Abstract

Excitation dynamics for pulsed optical excitation are described with the density-matrix equations and the rate equations for a two-level system. A critical comparison of the two descriptions is made with complete and consistent formalisms that are amenable to the modeling of applied laser-diagnostic techniques. General solutions, resulting from numerical integration of the differential equations describing the excitation process, are compared for collisional conditions that range from the completely coherent limit to the steady-state limit, for which the two formalisms are identical. This analysis demonstrates the failure of the rate equations to correctly describe the transient details of the excitation process outside the steady-state limit. However, reasonable estimates of the resultant population are obtained for nonsaturating (linear) excitation. This comparison provides the laser diagnostician with the means to evaluate the appropriate model for excitation through a simple picture of the breakdown of the rate-equation validity.

© 2002 Optical Society of America

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References

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  1. J. W. Daily, “Use of rate equations to describe laser excitation in flames,” Appl. Opt. 16, 2322–2327 (1977).
    [CrossRef] [PubMed]
  2. F. Bloch, “Nuclear induction,” Phys. Rev. 70, 460–474 (1946).
    [CrossRef]
  3. R. P. Feynman, F. L. Vernon, Jr., and R. W. Hellwarth, “Geometrical representation of the Schrödinger equation for solving laser problems,” J. Appl. Phys. 28, 49–52 (1957).
    [CrossRef]
  4. M. Sargent III, M. O. Scully, and W. E. Lamb, Jr., Laser Physics (Addison-Wesley, London, 1974).
  5. L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).
  6. B. W. Shore, The Theory of Coherent Atomic Excitation (Wiley, New York, 1990), Vols. 1 and 2.
  7. F. Ossler and M. Aldén, “Measurements of picosecond laser induced fluorescence from gas phase 3-pentanone and acetone: implications to combustion diagnostics,” Appl. Phys. B 64, 493–502 (1997).
    [CrossRef]
  8. M. W. Renfro, S. D. Pack, G. B. King, and N. M. Laurendeau, “A pulse-pileup correction procedure for rapid measurements of hydroxyl concentrations using picosecond time-resolved laser-induced fluorescence,” Appl. Phys. B 69, 137–146 (1999).
    [CrossRef]
  9. A. Brockhinke, A. Bülter, J. C. Rolon, and K. Kohse-Höinghaus, “ps-LIF measurements of minor species concentration in a counterflow diffusion flame interacting with a vortex,” Appl. Phys. B 72, 491–496 (2001).
    [CrossRef]
  10. R. Kienle, M. P. Lee, and K. Kohse-Höinghaus, “A detailed rate equation model for the simulation of energy transfer in OH laser-induced fluorescence,” Appl. Phys. B 62, 583–599 (1996).
    [CrossRef]
  11. W. P. Partridge, Jr., and N. M. Laurendeau, “Formulation of a dimensionless overlap fraction to account for spectrally distributed interactions in fluorescence studies,” Appl. Opt. 34, 2645–2647 (1995).
    [CrossRef] [PubMed]
  12. R. N. Brancewell, The Fourier Transform and Its Applications 2nd ed. (McGraw-Hill, New York, 1986), pp. 112–113.
  13. G. J. Blanchard and M. J. Wirth, “Transform-limited behavior from a synchronously pumped cw dye laser,” Opt. Commun. 53, 394–400 (1985).
    [CrossRef]
  14. R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992), pp. 116–123, 192–199.
  15. T. A. Reichardt and R. P. Lucht, “Resonant degenerate four-wave mixing spectroscopy of transitions with degenerate energy levels: saturation and polarization effects,” J. Chem. Phys. 111, 10008–10020 (1999).
    [CrossRef]
  16. R. P. Lucht, R. Trebino, and L. A. Rahn, “Resonant multiwave mixing spectra of gas-phase sodium: nonperturbative calculations,” Phys. Rev. A 45, 8209–8227 (1992).
    [CrossRef] [PubMed]
  17. R. P. Lucht, R. L. Farrow, and D. J. Rakestraw, “Saturation effects in gas-phase degenerate four-wave mixing spectroscopy: nonperturbative calculations,” J. Opt. Soc. Am. B 10, 1508–1520 (1993).
    [CrossRef]
  18. I. I. Rabi, “Space quantization in a gyrating magnetic field,” Phys. Rev. 51, 652–654 (1937).
    [CrossRef]
  19. M. D. DiRosa and R. L. Farrow, “Cross sections of photoionization and ac Stark shift measured from Doppler-free B← X(0, 0) excitation spectra of CO,” J. Opt. Soc. Am. B 16, 861–870 (1999).
    [CrossRef]
  20. J. W. Daily, “Laser induced fluorescence spectroscopy in flames,” Prog. Energy Combust. Sci. 23, 133–199 (1997).
    [CrossRef]
  21. W. E. Lamb, Jr., and T. M. Sanders, Jr., “Fine structure of short-lived states of hydrogen by a microwave-optical method. I,” Phys. Rev. 119, 1901–1914 (1960).
    [CrossRef]
  22. R. M. Measures, Laser Remote Sensing: Fundamentals and Applications (Wiley, New York, 1992), pp. 85–91.
  23. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), pp. 150–153.
  24. S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
    [CrossRef]

2001 (1)

A. Brockhinke, A. Bülter, J. C. Rolon, and K. Kohse-Höinghaus, “ps-LIF measurements of minor species concentration in a counterflow diffusion flame interacting with a vortex,” Appl. Phys. B 72, 491–496 (2001).
[CrossRef]

1999 (3)

T. A. Reichardt and R. P. Lucht, “Resonant degenerate four-wave mixing spectroscopy of transitions with degenerate energy levels: saturation and polarization effects,” J. Chem. Phys. 111, 10008–10020 (1999).
[CrossRef]

M. D. DiRosa and R. L. Farrow, “Cross sections of photoionization and ac Stark shift measured from Doppler-free B← X(0, 0) excitation spectra of CO,” J. Opt. Soc. Am. B 16, 861–870 (1999).
[CrossRef]

M. W. Renfro, S. D. Pack, G. B. King, and N. M. Laurendeau, “A pulse-pileup correction procedure for rapid measurements of hydroxyl concentrations using picosecond time-resolved laser-induced fluorescence,” Appl. Phys. B 69, 137–146 (1999).
[CrossRef]

1997 (2)

J. W. Daily, “Laser induced fluorescence spectroscopy in flames,” Prog. Energy Combust. Sci. 23, 133–199 (1997).
[CrossRef]

F. Ossler and M. Aldén, “Measurements of picosecond laser induced fluorescence from gas phase 3-pentanone and acetone: implications to combustion diagnostics,” Appl. Phys. B 64, 493–502 (1997).
[CrossRef]

1996 (1)

R. Kienle, M. P. Lee, and K. Kohse-Höinghaus, “A detailed rate equation model for the simulation of energy transfer in OH laser-induced fluorescence,” Appl. Phys. B 62, 583–599 (1996).
[CrossRef]

1995 (1)

1993 (1)

1992 (1)

R. P. Lucht, R. Trebino, and L. A. Rahn, “Resonant multiwave mixing spectra of gas-phase sodium: nonperturbative calculations,” Phys. Rev. A 45, 8209–8227 (1992).
[CrossRef] [PubMed]

1985 (1)

G. J. Blanchard and M. J. Wirth, “Transform-limited behavior from a synchronously pumped cw dye laser,” Opt. Commun. 53, 394–400 (1985).
[CrossRef]

1977 (1)

1969 (1)

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
[CrossRef]

1960 (1)

W. E. Lamb, Jr., and T. M. Sanders, Jr., “Fine structure of short-lived states of hydrogen by a microwave-optical method. I,” Phys. Rev. 119, 1901–1914 (1960).
[CrossRef]

1957 (1)

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

1946 (1)

F. Bloch, “Nuclear induction,” Phys. Rev. 70, 460–474 (1946).
[CrossRef]

1937 (1)

I. I. Rabi, “Space quantization in a gyrating magnetic field,” Phys. Rev. 51, 652–654 (1937).
[CrossRef]

Aldén, M.

F. Ossler and M. Aldén, “Measurements of picosecond laser induced fluorescence from gas phase 3-pentanone and acetone: implications to combustion diagnostics,” Appl. Phys. B 64, 493–502 (1997).
[CrossRef]

Blanchard, G. J.

G. J. Blanchard and M. J. Wirth, “Transform-limited behavior from a synchronously pumped cw dye laser,” Opt. Commun. 53, 394–400 (1985).
[CrossRef]

Bloch, F.

F. Bloch, “Nuclear induction,” Phys. Rev. 70, 460–474 (1946).
[CrossRef]

Brockhinke, A.

A. Brockhinke, A. Bülter, J. C. Rolon, and K. Kohse-Höinghaus, “ps-LIF measurements of minor species concentration in a counterflow diffusion flame interacting with a vortex,” Appl. Phys. B 72, 491–496 (2001).
[CrossRef]

Bülter, A.

A. Brockhinke, A. Bülter, J. C. Rolon, and K. Kohse-Höinghaus, “ps-LIF measurements of minor species concentration in a counterflow diffusion flame interacting with a vortex,” Appl. Phys. B 72, 491–496 (2001).
[CrossRef]

Daily, J. W.

J. W. Daily, “Laser induced fluorescence spectroscopy in flames,” Prog. Energy Combust. Sci. 23, 133–199 (1997).
[CrossRef]

J. W. Daily, “Use of rate equations to describe laser excitation in flames,” Appl. Opt. 16, 2322–2327 (1977).
[CrossRef] [PubMed]

DiRosa, M. D.

Farrow, R. L.

Feynman, R. P.

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

Hahn, E. L.

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
[CrossRef]

Hellwarth, R. W.

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

Kienle, R.

R. Kienle, M. P. Lee, and K. Kohse-Höinghaus, “A detailed rate equation model for the simulation of energy transfer in OH laser-induced fluorescence,” Appl. Phys. B 62, 583–599 (1996).
[CrossRef]

King, G. B.

M. W. Renfro, S. D. Pack, G. B. King, and N. M. Laurendeau, “A pulse-pileup correction procedure for rapid measurements of hydroxyl concentrations using picosecond time-resolved laser-induced fluorescence,” Appl. Phys. B 69, 137–146 (1999).
[CrossRef]

Kohse-Höinghaus, K.

A. Brockhinke, A. Bülter, J. C. Rolon, and K. Kohse-Höinghaus, “ps-LIF measurements of minor species concentration in a counterflow diffusion flame interacting with a vortex,” Appl. Phys. B 72, 491–496 (2001).
[CrossRef]

R. Kienle, M. P. Lee, and K. Kohse-Höinghaus, “A detailed rate equation model for the simulation of energy transfer in OH laser-induced fluorescence,” Appl. Phys. B 62, 583–599 (1996).
[CrossRef]

Lamb Jr., W. E.

W. E. Lamb, Jr., and T. M. Sanders, Jr., “Fine structure of short-lived states of hydrogen by a microwave-optical method. I,” Phys. Rev. 119, 1901–1914 (1960).
[CrossRef]

Laurendeau, N. M.

M. W. Renfro, S. D. Pack, G. B. King, and N. M. Laurendeau, “A pulse-pileup correction procedure for rapid measurements of hydroxyl concentrations using picosecond time-resolved laser-induced fluorescence,” Appl. Phys. B 69, 137–146 (1999).
[CrossRef]

W. P. Partridge, Jr., and N. M. Laurendeau, “Formulation of a dimensionless overlap fraction to account for spectrally distributed interactions in fluorescence studies,” Appl. Opt. 34, 2645–2647 (1995).
[CrossRef] [PubMed]

Lee, M. P.

R. Kienle, M. P. Lee, and K. Kohse-Höinghaus, “A detailed rate equation model for the simulation of energy transfer in OH laser-induced fluorescence,” Appl. Phys. B 62, 583–599 (1996).
[CrossRef]

Lucht, R. P.

T. A. Reichardt and R. P. Lucht, “Resonant degenerate four-wave mixing spectroscopy of transitions with degenerate energy levels: saturation and polarization effects,” J. Chem. Phys. 111, 10008–10020 (1999).
[CrossRef]

R. P. Lucht, R. L. Farrow, and D. J. Rakestraw, “Saturation effects in gas-phase degenerate four-wave mixing spectroscopy: nonperturbative calculations,” J. Opt. Soc. Am. B 10, 1508–1520 (1993).
[CrossRef]

R. P. Lucht, R. Trebino, and L. A. Rahn, “Resonant multiwave mixing spectra of gas-phase sodium: nonperturbative calculations,” Phys. Rev. A 45, 8209–8227 (1992).
[CrossRef] [PubMed]

McCall, S. L.

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
[CrossRef]

Ossler, F.

F. Ossler and M. Aldén, “Measurements of picosecond laser induced fluorescence from gas phase 3-pentanone and acetone: implications to combustion diagnostics,” Appl. Phys. B 64, 493–502 (1997).
[CrossRef]

Pack, S. D.

M. W. Renfro, S. D. Pack, G. B. King, and N. M. Laurendeau, “A pulse-pileup correction procedure for rapid measurements of hydroxyl concentrations using picosecond time-resolved laser-induced fluorescence,” Appl. Phys. B 69, 137–146 (1999).
[CrossRef]

Partridge Jr., W. P.

Rabi, I. I.

I. I. Rabi, “Space quantization in a gyrating magnetic field,” Phys. Rev. 51, 652–654 (1937).
[CrossRef]

Rahn, L. A.

R. P. Lucht, R. Trebino, and L. A. Rahn, “Resonant multiwave mixing spectra of gas-phase sodium: nonperturbative calculations,” Phys. Rev. A 45, 8209–8227 (1992).
[CrossRef] [PubMed]

Rakestraw, D. J.

Reichardt, T. A.

T. A. Reichardt and R. P. Lucht, “Resonant degenerate four-wave mixing spectroscopy of transitions with degenerate energy levels: saturation and polarization effects,” J. Chem. Phys. 111, 10008–10020 (1999).
[CrossRef]

Renfro, M. W.

M. W. Renfro, S. D. Pack, G. B. King, and N. M. Laurendeau, “A pulse-pileup correction procedure for rapid measurements of hydroxyl concentrations using picosecond time-resolved laser-induced fluorescence,” Appl. Phys. B 69, 137–146 (1999).
[CrossRef]

Rolon, J. C.

A. Brockhinke, A. Bülter, J. C. Rolon, and K. Kohse-Höinghaus, “ps-LIF measurements of minor species concentration in a counterflow diffusion flame interacting with a vortex,” Appl. Phys. B 72, 491–496 (2001).
[CrossRef]

Sanders Jr., T. M.

W. E. Lamb, Jr., and T. M. Sanders, Jr., “Fine structure of short-lived states of hydrogen by a microwave-optical method. I,” Phys. Rev. 119, 1901–1914 (1960).
[CrossRef]

Trebino, R.

R. P. Lucht, R. Trebino, and L. A. Rahn, “Resonant multiwave mixing spectra of gas-phase sodium: nonperturbative calculations,” Phys. Rev. A 45, 8209–8227 (1992).
[CrossRef] [PubMed]

Vernon Jr., F. L.

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

Wirth, M. J.

G. J. Blanchard and M. J. Wirth, “Transform-limited behavior from a synchronously pumped cw dye laser,” Opt. Commun. 53, 394–400 (1985).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. B (4)

F. Ossler and M. Aldén, “Measurements of picosecond laser induced fluorescence from gas phase 3-pentanone and acetone: implications to combustion diagnostics,” Appl. Phys. B 64, 493–502 (1997).
[CrossRef]

M. W. Renfro, S. D. Pack, G. B. King, and N. M. Laurendeau, “A pulse-pileup correction procedure for rapid measurements of hydroxyl concentrations using picosecond time-resolved laser-induced fluorescence,” Appl. Phys. B 69, 137–146 (1999).
[CrossRef]

A. Brockhinke, A. Bülter, J. C. Rolon, and K. Kohse-Höinghaus, “ps-LIF measurements of minor species concentration in a counterflow diffusion flame interacting with a vortex,” Appl. Phys. B 72, 491–496 (2001).
[CrossRef]

R. Kienle, M. P. Lee, and K. Kohse-Höinghaus, “A detailed rate equation model for the simulation of energy transfer in OH laser-induced fluorescence,” Appl. Phys. B 62, 583–599 (1996).
[CrossRef]

J. Appl. Phys. (1)

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

J. Chem. Phys. (1)

T. A. Reichardt and R. P. Lucht, “Resonant degenerate four-wave mixing spectroscopy of transitions with degenerate energy levels: saturation and polarization effects,” J. Chem. Phys. 111, 10008–10020 (1999).
[CrossRef]

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

Opt. Commun. (1)

G. J. Blanchard and M. J. Wirth, “Transform-limited behavior from a synchronously pumped cw dye laser,” Opt. Commun. 53, 394–400 (1985).
[CrossRef]

Phys. Rev. (4)

F. Bloch, “Nuclear induction,” Phys. Rev. 70, 460–474 (1946).
[CrossRef]

I. I. Rabi, “Space quantization in a gyrating magnetic field,” Phys. Rev. 51, 652–654 (1937).
[CrossRef]

W. E. Lamb, Jr., and T. M. Sanders, Jr., “Fine structure of short-lived states of hydrogen by a microwave-optical method. I,” Phys. Rev. 119, 1901–1914 (1960).
[CrossRef]

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
[CrossRef]

Phys. Rev. A (1)

R. P. Lucht, R. Trebino, and L. A. Rahn, “Resonant multiwave mixing spectra of gas-phase sodium: nonperturbative calculations,” Phys. Rev. A 45, 8209–8227 (1992).
[CrossRef] [PubMed]

Prog. Energy Combust. Sci. (1)

J. W. Daily, “Laser induced fluorescence spectroscopy in flames,” Prog. Energy Combust. Sci. 23, 133–199 (1997).
[CrossRef]

Other (7)

R. M. Measures, Laser Remote Sensing: Fundamentals and Applications (Wiley, New York, 1992), pp. 85–91.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), pp. 150–153.

R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992), pp. 116–123, 192–199.

R. N. Brancewell, The Fourier Transform and Its Applications 2nd ed. (McGraw-Hill, New York, 1986), pp. 112–113.

M. Sargent III, M. O. Scully, and W. E. Lamb, Jr., Laser Physics (Addison-Wesley, London, 1974).

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975).

B. W. Shore, The Theory of Coherent Atomic Excitation (Wiley, New York, 1990), Vols. 1 and 2.

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Figures (10)

Fig. 1
Fig. 1

Error associated with the RE-based solutions for the net excitation fraction in the collision-free limit with θ()π/2.

Fig. 2
Fig. 2

Excitation fraction as a function of time during a sech2 pulse with θ()=0.020π. The DME result is plotted as a solid curve and the RE result as a dashed curve.

Fig. 3
Fig. 3

Excitation fraction as a function of time during a sech2 pulse with θ()=0.205π, exciting 10% of the population. The DME result is plotted as a thick solid curve, the full RE result as a dashed curve, and the linearized RE result as a dotted curve.

Fig. 4
Fig. 4

Net excitation fraction as a function of pulse area for a sech2 pulse at resonance, with negligible quenching. Four dephasing cases are shown with values of T2 indicated. Circles indicate conditions referenced in Fig. 6.

Fig. 5
Fig. 5

Error associated with the RE as a function of the net excitation fraction for the cases shown in Fig. 4. Circles indicate conditions referenced in Fig. 6.

Fig. 6
Fig. 6

Net excitation fraction (top) and the corresponding RE error (bottom) as a function of detuning for a sech2 pulse with negligible quenching and moderate dephasing (T2=Δto). The three values of pulse area that produce peak net excitation fraction of 20%, 10%, and 1% are shown. These parameters correspond to the circles in Figs. 4 and 5. The shaded region corresponds to the laser spectral distribution function.

Fig. 7
Fig. 7

DME (solid curves) and RE (dashed curves) predictions of the net excitation fraction as a function of pulse area for a sech2 pulse at resonance with negligible quenching. The collisional dephasing time is T2=Δto. Three values of Doppler width are indicated.

Fig. 8
Fig. 8

DME (solid curves) and RE (dashed curves) predictions of the net excitation fraction for velocity groups u as a function of the corresponding Doppler shift. The three cases shown as circles in Fig. 7 are labeled (a) θ=0.5π, (b) θ=1.24π, and (c) θ=2.25π.

Fig. 9
Fig. 9

Excited-state population fraction as a function of time for efficient dephasing (T2=0.01Δto) at four quenching rates: (a) T1=0.1Δto, (b) T1=Δto, (c) T1=10Δto, and (d) T1=106Δto. The laser pulse is shown as the shaded region.

Fig. 10
Fig. 10

Error associated with the RE prediction of the maximum excitation fraction plotted as a function of ρ22max and T1 (=T2/2).

Equations (61)

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

E(t)=ε˜(t)exp(-i2πνot)+c.c.,
P(z, t)=p˜(z, t)exp(-i2πνot)+c.c.
Iν(ν, t)=EA T(t)L(ν)ΔtoΔνo.
T(t)=|ε˜(t)|2Δto-+|ε˜(t)|2dt.
L(ν)=|ε˜(ν-νo)|2Δνo-+|ε˜(ν-νo)|2dν.
EA=2oc -+|ε˜(t)|2dt,
Iν(ν, t)=2oc |ε˜(t)|2|εˆ(ν-νo)|2-+|εˆ(ν-νo)|2dν.
Iν(ν, t)=2oc |ε˜(t)|2|ε˜(ν-νo)|2-+|ε˜(t)|2dt.
ε(t)=εo sech2 ln(2+1) t-toΔto,
EA=2coεo2Δto 1ln(2+1).
T(t)=ln(2+1)sech22 ln(2+1) t-toΔto,
L(ν)=ln(2+1)sech22 ln(2+1) ν-νoΔνo,
ΔtoΔνo=4π2[ln(2+1)]2.
V˜12(t)=-μ˜12·E(t)
V˜21(t)=-μ˜21·E(t),
V12(t)=V21(t)=μ21ε˜(t)exp(-iωot)+c.c.
ρ22(t)t=-1T1ρ22+i(V12ρ˜21-ρ˜12V21),
ρ˜21(t)t=-iω21+1T2ρ˜21+iV21(2ρ22-1),
T1=(A21+Q21)-1.
T2=12T1+γc-1.
ρ˜21(t)=σ˜21(t)exp(-iωot).
ΩR(t)=2μ21εR(t),
ΩI(t)=2μ21εI(t).
ρ22(t)t=-1T1ρ22+ΩRσI-ΩIσR,
σR(t)t=-1T2σR+δσI+12ΩI(2ρ22-1),
σI(t)t=-1T2σI-δσR-12ΩR(2ρ22-1),
μ=ρ21μ21+ρ12μ21=μ21σ˜21 exp(-iωot)+c.c.
δ=ω21-ωo1-uc,
P(t)=Ntot-+μ(t)f(u)du.
p˜(t)=Ntotμ21-+σ˜21(z, t; u)f(u)du.
N2=Ntot-+ρ22(z, t; u)f(u)du.
dρ22,redt=W12(t)-[Q21+A21+W12(t)+W21(t)]ρ22,re(t).
W21(t)=-+ B21cIν(ν, t)g(ν)dν,
g(ν)=ΔνH2π (ν-ν21)2+ΔνH22-1.
ρ22ss=12 Ωo2T1T2 1δ2+(1/T2)2(1+Ωo2T1T2),
ΔωH=2T2 1+Ωo2T1T2.
ρ22,ress=W12Q21+A21+W21+W12.
Iν=2ocεo2δ(ν-νo).
W12=4πB12oεo2 1/T2(ω21-ωo)2+(1/T2)2.
ρ22,ress=128B12oεo2 T1T2 1δ2+(1/T2)2(1+8B12oεo20T1T2).
Ω2=8B12εoεo2.
B21=μ21222o.
ρ22(t)t=ΩRσ1,
σR(t)t=0,
σI(t)t=-12ΩR(2ρ22-1).
ρ22(t)=12[1-cos θ(t)],
σR(t)=0,
σI(t)=12 sin θ(t),
θ(t)=-tΩR(t)dt.
ρ22(t)μ2122 -t εR(t)dt2.
ρ22,re(t)=12 1-exp-2-tW12(t)dt.
ρ22,re(t)-tW12(t)dt.
W21(t)=μ2122 |ε˜(t)|2-+|ε˜(t)|2dt -+|εˆ(ν-νo)|2g(ν)dν.
W21(t)=μ2122 |εˆ(0)|2|ε˜(t)|2-+|ε˜(t)|2dt.
ρ22,re(t)μ2122 -t|ε˜(t)|2dt-+|ε˜(t)|2dt -+ ε˜0(t)dt2.
limt -tW21(t)dt=limt θ(t)22.
errorlimt ρ22-ρ22,reρ22.
θ(t)=ΩoΔtoΔνoΔto 1-2π arctan[(2+1)-2t/Δto],
-tW12(z, t)dt=14 (ΩoΔto)2(ΔνoΔto) (2+1)4t/Δto1+(2+1)4t/Δto.
error=exp(-12θ2)-cos (θ)1-cos(θ).
error=ρ22max-ρ22,remaxρ22max.

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