Abstract

We have employed chaotic and phase-diffusion models to study the effects of laser fourth-order coherence on polarization beats with phase-conjugation geometry in a four-level system (PBFS). We found that the temporal behavior of the beat signal depends on the stochastic properties of the lasers and the transverse relaxation rate of the transition. The modulation terms of the beat signal depend on the second-order coherence function, which is determined by the laser line shape. Inasmuch as different stochastic models of the laser field affect only the fourth-order coherence function, they have little influence on the general temporal modulation behavior of the beat signal. The different roles of phase fluctuation and amplitude fluctuation are pointed out. The cases that pump beams have either narrow-band or broadband linewidth are considered, and it is found that for both cases a Doppler-free precision in the measurement of the energy-level difference between two states that are dipolar forbidden from the ground state can be achieved. We also discuss the difference between the PBFS and ultrafast modulation spectroscopy from a physical viewpoint.

© 2000 Optical Society of America

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References

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  1. H. J. Eichler, U. Klein, and D. Langhans, “Coherence time measurement of picosecond pulses by a light-induced grating method,” Appl. Phys. 21, 215–218 (1980).
    [CrossRef]
  2. W. M. Grossman and D. M. Shemwell, “Coherence lengths and phase conjugation by degenerate four-wave mixing,” J. Appl. Phys. 51, 914–916 (1980).
    [CrossRef]
  3. R. Trebino, E. K. Gustafson, and A. E. Siegman, “Fourth-order partial-coherence effects in the formation of integrated-intensity gratings with pulsed light sources,” J. Opt. Soc. Am. B 3, 1295–1304 (1986).
    [CrossRef]
  4. D. DeBeer, L. G. Van Wagenen, R. Beach, and S. R. Hartmann, “Ultrafast modulation spectroscopy,” Phys. Rev. Lett. 56, 1128–1131 (1986).
    [CrossRef] [PubMed]
  5. P. M. Fu, Z. H. Yu, X. MI, X. F. Li, and Q. Jiang, “Doppler-free ultrafast modulation spectroscopy with phase-conjugation geometry,” Phys. Rev. A 50, 698–708 (1994).
    [CrossRef] [PubMed]
  6. P. M. Fu, X. Mi, Z. H. Yu, Q. Jiang, Y. P. Zhang, and X. F. Li, “Ultrafast modulation spectroscopy in a cascade three-level system,” Phys. Rev. A 52, 4867–4870 (1995).
    [CrossRef] [PubMed]
  7. N. Morita and T. Yajima, “Ultrahigh-time-resolution coherent transient spectroscopy with incoherent light,” Phys. Rev. A 30, 2525–2535 (1984).
    [CrossRef]
  8. S. Asaka, M. Nakatsuka, M. Fujiwara, and M. Matsuoke, “Accumulated photon echoes with incoherent light in Nd+3-doped silicate glass,” Phys. Rev. A 29, 2286–2289 (1984).
    [CrossRef]
  9. T. W. Mossberg, R. Kachru, S. R. Hartmann, and A. M. Flusberg, “Echoes in gaseous media: a generalized theory of rephasing phenomena,” Phys. Rev. A 20, 1976–1996 (1979).
    [CrossRef]
  10. B. Picinbono and E. Boileau, “Higher-order coherence functions of optical field and phase fluctuations,” J. Opt. Soc. Am. 58, 784–789 (1968).
    [CrossRef]
  11. X. Mi, Z. Yu, Q. Jiang, Z. Zhang, and P. Fu, “Time-delayed laser-induced double gratings,” J. Opt. Soc. Am. B 10, 725–732 (1993).
    [CrossRef]
  12. M. Mitsunaga and R. G. Brewer, “Generalized perturbation theory of coherent optical emission,” Phys. Rev. A 32, 1605–1613 (1985).
    [CrossRef] [PubMed]
  13. Y. P. Zhang, L. Q. Sun, T. T. Tang, and P. M. Fu, “Effects of field-correlation on polarization beats,” Phys. Rev. A (to be published).

1995 (1)

P. M. Fu, X. Mi, Z. H. Yu, Q. Jiang, Y. P. Zhang, and X. F. Li, “Ultrafast modulation spectroscopy in a cascade three-level system,” Phys. Rev. A 52, 4867–4870 (1995).
[CrossRef] [PubMed]

1994 (1)

P. M. Fu, Z. H. Yu, X. MI, X. F. Li, and Q. Jiang, “Doppler-free ultrafast modulation spectroscopy with phase-conjugation geometry,” Phys. Rev. A 50, 698–708 (1994).
[CrossRef] [PubMed]

1993 (1)

1986 (2)

1985 (1)

M. Mitsunaga and R. G. Brewer, “Generalized perturbation theory of coherent optical emission,” Phys. Rev. A 32, 1605–1613 (1985).
[CrossRef] [PubMed]

1984 (2)

N. Morita and T. Yajima, “Ultrahigh-time-resolution coherent transient spectroscopy with incoherent light,” Phys. Rev. A 30, 2525–2535 (1984).
[CrossRef]

S. Asaka, M. Nakatsuka, M. Fujiwara, and M. Matsuoke, “Accumulated photon echoes with incoherent light in Nd+3-doped silicate glass,” Phys. Rev. A 29, 2286–2289 (1984).
[CrossRef]

1980 (2)

H. J. Eichler, U. Klein, and D. Langhans, “Coherence time measurement of picosecond pulses by a light-induced grating method,” Appl. Phys. 21, 215–218 (1980).
[CrossRef]

W. M. Grossman and D. M. Shemwell, “Coherence lengths and phase conjugation by degenerate four-wave mixing,” J. Appl. Phys. 51, 914–916 (1980).
[CrossRef]

1979 (1)

T. W. Mossberg, R. Kachru, S. R. Hartmann, and A. M. Flusberg, “Echoes in gaseous media: a generalized theory of rephasing phenomena,” Phys. Rev. A 20, 1976–1996 (1979).
[CrossRef]

1968 (1)

Asaka, S.

S. Asaka, M. Nakatsuka, M. Fujiwara, and M. Matsuoke, “Accumulated photon echoes with incoherent light in Nd+3-doped silicate glass,” Phys. Rev. A 29, 2286–2289 (1984).
[CrossRef]

Beach, R.

D. DeBeer, L. G. Van Wagenen, R. Beach, and S. R. Hartmann, “Ultrafast modulation spectroscopy,” Phys. Rev. Lett. 56, 1128–1131 (1986).
[CrossRef] [PubMed]

Boileau, E.

Brewer, R. G.

M. Mitsunaga and R. G. Brewer, “Generalized perturbation theory of coherent optical emission,” Phys. Rev. A 32, 1605–1613 (1985).
[CrossRef] [PubMed]

DeBeer, D.

D. DeBeer, L. G. Van Wagenen, R. Beach, and S. R. Hartmann, “Ultrafast modulation spectroscopy,” Phys. Rev. Lett. 56, 1128–1131 (1986).
[CrossRef] [PubMed]

Eichler, H. J.

H. J. Eichler, U. Klein, and D. Langhans, “Coherence time measurement of picosecond pulses by a light-induced grating method,” Appl. Phys. 21, 215–218 (1980).
[CrossRef]

Flusberg, A. M.

T. W. Mossberg, R. Kachru, S. R. Hartmann, and A. M. Flusberg, “Echoes in gaseous media: a generalized theory of rephasing phenomena,” Phys. Rev. A 20, 1976–1996 (1979).
[CrossRef]

Fu, P.

Fu, P. M.

P. M. Fu, X. Mi, Z. H. Yu, Q. Jiang, Y. P. Zhang, and X. F. Li, “Ultrafast modulation spectroscopy in a cascade three-level system,” Phys. Rev. A 52, 4867–4870 (1995).
[CrossRef] [PubMed]

P. M. Fu, Z. H. Yu, X. MI, X. F. Li, and Q. Jiang, “Doppler-free ultrafast modulation spectroscopy with phase-conjugation geometry,” Phys. Rev. A 50, 698–708 (1994).
[CrossRef] [PubMed]

Fujiwara, M.

S. Asaka, M. Nakatsuka, M. Fujiwara, and M. Matsuoke, “Accumulated photon echoes with incoherent light in Nd+3-doped silicate glass,” Phys. Rev. A 29, 2286–2289 (1984).
[CrossRef]

Grossman, W. M.

W. M. Grossman and D. M. Shemwell, “Coherence lengths and phase conjugation by degenerate four-wave mixing,” J. Appl. Phys. 51, 914–916 (1980).
[CrossRef]

Gustafson, E. K.

Hartmann, S. R.

D. DeBeer, L. G. Van Wagenen, R. Beach, and S. R. Hartmann, “Ultrafast modulation spectroscopy,” Phys. Rev. Lett. 56, 1128–1131 (1986).
[CrossRef] [PubMed]

T. W. Mossberg, R. Kachru, S. R. Hartmann, and A. M. Flusberg, “Echoes in gaseous media: a generalized theory of rephasing phenomena,” Phys. Rev. A 20, 1976–1996 (1979).
[CrossRef]

Jiang, Q.

P. M. Fu, X. Mi, Z. H. Yu, Q. Jiang, Y. P. Zhang, and X. F. Li, “Ultrafast modulation spectroscopy in a cascade three-level system,” Phys. Rev. A 52, 4867–4870 (1995).
[CrossRef] [PubMed]

P. M. Fu, Z. H. Yu, X. MI, X. F. Li, and Q. Jiang, “Doppler-free ultrafast modulation spectroscopy with phase-conjugation geometry,” Phys. Rev. A 50, 698–708 (1994).
[CrossRef] [PubMed]

X. Mi, Z. Yu, Q. Jiang, Z. Zhang, and P. Fu, “Time-delayed laser-induced double gratings,” J. Opt. Soc. Am. B 10, 725–732 (1993).
[CrossRef]

Kachru, R.

T. W. Mossberg, R. Kachru, S. R. Hartmann, and A. M. Flusberg, “Echoes in gaseous media: a generalized theory of rephasing phenomena,” Phys. Rev. A 20, 1976–1996 (1979).
[CrossRef]

Klein, U.

H. J. Eichler, U. Klein, and D. Langhans, “Coherence time measurement of picosecond pulses by a light-induced grating method,” Appl. Phys. 21, 215–218 (1980).
[CrossRef]

Langhans, D.

H. J. Eichler, U. Klein, and D. Langhans, “Coherence time measurement of picosecond pulses by a light-induced grating method,” Appl. Phys. 21, 215–218 (1980).
[CrossRef]

Li, X. F.

P. M. Fu, X. Mi, Z. H. Yu, Q. Jiang, Y. P. Zhang, and X. F. Li, “Ultrafast modulation spectroscopy in a cascade three-level system,” Phys. Rev. A 52, 4867–4870 (1995).
[CrossRef] [PubMed]

P. M. Fu, Z. H. Yu, X. MI, X. F. Li, and Q. Jiang, “Doppler-free ultrafast modulation spectroscopy with phase-conjugation geometry,” Phys. Rev. A 50, 698–708 (1994).
[CrossRef] [PubMed]

Matsuoke, M.

S. Asaka, M. Nakatsuka, M. Fujiwara, and M. Matsuoke, “Accumulated photon echoes with incoherent light in Nd+3-doped silicate glass,” Phys. Rev. A 29, 2286–2289 (1984).
[CrossRef]

Mi, X.

P. M. Fu, X. Mi, Z. H. Yu, Q. Jiang, Y. P. Zhang, and X. F. Li, “Ultrafast modulation spectroscopy in a cascade three-level system,” Phys. Rev. A 52, 4867–4870 (1995).
[CrossRef] [PubMed]

P. M. Fu, Z. H. Yu, X. MI, X. F. Li, and Q. Jiang, “Doppler-free ultrafast modulation spectroscopy with phase-conjugation geometry,” Phys. Rev. A 50, 698–708 (1994).
[CrossRef] [PubMed]

X. Mi, Z. Yu, Q. Jiang, Z. Zhang, and P. Fu, “Time-delayed laser-induced double gratings,” J. Opt. Soc. Am. B 10, 725–732 (1993).
[CrossRef]

Mitsunaga, M.

M. Mitsunaga and R. G. Brewer, “Generalized perturbation theory of coherent optical emission,” Phys. Rev. A 32, 1605–1613 (1985).
[CrossRef] [PubMed]

Morita, N.

N. Morita and T. Yajima, “Ultrahigh-time-resolution coherent transient spectroscopy with incoherent light,” Phys. Rev. A 30, 2525–2535 (1984).
[CrossRef]

Mossberg, T. W.

T. W. Mossberg, R. Kachru, S. R. Hartmann, and A. M. Flusberg, “Echoes in gaseous media: a generalized theory of rephasing phenomena,” Phys. Rev. A 20, 1976–1996 (1979).
[CrossRef]

Nakatsuka, M.

S. Asaka, M. Nakatsuka, M. Fujiwara, and M. Matsuoke, “Accumulated photon echoes with incoherent light in Nd+3-doped silicate glass,” Phys. Rev. A 29, 2286–2289 (1984).
[CrossRef]

Picinbono, B.

Shemwell, D. M.

W. M. Grossman and D. M. Shemwell, “Coherence lengths and phase conjugation by degenerate four-wave mixing,” J. Appl. Phys. 51, 914–916 (1980).
[CrossRef]

Siegman, A. E.

Trebino, R.

Van Wagenen, L. G.

D. DeBeer, L. G. Van Wagenen, R. Beach, and S. R. Hartmann, “Ultrafast modulation spectroscopy,” Phys. Rev. Lett. 56, 1128–1131 (1986).
[CrossRef] [PubMed]

Yajima, T.

N. Morita and T. Yajima, “Ultrahigh-time-resolution coherent transient spectroscopy with incoherent light,” Phys. Rev. A 30, 2525–2535 (1984).
[CrossRef]

Yu, Z.

Yu, Z. H.

P. M. Fu, X. Mi, Z. H. Yu, Q. Jiang, Y. P. Zhang, and X. F. Li, “Ultrafast modulation spectroscopy in a cascade three-level system,” Phys. Rev. A 52, 4867–4870 (1995).
[CrossRef] [PubMed]

P. M. Fu, Z. H. Yu, X. MI, X. F. Li, and Q. Jiang, “Doppler-free ultrafast modulation spectroscopy with phase-conjugation geometry,” Phys. Rev. A 50, 698–708 (1994).
[CrossRef] [PubMed]

Zhang, Y. P.

P. M. Fu, X. Mi, Z. H. Yu, Q. Jiang, Y. P. Zhang, and X. F. Li, “Ultrafast modulation spectroscopy in a cascade three-level system,” Phys. Rev. A 52, 4867–4870 (1995).
[CrossRef] [PubMed]

Zhang, Z.

Appl. Phys. (1)

H. J. Eichler, U. Klein, and D. Langhans, “Coherence time measurement of picosecond pulses by a light-induced grating method,” Appl. Phys. 21, 215–218 (1980).
[CrossRef]

J. Appl. Phys. (1)

W. M. Grossman and D. M. Shemwell, “Coherence lengths and phase conjugation by degenerate four-wave mixing,” J. Appl. Phys. 51, 914–916 (1980).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Phys. Rev. A (6)

M. Mitsunaga and R. G. Brewer, “Generalized perturbation theory of coherent optical emission,” Phys. Rev. A 32, 1605–1613 (1985).
[CrossRef] [PubMed]

P. M. Fu, Z. H. Yu, X. MI, X. F. Li, and Q. Jiang, “Doppler-free ultrafast modulation spectroscopy with phase-conjugation geometry,” Phys. Rev. A 50, 698–708 (1994).
[CrossRef] [PubMed]

P. M. Fu, X. Mi, Z. H. Yu, Q. Jiang, Y. P. Zhang, and X. F. Li, “Ultrafast modulation spectroscopy in a cascade three-level system,” Phys. Rev. A 52, 4867–4870 (1995).
[CrossRef] [PubMed]

N. Morita and T. Yajima, “Ultrahigh-time-resolution coherent transient spectroscopy with incoherent light,” Phys. Rev. A 30, 2525–2535 (1984).
[CrossRef]

S. Asaka, M. Nakatsuka, M. Fujiwara, and M. Matsuoke, “Accumulated photon echoes with incoherent light in Nd+3-doped silicate glass,” Phys. Rev. A 29, 2286–2289 (1984).
[CrossRef]

T. W. Mossberg, R. Kachru, S. R. Hartmann, and A. M. Flusberg, “Echoes in gaseous media: a generalized theory of rephasing phenomena,” Phys. Rev. A 20, 1976–1996 (1979).
[CrossRef]

Phys. Rev. Lett. (1)

D. DeBeer, L. G. Van Wagenen, R. Beach, and S. R. Hartmann, “Ultrafast modulation spectroscopy,” Phys. Rev. Lett. 56, 1128–1131 (1986).
[CrossRef] [PubMed]

Other (1)

Y. P. Zhang, L. Q. Sun, T. T. Tang, and P. M. Fu, “Effects of field-correlation on polarization beats,” Phys. Rev. A (to be published).

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Figures (6)

Fig. 1
Fig. 1

Four-level configuration to be treated by PBFS.

Fig. 2
Fig. 2

Schematic diagram of the geometry of PBFS.

Fig. 3
Fig. 3

Signal intensity versus relative time delay with beams 2 and 3 consisting of only ω2.

Fig. 4
Fig. 4

Signal intensity versus relative time delay with beams 2 and 3 consisting of only ω3.

Fig. 5
Fig. 5

Beat signal intensity versus relative time delay in which τ is varied for a range of 5 ps.

Fig. 6
Fig. 6

Fourier spectrum of the PBFS data in which τ is varied for a range of 5 ps.

Equations (47)

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EP2=2u2(t)exp[i(k2·r-ω2t)]+3u3(t)exp[i(k3·r-ω3t)],
EP3=2u2(t-τ)exp[i(k2·r-ω2t+ω2τ)]+3u3(t-τ)exp[i(k3·r-ω3t+ω3τ)],
EP1=1 exp[i(k1·r-ω1t)],
ρ00(0)ω1ρ10(1)ω2ρ20(2)-ω3ρ10(3),
ρ00(0)ω1ρ10(1)ω3ρ30(2)-ω3ρ10(3),
ui(t1)ui(t2)ui*(t3)ui*(t4)
=ui(t1)ui*(t3)ui(t2)ui*(t4)+ui(t1)ui*(t4)
×ui(t2)ui*(t3),i=2,3.
ui(t1)ui*(t2)=exp(-αi|t1-t2|),i=2,3,
P(I)=S1(r)exp[-i(ω1t+ω2τ)]×-+ dvw(v)0 dt3 0 dt2 0dt1×exp[-iθI(v)]exp[-(Γ10+iΔ1)t3]×exp[-(Γ20+iΔ2+iΔ1)t2]×exp[-(Γ10+iΔ1)t1]exp(-α2|t2-τ|),
P(II)=S2(r)exp[-i(ω1t+ω3τ)]×-+ dvw(v)0 dt3 0 dt2 0 dt1×exp[-iθII(v)]exp[-(Γ10+iΔ1)t3]×exp[-(Γ30+iΔ3+iΔ1)t2]×exp[-(Γ10+iΔ1)t1]exp(-α3|t2-τ|),
S1(r)=-iNμ12μ22312(2)*×exp[i(k1+k2-k2)·r],
S2(r)=-iNμ12μ32312(3)*×exp[i(k1+k3-k3)·r],
θI(v)=v·[k1(t1+t2+t3)+k2(t2+t3)-k2t3],
θII(v)=v·[k1(t1+t2+t3)+k3(t2+t3)-k3t3];
I(τ)|P(3)|2|B1|2-B12Γ10Γ20(Γ10+Γ20+iΔ2)+|η|2|B2|2-B22Γ10Γ30(Γ10+Γ30+iΔ3)+|B1|2 exp(-2α2|τ|)+|ηB2|2 exp(-2α3|τ|)+exp[-(α2+α3)|τ|)]{ηB1*B2×exp[-i(ω3-ω2)τ]+η*B1B2*×exp[i(ω3-ω2)τ]},
B1=1(Γ20+iΔ1+iΔ2)(Γ102+Δ12),
B2=1(Γ30+iΔ1+iΔ3)(Γ102+Δ12),
η=S2(r)S1(r)=μ32μ223*32*2×exp{ir·[(k3-k3)-(k2-k2)]},
η=μ32μ223*32*2.
-+ dvw(v)exp[-iθI(v)]
2πk1uδ(t1+t2+t3-ξ1t2),
-+ dvw(v)exp[-iθII(v)]
2πk1uδ(t1+t2+t3-ξ2t2),
|P(3)|2=|P(I)+P(II)|2.
I(τ)|P(3)|2
=(ξ1-1)(Γ20a-Γ10+iΔ2a)2
×(ξ1-1)2+12(Γ20a-Γ10)[(ξ1+1)(Γ20a-Γ10)+iΔ2a(ξ1-1)]
+ξ12-ξ1+1(Γ20a-Γ10+iΔ2a)2
+12(Γ20a-Γ10)(Γ20a-Γ10-iΔ2a)-ξ1-12[(Γ20a-Γ10)2+(Δ2a)2]+|η|2(ξ2-1)(Γ30a-Γ10+iΔ3a)2(ξ2-1)2+12(Γ30a-Γ10)[(ξ2+1)(Γ30a-Γ10)+iΔ3a(ξ2-1)]+ξ22-ξ2+1(Γ30a-Γ10+iΔ3a)2+12(Γ30a-Γ10)(Γ30a-Γ10-iΔ3a)-ξ2-12[(Γ30a-Γ10)2+(Δ3a)2]+(ξ1-1)2[(Γ20a-Γ10)2+(Δ2a)2]2exp(-2α2|τ|)+|η|2(ξ2-1)2[(Γ30a-Γ10)2+(Δ3a)2]2exp(-2α3|τ|)+exp[-(α2+α3)|τ|]{q exp[-i(ω3-ω2)τ]+q* exp[i(ω3-ω2)τ]},
q=η(ξ1-1)(ξ2-1)[(Γ20a-Γ10)-iΔ2a]2[(Γ30a-Γ10)+iΔ3a]2,
Γ20a=Γ20+ξ1Γ10,Δ2a=Δ2+ξ1Δ1,
Γ30a=Γ30+ξ2Γ10,Δ3a=Δ3+ξ2Δ1.
I(τ)|P(3)|2
=(ξ1-1)[α22+(Δ2a)2-2iα2Δ2a]2(Γ20a-Γ10)2[α22+(Δ2a)2]
+(ξ2-1)[α32+(Δ3a)2-2iα3Δ3a]2(Γ30a-Γ10)2[α32+(Δ3a)2]|η|2+exp[-2(Γ20a-Γ10)|τ|]4(ξ1-1)2α22τ2[α22+(Δ2a)2]2+|η|24(ξ2-1)2α32τ2[α32+(Δ3a)2]2exp[-2(Γ30a-Γ10)|τ|]+4(ξ1-1)(ξ2-1)α3α2τ2[α22+(Δ2a)2][α32+(Δ3a)2]×exp[-(Γ30a+Γ20a-2Γ10)|τ|]×{η exp[-i(Ω3-Ω2)τ-i(ξ2-ξ1)Δ1τ]+η* exp[i(Ω3-Ω2)τ+i(ξ2-ξ1)Δ1τ]}.
I(τ)|P(3)|2
=(ξ1-1)[α22+(Δ2a)2-2iα2Δ2a]2(Γ20a-Γ10)2[α22+(Δ2a)2]+(ξ2-1)[α32+(Δ3a)2-2iα3Δ3a]2(Γ30a-Γ10)2[α32+(Δ3a)2]|η|2+(ξ1-1)2[α22+(Δ2a)2]2exp(-2α2|τ|)+(ξ2-1)2[α32+(Δ3a)2]2exp(-2α3|τ|)+exp[-(α2+α3)|τ|]{q exp[-i(ω3-ω2)τ]+(q)* exp[i(ω3-ω2)τ]},
q=η(ξ1-1)(ξ2-1)(α2-iΔ2a)2(α3+iΔ3a)2.
ui(t1)ui*(t2)=exp-αi2ln 2(t1-t2)2=exp{-[βi(t1-t2)]2},i=2,3.
ui(t1)ui*(t2)=exp[-βi2(t1-t2)2]πβiδ(t1-t2),i=2,3.
 I(τ)|P(3)|2=ξ1-14(Γ20a-Γ10)2+|η|2α22(ξ2-1)4α32(Γ30a-Γ10)2+τ2(ξ1-1)2 exp[-2(Γ20a-Γ10)|τ|]+|η|2τ2(ξ2-1)2α22α32exp[-2(Γ30a-Γ10)|τ|]+τ2(ξ1-1)(ξ2-1)α2α3×exp[-(Γ30a+Γ20a-2Γ10)|τ|]×{η exp[-i(Ω3-Ω2)τ-i(ξ2-ξ1)Δ1τ]+η* exp[i(Ω3-Ω2)τ+i(ξ2-ξ1)Δ1τ]}.
ui(t1)ui(t2)ui*(t3)ui*(t4)
=exp[-αi(|t1-t3|+|t1-t4|+|t2-t3|+|t2-t4|)]
×exp[αi(|t1-t2|+|t3-t4|)],i=2,3.
ui(t1)ui*(t2)=exp(-αi|t1-t2|)2αiδ(t1-t2),i=2,3,
I(τ)|P(3)|2=τ2(ξ1-1)2 exp[-2(Γ20a-Γ10)|τ|]+|η|2τ2(ξ2-1)2α22α32exp[-2(Γ30a-Γ10)|τ|]+τ2(ξ1-1)(ξ2-1)α2α3×exp[-(Γ30a+Γ20a-2Γ10)|τ|]×{η exp[-i(Ω3-Ω2)τ-i(ξ2-ξ1)Δ1τ]+η* exp[i(Ω3-Ω2)τ+i(ξ2-ξ1)Δ1τ]}.

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