Abstract

A time-dependent perturbational approach is used to study the Raman echo generated by an inhomogeneous ensemble of three-level atomic systems in the Λ configuration. The sixth-order perturbative solution for coherence between the two lower states contains eight echo terms that are visualized with double Feynman diagrams. The echo is interpreted as a response of the ensemble to six short single-frequency optical pulses whose timing satisfies a special condition. Nonperturbative numerical simulation confirms the predictions of the perturbation analysis about the main characteristics of the echo. When the excitation pulses vary adiabatically with respect to fast optical coherence decay the echo amplitude depends on interference of the eight terms. A computational scheme developed for the higher-order perturbative analysis is also discussed.

© 1998 Optical Society of America

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  1. S. R. Hartmann, “Photon, spin, and Raman echoes,” IEEE J. Quantum Electron. 4, 802–807 (1968).
    [CrossRef]
  2. P. Hu, S. Geschwind, and T. M. Jedju, “Spin-flip Raman echo in n-type CdS,” Phys. Rev. Lett. 37, 1357–1360 (1976).
    [CrossRef]
  3. K. P. Leung, T. W. Mossberg, and S. R. Hartmann, “Observation and density dependence of the Raman echo in atomic thallium vapor,” Opt. Commun. 43, 145–150 (1982).
    [CrossRef]
  4. M. Tanigawa, Y. Fukuda, T. Kohmoto, K. Sakuno, and T. Hashi, “Sublevel echoes selectively excited by light-pulse trains: synchronized-quantum-beat echoes,” Opt. Lett. 8, 620–622 (1983).
    [CrossRef] [PubMed]
  5. T. Mishina, M. Tanigawa, Y. Fukuda, and T. Hashi, “Synchronized quantum beat echoes in Cs vapor with diode lasers,” Opt. Commun. 62, 166–170 (1987).
    [CrossRef]
  6. S. Burschka and J. Mlynek, “Optically induced spin transients in the ground state of atomic sodium,” Opt. Commun. 66, 59–64 (1988).
    [CrossRef]
  7. M. Rosatzin, D. Suter, and J. Mlynek, “Light-shift-induced spin echoes in a J=1/2 atomic ground state,” Phys. Rev. A 42, 1839–1841 (1990).
    [CrossRef] [PubMed]
  8. P. R. Hemmer, K. Z. Cheng, J. Kierstead, M. S. Shahriar, and M. K. Kim, “Time-domain optical data storage by use of Raman coherent population trapping,” Opt. Lett. 19, 296–298 (1994).
    [CrossRef] [PubMed]
  9. B. S. Ham, M. S. Shahriar, M. K. Kim, and P. R. Hemmer, “Frequency-selective time-domain optical data storage by electromagnetically induced transparency in a rare-earth-doped solid,” Opt. Lett. 22, 1849–1851 (1997).
    [CrossRef]
  10. E. L. Hahn, “Spin echoes,” Phys. Rev. 80, 580–594 (1950).
    [CrossRef]
  11. I. D. Abella, N. A. Kurnit, and S. R. Hartmann, “Photon echoes,” Phys. Rev. 141, 391–406 (1966).
    [CrossRef]
  12. 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]
  13. T. W. Mossberg and S. R. Hartmann, “Diagrammatic representation of photon echoes and other laser-induced ordering processes in gases,” Phys. Rev. A 23, 1271–1280 (1981).
    [CrossRef]
  14. R. P. Feynman, F. L. Vernon, Jr., and R. W. Hellwarth, “Geometrical representation of the Schrödinger equation for solving maser problems,” J. Appl. Phys. 28, 49–52 (1957).
    [CrossRef]
  15. M. Takatsuji, “Theory of coherent two-photon resonance,” Phys. Rev. A 11, 619–624 (1975).
    [CrossRef]
  16. D. Grischkowsky, M. M. T. Loy, and P. F. Liao, “Adiabatic following model for two-photon transitions: nonlinear mixing and pulse propagation,” Phys. Rev. A 12, 2514–2533 (1975).
    [CrossRef]
  17. S. Aoki, “Double-quantum photon echo in an adiabatic-vector-model approximation,” Phys. Rev. A 14, 2258–2263 (1976).
    [CrossRef]
  18. H. R. Gray, R. M. Whitley, and C. R. Stroud, Jr., “Coherent trapping of atomic populations,” Opt. Lett. 3, 218–220 (1978).
    [CrossRef] [PubMed]
  19. M. S. Shahriar and P. R. Hemmer, “Direct excitation of microwave-spin dressed states using a laser-excited resonance Raman interaction,” Phys. Rev. Lett. 65, 1865–1868 (1990).
    [CrossRef] [PubMed]
  20. M. S. Shahriar, P. R. Hemmer, D. P. Katz, A. Lee, and M. G. Prentiss, “Dark-state-based three-element vector model for the stimulated Raman interaction,” Phys. Rev. A 55, 2272–2282 (1997).
    [CrossRef]
  21. P. R. Hemmer and M. G. Prentiss, “Coupled-pendulum model of the stimulated resonance Raman effect,” J. Opt. Soc. Am. B 5, 1613–1623 (1988).
    [CrossRef]
  22. A. Yariv, “The application of time evolution operators and Feynman diagrams to nonlinear optics,” IEEE J. Quantum Electron. 13, 943–950 (1977).
    [CrossRef]
  23. T. K. Yee and T. K. Gustafson, “Diagrammatic analysis of the density operator for nonlinear calculations: pulsed and cw responses,” Phys. Rev. A 18, 1597–1617 (1978).
    [CrossRef]
  24. A. Yariv, Quantum Electronics (Wiley, New York, 1989), Chaps. 3 and 16.
  25. Y. R. Shen, The Principles of Nonlinear Optics (Wiley-Interscience, New York, 1984), Chaps. 2 and 21.
  26. M. Mitsunaga and R. G. Brewer, “Generalized perturbation theory of coherent optical emission,” Phys. Rev. A 32, 1605–1613 (1985).
    [CrossRef] [PubMed]
  27. P. R. Hemmer, M. S. Shahriar, B. S. Ham, M. K. Kim, and Yu. Rozhdestvensky, “Optical spectral holeburning with Raman coherent population trapping,” Mol. Cryst. Liq. Cryst. 291, 287–294 (1996).
    [CrossRef]

1997 (2)

M. S. Shahriar, P. R. Hemmer, D. P. Katz, A. Lee, and M. G. Prentiss, “Dark-state-based three-element vector model for the stimulated Raman interaction,” Phys. Rev. A 55, 2272–2282 (1997).
[CrossRef]

B. S. Ham, M. S. Shahriar, M. K. Kim, and P. R. Hemmer, “Frequency-selective time-domain optical data storage by electromagnetically induced transparency in a rare-earth-doped solid,” Opt. Lett. 22, 1849–1851 (1997).
[CrossRef]

1996 (1)

P. R. Hemmer, M. S. Shahriar, B. S. Ham, M. K. Kim, and Yu. Rozhdestvensky, “Optical spectral holeburning with Raman coherent population trapping,” Mol. Cryst. Liq. Cryst. 291, 287–294 (1996).
[CrossRef]

1994 (1)

1990 (2)

M. S. Shahriar and P. R. Hemmer, “Direct excitation of microwave-spin dressed states using a laser-excited resonance Raman interaction,” Phys. Rev. Lett. 65, 1865–1868 (1990).
[CrossRef] [PubMed]

M. Rosatzin, D. Suter, and J. Mlynek, “Light-shift-induced spin echoes in a J=1/2 atomic ground state,” Phys. Rev. A 42, 1839–1841 (1990).
[CrossRef] [PubMed]

1988 (2)

S. Burschka and J. Mlynek, “Optically induced spin transients in the ground state of atomic sodium,” Opt. Commun. 66, 59–64 (1988).
[CrossRef]

P. R. Hemmer and M. G. Prentiss, “Coupled-pendulum model of the stimulated resonance Raman effect,” J. Opt. Soc. Am. B 5, 1613–1623 (1988).
[CrossRef]

1987 (1)

T. Mishina, M. Tanigawa, Y. Fukuda, and T. Hashi, “Synchronized quantum beat echoes in Cs vapor with diode lasers,” Opt. Commun. 62, 166–170 (1987).
[CrossRef]

1985 (1)

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

1983 (1)

1982 (1)

K. P. Leung, T. W. Mossberg, and S. R. Hartmann, “Observation and density dependence of the Raman echo in atomic thallium vapor,” Opt. Commun. 43, 145–150 (1982).
[CrossRef]

1981 (1)

T. W. Mossberg and S. R. Hartmann, “Diagrammatic representation of photon echoes and other laser-induced ordering processes in gases,” Phys. Rev. A 23, 1271–1280 (1981).
[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]

1978 (2)

T. K. Yee and T. K. Gustafson, “Diagrammatic analysis of the density operator for nonlinear calculations: pulsed and cw responses,” Phys. Rev. A 18, 1597–1617 (1978).
[CrossRef]

H. R. Gray, R. M. Whitley, and C. R. Stroud, Jr., “Coherent trapping of atomic populations,” Opt. Lett. 3, 218–220 (1978).
[CrossRef] [PubMed]

1977 (1)

A. Yariv, “The application of time evolution operators and Feynman diagrams to nonlinear optics,” IEEE J. Quantum Electron. 13, 943–950 (1977).
[CrossRef]

1976 (2)

S. Aoki, “Double-quantum photon echo in an adiabatic-vector-model approximation,” Phys. Rev. A 14, 2258–2263 (1976).
[CrossRef]

P. Hu, S. Geschwind, and T. M. Jedju, “Spin-flip Raman echo in n-type CdS,” Phys. Rev. Lett. 37, 1357–1360 (1976).
[CrossRef]

1975 (2)

M. Takatsuji, “Theory of coherent two-photon resonance,” Phys. Rev. A 11, 619–624 (1975).
[CrossRef]

D. Grischkowsky, M. M. T. Loy, and P. F. Liao, “Adiabatic following model for two-photon transitions: nonlinear mixing and pulse propagation,” Phys. Rev. A 12, 2514–2533 (1975).
[CrossRef]

1968 (1)

S. R. Hartmann, “Photon, spin, and Raman echoes,” IEEE J. Quantum Electron. 4, 802–807 (1968).
[CrossRef]

1966 (1)

I. D. Abella, N. A. Kurnit, and S. R. Hartmann, “Photon echoes,” Phys. Rev. 141, 391–406 (1966).
[CrossRef]

1957 (1)

R. P. Feynman, F. L. Vernon, Jr., and R. W. Hellwarth, “Geometrical representation of the Schrödinger equation for solving maser problems,” J. Appl. Phys. 28, 49–52 (1957).
[CrossRef]

1950 (1)

E. L. Hahn, “Spin echoes,” Phys. Rev. 80, 580–594 (1950).
[CrossRef]

IEEE J. Quantum Electron. (2)

A. Yariv, “The application of time evolution operators and Feynman diagrams to nonlinear optics,” IEEE J. Quantum Electron. 13, 943–950 (1977).
[CrossRef]

S. R. Hartmann, “Photon, spin, and Raman echoes,” IEEE J. Quantum Electron. 4, 802–807 (1968).
[CrossRef]

J. Appl. Phys. (1)

R. P. Feynman, F. L. Vernon, Jr., and R. W. Hellwarth, “Geometrical representation of the Schrödinger equation for solving maser problems,” J. Appl. Phys. 28, 49–52 (1957).
[CrossRef]

J. Opt. Soc. Am. B (1)

Mol. Cryst. Liq. Cryst. (1)

P. R. Hemmer, M. S. Shahriar, B. S. Ham, M. K. Kim, and Yu. Rozhdestvensky, “Optical spectral holeburning with Raman coherent population trapping,” Mol. Cryst. Liq. Cryst. 291, 287–294 (1996).
[CrossRef]

Opt. Commun. (3)

K. P. Leung, T. W. Mossberg, and S. R. Hartmann, “Observation and density dependence of the Raman echo in atomic thallium vapor,” Opt. Commun. 43, 145–150 (1982).
[CrossRef]

T. Mishina, M. Tanigawa, Y. Fukuda, and T. Hashi, “Synchronized quantum beat echoes in Cs vapor with diode lasers,” Opt. Commun. 62, 166–170 (1987).
[CrossRef]

S. Burschka and J. Mlynek, “Optically induced spin transients in the ground state of atomic sodium,” Opt. Commun. 66, 59–64 (1988).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. (2)

E. L. Hahn, “Spin echoes,” Phys. Rev. 80, 580–594 (1950).
[CrossRef]

I. D. Abella, N. A. Kurnit, and S. R. Hartmann, “Photon echoes,” Phys. Rev. 141, 391–406 (1966).
[CrossRef]

Phys. Rev. A (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]

T. W. Mossberg and S. R. Hartmann, “Diagrammatic representation of photon echoes and other laser-induced ordering processes in gases,” Phys. Rev. A 23, 1271–1280 (1981).
[CrossRef]

M. Rosatzin, D. Suter, and J. Mlynek, “Light-shift-induced spin echoes in a J=1/2 atomic ground state,” Phys. Rev. A 42, 1839–1841 (1990).
[CrossRef] [PubMed]

M. Takatsuji, “Theory of coherent two-photon resonance,” Phys. Rev. A 11, 619–624 (1975).
[CrossRef]

D. Grischkowsky, M. M. T. Loy, and P. F. Liao, “Adiabatic following model for two-photon transitions: nonlinear mixing and pulse propagation,” Phys. Rev. A 12, 2514–2533 (1975).
[CrossRef]

S. Aoki, “Double-quantum photon echo in an adiabatic-vector-model approximation,” Phys. Rev. A 14, 2258–2263 (1976).
[CrossRef]

T. K. Yee and T. K. Gustafson, “Diagrammatic analysis of the density operator for nonlinear calculations: pulsed and cw responses,” Phys. Rev. A 18, 1597–1617 (1978).
[CrossRef]

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

M. S. Shahriar, P. R. Hemmer, D. P. Katz, A. Lee, and M. G. Prentiss, “Dark-state-based three-element vector model for the stimulated Raman interaction,” Phys. Rev. A 55, 2272–2282 (1997).
[CrossRef]

Phys. Rev. Lett. (2)

M. S. Shahriar and P. R. Hemmer, “Direct excitation of microwave-spin dressed states using a laser-excited resonance Raman interaction,” Phys. Rev. Lett. 65, 1865–1868 (1990).
[CrossRef] [PubMed]

P. Hu, S. Geschwind, and T. M. Jedju, “Spin-flip Raman echo in n-type CdS,” Phys. Rev. Lett. 37, 1357–1360 (1976).
[CrossRef]

Other (2)

A. Yariv, Quantum Electronics (Wiley, New York, 1989), Chaps. 3 and 16.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley-Interscience, New York, 1984), Chaps. 2 and 21.

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Figures (9)

Fig. 1
Fig. 1

Energy levels of the atom in the Λ configuration. The upper state is coupled to the two ground states by laser radiation at frequencies ω1 and ω2. The detunings of the light-field components from the transition frequencies ω31 and ω32 are characterized by Δ1=ω31-ω1 and Δ2=ω32-ω2 or by δu=(Δ1+Δ2)/2 and δg=Δ1-Δ2. Spontaneous relaxation channels and constants are also indicated.

Fig. 2
Fig. 2

Double Feynman diagrams of the Raman-echo terms (21) with the characteristic density-matrix elements and the initial value of wg.

Fig. 3
Fig. 3

Time-scale of forming the Raman echo. The dephaser, stopper, and rephaser are short periods of optical coherence. Also shown are these characteristic density-matrix elements that are common to all eight diagrams in Fig. 2.

Fig. 4
Fig. 4

Time scale of the six-pulse echo with different excitation sequences.

Fig. 5
Fig. 5

Supplement to the Feynman diagrams in Fig. 2 to obtain the seventh-order diagrams of the optical-echo signal.

Fig. 6
Fig. 6

Examples of satisfying the phase matching condition (29). In the diagrams all angles are of arbitrary magnitude except for 0 and π rad.

Fig. 7
Fig. 7

Nonperturbative numerical simulation of the Raman echo and the corresponding optical signal. (a) Excitation pulses applied at t1=-0.04/γu, 0, 0.004/γu, and -0.01/γu in (b)–(e), respectively, and t2=0.04/γu, t3=10.00/γu, t4=10.03/γu, t5=30.00/γu, and t6=30.01/γu. The pulse at t1=-0.05/γu was present in (e) only. (b)–(e) Results of the simulation with the following parameters: excitation-pulse area Θ=1.70 except for the pulses at t1 and t1 in (e) with Θ=1.21; probe pulse at tp=40/γu, Θ=0.43; temporal width of all pulses τ=0.002/γu; Γu=0.5γu; Γg=γg=0; wgeq=0; χg=20γu; and χu=40γu. The expression in the timing condition j1(t2-t1)+j2(t4-t3)+j3(t6-t5)=-0.04/γu in (b), 0 in (c), and 0.004/γu in (d), where j1=-1, j2=j3=1. (b) The optical echo is expected at toe=40.04/γu, and in (c) the Raman echo is expected at te=tp=40.00/γu. Identical time-scale is used in (a)–(e). Note that the Raman echo |σg(t)| in (c) is not influenced by the optical fields at tp.

Fig. 8
Fig. 8

(a) Plot of Y(ξ, η)Y(χu/γu, δu/γu) shows the dependence of the averaged signal (30) on the normalized optical inhomogeneous broadening χu/γu and the normalized common-mode detuning δu/γu of the center of the inhomogeneous spectrum. Note that Y is an even function of the detuning ηδu/γu. (b) Cross section Y(0, η)Y(0, δu/γu) of the plot in (a) gives the δu dependence of the signal before the average is calculated.

Fig. 9
Fig. 9

(a) Structure of the matrix representing the nth-order encoded solution of a density-matrix element. (b) Structure of the segment for tq.

Equations (54)

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E(r, t)=12l=1, 2l(t)exp{iωl[nl(t)r/c-t]}+c.c.,
H=-ω310V1*(t)0-ω32V2*(t)V1(t)V2(t)0,
Vl(t)=-(pl/2){l(t)exp{iωl[nl(t)r/c-t]}+c.c.},
l=1, 2
ρt=i [ρ, H]+(decayterms).
UR=exp(-iω1t)000exp(-iω2t)0001,
ρR=URρUR-1=ρ1σg*σ1*σgρ2σ2*σ1σ2ρu
ρRt=i [ρR, HR]+(decayterms),
HR=URHUR-1+i URt UR-1=-Δ10-2 Ω1*0-Δ2-2 Ω2*-2 Ω1-2 Ω20.
Ωl(t)=-2 Vl(t)exp(iωlt)pll(t)exp[iωlnl(t)r/c],l=1, 2,
t wg=i2 (Ω1*σ1-Ω1σ1*)-i2 (Ω2*σ2-Ω2σ2*)-(Γ12+Γ21)wg+(Γ31-Γ32+Γ12-Γ21)ρu-(Γ12-Γ21).
t wg=i2 (Ω1*σ1-Ω1σ1*)-i2 (Ω2*σ2-Ω2σ2*)-Γgwg,
t ρu=-i2 (Ω1*σ1-Ω1σ1*)-i2 (Ω2*σ2-Ω2σ2*)-Γuρu,
t σg=i2 (Ω2*σ1-Ω1σ2*)-(γg+iδg)σg,
t σ1=i4 Ω1(1+wg-3ρu)+i2 Ω2σg-(γ1+iΔ1)σ1,
t σ2=i4 Ω2(1-wg-3ρu)+i2 Ω1σg*-(γ2+iΔ2)σ2.
σg(n)(t)=i2 -tdtn{Ω2*(tn)σ1(n-1)(tn)-Ω1(tn)[σ2(n-1)(tn)]*}×exp[-(γg+iδg)(t-tn)].
σ1(n+1)(t)=i2 -tdtn+1Ω2(tn+1)σg(n)(tn+1)×exp[-(γu+iΔ1)(t-tn+1)].
ρ(n)(t)=1πχuχg dδudδg×exp-δu-δuχu2-δg-δgχg2×ρ(n)(t, δu, δg),
σg(2)(t)=-18 -tdt2-t2dt1exp[-χg2(t-ts)2/4-γg(t-t2)-γu(t2-t1)]×exp[-χu2(t2-t1)2/4]×[Ω1(t1)Ω2*(t2)+Ω2*(t1)Ω1(t2)],
ts=(t1+t2)/2,
σg(4)(t)±-tdt4-t4dt3-t3dt2-t2dt1×Ω1,2(*)(t1)Ω1,2(*)(t2)Ω1,2(*)(t3)Ω1,2(*)(t4)×exp[-χg2(t-ts)2/4-γu(t2-t1+t4-t3)]×exp[-χu2[±(t4-t3)±(t2-t1)]2/4]×f(γg, Γg, Γu, t2, t3, t4, t),
ts=(t4+t3)/2±(t2-t1)/2
orts=(t4-t3)/2+(t2+t1)/2.
[σg(6)(t)]e
=-2-11-tdt6-t6dt5-t5dt4-t4dt3-t3dt2-t2dt1 ×exp-iδgt-t5+t62-t3+t42+t1+t22-γg(t-t6+t3-t2)-Γg(t5-t4)×exp[-γu(t2-t1+t4-t3+t6-t5)]×j1,j2,j3=±1exp{-iδu[j1(t2-t1)+j2(t4-t3)+j3(t6-t5)]}×[(1-j1)Ω1*(t1)Ω2(t2)+(1+j1)Ω2(t1)Ω1*(t2)]×[(1-j2)Ω2*(t3)Ω1(t4)-(1+j2)Ω1(t3)Ω2*(t4)]×[-(1-j3)Ω2*(t5)Ω1(t6)+(1+j3)Ω1(t5)Ω2*(t6)],
σg(6)(t)e
=-2-11-tdt6-t6dt5-t5dt4-t4dt3-t3dt2-t2dt1×exp-χg24t-t5+t62-t3+t42+t1+t22-γg(t-t6+t3-t2)-Γg(t5-t4)2
×exp[-γu(t2-t1+t4-t3+t6-t5)]×j1,j2,j3=±1exp{-χu2[j1(t2-t1)+j2(t4-t3)
+j3(t6-t5)]2/4}×[(1-j1)Ω1*(t1)Ω2(t2)+(1+j1)Ω2(t1)Ω1*(t2)]×[(1-j2)Ω2*(t3)Ω1(t4)-(1+j2)Ω1(t3)Ω2*(t4)]×[-(1-j3)Ω2*(t5)Ω1(t6)+(1+j3)Ω1(t5)Ω2*(t6)].
exp[-χg2(t-ts)2/4]
×exp[-γu(t2-t1+t4-t3+t6-t5)].
te=t5+t62+t3+t42-t1+t22.
j1(t2-t1)+j2(t4-t3)+j3(t6-t5)=0.
t7=t5+t62+t3+t42-t1+t22+12 [j1(t2-t1)+j2(t4-t3)+j3(t6-t5)],
toe=t5+t62+t3+t42-t1+t22-12 [j1(t2-t1)+j2(t4-t3)+j3(t6-t5)].
j1(t2-t1)+j2(t4-t3)+j3(t6-t5)<0.
k1e=k2p-Kd+Ks+Kr,
σg(6)(t)e=-2-7γu-3Yχuγu, δuγu×-tdtr-trdts-tsdtd×[Ω1(td)Ω2*(td)]*Ω1(ts)Ω2*(ts)×Ω1(tr)Ω2*(tr)×exp[-χg2(t-tr-ts+td)2/4-γg(t-tr+ts-td)-Γg(tr-ts)]×exp[-iδg(t-tr-ts+td)],
Y(ξ, η)=12 0(1+x-x2)×cos(ηx)exp(-x-ξ2x2/4)dx.
Ω1(tm)Ω2*(tm)=(p1p2*/2)1(tm)2*(tm)×exp{i[k1(tm)-k2(tm)]r},
sj1,j2,j3δuγu=-12 1(1+ij1δu/γu)j2(1+ij2δu/γu)×j3(1+ij3δu/γu).
Y0, δuγu=j1, j2, j3=±1sj1, j2, j3δuγu.
σ1(7)(t)
=-(i2-12)×-tdt7Ω2(t7)-t7dt6-t6dt5-t5dt4-t4dt3-t3dt2-t2dt1exp[-χg2(t+t7-t5-t6-t3-t4+t1+t2)2/16]exp[-γg(t7-t6+t3-t2)-Γg(t5-t4)]exp[-γu(t2-t1+t4-t3+t6-t5+t-t7)]j1,j2,j3=±1 exp{-χu2[t-t7+j1(t2-t1)+j2(t4-t3)+j3(t6-t5)]/4}×[(1-j1)Ω1*(t1)Ω2(t2)+(1+j1)Ω2(t1)Ω1*(t2)]×[(1-j2)Ω2*(t3)Ω1(t4)-(1+j2)Ω1(t3)Ω2*(t4)]×[-(1-j3)Ω2*(t5)Ω1(t6)+(1+j3)Ω1(t5)Ω2*(t6)].
σg(6)(t)e
=-2-11-tdt6-t6-τ3dt4-t4-τ2dt2×0dτ30dτ20dτ1exp[-χg2(t-t6+τ3/2-t4+τ2/2+t2-τ1/2)2/4]exp[-iδg(t-t6+τ3/2-t4+τ2/2+t2-τ1/2)]exp[-γg(t-t6+t4-τ2-t2)-Γg(t6-τ3-t4)-γu(τ1+τ2+τ3)]×j1,j2,j3=±1 exp[-χu2(j1τ1+j2τ2+j3τ3)2/4-iδu(j1τ1+j2τ2+j3τ3)]×[(1-j1)Ω1*(t2-τ1)Ω2(t2)+(1+j1)Ω2(t2-τ1)Ω1*(t2)]×[(1-j2)Ω2*(t4-τ2)Ω1(t4)-(1+j2)Ω1(t4-τ2)Ω2*(t4)][-(1-j3)Ω2*(t6-τ3)Ω1(t6)+(1+j3)Ω1(t6-τ3)Ω2*(t6)].
σg(6)(t)e=-2-7γu-3Yχuγu, δuγu×-tdt6-t6dt4-t4dt2Ω1*(t2)×Ω2(t2)Ω1(t4)Ω2*(t4)Ω1(t6)Ω2*(t6)×exp[-χg2(t-t6-t4+t2)2/4-γg(t-t6+t4-t2)-Γg(t6-t4)]×exp[-iδg(t-t6-t4+t2)].
Y1(ξ, η)=0dx30dx20dx1 exp-(x1+x2+x3)-ξ24 (x1+x2+x3)2-iη(x1+x2+x3)=120x2 exp(-x-ξ2x2/4-iηx)dx,
Y0(ξ, η)=0dx30dx20dx1 exp-(x1+x2+x3)-ξ24 (x1+x2-x3)2-iη(x1+x2-x3)=120[cos(ηx)+x exp(-iηx)]×exp(-x-ξ2x2/4)dx.
σ1(7)(t)e=-i2-9γu-4YχuγuΩ2(t)×-tdtr-trdts-tsdtd[Ω1(td)Ω2*(td)]*×Ω1(ts)Ω2*(ts)Ω1(tr)Ω2*(tr)×exp[-χg2(t-tr-ts+td)2/4-×γg(t-tr+ts-td)-Γg(tr-ts)],
Y(ξ)=Y0(ξ)-Y1(ξ)=120(1+x-x3/3)×exp(-x-ξ2x2/4)dx,
Y1(ξ)=0dx40dx30dx20dx1×exp-(x1+x2+x3+x4)-ξ24 (x1+x2+x3+x4)2=16 0x3 exp(-x-ξ2x2/4)dx,
Y0(ξ)=0dx40dx30dx20dx1×exp-(x1+x2+x3+x4)-ξ24 (x1+x2-x3-x4)2=12 0(1+x)exp(-x-ξ2x2/4)dx.

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