon, “The real symplectic groups in quantum mechanics and optics,” Pramana 45, 471–497 (1995).
[CrossRef]

Rev. Mod. Phys. (1)

M. Kolobov, “The spatial behavior of nonclassical light,” Rev. Mod. Phys. 71, 1539–1589 (1999).
[CrossRef]

Other (9)

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2013).

M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators, and Related Devices (Cambridge University, 2008).

R. Loudon, The Quantum Theory of Light (Oxford University, 2000).

G. Grynberg, A. Aspect, and C. Fabre, Introduction to Quantum Optics (Cambridge University, 2010).

See http://www.specialtyphotonics.com .

Note that since the expression of ϕ in terms of the fiber parameters reported in Eq. (A4) also involves the pump phases, the signal phase is in this way unambiguously defined.

E. Desurvire, Erbium-Doped Fiber Amplifiers (Wiley, 2002).

S. Haroche and J. M. Raymond, Exploring the Quantum (Oxford Graduate Texts, 2008).

A. Furusawa and P. Van Loock, Quantum Teleportation and Entanglement (Wiley-VCH, 2011).

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Figures (11)

Fig. 1.
Fig. 1.

Schematic representation of the two considered configurations for parametric amplification based on four-wave mixing. In configuration “A,” A2=As is the amplitude of the degenerate signal and idler while A1 and A3 are the amplitudes of the two pumps. Conversely, in configuration “B,” A2 holds for the two degenerate pumps while A1=As and A3=Ai are the amplitudes of the signal and idler, respectively.

Fig. 2.
Fig. 2.

Evolution of the field in the phasor representation along the three steps of the Bloch–Messiah decomposition of the parametric amplification in configuration “A.” The units for the quadratures are arbitrary. The incident field (1), which is in a coherent state |αs0 with αs0=0.8+i0.1, undergoes a first rotation by an angle ϕ, resulting in field (2). This field is transformed into field (3) by the squeezing operator Σ, before being rotated by an angle θ to give the output field (4). The calculations were performed for a fiber length z=300m, γ=11.3×103W1m1, Δβ=4.53×1011m1, and P1=P3=100mW.

Fig. 3.
Fig. 3.

Phasor representation for the input fields in the case of configuration “B.” The units for the quadratures are arbitrary. The incident signal and idler fields As0 and Ai0, which are in coherent states |αs0 with αs0=0.4exp(iπ/5) and |αi0 with αi0=0.54exp(iπ/5), respectively, are transformed into the fields A+0 and A0 according to Eqs. (30) and (31).

Fig. 4.
Fig. 4.

Evolution of the “+” (a) and “” (b) fields in the phasor representation along the three steps of the Bloch–Messiah decomposition of the parametric amplification in configuration “B.” The units for the quadratures are arbitrary. The incident coherent fields A±0 (1), which were already plotted in Fig. 3, undergo a first rotation by an angle ϕ, resulting in field (2). These fields are transformed into field (3) by the squeezing operators Σ±, before being rotated by an angle θ to give the output fields A± (4). The calculations were performed for a fiber length z=300m, γ=11.3×103W1m1, Δβ=4.54×1011m1, and P2=230mW.

Fig. 5.
Fig. 5.

Phasor representation for the output fields in the case of configuration “B.” The units for the quadratures are arbitrary. The output signal and idler fields As and Ai are obtained from the output fields A+ and A, which were obtained in Fig. 4, using Eqs. (30) and (31).

Fig. 6.
Fig. 6.

Evolution of the field in the phasor representation along the three steps of the Bloch–Messiah decomposition of the parametric amplification in configuration “A” in the case of optimal coupling. All parameters are the same as in Fig. 2, except for the phase of the input field which is chosen according to Eq. (56).

Fig. 7.
Fig. 7.

Phasor representation for the input fields in the case of configuration “B” for optimal coupling. The units for the quadratures are arbitrary. All parameters are the same as in Fig. 3. αs0=0.4exp(iπ/5) such as in Fig. 3. Only the idler input field is different from Fig. 3: it is chosen according to Eq. (63).

Fig. 8.
Fig. 8.

Same as Fig. 4 for the optimum input conditions of Fig. 7.

Fig. 9.
Fig. 9.

Same as Fig. 5 for the optimum input conditions of Fig. 7.

Fig. 10.
Fig. 10.

Left hand side of the Duan criterion in Eq. (91), given by the sum of the second and the third diagonal terms of the matrix given in Eq. (89) as a function of the pump power for parameters Δβ=4.53×1011m1, z=300m, and γ=11.3×103W1·m1.

Fig. 11.
Fig. 11.

Schematic representation of two possible links, respectively, composed either by an amplifier followed by a lossy transmission link (top) or, conversely, by a lossy link followed by an amplifier (bottom).

Equations (132)

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

dA1dz=iγ[(|A1|2+2|A2|2+2|A3|2)A1+A22A3*eiΔβz],
dA2dz=iγ[(2|A1|2+|A2|2+2|A3|2)A2+2A1A3A2*eiΔβz],
dA3dz=iγ[(2|A1|2+2|A2|2+|A3|2)A3+A1*A22eiΔβz],
As=μAs0+νAs0*,
μ=(coshgz+iκ2gsinhgz)eiδz,
ν=2iγgA1(0)A3(0)sinhgzeiδz,
κ=Δβ+γ(P1+P3)
g=4γ2P1P3(κ/2)2
δ=3γ(P1+P3)Δβ2
(AsAs*)=(μνν*μ*)(As0As0*).
G=|As|2|As0|2=|μAs0+νAs0*|2Ps0=|μ|2+|ν|2+2|μ||ν|cos(θμθν2θs0)=1+[1+κ2+16γ2P1P3+8κγP1P3cosξrel4g2]sinh2gz+2γgP1P3sinξrelsinh2gz,
ξrel=2θs0θ10θ30
Aj0=|Aj0|eiθj0,
Gmax=(|μ|+|ν|)2,forθμθν2θs0=2kπ,
Gmin=(|μ||ν|)2,forθμθν2θs0=(2k+1)π,
AsXs+iYs,
As0Xs0+iYs0
(XsYs)=(Re[μ+ν]Im[μν]Im[μ+ν]Re[μν])(Xs0Ys0)M(Xs0Ys0).
As=μAs0+νAi0*,
μ=(coshgz+iκ2gsinhgz)eiΔβz2,
ν=iγgA22(0)sinhgzeiΔβz2,
κ=2γP2Δβ
g=γ2P22(κ2)2,
Ai=νAs0*+μAi0.
G=|As|2|As0|2=|μAs0+νAi0*|2Ps0=|μ|2+η2|ν|2+2η|μ||ν|cos(θμθν+θs0+θi0)=1+[1+κ2+4γ2P22η2+4κγηP2cosξrel4g2]sinh2gz+γgηP2sinξrelsinh2gz,
ξrel=θs0+θi02θ20,
η=Pi0Ps0.
Gmax=(|μ|+|ν|)2,forθμθν+θs0+θi0=2kπ,
Gmin=(|μ||ν|)2,forθμθν+θs0+θi0=(2k+1)π,
A+(z)=As(z)+Ai(z)2,
A(z)=As(z)Ai(z)2.
A±=μA±0±νA±0*.
A±(z)X±(z)+iY±(z)
(X±Y±)=(Re[μ±ν]Im[μν]Im[μ±ν]Re[μν])(X±0Y±0)M±(X±0Y±0).
As=|μ|eiθμAs0+|ν|eiθνAs0*.
As=|μ|As0+|ν|As0*,
As0ei(θμθν2)As0,
Asei(θμ+θν2)As.
Xs=(|μ|+|ν|)Xs0,
Ys=(|μ||ν|)Ys0.
(Xs0Ys0)=(cos(θμθν2)sin(θμθν2)sin(θμθν2)cos(θμθν2))(Xs0Ys0);
(XsYs)=(|μ|+|ν|00|μ||ν|)(Xs0Ys0);
(XsYs)=(cos(θμ+θν2)sin(θμ+θν2)sin(θμ+θν2)cos(θμ+θν2))(XsYs).
M=CUΣWT=C(cosθsinθsinθcosθ)(|μ|+|ν|00|μ||ν|)×(cosϕsinϕsinϕcosϕ),
θ=θμ+θν2,
ϕ=θμθν2.
A+=|μ|A+0+|ν|A+0*,
A+0ei(θμθν2)A+0,
A+ei(θμ+θν2)A+.
A=|μ|A0|ν|A0*,
A0ei(θμθν2)A0,
Aei(θμ+θν2)A.
X±=(|μ|±|ν|)X±0,
Y±=(|μ||ν|)Y±0.
M±=C±UΣ±WT=C±(cosθsinθsinθcosθ)(|μ|±|ν|00|μ||ν|)×(cosϕsinϕsinϕcosϕ),
θs0=θνθμ2=ϕ,
Asei(θμθν2)=|As0|(|μ|+|ν|),
As=μAs0,
Ai=νAs0*.
GPIA=PsPs0=|μ|2,
As=|μ|eiθμAs0+|ν|eiθνAi0*,
Ai=|ν|eiθνAs0*+|μ|eiθμAi0,
Ai0=As0*ei(θμθν).
Aseiθμ=(|μ|+|ν|)As0,
Aieiθμ=(|μ|+|ν|)Ai0,
θμθν+θs0+θi0=0mod2π,
x^sθs0=(αs0eiθs0+αs0*eiθs0)/2=|αs0|,
Δ2xsφ0=14,
SNRin=x^sθs02Δ2xsθs0=4|αs0|2.
x^s,φ=|αs0|[|μ|cos(θμ+θs0φ)+|ν|cos(θνθs0φ)],
Δ2xsφ=2|μ||ν|cos(θμ+θν2φ)+|μ|2+|ν|24.
SNRout=4|αs0|2[|μ|cos(θμ+θs0φ)+|ν|cos(θνθs0φ)]22|μ||ν|cos(θμ+θν2φ)+|μ|2+|ν|2.
NF=SNRinSNRout=[2|μ||ν|cos(θμ+θν2φ)+(|μ|2+|ν|2)][|μ|cos(θμ+θs0φ)+|ν|cos(θνθs0φ)]2.
x^0=x^sθs0+x^sθi02.
x^0=|αs0|+|αi0|2=2|αs0|.
Δ2x0=Δ2xsθs0+Δ2xiθi02=14,
SNRs,in=x^02Δ2x0=8|αs0|2=8Ps.
x^=x^sθs+x^sθi2=2(|μ|+|ν|)|αs0|,
Δ2x=(|μ|+|ν|)24,
SNRs,out=x^2Δ2x=8|αs0|2=8Ps.
NF=SNRinSNRout=1,
SNRs,out=4(|μ|+|ν|)2|αs0|2(|μ|2+|ν|2),
NFs=SNRs,inSNRs,out=2(|μ|2+|ν|2)(|μ|+|ν|)2large gain=1,
NFs=|αs0|2(|μ|2+|ν|2)[|μ||αs0|cos(θμ+θs0φ)+|ν||αi0|cos(θνθi0φ)]2,
x^sφ=|μ|x^s0=|μ||αs0|cos(θs0φ),
NF=SNRinSNRout=|μ|2+|ν|2|μ|2large gain2|μ|2|μ|2=23dB,
(x^sx^iy^sy^i)=(Re[μ]Re[ν]Im[μ]Im[ν]Re[ν]Re[μ]Im[ν]Im[μ]Im[μ]Im[ν]Re[μ]Re[ν]Im[ν]Im[μ]Re[ν]Re[μ])(x^s0x^i0y^s0y^i0)Stot(x^s0x^i0y^s0y^i0).
Σsi=StotΣsi,0StotT=StotStotT/4=14(|μ|2+|ν|22Re[μν]02Im[μν]2Re[μν]|μ|2+|ν|22Im[μν]002Im[μν]|μ|2+|ν|22Re[μν]2Im[μν]02Re[μν]|μ|2+|ν|2),
Σ±=14((|μ|+|ν|)20000(|μ||ν|)20000(|μ||ν|)20000(|μ|+|ν|)2).
Δ2(x^1x^i2)+Δ2(p^s+p^i2)<12
Δ2x+Δ2p+<12
a^s=τμa^s0+τνa^s0+ρa^l,
x^sφ=12[eiφ(τμa^s0+τνa^s0+ρsa^l)+h.c.]=τ|αs0|[|μ|cos(θμ+θs0φ)+|ν|cos(θνθs0φ)].
Δ2xsφ=x^sφ2x^sφ2,
Δ2xsφ=τ2[2|μ||ν|cos(θμ+θν2φ)+(|μ|2+|ν|21)]+14.
SNRoutAL=x^sφ2Δ2xsφ=4τ2{|αs0|[|μ|cos(θμ+θs0φ)+|ν|cos(θνθs0φ)]}22τ2|μ||ν|cos(θμ+θν2φ)+2τ2|ν|2+1,
NFAL=SNRinSNRoutAL=NF(λφτ2τ2+1λφτ2),
λφ(2|μ||ν|cos(θμ+θν2φ)+|μ|2+|ν|2).
θs0=ϕ=(θμθν)2,φ=θ=θμ+θν2,
λopt=(|μ|+|ν|)2=Gmax,NFopt=1,
NFoptAL=(11Gmax+1Gmaxτ2).
a^s=τμa^s0+τνa^s0+μρa^l+νρa^l,
x^sφ=12[eiφ(τμa^s0+τνa^s0+μρa^l+νρa^l)+h.c.]=τ|μ||αs0|cos(θμ+θs0φ)+τ|ν||αi0|cos(θνθs0φ),
Δ2xsφ=|μ||ν|cos(θμ+θν2φ)2+|ν|22+14.
SNRoutLA=x^sφ2Δ2xsφ=4τ2|αs0|2{|μ|cos(θμ+θs0φ)+|ν|cos(θνθs0φ)}22|μ||ν|cos(θμ+θν2φ)+|μ|2+|ν|2,
NFLA=SNRinSNRoutLA=NFτ2.
NFoptLA=1τ2.
NFoptALNFoptLA=Gmaxτ2τ2+1Gmax.
a^s=τsμa^s0+τsνa^i0+ρsa^l1,a^i=τiνa^s0+τiμa^i0+ρia^l2,
x^sφ=12[eiφ(τsμa^s0+τsνa^i0+ρsa^l1)+h.c.]=τs[|μ||αs0|cos(θμ+θs0φ)+|ν||αi0|cos(θνθi0φ)].
Δ2xsφ=x^sφ2x^sφ2=τs2|ν|22+14,
SNRoutLA=x^sφ2Δ2xsφ=4(2τs2|ν|2+1)×{τs[|μ||αs0|cos(θμ+θs0φ)+|ν||αi0|cos(θνθi0φ)]}2.
NFLA=NF(G0τs2τs2+1G0τs2),
G0=(|μ|2+|ν|2).
θi0=θs0(θμθν),φ=θμ+θs0,
NFLA=(11G0+1G0τs2).
a^s=τsμa^s0+τiνa^i0+μρsa^l1+νρia^l2,a^i=τsνa^s0+τiμa^i0+νρsa^l1+μρia^l2,
x^sφ=12[eiφ(τsμa^s0+τiνa^i0+μρsa^l1+νρia^l2)+h.c.]=τs|μ||αs0|cos(θμ+θs0φ)+τi|ν||αi0|cos(θνθi0φ).
Δ2xsφ=|ν|22+14=|μ|2+|ν|24.
SNRout=x^sφ2Δ2xsφ=1(|μ|2+|ν|2){4[τs|μ||αs0|cos(θμ+θs0φ)+τi|ν||αi0|cos(θνθi0φ)]2},
NFLA=NFτ2,
NFoptLA=1τ2,
NFoptALNFoptLA=G0τ2τ2+1G0.
tan(θμ+θν)=κ2gtanhgztanθg1κ2gtanhgz+tanθg=tan2θ,
tan(θμθν)=κ2gtanhgztan(θ10+θ30)+1κ2gtanhgz+tan(θ10+θ30)=tan2ϕ,
θg=θ10+θ30+2δz.
tan(θμ+θν)=κ2gtanhgztanθg1κ2gtanhgz+tanθg=tan2θ,
tan(θμθν)=κ2gtanhgztan2θ20+1κ2gtanhgz+tan2θ20=tan2ϕ,
θg=2θ20+Δβz.
(x^sθsx^iθiy^sθsy^iθi)=(cosθs0sinθs00cosθi0sinθisinθs0cosθs00sinθi0cosθi)(x^sx^iy^sy^i);(x^x^diffy^y^diff)=12(1100110000110011)(x^sθsx^iθiy^sθsy^iθi),
(x^sθsx^iθiy^sθsy^iθi)=12(cosθscosθisinθssinθicosθscosθisinθssinθisinθssinθicosθscosθisinθssinθicosθscosθi)(x^sx^iy^sy^i)Stransf(x^sx^iy^sy^i).
{1.ddθs0NFAL(θs0,φ)=02.ddφNFAL(θs0,φ)=0.

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