Abstract

Theoretical and experimental investigations of continuous four-wave mixing and parametric amplification in a resonantly driven double-Λ scheme of Na2 molecules are reported. Cases of copropagating and counterpropagating pump waves are analyzed. Macroscopic propagation of the generated fields and the interplay of absorption, Raman gain, four-wave mixing, and parametric amplification are studied.

© 2000 Optical Society of America

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  1. K. S. E. Eikema, J. Walz, and T. W. Hänsch, “Continuous wave coherent Lyman-α radiation,” Phys. Rev. Lett. 83, 3828–3831 (1999).
    [CrossRef]
  2. S. Babin, U. Hinze, E. Tiemann, and B. Wellegehausen, “Continuous resonant four-wave mixing in double-Λ level configurations of Na2,” Opt. Lett. 21, 1186–1188 (1996); A. Apolonskii, S. Baluschev, U. Hinze, E. Tiemann, and B. Wellegehausen, “Continuous frequency up-conversion in a double-Λ scheme of Na2,” Appl. Phys. B 64, 435–442 (1997); S. Babin, E. V. Podivilov, D. A. Shapiro, U. Hinze, E. Tiemann, and B. Wellegehausen, “Effects of strong driving fields in resonant four-wave mixing schemes with down-conversion,” Phys. Rev. A PLRAAN 59, 1355–1366 (1999).
    [CrossRef] [PubMed]
  3. B. D. Agap’ev, M. B. Gornyi, B. G. Matisov, and Yu. V. Rozhdestvenskii, “Coherent population trapping in quantum systems,” Phys. Usp. 36, 763–793 (1993); E. Arimondo, “Coherent population trapping in laser spectroscopy,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1996), Vol. XXXV, pp. 257–354.
    [CrossRef]
  4. J. P. Marangos, “Electromagnetically induced transparency,” J. Mod. Opt. 45, 471–503 (1998); E. A. Korsunsky and D. V. Kosachiov, “Phase-dependent nonlinear optics with double-Λ atoms,” Phys. Rev. A 60, 4996–5009 (1999).
    [CrossRef]
  5. G. Müller, M. Müller, A. Wicht, R. H. Rinkleff, and K. Danzmann, “Optical resonantor with steep internal dispersion,” Phys. Rev. A 56, 2385–2389 (1997); M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229–5232 (1999).
    [CrossRef]
  6. A. Nagel, L. Graf, A. Naumov, E. Mariotti, V. Biancalana, D. Meschede, and R. Wynands, “Experimental realization of coherent dark-state magnetometers,” Europhys. Lett. 44, 31–36 (1998).
    [CrossRef]
  7. R. Wynands and A. Nagel, “Precision spectroscopy with coherent dark states,” Appl. Phys. B 68, 1–25 (1999).
    [CrossRef]
  8. A. N. Nesmeyanov, in Vapour Pressure of the Chemical Elements (Elsevier, Amsterdam, 1963), pp. 128–136.
  9. K. Schneider, S. Schiller, J. Mlynek, M. Bode, and I. Freitag, “1.1-W single-frequency 532-nm radiation by second-harmonic generation of a miniature Nd:YAG ring laser,” Opt. Lett. 21, 1999–2001 (1996).
    [CrossRef] [PubMed]
  10. B. Wellegehausen, “Optically pumped cw dimer lasers,” IEEE J. Quantum Electron. QE-15, 1108–1132 (1979).
    [CrossRef]
  11. D. Sarkisyan, U. Hinze, L. Meyer, and B. Wellegehausen, “Efficient cw sodium dimer Raman laser operation in a high temperature sapphire cell,” Appl. Phys. B 70, 351–354 (2000).
    [CrossRef]
  12. U. Hinze, L. Meyer, B. N. Chichkov, E. Tiemann, and B. Wellegehausen, “Continuous parametric amplification in a resonantly driven double-Λ system,” Opt. Commun. 166, 127–132 (1999).
    [CrossRef]
  13. Y. R. Shen, in The Principles of Nonlinear Optics (Wiley, New York, 1984), pp. 117–140.
  14. R. W. Boyd, in Nonlinear Optics (Academic, San Diego, Calif., 1992), pp. 380–389.
  15. S. E. Harris, “Proposed backward wave oscillation in the infrared,” Appl. Phys. Lett. 9, 114–116 (1966).
    [CrossRef]
  16. A. S. Zibrov, M. D. Lukin, and M. O. Scully, “Nondegenerate parametric self-oscillation via multiwave mixing in coherent atomic media,” Phys. Rev. Lett. 83, 4049–4052 (1999).
    [CrossRef]
  17. R. A. Fisher, ed., Optical Phase Conjugation (Academic, New York, 1983).
  18. At 720 K the densities of sodium dimers and atoms are nNa2≃5×1014 cm−3 and nNa≃1.5×1016 cm−3. Increasing the temperature to 1000 K results in nNa2≃3 × 1017 cm−3 and nNa≃1.2×1018 cm−3, which corresponds to an increase of the molecular density by a factor of 600 (see Ref. 8).

2000 (1)

D. Sarkisyan, U. Hinze, L. Meyer, and B. Wellegehausen, “Efficient cw sodium dimer Raman laser operation in a high temperature sapphire cell,” Appl. Phys. B 70, 351–354 (2000).
[CrossRef]

1999 (4)

U. Hinze, L. Meyer, B. N. Chichkov, E. Tiemann, and B. Wellegehausen, “Continuous parametric amplification in a resonantly driven double-Λ system,” Opt. Commun. 166, 127–132 (1999).
[CrossRef]

A. S. Zibrov, M. D. Lukin, and M. O. Scully, “Nondegenerate parametric self-oscillation via multiwave mixing in coherent atomic media,” Phys. Rev. Lett. 83, 4049–4052 (1999).
[CrossRef]

K. S. E. Eikema, J. Walz, and T. W. Hänsch, “Continuous wave coherent Lyman-α radiation,” Phys. Rev. Lett. 83, 3828–3831 (1999).
[CrossRef]

R. Wynands and A. Nagel, “Precision spectroscopy with coherent dark states,” Appl. Phys. B 68, 1–25 (1999).
[CrossRef]

1998 (1)

A. Nagel, L. Graf, A. Naumov, E. Mariotti, V. Biancalana, D. Meschede, and R. Wynands, “Experimental realization of coherent dark-state magnetometers,” Europhys. Lett. 44, 31–36 (1998).
[CrossRef]

1996 (1)

1979 (1)

B. Wellegehausen, “Optically pumped cw dimer lasers,” IEEE J. Quantum Electron. QE-15, 1108–1132 (1979).
[CrossRef]

1966 (1)

S. E. Harris, “Proposed backward wave oscillation in the infrared,” Appl. Phys. Lett. 9, 114–116 (1966).
[CrossRef]

Biancalana, V.

A. Nagel, L. Graf, A. Naumov, E. Mariotti, V. Biancalana, D. Meschede, and R. Wynands, “Experimental realization of coherent dark-state magnetometers,” Europhys. Lett. 44, 31–36 (1998).
[CrossRef]

Bode, M.

Chichkov, B. N.

U. Hinze, L. Meyer, B. N. Chichkov, E. Tiemann, and B. Wellegehausen, “Continuous parametric amplification in a resonantly driven double-Λ system,” Opt. Commun. 166, 127–132 (1999).
[CrossRef]

Eikema, K. S. E.

K. S. E. Eikema, J. Walz, and T. W. Hänsch, “Continuous wave coherent Lyman-α radiation,” Phys. Rev. Lett. 83, 3828–3831 (1999).
[CrossRef]

Freitag, I.

Graf, L.

A. Nagel, L. Graf, A. Naumov, E. Mariotti, V. Biancalana, D. Meschede, and R. Wynands, “Experimental realization of coherent dark-state magnetometers,” Europhys. Lett. 44, 31–36 (1998).
[CrossRef]

Hänsch, T. W.

K. S. E. Eikema, J. Walz, and T. W. Hänsch, “Continuous wave coherent Lyman-α radiation,” Phys. Rev. Lett. 83, 3828–3831 (1999).
[CrossRef]

Harris, S. E.

S. E. Harris, “Proposed backward wave oscillation in the infrared,” Appl. Phys. Lett. 9, 114–116 (1966).
[CrossRef]

Hinze, U.

D. Sarkisyan, U. Hinze, L. Meyer, and B. Wellegehausen, “Efficient cw sodium dimer Raman laser operation in a high temperature sapphire cell,” Appl. Phys. B 70, 351–354 (2000).
[CrossRef]

U. Hinze, L. Meyer, B. N. Chichkov, E. Tiemann, and B. Wellegehausen, “Continuous parametric amplification in a resonantly driven double-Λ system,” Opt. Commun. 166, 127–132 (1999).
[CrossRef]

Lukin, M. D.

A. S. Zibrov, M. D. Lukin, and M. O. Scully, “Nondegenerate parametric self-oscillation via multiwave mixing in coherent atomic media,” Phys. Rev. Lett. 83, 4049–4052 (1999).
[CrossRef]

Mariotti, E.

A. Nagel, L. Graf, A. Naumov, E. Mariotti, V. Biancalana, D. Meschede, and R. Wynands, “Experimental realization of coherent dark-state magnetometers,” Europhys. Lett. 44, 31–36 (1998).
[CrossRef]

Meschede, D.

A. Nagel, L. Graf, A. Naumov, E. Mariotti, V. Biancalana, D. Meschede, and R. Wynands, “Experimental realization of coherent dark-state magnetometers,” Europhys. Lett. 44, 31–36 (1998).
[CrossRef]

Meyer, L.

D. Sarkisyan, U. Hinze, L. Meyer, and B. Wellegehausen, “Efficient cw sodium dimer Raman laser operation in a high temperature sapphire cell,” Appl. Phys. B 70, 351–354 (2000).
[CrossRef]

U. Hinze, L. Meyer, B. N. Chichkov, E. Tiemann, and B. Wellegehausen, “Continuous parametric amplification in a resonantly driven double-Λ system,” Opt. Commun. 166, 127–132 (1999).
[CrossRef]

Mlynek, J.

Nagel, A.

R. Wynands and A. Nagel, “Precision spectroscopy with coherent dark states,” Appl. Phys. B 68, 1–25 (1999).
[CrossRef]

A. Nagel, L. Graf, A. Naumov, E. Mariotti, V. Biancalana, D. Meschede, and R. Wynands, “Experimental realization of coherent dark-state magnetometers,” Europhys. Lett. 44, 31–36 (1998).
[CrossRef]

Naumov, A.

A. Nagel, L. Graf, A. Naumov, E. Mariotti, V. Biancalana, D. Meschede, and R. Wynands, “Experimental realization of coherent dark-state magnetometers,” Europhys. Lett. 44, 31–36 (1998).
[CrossRef]

Sarkisyan, D.

D. Sarkisyan, U. Hinze, L. Meyer, and B. Wellegehausen, “Efficient cw sodium dimer Raman laser operation in a high temperature sapphire cell,” Appl. Phys. B 70, 351–354 (2000).
[CrossRef]

Schiller, S.

Schneider, K.

Scully, M. O.

A. S. Zibrov, M. D. Lukin, and M. O. Scully, “Nondegenerate parametric self-oscillation via multiwave mixing in coherent atomic media,” Phys. Rev. Lett. 83, 4049–4052 (1999).
[CrossRef]

Tiemann, E.

U. Hinze, L. Meyer, B. N. Chichkov, E. Tiemann, and B. Wellegehausen, “Continuous parametric amplification in a resonantly driven double-Λ system,” Opt. Commun. 166, 127–132 (1999).
[CrossRef]

Walz, J.

K. S. E. Eikema, J. Walz, and T. W. Hänsch, “Continuous wave coherent Lyman-α radiation,” Phys. Rev. Lett. 83, 3828–3831 (1999).
[CrossRef]

Wellegehausen, B.

D. Sarkisyan, U. Hinze, L. Meyer, and B. Wellegehausen, “Efficient cw sodium dimer Raman laser operation in a high temperature sapphire cell,” Appl. Phys. B 70, 351–354 (2000).
[CrossRef]

U. Hinze, L. Meyer, B. N. Chichkov, E. Tiemann, and B. Wellegehausen, “Continuous parametric amplification in a resonantly driven double-Λ system,” Opt. Commun. 166, 127–132 (1999).
[CrossRef]

B. Wellegehausen, “Optically pumped cw dimer lasers,” IEEE J. Quantum Electron. QE-15, 1108–1132 (1979).
[CrossRef]

Wynands, R.

R. Wynands and A. Nagel, “Precision spectroscopy with coherent dark states,” Appl. Phys. B 68, 1–25 (1999).
[CrossRef]

A. Nagel, L. Graf, A. Naumov, E. Mariotti, V. Biancalana, D. Meschede, and R. Wynands, “Experimental realization of coherent dark-state magnetometers,” Europhys. Lett. 44, 31–36 (1998).
[CrossRef]

Zibrov, A. S.

A. S. Zibrov, M. D. Lukin, and M. O. Scully, “Nondegenerate parametric self-oscillation via multiwave mixing in coherent atomic media,” Phys. Rev. Lett. 83, 4049–4052 (1999).
[CrossRef]

Appl. Phys. B (2)

R. Wynands and A. Nagel, “Precision spectroscopy with coherent dark states,” Appl. Phys. B 68, 1–25 (1999).
[CrossRef]

D. Sarkisyan, U. Hinze, L. Meyer, and B. Wellegehausen, “Efficient cw sodium dimer Raman laser operation in a high temperature sapphire cell,” Appl. Phys. B 70, 351–354 (2000).
[CrossRef]

Appl. Phys. Lett. (1)

S. E. Harris, “Proposed backward wave oscillation in the infrared,” Appl. Phys. Lett. 9, 114–116 (1966).
[CrossRef]

Europhys. Lett. (1)

A. Nagel, L. Graf, A. Naumov, E. Mariotti, V. Biancalana, D. Meschede, and R. Wynands, “Experimental realization of coherent dark-state magnetometers,” Europhys. Lett. 44, 31–36 (1998).
[CrossRef]

IEEE J. Quantum Electron. (1)

B. Wellegehausen, “Optically pumped cw dimer lasers,” IEEE J. Quantum Electron. QE-15, 1108–1132 (1979).
[CrossRef]

Opt. Commun. (1)

U. Hinze, L. Meyer, B. N. Chichkov, E. Tiemann, and B. Wellegehausen, “Continuous parametric amplification in a resonantly driven double-Λ system,” Opt. Commun. 166, 127–132 (1999).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. Lett. (2)

K. S. E. Eikema, J. Walz, and T. W. Hänsch, “Continuous wave coherent Lyman-α radiation,” Phys. Rev. Lett. 83, 3828–3831 (1999).
[CrossRef]

A. S. Zibrov, M. D. Lukin, and M. O. Scully, “Nondegenerate parametric self-oscillation via multiwave mixing in coherent atomic media,” Phys. Rev. Lett. 83, 4049–4052 (1999).
[CrossRef]

Other (9)

R. A. Fisher, ed., Optical Phase Conjugation (Academic, New York, 1983).

At 720 K the densities of sodium dimers and atoms are nNa2≃5×1014 cm−3 and nNa≃1.5×1016 cm−3. Increasing the temperature to 1000 K results in nNa2≃3 × 1017 cm−3 and nNa≃1.2×1018 cm−3, which corresponds to an increase of the molecular density by a factor of 600 (see Ref. 8).

Y. R. Shen, in The Principles of Nonlinear Optics (Wiley, New York, 1984), pp. 117–140.

R. W. Boyd, in Nonlinear Optics (Academic, San Diego, Calif., 1992), pp. 380–389.

S. Babin, U. Hinze, E. Tiemann, and B. Wellegehausen, “Continuous resonant four-wave mixing in double-Λ level configurations of Na2,” Opt. Lett. 21, 1186–1188 (1996); A. Apolonskii, S. Baluschev, U. Hinze, E. Tiemann, and B. Wellegehausen, “Continuous frequency up-conversion in a double-Λ scheme of Na2,” Appl. Phys. B 64, 435–442 (1997); S. Babin, E. V. Podivilov, D. A. Shapiro, U. Hinze, E. Tiemann, and B. Wellegehausen, “Effects of strong driving fields in resonant four-wave mixing schemes with down-conversion,” Phys. Rev. A PLRAAN 59, 1355–1366 (1999).
[CrossRef] [PubMed]

B. D. Agap’ev, M. B. Gornyi, B. G. Matisov, and Yu. V. Rozhdestvenskii, “Coherent population trapping in quantum systems,” Phys. Usp. 36, 763–793 (1993); E. Arimondo, “Coherent population trapping in laser spectroscopy,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1996), Vol. XXXV, pp. 257–354.
[CrossRef]

J. P. Marangos, “Electromagnetically induced transparency,” J. Mod. Opt. 45, 471–503 (1998); E. A. Korsunsky and D. V. Kosachiov, “Phase-dependent nonlinear optics with double-Λ atoms,” Phys. Rev. A 60, 4996–5009 (1999).
[CrossRef]

G. Müller, M. Müller, A. Wicht, R. H. Rinkleff, and K. Danzmann, “Optical resonantor with steep internal dispersion,” Phys. Rev. A 56, 2385–2389 (1997); M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229–5232 (1999).
[CrossRef]

A. N. Nesmeyanov, in Vapour Pressure of the Chemical Elements (Elsevier, Amsterdam, 1963), pp. 128–136.

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Figures (10)

Fig. 1
Fig. 1

Na2-level diagram: four vibrational–rotational levels of the X, A, and B bands are coupled by two pump (λ1 and λ3; thick lines) and two generated (λ2 and λ4; thinner lines) fields. Field λ2 is generated by the Raman process; λ4 is generated by four-wave mixing.

Fig. 2
Fig. 2

Schematic setup for four-wave mixing.

Fig. 3
Fig. 3

Central part of the 60-GHz broad tuning (generation) spectrum and transmission curve for the fourth field.

Fig. 4
Fig. 4

Generated field powers P2 and P4 as a function of heat-pipe temperature. What happens when the pump fields are tuned out of resonance is illustrated in the inset.

Fig. 5
Fig. 5

Schematic setup for investigation of parametric amplification with copropagating pump fields.

Fig. 6
Fig. 6

Modulated input and output signals for (a) the second and (b) the fourth radiation fields. Powers of the input fields are P1=25 mW,P2=80 µW,P3=30 mW, and P4=0.7 µW.

Fig. 7
Fig. 7

Calculated signals for (a) the second and (b) the fourth waves.

Fig. 8
Fig. 8

Theoretical and experimental dependencies of the output/input ratio on the input power for the fourth field.

Fig. 9
Fig. 9

Dependencies of generated field powers P2 and P4 on (a) the Raman gain and absorption coefficients and (b) the nonlinear coupling coefficients for relevant experimental conditions.

Fig. 10
Fig. 10

Schematic of the setup for investigations with counterpropagating pump waves (P1 and P3).

Equations (80)

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dI1dz=-α101+I1/IsI1,
I1(z)=I101-α10z2 Is/I102I10
dA2dz=-α22A2,
dA4dz=-α42A4+iγ4A2*exp(iΔkz),
αi=8πωicniN I(diR/Ai)=4πωicni I[χi(1)],
γ4=4πω4cn4N(d4F/A2*)=3πω4cn4χ4(3)A1A3
A2=exp(-α2z/2)A20,
A4=exp(-α4z/2)iγ4βA20*exp(βz)sinh(βz)+A40,
β=12α4-α22+iΔk
P2=P20 exp(-α2z),
P4=P20|γ4|2sin2(Δkz/2)+sinh2[(α4-α2)z/4](Δk/2)2+[(α4-α2)/4]2×exp[-(α2+α4)z/2]-2 Iγ4β sinh(βz)exp[i(Δkz/2-ϕ20-ϕ40)]×P20P40 exp[-(α2+3α4)z/4]+P40 exp(-α4z).
P2=P20 exp(g2L),
P4=P20|γ4|2sin2(ΔkL/2)+sinh2[(g2+α4)L/4](Δk/2)2+[(g2+α4)/4]2×exp[(g2-α4)L/2].
P4|γ4|2P20 exp(g2L)(Δk)2+[(α4+g2)/2]2.
P4=P4b+P4i,
P4b=P204|γ4|2(g2+α4)2[exp(g2L/2)-exp(-α4L/2)]2,
P4i=P20P404|γ4|g2+α4 exp(-α4L/2)[exp(g2L/2)-exp(-α4L/2)].
P20P40(g2+α4)216|γ4|2exp(α4L)[exp(g2L/2)-exp(-α4L/2)]2
dA2dz=-α22A2+iγ2A4*exp(iΔkz),
dA4dz=-α42A4+iγ4A2*exp(iΔkz).
γ2=4πω2cn2N(d2F/A4*)=3πω2cn2χ2(3)A1A3,
A2*=exp-α22z-βz-iγ2*GA40 sinh(Gz)+A20*cosh(Gz)+βG sinh(Gz),
A4=exp-α42z+βziγ4GA20* sinh(Gz)+A40cosh(Gz)-βG sinh(Gz),
P2=P20cosh(Gz)+βG sinh(Gz)2 exp[-(α2+α4)z/2],
P4=P20γ4G sinh(Gz)2 exp[-(α2+α4)z/2]×1+2 IG cosh(Gz)-β sinh(Gz)γ4 sinh(Gz)×exp[i(ϕ20+ϕ40)]P40/P20.
P4b=P20|γ4|2G2sinh2(Gz)exp[(g2-α4)z/2],
P4m=2P20P40|γ4|G sinh(Gz)cosh(Gz)-βG sinh(Gz)exp[(g2-α4)z/2].
P4bP204|γ4|2(g2+α4)2[exp(g2L/2)-exp(-α4L/2)]2,
P4mP20P404|γ4|g2+α4 exp(-α4L/2)[exp(g2L/2)-exp(-α4L/2)]+16P20P40|γ4|2|γ2|(g2+α4)3×[exp(g2L/2)-exp(-α4L/2)]2.
dA2dz=α22A2-iγ2A4*exp(iΔkz),
dA4dz=-α42A4+iγ4A2*exp(iΔkz),
A2*=exp(-α2dz/2+βdz)G cosh(GL)+β sinh(GL)×-iγ2*A40 expα22L-βLsinh(Gdz)+A2L*[G cosh(Gz)+β sinh(Gz)],
A4=exp(-α4z/2+βz)G cosh(GL)+β sinh(GL)×iγ4A2L*exp-α22L+βLsinh(Gz)+A40[G cosh(Gdz)+β sinh(Gdz)],
β=12α2+α42+iΔk.
L=L=12G lnβ-Gβ+G.
P2=P2LG cosh(Gz)+β sinh(Gz)G cosh(GL)+β sinh(GL)2×exp[-(α2-α4)dz/2],
P4=P2Lγ4 sinh(Gz)G cosh(GL)+β sinh(GL)2×exp[-(α2-α4)dz/2]×1+2 IG cosh(Gdz)+β sinh(Gdz)γ4 sinh(Gz)×exp[i(ϕ2L+ϕ40-ΔkL/2)]×exp[(α2-α4)L/4]P40/P2L.
P2(z=0)=P2L|G|2 exp[(g2+α4)L/2]|G cosh(GL)+β sinh(GL)|2,
P4(z=L)=P2Lγ4 sinh(GL)G cosh(GL)+β sinh(GL)2.
L=2g2-α4 ln(g2-α4)24γ2*γ4.
Δ21ρ21-Ω1(ρ22-ρ11)/2+Ω2ρ31/2-Ω4ρ24/2=0,
Δ23ρ23-Ω2(ρ22-ρ33)/2+Ω1ρ13/2-Ω3ρ24/2=0,
Δ43ρ43-Ω3(ρ44-ρ33)/2+Ω4ρ13/2-Ω2ρ42/2=0,
Δ41ρ41-Ω4(ρ44-ρ11)/2+Ω3ρ31/2-Ω1ρ42/2=0,
Δ31ρ31-Ω1ρ32/2+Ω2*ρ21/2+Ω3*ρ41/2-Ω4ρ34/2=0,
Δ42ρ42-Ω1*ρ41/2-Ω2*ρ43/2+Ω3ρ32/2+Ω4ρ12/2=0.
ρ23R=Ω22Z*4X1(ρ22-ρ33)-|Ω1|2X2(ρ11-ρ22)Δ12+|Ω3|2X3(ρ33-ρ44)Δ34,
ρ23F=Ω1Ω3Ω4*2Z*4(Δ13+Δ24)(ρ11-ρ44)-X3(ρ11-ρ22)Δ12+X2(ρ33-ρ44)Δ43,
ρ41R=Ω42Z4Y1(ρ11-ρ44)-|Ω1|2Y2(ρ11-ρ22)Δ12+|Ω3|2Y3(ρ33-ρ44)Δ34,
ρ41F=Ω1Ω3Ω2*2Z4(Δ31+Δ42)(ρ22-ρ33)-Y3(ρ11-ρ22)Δ21+Y2(ρ33-ρ44)Δ43,
X1=|Ω1|2Δ13+|Ω3|2Δ24-4Δ14Δ24Δ13,
X2=|Ω1|2-|Ω3|2-4Δ14Δ24,
X3=|Ω1|2-|Ω3|2+4Δ14Δ13;
Y1=|Ω1|2Δ42+|Ω3|2Δ31+4Δ32Δ42Δ31,
Y2=|Ω1|2-|Ω3|2+4Δ32Δ31,
Y3=|Ω1|2-|Ω3|2-4Δ32Δ42;
Z=(|Ω1|2-|Ω3|2)2+4|Ω1|2(Δ32Δ31-Δ42Δ41)+4|Ω3|2(Δ32Δ42-Δ31Δ41)-16Δ32Δ31Δ42Δ41.
ρ23R=Ω2Δ12(ρ22-ρ33)+Δ14(ρ11-ρ22)2(Δ23-Δ14)Δ12,
ρ23F=Ω1Ω3Ω4*|Ω1|2Δ43(ρ11-ρ44)-Δ14(ρ33-ρ44)2(Δ23-Δ14)Δ43,
ρ41R=Ω4Δ34(ρ11-ρ44)-Δ32(ρ33-ρ44)2(Δ32-Δ41)Δ34,
ρ41F=Ω1Ω3Ω2*|Ω1|2Δ21(ρ22-ρ33)+Δ32(ρ11-ρ22)2(Δ32-Δ41)Δ21,
Γ21ρ22+Γ41ρ44=i(Ω1ρ12-Ω1*ρ21)/2+i(Ω4ρ14-Ω4*ρ41)/2,
-Γ2ρ22=i(Ω1*ρ21-Ω1ρ12)/2+i(Ω2*ρ23-Ω2ρ32)/2,
Γ23ρ22+Γ43ρ44=i(Ω2ρ32-Ω2*ρ23)/2+i(Ω3ρ34-Ω3*ρ43)/2,
-Γ4ρ44=i(Ω4*ρ41-Ω4ρ14)/2+i(Ω3*ρ43-Ω3ρ34)/2,
ρ21=Ω1(ρ22-ρ11)2Δ21,ρ43=-Ω3(ρ33-ρ44)2Δ43.
ρ23R=iΩ2|Ω1|2(Γ2-Γ4)/2Z,
ρ23F=-iΩ1Ω3Ω4*|Ω1|2[Γ2Γ4(Γ2+Γ4)-(Γ2-Γ4)|Ω1|2]/2Z,
ρ41R=-iΩ4|Ω1|2(Γ2-Γ4)/2Z,
ρ41F=-iΩ1Ω3Ω2*|Ω1|2[Γ2Γ4(Γ2+Γ4)+(Γ2-Γ4)|Ω1|2]/2Z,
ρ21R=Ω12Δ21(ρ22-ρ11)+|Ω2|2Δ41Δ34[Δ21(ρ33-ρ22)+Δ32(ρ22-ρ11)]Z,
ρ21F=Ω2Ω3*Ω42ZΔ32[Δ34(ρ44-ρ11)+Δ41(ρ33-ρ44)],
ρ23R=Ω22Δ23(ρ22-ρ33)+|Ω1|2Δ14Δ43[Δ23(ρ22-ρ11)-Δ12(ρ22-ρ23)]Z*,
ρ23F=Ω1Ω3Ω4*2Z*Δ12[Δ14(ρ33-ρ44)-Δ43(ρ11-ρ44)],
ρ43R=Ω32Δ43(ρ44-ρ33)+|Ω4|2Δ12Δ23[Δ43(ρ44-ρ11)+Δ14(ρ33-ρ44)]Z*,
ρ43F=Ω1*Ω2Ω42Z*Δ14[Δ12(ρ33-ρ22)+Δ23(ρ22-ρ11)],
ρ41R=Ω42Δ41(ρ44-ρ11)+|Ω3|2Δ21Δ32[Δ41(ρ33-ρ44)-Δ34(ρ11-ρ44)]Z,
ρ41F=Ω1Ω2*Ω32ZΔ34[Δ32(ρ22-ρ11)-Δ21(ρ22-ρ33)],
Z=|Ω1|2Δ21Δ34Δ41-|Ω2|2Δ32Δ34Δ41-|Ω3|2Δ21Δ34Δ32
+|Ω4|2Δ21Δ32Δ41+4Δ21Δ32Δ31Δ34Δ41.

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