Abstract

We review soliton perturbation theory with renormalized soliton operators. We analytically evaluate the effects of the continuum on squeezing. Our results show that the contribution that is due to the continuum exhibits an oscillatory behavior and can be beneficial to squeezing.

© 2000 Optical Society of America

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References

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  1. P. D. Drummond and S. J. Carter, “Quantum-field theory of squeezing in solitons,” J. Opt. Soc. Am. B 4, 1565–1573 (1987).
    [CrossRef]
  2. Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. I. Time-dependent Hartree approximation,” Phys. Rev. A 40, 844–853 (1989); “Quantum theory of soli-tons in optical fibers. II. Exact solution,” Phys. Rev. A 40, 854–866 (1989).
    [CrossRef] [PubMed]
  3. H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing: a linearized approach,” J. Opt. Soc. Am. B 7, 386–392 (1990).
    [CrossRef]
  4. Y. Lai, “Quantum theory of soliton propagation: a unified approach based on the linearization approximation,” J. Opt. Soc. Am. B 10, 475–484 (1993).
    [CrossRef]
  5. F. X. Kärtner and L. Boivin, “Quantum noise of the fundamental soliton,” Phys. Rev. A 53, 454–466 (1996)
    [CrossRef] [PubMed]
  6. J. M. Fini, P. L. Hagelstein, and H. A. Haus, “Agreement of stochastic soliton formalism with second-quantized and configuration-space models,” Phys. Rev. A (to be published).
  7. S. J. Carter and P. D. Drummond, “Squeezed quantum solitons and Raman noise,” Phys. Rev. Lett. 67, 3757–3760 (1991).
    [CrossRef] [PubMed]
  8. D. Yao, “Quantum fluctuations of optical solitons with optical pulses,” Phys. Rev. A 52, 4871–4881 (1995).
    [CrossRef] [PubMed]
  9. E. Wright, “Quantum theory of soliton propagation in an optical fiber using the Hartree approximation,” Phys. Rev. A 43, 3836–3844 (1991).
    [CrossRef] [PubMed]
  10. P. Hagelstein, “Application of a photon configuration-space model to soliton propagation in a fiber,” Phys. Rev. A 54, 2426–2438 (1996).
    [CrossRef] [PubMed]
  11. S. Carter, P. Drummond, M. Reid, and R. Shelby, “Squeezing of quantum solitons,” Phys. Rev. Lett. 58, 1841–1844 (1987).
    [CrossRef] [PubMed]
  12. Y. Lai and S. Yu, “General quantum theory of nonlinear optical pulse propagation,” Phys. Rev. A 51, 817–829 (1995).
    [CrossRef] [PubMed]
  13. M. Rosenbluh and R. M. Shelby, “Squeezed optical solitons,” Phys. Rev. Lett. 66, 153–156 (1991).
    [CrossRef] [PubMed]
  14. H. A. Haus and J. A. Mullen, “Quantum noise in linear amplifiers,” Phys. Rev. 128, 2407–2415 (1962).
    [CrossRef]
  15. H. A. Haus and Y. Yamamoto, “Quantum circuit theory of phase-sensitive linear systems,” QE-23, 212–221 (1987).
  16. M. Margalit and H. A. Haus, “Accounting for the continuum in the analysis of squeezing with solitons,” J. Opt. Soc. Am. B 15, 1387–1391 (1998).
    [CrossRef]
  17. D. Krylov and K. Bergman, “Amplitude-squeezed solitons from an asymmetric fiber interferometer,” Opt. Lett. 23, 1390–1392 (1998).
    [CrossRef]
  18. S. Spalter, N. Korolkova, F. Konig, A. Sizmann, and G. Leuchs, “Observation of multimode quantum correlations in fiber optical solitons,” Phys. Rev. Lett. 81, 786–788 (1998).
    [CrossRef]
  19. S. Schmitt, J. Ficker, M. Wolff, F. Konig, A. Sizmann, and G. Leuchs, “Photon-number squeezed solitons from an asymmetric fiber-optic Sagnac interferometer,” Phys. Rev. Lett. 81, 2446–2449 (1998).
    [CrossRef]
  20. D. Levandovsky, M. Vasilyev, and P. Kumar, “Perturbation theory of quantum solitons: continuum evolution and optimum squeezing by spectral filtering,” Opt. Lett. 24, 43–45 (1999).
    [CrossRef]
  21. D. Levandovsky, M. Vasilyev, and P. Kumar, “Soliton squeezing in a highly transmissive nonlinear optical loop mirror,” Opt. Lett. 24, 89–91 (1999).
    [CrossRef]
  22. J. P. Gordon, “Dispersive perturbations of the nonlinear Schrödinger equations,” J. Opt. Soc. Am. B 9, 91–97 (1992).
    [CrossRef]
  23. L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Nonrelativistic Theory, translated by J. B. Sakes and J. S. Bell (Pergamon, London, 1958).
  24. K. Bergman and H. A. Haus, “Squeezing in fibers with optical pulses,” Opt. Lett. 16, 373–375 (1989).
  25. C. Doerr, M. Shirasaki, and F. Khatri, “Simulation of pulsed squeezing in optical fiber with chromatic dispersion,” J. Opt. Soc. Am. B 11, 143–149 (1994).
    [CrossRef]
  26. A. M. Weiner, J. P. Heritage, and E. M. Kirschner, “High resolution femtosecond pulse shaping,” J. Opt. Soc. Am. B 5, 1563–1572 (1988).
    [CrossRef]
  27. H. A. Haus, W. S. Wong, and F. I. Khatri, “Continuum generation by perturbation of solitons,” J. Opt. Soc. Am. B 14, 304–313 (1997).
    [CrossRef]

1999 (2)

1998 (4)

M. Margalit and H. A. Haus, “Accounting for the continuum in the analysis of squeezing with solitons,” J. Opt. Soc. Am. B 15, 1387–1391 (1998).
[CrossRef]

D. Krylov and K. Bergman, “Amplitude-squeezed solitons from an asymmetric fiber interferometer,” Opt. Lett. 23, 1390–1392 (1998).
[CrossRef]

S. Spalter, N. Korolkova, F. Konig, A. Sizmann, and G. Leuchs, “Observation of multimode quantum correlations in fiber optical solitons,” Phys. Rev. Lett. 81, 786–788 (1998).
[CrossRef]

S. Schmitt, J. Ficker, M. Wolff, F. Konig, A. Sizmann, and G. Leuchs, “Photon-number squeezed solitons from an asymmetric fiber-optic Sagnac interferometer,” Phys. Rev. Lett. 81, 2446–2449 (1998).
[CrossRef]

1997 (1)

1996 (2)

F. X. Kärtner and L. Boivin, “Quantum noise of the fundamental soliton,” Phys. Rev. A 53, 454–466 (1996)
[CrossRef] [PubMed]

P. Hagelstein, “Application of a photon configuration-space model to soliton propagation in a fiber,” Phys. Rev. A 54, 2426–2438 (1996).
[CrossRef] [PubMed]

1995 (2)

D. Yao, “Quantum fluctuations of optical solitons with optical pulses,” Phys. Rev. A 52, 4871–4881 (1995).
[CrossRef] [PubMed]

Y. Lai and S. Yu, “General quantum theory of nonlinear optical pulse propagation,” Phys. Rev. A 51, 817–829 (1995).
[CrossRef] [PubMed]

1994 (1)

1993 (1)

1992 (1)

1991 (3)

E. Wright, “Quantum theory of soliton propagation in an optical fiber using the Hartree approximation,” Phys. Rev. A 43, 3836–3844 (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]

M. Rosenbluh and R. M. Shelby, “Squeezed optical solitons,” Phys. Rev. Lett. 66, 153–156 (1991).
[CrossRef] [PubMed]

1990 (1)

1989 (1)

K. Bergman and H. A. Haus, “Squeezing in fibers with optical pulses,” Opt. Lett. 16, 373–375 (1989).

1988 (1)

1987 (2)

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

1962 (1)

H. A. Haus and J. A. Mullen, “Quantum noise in linear amplifiers,” Phys. Rev. 128, 2407–2415 (1962).
[CrossRef]

Bergman, K.

D. Krylov and K. Bergman, “Amplitude-squeezed solitons from an asymmetric fiber interferometer,” Opt. Lett. 23, 1390–1392 (1998).
[CrossRef]

K. Bergman and H. A. Haus, “Squeezing in fibers with optical pulses,” Opt. Lett. 16, 373–375 (1989).

Boivin, L.

F. X. Kärtner and L. Boivin, “Quantum noise of the fundamental soliton,” Phys. Rev. A 53, 454–466 (1996)
[CrossRef] [PubMed]

Carter, S.

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

Carter, S. J.

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

Doerr, C.

Drummond, P.

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

Drummond, P. D.

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

Ficker, J.

S. Schmitt, J. Ficker, M. Wolff, F. Konig, A. Sizmann, and G. Leuchs, “Photon-number squeezed solitons from an asymmetric fiber-optic Sagnac interferometer,” Phys. Rev. Lett. 81, 2446–2449 (1998).
[CrossRef]

Gordon, J. P.

Hagelstein, P.

P. Hagelstein, “Application of a photon configuration-space model to soliton propagation in a fiber,” Phys. Rev. A 54, 2426–2438 (1996).
[CrossRef] [PubMed]

Haus, H. A.

Heritage, J. P.

Kärtner, F. X.

F. X. Kärtner and L. Boivin, “Quantum noise of the fundamental soliton,” Phys. Rev. A 53, 454–466 (1996)
[CrossRef] [PubMed]

Khatri, F.

Khatri, F. I.

Kirschner, E. M.

Konig, F.

S. Schmitt, J. Ficker, M. Wolff, F. Konig, A. Sizmann, and G. Leuchs, “Photon-number squeezed solitons from an asymmetric fiber-optic Sagnac interferometer,” Phys. Rev. Lett. 81, 2446–2449 (1998).
[CrossRef]

S. Spalter, N. Korolkova, F. Konig, A. Sizmann, and G. Leuchs, “Observation of multimode quantum correlations in fiber optical solitons,” Phys. Rev. Lett. 81, 786–788 (1998).
[CrossRef]

Korolkova, N.

S. Spalter, N. Korolkova, F. Konig, A. Sizmann, and G. Leuchs, “Observation of multimode quantum correlations in fiber optical solitons,” Phys. Rev. Lett. 81, 786–788 (1998).
[CrossRef]

Krylov, D.

Kumar, P.

Lai, Y.

Leuchs, G.

S. Spalter, N. Korolkova, F. Konig, A. Sizmann, and G. Leuchs, “Observation of multimode quantum correlations in fiber optical solitons,” Phys. Rev. Lett. 81, 786–788 (1998).
[CrossRef]

S. Schmitt, J. Ficker, M. Wolff, F. Konig, A. Sizmann, and G. Leuchs, “Photon-number squeezed solitons from an asymmetric fiber-optic Sagnac interferometer,” Phys. Rev. Lett. 81, 2446–2449 (1998).
[CrossRef]

Levandovsky, D.

Margalit, M.

Mullen, J. A.

H. A. Haus and J. A. Mullen, “Quantum noise in linear amplifiers,” Phys. Rev. 128, 2407–2415 (1962).
[CrossRef]

Reid, M.

S. Carter, P. Drummond, M. Reid, and R. 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).
[CrossRef] [PubMed]

Schmitt, S.

S. Schmitt, J. Ficker, M. Wolff, F. Konig, A. Sizmann, and G. Leuchs, “Photon-number squeezed solitons from an asymmetric fiber-optic Sagnac interferometer,” Phys. Rev. Lett. 81, 2446–2449 (1998).
[CrossRef]

Shelby, R.

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

Shelby, R. M.

M. Rosenbluh and R. M. Shelby, “Squeezed optical solitons,” Phys. Rev. Lett. 66, 153–156 (1991).
[CrossRef] [PubMed]

Shirasaki, M.

Sizmann, A.

S. Schmitt, J. Ficker, M. Wolff, F. Konig, A. Sizmann, and G. Leuchs, “Photon-number squeezed solitons from an asymmetric fiber-optic Sagnac interferometer,” Phys. Rev. Lett. 81, 2446–2449 (1998).
[CrossRef]

S. Spalter, N. Korolkova, F. Konig, A. Sizmann, and G. Leuchs, “Observation of multimode quantum correlations in fiber optical solitons,” Phys. Rev. Lett. 81, 786–788 (1998).
[CrossRef]

Spalter, S.

S. Spalter, N. Korolkova, F. Konig, A. Sizmann, and G. Leuchs, “Observation of multimode quantum correlations in fiber optical solitons,” Phys. Rev. Lett. 81, 786–788 (1998).
[CrossRef]

Vasilyev, M.

Weiner, A. M.

Wolff, M.

S. Schmitt, J. Ficker, M. Wolff, F. Konig, A. Sizmann, and G. Leuchs, “Photon-number squeezed solitons from an asymmetric fiber-optic Sagnac interferometer,” Phys. Rev. Lett. 81, 2446–2449 (1998).
[CrossRef]

Wong, W. S.

Wright, E.

E. Wright, “Quantum theory of soliton propagation in an optical fiber using the Hartree approximation,” Phys. Rev. A 43, 3836–3844 (1991).
[CrossRef] [PubMed]

Yao, D.

D. Yao, “Quantum fluctuations of optical solitons with optical pulses,” Phys. Rev. A 52, 4871–4881 (1995).
[CrossRef] [PubMed]

Yu, S.

Y. Lai and S. Yu, “General quantum theory of nonlinear optical pulse propagation,” Phys. Rev. A 51, 817–829 (1995).
[CrossRef] [PubMed]

J. Opt. Soc. Am. B (8)

Opt. Lett. (4)

Phys. Rev. (1)

H. A. Haus and J. A. Mullen, “Quantum noise in linear amplifiers,” Phys. Rev. 128, 2407–2415 (1962).
[CrossRef]

Phys. Rev. A (5)

Y. Lai and S. Yu, “General quantum theory of nonlinear optical pulse propagation,” Phys. Rev. A 51, 817–829 (1995).
[CrossRef] [PubMed]

F. X. Kärtner and L. Boivin, “Quantum noise of the fundamental soliton,” Phys. Rev. A 53, 454–466 (1996)
[CrossRef] [PubMed]

D. Yao, “Quantum fluctuations of optical solitons with optical pulses,” Phys. Rev. A 52, 4871–4881 (1995).
[CrossRef] [PubMed]

E. Wright, “Quantum theory of soliton propagation in an optical fiber using the Hartree approximation,” Phys. Rev. A 43, 3836–3844 (1991).
[CrossRef] [PubMed]

P. Hagelstein, “Application of a photon configuration-space model to soliton propagation in a fiber,” Phys. Rev. A 54, 2426–2438 (1996).
[CrossRef] [PubMed]

Phys. Rev. Lett. (5)

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

M. Rosenbluh and R. M. Shelby, “Squeezed optical solitons,” Phys. Rev. Lett. 66, 153–156 (1991).
[CrossRef] [PubMed]

S. Spalter, N. Korolkova, F. Konig, A. Sizmann, and G. Leuchs, “Observation of multimode quantum correlations in fiber optical solitons,” Phys. Rev. Lett. 81, 786–788 (1998).
[CrossRef]

S. Schmitt, J. Ficker, M. Wolff, F. Konig, A. Sizmann, and G. Leuchs, “Photon-number squeezed solitons from an asymmetric fiber-optic Sagnac interferometer,” Phys. Rev. Lett. 81, 2446–2449 (1998).
[CrossRef]

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

Other (4)

L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Nonrelativistic Theory, translated by J. B. Sakes and J. S. Bell (Pergamon, London, 1958).

H. A. Haus and Y. Yamamoto, “Quantum circuit theory of phase-sensitive linear systems,” QE-23, 212–221 (1987).

Y. Lai and H. A. Haus, “Quantum theory of solitons in optical fibers. I. Time-dependent Hartree approximation,” Phys. Rev. A 40, 844–853 (1989); “Quantum theory of soli-tons in optical fibers. II. Exact solution,” Phys. Rev. A 40, 854–866 (1989).
[CrossRef] [PubMed]

J. M. Fini, P. L. Hagelstein, and H. A. Haus, “Agreement of stochastic soliton formalism with second-quantized and configuration-space models,” Phys. Rev. A (to be published).

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Figures (8)

Fig. 1
Fig. 1

Deformation of the uncertainty ellipse.

Fig. 2
Fig. 2

Squeezing setup; (a), (b), input and output ports, respectively.

Fig. 3
Fig. 3

Squeezing and antisqueezing (the minor and major axes of the squeezing ellipse) as functions of 2Φ.

Fig. 4
Fig. 4

Root-mean-square fluctuations as a function of the phase angle with respect to the LO: 2Φ=2, 4, 8.

Fig. 5
Fig. 5

Minimum and maximum fluctuations of the soliton alone as detected by a LO of hyperbolic secant shape. Comparison with ideal LO use.

Fig. 6
Fig. 6

Fluctuations of the soliton alone as detected by a LO of hyperbolic secant shape as a function of phase angle with respect to the LO, Φ=1.

Fig. 7
Fig. 7

Matrix elements (a) σ22(cont, cont) and (b) σ22(sol, cont) as functions of phase Φ. Note the beats.

Fig. 8
Fig. 8

Minimum fluctuations detected by a LO that is orthogonal to the continuum and by a hyperbolic secant LO with and without a continuum.

Equations (103)

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taˆ(x)=i2d2ωdβ22x2aˆ(x)+iKaˆ(x)aˆ(x)aˆ(x).
aˆ(x)=a0(x)+Δaˆ(x),
[Δaˆ(x), Δaˆ(x)]=δ(x-x).
-i ta0=C22x2a0+Ka0*a0a0,C=d2ωdβ2.
a0(t, x)=A0 expiKA022t-C2p02t+p0x+θ0×sechx-x0-Cp0tξ,
A02ξ2=C/K.
 dx|a0(t, x)|2=|A0|2 sech2x-x0-Cp0tξdx=2|A0|2ξ=n0.
-i tΔaˆ=C22x2Δaˆ+2K|a0|2Δaˆ+Ka02Δaˆ.
Δaˆ=Δaˆsol+Δaˆcont.
Δaˆsol=[Δnˆ(t)fn(x)+Δθˆ(t)fθ(x)+Δxˆ(t)fx(x)+n0Δpˆ(t)fp(x)]expi K|A0|22t.
fn(x)=12A0ξ1-xξtanh(x/ξ)sech(x/ξ),
fθ(x)=iA0 sech(x/ξ),
fx(x)=A0ξtanh(x/ξ)sech(x/ξ),
fp(x)=i 12A0ξx sech(x/ξ),
ddtΔnˆ=0,
ddtΔθˆ=12K Δnˆξ,
ddtΔxˆξ=-Cξ2Δpˆ,
ddtΔpˆ=0.
ΔA0ξΔAˆ1.
f1(x)=1ξ1-xξtanhxξsechxξ.
A0ΔθˆξΔAˆ2.
f2(x)=iξsechxξ.
[ΔXˆ, ΔPˆ]=i/2.
fX(x)=1ξtanhxξsechxξ,
fP(x)=iξxξsechxξ.
Δaˆsol=[ΔAˆ1(t)f1(x)+ΔAˆ2(t)f2(x)+ΔXˆ(t)fX(t)+ΔPˆ(t)fP(x)]expi K|A0|22t.
Re dxf¯m*(x)fn(x)=δmn
i tΔaˆ=C22x2Δaˆ+2K|a0|2Δaˆ-Ka02Δaˆ.
f¯1(x)=1ξsechxξ,
f¯2(x)=iξ1-xξtanhxξsechxξ,
f¯X(x)=1ξxξsechxξ,
f¯P(x)=iξtanhxξsechxξ.
[ΔAˆ1, ΔAˆ2]=i dxf¯1*(x) dxf¯2*(x)×[Δaˆ(1)(x), Δaˆ(2)(x)]=-½  dxf¯1*(x) dxf¯2*(x)δ(x-x)=i/2
|(ΔAˆ1)2|= dxf¯1*(x) dxf¯1(x)[Δaˆ(1)(x)Δaˆ(1)(x)]=¼  dxf¯1*(x) dxf¯1(x)δ(x-x)=¼  dx|f¯1*(x)|2=1/2.
 dx|f¯Q(x)|2=2, (1.214), π2/6, 2/3;Q=1, 2, X, P.
|(ΔAˆ1)2||(ΔAˆ2)2|=2.43/16,
|ΔXˆ2||ΔPˆ2|=1.09/16.
|ΔAˆ1ΔAˆ2+ΔAˆ2ΔAˆ1|=0.
Δn=2A0ΔA0ξ2A0ξΔAˆ1.
Δnˆ2=2A02ξ2ΔAˆ12=n.
ddtΔAˆ1=0,
ddtΔAˆ2=Cξ2ΔAˆ1,
ddtΔXˆ=Cξ2ΔPˆ,
ddtΔPˆ=0,
fc,s=c[Ω2-2iΩ tanh(x)-tanh2(x)]exp-i(Ωx+Ω2t/2)+c* sech2(x)exp(it)exp iΩx+Ω22t,
c=1forthein-phasecomponent,
c=iforthequadraturecomponent.
f¯c,s=c[Ω2-2iΩ tanh(x)-tanh2(x)]exp-i(Ωx+Ω2t/2)-c* sech2(x)exp(it)exp iΩx+Ω22t,
c¯=1(1+Ω2)forthein-phasecomponent,
c¯=i(1+Ω2)forthequadraturecomponent.
Re dxfQ(x) f¯c,s*(x, t=0)=0,Q=1, 2, X, P.
Re dxf¯c*(Ω, x, t)fc(Ω, x, t)
=Re dxf¯s*(Ω, x, t)fs(Ω, x, t)
=2πδ(Ω-Ω).
Δaˆcont=- dΩ2π[Fˆc(Ω)fc(x, Ω, t)+Fˆs(Ω)fs(x, Ω, t)],
Fˆc,s(Ω)=12- dx[Δaˆ(x)f¯c,s*(x, Ω, t)+Δaˆ(x)f¯c,s(x, Ω, t)].
ΔAˆ1(t)=ΔAˆ1(0),
ΔAˆ2(t)=ΔAˆ2(0)+2Φ(t)ΔAˆ1(0),
[ΔAˆ1(t)]2=[ΔAˆ1(0)]2,
[ΔAˆ2(t)]2=[ΔAˆ2(0)]2+4Φ2[ΔAˆ1(0)]2,
½ΔAˆ1(0)ΔAˆ2(t)+ΔAˆ2(0)ΔAˆ1(t)=2ΦΔAˆ12(0).
ΔAˆ(t)ΔAˆ1(t)+iΔAˆ2(t)=μ(t)ΔAˆ(0)+ν(t)ΔAˆ(0),
μ=1+i2Φ(t),ν=i2Φ(t).
ifL(x)=½[cos ψf¯1(x)+sin ψf¯2(x)]exp(iψ),
ΔQˆq=i  dx[fL*(x)Δaˆˆsol-fL(x)Δaˆˆsol]=[cos ψΔAˆ1(t)+sin ψΔAˆ2(t)]=½[exp(-iψ)ΔAˆ(t)+exp(iψ)ΔAˆ(t)]=½{exp(-iψ)[μΔAˆ(0)+νΔAˆ(0)]+exp(iψ)[μ*ΔAˆ(0)+ν*ΔAˆ(0)]},
|ΔQˆ2(t)|q2=cos2 ψ|ΔAˆ12(t)|+sin2 ψ|ΔAˆ22(t)|+sin(2ψ)½|ΔAˆ1(t)ΔAˆ2(t)+ΔAˆ2(t)ΔAˆ1(t)|.
|ΔAˆ12(0)|,|ΔAˆ22(0)|+|ΔAˆ12(0)|4Φ2.
½|ΔAˆ1ΔAˆ2+ΔAˆ2ΔAˆ1|=2Φ|ΔAˆ12|.
p(ξ1, ξ2, t)exp-12ξ12σ11(t)+ξ22σ22(t)+2ξ1ξ2σ12(t),
σ11(t)σ12(t)σ21(t)σ22(t)=|ΔAˆ12(t)|½|ΔAˆ1(t)ΔAˆ2(t)+ΔAˆ2(t)ΔAˆ1(t)|1/2|ΔAˆ1(t)ΔAˆ2(t)+ΔAˆ2(t)ΔAˆ1(t)||ΔA22(t)|=|ΔAˆ12(0)|12Φ(t)2Φ(t)η+4Φ2(t)
C(k1, k2, t)exp-½[σ11(t)k12+σ22(t)k22+2σ12(t)k1k2].
λ±=|ΔAˆ12(0)|1+η+4Φ22±1+η+4Φ222-η1/2.
λ+λ-=η|ΔAˆ12(0)|2
Δaˆ=Δaˆsol+Δaˆcont.
ifL(x, t)=(1/2)sechxξexp(it/2)exp(iψ)=(1/2)f¯1(x, t)exp(iψ).
ΔQˆsolq=i  dx[fL*(x, t)Δaˆsol-fL(x, t)Δaˆsol]=[cos ψΔAˆ1(t)+2 sin ψΔAˆ2(t)].
|ΔQˆ2(t)|solq2=cos2 ψ|ΔAˆ12(t)|+4 sin2 ψ|ΔAˆ22(t)|+sin(2ψ)|ΔAˆ1(t)ΔAˆ2(t)+ΔAˆ2(t)ΔAˆ1(t)|,
σ11(t)σ12(t)σ21(t)σ22(t)sol,sol
=|ΔAˆ12(0)|14Φ(t)4Φ(t)4[η+4Φ2(t)],
[Δaˆsol, Δaˆcont]+[Δaˆcont, Δaˆsol]0.
σ22(cont, cont)=π8 dΩ sech2π2Ωcos2[(1+Ω2)t/2],
σ22(sol, cont)=14 dΩ 1(1+Ω2)23π+π26Ω tanhπ2Ω+π2Ω36tanhπ2Ωsech2π2Ω×cos[(1+Ω2)t/2].
ΔQˆcontq=-12 dx sech(x)c,s  dΩ2π×Fˆc,s[exp(-iψ-it/2)fc,s(x, Ω, t)+exp(iψ+it/2)fc,s*(x, Ω, t)]=-12cos ψ c,s  dΩ2πFˆc,s  dx[f¯1*(x, t)×fc,s(x, Ω, t)+f¯1(x, t)fc,s*(x, Ω, t)]+i2sin ψ c,s  dΩ2πFˆc,s  dx[f¯1*(x, t)×fc,s(x, Ω, t)-f¯1(x, t)fc,s*(x, Ω, t)].
ΔQˆcontq= dΩ2π[cos ψFˆc(Ω)Ic(Ω, t)+sin ψFˆs(Ω)Is(Ω, t)]
Ic(Ω, t)=Re dx[f¯1(x, t)fc*(x, Ω, t)],
Is(Ω, t)=Im dx[f¯1(x, t)fs*(x, Ω, t)].
Ic(Ω, t)=0,
Is(Ω, t)=π(1+Ω2)sechπ2Ωcos[(1+Ω2)t/2].
|Fˆc(Ω)Fˆc(Ω)|
=¼ dx[Δaˆ(x)f¯c*(x, Ω, 0)+Δaˆˆ(x)f¯c(x, Ω, 0)]× dx[Δaˆ(x)f¯c*(x, Ω, 0)+Δaˆ(x)f¯c(x, Ω, 0)]
=¼  dx  dx|Δaˆˆ(x)Δaˆ(x)|×f¯c*(x, Ω, 0)f¯c(x, Ω, 0)=¼ dxf¯c*(x, Ω, 0)f¯c(x, Ω, 0)=142π[1+Ω2]2δ(Ω-Ω).
|Fˆs(Ω)Fˆs(Ω)|=¼  dxf¯s*(x, Ω, 0)f¯s(x, Ω, 0)=142π[1+Ω2]2δ(Ω-Ω).
½|Fˆc(Ω)Fˆs(Ω)+Fˆs(Ω)Fˆc(Ω)|= dx[Δaˆ(x)f¯jc(x, Ω, 0)+Δaˆ(x)f¯c(x, Ω, 0)] dx[Δaˆ(x)f¯s*(x, Ω, 0)+Δaˆ(x)f¯s(x, Ω, 0)]+h.c.=  dx  dx|Δaˆ(x)Δaˆ(x)|×f¯c*(x, Ω, 0)f¯s(x, Ω, 0)+h.c.= dxf¯c*(x, Ω, 0)f¯s(x, Ω, 0)+ dxf¯s*(x, Ω, 0)f¯c(x, Ω, 0).
σ22(cont, cont)= dΩ2πFˆs(Ω)Is(Ω, t)× dΩ2πFˆs(Ω)Is(Ω, t)=1412π dΩ Is2(Ω, t)(1+Ω2)2=π8 dΩ sech2π2Ωcos2[(1+Ω2)t/2].
σ22(sol, cont)=2ΔAˆ2(t)= dΩ2πFˆs(Ω)Is(Ω, t).
|ΔAˆ1(0)Fˆs(Ω)|
=¼ dx[Δaˆ(x)f¯1*(x)+Δaˆ(x)f¯1(x)] dx[Δaˆ(x)f¯*(x, Ω, 0)+Δaˆ(x)f¯s(x, Ω, 0)]=¼  dx  dx|Δaˆ(x)Δaˆ(x)|f¯1*(x)f¯s(x, Ω, 0)=¼  dx  dxδ(x-x)f¯1*(x)f¯s(x, Ω, 0)=¼  dxf¯1*(x)f¯s(x, Ω, 0),
½|ΔAˆ1(0)Fˆs(Ω)+Fˆs(Ω)ΔAˆ1(0)|
= dxf¯1*(x)f¯s(x, Ω, 0)+ dxf¯1(x)f¯s*(x, Ω, 0)C1s(Ω).
½|ΔAˆ2(0)Fˆs(Ω)+Fˆs(Ω)ΔAˆ2(0)|
= dxf¯2*(x)f¯s(x, Ω, 0)+ dxf¯2(x)f¯s*(x, Ω, 0)C2s(Ω).
C2s=14(1+Ω2)223π+π26Ω tanhπ2Ω+π26Ω3 tanhπ2Ωsechπ2Ω,
σ22(sol, cont)=2  dΩ2π[C2s(Ω)+2Φ(t)C1s(Ω)]Is(Ω, t)=2  dΩ2πC2s(Ω)Is(Ω, t)=14 dΩ 1(1+Ω2)23π+π26Ω tanhπ2Ω+π2Ω36tanhπ2Ωsech2π2Ω×cos(1+Ω2)t2.

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