Abstract

We develop a quantum theory of propagation in dispersive nonlinear media from the foundations of a correctly quantized field theory. Quantum fluctuations are handled by coherent-state expansions of localized field states. A stochastic nonlinear Schrödinger equation in the field variables is obtained for media with an intensity-dependent refractive index. This predicts squeezing for a continuous-wave input, over a wide bandwidth with anomalous dispersion, and over a gradually reducing bandwidth with normal dispersion. The equation is easily modified to include thermal-noise sources as well. For solitons, fluctuations are reduced over the soliton bandwidth. This leads to quantum solitons that have quadrature fluctuations less than the level of vacuum fluctuations. The complementary quadrature has a correspondingly increased fluctuation level.

© 1987 Optical Society of America

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  1. D. Stoler, Phys. Rev. D 1, 3217 (1970); Phys. Rev. D 4, 1935 (1971). For a review see D. F. Walls, Nature 306, 141 (1983).
    [CrossRef]
  2. C. M. Caves, Phys. Rev. D 26, 1817 (1980).
    [CrossRef]
  3. H. P. Yuen, Phys. Rev. A 13, 2226 (1976); H. P. Yuen and J. H. Shapiro, IEEE Trans. Inf. Theory IT-24, 657 (1978); IEEE Trans. Inf. Theory 26, 78 (1980).
    [CrossRef]
  4. R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
    [CrossRef] [PubMed]
  5. R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. S. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
    [CrossRef] [PubMed]
  6. M. W. Leade, P. Kumar, and J. H. Shapiro, in Digest of Fourteenth Quantum Electronic Conference (Optical Society of America, Washington, D.C., 1986).
  7. L. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
    [CrossRef] [PubMed]
  8. P. Kumar and J. H. Shapiro, Phys. Rev. A 30, 1568 (1984).
    [CrossRef]
  9. M. D. Levenson, R. M. Shelby, M. D. Reid, D. F. Walls, and A. Aspect, Phys. Rev. A 32, 1550 (1985).
    [CrossRef] [PubMed]
  10. P. D. Drummond and C. W. Gardiner, J. Phys. A 13, 2353 (1980), P. D. Drummond, C. W. Gardiner, and D. F. Walls, Phys. Rev. A 24, 914 (1981).
    [CrossRef]
  11. For recent reviews, see R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations (Academic, London, 1982); P. Garbaczewski, Classical and Quantum Field Theory of Exactly Soluble Nonlinear Systems (World Scientific, Singapore, 1985).
    [CrossRef]
  12. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980). We use the normalization of x0, t0as in L. F. Mollenauer, Philos. Trans. R. Soc. London A Ser. 315, 435 (1985). In SI units, χ(3)≈ 2.4 × 10−22m2/V2, n2≈ 3.2 × 10−20m2/W for silica.
    [CrossRef]
  13. W. J. Tomlinson, R. H. Stolen, and C. V. Shank, J. Opt. Soc. Am. B 1, 139 (1984).
    [CrossRef]
  14. R. Loudon, The Quantum Theory of Light (Oxford U. Press, Oxford, 1983); N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965). For nonlinear quantization, see M. Hillery and L. D. Mlodinow, Phys. Rev. A 30, 1860 (1984).
    [CrossRef]
  15. P. D. Drummond and D. F. Walls, J. Phys. A 13, 725 (1980), P. D. Drummond, K. J. McNeil, and D. F. Walls, Opt. Acta 2, 211 (1981).
    [CrossRef]
  16. R. J. Glauber, Phys. Rev. 131, 2766 (1963), E. C. G. Sudarshan, Phys. Rev. Lett. 10, 277 (1963).
    [CrossRef]
  17. C. W. Gardiner, Handbook of Stochastic Methods (Springer-Verlag, Berlin, 1983); L. Arnold, Stochastic Differential Equations (Wiley, New York, 1973).
    [CrossRef]
  18. B. Yurke and J. S. Denker, Phys. Rev. A 29, 1419 (1984).
    [CrossRef]
  19. C. W. Gardiner and C. M. Savage, Opt. Commun. 50, 173 (1984); M. J. Collet and C. W. Gardiner, Phys. Rev. A 31, 3761 (1985).
    [CrossRef]
  20. A. Hasegawa and W. Brinkman. IEEE J. Quantum Electron. QE-16, 694 (1980).
    [CrossRef]

1986 (2)

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. S. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
[CrossRef] [PubMed]

L. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

1985 (2)

M. D. Levenson, R. M. Shelby, M. D. Reid, D. F. Walls, and A. Aspect, Phys. Rev. A 32, 1550 (1985).
[CrossRef] [PubMed]

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
[CrossRef] [PubMed]

1984 (4)

P. Kumar and J. H. Shapiro, Phys. Rev. A 30, 1568 (1984).
[CrossRef]

W. J. Tomlinson, R. H. Stolen, and C. V. Shank, J. Opt. Soc. Am. B 1, 139 (1984).
[CrossRef]

B. Yurke and J. S. Denker, Phys. Rev. A 29, 1419 (1984).
[CrossRef]

C. W. Gardiner and C. M. Savage, Opt. Commun. 50, 173 (1984); M. J. Collet and C. W. Gardiner, Phys. Rev. A 31, 3761 (1985).
[CrossRef]

1980 (5)

A. Hasegawa and W. Brinkman. IEEE J. Quantum Electron. QE-16, 694 (1980).
[CrossRef]

P. D. Drummond and D. F. Walls, J. Phys. A 13, 725 (1980), P. D. Drummond, K. J. McNeil, and D. F. Walls, Opt. Acta 2, 211 (1981).
[CrossRef]

P. D. Drummond and C. W. Gardiner, J. Phys. A 13, 2353 (1980), P. D. Drummond, C. W. Gardiner, and D. F. Walls, Phys. Rev. A 24, 914 (1981).
[CrossRef]

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980). We use the normalization of x0, t0as in L. F. Mollenauer, Philos. Trans. R. Soc. London A Ser. 315, 435 (1985). In SI units, χ(3)≈ 2.4 × 10−22m2/V2, n2≈ 3.2 × 10−20m2/W for silica.
[CrossRef]

C. M. Caves, Phys. Rev. D 26, 1817 (1980).
[CrossRef]

1976 (1)

H. P. Yuen, Phys. Rev. A 13, 2226 (1976); H. P. Yuen and J. H. Shapiro, IEEE Trans. Inf. Theory IT-24, 657 (1978); IEEE Trans. Inf. Theory 26, 78 (1980).
[CrossRef]

1970 (1)

D. Stoler, Phys. Rev. D 1, 3217 (1970); Phys. Rev. D 4, 1935 (1971). For a review see D. F. Walls, Nature 306, 141 (1983).
[CrossRef]

1963 (1)

R. J. Glauber, Phys. Rev. 131, 2766 (1963), E. C. G. Sudarshan, Phys. Rev. Lett. 10, 277 (1963).
[CrossRef]

Aspect, A.

M. D. Levenson, R. M. Shelby, M. D. Reid, D. F. Walls, and A. Aspect, Phys. Rev. A 32, 1550 (1985).
[CrossRef] [PubMed]

Brinkman, W.

A. Hasegawa and W. Brinkman. IEEE J. Quantum Electron. QE-16, 694 (1980).
[CrossRef]

Caves, C. M.

C. M. Caves, Phys. Rev. D 26, 1817 (1980).
[CrossRef]

Denker, J. S.

B. Yurke and J. S. Denker, Phys. Rev. A 29, 1419 (1984).
[CrossRef]

DeVoe, R. S.

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. S. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
[CrossRef] [PubMed]

Dodd, R. K.

For recent reviews, see R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations (Academic, London, 1982); P. Garbaczewski, Classical and Quantum Field Theory of Exactly Soluble Nonlinear Systems (World Scientific, Singapore, 1985).
[CrossRef]

Drummond, P. D.

P. D. Drummond and C. W. Gardiner, J. Phys. A 13, 2353 (1980), P. D. Drummond, C. W. Gardiner, and D. F. Walls, Phys. Rev. A 24, 914 (1981).
[CrossRef]

P. D. Drummond and D. F. Walls, J. Phys. A 13, 725 (1980), P. D. Drummond, K. J. McNeil, and D. F. Walls, Opt. Acta 2, 211 (1981).
[CrossRef]

Eilbeck, J. C.

For recent reviews, see R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations (Academic, London, 1982); P. Garbaczewski, Classical and Quantum Field Theory of Exactly Soluble Nonlinear Systems (World Scientific, Singapore, 1985).
[CrossRef]

Gardiner, C. W.

C. W. Gardiner and C. M. Savage, Opt. Commun. 50, 173 (1984); M. J. Collet and C. W. Gardiner, Phys. Rev. A 31, 3761 (1985).
[CrossRef]

P. D. Drummond and C. W. Gardiner, J. Phys. A 13, 2353 (1980), P. D. Drummond, C. W. Gardiner, and D. F. Walls, Phys. Rev. A 24, 914 (1981).
[CrossRef]

C. W. Gardiner, Handbook of Stochastic Methods (Springer-Verlag, Berlin, 1983); L. Arnold, Stochastic Differential Equations (Wiley, New York, 1973).
[CrossRef]

Gibbon, J. D.

For recent reviews, see R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations (Academic, London, 1982); P. Garbaczewski, Classical and Quantum Field Theory of Exactly Soluble Nonlinear Systems (World Scientific, Singapore, 1985).
[CrossRef]

Glauber, R. J.

R. J. Glauber, Phys. Rev. 131, 2766 (1963), E. C. G. Sudarshan, Phys. Rev. Lett. 10, 277 (1963).
[CrossRef]

Gordon, J. P.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980). We use the normalization of x0, t0as in L. F. Mollenauer, Philos. Trans. R. Soc. London A Ser. 315, 435 (1985). In SI units, χ(3)≈ 2.4 × 10−22m2/V2, n2≈ 3.2 × 10−20m2/W for silica.
[CrossRef]

Hall, J. L.

L. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

Hasegawa, A.

A. Hasegawa and W. Brinkman. IEEE J. Quantum Electron. QE-16, 694 (1980).
[CrossRef]

Hollberg, L. W.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
[CrossRef] [PubMed]

Kimble, H. J.

L. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

Kumar, P.

P. Kumar and J. H. Shapiro, Phys. Rev. A 30, 1568 (1984).
[CrossRef]

M. W. Leade, P. Kumar, and J. H. Shapiro, in Digest of Fourteenth Quantum Electronic Conference (Optical Society of America, Washington, D.C., 1986).

Leade, M. W.

M. W. Leade, P. Kumar, and J. H. Shapiro, in Digest of Fourteenth Quantum Electronic Conference (Optical Society of America, Washington, D.C., 1986).

Levenson, M. D.

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. S. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
[CrossRef] [PubMed]

M. D. Levenson, R. M. Shelby, M. D. Reid, D. F. Walls, and A. Aspect, Phys. Rev. A 32, 1550 (1985).
[CrossRef] [PubMed]

Loudon, R.

R. Loudon, The Quantum Theory of Light (Oxford U. Press, Oxford, 1983); N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965). For nonlinear quantization, see M. Hillery and L. D. Mlodinow, Phys. Rev. A 30, 1860 (1984).
[CrossRef]

Mertz, J. C.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
[CrossRef] [PubMed]

Mollenauer, L. F.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980). We use the normalization of x0, t0as in L. F. Mollenauer, Philos. Trans. R. Soc. London A Ser. 315, 435 (1985). In SI units, χ(3)≈ 2.4 × 10−22m2/V2, n2≈ 3.2 × 10−20m2/W for silica.
[CrossRef]

Morris, H. C.

For recent reviews, see R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations (Academic, London, 1982); P. Garbaczewski, Classical and Quantum Field Theory of Exactly Soluble Nonlinear Systems (World Scientific, Singapore, 1985).
[CrossRef]

Perlmutter, S. H.

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. S. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
[CrossRef] [PubMed]

Reid, M. D.

M. D. Levenson, R. M. Shelby, M. D. Reid, D. F. Walls, and A. Aspect, Phys. Rev. A 32, 1550 (1985).
[CrossRef] [PubMed]

Savage, C. M.

C. W. Gardiner and C. M. Savage, Opt. Commun. 50, 173 (1984); M. J. Collet and C. W. Gardiner, Phys. Rev. A 31, 3761 (1985).
[CrossRef]

Shank, C. V.

Shapiro, J. H.

P. Kumar and J. H. Shapiro, Phys. Rev. A 30, 1568 (1984).
[CrossRef]

M. W. Leade, P. Kumar, and J. H. Shapiro, in Digest of Fourteenth Quantum Electronic Conference (Optical Society of America, Washington, D.C., 1986).

Shelby, R. M.

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. S. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
[CrossRef] [PubMed]

M. D. Levenson, R. M. Shelby, M. D. Reid, D. F. Walls, and A. Aspect, Phys. Rev. A 32, 1550 (1985).
[CrossRef] [PubMed]

Slusher, R. E.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
[CrossRef] [PubMed]

Stolen, R. H.

W. J. Tomlinson, R. H. Stolen, and C. V. Shank, J. Opt. Soc. Am. B 1, 139 (1984).
[CrossRef]

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980). We use the normalization of x0, t0as in L. F. Mollenauer, Philos. Trans. R. Soc. London A Ser. 315, 435 (1985). In SI units, χ(3)≈ 2.4 × 10−22m2/V2, n2≈ 3.2 × 10−20m2/W for silica.
[CrossRef]

Stoler, D.

D. Stoler, Phys. Rev. D 1, 3217 (1970); Phys. Rev. D 4, 1935 (1971). For a review see D. F. Walls, Nature 306, 141 (1983).
[CrossRef]

Tomlinson, W. J.

Valley, J. F.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
[CrossRef] [PubMed]

Walls, D. F.

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. S. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
[CrossRef] [PubMed]

M. D. Levenson, R. M. Shelby, M. D. Reid, D. F. Walls, and A. Aspect, Phys. Rev. A 32, 1550 (1985).
[CrossRef] [PubMed]

P. D. Drummond and D. F. Walls, J. Phys. A 13, 725 (1980), P. D. Drummond, K. J. McNeil, and D. F. Walls, Opt. Acta 2, 211 (1981).
[CrossRef]

Wu, H.

L. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

Wu, L.

L. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

Yuen, H. P.

H. P. Yuen, Phys. Rev. A 13, 2226 (1976); H. P. Yuen and J. H. Shapiro, IEEE Trans. Inf. Theory IT-24, 657 (1978); IEEE Trans. Inf. Theory 26, 78 (1980).
[CrossRef]

Yurke, B.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
[CrossRef] [PubMed]

B. Yurke and J. S. Denker, Phys. Rev. A 29, 1419 (1984).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. Hasegawa and W. Brinkman. IEEE J. Quantum Electron. QE-16, 694 (1980).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Phys. A (2)

P. D. Drummond and D. F. Walls, J. Phys. A 13, 725 (1980), P. D. Drummond, K. J. McNeil, and D. F. Walls, Opt. Acta 2, 211 (1981).
[CrossRef]

P. D. Drummond and C. W. Gardiner, J. Phys. A 13, 2353 (1980), P. D. Drummond, C. W. Gardiner, and D. F. Walls, Phys. Rev. A 24, 914 (1981).
[CrossRef]

Opt. Commun. (1)

C. W. Gardiner and C. M. Savage, Opt. Commun. 50, 173 (1984); M. J. Collet and C. W. Gardiner, Phys. Rev. A 31, 3761 (1985).
[CrossRef]

Phys. Rev. (1)

R. J. Glauber, Phys. Rev. 131, 2766 (1963), E. C. G. Sudarshan, Phys. Rev. Lett. 10, 277 (1963).
[CrossRef]

Phys. Rev. A (4)

H. P. Yuen, Phys. Rev. A 13, 2226 (1976); H. P. Yuen and J. H. Shapiro, IEEE Trans. Inf. Theory IT-24, 657 (1978); IEEE Trans. Inf. Theory 26, 78 (1980).
[CrossRef]

P. Kumar and J. H. Shapiro, Phys. Rev. A 30, 1568 (1984).
[CrossRef]

M. D. Levenson, R. M. Shelby, M. D. Reid, D. F. Walls, and A. Aspect, Phys. Rev. A 32, 1550 (1985).
[CrossRef] [PubMed]

B. Yurke and J. S. Denker, Phys. Rev. A 29, 1419 (1984).
[CrossRef]

Phys. Rev. D (2)

D. Stoler, Phys. Rev. D 1, 3217 (1970); Phys. Rev. D 4, 1935 (1971). For a review see D. F. Walls, Nature 306, 141 (1983).
[CrossRef]

C. M. Caves, Phys. Rev. D 26, 1817 (1980).
[CrossRef]

Phys. Rev. Lett. (4)

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, Phys. Rev. Lett. 55, 2409 (1985).
[CrossRef] [PubMed]

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. S. DeVoe, and D. F. Walls, Phys. Rev. Lett. 57, 691 (1986).
[CrossRef] [PubMed]

L. Wu, H. J. Kimble, J. L. Hall, and H. Wu, Phys. Rev. Lett. 57, 2520 (1986).
[CrossRef] [PubMed]

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980). We use the normalization of x0, t0as in L. F. Mollenauer, Philos. Trans. R. Soc. London A Ser. 315, 435 (1985). In SI units, χ(3)≈ 2.4 × 10−22m2/V2, n2≈ 3.2 × 10−20m2/W for silica.
[CrossRef]

Other (4)

For recent reviews, see R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations (Academic, London, 1982); P. Garbaczewski, Classical and Quantum Field Theory of Exactly Soluble Nonlinear Systems (World Scientific, Singapore, 1985).
[CrossRef]

C. W. Gardiner, Handbook of Stochastic Methods (Springer-Verlag, Berlin, 1983); L. Arnold, Stochastic Differential Equations (Wiley, New York, 1973).
[CrossRef]

R. Loudon, The Quantum Theory of Light (Oxford U. Press, Oxford, 1983); N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965). For nonlinear quantization, see M. Hillery and L. D. Mlodinow, Phys. Rev. A 30, 1860 (1984).
[CrossRef]

M. W. Leade, P. Kumar, and J. H. Shapiro, in Digest of Fourteenth Quantum Electronic Conference (Optical Society of America, Washington, D.C., 1986).

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Figures (6)

Fig. 1
Fig. 1

Plot of the logarithmic spectrum, ln[1 + S(ζ, ω ¯)max], of maximum squeezing in the anomalous dispersion, continuous-wave case.

Fig. 2
Fig. 2

Plot of the logarithmic spectrum of maximum squeezing in the normal dispersion cw case.

Fig. 3
Fig. 3

Plot of the phase angle of maximum squeezing in the anomalous dispersion cw case.

Fig. 4
Fig. 4

Plot of the phase angle of maximum squeezing in the normal dispersion cw case.

Fig. 5
Fig. 5

Plot of the maximum squeezing [S(ζ, ω ¯)max] in the soliton case.

Fig. 6
Fig. 6

Complementary fluctuations [S(ζ, ω ¯)orthog] in the soliton case.

Equations (73)

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

P T = 0 [ χ ( 1 ) : E T + χ ( 2 ) : E T E T + ] = n P ( n ) .
H E = 0 D E · d D ,
D = : E T + n > 1 P ( n ) .
H = v [ 1 2 E T · : E T + n > 1 ( n n + 1 ) E T · P ( n ) + 1 2 μ B ] d 3 x .
H = v ( 1 2 { D 2 - [ n > 1 P ( n ) ] 2 } - n > 1 1 n + 1 E T · P ( n ) + 1 2 μ B 2 ) d 3 x .
L 0 ( A , A ˙ ) = ( 2 E 2 - 1 2 μ B 2 ) .
( 2 - 1 c 2 2 t 2 ) A = - μ P ˙ T .
L = [ U ( A ˙ ) - 1 2 μ k ( A x k ) 2 ] .
2 A j = μ t [ U ( A ˙ ) A ˙ j ] .
E T = - A ˙ ,
U ( A ˙ ) = 0 D D · d E T = E T · D - H E .
[ D ^ j ( x ) , A ^ k ( x ) ] = i δ j k T ( x - x ) .
H ^ = v [ 1 2 D ^ 2 ( x ) + 1 2 μ B ^ 2 ( x ) ] d 2 x - 1 4 0 · χ ( 3 ) × v E T 4 ( x ) d 2 x + O ( E ^ T 6 ) .
H ^ = n ω n a ^ n a ^ n - 0 χ ( 3 ) 4 4 v D ^ T 4 ( x ) d 3 x .
D ^ ( x ) = i n ( ω k n 2 ) 1 / 2 a ^ n u n ( r ) exp ( i k n z ) + h . c . ,
k n = k 0 + n Δ k , v u n * ( r ) u n d 3 x = 1.
α ^ l 1 ( 2 N + 1 ) 1 / 2 n = - N + N a ^ n exp ( i 2 π n l 2 N + 1 )
[ α ^ k , α ^ j + ] = δ k j , [ α ^ k , α ^ j ] = 0.
ω n = ω ( k 0 + n Δ k ) ω ( k 0 ) + n Δ k d ω d k | k 0 + ( n Δ k ) 2 2 d 2 ω d k 2 | k 0 ω 0 + Δ ω + n Δ k ω + ( n Δ k ) 2 2 ω .
n ω n α ^ n α ^ n = [ ω 0 l α ^ l + α ^ l + l l ω l l α ^ l + α ^ l ,
ω l l = n [ ( n Δ k ) ω + ½ ( n Δ k ) 2 ω ( 2 N + 1 ) ] × exp [ i 2 π n ( l - l ) 2 N + 1 ] + Δ ω δ l l .
H ^ = H ^ 0 + H ^ I ,
H ^ 0 = ω 0 l α ^ l + α ^ l = ω 0 l α ^ l + ( t ) α ^ l ( t ) , α ^ l ( t ) = α ^ l exp ( - i ω 0 t ) ,
- 0 χ ( 3 ) 4 4 v : D ^ 4 : d 3 x - χ α l α ^ l + 2 α l 2 .
χ α = 3 0 χ ( 3 ) ( ω k 0 ) 2 8 2 Δ V ,
Δ V = L A / ( 2 N + 1 )
L A = [ v u ( r ) 4 d 3 x ] - 1 = ( 2 N + 1 ) Δ V .
H ^ I = ( l l ω l l α ^ l + α ^ l - χ α l α ^ l + 2 α l 2 ) .
ψ ( t ) ψ ( t ) = ρ ^ ( t ) = P ( α ) Λ ^ ( α ) d μ ( α ) ,
α = α - N , α - N + , , α N , α N + , Λ ( α ) = α - N , α - N + 1 , , α N ( α - N + ) * , ( α - N + 1 + ) * , , ( α N + ) * ( α - N + ) * , ( α - N + 1 + ) * , , ( α N + ) * α - N , α - N + 1 , , α N , d μ ( α ) = d 2 α - N d + 2 α - N d 2 α - N + 1 d + 2 α N .
α ^ 1 Λ ^ ( α ) = α l Λ ^ ( α ) , α ^ l + Λ ^ ( α ) = ( α l + + α l ) Λ ^ ( α ) , Λ ^ ( α ) α ^ l + = Λ ^ ( α ) α l + , Λ ^ ( α ) α ^ l = ( α l + + α l ) Λ ^ ( α ) .
i Λ ^ ( α ) t P ( α ) d μ ( α ) = P ( α ) [ H ^ I , Λ ^ ( α ) ] d μ ( α ) .
t P ( α ) = i l [ α l ( l ω l l α l - 2 χ α α l 2 α l + + 2 α l 2 χ α α l 2 ) ] P ( α ) + h . c .
t P ( α ) = [ i α i A i ( α ) + 1 2 i j α i α j d i j ( α ) ] P ( α ) ,
d = χ α [ - 2 α 1 α 2 α 1 2 - α 2 2 α 1 2 - α 2 2 2 α 1 α 2 ] .
i α 2 = ( u 1 + i u 2 ) 2 ,             - i α + 2 = ( v 1 + i v 2 ) 2 ,
2 α 2 2 α x 2 - 2 α y 2 - i 2 α x α y ,
d = χ α [ u x 2 0 u x u y 0 0 v x 2 0 v x v y u x u y 0 u y 2 0 0 v x v y 0 v y 2 ]
α l t = i ( 2 χ α α l 2 α l + - l ω l l α l ) + ( 2 i χ α ) 1 / 2 α l ξ l ( t ) , α l + t = - i ( 2 χ α α l + 2 α l - l ω l l α l + ) + ( - 2 i χ α ) 1 / 2 α l + ξ l + ( t ) ,
ξ l ( t 1 ) ξ l ( t 2 ) = ξ l + ( t 1 ) ξ l + ( t 2 ) δ l l δ ( t l - t 2 ) , ξ l ( t 1 ) ξ l + ( t 2 ) = 0.
Φ ( z ) lim Δ z 0 ( α l ) ( ω Δ z ) 1 / 2 ,
Φ ^ ( z ) = ( ω L ) 1 / 2 n a ^ n exp [ i ( k n - k 0 ) z ] .
t Φ ( t , z ) = ( - i Δ ω - ω z + i ω 2 2 z 2 ) Φ ( t , z ) + i χ Φ ω Φ 2 ( t , z ) Φ + ( t , z ) ,
χ Φ = 3 χ ( 3 ) ω 0 2 4 A c 2 .
z Φ ( t , z ) = ( - i Δ ω ω + i ω 2 ω 3 2 t 2 ) Φ + i χ Φ Φ 2 Φ + .
ξ ( t 1 , z 1 ) ξ ( t 2 , z 2 ) = δ ( t 1 - t 2 ) δ ( z 1 - z 2 ) , ξ + ( t 1 , z 1 ) ξ + ( t 2 , z 2 ) = δ ( t 1 - t 2 ) δ ( z 1 - z 2 ) .
z Φ ( t , z ) = - i k 2 ( ± 1 t 0 2 + 2 t 2 ) Φ + i χ Φ Φ 2 Φ + + ( i χ Φ ) 1 / 2 Φ ξ ( t , z ) ,
t 0 = [ ω k / ( 2 Δ ω ) ] 1 / 2 , k = d 2 k d ω 2 .
ξ ¯ ( t , z ) = ξ ( t , z ) + ( i ω 0 2 χ Φ n 2 ω 2 ) 1 / 2 δ n ( t , z ) , ξ ¯ + ( t , z ) = ξ + ( t , z ) + [ - i ω 0 2 χ Φ n 2 ω 2 ] 1 / 2 δ n ( t , z ) ,
δ n ( t , z ) = [ n ( t , x ) - n ¯ ] u ( r ) 2 d 2 r .
z 0 = t 0 2 / k , p 0 = k / ( χ Φ t 0 ) .
k = - ω / ( ω ) 3 , χ Φ = ( n 2 ω 0 2 ) / ( A c ) ,
ϕ ( τ , ζ ) = ( t 0 / p 0 ) 1 / 2 Φ ( τ t 0 , ζ z 0 ) , ϕ + ( τ , ζ ) = ( t 0 / p 0 ) 1 / 2 Φ + ( τ 0 , ζ z 0 ) .
P = ω 0 p 0 ϕ + ( t , ζ ) ϕ ( τ , ζ ) / t 0 .
t 0 = ( ω 0 k χ Φ P 0 ) 1 / 2 ϕ 0 .
Δ ω = ( ω 0 n 2 P 0 2 n A ϕ 0 2 ) .
ζ ϕ ( τ , ζ ) = [ - i 2 ( 1 ± 2 τ 2 ) + i ϕ + ϕ + ( i / p 0 ) 1 / 2 η ( τ , ζ ) ] ϕ ( τ , ζ ) , ζ ϕ + ( τ , ζ ) = [ i 2 ( 1 ± 2 τ 2 ) - i ϕ + ϕ + ( - i / p 0 ) 1 / 2 η + ( τ , ζ ) ] ϕ + ( τ , ζ ) ,
ζ = z / z 0 ,             τ = t / t 0 .
δ n ( t 1 , z 1 ) δ n ( t 2 , z 2 ) = Γ δ ( t 1 - t 2 ) δ ( z 1 - z 2 ) .
η ( τ 1 , ζ 1 ) η ( τ 2 , ζ 2 ) = ( 1 + i g ) δ ( τ 1 - τ 2 ) δ ( ζ 1 - ζ 2 ) , η ( τ 1 , ζ 1 ) η + ( τ 2 , ζ 2 ) = g δ ( τ 1 - τ 2 ) δ ( ζ 1 - ζ 2 ) , η + ( τ 1 , ζ 1 ) η + ( τ 2 , ζ 2 ) = ( 1 - i g ) δ ( τ 1 - τ 2 ) δ ( ζ 1 - ζ 2 ) ,
g = ( Γ ω 0 2 χ Φ n 2 ω 2 ) = 4 Γ A 3 0 χ ( 3 ) ( c n 2 ω ) 2 .
S ( ω ¯ , ζ , θ ) 4 π t 0 p 0 T p Re [ e - 2 i θ δ ϕ ˜ ( ω ¯ , ζ ) δ ϕ ˜ ( - ω ¯ , ζ ) + δ ϕ ˜ ( ω ¯ , ζ ) δ ϕ ˜ + ( - ω ¯ , ζ ) ] ,
δ ϕ ˜ ( ω ¯ , ζ ) 1 2 π - T / 2 t 0 T / 2 t 0 δ ϕ ( τ , ζ ) e i ω ¯ τ d τ , δ ϕ ( τ , ζ ) ϕ ( τ , ζ ) - ϕ ( τ , ζ ) .
ϕ 0 2 = 1 / 2
P 0 = ω 0 p 0 / ( 2 t 0 ) = ω 0 k / ( 2 χ Φ t 0 2 ) .
ζ δ ϕ ( τ , ζ ) = - i 2 ( ± 2 τ 2 - 1 ) δ ϕ + i 2 δ ϕ + + [ i / 2 p 0 ] 1 / 2 η ( τ , ζ ) ,
ζ [ δ ϕ ˜ ( ω ¯ , ζ ) δ ϕ ˜ + ( ω ¯ , ζ ) ] = A ( ω ¯ ) [ δ ϕ ˜ ( ω ¯ , ζ ) δ ϕ ˜ + ( ω ¯ , ζ ) ] + [ i 2 p 0 ] 1 / 2 [ η ˜ ( ω ¯ , ζ ) i η ˜ + ( ω ¯ , ζ ) ] ,
A ( ω ¯ ) = i 2 [ ( 1 ± ω ¯ 2 ) 1 - 1 - ( 1 ± ω ¯ 2 ) ] .
[ δ ϕ ˜ ( ω ¯ , ζ ) δ ϕ ˜ + ( ω ¯ , ζ ) ] = 0 ζ d ζ exp [ A ( ω ¯ ) ( ζ - ζ ) ] [ i 2 p 0 ] 1 / 2 [ η ˜ ( ω ¯ , ζ ) i η ˜ + ( ω ¯ , ζ ) ] .
S ( ω ¯ , ζ ) max = [ 1 - cos ( γ ζ ) ] / γ 2 - i sin ( γ ζ ) / γ + β [ cos ( γ ζ ) - 1 ] / γ 2 ,
β = ( 1 ± ω ¯ 2 ) ,             γ = ω ¯ [ ω ¯ 2 ± 2 ] 1 / 2 .
ϕ 0 ( τ ) = sech ( τ ) .
ζ δ ϕ ( τ , ζ ) = i 2 [ 2 τ 2 + 4 ϕ 0 2 ( τ ) - 1 ] δ ϕ + i ϕ 0 2 ( τ ) δ ϕ + + [ i / p 0 ] 1 / 2 η ( τ , ζ ) ϕ 0 ( τ ) .

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