Abstract

A quantum-mechanical description of a vibronic multimode standing-wave transition-metal ion laser is presented. The impurity ion levels are coupled both by the radiation field and by the phonons of the host lattice. A dynamic Heisenberg–Langevin set of equations for the material system and the photon and phonon operators, including radiative and nonradiative damping terms and quantum-stochastic forces, has been derived. The numerical solutions of those equations show the crucial role of the phonons in the laser dynamics. The competition between the photons and the phonons leads to regular stable self-pulsation for certain sets of parameters.

© 1999 Optical Society of America

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References

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  1. Hong Fu and H. Haken, “Semiclassical dye-laser equations and the unidirectional single-frequency operation,” Phys. Rev. A 36, 4802 (1987); “Multichromatic operations in cw dye lasers,” Phys. Rev. Lett. 60, 2614 (1988).
    [CrossRef] [PubMed]
  2. S. A. Kovalenko, “Quantum intensity fluctuations in multimode cw lasers and maximum sensitivity of intracavity laser spectroscopy,” Sov. J. Quantum Electron. 11, 759 (1981).
    [CrossRef]
  3. M. G. Raymer, Z. Deng, and M. Beck, “Strong-field dynamics of a multimode, standing-wave dye laser,” J. Opt. Soc. Am. B 5, 1588 (1988).
    [CrossRef]
  4. M. Beck, I. McMackin, and M. G. Raymer, “Transition from quantum-noise-driven dynamics to deterministic dynamics in a multimode laser,” Phys. Rev. A 40, 2410 (1989).
    [CrossRef] [PubMed]
  5. S. E. Hodges, M. Munroe, J. Cooper, and M. G. Raymer, “Multimode laser model with coupled cavities and quantum noise,” J. Opt. Soc. Am. B 14, 191 (1997).
    [CrossRef]
  6. D. E. McCumber, “Einstein relations connecting broadband emission and absorption spectra,” Phys. Rev. A 136, 954 (1964).
    [CrossRef]
  7. J. C. Walling, O. G. Peterson, H. P. Jenssen, R. C. Morris, and E. W. O’Dell, “Tunable alexandrite lasers,” IEEE J. Quantum Electron. QE-16, 1302 (1980).
    [CrossRef]
  8. J. C. Walling, “Tunable paramagnetic solid state lasers,” in Tunable Lasers, L. F. Mollenauer and J. C. White, eds. (Springer-Verlag, Berlin, 1987), p. 344.
  9. J. C. Walling, D. F. Heller, H. Samelson, D. J. Harter, J. A. Pete, and R. C. Morris, “Tunable alexandrite lasers: development and performance,” IEEE J. Quantum Electron. QE-21, 1568 (1985).
    [CrossRef]
  10. W. S. Mashkievitch, Kinetic Theory of Lasers (Nauka, Moscow, 1971; in Russian).
  11. S. G. Demos, J. M. Buchert, and R. R. Alfano, “Time resolved nonequilibrium phonon dynamics in the nonradiative decay of photoexcited forsterite,” Appl. Phys. Lett. 61, 660 (1992); S. G. Demos and R. R. Alfano, “Subpicosecond time-resolved Raman investigation of optical phonon modes in Cr-doped forsterite,” Phys. Rev. B 52, 987 (1995).
    [CrossRef]
  12. S. K. Gayen, W. B. Wang, V. Petricević, and R. R. Alfano, “Nonradiative transition dynamics in alexandrite,” Appl. Phys. Lett. 49, 437 (1986); S. K. Gayen, W. B. Wang, V. Petricević, K. M. Yoo, and R. R. Alfano, “Picosecond excite-and-probe absorption measurement of the intra-2EgE3/2-state vibrational relaxation time in Ti3+:Al2O3,” Appl. Phys. Lett. 50, 1494 (1987); S. K. Gayen, W. B. Wang, V. Petricević, R. Dorsinville, and R. R. Alfano, “Picosecond excite-and-probe absorption measurement of the 4T2 state nonradiative lifetime in ruby,” Appl. Phys. Lett. APPLAB 47, 454 (1985).
    [CrossRef]
  13. R. Englman, Non-Radiative Decay of Ions and Molecules in Solids (North-Holland, Amsterdam, 1979).
  14. M. H. L. Pryce, “Interaction of vibrations with electrons at point defects,” in Phonons in Perfect Lattices and in Lattices with Point Imperfections, R. W. H. Sevenson, ed. (Oliver & Boyd, Edinburgh, 1966).
  15. H. Haken, Light (North-Holland, Amsterdam, 1981), Vol. 1; Quantum Field Theory of Solids (North-Holland, Amsterdam, 1976).
  16. W. Gadomski and B. Ratajska-Gadomska, “Parametric bistable resonance in coherent Raman scattering in crystals,” Phys. Rev. A 34, 1277 (1986).
    [CrossRef] [PubMed]
  17. P. Meystre and M. Sargent III, Elements of Quantum Optics (Springer-Verlag, Berlin, 1991).
  18. W. Gadomski and B. Gadomska, “Self-pulsations in phonon-assisted lasers,” J. Opt. Soc. Am. B 15, 2689 (1998).
    [CrossRef]

1998 (1)

1997 (1)

1989 (1)

M. Beck, I. McMackin, and M. G. Raymer, “Transition from quantum-noise-driven dynamics to deterministic dynamics in a multimode laser,” Phys. Rev. A 40, 2410 (1989).
[CrossRef] [PubMed]

1988 (1)

1986 (1)

W. Gadomski and B. Ratajska-Gadomska, “Parametric bistable resonance in coherent Raman scattering in crystals,” Phys. Rev. A 34, 1277 (1986).
[CrossRef] [PubMed]

1985 (1)

J. C. Walling, D. F. Heller, H. Samelson, D. J. Harter, J. A. Pete, and R. C. Morris, “Tunable alexandrite lasers: development and performance,” IEEE J. Quantum Electron. QE-21, 1568 (1985).
[CrossRef]

1981 (1)

S. A. Kovalenko, “Quantum intensity fluctuations in multimode cw lasers and maximum sensitivity of intracavity laser spectroscopy,” Sov. J. Quantum Electron. 11, 759 (1981).
[CrossRef]

1980 (1)

J. C. Walling, O. G. Peterson, H. P. Jenssen, R. C. Morris, and E. W. O’Dell, “Tunable alexandrite lasers,” IEEE J. Quantum Electron. QE-16, 1302 (1980).
[CrossRef]

1964 (1)

D. E. McCumber, “Einstein relations connecting broadband emission and absorption spectra,” Phys. Rev. A 136, 954 (1964).
[CrossRef]

Beck, M.

M. Beck, I. McMackin, and M. G. Raymer, “Transition from quantum-noise-driven dynamics to deterministic dynamics in a multimode laser,” Phys. Rev. A 40, 2410 (1989).
[CrossRef] [PubMed]

M. G. Raymer, Z. Deng, and M. Beck, “Strong-field dynamics of a multimode, standing-wave dye laser,” J. Opt. Soc. Am. B 5, 1588 (1988).
[CrossRef]

Cooper, J.

Deng, Z.

Gadomska, B.

Gadomski, W.

W. Gadomski and B. Gadomska, “Self-pulsations in phonon-assisted lasers,” J. Opt. Soc. Am. B 15, 2689 (1998).
[CrossRef]

W. Gadomski and B. Ratajska-Gadomska, “Parametric bistable resonance in coherent Raman scattering in crystals,” Phys. Rev. A 34, 1277 (1986).
[CrossRef] [PubMed]

Harter, D. J.

J. C. Walling, D. F. Heller, H. Samelson, D. J. Harter, J. A. Pete, and R. C. Morris, “Tunable alexandrite lasers: development and performance,” IEEE J. Quantum Electron. QE-21, 1568 (1985).
[CrossRef]

Heller, D. F.

J. C. Walling, D. F. Heller, H. Samelson, D. J. Harter, J. A. Pete, and R. C. Morris, “Tunable alexandrite lasers: development and performance,” IEEE J. Quantum Electron. QE-21, 1568 (1985).
[CrossRef]

Hodges, S. E.

Jenssen, H. P.

J. C. Walling, O. G. Peterson, H. P. Jenssen, R. C. Morris, and E. W. O’Dell, “Tunable alexandrite lasers,” IEEE J. Quantum Electron. QE-16, 1302 (1980).
[CrossRef]

Kovalenko, S. A.

S. A. Kovalenko, “Quantum intensity fluctuations in multimode cw lasers and maximum sensitivity of intracavity laser spectroscopy,” Sov. J. Quantum Electron. 11, 759 (1981).
[CrossRef]

McCumber, D. E.

D. E. McCumber, “Einstein relations connecting broadband emission and absorption spectra,” Phys. Rev. A 136, 954 (1964).
[CrossRef]

McMackin, I.

M. Beck, I. McMackin, and M. G. Raymer, “Transition from quantum-noise-driven dynamics to deterministic dynamics in a multimode laser,” Phys. Rev. A 40, 2410 (1989).
[CrossRef] [PubMed]

Morris, R. C.

J. C. Walling, D. F. Heller, H. Samelson, D. J. Harter, J. A. Pete, and R. C. Morris, “Tunable alexandrite lasers: development and performance,” IEEE J. Quantum Electron. QE-21, 1568 (1985).
[CrossRef]

J. C. Walling, O. G. Peterson, H. P. Jenssen, R. C. Morris, and E. W. O’Dell, “Tunable alexandrite lasers,” IEEE J. Quantum Electron. QE-16, 1302 (1980).
[CrossRef]

Munroe, M.

O’Dell, E. W.

J. C. Walling, O. G. Peterson, H. P. Jenssen, R. C. Morris, and E. W. O’Dell, “Tunable alexandrite lasers,” IEEE J. Quantum Electron. QE-16, 1302 (1980).
[CrossRef]

Pete, J. A.

J. C. Walling, D. F. Heller, H. Samelson, D. J. Harter, J. A. Pete, and R. C. Morris, “Tunable alexandrite lasers: development and performance,” IEEE J. Quantum Electron. QE-21, 1568 (1985).
[CrossRef]

Peterson, O. G.

J. C. Walling, O. G. Peterson, H. P. Jenssen, R. C. Morris, and E. W. O’Dell, “Tunable alexandrite lasers,” IEEE J. Quantum Electron. QE-16, 1302 (1980).
[CrossRef]

Ratajska-Gadomska, B.

W. Gadomski and B. Ratajska-Gadomska, “Parametric bistable resonance in coherent Raman scattering in crystals,” Phys. Rev. A 34, 1277 (1986).
[CrossRef] [PubMed]

Raymer, M. G.

Samelson, H.

J. C. Walling, D. F. Heller, H. Samelson, D. J. Harter, J. A. Pete, and R. C. Morris, “Tunable alexandrite lasers: development and performance,” IEEE J. Quantum Electron. QE-21, 1568 (1985).
[CrossRef]

Walling, J. C.

J. C. Walling, D. F. Heller, H. Samelson, D. J. Harter, J. A. Pete, and R. C. Morris, “Tunable alexandrite lasers: development and performance,” IEEE J. Quantum Electron. QE-21, 1568 (1985).
[CrossRef]

J. C. Walling, O. G. Peterson, H. P. Jenssen, R. C. Morris, and E. W. O’Dell, “Tunable alexandrite lasers,” IEEE J. Quantum Electron. QE-16, 1302 (1980).
[CrossRef]

IEEE J. Quantum Electron. (2)

J. C. Walling, O. G. Peterson, H. P. Jenssen, R. C. Morris, and E. W. O’Dell, “Tunable alexandrite lasers,” IEEE J. Quantum Electron. QE-16, 1302 (1980).
[CrossRef]

J. C. Walling, D. F. Heller, H. Samelson, D. J. Harter, J. A. Pete, and R. C. Morris, “Tunable alexandrite lasers: development and performance,” IEEE J. Quantum Electron. QE-21, 1568 (1985).
[CrossRef]

J. Opt. Soc. Am. B (3)

Phys. Rev. A (3)

M. Beck, I. McMackin, and M. G. Raymer, “Transition from quantum-noise-driven dynamics to deterministic dynamics in a multimode laser,” Phys. Rev. A 40, 2410 (1989).
[CrossRef] [PubMed]

D. E. McCumber, “Einstein relations connecting broadband emission and absorption spectra,” Phys. Rev. A 136, 954 (1964).
[CrossRef]

W. Gadomski and B. Ratajska-Gadomska, “Parametric bistable resonance in coherent Raman scattering in crystals,” Phys. Rev. A 34, 1277 (1986).
[CrossRef] [PubMed]

Sov. J. Quantum Electron. (1)

S. A. Kovalenko, “Quantum intensity fluctuations in multimode cw lasers and maximum sensitivity of intracavity laser spectroscopy,” Sov. J. Quantum Electron. 11, 759 (1981).
[CrossRef]

Other (9)

Hong Fu and H. Haken, “Semiclassical dye-laser equations and the unidirectional single-frequency operation,” Phys. Rev. A 36, 4802 (1987); “Multichromatic operations in cw dye lasers,” Phys. Rev. Lett. 60, 2614 (1988).
[CrossRef] [PubMed]

J. C. Walling, “Tunable paramagnetic solid state lasers,” in Tunable Lasers, L. F. Mollenauer and J. C. White, eds. (Springer-Verlag, Berlin, 1987), p. 344.

P. Meystre and M. Sargent III, Elements of Quantum Optics (Springer-Verlag, Berlin, 1991).

W. S. Mashkievitch, Kinetic Theory of Lasers (Nauka, Moscow, 1971; in Russian).

S. G. Demos, J. M. Buchert, and R. R. Alfano, “Time resolved nonequilibrium phonon dynamics in the nonradiative decay of photoexcited forsterite,” Appl. Phys. Lett. 61, 660 (1992); S. G. Demos and R. R. Alfano, “Subpicosecond time-resolved Raman investigation of optical phonon modes in Cr-doped forsterite,” Phys. Rev. B 52, 987 (1995).
[CrossRef]

S. K. Gayen, W. B. Wang, V. Petricević, and R. R. Alfano, “Nonradiative transition dynamics in alexandrite,” Appl. Phys. Lett. 49, 437 (1986); S. K. Gayen, W. B. Wang, V. Petricević, K. M. Yoo, and R. R. Alfano, “Picosecond excite-and-probe absorption measurement of the intra-2EgE3/2-state vibrational relaxation time in Ti3+:Al2O3,” Appl. Phys. Lett. 50, 1494 (1987); S. K. Gayen, W. B. Wang, V. Petricević, R. Dorsinville, and R. R. Alfano, “Picosecond excite-and-probe absorption measurement of the 4T2 state nonradiative lifetime in ruby,” Appl. Phys. Lett. APPLAB 47, 454 (1985).
[CrossRef]

R. Englman, Non-Radiative Decay of Ions and Molecules in Solids (North-Holland, Amsterdam, 1979).

M. H. L. Pryce, “Interaction of vibrations with electrons at point defects,” in Phonons in Perfect Lattices and in Lattices with Point Imperfections, R. W. H. Sevenson, ed. (Oliver & Boyd, Edinburgh, 1966).

H. Haken, Light (North-Holland, Amsterdam, 1981), Vol. 1; Quantum Field Theory of Solids (North-Holland, Amsterdam, 1976).

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Figures (3)

Fig. 1
Fig. 1

Energy-level diagram of the impurity ion in the transition-metal ion laser.

Fig. 2
Fig. 2

Transient state; the system tends to stable output. (a) The stable part of population inversion and the phonon number, (b) photon numbers of the I0, I1, and I2 modes.

Fig. 3
Fig. 3

Regular stable pulsation. (a) The stable part of population inversion and the phonon number, (b) the total photon numbers in the I and the I1 modes.

Equations (135)

Equations on this page are rendered with MathJax. Learn more.

|i,{ni}, nα=(nα!)-1/2bk1i+bknii+(Aαi+)nαΦ0i,
|i{ni}=bk1i+bknii+φ0e,
|nαi=(nαi!)-1/2(Aαi)nαiΨ0i,
i, {ni}|j, {nj}=δijδ{ni}{nj},
nα(i)|mα(i)=δnm,
nα(i)|mα(j)=exp[-|Aαi(o)-Aαj(o)|2/2]×-[Aαi(o)*-Aαj(o)*]mα[Aαi(o)-Aαj(o)]nα+rmα!nα!(-1)mα-r(mα-r)!(nα-r)!r!×[Aαi(o)*-Aαj(o)*]mα-r×[Aαi(o)-Aαj(o)]nα-r.
bμinα+bμjnα=bμi+bμj|nα(i)nα(j)|=(nα!nα!)-1/2bμi+bμj(Aαi+)nα|Ψ0i×Ψ0j|(Aαi)nα.
[bμinα+bμjnαbνkmα+bνpmα]=δμν(bμinα+bμpmαδkjδnαmα-bμkmα+bμjnαδipδmαnα).
binα+bjnα(z)=μδ(z-zμ)bμinα+bμjnα.
binα+binα=μδ(z-zμ)i, ni, nα|bμinα+bμinα|i, ni, nα=ni.
E0=E0e0{a0 exp[i(ω0t-K0z)]+a0+ exp[-i(ω0t-K0z)]},
E=eˆλi(aλ+-aλ)Eλuλ(z),
d2dz2uλ+Kλ2uλ=0,
ωλ=ω¯-Δλ=ω¯-λπcL.
EB=ri(Br+-Br)Erurreˆr.
ρF=ρFLρB,
ρFL={nk}ρ{nk}{nk}(0)|{nk}{nk}|,
ρB=1Z{nr} exp-1kTrnrωr|{nr}{nr}|.
H=HM+HF+HB+HMF+HMB,
HE=μiEibμi+bμi
Hp=iΩα(Aαi+Aαi+1/2)+βΩβ(Aβ+Aβ+1/2)+χ,β,β[H(3)(χ, β, β)Aχ+AβAβ+c.c.]
HPE=μsβββnβ[F3,5s(βββ)(Aβ+)nβAβ+Aβ+bμ3+bμ5s+F5s,3(βββ)(Aβ)nβAβAβbμ5s+bμ3]+μmβ[F4m,3(β)Aβbμ4m+bμ3+F3,4m(β)Aβ+bμ3+bμ4m]+pββ[F2p,1*bμ1+bμ2pAβ+Aβ++F2p,1bμ2p+bμ1AβAβ],
Finα,jnα(β1βk)=Fjnα,inα*(β1βk)=kMkΩ1Ωk1/2×Jnαnαi|HPE(k)(β1βk)|j
F2p,1=H(3)(α, β, β)Aα1(0)Jp0.
HF=λωλaλ+aλ+ω0a0+a0,
HFM=μV5s,10bμ5s+bμ1a0-V1,5s0bμ1+bμ5sa0++mpλ(V4m,2pλbμ4m+bμ2paλ-V2p,4mλbμ2p+bμ4maλ+),
Vinα,jnαλ=iμijJnαnαEλeλuλ(z),Vinα,jnαλ*=-Vjnα,inαλ
HB=rωrBr+Br,
HBM=μrijαα[Viα,jαrbμiα+bμjαBr+Viα,jαr*bμiα+bμjαBr+].
ddtbkmα+blmα=i(Ekmα-Elmα)bkmα+blmα+iλ,i,nα,(Vinα,kmαλbinα+blmα-Vlmα,inαλbkmα+binα)(aλ-aλ+)+iβ1β2β3βki,nα(Finα,kmαbinα+blmα-Flmα,inαbkmα+binα)kAβk+kAβk++ir,i,nα(Vinα,kmαrbinα+blmα-Vlmα,inαrbkmα+binα)(Br-Br+),
ddtbkmα+blmα=-ddtblmα+bkmα.
ddtaλ=-iωλaλ-κλaλ+ip,mV2p,4mλb2p+b4m,
ddtBr=-iωrBr+ii,j,nα,nαVinα,jnαrbinα+bjnα.
ddtAβ=-iΩβAβ-ΓβAβ-iβ1β2β3βknβ..nβki,j,na,nanβ×Fina,jna(ββ1β2β3βk)bina+bjna(Aβ+)nβ-1×k(Aβk+)nβk+Φβ(t),
Γβ=12β,β|H(3)(β,β,β)|2×|Aβ(0)|2|Aβ(0)|2δ(Ωβ±Ωβ±Ωβ)
Φβ(t)Φβ*(t)=ΩβΓβ|Aβ(0)|2exp[iΩβ(t-t)];
ddtb5s+b1=-γ51+i(E5s-E1)b5s+b1+iV1,5s0(b1+b1-b5s+b5s)a0++Φ51(t),
ddtb5s+b5s=-γ5b5s+b5s+i(V1,5s0b1+b5sa0++V5s,10b5s+b1a0)+Φ55(t)+iβββnβ[F3,5sb3+b5s(Aβ+)nβAβ+Aβ+-F5s,3b5s+b3(Aβ)nβAβAβ],
ddtb3+b5s=i(E3-E5s)-γ35b3+b5s+Φ35(t)+iβββnβF5s,3(b5s+b5s-b3+b3)×(Aβ)nβAβAβ+imβF4m,3b4m+b5sAβ,
ddtb3+b3=-γ3b3+b3-imβ(F3,4mb3+b4mAβ+-F4m,3b4m+b3Aβ)+Φ33(t)-iβββnβ[F3,5sb3+b5s(Aβ+)nβAβ+Aβ+-F5s,3b5s+b3(Aβ)nβ],
ddtb4m+b3=i(E4m-E3)-γ43b4m+b3-iβββnβF3,5sb4m+b5s(Aβ+)nβAβ+Aβ+-iβF3,4m(b4m+b4m-b3+b3)Aβ+-pλV2p,4mλb2p+b3+Φ43(t),
ddtb4m+b4m=-γ4b4m+b4m+imβ(F3,4mb3+b4mAβ+-F4m,3b4m+b3Aβ)-ipλ(V2p,4mλb2p+b4maλ++V4m,2pλb4m+b2paλ)+Φ44(t),
ddtb2p+b4m=i(E2p-E4m)-γ24b2p+b4m-iβF4m,3b2p+b3Aβ+iββF1,2pb1+b4mAβ+Aβ++iλV4µ,2p(b4m+b4m-b2p+b2p)aλ+Φ24(t),
ddtb2p+b2p=iββ(F1,2pb1+b2pAβ+Aβ+-F2p,1b2p+b1AβAβ)+mW2p,4mb4m+b4m+imλ(V4m,2pλb4m+b2paλ+V2p,4mλb2p+b4maλ+)+Φ22(t),
ddtb1+b2p=i(E1-E2p)+iββF2p,1(b2p+b2p-b1+b1)AβAβ-imλV2p,4mλb1+b4maλ++isV5s,10b5s+b2pa0,
ddtb1+b1=iββ(F2p,1b2p+b1AβAβ-F1,2pb1+b2pAβ+Aβ+)+Φ11(t)-is(V5s,10b5s+b1a0+V1,5s0b1+b5sa0+)+W1,3b3+b3+mW1,4mb4m+b4m+sW1,5sb5s+b5s.
ddtbi+b1=i(Ei-E1)-γi1bi+b1+Φi1(t),
γk=i,naWkma,ina,
Wkma,ina
=(Eina-Ekma)334c3e0|μki|2|Jnama|2(Eina-Ekma)>00(Eina-Ekma)<0,
ΦklΦkmαlmα(t)=ir,i,nα(Vinα,kmαrbinα+blmα-Vlmα,inαrbkmα+binα)×[Br(0)exp(-iωrt)-Br+(0)exp(iωrt)]
Aβ+(t)=A˜β+(t)exp[(-Γ+iΩβ)t],
bin+bjm(t)=b˜in+b˜jm(t)exp1(Ein-Ejm)t,
Aβ(t)=-iΓββnβ1Γ+i[(nβΩβ+Ωβ+Ωβ)-(E5s-E3)/]×F3,5n(βββ)b3+b5s(nβ!)-1/2[A˜β+(t)]nβA˜β(t)exp{i[(nβΩβ+Ωβ+Ωβ)t]}+1ΓΦβ(0)exp(-iΩβt)
Aβ(t)=-iΓβ1Γ+i(Ωβ+Ωβ-pΩα)×F1,2ρ(β, β)b1+b2ρA˜β+(t)exp[i(Ωβ+Ωβ)t]+1ΓΦβ(0)exp(-iΩβt)
bi+b1(t)=iV1,i0(b1+b1-bi+bi)a0+(t)+Φi1(t)×exp{[i/(Ei-E1)-γi](t-t)}dt,
(V1,i0=0fori=3, 4),
b5s+b1(t)=iγ51V1,5s0b1+b1(t)a0+×f(E5s-E1-ω0)+G51(t),
bi+b1=Gi1(t),
Gj1(t)=iγj1rV1jr(b1+b1-bj+bj)Br+(0)×f(Ej-E1-ωr)exp(iωrt),
j=3, 4m, 5s,
ddtb5s+b5s=-γ5b5s+b5s+12γ5|V5s,10|2I0b1+b1+βnβ[B3,5snβb3+b3(Aβ+Aβ)nβ-B5s,3nβb5s+b5s(Aβ+Aβ+1)nβ]+Φ55S+Φ¯55(t),
ddtb3+b3=-γ3b3+b3-βnβB3,5snβb3+b3(Aβ+Aβ)nβ-sB5s,3nβb5s+b5s(Aβ+Aβ+1)nβ-imβ(F3,4mb3+b4mAβ+-F4m,3b4m+b3Aβ)+Φ33S,
ddtb4m+b3=i(E4m-E3)-γ43b4m+b3-iβF3,4m(b4m+b4m-b3+b3)Aβ+-ipλV2p,4mλb2p+b3aλ++Φ43S,
ddtb4m+b4m=-γ4b4m+b4m+imβ(F3,4mb3+b4mAβ+-F4m,3b4m+b3Aβ)-ipλ(V2p,4mλb2p+b4maλ++V4m,2pλb4m+b2paλ)+Φ44S+Φ¯44(t),
ddtb2p+b4m=i(E2p-E4m)-γ4-γ2p1NRb2p+b4m-iβF4m,3b2p+b3Aβ+iλV4µ,2p(b4m+b4m-b2p+b2p)aλ+Φ24(t),
ddtb2p+b2p=-γ2p1NRb2p+b2p+γ12pNRb1+b1+mW2p,4mb4m+b4m+imλ(V4m,2pλb4m+b2paλ+V2p,4mλb2p+b4maλ+)+Φ55S+Φ¯22(t),
ddtb1+b1=-12γ51|V5n,10|2I0+ργ12pNRb1+b1+γ3b3+b3+mγ4b4m+b4m+sγ5b5s+b5s+ργ2p1NRb2p+b2p+Φ11S+Φ¯11(t),
γ2p1NR=12ββ|F2p1|2Γβ1Γβ12+(Ωβ+Ωβ-pΩα)2×[(AβAβ+)+(AβAβ+)],
γ12pNR=12ββ|F2p1|2Γβ1Γβ12+(Ωβ+Ωβ-pΩα)2×[(Aβ+Aβ)+(Aβ+Aβ)],
γ5s3NR(Nβ)=B5s,3nβ(Aβ+Aβ+1)nβ,
B5s,3nβ=12ββ|F53|2×ΓβΓβ2+Ωβ+Ωβ+nβΩβ-1(E5s-E3)2×[(AβAβ+)+(AβAβ+)].
Φ55S=12γ5r|V5s,1r|2b1+b1Br+Brf(E5s-E1-ωr)+12γ53NRββ|F3,5s|2b3+b3(Nβ+1)nβ×Aβ+Aβ(0)Aβ+Aβ(0)×f(E5s-E1-nβΩβ-Ωβ-Ωβ),
Φ33S=12γ3r|V3,1r|2b1+b1Br+Brf(E3-E1-ωr)-12γ53NRββ|F3,5s|2b3+b3(Nβ+1)nβ×Aβ+Aβ(0)Aβ+Aβ(0)×f(E5s-E1-nβΩβ-Ωβ-Ωβ),
Φ44S=12γ4r|V4m,1r|2b1+b1Br+Brf(E4m-E1-ωr),
Φ22S=12γ53NRββ|F2ρ,1|2b1+b1Aβ+AβAβ+Aβ×f(E2ρ-E1-Ωβ-Ωβ-Ωβ),
Φ11S=-Φ33S-Φ44S-Φ55S-Φ22S,
Φ34S=12rV14mrV31r[(b1+b1-b3+b3)BrBr+×f(E3-E1-ωr)/γ3+(b1+b1-b4m+b4m)Br+Br×f(E4-E1-ωr)/γ4].
Φ¯55(t)=12γ51r{V15srV5s10a0+Br(0)×exp[i(E5s-E1-ωr)t/]+hc}b1+b1,
Φ¯44(t)=-Φ¯22(t)=ir(V2ρ,4mrb2ρ+b4m-V4m,2ρrb4m+b2ρ)×[Br(0)exp(-iωrt)-Br+(0)exp(iωrt)],
Φ¯24(t)=irV4m,2ρr(b4m+b4m-b2ρ+b2ρ)×[Br(0)exp(-iωrt)-Br+(0)exp(iωrt)],
Φ¯11(t)=-Φ¯55(t).
dNβdt=-2Γβ[Nβ-Nβ(0)]+s,nβnβB5s,3nβb5s+b5s(Nβ+1)nβ+im(F4m3b4m+b3Aβ-F34mb3+b4mAβ+).
b5s+b5s=1γ5+βnβB5s,3nβ(Aβ+Aβ+1)nβ×12γ51|V5s,10|2I0b1+b1+βB3,5snβb3+b3(Aβ+Aβ)nβ+Φ55S+Φ¯55(t),
b2p+b2p=1γ21NRγ12NRb1+b1+mW2p,4mb4m+b4m+imλ(V4m,2pλb4m+b2paλ-V2p,4mλb2p+b4maλ+)+Φ22S+Φ¯22(t).
b4m+b3=-iγ43βF3,4m[b4m+b4m-b3+b3]Aβ+×f(E4m-E3-Ωβ)+Φ4m,3Sf(E4m-E3)-ipλV2p,4mλb2p+b4maλ+f(E4m-E3),
ddtb3+b3=-γ3b3+b3+12γ51|V5s,10|2I0b1+b1+Φ33S+Φ55S+Φ¯55(t)+12mβ|F3,4m|2(b4m+b4m-b3+b3)×f(E4m-E3-Ωβ)(Aβ+Aβ),
ddtb4m+b4m=-γ˜4b4m+b4m-ipλ(V2p,4mλb2p+b4maλ++V4m,2pλb4m+b2paλ)×1+12(γ43)2β|F3,4m|2Aβ+Aβf(E4m-E3)+Φ44S-12γ43β|F3,4m|2(b4m+b4m-b3+b3)×(Aβ+Aβ)f(E4m-E3-Ωβ)+Φ¯44(t),
ddtb2p+b4m=i(E2p-E4m)-γ24-γ21NR-12γ24λλV4m,2pλV4m,2pλ*aλ+aλb2p+b4m-12γ43β|F3,4m|2b2p+b4m(Aβ+Aβ)×f(E4m-E3-Ωβ)+iλV4m,2pλb4m+b4m1-W2p,4mγ21NR-γ12NRγ21NRb1+b1aλ+Φ¯¯24(t),
ddtb1+b1=-12γ51|V5n,10|2I0b1+b1+γ3b3+b3+mγ4b4m+b4m+impλ(V4m,2pλb4m+b2paλ+V2p,4mλb2p+b4maλ+)+Φ¯11(t),
Φ¯¯24(t)=irV4m,2prb4m+b4m1-W2p,4mγ21NR-γ12NRγ21NRb1+b1Br-12γ21NR×λr[V4m,2pλV4m,2pr*Br+aλ+hc]b2p+b4m
γ˜4=γ4+12γ43β|F3,4m|2f(E4m-E3-Ωβ),
P(z, t)=p,m(μ24Jpmb2p+b4m+μ42Jmpb4m+b2p).
b2p+b4m=iγˆ24λV4m,2pλb4m+b4m1-W2p,4mγ21NR-γ12NRγ21NRb1+b1×aλfˆ(E4m-E2p-ωλ)+G24(t),
γˆ24=γ24+γ21NR+12γ21NRλ|V4m,2pλ|2aλ+aλ+12γ21NRr|V4m,2pr|2Br+Br+12γ43β|F3,4m|2(Aβ+Aβ)f(E4m-E3-Ωβ)
G24(t)=iγˆ24rV4m,2prb4m+b4m1-W2p,4mγ21NR-γ12NRγ21NRb1+b1Brfˆ(E4m-E2p-ωr).
ρ=11-exp-ΩαkTm exp-mΩαkT|mm|.
ddtb3+b3=-Γn3b3+b3+H3WW+H30n0+Φ33SB+Φ55SB,
ddtW=-ΓWW-2λλDλλ(I)IλλW+HW3b3+b3+HW0n0+ΦWSB,
ddtIλλ=-iΔλλIλλ-2κλIλλ+λ[D˜λλ(I, N)Iλλ+D˜λλ(I, N)Iλλ]W,
ddtNβ=-Γβ[Nβ-Nβ(0)]-HN3b3+b3+HNWW+HN0n0+ηΦ55SB,
g1g1λλ=1Dλλ(I)pmCλλpmγ12pNRγ2p1NR+λCλλIλλ×f(E4m-E2p-ωλ)
D˜λλ(I, N)=11-exp-ΩαkTpmCλλpmγ2p1NRγˆ241-W2p,4mγ2p1NR×f(E4m-E2p-ωλ)exp-mΩαkT,
Cλλpm=12γ21NRV2p,4mλV2p,4mλ*,
ΓW=(2γ˜4+FβNβ+C00I0)/(1+g1),
HW3=FβNβ[g3β+g1/(1+g1)]+C00I0/(1+g1)+2γ˜4g1/(1+g1)-γ3,
HW0=[C00I0-g1FβNβ-2γ˜4g1]/(1+g1),
Γn3=γ3+C00I0/(1+g1)+FβNβ[g3β+g1/(1+g1)],
H3W=(FβNβ-C00I0)/(1+g1),
H30=(C00I0+g1FβNβ)/(1+g1),
HN3=ηC00I0/(1+g1)+FβNβ[g3β+g1/(1+g1)],
HNW=(FβNβ-ηC00I0)/(1+g1),
HN0=(ηC00I0+g1FβNβ)/(1+g1),
C00=12γ51|V5n,10|2,
g3β=1Fβ2γ53NRm|F3,4m|2f(E4m-E3-Ωβ),
Fβ=12γ53NR11-exp-ΩαkT×m|F3,4m|2f(E4m-E3-Ωβ)×exp-mΩαkT.
η=nβnβB53nβ(Nβ+1)nβγ5+nβB53nβ(Nβ+1)nβ,
B53nβ(Nβ+1)nβγ5+nβB53nβ(Nβ+1)nβ
Φ55SB=12γ5r|V5s,1r|2b1+b1nrf(E5s-E1-ωr)+12γ53NRββ|F3,5s|2b3+b3(Nβ+1)nβ×Aβ+Aβ(0)Aβ+Aβ(0)×f(E5s-E1-nβΩβ-Ωβ-Ωβ),
Φ33SB=12γ3r|V3,1r|2b1+b1nrf(E3-E1-ωr)-12γ53NRββ|F3,5s|2b3+b3(Nβ+1)nβ×Aβ+Aβ(0)Aβ+Aβ(0)×f(E5s-E1-nβΩβ-Ωβ-Ωβ),
Φˆ44B=-r,rDrr(I)nr(b4+b4-g1λλb1+b1),
Φ11SB=-Φ33SB-Φˆ44B-Φ55SB,
ΦWSB=Φˆ44B-g1Φ11SB,
Wm=0ΔLdzW(z)exp(imKz),
Um=0ΔLdzW(z)exp[i(Kmax+mK)z],
(b3+b3)m=0ΔLdzb3+b3 exp(imKz),
(b¯3+b¯3)m=0ΔLdzb3+b3 exp[i(Kmax+mK)z],
ddt(b3+b3)m=-Γn3(b3+b3)m+H3WWm+H30n0m+Φ˜33m(t)+Φ˜55m(t),
ddt(b¯3+b¯3)m=-Γn3(b¯3+b¯3)m+H3WUm+HN0n¯0m+Φ˜¯33m(t)+Φ˜¯55m(t),
ddtWm=-ΓWWm+HW3(b3+b3)m+HW0n0mΦˆWm(t)-12λλDλλ(I)Iλλ(Wm-λ+λ+Wm+λ-λ-Um+λ+λ-Um-λ-λ),
ddtUm=-ΓWUm+HW3(b¯3+b¯3)m+HW0n¯0mΦˆ¯Wm(t)-12λλDλλ(I)Iλλ(Um-λ+λ+Um+λ-λ-Wm+λ+λ-Wm-λ-λ),
ddtIλλ=-iΔλλIλλ-2κλIλλ+1ΔLλD˜λλ(I, N)Iλλ(W-λ+λ+Wλ-λ-Uλ+λ-U-λ-λ)+1ΔLλD˜λλ(I, N)Iλλ(W-λ+λ+Wλ-λ-Uλ+λ-U-λ-λ),
ddtNβ=-Γβ[Nβ-Nβ(0)]-1ΔL[HN3(b3+b3)0-HNWW0+HN0n0]+ηΦ¯550,
Φ¯jjm=0ΔLdzΦjjm(z, t)B exp[i(Kmax+mK)z],
ρm=0ΔLdzρ(z)exp(imKz).

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