Abstract

Making use of c-number stochastic theory and soliton perturbation theory, we study the quantum fluctuations of a self-induced transparency (SIT) soliton propagating through a lossless two-level medium. Considering the fluctuations as small corrections to the classical soliton, we are able to construct and solve four stochastic equations that govern the evolution of four soliton parameters: photon number (intensity), phase, timing, and momentum (frequency). We find excellent agreement between our stochastic theory of SIT solitons and the second-quantized theory of Lai and Haus [Phys. Rev. A 42, 2925 (1990)].

© 2000 Optical Society of America

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  1. S. L. McCall and E. L. Hahn, “Self-induced transparency by pulsed coherent light,” Phys. Rev. Lett. 18, 908–911 (1967); “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969); G. L. Lamb, Jr., “Analytical descriptions of ultrashort optical pulse propagation in a resonant medium,” Rev. Mod. Phys. RMPHAT 43, 99–124 (1971); L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, New York, 1987); A. I. Maimistov, A. M. Basharov, S. O. Elyutin, and Yu. M. Sklyarov, “Present state of self-induced transparency theory,” Phys. Rep. PRPLCM 191, 1–108 (1990).
    [CrossRef]
  2. H. A. Haus, “Physical interpretation of inverse scattering formalism applied to self-induced transparency,” Rev. Mod. Phys. 51, 331–339 (1979).
    [CrossRef]
  3. Y. Lai and H. A. Haus, “Quantum theory of self-induced transparency solitons—a linearized approach,” Phys. Rev. A 42, 2925–2934 (1990).
    [CrossRef] [PubMed]
  4. K. Watanabe, H. Nakano, A. Honold, and Y. Yamamoto, “Optical nonlinearities of excitonic self-induced transparency solitons—toward ultimate realization of squeezed states and quantum nondemolition measurements,” Phys. Rev. Lett. 62, 2257–2260 (1989).
    [CrossRef] [PubMed]
  5. A. B. Matsko, V. V. Kozlov, and M. O. Scully, “Backaction cancellation in quantum nondemolition measurement of optical solitons,” Phys. Rev. Lett. 82, 3244–3247 (1999).
    [CrossRef]
  6. K. P. Komarov and V. D. Ugozhaev, “Steady state 2π pulses under passive laser mode-locking,” Sov. J. Quantum Electron. 14, 787–792 (1984).
    [CrossRef]
  7. V. V. Kozlov and E. E. Fradkin, “Theory of mode synchronization with coherent absorber—generation of soliton-like 2π pulses,” Sov. Phys. JETP 80, 32–40 (1995).
  8. V. P. Kalosha, M. Müller, and J. Herrmann, “Coherent-absorber mode locking of solid-state lasers,” Opt. Lett. 23, 117–119 (1998).
    [CrossRef]
  9. V. V. Kozlov, “Self-induced transparency soliton laser via coherent mode locking,” Phys. Rev. A 56, 1607–1612 (1997).
    [CrossRef]
  10. B. Huttner and S. M. Barnett, “Quantization of the electromagnetic field in dielectric,” Phys. Rev. A 46, 4306–4322 (1992).
    [CrossRef] [PubMed]
  11. M. Hillery and L. Mlodinow, “Quantized fields in a nonlinear dielectric medium: a microscopic approach,” Phys. Rev. A 55, 678–689 (1997).
    [CrossRef]
  12. L.-M. Duan and G.-C. Guo, “Alternative approach to electromagnetic field quantization in nonlinear and inhomogeneous media,” Phys. Rev. A 56, 925–930 (1997); E. Schmidt, J. Jeffers, S. M. Barnett, L. Knoll, and D. G. Welsch, “Quantum theory of light in nonlinear media with dispersion and absorption,” J. Mod. Opt. 45, 377–401 (1998); also see references therein.
    [CrossRef]
  13. P. D. Drummond, C. W. Gardiner, and D. F. Walls, “Quasiprobability methods for nonlinear chemical and optical systems,” Phys. Rev. A 24, 914–926 (1981).
    [CrossRef]
  14. P. D. Drummond and S. J. Carter, “Quantum-field theory of squeezing in solitons,” J. Opt. Soc. Am. B 4, 1565–1573 (1987).
    [CrossRef]
  15. C. W. Gardiner, Quantum Noise (Springer, Berlin, 1991).
  16. 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); for three-level systems see M. Fleischhauer and T. Richter, “Pulse matching and correlation of phase fluctuations in Λ-systems,” Phys. Rev. A 51, 2430–2442 (1995).
    [CrossRef] [PubMed]
  17. 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 57, 4842–4853 (1998).
    [CrossRef]
  18. H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing—a linearized approach,” J. Opt. Soc. Am. B 7, 386–392 (1990).
    [CrossRef]
  19. Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989), and references therein.
    [CrossRef]

1999

A. B. Matsko, V. V. Kozlov, and M. O. Scully, “Backaction cancellation in quantum nondemolition measurement of optical solitons,” Phys. Rev. Lett. 82, 3244–3247 (1999).
[CrossRef]

1998

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 57, 4842–4853 (1998).
[CrossRef]

V. P. Kalosha, M. Müller, and J. Herrmann, “Coherent-absorber mode locking of solid-state lasers,” Opt. Lett. 23, 117–119 (1998).
[CrossRef]

1997

M. Hillery and L. Mlodinow, “Quantized fields in a nonlinear dielectric medium: a microscopic approach,” Phys. Rev. A 55, 678–689 (1997).
[CrossRef]

V. V. Kozlov, “Self-induced transparency soliton laser via coherent mode locking,” Phys. Rev. A 56, 1607–1612 (1997).
[CrossRef]

1995

V. V. Kozlov and E. E. Fradkin, “Theory of mode synchronization with coherent absorber—generation of soliton-like 2π pulses,” Sov. Phys. JETP 80, 32–40 (1995).

1992

B. Huttner and S. M. Barnett, “Quantization of the electromagnetic field in dielectric,” Phys. Rev. A 46, 4306–4322 (1992).
[CrossRef] [PubMed]

1990

Y. Lai and H. A. Haus, “Quantum theory of self-induced transparency solitons—a linearized approach,” Phys. Rev. A 42, 2925–2934 (1990).
[CrossRef] [PubMed]

H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing—a linearized approach,” J. Opt. Soc. Am. B 7, 386–392 (1990).
[CrossRef]

1989

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989), and references therein.
[CrossRef]

K. Watanabe, H. Nakano, A. Honold, and Y. Yamamoto, “Optical nonlinearities of excitonic self-induced transparency solitons—toward ultimate realization of squeezed states and quantum nondemolition measurements,” Phys. Rev. Lett. 62, 2257–2260 (1989).
[CrossRef] [PubMed]

1987

1984

K. P. Komarov and V. D. Ugozhaev, “Steady state 2π pulses under passive laser mode-locking,” Sov. J. Quantum Electron. 14, 787–792 (1984).
[CrossRef]

1981

P. D. Drummond, C. W. Gardiner, and D. F. Walls, “Quasiprobability methods for nonlinear chemical and optical systems,” Phys. Rev. A 24, 914–926 (1981).
[CrossRef]

1979

H. A. Haus, “Physical interpretation of inverse scattering formalism applied to self-induced transparency,” Rev. Mod. Phys. 51, 331–339 (1979).
[CrossRef]

Barnett, S. M.

B. Huttner and S. M. Barnett, “Quantization of the electromagnetic field in dielectric,” Phys. Rev. A 46, 4306–4322 (1992).
[CrossRef] [PubMed]

Carter, S. J.

Drummond, P. D.

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, C. W. Gardiner, and D. F. Walls, “Quasiprobability methods for nonlinear chemical and optical systems,” Phys. Rev. A 24, 914–926 (1981).
[CrossRef]

Fini, J. M.

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 57, 4842–4853 (1998).
[CrossRef]

Fradkin, E. E.

V. V. Kozlov and E. E. Fradkin, “Theory of mode synchronization with coherent absorber—generation of soliton-like 2π pulses,” Sov. Phys. JETP 80, 32–40 (1995).

Gardiner, C. W.

P. D. Drummond, C. W. Gardiner, and D. F. Walls, “Quasiprobability methods for nonlinear chemical and optical systems,” Phys. Rev. A 24, 914–926 (1981).
[CrossRef]

Hagelstein, P. L.

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 57, 4842–4853 (1998).
[CrossRef]

Haus, H. A.

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 57, 4842–4853 (1998).
[CrossRef]

Y. Lai and H. A. Haus, “Quantum theory of self-induced transparency solitons—a linearized approach,” Phys. Rev. A 42, 2925–2934 (1990).
[CrossRef] [PubMed]

H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing—a linearized approach,” J. Opt. Soc. Am. B 7, 386–392 (1990).
[CrossRef]

H. A. Haus, “Physical interpretation of inverse scattering formalism applied to self-induced transparency,” Rev. Mod. Phys. 51, 331–339 (1979).
[CrossRef]

Herrmann, J.

Hillery, M.

M. Hillery and L. Mlodinow, “Quantized fields in a nonlinear dielectric medium: a microscopic approach,” Phys. Rev. A 55, 678–689 (1997).
[CrossRef]

Honold, A.

K. Watanabe, H. Nakano, A. Honold, and Y. Yamamoto, “Optical nonlinearities of excitonic self-induced transparency solitons—toward ultimate realization of squeezed states and quantum nondemolition measurements,” Phys. Rev. Lett. 62, 2257–2260 (1989).
[CrossRef] [PubMed]

Huttner, B.

B. Huttner and S. M. Barnett, “Quantization of the electromagnetic field in dielectric,” Phys. Rev. A 46, 4306–4322 (1992).
[CrossRef] [PubMed]

Kalosha, V. P.

Kivshar, Y. S.

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989), and references therein.
[CrossRef]

Komarov, K. P.

K. P. Komarov and V. D. Ugozhaev, “Steady state 2π pulses under passive laser mode-locking,” Sov. J. Quantum Electron. 14, 787–792 (1984).
[CrossRef]

Kozlov, V. V.

A. B. Matsko, V. V. Kozlov, and M. O. Scully, “Backaction cancellation in quantum nondemolition measurement of optical solitons,” Phys. Rev. Lett. 82, 3244–3247 (1999).
[CrossRef]

V. V. Kozlov, “Self-induced transparency soliton laser via coherent mode locking,” Phys. Rev. A 56, 1607–1612 (1997).
[CrossRef]

V. V. Kozlov and E. E. Fradkin, “Theory of mode synchronization with coherent absorber—generation of soliton-like 2π pulses,” Sov. Phys. JETP 80, 32–40 (1995).

Lai, Y.

Y. Lai and H. A. Haus, “Quantum theory of self-induced transparency solitons—a linearized approach,” Phys. Rev. A 42, 2925–2934 (1990).
[CrossRef] [PubMed]

H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing—a linearized approach,” J. Opt. Soc. Am. B 7, 386–392 (1990).
[CrossRef]

Malomed, B. A.

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989), and references therein.
[CrossRef]

Matsko, A. B.

A. B. Matsko, V. V. Kozlov, and M. O. Scully, “Backaction cancellation in quantum nondemolition measurement of optical solitons,” Phys. Rev. Lett. 82, 3244–3247 (1999).
[CrossRef]

Mlodinow, L.

M. Hillery and L. Mlodinow, “Quantized fields in a nonlinear dielectric medium: a microscopic approach,” Phys. Rev. A 55, 678–689 (1997).
[CrossRef]

Müller, M.

Nakano, H.

K. Watanabe, H. Nakano, A. Honold, and Y. Yamamoto, “Optical nonlinearities of excitonic self-induced transparency solitons—toward ultimate realization of squeezed states and quantum nondemolition measurements,” Phys. Rev. Lett. 62, 2257–2260 (1989).
[CrossRef] [PubMed]

Scully, M. O.

A. B. Matsko, V. V. Kozlov, and M. O. Scully, “Backaction cancellation in quantum nondemolition measurement of optical solitons,” Phys. Rev. Lett. 82, 3244–3247 (1999).
[CrossRef]

Ugozhaev, V. D.

K. P. Komarov and V. D. Ugozhaev, “Steady state 2π pulses under passive laser mode-locking,” Sov. J. Quantum Electron. 14, 787–792 (1984).
[CrossRef]

Walls, D. F.

P. D. Drummond, C. W. Gardiner, and D. F. Walls, “Quasiprobability methods for nonlinear chemical and optical systems,” Phys. Rev. A 24, 914–926 (1981).
[CrossRef]

Watanabe, K.

K. Watanabe, H. Nakano, A. Honold, and Y. Yamamoto, “Optical nonlinearities of excitonic self-induced transparency solitons—toward ultimate realization of squeezed states and quantum nondemolition measurements,” Phys. Rev. Lett. 62, 2257–2260 (1989).
[CrossRef] [PubMed]

Yamamoto, Y.

K. Watanabe, H. Nakano, A. Honold, and Y. Yamamoto, “Optical nonlinearities of excitonic self-induced transparency solitons—toward ultimate realization of squeezed states and quantum nondemolition measurements,” Phys. Rev. Lett. 62, 2257–2260 (1989).
[CrossRef] [PubMed]

J. Opt. Soc. Am. B

Opt. Lett.

Phys. Rev. A

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 57, 4842–4853 (1998).
[CrossRef]

P. D. Drummond, C. W. Gardiner, and D. F. Walls, “Quasiprobability methods for nonlinear chemical and optical systems,” Phys. Rev. A 24, 914–926 (1981).
[CrossRef]

Y. Lai and H. A. Haus, “Quantum theory of self-induced transparency solitons—a linearized approach,” Phys. Rev. A 42, 2925–2934 (1990).
[CrossRef] [PubMed]

V. V. Kozlov, “Self-induced transparency soliton laser via coherent mode locking,” Phys. Rev. A 56, 1607–1612 (1997).
[CrossRef]

B. Huttner and S. M. Barnett, “Quantization of the electromagnetic field in dielectric,” Phys. Rev. A 46, 4306–4322 (1992).
[CrossRef] [PubMed]

M. Hillery and L. Mlodinow, “Quantized fields in a nonlinear dielectric medium: a microscopic approach,” Phys. Rev. A 55, 678–689 (1997).
[CrossRef]

Phys. Rev. Lett.

K. Watanabe, H. Nakano, A. Honold, and Y. Yamamoto, “Optical nonlinearities of excitonic self-induced transparency solitons—toward ultimate realization of squeezed states and quantum nondemolition measurements,” Phys. Rev. Lett. 62, 2257–2260 (1989).
[CrossRef] [PubMed]

A. B. Matsko, V. V. Kozlov, and M. O. Scully, “Backaction cancellation in quantum nondemolition measurement of optical solitons,” Phys. Rev. Lett. 82, 3244–3247 (1999).
[CrossRef]

Rev. Mod. Phys.

H. A. Haus, “Physical interpretation of inverse scattering formalism applied to self-induced transparency,” Rev. Mod. Phys. 51, 331–339 (1979).
[CrossRef]

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989), and references therein.
[CrossRef]

Sov. J. Quantum Electron.

K. P. Komarov and V. D. Ugozhaev, “Steady state 2π pulses under passive laser mode-locking,” Sov. J. Quantum Electron. 14, 787–792 (1984).
[CrossRef]

Sov. Phys. JETP

V. V. Kozlov and E. E. Fradkin, “Theory of mode synchronization with coherent absorber—generation of soliton-like 2π pulses,” Sov. Phys. JETP 80, 32–40 (1995).

Other

L.-M. Duan and G.-C. Guo, “Alternative approach to electromagnetic field quantization in nonlinear and inhomogeneous media,” Phys. Rev. A 56, 925–930 (1997); E. Schmidt, J. Jeffers, S. M. Barnett, L. Knoll, and D. G. Welsch, “Quantum theory of light in nonlinear media with dispersion and absorption,” J. Mod. Opt. 45, 377–401 (1998); also see references therein.
[CrossRef]

S. L. McCall and E. L. Hahn, “Self-induced transparency by pulsed coherent light,” Phys. Rev. Lett. 18, 908–911 (1967); “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969); G. L. Lamb, Jr., “Analytical descriptions of ultrashort optical pulse propagation in a resonant medium,” Rev. Mod. Phys. RMPHAT 43, 99–124 (1971); L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, New York, 1987); A. I. Maimistov, A. M. Basharov, S. O. Elyutin, and Yu. M. Sklyarov, “Present state of self-induced transparency theory,” Phys. Rep. PRPLCM 191, 1–108 (1990).
[CrossRef]

C. W. Gardiner, Quantum Noise (Springer, Berlin, 1991).

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); for three-level systems see M. Fleischhauer and T. Richter, “Pulse matching and correlation of phase fluctuations in Λ-systems,” Phys. Rev. A 51, 2430–2442 (1995).
[CrossRef] [PubMed]

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Equations (73)

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σˆ12(z, t)=1NVlimM(2M+1)lσˆ12l(t)|zkz,
σˆ22(z, t)-σˆ11(z, t)
=1NVlimM(2M+1)×l[σˆ22l(t)-σˆ11l(t)]|zkz,
σˆ12l=|12|l,σˆ11l=|11|l,σˆ22l=|22|l.
Eˆ, σˆ21, σˆ22, σˆ11, σˆ12,Eˆ,
z+1c-1v¯g ξE=κP,
ξP=12EN+fP,
ξN=-(P*E+PE*)+fN,
fμ(z, ξ) fν(z, ξ)=δ(z-z)δ(ξ-ξ)Dμν(z, ξ).
DNN=2SNN¯ξ,DPP=[DP*P*]*=1SNE¯P¯,
E¯=2A¯cosh-1[A¯(ξ-τ¯)]exp(iδ¯ξ+iϕ¯).
ϕ¯=δ¯A¯2+δ¯2ζ.
1v¯g=1c+κ/2A¯2+δ¯2.
P¯=A¯2A¯2+δ¯2tanh[A¯(ξ-τ¯)]+i δ¯A¯ exp(iδ¯ξ+iϕ¯)cosh[A¯(ξ-τ¯)],
N¯=-1+2A¯2A¯2+δ¯21cosh2[A¯(ξ-τ¯)].
I1-+E* Pξ-E P*ξdξ,
I2-+ξE* Pξ-E P*ξdξ,
I3-+ |E|2ζdξ+[N(+)-N(-)],
I4-+ξ|E|2ζ-α |E|2ξ+Nξdξ,
I1=-+(E*fP-EfP*)dξ,
I2=-+ξ(E*fP-EfP*)dξ,
I3=-+fN dξ,
I4=-+ξ fN dξ.
E=E¯(ξ, ζ)+[ΔE(1)(ξ, ζ)+iΔE(2)(ξ, ζ)].
Δδζ=Fδ,
Δϕζ=1A¯2Δδ+Fϕ,
ΔAζ=FA,
Δτζ=-1A¯3Δ A+Fτ
Δδ-+Pδ ΔE(2)dξ,Δϕ-+PϕΔE(2)dξ,
ΔA-+PAΔE(1)dξ,Δτ-+PτΔE(1)dξ.
Pδ=-14A¯E¯ξ,Pϕ=14A¯(ξE¯)ξ,
PA=14E¯,Pτ=14A¯(ξE¯).
Δδ2=Δδ20,
Δϕ2=Δϕ20+13βA¯3ζ2,
ΔA2=ΔA20,
Δτ2=Δτ20+1βA¯5ζ2.
n0=1ω0Sc2π-+E*Ed t=1ω0Sc2πd2A¯=βA¯.
ΔEˆ=EˆA¯ΔAˆ+E¯δ¯Δδˆ+E¯ϕ¯Δϕˆ+E¯τ¯Δτˆ,
E¯A¯=1A¯(ξE¯)ξ,E¯δ¯=i(ξE¯),
E¯ϕ¯=iE¯,E¯τ¯=-E¯ξ.
Δδˆζ=0,
Δϕˆζ=1A¯2Δδˆ,
ΔAˆζ=0,
Δτˆζ=-1A¯3ΔAˆ.
[ΔEˆ(ζ, ξ), ΔEˆ(ζ, ξ)]=8βδ(ξ-ξ),
[ΔEˆ(ζ, ξ), ΔEˆ(ζ, ξ)]=[ΔEˆ(ζ, ξ), ΔEˆ(ζ, ξ)]=0.
[ΔAˆ, Δϕˆ]=i 1β,[Δδˆ, Δτˆ]=i 1βA¯.
Δδˆ2=Δδˆ20,
Δϕˆ2=Δϕˆ20+1A¯4Δδˆ20ζ2,
ΔAˆ2=ΔAˆ20,
Δτˆ2=Δτˆ20+1A¯6ΔAˆ20ζ2.
ΔLˆ20=2A¯n0-+|PL|2dξ,
Δδˆ20=13A¯β,Δϕˆ20=131+π212 1βA¯,
Δ Aˆ20=A¯β,Δτˆ20=π2121βA¯3.
P=P¯(ξ, ζ)+[ΔP(1)(ξ, ζ)+iΔP(2)(ξ, ζ)],
N=N¯(ξ, ζ)+ΔN(ξ, ζ).
ΔP(1)(ξ, ζ)=ΔEζ(1)-αΔE˙(1),
ΔP(2)(ξ, ζ)=ΔEζ(2)-αΔE˙(2),
ΔN˙=-2[E¯ΔP(1)+P¯ΔE(1)]+fN.
12ξ(|2P|2+N2)=NfN+2(P*fP+PfP*).
ΔP(1)=-N¯4P¯ΔN+14P¯-ξ[N¯fN+2(P¯*fP+P¯fP*)]dξ.
ΔN=-2P¯-ξΔE(1)dξ+P¯-ξfNP¯-E¯P¯2×-ξ(N¯fN+2P¯*fP+2P¯fP*)dξdξ.
8iA¯ζ-+PδΔE(2)dξ=8iA¯Fδ,
-8iA¯ζ-+PϕΔE(2)dξ+8iA¯-+PδΔE(2)dξ
=-8iA¯Fϕ,
8 ζ-+PAΔE(1)dξ=8FA,
8A¯ζ-+PτΔE(1)dξ+8A¯2-+PAΔE(1)dξ=8A¯Fτ,
Fδ=-i8A¯-+(E¯*fP-E¯fP*)dξ,
Fϕ=i8A¯-+ξ(E¯*fP-E¯fP*)dξ,
FA=18-+fNdξ,
Fτ=18A¯-+ξ+α E¯P¯fN-α E¯2P¯2-ξ12N¯fN+P¯*fP+P¯fP*dξdξ.
Fμ(ζ)Fν(ζ)=δ(ζ-ζ)Dμν(ζ).
Dδϕ=κSN112 A¯,DAτ=-κSN14 A¯2.

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