Abstract

It is shown theoretically that when the light produced by spontaneous parametric downconversion is mixed with a stationary, narrow-band local oscillator field centered on the mid-frequency, the resulting optical field exhibits photon antibunching in the steady state. The treatment is based on the integration of the coupled Heisenberg equations of motion for the field modes.

© 1997 Optical Society of America

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References

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  1. D. Stoler, “Equivalence classes of minimum-uncertainty packets,” Phys. Rev. D 1, 3217–3219 (1970); “Equivalence classes of minimum-uncertainty packets. II,” 4, 1925–1926 (1974).
    [CrossRef]
  2. H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13, 2226–2243 (1976).
    [CrossRef]
  3. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
    [CrossRef]
  4. L. Mandel, “Squeezed states and sub-Poissonian photon statistics,” Phys. Rev. Lett. 49, 136–138 (1982).
    [CrossRef]
  5. Z. Y. Ou, C. K. Hong, and L. Mandel, “Coherence properties of squeezed light and the degree of squeezing,” J. Opt. Soc. Am. B 4, 1574–1587 (1987).
    [CrossRef]
  6. H. P. Yuen and J. H. Shapiro, “Optical communications with two-photon coherent states. III. Quantummeasurements realizable with photoemissive detectors,” IEEE Trans. Inf. Theory IT-26, 78–92 (1980).
    [CrossRef]
  7. H. P. Yuen and V. W. S. Chan, “Noise in homodyne and heterodyne detection,” Opt. Lett. 8, 177–179 (1983).
    [CrossRef] [PubMed]
  8. Z. Y. Ou, C. H. Hong, and L. Mandel, “Detection of squeezed states by cross correlation,” Phys. Rev. A 36, 192–196 (1987).
    [CrossRef] [PubMed]
  9. X. Y. Zou and L. Mandel, “Photon-antibunching and sub-Poissonian photon statistics,” Phys. Rev. A 41, 475–476 (1990).
    [CrossRef] [PubMed]
  10. M. Koashi, M. Matsuoka, and T. Hirano, “Photon antibunching by destructive two-photon interference,” Phys. Rev. A 53, 3621–3624 (1996).
    [CrossRef] [PubMed]
  11. R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in anoptical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
    [CrossRef] [PubMed]
  12. L. A. Wu, H. J. Kimble, J. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520–2523 (1986).
    [CrossRef] [PubMed]
  13. R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 13, 2529–2539 (1963).
    [CrossRef]
  14. Z. Y. Ou, L. J. Wang, and L. Mandel, “Vacuum effects on interference in two-photon down conversion,” Phys. Rev. A 40, 1428–1435 (1989).
    [CrossRef] [PubMed]
  15. L. J. Wang, X. Y. Zou, and L. Mandel, “Time-varying induced coherence,” J. Opt. Soc. Am. B 9, 605–609 (1992).
    [CrossRef]
  16. R. Boyd, Nonlinear Optics (Academic, Boston, 1992).

1996 (1)

M. Koashi, M. Matsuoka, and T. Hirano, “Photon antibunching by destructive two-photon interference,” Phys. Rev. A 53, 3621–3624 (1996).
[CrossRef] [PubMed]

1992 (1)

1990 (1)

X. Y. Zou and L. Mandel, “Photon-antibunching and sub-Poissonian photon statistics,” Phys. Rev. A 41, 475–476 (1990).
[CrossRef] [PubMed]

1989 (1)

Z. Y. Ou, L. J. Wang, and L. Mandel, “Vacuum effects on interference in two-photon down conversion,” Phys. Rev. A 40, 1428–1435 (1989).
[CrossRef] [PubMed]

1987 (2)

Z. Y. Ou, C. H. Hong, and L. Mandel, “Detection of squeezed states by cross correlation,” Phys. Rev. A 36, 192–196 (1987).
[CrossRef] [PubMed]

Z. Y. Ou, C. K. Hong, and L. Mandel, “Coherence properties of squeezed light and the degree of squeezing,” J. Opt. Soc. Am. B 4, 1574–1587 (1987).
[CrossRef]

1986 (1)

L. A. Wu, H. J. Kimble, J. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520–2523 (1986).
[CrossRef] [PubMed]

1985 (1)

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in anoptical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[CrossRef] [PubMed]

1983 (1)

1982 (1)

L. Mandel, “Squeezed states and sub-Poissonian photon statistics,” Phys. Rev. Lett. 49, 136–138 (1982).
[CrossRef]

1981 (1)

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[CrossRef]

1980 (1)

H. P. Yuen and J. H. Shapiro, “Optical communications with two-photon coherent states. III. Quantummeasurements realizable with photoemissive detectors,” IEEE Trans. Inf. Theory IT-26, 78–92 (1980).
[CrossRef]

1976 (1)

H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13, 2226–2243 (1976).
[CrossRef]

1963 (1)

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 13, 2529–2539 (1963).
[CrossRef]

Caves, C. M.

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[CrossRef]

Chan, V. W. S.

Glauber, R. J.

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 13, 2529–2539 (1963).
[CrossRef]

Hall, J.

L. A. Wu, H. J. Kimble, J. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520–2523 (1986).
[CrossRef] [PubMed]

Hirano, T.

M. Koashi, M. Matsuoka, and T. Hirano, “Photon antibunching by destructive two-photon interference,” Phys. Rev. A 53, 3621–3624 (1996).
[CrossRef] [PubMed]

Hollberg, L. W.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in anoptical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[CrossRef] [PubMed]

Hong, C. H.

Z. Y. Ou, C. H. Hong, and L. Mandel, “Detection of squeezed states by cross correlation,” Phys. Rev. A 36, 192–196 (1987).
[CrossRef] [PubMed]

Hong, C. K.

Kimble, H. J.

L. A. Wu, H. J. Kimble, J. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520–2523 (1986).
[CrossRef] [PubMed]

Koashi, M.

M. Koashi, M. Matsuoka, and T. Hirano, “Photon antibunching by destructive two-photon interference,” Phys. Rev. A 53, 3621–3624 (1996).
[CrossRef] [PubMed]

Mandel, L.

L. J. Wang, X. Y. Zou, and L. Mandel, “Time-varying induced coherence,” J. Opt. Soc. Am. B 9, 605–609 (1992).
[CrossRef]

X. Y. Zou and L. Mandel, “Photon-antibunching and sub-Poissonian photon statistics,” Phys. Rev. A 41, 475–476 (1990).
[CrossRef] [PubMed]

Z. Y. Ou, L. J. Wang, and L. Mandel, “Vacuum effects on interference in two-photon down conversion,” Phys. Rev. A 40, 1428–1435 (1989).
[CrossRef] [PubMed]

Z. Y. Ou, C. H. Hong, and L. Mandel, “Detection of squeezed states by cross correlation,” Phys. Rev. A 36, 192–196 (1987).
[CrossRef] [PubMed]

Z. Y. Ou, C. K. Hong, and L. Mandel, “Coherence properties of squeezed light and the degree of squeezing,” J. Opt. Soc. Am. B 4, 1574–1587 (1987).
[CrossRef]

L. Mandel, “Squeezed states and sub-Poissonian photon statistics,” Phys. Rev. Lett. 49, 136–138 (1982).
[CrossRef]

Matsuoka, M.

M. Koashi, M. Matsuoka, and T. Hirano, “Photon antibunching by destructive two-photon interference,” Phys. Rev. A 53, 3621–3624 (1996).
[CrossRef] [PubMed]

Mertz, J. C.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in anoptical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[CrossRef] [PubMed]

Ou, Z. Y.

Z. Y. Ou, L. J. Wang, and L. Mandel, “Vacuum effects on interference in two-photon down conversion,” Phys. Rev. A 40, 1428–1435 (1989).
[CrossRef] [PubMed]

Z. Y. Ou, C. H. Hong, and L. Mandel, “Detection of squeezed states by cross correlation,” Phys. Rev. A 36, 192–196 (1987).
[CrossRef] [PubMed]

Z. Y. Ou, C. K. Hong, and L. Mandel, “Coherence properties of squeezed light and the degree of squeezing,” J. Opt. Soc. Am. B 4, 1574–1587 (1987).
[CrossRef]

Shapiro, J. H.

H. P. Yuen and J. H. Shapiro, “Optical communications with two-photon coherent states. III. Quantummeasurements realizable with photoemissive detectors,” IEEE Trans. Inf. Theory IT-26, 78–92 (1980).
[CrossRef]

Slusher, R. E.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in anoptical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[CrossRef] [PubMed]

Valley, J. F.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in anoptical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[CrossRef] [PubMed]

Wang, L. J.

L. J. Wang, X. Y. Zou, and L. Mandel, “Time-varying induced coherence,” J. Opt. Soc. Am. B 9, 605–609 (1992).
[CrossRef]

Z. Y. Ou, L. J. Wang, and L. Mandel, “Vacuum effects on interference in two-photon down conversion,” Phys. Rev. A 40, 1428–1435 (1989).
[CrossRef] [PubMed]

Wu, H.

L. A. Wu, H. J. Kimble, J. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520–2523 (1986).
[CrossRef] [PubMed]

Wu, L. A.

L. A. Wu, H. J. Kimble, J. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520–2523 (1986).
[CrossRef] [PubMed]

Yuen, H. P.

H. P. Yuen and V. W. S. Chan, “Noise in homodyne and heterodyne detection,” Opt. Lett. 8, 177–179 (1983).
[CrossRef] [PubMed]

H. P. Yuen and J. H. Shapiro, “Optical communications with two-photon coherent states. III. Quantummeasurements realizable with photoemissive detectors,” IEEE Trans. Inf. Theory IT-26, 78–92 (1980).
[CrossRef]

H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13, 2226–2243 (1976).
[CrossRef]

Yurke, B.

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in anoptical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[CrossRef] [PubMed]

Zou, X. Y.

L. J. Wang, X. Y. Zou, and L. Mandel, “Time-varying induced coherence,” J. Opt. Soc. Am. B 9, 605–609 (1992).
[CrossRef]

X. Y. Zou and L. Mandel, “Photon-antibunching and sub-Poissonian photon statistics,” Phys. Rev. A 41, 475–476 (1990).
[CrossRef] [PubMed]

IEEE Trans. Inf. Theory (1)

H. P. Yuen and J. H. Shapiro, “Optical communications with two-photon coherent states. III. Quantummeasurements realizable with photoemissive detectors,” IEEE Trans. Inf. Theory IT-26, 78–92 (1980).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Lett. (1)

Phys. Rev. (1)

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 13, 2529–2539 (1963).
[CrossRef]

Phys. Rev. A (5)

Z. Y. Ou, L. J. Wang, and L. Mandel, “Vacuum effects on interference in two-photon down conversion,” Phys. Rev. A 40, 1428–1435 (1989).
[CrossRef] [PubMed]

H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13, 2226–2243 (1976).
[CrossRef]

Z. Y. Ou, C. H. Hong, and L. Mandel, “Detection of squeezed states by cross correlation,” Phys. Rev. A 36, 192–196 (1987).
[CrossRef] [PubMed]

X. Y. Zou and L. Mandel, “Photon-antibunching and sub-Poissonian photon statistics,” Phys. Rev. A 41, 475–476 (1990).
[CrossRef] [PubMed]

M. Koashi, M. Matsuoka, and T. Hirano, “Photon antibunching by destructive two-photon interference,” Phys. Rev. A 53, 3621–3624 (1996).
[CrossRef] [PubMed]

Phys. Rev. D (1)

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[CrossRef]

Phys. Rev. Lett. (3)

L. Mandel, “Squeezed states and sub-Poissonian photon statistics,” Phys. Rev. Lett. 49, 136–138 (1982).
[CrossRef]

R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by four-wave mixing in anoptical cavity,” Phys. Rev. Lett. 55, 2409–2412 (1985).
[CrossRef] [PubMed]

L. A. Wu, H. J. Kimble, J. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. 57, 2520–2523 (1986).
[CrossRef] [PubMed]

Other (2)

D. Stoler, “Equivalence classes of minimum-uncertainty packets,” Phys. Rev. D 1, 3217–3219 (1970); “Equivalence classes of minimum-uncertainty packets. II,” 4, 1925–1926 (1974).
[CrossRef]

R. Boyd, Nonlinear Optics (Academic, Boston, 1992).

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Figures (1)

Fig. 1
Fig. 1

Proposed experimental setup that underlies the calculations in this paper.

Equations (82)

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EˆI(r, t)=1L3/2kl(ω)aˆk×exp[i(k·r-ωt-χ)]+H.c.,
EˆII(r, t)=1L3/2kl(ω)aˆk×exp[i(k·r-ωt-χ-π/2)]+H.c.,
λR(t, t+τ)EˆR(-)(t)EˆR(-)(t+τ)EˆR(+)(t+τ)EˆR(+)(t)EˆR(-)(t)EˆR(+)(t)EˆR(-)(t+τ)EˆR(+)(t+τ)-1,
λ(t, t+τ)>λ(t, t).
Eˆ2(+)(t)=T2EˆsF(+)(t-τ2-τs)+R2Eˆ0(+)(t-τ2-τ0),
Eˆ3(+)(t)=T3Eˆ2(+)(t-τ3)+R3Eˆv3(+)(t-τ3),
Eˆ4(+)(t)=R3Eˆ2(+)(t-τ4)+T3Eˆv3(+)(t-τ4).
Eˆ3(+)(t)=T3T2EˆsF(+)(t-τ5)+T3R2Eˆ0(+)(t-τ5-τ0+τs)+R3Eˆv3(+)(t-τ3),
Eˆ4(+)(t)=R3T2EˆsF(+)(t-τ6)+R3R2Eˆ0(+)(t-τ6-τ0+τs)+T3Eˆv3(+)(t-τ4),
τ5τ2+τ3+τs,
τ6τ2+τ4+τs.
P34(t, t+τ)=Eˆ3(-)(t)Eˆ4(-)(t+τ)Eˆ4(+)(t+τ)Eˆ3(+)(t)
=Iˆ3(t)Iˆ4(t+τ)[1+λ34(t, t+τ)],
Iˆ3(t)=Eˆ3(-)(t)Eˆ3(+)(t)=|T3|2[|R2|2Iˆ0+|T2|2IˆsF(t)+T2*R2EˆsF(-)(t)Eˆ0(+)(t)+c.c.],
Iˆ4(t)=Eˆ4(-)(t)Eˆ4(+)(t)=|R3|2[|R2|2Iˆ0+|T2|2IˆsF(t)+T2*R2EˆsF(-)(t)Eˆ0(+)(t)+c.c.],
P34(t, t+τ)-Iˆ3Iˆ4
=T:ΔIˆ3(t)ΔIˆ4(t+τ):=|R3T3|2{|T2|2|R2|2EˆsF(-)(t-τ5)×Eˆ0(-)(t+τ-τ6-τ0+τs)×EˆsF(+)(t+τ-τ6)Eˆ0(+)(t-τ5-τ0+τs)+c.c.,+T2*2R22EˆsF(-)(t-τ5)EˆsF(-)(t+τ-τ6)×Eˆ0(+)(t+τ-τ6-τ0+τs)Eˆ0(+)(t-τ5-τ0+τs)+c.c.
Hˆ=j=1sωjnˆj+12+ηδω2π2ω1ω1ωsωs×Φ(ω1,ω1, ωs,ωs)aˆs(ωs)aˆs(ωs)aˆ1(ω1)a1(ω1)×exp[i(ω1+ω1)τ1]+H.c.,
Φ(ω1, ω1, ωs, ωs)=Φ(ω1, ω1, ωs, ωs)=Φ(ω1, ω1, ωs, ωs)=Φ(ω1, ω1, ωs, ωs).
Φ(ω1, ω1, ωs, ωs)
=δω2πωΦSHG(ω1, ω1; ω)ΦDC(ωs, ωs; ω).
aˆ˙s(ω, t)=1i[aˆs(ω, t),Hˆ],
aˆ˙s(ω, t)=-iωaˆs(ω, t)-2iηδω2π2ω1ω1ωs×Φ(ω1, ω1, ωs, ω)aˆs(ωs, t)aˆ1(ω1, t)×aˆ1(ω1, t)exp[i(ω1+ω1)τ1].
Aˆ(ω, t)aˆ(ω, t)exp(iωt),
Aˆ˙s(ω, t)=-2iηδω2π2ω1ω1ωsΦ(ω1, ω1, ωs, ω)×Aˆs(ωs, t)Aˆ1(ω1, t)Aˆ1(ω1, t)×exp[i(ω+ωs-ω1-ω1)t]×exp[i(ω1+ω1)τ1].
Aˆ¨s(ω, t)=1i[Aˆ˙s(ω, t), Hˆ]+Aˆ˙s(ω, t)t=2ηδω2π2ω1ω1ωsΦ(ω1, ω1, ωs, ω)×[ω+ωs-ω1-ω1]×exp[i(ω+ωs-ω1-ω1)t]×Aˆs(ωs, t)Aˆ1(ω1, t)Aˆ1(ω1, t)×exp[i(ω1+ω1)τ1]+0(|η|2),
Aˆs(ω, t)=2iηδω2π2ω1ω1ωsΦ(ω1, ω1, ωs, ω)×[ω+ωs-ω1-ω1]2×exp[i(ω+ωs-ω1-ω1)t]×Aˆs(ωs, t)Aˆ1(ω1, t)Aˆ1(ω1, t)×exp[i(ω1+ω1)τ1]+0(|η|2),
Aˆs(ω, t)=Aˆs(ω, 0)+tAˆ˙s(ω, 0)+t22!Aˆ¨s(ω, 0)+=Aˆs(ω, 0)-2iηδω2π2×ω1ω1ωsΦ(ω1, ω1, ωs, ω)×Aˆs(ωs, 0)Aˆ1(ω1, 0)Aˆ1(ω1, 0)×exp[i(ω+ωs-ω1-ω1)t/2]×exp[i(ω1+ω1)τ1]×sin(ω+ωs-ω1-ω1)t/2(ω+ωs-ω1-ω1)/2+0(|η|2),
aˆs(ω, t)=aˆs(ω, 0)exp(-iωt)-2iηδω2π2×ω1ω1ωsΦ(ω1, ω1, ωs, ω)×S(ω+ωs-ω1-ω1, t)×aˆs(ωs, 0)aˆ1(ω1, 0)aˆ1(ω1, 0)×exp(-iωt)exp[i(ω1+ω1)τ1]+0(|η|2),
S(ω, t)exp(iωt/2) sin(ωt/2)ω/2.
aˆsF(ω)=aˆs(ω)f(ω)
f(ω0-ω)=f*(ω0+ω).
Eˆ00(+)(t)|{v}=E00(+)(t)|{v},
Eˆ1(+)=R1Eˆ00(+)+T1Eˆv1(+).
aˆs(ω, 0)|vac=0=vac|aˆs(ω, 0).
EˆsF(+)(t)=δω2π1/2ωaˆsF(ω, t),
EˆsF(+)(t)=0,
EˆsF(-)(t)Eˆ1(+)(t)=0=E1(+)(t)EˆsF(-)(t).
EˆsF(-)(t-τ5)Eˆ0(-)(t+τ-τ6-τ0+τs)EˆsF(+)(t+τ-τ6)×Eˆ0(+)(t-τ5-τ0+τs)=Eˆ0(-)(t+τ-τ6-τ0+τs)EˆsF(-)(t-τ5)×EˆsF(+)(t+τ-τ6)Eˆ0(+)(t-τ5-τ0+τs),
=E0(-)(t+τ-τ6-τ0+τs)E0(+)(t-τ5-τ0+τs)×EˆsF(-)(t-τ5)EˆsF(+)(t+τ-τ6).
EˆsF(-)(t-τ5)EˆsF(+)(t+τ-τ6)
=δω2π54|η|2ωωω1ω1ωsω1ω1ωs×Φ(ω1, ω1, ωs, ω)f(ω)f*(ω)Φ(ω1, ω1, ωs, ω)×S*(ω+ωs-ω1-ω1, t-τ5)×S(ω+ωs-ω1-ω1, t+τ-τ6)×exp[iω(t-τ5)]exp[-iω(t+τ-τ6)]×v1*(ω1)v1*(ω1)v1(ω1)v1(ω1)δωsωs×exp[i(ω1+ω1-ω1-ω1)τ1].
ω+ωs-ω-ω0Ω,
ω+ωs-ω0-ω0Ω,
Φ(ω1, ω1, ωs, ω1+ω1-ωs+Ω)
×f(ω1+ω1-ωs+Ω),
Φ(ω1, ω1, ωs, ω1+ω1-ωs+Ω1)
×f(ω1+ω1-ωs+Ω1),
Φ(ω1, ω1, ωs, ω1+ω1-ωs)f(ω1+ω1-ωs),
Φ(ω1, ω1, ωs, ω1+ω1-ωs)f(ω1+ω1-ωs),
12π-dΩ sin(Ωt/2)Ω/2exp(iΩt/2-τs)=1
for0<τs<t=0otherwise.
EˆsF(-)(t-τ5)EˆsF(+)(t+τ-τ6)=δω2π34|η|2ω1ω1ω1ω1ωsΦ(ω1, ω1, ωs, ω1+ω1-ωs)Φ(ω1, ω1, ωs, ω1+ω1-ωs)×f(ω1+ω1-ωs)f(ω1+ω1-ωs)×exp[i(ω1+ω1-ωs)(t-τ5)]×exp[-i(ω1+ω1-ωs)(t+τ-τ6)]×v1*(ω1)v1*(ω1)v1(ω1)v1(ω1)×exp[i(ω1+ω1-ω1-ω1)τ1]=4|η|2E1(-)2(t-τ5-τ1)E1(+)2(t+τ-τ6-τ1) ×12π0dωs|Φ(ω0, ω0, ωs, 2ω0-ωs)|2×|f(ωs)|2 exp[-iωs(τ6-τ5-τ)].
E0(+)(t)δω2π1/2ω0v0(ω0)exp(-iω0t),
μ(τ)12π-ω0dω|Φ(ω0, ω0, ω0-ω, ω0+ω)|2×|f(ω0+ω)|2 exp(+iωτ),
f(E(+), E(-))f(E(+), E(-))CE,
EˆsF(-)(t-τ5)Eˆ0(-)(t+τ-τ6-τ0+τs)
×EˆsF(+)(t+τ-τ6)Eˆ0(+)(t-τ5-τ0+τs)=4|η|2E0(-)(t+τ-τ6-τ0+τs)×E0(+)(t-τ5-τ0+τs)E1(-)2(t-τ5)×E1(+)2(t+τ-τ6)CEμ(τ6-τ5-τ)×exp[-iω0(τ-τ6+τ5)],
=4|η|2|R1|2|T1|4E00(-)2(t-τ5)×E00(-)(t+τ-τ6-τ0+τs)×E00(+)(t-τ5-τ0+τs)×E00(+)2(t+τ-τ6)CEμ(τ+τ5-τ6)×exp[-iω0(τ-τ6+τ5)].
EˆsF(-)(t-τ5)EˆsF(-)(t+τ-τ6)Eˆ0(+)(t+τ-τ6-τ0+τs)
×Eˆ0(+)(t-τ5-τ0+τs)
=EˆsF(-)(t-τ5)EˆsF(-)(t+τ-τ6)QM×E0(+)(t+τ-τ6-τ0+τs)×E0(+)(t-τ5-τ0+τs)CE=2iη*E00(-)2(t+τ-τ6-τ1)E0(+)(t+τ-τ6-τ0+τs)E0(+)(t-τ5-τ0+τs)CE×g(τ+τ5-τ6)exp[-iω0(τ-τ6+τ5)]=2iη*T1*2R1*2E00(-)2(t+τ-τ6-τ1)×E00(+)(t+τ-τ6-τ0+τs)×E00(+)(t-τ5-τ0+τs)CE g(τ+τ5-τ6)×exp[-iω0(τ-τ6+τ5)],
g(τ)12π-ω0dωΦ(ω0, ω0, ω0-ω, ω0+ω)×f2(ω0+ω)exp(iωτ),
E00(-)(t)E00(+)(t+τ)CE=I00(t)exp(-σLτ)exp(-iω0τ),
T:ΔIˆ3(t)ΔIˆ4(t+τ):
Iˆ3Iˆ4λ34(τ)=2|R3T3|2|R2T2|2|R1|2I00[4|T1|4|η|2×I002μ(τ+τ3-τ4)+2|T1|2|η|×I00g(τ+τ3-τ4)cos Θ],
Θ2 arg(R2)-2 arg(T2)-2 arg(R1)-2 arg(T1)-arg(η)+π/2+2ω0(τ0-τ1-τs).
:Iˆ3(t)Iˆ4(t):<T:Iˆ3(t)Iˆ4(t+τ):,
μ(τ)|Φ(ω0, ω0, ω0, ω0)|2 12π×ω0dω|f(ω0+ω)|2 exp(iωτ),
g(τ)Φ(ω0, ω0, ω0, ω0) 12π×-ω0dωf2(ω0+ω)exp(iωτ).
1=-ω0dωΦ(ω0, ω0, ω0-ω, ω0+ω),
Φ(ω0ω0, ω0, ω0)1ΔωDC.
Iˆ3Iˆ4λ34(τ)=4|R3T3|2|R2T2|2|R1T1|2 |η|I00ΔωDC×2|T1|2|η|I00ΔωDCF1(τ)+cos ΘF2(τ),
F1(τ)μ(τ)Φ2(ω0, ω0, ω0, ω0)=12π-dω|f(ω0+ω)|2 exp(iωτ),
F2(τ)g(τ)Φ(ω0, ω0, ω0, ω0)=12π-dω[f(ω0+ω)]2 exp(iωτ).
F1(τ)=F1(0)+τ1!F˙1(0)+τ22!F¨1(0)+,
F2(τ)F2(0)+τ1!F˙2(0)+τ22!F¨2(0)+.
F1(τ)-F1(0)τ22!F¨1(0)=-τ22!12π-ω2|f(ω0+ω)|2dω,
F2(τ)-F2(0)τ22!F¨2(0)=-τ22!12π-ω2[f(ω0+ω)]2dω.
Iˆ1Iˆ4[λ34(τ)-λ34(0)]
=2|R3T3|2|R2T2|2|R1T1|2 |η|I00ΔωDCτ2×-2|T1|2|η|I00ΔωDC12π-ω2|f(ω0+ω)|2dω-cos Θ2π-ω2[f(ω0+ω)]2dω.
2|T1|2|η|I00ΔωDC<1.

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