Abstract

We analyze the quantum dynamics of radiation propagating in a single-mode optical fiber with dispersion, nonlinearity, and Raman coupling to thermal phonons. We start from a fundamental Hamiltonian that includes the principal known nonlinear effects and quantum-noise sources, including linear gain and loss. Both Markovian and frequency-dependent, non-Markovian reservoirs are treated. This treatment allows quantum Langevin equations, which have a classical form except for additional quantum-noise terms, to be calculated. In practical calculations, it is more useful to transform to Wigner or +P quasi-probability operator representations. These transformations result in stochastic equations that can be analyzed by use of perturbation theory or exact numerical techniques. The results have applications to fiber-optics communications, networking, and sensor technology.

© 2001 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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  6. J. F. Corney and P. D. Drummond, “Quantum noise in optical fibers. II. Raman jitter in soliton communications,” J. Opt. Soc. Am. B 18, 153–161 (2001).
    [CrossRef]
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  20. M. D. Levenson, Introduction to Nonlinear Laser Spectroscopy (Academic, New York, 1982).
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  28. R. J. Mears, L. Reekie, I. M. Jauncey, and D. N. Payne, “Low-noise erbium-doped fibre amplifier operating at 1.54 µm,” Electron. Lett. 23, 1026–1028 (1987).
    [CrossRef]
  29. E. Desurvire, Erbium-Doped Fiber Amplifiers, Principles and Applications (Wiley, New York, 1993).
  30. P. D. Drummond and M. G. Raymer, “Quantum theory of propagation of nonclassical radiation in a near-resonant medium,” Phys. Rev. A 44, 2072–2085 (1991).
    [CrossRef] [PubMed]
  31. L. F. Mollenauer, “Solitons in optical fibers and the soliton laser,” Philos. Trans. R. Soc. London 15, 437–450 (1985); L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
    [CrossRef]
  32. J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986); F. M. Mitschjke and L. F. Mollenauer, “Discovery of the soliton self-frequency shift,” Opt. Lett. 11, 659–661 (1986).
    [CrossRef] [PubMed]
  33. P. D. Drummond and A. D. Hardman, “Simulation of quantum effects in Raman-active waveguides,” Europhys. Lett. 21, 279–284 (1993); P. D. Drummond and W. Man, “Quantum noise in reversible soliton logic,” Opt. Commun. 105, 99–103 (1994).
    [CrossRef]
  34. S. J. Carter, “Quantum theory of nonlinear fiber optics: phase-space representations,” Phys. Rev. A 51, 3274–3301 (1995).
    [CrossRef] [PubMed]
  35. M. J. Werner and P. D. Drummond, “Robust algorithms for solving stochastic partial differential equations,” J. Comput. Phys. 132, 312–326 (1997).
    [CrossRef]
  36. N. J. Smith, N. J. Doran, W. Forysiak, and F. M. Knox, “Soliton transmission using periodic dispersion compensation,” J. Lightwave Technol. 15, 1808–1822 (1997).
    [CrossRef]
  37. T. I. Lakoba and D. J. Kaup, “Influence of the Raman effect on dispersion-managed solitons and their interchannel collisions,” Opt. Lett. 24, 808–810 (1999).
    [CrossRef]
  38. S. Namiki, C. X. Yu, and H. A. Haus, “Observation of nearly quantum-limited timing jitter in an all-fiber ring laser,” J. Opt. Soc. Am. B 13, 2817–2823 (1996); B. C. Collings, K. Bergman, and W. H. Knox, “Stable multigigahertz pulse-train formation in a short-cavity passively harmonic mode-locked erbium/ytterbium fiber laser,” Opt. Lett. 23, 123–125 (1998).
    [CrossRef]
  39. S. R. Friberg, S. Machida, M. J. Werner, A. Levanon, and T. Mukai, “Observation of optical soliton photon-number squeezing,” Phys. Rev. Lett. 77, 3775–3778 (1996); S. Spalter, M. Burk, U. Strossner, M. Bohm, A. Sizmann, and G. Leuchs, “Photon number squeezing of spectrally filtered sub-picosecond optical solitons,” Europhys. Lett. 38, 335–340 (1997); D. Krylov and K. Bergman, “Amplitude-squeezed solitons from an asymmetric fiber interferometer,” Opt. Lett. 23, 1390–1392 (1998).
    [CrossRef] [PubMed]
  40. M. J. Werner, “Raman-induced photon correlations in optical fiber solitons,” Phys. Rev. A 60, R781–R784 (1999).
    [CrossRef]

2001 (1)

1999 (3)

T. I. Lakoba and D. J. Kaup, “Influence of the Raman effect on dispersion-managed solitons and their interchannel collisions,” Opt. Lett. 24, 808–810 (1999).
[CrossRef]

M. J. Werner, “Raman-induced photon correlations in optical fiber solitons,” Phys. Rev. A 60, R781–R784 (1999).
[CrossRef]

P. D. Drummond and M. Hillery, “Quantum theory of dispersive electromagnetic modes,” Phys. Rev. A 59, 691–707 (1999).
[CrossRef]

1997 (2)

M. J. Werner and P. D. Drummond, “Robust algorithms for solving stochastic partial differential equations,” J. Comput. Phys. 132, 312–326 (1997).
[CrossRef]

N. J. Smith, N. J. Doran, W. Forysiak, and F. M. Knox, “Soliton transmission using periodic dispersion compensation,” J. Lightwave Technol. 15, 1808–1822 (1997).
[CrossRef]

1996 (3)

S. R. Friberg, S. Machida, M. J. Werner, A. Levanon, and T. Mukai, “Observation of optical soliton photon-number squeezing,” Phys. Rev. Lett. 77, 3775–3778 (1996); S. Spalter, M. Burk, U. Strossner, M. Bohm, A. Sizmann, and G. Leuchs, “Photon number squeezing of spectrally filtered sub-picosecond optical solitons,” Europhys. Lett. 38, 335–340 (1997); D. Krylov and K. Bergman, “Amplitude-squeezed solitons from an asymmetric fiber interferometer,” Opt. Lett. 23, 1390–1392 (1998).
[CrossRef] [PubMed]

S. Namiki, C. X. Yu, and H. A. Haus, “Observation of nearly quantum-limited timing jitter in an all-fiber ring laser,” J. Opt. Soc. Am. B 13, 2817–2823 (1996); B. C. Collings, K. Bergman, and W. H. Knox, “Stable multigigahertz pulse-train formation in a short-cavity passively harmonic mode-locked erbium/ytterbium fiber laser,” Opt. Lett. 23, 123–125 (1998).
[CrossRef]

H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996).
[CrossRef]

1995 (2)

Y. Lai and S.-S. Yu, “General quantum theory of nonlinear optical-pulse propagation,” Phys. Rev. A 51, 817–829 (1995); S.-S. Yu and Y. Lai, “Impacts of self-Raman effect and third-order dispersion on pulse squeezed state generation using optical fibers,” J. Opt. Soc. Am. B 12, 2340–2346 (1995).
[CrossRef] [PubMed]

S. J. Carter, “Quantum theory of nonlinear fiber optics: phase-space representations,” Phys. Rev. A 51, 3274–3301 (1995).
[CrossRef] [PubMed]

1994 (1)

1993 (1)

P. D. Drummond and A. D. Hardman, “Simulation of quantum effects in Raman-active waveguides,” Europhys. Lett. 21, 279–284 (1993); P. D. Drummond and W. Man, “Quantum noise in reversible soliton logic,” Opt. Commun. 105, 99–103 (1994).
[CrossRef]

1992 (1)

K. Bergman, H. A. Haus, and M. Shirasaki, “Analysis and measurement of GAWBS spectrum in a nonlinear fiber ring,” Appl. Phys. B 55, 242–249 (1992).
[CrossRef]

1991 (3)

P. D. Drummond and M. G. Raymer, “Quantum theory of propagation of nonclassical radiation in a near-resonant medium,” Phys. Rev. A 44, 2072–2085 (1991).
[CrossRef] [PubMed]

M. Rosenbluh and R. M. Shelby, “Squeezed optical solitons,” Phys. Rev. Lett. 66, 153–156 (1991); P. D. Drummond, R. M. Shelby, S. R. Friberg, and Y. Yamamoto, “Quantum solitons in optical fibres,” Nature 365, 307–313 (1993).
[CrossRef] [PubMed]

S. J. Carter and P. D. Drummond, “Squeezed quantum solitons and Raman noise,” Phys. Rev. Lett. 67, 3757–3760 (1991).
[CrossRef] [PubMed]

1990 (2)

P. D. Drummond, “Electromagnetic quantization in dispersive inhomogeneous nonlinear dielectrics,” Phys. Rev. A 42, 6845–6857 (1990).
[CrossRef] [PubMed]

R. M. Shelby, P. D. Drummond, and S. J. Carter, “Phase-noise scaling in quantum soliton propagation,” Phys. Rev. A 42, 2966–2796 (1990).
[CrossRef] [PubMed]

1989 (2)

1988 (1)

S. H. Perlmutter, M. D. Levenson, R. M. Shelby, and M. B. Weissman, “Inverse-power-law light scattering in fused-silica optical fiber,” Phys. Rev. Lett. 61, 1388–1391 (1988); “Polarization of quasielastic light scattering in fused-silica optical fiber,” Phys. Rev. B 42, 5294–5305 (1990).
[CrossRef] [PubMed]

1987 (4)

R. J. Mears, L. Reekie, I. M. Jauncey, and D. N. Payne, “Low-noise erbium-doped fibre amplifier operating at 1.54 µm,” Electron. Lett. 23, 1026–1028 (1987).
[CrossRef]

S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, “Squeezing of quantum solitons,” Phys. Rev. Lett. 58, 1841–1844 (1987).
[CrossRef] [PubMed]

P. D. Drummond and S. J. Carter, “Quantum-field theory of squeezing in solitons,” J. Opt. Soc. Am. B 4, 1565–1573 (1987); P. D. Drummond, S. J. Carter, and R. M. Shelby, “Time dependence of quantum fluctuations in solitons,” Opt. Lett. 14, 373–375 (1989).
[CrossRef] [PubMed]

P. D. Drummond and S. J. Carter, “Quantum-field theory of squeezing in solitons,” J. Opt. Soc. Am. B 4, 1565–1573 (1987).
[CrossRef]

1986 (2)

1985 (2)

L. F. Mollenauer, “Solitons in optical fibers and the soliton laser,” Philos. Trans. R. Soc. London 15, 437–450 (1985); L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
[CrossRef]

R. M. Shelby, M. D. Levenson, and P. W. Bayer, “Guided acoustic-wave Brillouin scattering,” Phys. Rev. B 31, 5244–5252 (1985).
[CrossRef]

1984 (2)

1972 (1)

P. Dean, “The vibrational properties of disordered systems: numerical studies,” Rev. Mod. Phys. 44, 127–168 (1972).
[CrossRef]

1971 (1)

T. von Foerster and R. J. Glauber, “Quantum theory of light propagation in amplifying media,” Phys. Rev. A 3, 1484–1511 (1971); I. A. Walmsley and M. G. Raymer, “Observation of macroscopic quantum fluctuations in stimulated Raman scattering,” Phys. Rev. Lett. 50, 962–965 (1983).
[CrossRef]

1959 (1)

E. Power and S. Zienau, “Coulomb gauge in nonrelativistic quantum electrodynamics and the shape of spectral lines,” Philos. Trans. R. Soc. London, Ser. A 251, 427–454 (1959); R. Loudon, The Quantum Theory of Light (Clarendon, Oxford, 1983).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995), pp. 28–59.

Bayer, P. W.

R. M. Shelby, M. D. Levenson, and P. W. Bayer, “Guided acoustic-wave Brillouin scattering,” Phys. Rev. B 31, 5244–5252 (1985).
[CrossRef]

Bergman, K.

K. Bergman, H. A. Haus, and M. Shirasaki, “Analysis and measurement of GAWBS spectrum in a nonlinear fiber ring,” Appl. Phys. B 55, 242–249 (1992).
[CrossRef]

Bloembergen, N.

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).

Carter, S. J.

S. J. Carter, “Quantum theory of nonlinear fiber optics: phase-space representations,” Phys. Rev. A 51, 3274–3301 (1995).
[CrossRef] [PubMed]

S. J. Carter and P. D. Drummond, “Squeezed quantum solitons and Raman noise,” Phys. Rev. Lett. 67, 3757–3760 (1991).
[CrossRef] [PubMed]

R. M. Shelby, P. D. Drummond, and S. J. Carter, “Phase-noise scaling in quantum soliton propagation,” Phys. Rev. A 42, 2966–2796 (1990).
[CrossRef] [PubMed]

P. D. Drummond and S. J. Carter, “Quantum-field theory of squeezing in solitons,” J. Opt. Soc. Am. B 4, 1565–1573 (1987); P. D. Drummond, S. J. Carter, and R. M. Shelby, “Time dependence of quantum fluctuations in solitons,” Opt. Lett. 14, 373–375 (1989).
[CrossRef] [PubMed]

P. D. Drummond and S. J. Carter, “Quantum-field theory of squeezing in solitons,” J. Opt. Soc. Am. B 4, 1565–1573 (1987).
[CrossRef]

S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, “Squeezing of quantum solitons,” Phys. Rev. Lett. 58, 1841–1844 (1987).
[CrossRef] [PubMed]

Corney, J. F.

Dean, P.

P. Dean, “The vibrational properties of disordered systems: numerical studies,” Rev. Mod. Phys. 44, 127–168 (1972).
[CrossRef]

Desurvire, E.

E. Desurvire, Erbium-Doped Fiber Amplifiers, Principles and Applications (Wiley, New York, 1993).

Doran, N. J.

N. J. Smith, N. J. Doran, W. Forysiak, and F. M. Knox, “Soliton transmission using periodic dispersion compensation,” J. Lightwave Technol. 15, 1808–1822 (1997).
[CrossRef]

Dougherty, D. J.

Drummond, P. D.

J. F. Corney and P. D. Drummond, “Quantum noise in optical fibers. II. Raman jitter in soliton communications,” J. Opt. Soc. Am. B 18, 153–161 (2001).
[CrossRef]

P. D. Drummond and M. Hillery, “Quantum theory of dispersive electromagnetic modes,” Phys. Rev. A 59, 691–707 (1999).
[CrossRef]

M. J. Werner and P. D. Drummond, “Robust algorithms for solving stochastic partial differential equations,” J. Comput. Phys. 132, 312–326 (1997).
[CrossRef]

P. D. Drummond and A. D. Hardman, “Simulation of quantum effects in Raman-active waveguides,” Europhys. Lett. 21, 279–284 (1993); P. D. Drummond and W. Man, “Quantum noise in reversible soliton logic,” Opt. Commun. 105, 99–103 (1994).
[CrossRef]

P. D. Drummond and M. G. Raymer, “Quantum theory of propagation of nonclassical radiation in a near-resonant medium,” Phys. Rev. A 44, 2072–2085 (1991).
[CrossRef] [PubMed]

S. J. Carter and P. D. Drummond, “Squeezed quantum solitons and Raman noise,” Phys. Rev. Lett. 67, 3757–3760 (1991).
[CrossRef] [PubMed]

P. D. Drummond, “Electromagnetic quantization in dispersive inhomogeneous nonlinear dielectrics,” Phys. Rev. A 42, 6845–6857 (1990).
[CrossRef] [PubMed]

R. M. Shelby, P. D. Drummond, and S. J. Carter, “Phase-noise scaling in quantum soliton propagation,” Phys. Rev. A 42, 2966–2796 (1990).
[CrossRef] [PubMed]

S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, “Squeezing of quantum solitons,” Phys. Rev. Lett. 58, 1841–1844 (1987).
[CrossRef] [PubMed]

P. D. Drummond and S. J. Carter, “Quantum-field theory of squeezing in solitons,” J. Opt. Soc. Am. B 4, 1565–1573 (1987); P. D. Drummond, S. J. Carter, and R. M. Shelby, “Time dependence of quantum fluctuations in solitons,” Opt. Lett. 14, 373–375 (1989).
[CrossRef] [PubMed]

P. D. Drummond and S. J. Carter, “Quantum-field theory of squeezing in solitons,” J. Opt. Soc. Am. B 4, 1565–1573 (1987).
[CrossRef]

P. D. Drummond, “Quantum theory of fiber-optics and solitons,” in Coherence and Quantum Optics VII, J. Eberly, L. Mandel, and E. Wolf, eds. (Plenum, New York, 1996), pp. 323–332.

Forysiak, W.

N. J. Smith, N. J. Doran, W. Forysiak, and F. M. Knox, “Soliton transmission using periodic dispersion compensation,” J. Lightwave Technol. 15, 1808–1822 (1997).
[CrossRef]

Friberg, S. R.

S. R. Friberg, S. Machida, M. J. Werner, A. Levanon, and T. Mukai, “Observation of optical soliton photon-number squeezing,” Phys. Rev. Lett. 77, 3775–3778 (1996); S. Spalter, M. Burk, U. Strossner, M. Bohm, A. Sizmann, and G. Leuchs, “Photon number squeezing of spectrally filtered sub-picosecond optical solitons,” Europhys. Lett. 38, 335–340 (1997); D. Krylov and K. Bergman, “Amplitude-squeezed solitons from an asymmetric fiber interferometer,” Opt. Lett. 23, 1390–1392 (1998).
[CrossRef] [PubMed]

Glauber, R. J.

T. von Foerster and R. J. Glauber, “Quantum theory of light propagation in amplifying media,” Phys. Rev. A 3, 1484–1511 (1971); I. A. Walmsley and M. G. Raymer, “Observation of macroscopic quantum fluctuations in stimulated Raman scattering,” Phys. Rev. Lett. 50, 962–965 (1983).
[CrossRef]

Gordon, J. P.

Hardman, A. D.

P. D. Drummond and A. D. Hardman, “Simulation of quantum effects in Raman-active waveguides,” Europhys. Lett. 21, 279–284 (1993); P. D. Drummond and W. Man, “Quantum noise in reversible soliton logic,” Opt. Commun. 105, 99–103 (1994).
[CrossRef]

Haus, H. A.

Hillery, M.

P. D. Drummond and M. Hillery, “Quantum theory of dispersive electromagnetic modes,” Phys. Rev. A 59, 691–707 (1999).
[CrossRef]

M. Hillery and L. D. Mlodinow, “Quantization of electrodynamics in nonlinear dielectric media,” Phys. Rev. A 30, 1860–1865 (1984).
[CrossRef]

Ippen, E. P.

Jain, R. K.

Jauncey, I. M.

R. J. Mears, L. Reekie, I. M. Jauncey, and D. N. Payne, “Low-noise erbium-doped fibre amplifier operating at 1.54 µm,” Electron. Lett. 23, 1026–1028 (1987).
[CrossRef]

Kartner, F. X.

Kaup, D. J.

Knox, F. M.

N. J. Smith, N. J. Doran, W. Forysiak, and F. M. Knox, “Soliton transmission using periodic dispersion compensation,” J. Lightwave Technol. 15, 1808–1822 (1997).
[CrossRef]

Lai, Y.

Y. Lai and S.-S. Yu, “General quantum theory of nonlinear optical-pulse propagation,” Phys. Rev. A 51, 817–829 (1995); S.-S. Yu and Y. Lai, “Impacts of self-Raman effect and third-order dispersion on pulse squeezed state generation using optical fibers,” J. Opt. Soc. Am. B 12, 2340–2346 (1995).
[CrossRef] [PubMed]

Lakoba, T. I.

Lee, C.

Levanon, A.

S. R. Friberg, S. Machida, M. J. Werner, A. Levanon, and T. Mukai, “Observation of optical soliton photon-number squeezing,” Phys. Rev. Lett. 77, 3775–3778 (1996); S. Spalter, M. Burk, U. Strossner, M. Bohm, A. Sizmann, and G. Leuchs, “Photon number squeezing of spectrally filtered sub-picosecond optical solitons,” Europhys. Lett. 38, 335–340 (1997); D. Krylov and K. Bergman, “Amplitude-squeezed solitons from an asymmetric fiber interferometer,” Opt. Lett. 23, 1390–1392 (1998).
[CrossRef] [PubMed]

Levenson, M. D.

S. H. Perlmutter, M. D. Levenson, R. M. Shelby, and M. B. Weissman, “Inverse-power-law light scattering in fused-silica optical fiber,” Phys. Rev. Lett. 61, 1388–1391 (1988); “Polarization of quasielastic light scattering in fused-silica optical fiber,” Phys. Rev. B 42, 5294–5305 (1990).
[CrossRef] [PubMed]

R. M. Shelby, M. D. Levenson, and P. W. Bayer, “Guided acoustic-wave Brillouin scattering,” Phys. Rev. B 31, 5244–5252 (1985).
[CrossRef]

M. D. Levenson, Introduction to Nonlinear Laser Spectroscopy (Academic, New York, 1982).

Machida, S.

S. R. Friberg, S. Machida, M. J. Werner, A. Levanon, and T. Mukai, “Observation of optical soliton photon-number squeezing,” Phys. Rev. Lett. 77, 3775–3778 (1996); S. Spalter, M. Burk, U. Strossner, M. Bohm, A. Sizmann, and G. Leuchs, “Photon number squeezing of spectrally filtered sub-picosecond optical solitons,” Europhys. Lett. 38, 335–340 (1997); D. Krylov and K. Bergman, “Amplitude-squeezed solitons from an asymmetric fiber interferometer,” Opt. Lett. 23, 1390–1392 (1998).
[CrossRef] [PubMed]

Mears, R. J.

R. J. Mears, L. Reekie, I. M. Jauncey, and D. N. Payne, “Low-noise erbium-doped fibre amplifier operating at 1.54 µm,” Electron. Lett. 23, 1026–1028 (1987).
[CrossRef]

Mlodinow, L. D.

M. Hillery and L. D. Mlodinow, “Quantization of electrodynamics in nonlinear dielectric media,” Phys. Rev. A 30, 1860–1865 (1984).
[CrossRef]

Mollenauer, L. F.

K. Smith and L. F. Mollenauer, “Experimental observation of soliton interaction over long fiber paths: discovery of a long-range interaction,” Opt. Lett. 14, 1284–1286 (1989); E. M. Dianov, A. V. Luchnikov, A. N. Pilipetskii, and A. M. Prokhorov, “Long-range interaction of picosecond solitons through excitation of acoustic waves in optical fibers,” Appl. Phys. B 54, 175–180 (1992).
[CrossRef] [PubMed]

L. F. Mollenauer, “Solitons in optical fibers and the soliton laser,” Philos. Trans. R. Soc. London 15, 437–450 (1985); L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
[CrossRef]

Mukai, T.

S. R. Friberg, S. Machida, M. J. Werner, A. Levanon, and T. Mukai, “Observation of optical soliton photon-number squeezing,” Phys. Rev. Lett. 77, 3775–3778 (1996); S. Spalter, M. Burk, U. Strossner, M. Bohm, A. Sizmann, and G. Leuchs, “Photon number squeezing of spectrally filtered sub-picosecond optical solitons,” Europhys. Lett. 38, 335–340 (1997); D. Krylov and K. Bergman, “Amplitude-squeezed solitons from an asymmetric fiber interferometer,” Opt. Lett. 23, 1390–1392 (1998).
[CrossRef] [PubMed]

Namiki, S.

Payne, D. N.

R. J. Mears, L. Reekie, I. M. Jauncey, and D. N. Payne, “Low-noise erbium-doped fibre amplifier operating at 1.54 µm,” Electron. Lett. 23, 1026–1028 (1987).
[CrossRef]

Perlmutter, S. H.

S. H. Perlmutter, M. D. Levenson, R. M. Shelby, and M. B. Weissman, “Inverse-power-law light scattering in fused-silica optical fiber,” Phys. Rev. Lett. 61, 1388–1391 (1988); “Polarization of quasielastic light scattering in fused-silica optical fiber,” Phys. Rev. B 42, 5294–5305 (1990).
[CrossRef] [PubMed]

Potasek, M. J.

Power, E.

E. Power and S. Zienau, “Coulomb gauge in nonrelativistic quantum electrodynamics and the shape of spectral lines,” Philos. Trans. R. Soc. London, Ser. A 251, 427–454 (1959); R. Loudon, The Quantum Theory of Light (Clarendon, Oxford, 1983).
[CrossRef]

Raymer, M. G.

P. D. Drummond and M. G. Raymer, “Quantum theory of propagation of nonclassical radiation in a near-resonant medium,” Phys. Rev. A 44, 2072–2085 (1991).
[CrossRef] [PubMed]

Reekie, L.

R. J. Mears, L. Reekie, I. M. Jauncey, and D. N. Payne, “Low-noise erbium-doped fibre amplifier operating at 1.54 µm,” Electron. Lett. 23, 1026–1028 (1987).
[CrossRef]

Reid, M. D.

S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, “Squeezing of quantum solitons,” Phys. Rev. Lett. 58, 1841–1844 (1987).
[CrossRef] [PubMed]

Rosenbluh, M.

M. Rosenbluh and R. M. Shelby, “Squeezed optical solitons,” Phys. Rev. Lett. 66, 153–156 (1991); P. D. Drummond, R. M. Shelby, S. R. Friberg, and Y. Yamamoto, “Quantum solitons in optical fibres,” Nature 365, 307–313 (1993).
[CrossRef] [PubMed]

Shelby, R. M.

M. Rosenbluh and R. M. Shelby, “Squeezed optical solitons,” Phys. Rev. Lett. 66, 153–156 (1991); P. D. Drummond, R. M. Shelby, S. R. Friberg, and Y. Yamamoto, “Quantum solitons in optical fibres,” Nature 365, 307–313 (1993).
[CrossRef] [PubMed]

R. M. Shelby, P. D. Drummond, and S. J. Carter, “Phase-noise scaling in quantum soliton propagation,” Phys. Rev. A 42, 2966–2796 (1990).
[CrossRef] [PubMed]

S. H. Perlmutter, M. D. Levenson, R. M. Shelby, and M. B. Weissman, “Inverse-power-law light scattering in fused-silica optical fiber,” Phys. Rev. Lett. 61, 1388–1391 (1988); “Polarization of quasielastic light scattering in fused-silica optical fiber,” Phys. Rev. B 42, 5294–5305 (1990).
[CrossRef] [PubMed]

S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, “Squeezing of quantum solitons,” Phys. Rev. Lett. 58, 1841–1844 (1987).
[CrossRef] [PubMed]

R. M. Shelby, M. D. Levenson, and P. W. Bayer, “Guided acoustic-wave Brillouin scattering,” Phys. Rev. B 31, 5244–5252 (1985).
[CrossRef]

Shirasaki, M.

K. Bergman, H. A. Haus, and M. Shirasaki, “Analysis and measurement of GAWBS spectrum in a nonlinear fiber ring,” Appl. Phys. B 55, 242–249 (1992).
[CrossRef]

Smith, K.

Smith, N. J.

N. J. Smith, N. J. Doran, W. Forysiak, and F. M. Knox, “Soliton transmission using periodic dispersion compensation,” J. Lightwave Technol. 15, 1808–1822 (1997).
[CrossRef]

Stolen, R. H.

von Foerster, T.

T. von Foerster and R. J. Glauber, “Quantum theory of light propagation in amplifying media,” Phys. Rev. A 3, 1484–1511 (1971); I. A. Walmsley and M. G. Raymer, “Observation of macroscopic quantum fluctuations in stimulated Raman scattering,” Phys. Rev. Lett. 50, 962–965 (1983).
[CrossRef]

Weissman, M. B.

S. H. Perlmutter, M. D. Levenson, R. M. Shelby, and M. B. Weissman, “Inverse-power-law light scattering in fused-silica optical fiber,” Phys. Rev. Lett. 61, 1388–1391 (1988); “Polarization of quasielastic light scattering in fused-silica optical fiber,” Phys. Rev. B 42, 5294–5305 (1990).
[CrossRef] [PubMed]

Werner, M. J.

M. J. Werner, “Raman-induced photon correlations in optical fiber solitons,” Phys. Rev. A 60, R781–R784 (1999).
[CrossRef]

M. J. Werner and P. D. Drummond, “Robust algorithms for solving stochastic partial differential equations,” J. Comput. Phys. 132, 312–326 (1997).
[CrossRef]

S. R. Friberg, S. Machida, M. J. Werner, A. Levanon, and T. Mukai, “Observation of optical soliton photon-number squeezing,” Phys. Rev. Lett. 77, 3775–3778 (1996); S. Spalter, M. Burk, U. Strossner, M. Bohm, A. Sizmann, and G. Leuchs, “Photon number squeezing of spectrally filtered sub-picosecond optical solitons,” Europhys. Lett. 38, 335–340 (1997); D. Krylov and K. Bergman, “Amplitude-squeezed solitons from an asymmetric fiber interferometer,” Opt. Lett. 23, 1390–1392 (1998).
[CrossRef] [PubMed]

Wong, W. S.

H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996).
[CrossRef]

Yu, C. X.

Yu, S.-S.

Y. Lai and S.-S. Yu, “General quantum theory of nonlinear optical-pulse propagation,” Phys. Rev. A 51, 817–829 (1995); S.-S. Yu and Y. Lai, “Impacts of self-Raman effect and third-order dispersion on pulse squeezed state generation using optical fibers,” J. Opt. Soc. Am. B 12, 2340–2346 (1995).
[CrossRef] [PubMed]

Yurke, B.

Zienau, S.

E. Power and S. Zienau, “Coulomb gauge in nonrelativistic quantum electrodynamics and the shape of spectral lines,” Philos. Trans. R. Soc. London, Ser. A 251, 427–454 (1959); R. Loudon, The Quantum Theory of Light (Clarendon, Oxford, 1983).
[CrossRef]

Appl. Phys. B (1)

K. Bergman, H. A. Haus, and M. Shirasaki, “Analysis and measurement of GAWBS spectrum in a nonlinear fiber ring,” Appl. Phys. B 55, 242–249 (1992).
[CrossRef]

Electron. Lett. (1)

R. J. Mears, L. Reekie, I. M. Jauncey, and D. N. Payne, “Low-noise erbium-doped fibre amplifier operating at 1.54 µm,” Electron. Lett. 23, 1026–1028 (1987).
[CrossRef]

Europhys. Lett. (1)

P. D. Drummond and A. D. Hardman, “Simulation of quantum effects in Raman-active waveguides,” Europhys. Lett. 21, 279–284 (1993); P. D. Drummond and W. Man, “Quantum noise in reversible soliton logic,” Opt. Commun. 105, 99–103 (1994).
[CrossRef]

J. Comput. Phys. (1)

M. J. Werner and P. D. Drummond, “Robust algorithms for solving stochastic partial differential equations,” J. Comput. Phys. 132, 312–326 (1997).
[CrossRef]

J. Lightwave Technol. (1)

N. J. Smith, N. J. Doran, W. Forysiak, and F. M. Knox, “Soliton transmission using periodic dispersion compensation,” J. Lightwave Technol. 15, 1808–1822 (1997).
[CrossRef]

J. Opt. Soc. Am. B (7)

R. H. Stolen, C. Lee, and R. K. Jain, “Development of the stimulated Raman spectrum in single-mode silica fibers,” J. Opt. Soc. Am. B 1, 652–657 (1984); D. J. Dougherty, F. X. Kartner, H. A. Haus, and E. P. Ippen, “Measurement of the Raman gain spectrum of optical fibers,” Opt. Lett. 20, 31–33 (1995); R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, “Raman response function of silica-core fibers,” J. Opt. Soc. Am. B 6, 1159–1166 (1989).
[CrossRef] [PubMed]

P. D. Drummond and S. J. Carter, “Quantum-field theory of squeezing in solitons,” J. Opt. Soc. Am. B 4, 1565–1573 (1987).
[CrossRef]

J. F. Corney and P. D. Drummond, “Quantum noise in optical fibers. II. Raman jitter in soliton communications,” J. Opt. Soc. Am. B 18, 153–161 (2001).
[CrossRef]

P. D. Drummond and S. J. Carter, “Quantum-field theory of squeezing in solitons,” J. Opt. Soc. Am. B 4, 1565–1573 (1987); P. D. Drummond, S. J. Carter, and R. M. Shelby, “Time dependence of quantum fluctuations in solitons,” Opt. Lett. 14, 373–375 (1989).
[CrossRef] [PubMed]

B. Yurke and M. J. Potasek, “Solution to the initial value problem for the quantum nonlinear Schrödinger equation,” J. Opt. Soc. Am. B 6, 1227–1238 (1989).
[CrossRef]

F. X. Kartner, D. J. Dougherty, H. A. Haus, and E. P. Ippen, “Raman noise and soliton squeezing,” J. Opt. Soc. Am. B 11, 1267–1276 (1994).
[CrossRef]

S. Namiki, C. X. Yu, and H. A. Haus, “Observation of nearly quantum-limited timing jitter in an all-fiber ring laser,” J. Opt. Soc. Am. B 13, 2817–2823 (1996); B. C. Collings, K. Bergman, and W. H. Knox, “Stable multigigahertz pulse-train formation in a short-cavity passively harmonic mode-locked erbium/ytterbium fiber laser,” Opt. Lett. 23, 123–125 (1998).
[CrossRef]

Opt. Lett. (4)

Philos. Trans. R. Soc. London (1)

L. F. Mollenauer, “Solitons in optical fibers and the soliton laser,” Philos. Trans. R. Soc. London 15, 437–450 (1985); L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095–1098 (1980).
[CrossRef]

Philos. Trans. R. Soc. London, Ser. A (1)

E. Power and S. Zienau, “Coulomb gauge in nonrelativistic quantum electrodynamics and the shape of spectral lines,” Philos. Trans. R. Soc. London, Ser. A 251, 427–454 (1959); R. Loudon, The Quantum Theory of Light (Clarendon, Oxford, 1983).
[CrossRef]

Phys. Rev. A (9)

M. Hillery and L. D. Mlodinow, “Quantization of electrodynamics in nonlinear dielectric media,” Phys. Rev. A 30, 1860–1865 (1984).
[CrossRef]

Y. Lai and S.-S. Yu, “General quantum theory of nonlinear optical-pulse propagation,” Phys. Rev. A 51, 817–829 (1995); S.-S. Yu and Y. Lai, “Impacts of self-Raman effect and third-order dispersion on pulse squeezed state generation using optical fibers,” J. Opt. Soc. Am. B 12, 2340–2346 (1995).
[CrossRef] [PubMed]

T. von Foerster and R. J. Glauber, “Quantum theory of light propagation in amplifying media,” Phys. Rev. A 3, 1484–1511 (1971); I. A. Walmsley and M. G. Raymer, “Observation of macroscopic quantum fluctuations in stimulated Raman scattering,” Phys. Rev. Lett. 50, 962–965 (1983).
[CrossRef]

P. D. Drummond and M. Hillery, “Quantum theory of dispersive electromagnetic modes,” Phys. Rev. A 59, 691–707 (1999).
[CrossRef]

P. D. Drummond, “Electromagnetic quantization in dispersive inhomogeneous nonlinear dielectrics,” Phys. Rev. A 42, 6845–6857 (1990).
[CrossRef] [PubMed]

P. D. Drummond and M. G. Raymer, “Quantum theory of propagation of nonclassical radiation in a near-resonant medium,” Phys. Rev. A 44, 2072–2085 (1991).
[CrossRef] [PubMed]

S. J. Carter, “Quantum theory of nonlinear fiber optics: phase-space representations,” Phys. Rev. A 51, 3274–3301 (1995).
[CrossRef] [PubMed]

R. M. Shelby, P. D. Drummond, and S. J. Carter, “Phase-noise scaling in quantum soliton propagation,” Phys. Rev. A 42, 2966–2796 (1990).
[CrossRef] [PubMed]

M. J. Werner, “Raman-induced photon correlations in optical fiber solitons,” Phys. Rev. A 60, R781–R784 (1999).
[CrossRef]

Phys. Rev. B (1)

R. M. Shelby, M. D. Levenson, and P. W. Bayer, “Guided acoustic-wave Brillouin scattering,” Phys. Rev. B 31, 5244–5252 (1985).
[CrossRef]

Phys. Rev. Lett. (5)

S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, “Squeezing of quantum solitons,” Phys. Rev. Lett. 58, 1841–1844 (1987).
[CrossRef] [PubMed]

S. J. Carter and P. D. Drummond, “Squeezed quantum solitons and Raman noise,” Phys. Rev. Lett. 67, 3757–3760 (1991).
[CrossRef] [PubMed]

S. H. Perlmutter, M. D. Levenson, R. M. Shelby, and M. B. Weissman, “Inverse-power-law light scattering in fused-silica optical fiber,” Phys. Rev. Lett. 61, 1388–1391 (1988); “Polarization of quasielastic light scattering in fused-silica optical fiber,” Phys. Rev. B 42, 5294–5305 (1990).
[CrossRef] [PubMed]

M. Rosenbluh and R. M. Shelby, “Squeezed optical solitons,” Phys. Rev. Lett. 66, 153–156 (1991); P. D. Drummond, R. M. Shelby, S. R. Friberg, and Y. Yamamoto, “Quantum solitons in optical fibres,” Nature 365, 307–313 (1993).
[CrossRef] [PubMed]

S. R. Friberg, S. Machida, M. J. Werner, A. Levanon, and T. Mukai, “Observation of optical soliton photon-number squeezing,” Phys. Rev. Lett. 77, 3775–3778 (1996); S. Spalter, M. Burk, U. Strossner, M. Bohm, A. Sizmann, and G. Leuchs, “Photon number squeezing of spectrally filtered sub-picosecond optical solitons,” Europhys. Lett. 38, 335–340 (1997); D. Krylov and K. Bergman, “Amplitude-squeezed solitons from an asymmetric fiber interferometer,” Opt. Lett. 23, 1390–1392 (1998).
[CrossRef] [PubMed]

Rev. Mod. Phys. (2)

H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996).
[CrossRef]

P. Dean, “The vibrational properties of disordered systems: numerical studies,” Rev. Mod. Phys. 44, 127–168 (1972).
[CrossRef]

Other (5)

E. Desurvire, Erbium-Doped Fiber Amplifiers, Principles and Applications (Wiley, New York, 1993).

M. D. Levenson, Introduction to Nonlinear Laser Spectroscopy (Academic, New York, 1982).

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).

P. D. Drummond, “Quantum theory of fiber-optics and solitons,” in Coherence and Quantum Optics VII, J. Eberly, L. Mandel, and E. Wolf, eds. (Plenum, New York, 1996), pp. 323–332.

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995), pp. 28–59.

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Figures (1)

Fig. 1
Fig. 1

Parallel polarization Raman gain |I{h˜(ωt0)}|=|h(ωt0)| for the 11-Lorentzian model (continuous curve) and the single-Lorentzian model (dashed curve) for a temperature of T=300 K.

Tables (1)

Tables Icon

Table 1 Fitting Parameters for the 11-Lorentzian Model of the Raman Gain Function hR(t/t0)a

Equations (109)

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

HD=dV12µ|B|2+t0tE(t)·D˙(t)dt,
HˆF= dkω(k)aˆ(k)aˆ(k)- d3xΔχ(1)(x)2(ω0):|Dˆ|2(x):+χ(3)(x)43(ω0):|Dˆ|4(x):.
[aˆ(k), aˆ(k)]=δ(k-k).
Dˆ(x)=i  dkk[ω(k)]v(k)4π1/2aˆ(k)u(r)×exp(ikx)+h.c.,
 d2r|u(r)|2=1.
Ψˆ(t, x)=12π  dkaˆ(t, k)exp[i(k-k0)x+iω0t].
[Ψˆ(t, x),Ψˆ(t, x)]=δ(x-x).
HˆF=  dx  dxω(x, x)Ψˆ(t, x)Ψˆ(t, x)-2  dxχE(x)Ψˆ2(t, x)Ψˆ2(t, x).
ω(x, x)= dk2πω(k)exp[i(k-k0)(x-x)]-12k0v(k0) d2rΔχ(1)(x)|u(r)|2δ(x-x)
[ω0+Δω(x)]δ(x-x)+ dk4π[iω0(x-x)+ω0(xx)+]exp[ik(x-x)].
χE(x)3ω02v(k0)24(ω0)c2 d2rχ(3)(x)|u(r)|4n2e(x)ω02v2Ac.
HˆF=HˆF- dkω0aˆ(k)aˆ(k)
=2 - dxΔω(x)ΨˆΨˆ+iv2(ΨˆΨˆ-ΨˆΨˆ)+ω2ΨˆΨˆ-χE(x)2 Ψˆ2Ψˆ2.
v x+tΨˆ(t, x)=-iΔω(x)+iω2 2x2+iχE(x)Ψˆ(t, x)Ψˆ(t, x)Ψˆ(t, x).
HR=12 j ηjRD(x¯j)D(x¯j)δxj+12 ijκij:δxiδxj.
HˆR=- dx 0dω{Ψˆ(x)Ψˆ(x)R(ω, x)×[bˆ(ω, x)+bˆ(ω, x)]+ωbˆ(ω, x)bˆ(ω, x)}.
[bˆ(t, ω, x), bˆ(t, ω, x)]=δ(x-x)δ(ω-ω).
v x+tΨˆ(t, x)=i-Δω(x)+ω2 2x2+χE(x)Ψˆ(t, x)Ψˆ(t, x)Ψˆ(t, x)-i0 R(ω, x)[bˆ(t, ω, x)+bˆ(t, ω, x)]dωΨˆ(t, x),
tbˆ(t, ω, x)=-iωbˆ(t, ω, x)-iR(ω, x)Ψˆ(t, x)Ψˆ(t, x).
v x+tΨˆ(t, x)=i-Δω(x)+ω2 2x2+0- dtχ(t, x)×[ΨˆΨˆ](t-t, x)+ΓˆR(t, x)Ψˆ(t, x),
χ(t, x)=χE(x)δ(t)+2Θ(t)0 R2(ω, x)sin(ωt)dω,
ΓˆR(t, x)=-0 R(ω, x)[bˆ(t, ω, x)+bˆ(t, ω, x)]dω,
ΓˆR(ω, x)=12π  dt exp(iωt)ΓˆR(t, x),
ΓˆR(ω, x)=12π  dt exp(-iωt)ΓˆR(t, x).
ΓˆR(ω, x)ΓˆR(ω, x)=2χ(x, |ω|)×[nth(|ω|)+Θ(-ω)]×δ(x-x)δ(ω-ω).
χ˜(ω, x)= dt exp(iωt)χ(t, x),
1I0  ln Ix=-2χ(ω, x)/v2.
χ(t, x)=χE(x)δ(t)+χ(x)Θ(t)j=0n Fjδj exp(-δjt)sin(ωjt).
χ(x)=χE(x)+200 R2(ω, x)sin(ωt)dωdt.
χE(x)=(1-f)n2ω02v2Ac,
f=χRχ=2χ 0 dt 0 dωR2(ω, x)sin(ωt)0.2.
HˆA=- dx0 dω{[Ψˆ(x)aˆ(ω, x)A(ω, x)+h.c.]+(ω-ω0)aˆaˆ(ω, x)},
[aˆ(ω, x),aˆ(ω, x)]=δ(x-x)δ(ω-ω).
taˆ(t, ω, x)=-i(ω-ω0)aˆ(t, ω, x)-iA(ω, x)Ψˆ(t, x).
aˆ(t, ω, x)=aˆ(t0, ω, x)exp[-i(ω-ω0)(t-t0)]-iA(ω, x)t0t exp[-i(ω-ω0)×(t-t)]Ψˆ(t, x)dt,
aˆ(t0, ω, x)aˆ(t0, ω, x)=nth(ω)δ(x-x)δ(ω-ω),
aˆ(t0, ω, x)aˆ(t0, ω, x)=[nth(ω)+1]×δ(x-x)δ(ω-ω).
-i0 A*(ω, x)aˆ(t, ω, x)dω
=-0 dω|A(ω, x)|2 t0tdt exp[-i(ω-ω0)(t-t)]×Ψˆ(t, ω)-i 0 dωA*(ω, x)exp[-i(ω-ω0)×(t-t0)]aˆ(t0, ω, x)
=-0 dtγA(t, x)Ψˆ(t-t, x)+ΓˆA(t, x),
γA(t, x)Θ(t)-+ dω|A(ω, x)|2 exp[-i(ω-ω0)t],
ΓˆA(t, x)=-i 0 dωA*(ω, x)exp[-i(ω-ω0)(t-t0)]×aˆ(t0, ω, x).
γ˜A(ω, x)= γA(t, x)exp(iωt)dt=γA(ω, x)+iγA(ω, x),
γA(ω, x)=π|A(ω0+ω, x)|2.
γA(t)γ˜Aδ(t),
γ˜A=γ˜A(0)=0-+ dtdω|A(ω)|2 exp[-i(ω-ω0)t]
=γA+iγA.
ΓˆA(t, x)ΓˆA(t, x)
=0 dω|A(ω, x)|2 exp[-i(ω-ω0)(t-t)]×[nth(ω)+1]δ(x-x)
[γA(t-t, x)+γA*(t-t, x)]×[nth(ω0)+1]δ(x-x),
ΓˆA(t, x)ΓˆA(t, x)
=0 dω|A(ω, x)|2 exp[-i(ω-ω0)×(t-t)]nth(ω)δ(x-x)
[γA(t-t, x)+γA*(t-t, x)]×nth(ω0)δ(x-x).
ΓˆA(ω, x)ΓˆA(ω, x)=2γA(ω, x)δ(x-x)δ(ω-ω).
ΓˆA(t, x)ΓˆA(t, x)=2γAδ(t-t)δ(x-x),
ΓˆA(t, x)ΓˆA(t, x)=0.
HˆG= - dx 0 dω[Ψˆσˆ+(ω, x)G(ω, x)+h.c.]+ω-ω02σz(ω, x),
σˆ+(ω, x, t)=1ρ(ω, x) μ|21|μ exp(-iω0t)×δ(x-xμ)δ(ω-ωμ),
σˆ-(ω, x, t)=1ρ(ω, x) μ|12|μ exp(iω0t)×δ(x-xμ)δ(ω-ωμ),
σˆz(ω, x, t)=1ρ(ω, x) μ[|22|-|11|]μ×δ(x-xμ)δ(ω-ωμ).
[σˆ+(t, ω, x),σˆ-(t, ω, x)]=σˆz(t, ω, x)×δ(x-x)δ(ω-ω).
tσˆ-(t, ω, x)=-i(ω-ω0)σˆ-(t, ω, x)+iσˆz(t, ω, x)G(ω, x)Ψˆ(t, x).
σˆ-(t, ω, x)=σˆ-(t0, ω, x)exp[-i(ω-ω0)(t-t0)]+iG(ω, x)t0t exp[-i(ω-ω0)×(t-t)]σˆz(tω, x)Ψˆ(t, x)dt.
σˆ+(t0, ω, x)σˆ-(t0, ω, x)=δ(x-x)δ(ω-ω),
σˆ-(t0, ω, x)σˆ+(t0, ω x)=0.
-i 0 G*(ω, x)σˆ-(t, ω, x)dω
=0 dω|G(ω, x)|2 t0t dt exp[-i(ω-ω0)×(t-t)]Ψˆ(t, x)-i 0 dωG*(ω, x)×exp[-i(ω-ω0)(t-t0)]σ-(t0, ω, x)=0 dtγG(t, x)Ψˆ(t-t, x)+ΓˆG(t, x),
γG(t, x)Θ(t)-+ dω|G(ω, x)|2 exp[-i(ω-ω0)t],
ΓˆG(t, x)-i - dωG*(ω, x)exp[-i(ω-ω0)×(t-t0)]σ-(t0, ω, x).
γ˜G(ω, x)= γG(t, x)exp(iωt)dt=γG(ω, x)+iγG(ω, x),
γG(ω, x)=π|G(ω+ω0, x)|2.
ΓˆG(t, x)ΓˆG(t, x)
=0 dω|G(ω, x)|2 exp[i(ω-ω0)(t-t)]δ(x-x)=[γG(t-t, x)+γG*(t-t, x)]δ(x-x).
ΓˆG(ω, x)ΓˆG(ω, x)=2γG(ω, x)δ(x-x)δ(ω-ω).
ΓˆG(t, x)ΓˆG(t, x)=0,
ΓˆG(t, x)ΓˆG(t, x)=2γGδ(t-t)δ(x-x).
v x+tΨˆ(t, x)
=-0 dtγ(t, x)Ψˆ(t-t, x)+Γˆ(t, x)
+iω2 2x2+0- dtχ(t)×[ΨˆΨˆ](t-t, x)+ΓˆR(t, x)Ψˆ(t, x).
γ(t, x)=γA(t, x)-γG(t, x)+iΔω(x)δ(t)
 ln Ix=2[γG(ω, x)+γA(ω, x)]/v.
ΓˆR(ω, x)ΓˆR(ω, x)=2χ(x, |ω|)[nth(|ω|)+Θ(-ω)]δ(x-x)×δ(ω-ω),
Γˆ(ω, x)Γˆ(ω, x)=2γG(ω, x)δ(x-x)δ(ω-ω),
Γˆ(ω, x)Γˆ(ω, x)=2γA(ω, x)δ(x-x)δ(ω-ω),
Φˆ(tv, x)=vΨˆ(t, x).
x Φˆ(tv, x)=-0 dtv γ(tv, x)v Φˆ(tv-tv, x)+Γˆ(t)v+i-k2 2tv2+0 dt χ(tv)v2×[ΦˆΦˆ](tv-tv, x)+1v ΓˆRΦˆ(tv, x).
Γˆ(t, xv)Γˆ(t, xv)=2γGδ(xv-xv)×δ(t-t)×Γˆ(t, xv)Γˆ(t, xv)=2γAδ(xv-xv)δ(t-t).
ρ˙ˆΨ=Tr Rρ˙ˆΨ=Tr R1i[Hˆ, ρˆ],
ρˆΨ(t)= P(t, Ψ, Ψ¯) |ΨΨ¯|Ψ¯|Ψ d[Ψ]d[Ψ¯].
ζϕ(τ, ζ)=-- dτg(ττ)ϕ(τ, ζ)+Γ(τ, ζ)+±i2 2ϕτ2+i - dτh(τ-τ)×ϕ*(τ, ζ)ϕ(τ, ζ)+ΓR(τ, ζ)ϕ(τ, ζ),
ζϕ+(τ, ζ)=-- dτg*(τ-τ)ϕ+(τ, ζ)+Γ+(τ, ζ)+i2 2ϕ+τ2-i - dτh*(τ-τ)×ϕ(τ, ζ)ϕ+(τ, ζ)+Γ+R(τ, ζ)ϕ+(τ, ζ).
g(τ)=γ(τt0)x0v.
2g(Ω)=αA(Ω)-αG(Ω).
h(τ)=hE(τ)+hR(τ)=n¯x0χ(τt0)v2.
h˜(Ω)= dt exp(iΩτ)h(τ)=h(Ω)+ih(Ω).
hR(t/t0)=Θ(t)j=0n Fjδjt0 exp(-δjt)sin(ωjt).
hR(τ)=Θ(τ)j=0n FjΔj exp(-Δjτ)sin(Ωjτ).
αR(Ω)=2|h(Ω)|.
ϕP(τ, 0)=[ϕP+(τ, 0)]*=ϕˆ(τ, 0).
ϕW(τ, 0)=ϕˆ(τ, 0),
ΔϕW(τ, 0)ΔϕW*(τ, 0)=12n¯δ(τ-τ).
Γ(Ω, ζ)Γ*(Ω, ζ)=[αG(Ω)+αA(Ω)]2n¯×δ(ζ-ζ)δ(Ω-Ω),
Γ(Ω, ζ)=12π - dτΓ(τ, ζ)exp(iΩτ),
Γ*(Ω, ζ)=12π - dτΓ*(τ, ζ)exp(-iΩτ).
ΓR(Ω, ζ)ΓR*(Ω, ζ)=αR(|Ω|)n¯[nth(|Ω|/t0)+(1/2)]×δ(ζ-ζ)δ(Ω-Ω).
Γ(Ω, ζ)Γ*(Ω, ζ)=αG(Ω)n¯δ(ζ-ζ)δ(Ω-Ω).
Γ+(Ω, ζ)=12π -dτΓ+(τ, ζ)exp(-iΩτ).
ΓR(Ω, ζ)ΓR(Ω, ζ)=δ(ζ-ζ)δ(Ω+Ω)×{[nth(|Ω|/t0)+1/2]×αR(|Ω|)-ih(Ω)}/n¯,
ΓR+(Ω, ζ)ΓR(Ω, ζ)=δ(ζ-ζ)δ(Ω-Ω)×[nth(|Ω|/t0)+Θ(-Ω)]αR(|Ω|)/n¯.

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