Abstract

Two sets of formulas are derived for the field-quadrature and photon-number fluctuations (variances and correlations) produced by parametric amplifiers and frequency convertors that are driven by pulsed pumps and act on pulsed signals. The first set is based on the Green functions for the underlying parametric processes, whereas the second is based on the associated Schmidt coefficients and modes. These formulas facilitate the modeling and performance optimization of parametric devices used in a wide variety of applications.

© 2013 Optical Society of America

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References

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    [Crossref]
  3. J. H. Lee, “All-optical signal processing devices based on holey fiber,” IEICE Trans. Electron. E88-C, 327–334 (2005).
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  5. P. A. Andrekson and M. Westlund, “Nonlinear optical fiber based high resolution all-optical waveform sampling,” Laser Photon. Rev. 1, 231–248 (2007).
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  39. T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,” IEEE Photon. Technol. Lett. 16, 551–553 (2004).
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  45. T. Torounidis and P. Andrekson, “Broadband single-pumped fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. 19, 650–652 (2007).
    [Crossref]
  46. J. M. Chavez Boggio, S. Moro, E. Myslivets, J. R. Windmiller, N. Alic, and S. Radic, “155-nm continuous-wave two-pump parametric amplification,” IEEE Photon. Technol. Lett. 21, 612–614 (2009).
    [Crossref]
  47. M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. 14, 983–985 (2002).
    [Crossref]
  48. J. E. Sharping, J. Chen, X. Li, and P. Kumar, “Quantum-correlated twin photons from microstructure fiber,” Opt. Express 12, 3086–3094 (2004).
    [Crossref] [PubMed]
  49. J. G. Rarity, J. Fulconis, J. Duligall, W. J. Wadsworth, and P. S. J. Russell, “Photonic crystal fiber source of correlated photon pairs,” Opt. Express 13, 534–544 (2005).
    [Crossref] [PubMed]
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    [Crossref]
  51. M. Halder, J. Fulconis, B. Cemlyn, A. Clark, C. Xiong, W. J. Wadsworth, and J. G. Rarity, “Nonclassical 2-photon interference with separate intrinsically narrowband fibre sources,” Opt. Express 17, 4670–4676 (2009).
    [Crossref] [PubMed]
  52. O. Cohen, J. S. Lundeen, B. J. Smith, G. Puentes, P. J. Mosley, and I. A. Walmsley, “Tailored photon-pair generation in optical fibers,” Phys. Rev. Lett. 102, 123603 (2009).
    [Crossref] [PubMed]
  53. C. J. McKinstrie and S. Radic, “Phase-sensitive amplification in a fiber,” Opt. Express 12, 4973–4979 (2004).
    [Crossref] [PubMed]
  54. K. Croussore and G. Li, “Phase and amplitude regeneration of differential phase-shift keyed signals using phase-sensitive amplification,” J. Sel. Top. Quantum Electron. 14648–658 (2008).
    [Crossref]
  55. R. Slavik, A. Bogris, F. Parmigiani, J. Kakande, M. Westlund, M. Sköld, L. Grüner-Nielsen, R. Phelan, D. Syvridis, P. Petropoulos, and D. J. Richardson, “Coherent all-optical phase and amplitude regenerator of binary phase-encoded signals,” IEEE J. Sel. Top. Quantum Electron. 18, 859–869 (2012).
    [Crossref]
  56. R. Tang, J. Lasri, P. S. Devgan, V. Grigoryan, P. Kumar, and M. Vasilyev, “Gain characteristics of a frequency nondegenerate phase-sensitive fiber-optic parametric amplifier with phase self-stabilized input,” Opt. Express 13, 10483–10493 (2005).
    [Crossref] [PubMed]
  57. J. Kakande, F. Parmigiani, M. Ibsen, P. Petropoulos, and D. J. Richardson, “Wide bandwidth experimental study of nondegenerate phase-sensitive amplifiers in single- and dual-pump configurations,” IEEE Photon. Technol. Lett. 22, 1781–1783 (2010).
    [Crossref]
  58. 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]
  59. S. Prasad, M. O. Scully, and W. Martienssen, “A quantum description of the beam splitter,” Opt. Commun. 62, 139–145 (1987).
    [Crossref]
  60. R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) summetry and photon statistics,” Phys. Rev. A 40, 1371–1384 (1989).
    [Crossref] [PubMed]
  61. C. M. Caves, C. Zhu, G. J. Milburn, and W. Schleich, “Photon statistics of two-mode squeezed states and interference in four-dimensional phase space,” Phys. Rev. A 43, 3854–3861 (1991).
    [Crossref] [PubMed]
  62. M. Artoni, U. P. Ortiz, and J. L. Birman, “Photocount distribution of two-mode squeezed states,” Phys. Rev. A 43, 3954–3965 (1991).
    [Crossref] [PubMed]
  63. A. I. Lvovsky, W. Wasilewski, and K. Banaszek, “Decomposing a pulsed optical parametric amplifier into independent squeezers,” J. Mod. Opt. 54, 721–733 (2007).
    [Crossref]
  64. M. Vasilyev, M. Annamalai, N. Stelmakh, and P. Kumar, “Quantum properties of a spatially-broadband traveling-wave phase-sensitive optical parametric amplifier,” J. Mod. Opt. 57, 1908–1915 (2010).
    [Crossref]
  65. A. K. Ekert and P. L. Knight, “Relationship between semiclassical and quantum-mechanical input-output theories of optical response,” Phys. Rev. A 43, 3934–3938 (1991).
    [Crossref] [PubMed]
  66. C. J. McKinstrie, “Unitary and singular value decompositions of parametric processes in fibers,” Opt. Commun. 282, 583–593 (2009).
    [Crossref]

2013 (3)

2012 (6)

R. Slavik, A. Bogris, F. Parmigiani, J. Kakande, M. Westlund, M. Sköld, L. Grüner-Nielsen, R. Phelan, D. Syvridis, P. Petropoulos, and D. J. Richardson, “Coherent all-optical phase and amplitude regenerator of binary phase-encoded signals,” IEEE J. Sel. Top. Quantum Electron. 18, 859–869 (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]

L. Mejling, C. J. McKinstrie, M. G. Raymer, and K. Rottwitt, “Quantum frequency translation by four-wave mixing in a fiber: Low-conversion regime,” Opt. Express 20, 8367–8396 (2012).
[Crossref] [PubMed]

C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, “Quantum-state-preserving optical frequency conversion and pulse reshaping by four-wave mixing,” Phys. Rev. A 85, 053829 (2012).
[Crossref]

S. Radic, “Parametric signal processing,” IEEE J. Sel. Top. Quantum Electron. 18, 670–680 (2012).
[Crossref]

M. G. Raymer and K. Srinivasan, “Manipulating the color and shape of single photons,” Phys. Today 65(11), 32–37 (2012).
[Crossref]

2011 (2)

H. Cruz-Ramirez, R. Ramirez-Alarcon, M. Corona, K. Garay-Palmett, and A. B. U’Ren, “Spontaneous parametric processes in quantum optics,” Opt. Photon. News 22(11), 37–41 (2011).
[Crossref]

A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on spectrally engineered sum frequency generation,” Opt. Express 19, 13770–13778 (2011).
[Crossref] [PubMed]

2010 (5)

M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun. 283, 747–752 (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] [PubMed]

J. Kakande, F. Parmigiani, M. Ibsen, P. Petropoulos, and D. J. Richardson, “Wide bandwidth experimental study of nondegenerate phase-sensitive amplifiers in single- and dual-pump configurations,” IEEE Photon. Technol. Lett. 22, 1781–1783 (2010).
[Crossref]

M. Vasilyev, M. Annamalai, N. Stelmakh, and P. Kumar, “Quantum properties of a spatially-broadband traveling-wave phase-sensitive optical parametric amplifier,” J. Mod. Opt. 57, 1908–1915 (2010).
[Crossref]

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105, 093604 (2010).
[Crossref] [PubMed]

2009 (4)

J. M. Chavez Boggio, S. Moro, E. Myslivets, J. R. Windmiller, N. Alic, and S. Radic, “155-nm continuous-wave two-pump parametric amplification,” IEEE Photon. Technol. Lett. 21, 612–614 (2009).
[Crossref]

M. Halder, J. Fulconis, B. Cemlyn, A. Clark, C. Xiong, W. J. Wadsworth, and J. G. Rarity, “Nonclassical 2-photon interference with separate intrinsically narrowband fibre sources,” Opt. Express 17, 4670–4676 (2009).
[Crossref] [PubMed]

O. Cohen, J. S. Lundeen, B. J. Smith, G. Puentes, P. J. Mosley, and I. A. Walmsley, “Tailored photon-pair generation in optical fibers,” Phys. Rev. Lett. 102, 123603 (2009).
[Crossref] [PubMed]

C. J. McKinstrie, “Unitary and singular value decompositions of parametric processes in fibers,” Opt. Commun. 282, 583–593 (2009).
[Crossref]

2008 (1)

K. Croussore and G. Li, “Phase and amplitude regeneration of differential phase-shift keyed signals using phase-sensitive amplification,” J. Sel. Top. Quantum Electron. 14648–658 (2008).
[Crossref]

2007 (6)

T. Torounidis and P. Andrekson, “Broadband single-pumped fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. 19, 650–652 (2007).
[Crossref]

A. I. Lvovsky, W. Wasilewski, and K. Banaszek, “Decomposing a pulsed optical parametric amplifier into independent squeezers,” J. Mod. Opt. 54, 721–733 (2007).
[Crossref]

K. Garay-Palmett, H. J. McGuinness, O. Cohen, J. S. Lundeen, R. Rangel-Rojo, A. B. U’Ren, M. G. Raymer, C. J. McKinstrie, S. Radic, and I. A. Walmsley, “Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber,” Opt. Express 15, 14870–14886 (2007).
[Crossref] [PubMed]

J. Fan, A. Migdall, and L. Wang, “A twin photon source,” Opt. Photon. News 18(3), 26–33 (2007).
[Crossref]

C. J. McKinstrie, S. Radic, and A. H. Gnauck, “All-optical signal processing by fiber-based parametric devices,” Opt. Photon. News 18(3), 34–40 (2007).
[Crossref]

P. A. Andrekson and M. Westlund, “Nonlinear optical fiber based high resolution all-optical waveform sampling,” Laser Photon. Rev. 1, 231–248 (2007).
[Crossref]

2006 (1)

2005 (10)

R. Tang, J. Lasri, P. S. Devgan, V. Grigoryan, P. Kumar, and M. Vasilyev, “Gain characteristics of a frequency nondegenerate phase-sensitive fiber-optic parametric amplifier with phase self-stabilized input,” Opt. Express 13, 10483–10493 (2005).
[Crossref] [PubMed]

J. G. Rarity, J. Fulconis, J. Duligall, W. J. Wadsworth, and P. S. J. Russell, “Photonic crystal fiber source of correlated photon pairs,” Opt. Express 13, 534–544 (2005).
[Crossref] [PubMed]

J. Fan, A. Migdall, and L. J. Wang, “Efficient generation of correlated photon pairs in a microstructure fiber,” Opt. Lett. 30, 3368–3370 (2005).
[Crossref]

J. H. Lee, “All-optical signal processing devices based on holey fiber,” IEICE Trans. Electron. E88-C, 327–334 (2005).
[Crossref]

I. A. Walmsley and M. G. Raymer, “Toward quantum information processing with photons,” Science 307, 1733–1734 (2005).
[Crossref] [PubMed]

S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature 437, 116–120 (2005).
[Crossref] [PubMed]

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

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

A. B. U’Ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146–161 (2005).

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

2004 (3)

T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,” IEEE Photon. Technol. Lett. 16, 551–553 (2004).
[Crossref]

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

J. E. Sharping, J. Chen, X. Li, and P. Kumar, “Quantum-correlated twin photons from microstructure fiber,” Opt. Express 12, 3086–3094 (2004).
[Crossref] [PubMed]

2002 (3)

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. 14, 983–985 (2002).
[Crossref]

K. Uesaka, K. K. Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: Theory and experiments,” IEEE J. Sel. Top. Quantum Electron. 8, 560–568 (2002).
[Crossref]

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

2001 (1)

W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A 64, 063815 (2001).
[Crossref]

2000 (1)

C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: Effective finite Hilbert space and entropy control,” Phys. Rev. Lett. 84, 5304–5307 (2000).
[Crossref] [PubMed]

1994 (1)

K. Inoue, “Tunable and selective wavelength conversion using fiber four-wave mixing with two pump lights,” IEEE Photon. Technol. Lett. 6, 1451–1453 (1994).
[Crossref]

1993 (1)

G. W. Stewart, “On the early history of the singular value decomposition,” SIAM Rev. 35, 551–566 (1993).
[Crossref]

1991 (3)

A. K. Ekert and P. L. Knight, “Relationship between semiclassical and quantum-mechanical input-output theories of optical response,” Phys. Rev. A 43, 3934–3938 (1991).
[Crossref] [PubMed]

C. M. Caves, C. Zhu, G. J. Milburn, and W. Schleich, “Photon statistics of two-mode squeezed states and interference in four-dimensional phase space,” Phys. Rev. A 43, 3854–3861 (1991).
[Crossref] [PubMed]

M. Artoni, U. P. Ortiz, and J. L. Birman, “Photocount distribution of two-mode squeezed states,” Phys. Rev. A 43, 3954–3965 (1991).
[Crossref] [PubMed]

1989 (2)

R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) summetry and photon statistics,” Phys. Rev. A 40, 1371–1384 (1989).
[Crossref] [PubMed]

H. P. Yuen, “Multimode two-photon coherent states and unitary representation of the symplectic group,” Nucl. Phys. B (Proc. Supp.) 6, 309–313 (1989).
[Crossref]

1987 (1)

S. Prasad, M. O. Scully, and W. Martienssen, “A quantum description of the beam splitter,” Opt. Commun. 62, 139–145 (1987).
[Crossref]

1985 (1)

R. Loudon, “Theory of noise accumulation in optical-amplifier chains,” IEEE J. Quantum Electron. 21, 766–773 (1985).
[Crossref]

1982 (1)

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

1976 (1)

H. P. Yuen, “Two-photon states of the radiation field,” Phys. Rev. A 13, 2226–2243 (1976).
[Crossref]

1963 (1)

J. P. Gordon, W. H. Louisell, and L. R. Walker, “Quantum fluctuations and noise in parametric processes II,” Phys. Rev. 129, 481–485 (1963).
[Crossref]

1961 (1)

W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes I,” Phys. Rev. 124, 1646–1654 (1961).

1957 (1)

M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc. IRE 45, 1012–1013 (1957).

1956 (1)

J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements,” Proc. IRE 44, 904–913 (1956).
[Crossref]

Alibart, O.

S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature 437, 116–120 (2005).
[Crossref] [PubMed]

Alic, N.

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A. B. U’Ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146–161 (2005).

I. A. Walmsley and M. G. Raymer, “Toward quantum information processing with photons,” Science 307, 1733–1734 (2005).
[Crossref] [PubMed]

Richardson, D. J.

R. Slavik, A. Bogris, F. Parmigiani, J. Kakande, M. Westlund, M. Sköld, L. Grüner-Nielsen, R. Phelan, D. Syvridis, P. Petropoulos, and D. J. Richardson, “Coherent all-optical phase and amplitude regenerator of binary phase-encoded signals,” IEEE J. Sel. Top. Quantum Electron. 18, 859–869 (2012).
[Crossref]

J. Kakande, F. Parmigiani, M. Ibsen, P. Petropoulos, and D. J. Richardson, “Wide bandwidth experimental study of nondegenerate phase-sensitive amplifiers in single- and dual-pump configurations,” IEEE Photon. Technol. Lett. 22, 1781–1783 (2010).
[Crossref]

Rottwitt, K.

C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, “Quantum-state-preserving optical frequency conversion and pulse reshaping by four-wave mixing,” Phys. Rev. A 85, 053829 (2012).
[Crossref]

L. Mejling, C. J. McKinstrie, M. G. Raymer, and K. Rottwitt, “Quantum frequency translation by four-wave mixing in a fiber: Low-conversion regime,” Opt. Express 20, 8367–8396 (2012).
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J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements,” Proc. IRE 44, 904–913 (1956).
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Saleh, B. E. A.

R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) summetry and photon statistics,” Phys. Rev. A 40, 1371–1384 (1989).
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Schleich, W.

C. M. Caves, C. Zhu, G. J. Milburn, and W. Schleich, “Photon statistics of two-mode squeezed states and interference in four-dimensional phase space,” Phys. Rev. A 43, 3854–3861 (1991).
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S. Prasad, M. O. Scully, and W. Martienssen, “A quantum description of the beam splitter,” Opt. Commun. 62, 139–145 (1987).
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T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,” IEEE Photon. Technol. Lett. 16, 551–553 (2004).
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Shahnia, S.

Sharping, J. E.

J. E. Sharping, J. Chen, X. Li, and P. Kumar, “Quantum-correlated twin photons from microstructure fiber,” Opt. Express 12, 3086–3094 (2004).
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M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. 14, 983–985 (2002).
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W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes I,” Phys. Rev. 124, 1646–1654 (1961).

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A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on spectrally engineered sum frequency generation,” Opt. Express 19, 13770–13778 (2011).
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A. B. U’Ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146–161 (2005).

Sköld, M.

R. Slavik, A. Bogris, F. Parmigiani, J. Kakande, M. Westlund, M. Sköld, L. Grüner-Nielsen, R. Phelan, D. Syvridis, P. Petropoulos, and D. J. Richardson, “Coherent all-optical phase and amplitude regenerator of binary phase-encoded signals,” IEEE J. Sel. Top. Quantum Electron. 18, 859–869 (2012).
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Slavik, R.

R. Slavik, A. Bogris, F. Parmigiani, J. Kakande, M. Westlund, M. Sköld, L. Grüner-Nielsen, R. Phelan, D. Syvridis, P. Petropoulos, and D. J. Richardson, “Coherent all-optical phase and amplitude regenerator of binary phase-encoded signals,” IEEE J. Sel. Top. Quantum Electron. 18, 859–869 (2012).
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O. Cohen, J. S. Lundeen, B. J. Smith, G. Puentes, P. J. Mosley, and I. A. Walmsley, “Tailored photon-pair generation in optical fibers,” Phys. Rev. Lett. 102, 123603 (2009).
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Srinivasan, K.

M. G. Raymer and K. Srinivasan, “Manipulating the color and shape of single photons,” Phys. Today 65(11), 32–37 (2012).
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Stelmakh, N.

M. Vasilyev, M. Annamalai, N. Stelmakh, and P. Kumar, “Quantum properties of a spatially-broadband traveling-wave phase-sensitive optical parametric amplifier,” J. Mod. Opt. 57, 1908–1915 (2010).
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G. W. Stewart, “On the early history of the singular value decomposition,” SIAM Rev. 35, 551–566 (1993).
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R. Slavik, A. Bogris, F. Parmigiani, J. Kakande, M. Westlund, M. Sköld, L. Grüner-Nielsen, R. Phelan, D. Syvridis, P. Petropoulos, and D. J. Richardson, “Coherent all-optical phase and amplitude regenerator of binary phase-encoded signals,” IEEE J. Sel. Top. Quantum Electron. 18, 859–869 (2012).
[Crossref]

Tanemura, T.

T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,” IEEE Photon. Technol. Lett. 16, 551–553 (2004).
[Crossref]

Tang, R.

Tanzilli, S.

S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature 437, 116–120 (2005).
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Teich, M. C.

R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) summetry and photon statistics,” Phys. Rev. A 40, 1371–1384 (1989).
[Crossref] [PubMed]

Tittel, W.

S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature 437, 116–120 (2005).
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Tong, Z.

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).
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C. J. McKinstrie, M. Karlsson, and Z. Tong, “Field-quadrature and photon-number correlations produced by parametric processes,” Opt. Express 18, 19792–19823 (2010).
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T. Torounidis and P. Andrekson, “Broadband single-pumped fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. 19, 650–652 (2007).
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H. Cruz-Ramirez, R. Ramirez-Alarcon, M. Corona, K. Garay-Palmett, and A. B. U’Ren, “Spontaneous parametric processes in quantum optics,” Opt. Photon. News 22(11), 37–41 (2011).
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K. Garay-Palmett, H. J. McGuinness, O. Cohen, J. S. Lundeen, R. Rangel-Rojo, A. B. U’Ren, M. G. Raymer, C. J. McKinstrie, S. Radic, and I. A. Walmsley, “Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber,” Opt. Express 15, 14870–14886 (2007).
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A. B. U’Ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146–161 (2005).

W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A 64, 063815 (2001).
[Crossref]

Uesaka, K.

K. Uesaka, K. K. Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: Theory and experiments,” IEEE J. Sel. Top. Quantum Electron. 8, 560–568 (2002).
[Crossref]

van Enk, S. J.

M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun. 283, 747–752 (2010).
[Crossref]

Vasilyev, M.

M. Vasilyev, M. Annamalai, N. Stelmakh, and P. Kumar, “Quantum properties of a spatially-broadband traveling-wave phase-sensitive optical parametric amplifier,” J. Mod. Opt. 57, 1908–1915 (2010).
[Crossref]

R. Tang, J. Lasri, P. S. Devgan, V. Grigoryan, P. Kumar, and M. Vasilyev, “Gain characteristics of a frequency nondegenerate phase-sensitive fiber-optic parametric amplifier with phase self-stabilized input,” Opt. Express 13, 10483–10493 (2005).
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Vasilyev, M. V.

Voss, P. L.

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. 14, 983–985 (2002).
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Wadsworth, W. J.

Walker, L. R.

J. P. Gordon, W. H. Louisell, and L. R. Walker, “Quantum fluctuations and noise in parametric processes II,” Phys. Rev. 129, 481–485 (1963).
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Walmsley, I. A.

O. Cohen, J. S. Lundeen, B. J. Smith, G. Puentes, P. J. Mosley, and I. A. Walmsley, “Tailored photon-pair generation in optical fibers,” Phys. Rev. Lett. 102, 123603 (2009).
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K. Garay-Palmett, H. J. McGuinness, O. Cohen, J. S. Lundeen, R. Rangel-Rojo, A. B. U’Ren, M. G. Raymer, C. J. McKinstrie, S. Radic, and I. A. Walmsley, “Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber,” Opt. Express 15, 14870–14886 (2007).
[Crossref] [PubMed]

A. B. U’Ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146–161 (2005).

I. A. Walmsley and M. G. Raymer, “Toward quantum information processing with photons,” Science 307, 1733–1734 (2005).
[Crossref] [PubMed]

W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A 64, 063815 (2001).
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C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: Effective finite Hilbert space and entropy control,” Phys. Rev. Lett. 84, 5304–5307 (2000).
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J. Fan, A. Migdall, and L. Wang, “A twin photon source,” Opt. Photon. News 18(3), 26–33 (2007).
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Wang, L. J.

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A. I. Lvovsky, W. Wasilewski, and K. Banaszek, “Decomposing a pulsed optical parametric amplifier into independent squeezers,” J. Mod. Opt. 54, 721–733 (2007).
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M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc. IRE 45, 1012–1013 (1957).

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R. Slavik, A. Bogris, F. Parmigiani, J. Kakande, M. Westlund, M. Sköld, L. Grüner-Nielsen, R. Phelan, D. Syvridis, P. Petropoulos, and D. J. Richardson, “Coherent all-optical phase and amplitude regenerator of binary phase-encoded signals,” IEEE J. Sel. Top. Quantum Electron. 18, 859–869 (2012).
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P. A. Andrekson and M. Westlund, “Nonlinear optical fiber based high resolution all-optical waveform sampling,” Laser Photon. Rev. 1, 231–248 (2007).
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J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
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Windmiller, J. R.

J. M. Chavez Boggio, S. Moro, E. Myslivets, J. R. Windmiller, N. Alic, and S. Radic, “155-nm continuous-wave two-pump parametric amplification,” IEEE Photon. Technol. Lett. 21, 612–614 (2009).
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Wong, K. K. Y.

K. Uesaka, K. K. Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: Theory and experiments,” IEEE J. Sel. Top. Quantum Electron. 8, 560–568 (2002).
[Crossref]

Xiong, C.

Yariv, A.

W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes I,” Phys. Rev. 124, 1646–1654 (1961).

Yu, M.

Yuen, H. P.

H. P. Yuen, “Multimode two-photon coherent states and unitary representation of the symplectic group,” Nucl. Phys. B (Proc. Supp.) 6, 309–313 (1989).
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H. P. Yuen, “Two-photon states of the radiation field,” Phys. Rev. A 13, 2226–2243 (1976).
[Crossref]

Zbinden, H.

S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature 437, 116–120 (2005).
[Crossref] [PubMed]

Zhu, C.

C. M. Caves, C. Zhu, G. J. Milburn, and W. Schleich, “Photon statistics of two-mode squeezed states and interference in four-dimensional phase space,” Phys. Rev. A 43, 3854–3861 (1991).
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IEEE J. Quantum Electron. (1)

R. Loudon, “Theory of noise accumulation in optical-amplifier chains,” IEEE J. Quantum Electron. 21, 766–773 (1985).
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IEEE J. Sel. Top. Quantum Electron (1)

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).
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IEEE J. Sel. Top. Quantum Electron. (4)

K. Uesaka, K. K. Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: Theory and experiments,” IEEE J. Sel. Top. Quantum Electron. 8, 560–568 (2002).
[Crossref]

R. Slavik, A. Bogris, F. Parmigiani, J. Kakande, M. Westlund, M. Sköld, L. Grüner-Nielsen, R. Phelan, D. Syvridis, P. Petropoulos, and D. J. Richardson, “Coherent all-optical phase and amplitude regenerator of binary phase-encoded signals,” IEEE J. Sel. Top. Quantum Electron. 18, 859–869 (2012).
[Crossref]

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
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S. Radic, “Parametric signal processing,” IEEE J. Sel. Top. Quantum Electron. 18, 670–680 (2012).
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IEEE Photon. Technol. Lett. (6)

K. Inoue, “Tunable and selective wavelength conversion using fiber four-wave mixing with two pump lights,” IEEE Photon. Technol. Lett. 6, 1451–1453 (1994).
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T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,” IEEE Photon. Technol. Lett. 16, 551–553 (2004).
[Crossref]

T. Torounidis and P. Andrekson, “Broadband single-pumped fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. 19, 650–652 (2007).
[Crossref]

J. M. Chavez Boggio, S. Moro, E. Myslivets, J. R. Windmiller, N. Alic, and S. Radic, “155-nm continuous-wave two-pump parametric amplification,” IEEE Photon. Technol. Lett. 21, 612–614 (2009).
[Crossref]

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. 14, 983–985 (2002).
[Crossref]

J. Kakande, F. Parmigiani, M. Ibsen, P. Petropoulos, and D. J. Richardson, “Wide bandwidth experimental study of nondegenerate phase-sensitive amplifiers in single- and dual-pump configurations,” IEEE Photon. Technol. Lett. 22, 1781–1783 (2010).
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IEICE Trans. Electron. (1)

J. H. Lee, “All-optical signal processing devices based on holey fiber,” IEICE Trans. Electron. E88-C, 327–334 (2005).
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J. Mod. Opt. (2)

A. I. Lvovsky, W. Wasilewski, and K. Banaszek, “Decomposing a pulsed optical parametric amplifier into independent squeezers,” J. Mod. Opt. 54, 721–733 (2007).
[Crossref]

M. Vasilyev, M. Annamalai, N. Stelmakh, and P. Kumar, “Quantum properties of a spatially-broadband traveling-wave phase-sensitive optical parametric amplifier,” J. Mod. Opt. 57, 1908–1915 (2010).
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J. Opt. Soc. Am. B (1)

J. Sel. Top. Quantum Electron. (1)

K. Croussore and G. Li, “Phase and amplitude regeneration of differential phase-shift keyed signals using phase-sensitive amplification,” J. Sel. Top. Quantum Electron. 14648–658 (2008).
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Laser Photon. Rev. (1)

P. A. Andrekson and M. Westlund, “Nonlinear optical fiber based high resolution all-optical waveform sampling,” Laser Photon. Rev. 1, 231–248 (2007).
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Laser Phys. (1)

A. B. U’Ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146–161 (2005).

Nature (1)

S. Tanzilli, W. Tittel, M. Halder, O. Alibart, P. Baldi, N. Gisin, and H. Zbinden, “A photonic quantum information interface,” Nature 437, 116–120 (2005).
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Nucl. Phys. B (Proc. Supp.) (1)

H. P. Yuen, “Multimode two-photon coherent states and unitary representation of the symplectic group,” Nucl. Phys. B (Proc. Supp.) 6, 309–313 (1989).
[Crossref]

Opt. Commun. (3)

M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun. 283, 747–752 (2010).
[Crossref]

S. Prasad, M. O. Scully, and W. Martienssen, “A quantum description of the beam splitter,” Opt. Commun. 62, 139–145 (1987).
[Crossref]

C. J. McKinstrie, “Unitary and singular value decompositions of parametric processes in fibers,” Opt. Commun. 282, 583–593 (2009).
[Crossref]

Opt. Express (13)

J. E. Sharping, J. Chen, X. Li, and P. Kumar, “Quantum-correlated twin photons from microstructure fiber,” Opt. Express 12, 3086–3094 (2004).
[Crossref] [PubMed]

J. G. Rarity, J. Fulconis, J. Duligall, W. J. Wadsworth, and P. S. J. Russell, “Photonic crystal fiber source of correlated photon pairs,” Opt. Express 13, 534–544 (2005).
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D. Méchin, R. Provo, J. D. Harvey, and C. J. McKinstrie, “180-nm wavelength conversion based on Bragg scattering in an optical fiber,” Opt. Express 14, 8995–8999 (2006).
[Crossref] [PubMed]

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

R. Tang, J. Lasri, P. S. Devgan, V. Grigoryan, P. Kumar, and M. Vasilyev, “Gain characteristics of a frequency nondegenerate phase-sensitive fiber-optic parametric amplifier with phase self-stabilized input,” Opt. Express 13, 10483–10493 (2005).
[Crossref] [PubMed]

M. Halder, J. Fulconis, B. Cemlyn, A. Clark, C. Xiong, W. J. Wadsworth, and J. G. Rarity, “Nonclassical 2-photon interference with separate intrinsically narrowband fibre sources,” Opt. Express 17, 4670–4676 (2009).
[Crossref] [PubMed]

A. Eckstein, B. Brecht, and C. Silberhorn, “A quantum pulse gate based on spectrally engineered sum frequency generation,” Opt. Express 19, 13770–13778 (2011).
[Crossref] [PubMed]

L. Mejling, C. J. McKinstrie, M. G. Raymer, and K. Rottwitt, “Quantum frequency translation by four-wave mixing in a fiber: Low-conversion regime,” Opt. Express 20, 8367–8396 (2012).
[Crossref] [PubMed]

K. Garay-Palmett, H. J. McGuinness, O. Cohen, J. S. Lundeen, R. Rangel-Rojo, A. B. U’Ren, M. G. Raymer, C. J. McKinstrie, S. Radic, and I. A. Walmsley, “Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber,” Opt. Express 15, 14870–14886 (2007).
[Crossref] [PubMed]

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

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

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

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] [PubMed]

Opt. Lett. (2)

Opt. Photon. News (3)

J. Fan, A. Migdall, and L. Wang, “A twin photon source,” Opt. Photon. News 18(3), 26–33 (2007).
[Crossref]

H. Cruz-Ramirez, R. Ramirez-Alarcon, M. Corona, K. Garay-Palmett, and A. B. U’Ren, “Spontaneous parametric processes in quantum optics,” Opt. Photon. News 22(11), 37–41 (2011).
[Crossref]

C. J. McKinstrie, S. Radic, and A. H. Gnauck, “All-optical signal processing by fiber-based parametric devices,” Opt. Photon. News 18(3), 34–40 (2007).
[Crossref]

Phys. Rev. (2)

W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes I,” Phys. Rev. 124, 1646–1654 (1961).

J. P. Gordon, W. H. Louisell, and L. R. Walker, “Quantum fluctuations and noise in parametric processes II,” Phys. Rev. 129, 481–485 (1963).
[Crossref]

Phys. Rev. A (8)

H. P. Yuen, “Two-photon states of the radiation field,” Phys. Rev. A 13, 2226–2243 (1976).
[Crossref]

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

W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A 64, 063815 (2001).
[Crossref]

C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, “Quantum-state-preserving optical frequency conversion and pulse reshaping by four-wave mixing,” Phys. Rev. A 85, 053829 (2012).
[Crossref]

R. A. Campos, B. E. A. Saleh, and M. C. Teich, “Quantum-mechanical lossless beam splitter: SU(2) summetry and photon statistics,” Phys. Rev. A 40, 1371–1384 (1989).
[Crossref] [PubMed]

C. M. Caves, C. Zhu, G. J. Milburn, and W. Schleich, “Photon statistics of two-mode squeezed states and interference in four-dimensional phase space,” Phys. Rev. A 43, 3854–3861 (1991).
[Crossref] [PubMed]

M. Artoni, U. P. Ortiz, and J. L. Birman, “Photocount distribution of two-mode squeezed states,” Phys. Rev. A 43, 3954–3965 (1991).
[Crossref] [PubMed]

A. K. Ekert and P. L. Knight, “Relationship between semiclassical and quantum-mechanical input-output theories of optical response,” Phys. Rev. A 43, 3934–3938 (1991).
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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. (3)

C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: Effective finite Hilbert space and entropy control,” Phys. Rev. Lett. 84, 5304–5307 (2000).
[Crossref] [PubMed]

H. J. McGuinness, M. G. Raymer, C. J. McKinstrie, and S. Radic, “Quantum frequency translation of single-photon states in a photonic crystal fiber,” Phys. Rev. Lett. 105, 093604 (2010).
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Figures (3)

Fig. 1
Fig. 1

In homodyne detection, a beam splitter (partially-reflecting mirror) is used to combine a signal mode (s) with a local-oscillator mode (l). Each output mode is a superposition of both input modes. In attenuation (modeled as beam splitting), a signal mode interacts with a loss-mode (l) at a virtual mirror. The input and output loss modes are inaccessible.

Fig. 2
Fig. 2

Frequency diagrams for (a) distant and (b) nearby Bragg scattering. Long arrows denote pumps (p1 and p2), whereas short arrows denote signal and idler sidebands (s1 and s2). Downward arrows denote modes that lose photons, whereas upward arrows denote modes that gain photons. The directions of the arrows are reversible.

Fig. 3
Fig. 3

Frequency diagrams for (a) modulation interaction, (b) inverse modulation interaction, and (c) outer-band and (d) inner-band phase conjugation. Long arrows denote pumps (p1 and p2), whereas short arrows denote sidebands (s1 and s2). Downward arrows denote modes that lose photons, whereas upward arrows denote modes that gain photons.

Equations (207)

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[ a j ( t 1 ) , a k ( t 2 ) ] = 0 , [ a j ( t 1 ) , a k ( t 2 ) ] = δ j k δ ( t 1 t 2 ) ,
δ N j 2 = ( N j N j ) 2 = N j 2 N j 2 .
δ N j δ N k = ( N j N j ) ( N k N k ) = N j N k N j N k .
n j ( t ) = a j ( t ) a j ( t )
N j = n j ( t ) d t ,
N j N k = n j ( t 1 ) n k ( t 2 ) d t 1 d t 2 .
a j ( t ) | α j = α j ( t ) | α j , α j | a j ( t ) = α j | α j * ( t ) ,
n j ( t ) = a j ( t ) a j ( t ) = | α j ( t ) 2 .
n j ( t 1 ) n j ( t 2 ) = a j ( t 1 ) a j ( t 1 ) a j ( t 2 ) a j ( t 2 ) = a j ( t 1 ) [ a j ( t 2 ) a j ( t 1 ) + δ ( t 1 t 2 ) ] a j ( t 2 ) = | α j ( t 1 ) | 2 | α j ( t 2 ) | 2 + α j * ( t 1 ) α j ( t 2 ) δ ( t 1 t 2 ) .
N j = | α j ( t ) | 2 d t ,
δ N j 2 = N j .
n j ( t 1 ) n k ( t 2 ) = a j ( t 1 ) a j ( t 1 ) a k ( t 2 ) a k ( t 2 ) = | α j ( t 1 ) | 2 | α k ( t 2 ) | 2 ,
δ N j δ N k = 0 .
b j ( t ) = τ a j ( t ) + ρ a k ( t ) ,
b k ( t ) = ρ * a j ( t ) + τ * a k ( t ) .
n j k ( t ) = ( | τ | 2 | ρ | 2 ) [ a j ( t ) a j ( t ) a k ( t ) a k ( t ) ] + 2 [ τ ρ * a j ( t ) a k ( t ) + τ * ρ a j ( t ) a k ( t ) ] .
n j k ( t ) = a j ( t ) a k ( t ) + a j ( t ) a k ( t ) ,
n j k ( t ) = α j ( t ) α k * ( t ) + α j ( t ) α k ( t ) ,
n j k ( t 1 ) n j k ( t 2 ) = [ a j ( t 1 ) a k ( t 1 ) + a j ( t 1 ) a k ( t 1 ) ] [ a j ( t 2 ) a k ( t 2 ) + a j ( t 2 ) a k ( t 2 ) ] = [ α j ( t 1 ) α k * ( t 1 ) + α j * ( t 1 ) α k ( t 1 ) ] [ α j ( t 2 ) α k * ( t 2 ) + α j * ( t 2 ) α k ( t 2 ) ] + α k * ( t 1 ) α k ( t 2 ) δ j ( t 1 t 2 ) + α j * ( t 1 ) α j ( t 2 ) δ k ( t 1 t 2 ) ,
N j k = [ α j ( t ) α k * ( t ) + α j * ( t ) α k ( t ) ] d t ,
δ N j k 2 = N k 1 j + N j 1 k ,
q j ( t ) = [ α p * ( t ) a j ( t ) + α p ( t ) a j ( t ) ] / 2 1 / 2 ,
q j ( t ) = [ α p * ( t ) α j ( t ) + α p ( t ) α j * ( t ) ] / 2 1 / 2 ,
q j ( t 1 ) q j ( t 2 ) = [ α p * ( t 1 ) a j ( t 1 ) + α p ( t 1 ) a j ( t 1 ) ] [ α p * ( t 2 ) a j ( t 2 ) + α p ( t 2 ) a j ( t 2 ) ] / 2 = [ α p * ( t 1 ) α j ( t 1 ) + α p ( t 1 ) α j * ( t 1 ) ] [ α p * ( t 2 ) α j ( t 2 ) + α p ( t 2 ) α j * ( t 2 ) ] / 2 + α p * ( t 1 ) α p ( t 2 ) δ ( t 1 t 2 ) / 2 .
Q j = [ α p * ( t ) α j ( t ) + α p ( t ) α j * ( t ) ] d t / 2 1 / 2 ,
δ Q j 2 = 1 / 2 .
q j ( t 1 ) q k ( t 2 ) = [ α p * ( t 1 ) a j ( t 1 ) + α p ( t 1 ) a j ( t 1 ) ] [ α q * ( t 2 ) a k ( t 2 ) + α q ( t 2 ) a k ( t 2 ) ] / 2 = [ α p * ( t 1 ) α j ( t 1 ) + α p ( t 1 ) α j * ( t 1 ) ] [ α q * ( t 2 ) α k ( t 2 ) + α q ( t 2 ) α k * ( t 2 ) ] / 2 ,
δ Q j δ Q k = 0 .
b 1 ( t ) = [ g 11 ( t , t ) a 1 ( t ) + g 12 ( t , t ) a 2 ( t ) ] d t ,
b 2 ( t ) = [ g 21 ( t , t ) a 1 ( t ) + g 22 ( t , t ) a 2 ( t ) ] d t ,
q 1 ( t ) = [ β p * ( t ) β 1 ( t ) + β p ( t ) β 1 * ( t ) ] / 2 1 / 2 ,
Q 1 = [ β p * ( t ) β 1 ( t ) + β p ( t ) β 1 * ( t ) ] d t / 2 1 / 2 .
q 1 ( t 1 ) q 1 ( t 2 ) = [ β p * ( t 1 ) b 1 ( t 1 ) + β p ( t 1 ) b 1 ( t 1 ) ] [ β p * ( t 2 ) b 1 ( t 2 ) + β p ( t 2 ) b 1 ( t 2 ) ] / 2 = [ β p * ( t 1 ) β 1 ( t 1 ) + β p ( t 1 ) β 1 * ( t 1 ) ] [ β p * ( t 2 ) β 1 ( t 2 ) + β p ( t 2 ) β 1 * ( t 2 ) ] / 2 + β p * ( t 1 ) β p ( t 2 ) δ ( t 1 t 2 ) / 2 ,
q 1 ( t 1 ) q 2 ( t 2 ) = [ β p * ( t 1 ) b 1 ( t 1 ) + β p ( t 1 ) b 1 ( t 1 ) ] [ β q * ( t 2 ) b 2 ( t 2 ) + β q ( t 2 ) b 2 ( t 2 ) ] / 2 = [ β p * ( t 1 ) β 1 ( t 1 ) + β p ( t 1 ) β 1 * ( t 1 ) ] [ β q * ( t 2 ) β 2 ( t 2 ) + β q ( t 2 ) β 2 * ( t 2 ) ] / 2 ,
δ Q 1 2 = 1 / 2 ,
δ Q 1 δ Q 2 = 0 .
β p * ( t 1 ) β p ( t 2 ) [ g 11 ( t 1 , t ) g 11 * ( t 2 , t ) + g 12 ( t 1 , t ) g 12 * ( t 2 , t ) ] d t / 2 ,
β p * ( t 1 ) β q ( t 2 ) [ g 11 ( t 1 , t ) g 21 * ( t 2 , t ) + g 12 ( t 1 , t ) g 22 * ( t 2 , t ) ] d t / 2
n 1 ( t ) = b 1 ( t 1 ) b 1 ( t ) = | β 1 ( t ) | 2 .
N 1 = | β 1 ( t ) | 2 d t .
n 1 ( t 1 ) n 1 ( t 2 ) = b 1 ( t 1 ) b 1 ( t 1 ) b 1 ( t 2 ) b 1 ( t 2 ) = | β 1 ( t 1 ) | 2 | β 1 ( t 2 ) | 2 + β 1 * ( t 1 ) β 1 ( t 2 ) δ ( t 1 t 2 ) ,
n 1 ( t 1 ) n 2 ( t 2 ) = b 1 ( t 1 ) b 1 ( t 1 ) b 2 ( t 2 ) b 2 ( t 2 ) = | β 1 ( t 1 ) | 2 | β 2 ( t 2 ) | 2 ,
δ N 1 2 = N 1 ,
δ N 1 δ N 2 = 0 .
β 1 * ( t 1 ) β 1 ( t 2 ) [ g 11 ( t 1 , t ) g 11 * ( t 2 , t ) + g 12 ( t 1 , t ) g 12 * ( t 2 , t ) ] d t ,
β 1 * ( t 1 ) β 2 ( t 2 ) [ g 11 ( t 1 , t ) g 21 * ( t 2 , t ) + g 12 ( t 1 , t ) g 22 * ( t 2 , t ) ] d t
b 1 ( t ) = [ g 11 ( t , t ) a 1 ( t ) + g 12 ( t , t ) a 2 ( t ) ] d t ,
b 2 ( t ) = [ g 22 ( t , t ) a 2 ( t ) + g 21 ( t , t ) a 1 ( t ) ] d t ,
q 1 ( t ) = [ β p * ( t ) β 1 ( t ) + β p ( t ) β 1 * ( t ) ] / 2 1 / 2 ,
Q 1 = [ β p * ( t ) β 1 ( t ) + β p ( t ) β 1 * ( t ) ] d t / 2 1 / 2 .
q 1 ( t 1 ) q 1 ( t 2 ) = [ β p * ( g 11 a 1 + g 12 a 2 ) d t + β p ( g 11 * a 1 + g 12 * a 2 ) d t ] t 1 × [ β p * ( g 11 a 1 + g 12 a 2 ) d t + β p ( g 11 * a 1 + g 12 * a 2 ) d t ] t 2 / 2 ,
δ q 1 ( t 1 ) δ q 1 ( t 2 ) = β p * ( t 1 ) β p ( t 2 ) g 11 ( t 1 , t ) g 11 * ( t 2 , t ) d t / 2 + β p ( t 1 ) β p * ( t 2 ) g 12 * ( t 1 , t ) g 12 ( t 2 , t ) d t / 2 .
δ q 1 ( t 1 ) δ q 1 ( t 2 ) = β p * ( t 1 ) β q ( t 2 ) δ ( t 1 t 2 ) / 2 + β p * ( t 1 ) β p ( t 2 ) g 12 ( t 1 , t ) g 12 * ( t 2 , t ) d t / 2 + β p ( t 1 ) β p * ( t 2 ) g 12 * ( t 1 , t ) g 12 ( t 2 , t ) d t / 2 ,
q 1 ( t 1 ) q 2 ( t 2 ) = [ β p * ( g 11 a 1 + g 12 a 2 ) d t + β p ( g 11 * a 1 + g 12 * a 2 ) d t ] t 1 × [ β q * ( g 22 a 2 + g 21 a 1 ) d t + β q ( g 22 * a 2 + g 21 * a 1 ) d t ] t 2 / 2 .
δ q 1 ( t 1 ) δ q 2 ( t 2 ) = β p * ( t 1 ) β q * ( t 2 ) g 11 ( t 1 , t ) g 21 ( t 2 , t ) d t / 2 + β p ( t 1 ) β q ( t 2 ) g 12 * ( t 1 , t ) g 22 * ( t 2 , t ) d t / 2 .
δ q 1 ( t 1 ) δ q 2 ( t 2 ) = β p * ( t 1 ) β q * ( t 2 ) g 11 ( t 1 , t ) g 21 ( t 2 , t ) d t / 2 + β p ( t 1 ) β q ( t 2 ) g 11 * ( t 1 , t ) g 21 * ( t 2 , t ) d t / 2 ,
δ Q 1 2 = [ | β p * ( t ) g 11 ( t , t ) d t | 2 + | β p * ( t ) g 12 ( t , t ) d t | 2 ] d t / 2 = 1 / 2 + | β p * ( t ) g 12 ( t , t ) d t | 2 d t ,
δ Q 1 δ Q 2 = Re [ β p * ( t 1 ) g 11 ( t 1 , t ) d t 1 β q * ( t 2 ) g 21 ( t 2 , t ) d t 2 ] d t ,
n 1 ( t ) = [ g 11 * ( t , t ) a 1 ( t ) + g 12 * ( t , t ) a 2 ( t ) ] d t × [ g 11 ( t , t ) a 1 ( t ) + g 12 ( t , t ) a 2 ( t ) ] d t = | β 1 ( t ) | 2 + | g 12 ( t , t ) | 2 d t .
N 1 = | β 1 ( t ) | 2 d t + | g 12 ( t , t ) | 2 d t d t .
n 1 ( t ) = | β 1 ( t ) | 2 + β 1 * ( t ) w 1 ( t ) + β 1 ( t ) w 1 ( t ) + w 1 ( t ) w 1 ( t ) .
n 1 l ( t ) = | β 1 ( t ) | 2 + β 1 * ( t ) g 11 ( t , t ) v 1 ( t ) d t + β 1 ( t ) g 12 * ( t , t ) v 2 ( t ) d t + g 12 * ( t , t ) v 2 ( t ) d t [ g 11 ( t , t ) v 1 ( t ) + g 12 ( t , t ) v 2 ( t ) ] d t ,
n 1 r ( t ) = | β 1 ( t ) | 2 + β 1 * ( t ) g 12 ( t , t ) v 2 ( t ) d t + β 1 ( t ) g 11 * ( t , t ) v 1 ( t ) d t + [ g 11 * ( t , t ) v 1 ( t ) + g 12 * ( t , t ) v 2 ( t ) ] d t g 12 ( t , t ) v 2 ( t ) d t ,
n 1 l ( t 1 ) n 1 r ( t 2 ) = [ | β 1 | 2 + β 1 * g 11 v 1 + β 1 g 12 * v 2 + g 12 * v 2 ( g 11 v 1 + g 12 v 2 ) ] t 1 × [ | β 1 | 2 + β 1 * g 12 v 2 + β 1 g 11 * v 1 + ( g 11 * v 1 + g 12 * v 2 ) g 12 v 2 ] t 2 ,
n 1 ( t 1 ) n 1 ( t 2 ) = | β 1 ( t 1 ) β 1 ( t 2 ) | 2 + | β 1 ( t 1 ) | 2 | g 12 ( t 2 , t ) | 2 d t + | β 1 ( t 2 ) | 2 | g 12 ( t 1 , t ) | 2 d t + β 1 * ( t 1 ) β 1 ( t 2 ) g 11 ( t 1 , t ) g 11 * ( t 2 , t ) d t + β 1 ( t 1 ) β 1 * ( t 2 ) g 12 * ( t 1 , t ) g 12 ( t 2 , t ) d t + g 11 ( t 1 , t ) g 11 * ( t 2 , t ) d t g 12 * ( t 1 , t ) g 12 ( t 2 , t ) d t + | g 12 ( t 1 , t ) | 2 d t | g 12 ( t 2 , t ) | 2 d t .
δ n 1 ( t 1 ) δ n 1 ( t 2 ) = β 1 * ( t 1 ) β 1 ( t 2 ) g 11 ( t 1 , t ) g 11 * ( t 2 , t ) d t + β 1 ( t 1 ) β 1 * ( t 2 ) g 12 * ( t 1 , t ) g 12 ( t 2 , t ) d t + g 11 ( t 1 , t ) g 11 * ( t 2 , t ) d t g 12 * ( t 1 , t ) g 12 ( t 2 , t ) d t .
δ n 1 ( t 1 ) δ n 1 ( t 2 ) = 2 Re [ β 1 * ( t 1 ) β 1 ( t 2 ) g 12 ( t 1 , t ) g 12 * ( t 2 , t ) d t ] + β 1 * ( t 1 ) β 1 ( t 2 ) δ ( t 1 t 2 ) + | g 12 ( t 1 , t ) g 12 * ( t 2 , t ) d t | 2 + g 12 * ( t 1 , t ) g 12 ( t 2 , t ) d t δ ( t 1 t 2 ) ,
n 1 l ( t 1 ) n 2 r ( t 2 ) = [ | β 1 | 2 + β 1 * g 11 v 1 + β 1 g 12 * v 2 + g 12 * v 2 ( g 11 v 1 + g 12 v 2 ) ] t 1 × [ | β 2 | 2 + β 2 * g 21 v 1 + β 2 g 22 * v 2 + ( g 22 * v 2 + g 21 * v 1 ) g 21 v 1 ] t 2 .
n 1 ( t 1 ) n 2 ( t 2 ) = | β 1 ( t 1 ) β 2 ( t 2 ) | 2 + | β 1 ( t 1 ) | 2 | g 21 ( t 2 , t ) | 2 d t + | β 2 ( t 2 ) | 2 | g 12 ( t 1 , t ) | 2 d t + β 1 * ( t 1 ) β 2 * ( t 2 ) g 11 ( t 1 , t ) g 21 ( t 2 , t ) d t + β 1 ( t 1 ) β 2 ( t 2 ) g 12 * ( t 1 , t ) g 22 * ( t 2 , t ) d t + g 11 ( t 1 , t ) g 21 ( t 2 , t ) d t g 12 * ( t 1 , t ) g 22 * ( t 2 , t ) d t + | g 12 ( t 1 , t ) | 2 d t | g 21 ( t 2 , t ) | 2 d t .
δ n 1 ( t 1 ) δ n 2 ( t 2 ) = β 1 * ( t 2 ) β 2 * ( t 2 ) g 11 ( t 1 , t ) g 21 ( t 2 , t ) d t + β 1 ( t 1 ) β 2 ( t 2 ) g 12 * ( t 1 , t ) g 22 * ( t 2 , t ) d t + g 11 ( t 1 , t ) g 21 ( t 2 , t ) d t g 12 * ( t 1 , t ) g 22 * ( t 2 , t ) d t .
δ n 1 ( t 1 ) δ n 2 ( t 2 ) = 2 Re [ β 1 * ( t 1 ) β 2 * ( t 2 ) g 11 ( t 1 , t ) g 21 ( t 2 , t ) d t ] + | g 11 ( t 1 , t ) g 21 ( t 2 , t ) d t | 2 ,
δ N 1 2 = [ | β 1 * ( t ) g 11 ( t , t ) d t | 2 + | β 1 * ( t ) g 12 ( t , t ) d t | 2 ] d t + | g 11 ( t , t ) g 12 * ( t , t ) d t | 2 d t d t = | β 1 ( t ) | 2 d t + 2 | β 1 * ( t ) g 12 ( t , t ) d t | 2 d t + | g 12 ( t , t ) g 12 * ( t , t ) d t | 2 d t d t + | g 12 ( t , t ) | 2 d t d t ,
δ N 1 δ N 2 = 2 Re [ β 1 * ( t 1 ) g 11 ( t 1 , t ) d t 1 β 2 * ( t 2 ) g 21 ( t 2 , t ) d t 2 ] d t + | g 11 ( t , t ) g 12 * ( t , t ) d t | 2 d t d t ,
b ( t ) = [ g s ( t , t ) a ( t ) + g c ( t , t ) a ( t ) ] d t ,
δ q ( t 1 ) δ q ( t 2 ) = [ β p * ( t 1 ) g s ( t 1 , t ) + β p ( t 1 ) g c * ( t 1 , t ) ] × [ β p ( t 2 ) g s * ( t 2 , t ) + β p * ( t 2 ) g c ( t 2 , t ) ] d t .
δ Q 2 = | [ β p * ( t ) g s ( t , t ) + β p ( t ) g c * ( t , t ) ] d t | 2 d t / 2 ,
δ n ( t 1 ) δ n ( t 2 ) = [ β * ( t 1 ) g s ( t 1 , t ) + β ( t 1 ) g c * ( t 1 , t ) ] × [ β ( t 2 ) g s ( t 2 , t ) + β * ( t 2 ) g c ( t 2 , t ) ] d t + g s ( t 1 , t ) g s * ( t 2 , t ) d t g c * ( t 1 , t ) g c ( t 2 , t ) d t + g s ( t 1 , t ) g c ( t 2 , t ) d t g c * ( t 1 , t ) g s * ( t 2 , t ) d t .
δ N 2 = | [ β * ( t ) g s ( t , t ) + β ( t ) g c * ( t , t ) ] d t | 2 d t + 2 | g s ( t , t ) g c * ( t , t ) d t | 2 d t d t ,
g ( t , t ) = j v j ( t ) σ j u j * ( t ) ,
κ ( t , t ) = g * ( t , t ) g ( t , t ) d t ,
λ ( t , t ) = g ( t , t ) g * ( t , t ) d t ,
u j * ( t ) u k ( t ) d t = δ j k , j u j ( t 1 ) u j * ( t 2 ) = δ ( t 1 t 2 ) ,
[ g 11 ( t , t ) g 12 ( t , t ) g 21 ( t , t ) g 22 ( t , t ) ] = j [ v 1 j ( t ) τ j u 1 j * ( t ) v 1 j ( t ) ρ j u 2 j * ( t ) v 2 j ( t ) ρ j * u 1 j * ( t ) v 2 j ( t ) τ j * u 2 j * ( t ) ] ,
a i ( t ) = j a i j u i j ( t ) , a i j = u i j * ( t ) a i ( t ) d t .
[ a i k , a j l ] = 0 , [ a i k , a j l ] = δ i j δ k l .
b 1 j = τ j a 1 j + ρ j a 2 j ,
b 2 j = ρ j * a 1 j + τ j * a 2 j .
q 1 j = ( β ¯ p j * β 1 j + β ¯ p j β 1 j * ) / 2 1 / 2 ,
δ q 1 j 2 = | β ¯ p j | 2 / 2 ,
δ q 1 j δ q 2 j = 0 ,
n 1 j = | β 1 j | 2 ,
δ n 1 j 2 = | β 1 j | 2 ,
δ n 1 j δ n 2 j = 0 .
q 1 ( t ) = j k [ β p j * v 1 j * ( t ) β 1 k v 1 k ( t ) + β p j v 1 j ( t ) β 1 k * v 1 k * ( t ) ] / 2 1 / 2 .
Q 1 = j ( β p j * β 1 j + β p j β 1 j * ) / 2 1 / 2 ,
δ q 1 ( t 1 ) δ q 1 ( t 2 ) = i j k β p l * v 1 i * ( t 1 ) v 1 j ( t 1 ) v 1 j * ( t 2 ) β p k v 1 k ( t 2 ) / 2 .
δ Q 1 2 = j | β p j | 2 / 2 = 1 / 2 ,
[ g 11 ( t , t ) g 12 ( t , t ) g 21 ( t , t ) g 22 ( t , t ) ] = j [ v 1 j ( t ) μ j u 1 j * ( t ) v 1 j ( t ) ν j u 2 j ( t ) v 2 j ( t ) ν j u 1 j ( t ) v 2 j ( t ) μ j u 2 j * ( t ) ] ,
b 1 j = μ j a 1 j + ν j a 2 j ,
b 2 j = μ j a 2 j + ν j a 1 j .
q 1 j = ( β ¯ p j * β 1 j + β ¯ p j β 1 j * ) / 2 1 / 2 ,
δ q 1 j 2 = | β ¯ p j | 2 ( | μ j | 2 + | ν j | 2 ) / 2 ,
δ q 1 j δ q 2 j = ( β ¯ p j * β ¯ q j * μ j ν j + β ¯ p j β ¯ q j μ j * ν j * ) / 2 .
n 1 j = | β 1 j | 2 + | ν j | 2 ,
δ n 1 j 2 = | β 1 j | 2 ( | μ j | 2 + | ν j | 2 ) + | μ j ν j | 2 ,
δ n 1 j δ n 2 j = β 1 j * β 2 j * μ j ν j + β 1 j β 2 j μ j * ν j * + | μ j ν j | 2 .
δ q 1 ( t 1 ) δ q 1 ( t 2 ) = i j k [ β p i * v 1 i * ( t 1 ) v 1 j ( t 1 ) | μ j | 2 v 1 j * ( t 2 ) β p k v 1 k ( t 2 ) + β p i v 1 i ( t 1 ) v 1 j * ( t 1 ) | ν j | 2 v 1 j ( t 2 ) β p k * v 1 k * ( t 2 ) ] / 2 ,
δ q 1 ( t 1 ) δ q 2 ( t 2 ) = i j k [ β p i * v 1 i * ( t 1 ) v 1 j ( t 1 ) μ j ν j v 2 j ( t 2 ) β q k * v 2 k * ( t 2 ) + β p i v 1 i ( t 1 ) v 1 j * ( t 1 ) μ j * ν j * v 2 j * ( t 2 ) β q k v 2 k ( t 2 ) ] / 2 .
δ Q 1 2 = j | β p j | 2 ( | μ j | 2 + | ν j | 2 ) / 2 ,
δ Q 1 δ Q 2 = j ( β p j * β q j * μ j ν j + β p j β q j μ j * ν j * ) / 2 .
n 1 ( t ) = j k β 1 j * v 1 j * ( t ) β 1 k v 1 k ( t ) + j | ν j | 2 | v 1 j ( t ) | 2 ,
N 1 = j ( | β 1 j | 2 + | ν j | 2 ) ,
δ n 1 ( t 1 ) δ n 1 ( t 2 ) nn = [ j v 1 j ( t 1 ) | μ j | 2 v 1 j * ( t 2 ) ] [ k v 1 k * ( t 1 ) | ν k | 2 v 1 k ( t 2 ) ] ,
δ n 1 ( t 1 ) δ n 2 ( t 2 ) nn = [ j v 1 j ( t 1 ) μ j ν j v 2 j ( t 2 ) ] [ k v 1 k * ( t 1 ) μ k * ν k * v 2 k * ( t 2 ) ] .
δ N 1 2 nn = j | μ j ν j | 2 = δ N 1 δ N 2 nn .
n 1 ( t 1 ) n 2 ( t 2 ) nn = [ j μ j ν j v 1 j ( t 1 ) v 2 j ( t 2 ) ] [ k μ k * ν k * v 1 k * ( t 1 ) v 2 k * ( t 2 ) ] + [ j | ν j | 2 | v 1 j ( t 1 ) | 2 ] [ k | ν k | 2 | v 2 k ( t 2 ) | 2 ] .
n 1 ( t 1 ) n 2 ( t 2 ) nn = ( | μ 1 | 2 + | ν 1 | 2 ) | ν 1 | 2 | v 11 ( t 1 ) v 21 ( t 2 ) | 2 .
g s ( t , t ) = j v j ( t ) μ j u j * ( t ) , g c ( t , t ) = j v j ( t ) ν j u j ( t ) ,
δ q ( t 1 ) δ q ( t 2 ) = i j k [ β p i * v i * ( t 1 ) μ j v j ( t 1 ) + β p i v i ( t 1 ) ν j * v j * ( t 1 ) ] × [ μ j * v j * ( t 2 ) β p k v k ( t 2 ) + ν j v j ( t 2 ) β p k * v k * ( t 2 ) ] / 2 ,
δ Q 2 = j | β p j * μ j + β p j ν j * | 2 / 2 ,
δ n ( t 1 ) δ n ( t 2 ) = i j k [ β i * v i * ( t 1 ) μ j v j ( t 1 ) + β i v i ( t 1 ) ν j * v j * ( t 1 ) ] × [ μ j * v j * ( t 2 ) β k v k ( t 2 ) + ν j v j ( t 2 ) β k * v k * ( t 2 ) ] + j k | μ j | 2 v j ( t 1 ) v j * ( t 2 ) | ν k | 2 v k * ( t 1 ) v k ( t 2 ) + j k μ j ν j v j ( t 1 ) v j ( t 2 ) μ k * ν k * v k * ( t 1 ) v k * ( t 2 ) ,
δ N 2 = j | β j * μ j + β j ν j * | 2 + 2 j | μ j ν j | 2 ,
[ b 1 ( t 1 ) , b 1 ( t 2 ) ] = [ g 11 ( t 1 , t ) g 11 * ( t 2 , t ) + g 12 ( t 1 , t ) g 12 * ( t 2 , t ) ] d t = δ ( t 1 t 2 ) ,
[ b 1 ( t 1 ) , b 2 ( t 2 ) ] = [ g 11 ( t 1 , t ) g 21 * ( t 2 , t ) + g 12 ( t 1 , t ) g 22 * ( t 2 , t ) ] d t = 0 .
a 1 ( t ) = [ h 11 ( t , t ) b 1 ( t ) + h 12 ( t , t ) b 2 ( t ) ] d t ,
a 2 ( t ) = [ h 21 ( t , t ) b 1 ( t ) + h 22 ( t , t ) b 2 ( t ) ] d t ,
[ a 1 ( t 1 ) , a 1 ( t 2 ) ] = [ h 11 ( t 11 , t ) h 11 * ( t 2 , t ) + h 12 ( t 1 , t ) h 12 * ( t 2 , t ) ] d t = δ ( t 1 t 2 ) ,
[ a 1 ( t 1 ) , a 2 ( t 2 ) ] = [ h 11 ( t 1 , t ) h 21 * ( t 2 , t ) + h 12 ( t 1 , t ) h 22 * ( t 2 , t ) ] d t = 0 .
b 1 ( t ) = [ g 11 ( t , t ) h 11 ( t , t ) + g 12 ( t , t ) h 21 ( t , t ) ] b 1 ( t ) d t d t + [ g 11 ( t , t ) h 12 ( t , t ) + g 12 ( t , t ) h 22 ( t , t ) ] b 2 ( t ) d t d t ,
[ g 11 ( t , t ) h 11 ( t , t ) + g 12 ( t , t ) h 21 ( t , t ) ] d t = δ ( t t ) ,
[ g 11 ( t , t ) h 12 ( t , t ) + g 12 ( t , t ) h 22 ( t , t ) ] d t = 0 .
h 11 ( t 1 , t 2 ) = g 11 * ( t 2 , t 1 ) , h 21 ( t 1 , t 2 ) = g 12 * ( t 2 , t 1 ) .
[ g 11 * ( t , t 1 ) g 11 ( t , t 2 ) + g 21 * ( t , t 1 ) g 21 ( t , t 2 ) ] d t = δ ( t 1 t 2 ) ,
[ g 11 * ( t , t 1 ) g 12 ( t , t 2 ) + g 21 * ( t , t 1 ) g 22 ( t , t 2 ) ] d t = 0 .
n 1 ( t ) = [ g 11 * ( t , t ) a 1 ( t ) g 11 ( t , t ) a 1 ( t ) + g 11 * ( t , t ) a 1 ( t ) g 12 ( t , t ) a 2 ( t ) + g 12 * ( t , t ) a 2 ( t ) g 11 ( t , t ) a 1 ( t ) + g 12 * ( t , t ) a 2 ( t ) g 12 ( t , t ) a 2 ( t ) ] d t d t .
n 1 ( t ) + n 2 ( t ) = [ g 11 * ( t , t ) g 11 ( t , t ) + g 21 * ( t , t ) g 21 ( t , t ) ] a 1 ( t ) a 1 ( t ) d t d t + [ g 11 * ( t , t ) g 12 ( t , t ) + g 21 * ( t , t ) g 22 ( t , t ) ] a 1 ( t ) a 2 ( t ) d t d t + [ g 22 * ( t , t ) g 21 ( t , t ) + g 12 * ( t , t ) g 11 ( t , t ) ] a 2 ( t ) a 1 ( t ) d t d t + [ g 22 * ( t , t ) g 22 ( t , t ) + g 12 * ( t , t ) g 12 ( t , t ) ] a 2 ( t ) a 2 ( t ) d t d t .
N 1 + N 2 = M 1 + M 2 ,
N 1 + N 2 = M 1 + M 2 ,
δ N 1 2 + 2 δ N 1 δ N 2 + δ N 2 2 = δ M 1 2 + 2 δ M 1 δ M 2 + δ M 2 2 .
δ N 1 2 + 2 δ N 1 δ N 2 + δ N 2 2 = M 1 + M 2 .
[ b 1 ( t 1 ) , b 1 ( t 2 ) ] = [ g 11 ( t 1 , t ) g 11 * ( t 2 , t ) g 12 ( t 1 , t ) g 12 * ( t 2 , t ) ] d t = δ ( t 1 t 2 ) ,
[ b 1 ( t 1 ) , b 2 ( t 2 ) ] = [ g 11 ( t 1 , t ) g 21 ( t 2 , t ) g 12 ( t 1 , t ) g 22 ( t 2 , t ) ] d t = 0 .
a 1 ( t ) = [ h 11 ( t , t ) b 1 ( t ) + h 12 ( t , t ) b 2 ( t ) ] d t ,
a 2 ( t ) = [ h 22 ( t , t ) b 2 ( t ) + h 21 ( t , t ) b 1 ( t ) ] d t .
[ a 1 ( t 1 ) , a 1 ( t 2 ) ] = [ h 11 ( t 1 , t ) h 11 * ( t 2 , t ) h 12 ( t 1 , t ) h 12 * ( t 2 , t ) ] d t = δ ( t 1 t 2 ) ,
[ a 1 ( t 1 ) , a 2 ( t 2 ) ] = [ h 11 ( t 1 , t ) h 21 ( t 2 , t ) h 12 ( t 1 , t ) h 22 ( t 2 , t ) ] d t = 0 .
b 1 ( t ) = [ g 11 ( t , t ) h 11 ( t , t ) + g 12 ( t , t ) h 21 * ( t , t ) ] b 1 ( t ) d t d t + [ g 11 ( t , t ) h 12 ( t , t ) + g 12 ( t , t ) h 22 * ( t , t ) ] b 2 ( t ) d t d t ,
[ g 11 ( t , t ) h 11 ( t , t ) + g 12 ( t , t ) h 21 * ( t , t ) ] d t = δ ( t t ) ,
[ g 11 ( t , t ) h 12 ( t , t ) + g 12 ( t , t ) h 22 * ( t , t ) ] d t = 0 .
h 11 ( t 1 , t 2 ) = g 11 * ( t 2 , t 1 ) , h 21 ( t 1 , t 2 ) = g 12 ( t 2 , t 1 ) .
[ g 11 * ( t , t 1 ) g 11 ( t , t 2 ) g 21 ( t , t 1 ) g 21 * ( t , t 2 ) ] d t = δ ( t 1 t 2 ) ,
[ g 11 * ( t , t 1 ) g 12 ( t , t 2 ) g 21 ( t , t 1 ) g 22 * ( t , t 2 ) ] d t = 0 .
n 1 ( t ) = [ g 11 * ( t , t ) a 1 ( t ) g 11 ( t , t ) a 1 ( t ) + g 11 * ( t , t ) a 1 ( t ) g 12 ( t , t ) a 2 ( t ) + g 12 * ( t , t ) a 2 ( t ) g 11 ( t , t ) a 1 ( t ) + g 12 * ( t , t ) a 2 ( t ) g 12 ( t , t ) a 2 ( t ) ] d t d t ,
n 2 ( t ) = [ g 22 * ( t , t ) a 2 ( t ) g 22 ( t , t ) a 2 ( t ) + g 22 * ( t , t ) a 2 ( t ) g 21 ( t , t ) a 1 ( t ) + g 21 * ( t , t ) a 1 ( t ) g 22 ( t , t ) a 2 ( t ) + g 21 * ( t , t ) a 1 ( t ) g 21 ( t , t ) a 1 ( t ) ] d t d t .
n 1 ( t ) n 2 ( t ) = [ g 11 * ( t , t ) g 11 ( t , t ) g 21 ( t , t ) g 21 * ( t , t ) ] a 1 ( t ) a 1 ( t ) d t d t + [ g 11 * ( t , t ) g 12 ( t , t ) g 21 ( t , t ) g 22 * ( t , t ) ] a 1 ( t ) a 2 ( t ) d t d t + [ g 11 ( t , t ) g 12 * ( t , t ) g 21 * ( t , t ) g 22 ( t , t ) ] a 1 ( t ) a 2 ( t ) d t d t + [ g 12 ( t , t ) g 12 * ( t , t ) g 22 * ( t , t ) g 22 ( t , t ) ] a 2 ( t ) a 2 ( t ) d t d t + [ | g 12 ( t , t ) | 2 | g 21 ( t , t ) | 2 ] d t .
N 1 N 2 = M 1 M 2 + [ | g 12 ( t , t ) | 2 | g 21 ( t , t ) | 2 ] d t d t .
N 1 N 2 = M 1 M 2 ,
δ N 1 2 2 δ N 1 δ N 2 + δ N 2 2 = δ M 1 2 2 δ M 1 δ M 2 + δ M 2 2 .
δ N 1 2 2 δ N 1 δ N 2 + δ N 2 2 = M 1 + M 2 .
q j ( t ) = [ β p * ( t ) β j ( t ) + β p ( t ) β j * ( t ) ] / 2 1 / 2 ,
δ q j ( t ) = [ β p * ( t ) w j ( t ) + β p ( t ) w j ( t ) ] / 2 1 / 2 ,
δ q j ( t 1 ) δ q k ( t 2 ) = [ β p * ( t 1 ) w j ( t 1 ) + β p ( t 1 ) w j ( t 1 ) ] [ β q * ( t 2 ) w k ( t 2 ) + β q ( t 2 ) w k ( t 2 ) ] / 2 ,
n j ( t ) = | β j ( t ) | 2 + w j ( t ) w j ( t ) ,
δ n j ( t ) = β j * ( t ) w j ( t ) + β j ( t ) w j ( t ) + w j ( t ) w j ( t ) w j ( t ) w j ( t ) ,
δ n j ( t 1 ) δ n k ( t 2 ) = [ β j * ( t 1 ) w j ( t 1 ) + β j ( t 1 ) w j ( t 1 ) + w j ( t 1 ) w j ( t 1 ) w j ( t 1 ) w j ( t 1 ) ] × [ β k * ( t 2 ) w k ( t 2 ) + β k ( t 2 ) w k ( t 2 ) + w k ( t 2 ) w k ( t 2 ) w k ( t 2 ) w k ( t 2 ) ] ,
δ n j ( t 1 ) δ n k ( t 2 ) = [ β j * ( t 1 ) w j ( t 1 ) + β j ( t 1 ) w j ( t 1 ) ] [ β k * ( t 2 ) w k ( t 2 ) + β k ( t 2 ) w k ( t 2 ) ] + w j ( t 1 ) w j ( t 1 ) w k ( t 2 ) w k ( t 2 ) w j ( t 1 ) w j ( t 1 ) w k ( t 2 ) w k ( t 2 ) .
δ n j ( t 1 ) δ n k ( t 2 ) sn = [ β j * ( t 1 ) w j ( t 1 ) + β j ( t 1 ) w j ( t 1 ) ] [ β k * ( t 2 ) w k ( t 2 ) + β k ( t 2 ) w k ( t 2 ) ] .
b i ( t ) = k d t [ μ i k ( t , t ) a k ( t ) + ν i k ( t , t ) a k ( t ) ] ,
[ b i ( t 1 ) , b j ( t 2 ) ] = k d t [ μ i k ( t 1 , t ) μ j k * ( t 2 , t ) ν i k ( t 1 , t ) ν j k * ( t 2 , t ) ] = δ i j δ ( t 1 t 2 ) ,
[ b i ( t 1 ) , b j ( t 2 ) ] = k d t [ μ i k ( t 1 , t ) ν j k ( t 2 , t ) ν i k ( t 1 , t ) μ j k ( t 2 , t ) ] = 0 .
a i ( t ) = k d t [ μ ¯ i k ( t , t ) b k ( t ) + ν ¯ i k ( t , t ) b k ( t ) ] ,
b i ( t ) = j k d t d t { [ μ i j ( t , t ) μ ¯ j k ( t , t ) + ν i j ( t , t ) ν ¯ j k * ( t , t ) ] b k ( t ) + [ μ i j ( t , t ) ν ¯ j k ( t , t ) + ν i j ( t , t ) μ ¯ j k * ( t , t ) ] b k ( t ) } ,
j d t [ μ i j ( t , t ) μ ¯ j k ( t , t ) + ν i j ( t , t ) ν ¯ j k * ( t , t ) ] = δ i k δ ( t t ) ,
j d t [ μ i j ( t , t ) ν ¯ j k ( t , t ) + ν i j ( t , t ) μ ¯ j k * ( t , t ) ] = 0 .
μ ¯ i j ( t 1 , t 2 ) = μ j i * ( t 2 , t 1 ) , ν ¯ i j ( t 1 , t 2 ) = ν j i ( t 2 , t 1 ) .
k d t [ μ k i * ( t , t 1 ) μ k j ( t , t 2 ) ν k i ( t , t 1 ) ν k j * ( t , t 2 ) ] = δ i j δ ( t 1 t 2 ) ,
k d t [ μ k i * ( t , t 1 ) ν k j ( t , t 2 ) ν k i ( t , t 1 ) μ k j * ( t , t 2 ) ] = 0 .
q i ( t ) = [ β p * ( t ) β i ( t ) + β p ( t ) β i * ( t ) ] / 2 1 / 2 ,
δ q i ( t ) = [ β p * ( t ) w i ( t ) + β p ( t ) w i ( t ) ] / 2 1 / 2 ,
Q i = d t [ β p * ( t ) β i ( t ) + β p ( t ) β i * ( t ) ] / 2 1 / 2 .
δ q i ( t ) = k d t { β p * ( t ) [ μ i k ( t , t ) v k ( t ) + ν i k ( t , t ) v k ( t ) ] + β p ( t ) [ μ i k * ( t , t ) v k ( t ) + ν i k * ( t , t ) v k ( t ) ] } / 2 1 / 2 .
[ δ q i ( t ) ] l = k d t [ β p * ( t ) μ i k ( t , t ) + β p ( t ) ν i k * ( t , t ) ] v k ( t ) / 2 1 / 2 ,
[ δ q i ( t ) ] r = k d t [ β p * ( t ) ν i k ( t , t ) + β p ( t ) μ i k * ( t , t ) ] v k ( t ) / 2 1 / 2 ,
δ q i ( t 1 ) δ q j ( t 2 ) = k d t [ β p * ( t 1 ) μ i k ( t 1 , t ) + β p ( t 1 ) ν i k * ( t 1 , t ) ] × [ β q * ( t 2 ) ν j k ( t 2 , t ) + β q ( t 2 ) μ j k * ( t 2 , t ) ] / 2 ,
δ Q i δ Q j = k d t d t 1 [ β p * ( t 1 ) μ i k ( t 1 , t ) + β p ( t 1 ) ν i k * ( t 1 , t ) ] × d t 2 [ β q * ( t 2 ) ν j k ( t 2 , t ) + β q ( t 2 ) μ j k * ( t 2 , t ) ] / 2 .
δ q i ( t 1 ) δ q j ( t 2 ) ( β p * μ i k ) 1 ( β q * ν j k ) 2 + ( β p * μ i k ) 1 ( β q μ j k * ) 2 + ( β p ν i k * ) 1 ( β q * ν j k ) 2 + ( β p ν i k * ) 1 ( β q μ j k * ) 2 ,
( β p * μ i k ) 1 ( β q * ν j k ) 2 + ( β p ν i k * ) 1 ( β q μ j k * ) 2 = ( β p * μ i k ) 1 ( β q * ν j k ) 2 + ( β p μ i k * ) 1 ( β q ν j k * ) 2 = ( β p * ν i k ) 1 ( β q * μ j k ) 2 + ( β p μ i k * ) 1 ( β q ν j k * ) 2 = ( β q * μ j k ) 2 ( β p * ν i k ) 1 + ( β q ν j k * ) 2 ( β p μ i k * ) 1 .
( β p * μ i k ) 1 ( β q μ j k * ) 2 + ( β p ν i k * ) 1 ( β q * ν j k ) 2 = ( β p * ν i k ) 1 ( β q ν j k * ) 2 + ( β p ν i k * ) 1 ( β q * ν j k ) 2 + ( | β p | 2 ) 1 δ i j δ 12 = ( β p * ν i k ) 1 ( β q ν j k * ) 2 + ( β p μ i k * ) 1 ( β q * μ j k ) 2 = ( β q * μ j k ) 2 ( β p μ i k * ) 1 + ( β q ν j k * ) 2 ( β p * ν i k ) 1 .
δ Q i 2 = k d t | d t [ β p * ( t ) μ i k ( t , t ) + β p ( t ) ν i k * ( t , t ) ] | 2 / 2 ,
n i ( t ) = | β i ( t ) | 2 + w i ( t ) w i ( t ) ,
δ n i ( t ) = β p * ( t ) w i ( t ) + β p ( t ) w i ( t ) + w i ( t ) w i ( t ) w i ( t ) w i ( t ) ,
N i = d t | β i ( t ) | 2 + k d t d t | ν i k ( t , t ) | 2 .
δ n i ( t 1 ) δ n j ( t 2 ) sn = k d t [ β i * ( t 1 ) μ i k ( t 1 , t ) + β i ( t 1 ) ν i k * ( t 1 , t ) ] × [ β j * ( t 2 ) ν j k ( t 2 , t ) + β j ( t 2 ) μ j k * ( t 2 , t ) ] ,
δ N i δ N j sn = k d t d t 1 [ β i * ( t 1 ) μ i k ( t 1 , t ) + β i ( t 1 ) ν i k * ( t 1 , t ) ] × d t 2 [ β j * ( t 2 ) ν j k ( t 2 , t ) + β j ( t 2 ) μ j k * ( t 2 , t ) ] .
δ N i 2 sn = k d t | d t [ β i * ( t ) μ i k ( t , t ) + β i ( t ) ν i k * ( t , t ) ] | 2 ,
[ w i ( t ) w i ( t ) ] 1 = k l d t d t ν i k * ( t , t ) v k ( t ) [ μ i l ( t , t ) v l ( t ) + ν i l ( t , t ) v l ( t ) ] = k l d t d t [ ν i k * ( t , t ) μ i l ( t , t ) v k ( t ) v l ( t ) + ν i k * ( t , t ) ν i l ( t , t ) v k ( t ) v l ( t ) ] ,
[ w i ( t ) w i ( t ) ] r = k l d t d t [ μ i k * ( t , t ) v k ( t ) + ν i k * ( t , t ) v k ( t ) ] ν i l ( t , t ) v l ( t ) = k l d t d t [ μ i k * ( t , t ) ν i l ( t , t ) v k ( t ) v l ( t ) + ν i k * ( t , t ) ν i l ( t , t ) v k ( t ) v l ( t ) ] ,
v k ( t k ) v l ( t l ) v m ( t m ) v n ( t n ) = δ k l δ ( t k t l ) δ m n δ ( t m t n ) ,
v k ( t k ) v l ( t l ) v m ( t m ) v n ( t n ) = δ k m δ ( t k t m ) δ ln δ ( t l t n ) + δ k n δ ( t k t n ) δ l m δ ( t l t m ) .
w i ( t 1 ) w i ( t 1 ) w j ( t 2 ) w j ( t 2 ) = k l d t d t [ | ν i k ( t 1 , t ) | 2 | ν j l ( t 2 , t ) | 2 + ν i k * ( t 1 , t ) μ j k * ( t 2 , t ) μ i l ( t 1 , t ) ν j l ( t 2 , t ) + ν i k * ( t 1 , t ) ν j k ( t 2 , t ) μ i l ( t 1 , t ) μ j l * ( t 2 , t ) ] .
δ n i ( t 1 ) δ n j ( t 2 ) nn = k l d t d t [ μ i k ( t 1 , t ) μ j k * ( t 2 , t ) ν i l * ( t 1 , t ) ν j l ( t 2 , t ) + μ i k ( t 1 , t ) ν j k ( t 2 , t ) ν i l * ( t 1 , t ) μ j l * ( t 2 , t ) ] .
δ N i δ N j nn = k l d t d t [ d t 1 μ i k ( t 1 , t ) ν i l * ( t 1 , t ) d t 2 μ j k * ( t 2 , t ) ν j l ( t 2 , t ) + d t 1 μ i k ( t 1 , t ) ν i l * ( t 1 , t ) d t 2 ν j k ( t 2 , t ) μ j l * ( t 2 , t ) ] .
δ n i ( t 1 ) δ n j ( t 2 ) nn k l [ ( μ i k ) 1 ( μ j k * ) 2 ( ν i l * ) 1 ( ν j l ) 2 + ( μ i k ) 1 ( ν j k ) 2 ( ν i l * ) 1 ( μ j l * ) 2 ] ,
k l ( μ i k ) 1 ( μ j k * ) 2 ( ν i l * ) 1 ( ν j l ) 2 = k l ( ν i k ) 1 ( ν j k * ) 2 ( ν i l * ) 1 ( ν j l ) 2 + δ i j l ( | ν i l | 2 ) 1 = k l ( ν i k ) 1 ( ν j k * ) 2 ( μ i l * ) 1 ( μ j l ) 2 δ i j k ( | ν i k | 2 ) 1 + δ i j l ( | ν i l | 2 ) 1 = k l ( μ j l ) 2 ( μ i l * ) 1 ( ν j k * ) 2 ( ν i k ) 1 .
k l ( μ i k ) 1 ( ν j k ) 2 ( ν i l * ) 1 ( μ j l * ) 2 = k l ( μ i k ) 1 ( ν j k ) 2 ( μ i l * ) 1 ( ν j l * ) 2 = k l ( ν i k ) 1 ( μ j k ) 2 ( μ i l * ) 1 ( ν j l * ) 2 = k l ( μ j k ) 2 ( ν i k ) 1 ( ν j l * ) 2 ( μ i l * ) 1 .
δ N i δ N j nn 2 k ( μ i k ν i k * ) 1 ( μ j k * ν j k ) 2 + k l > k ( μ i k ν i l * + μ i l ν i k * ) 1 ( μ j k * ν j l + μ j l * ν j k ) 2 ,
δ N i 2 nn 2 k | μ i k ν i k * | 2 + k l > k | μ i k ν i l * + μ i l ν i k * | 2 ,

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