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] [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]
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    [CrossRef]
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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]

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

M. J. Werner, “Raman-induced photon correlations in optical fiber solitons,” Phys. Rev. A 60, R781–R784 (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 (1)

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

1995 (1)

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

1994 (1)

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 (2)

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]

1990 (2)

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

1989 (1)

1987 (3)

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

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]

1986 (1)

1985 (1)

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

1984 (1)

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

1972 (1)

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

Bayer, P. W.

R. M. Shelby, M. D. Levenson, and P. W. Bayer, “Guidedacoustic-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]

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).
[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]

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]

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 M. G. Raymer, “Quantum theory of propagation of nonclassical radiation in a near-resonant medium,” Phys. Rev. A 44, 2072–2085 (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]

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]

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]

Gordon, J. P.

Haus, H. A.

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

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]

J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. 11, 665–667 (1986).
[CrossRef] [PubMed]

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.

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]

Lakoba, T. I.

Levenson, M. D.

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

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]

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]

Potasek, M. J.

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]

Shelby, R. M.

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]

R. M. Shelby, M. D. Levenson, and P. W. Bayer, “Guidedacoustic-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, 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]

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]

Wong, W. S.

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

Yurke, B.

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]

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 (4)

Opt. Lett. (2)

Phys. Rev. A (7)

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

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

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

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

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

Phys. Rev. B (1)

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

Phys. Rev. Lett. (2)

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

Rev. Mod. Phys. (2)

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

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

Other (18)

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]

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]

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]

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]

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]

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

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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)

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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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