Abstract

The results of a study of molecular oxygen by magnetic rotation spectroscopy are presented. The experiment is based on the use of a semiconductor diode laser operating near room temperature at λ=762 nm. The dependence of the signal on a variety of parameters including magnetic-field strength, oxygen pressure, and analyzing polarizer offset angle is investigated both theoretically and experimentally.

© 1997 Optical Society of America

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References

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  1. A. D. Buckingham and P. J. Stephens, “Magnetic optical activity,” Ann. Rev. Phys. Chem. 17, 399–432 (1966).
    [CrossRef]
  2. G. A. Mann and C. D. Hause, “Magnetic rotation spectra of nitric oxide in the near infrared,” J. Chem. Phys. 33, 1117–1123 (1960).
    [CrossRef]
  3. J. L. Aubel and C. D. Hause, “Magnetic rotation spectra of the 2–0 vibration–rotation band of NO,” J. Chem. Phys. 44, 2659–2664 (1966).
    [CrossRef]
  4. D. B. Keck and C. D. Hause, “Spectra of the 1–0 vibration–rotation band of nitric oxide,” J. Chem. Phys. 49, 3458–3464 (1968).
    [CrossRef]
  5. A. D. Buckingham and G. A. Segal, “Calculation of the magnetic rotation spectra of NO in the near infrared,” J. Chem. Phys. 49, 1964–1966 (1968).
    [CrossRef]
  6. F. A. Blum, K. W. Nill, and A. J. Strauss, “Line shape of the Doppler-limited infrared magnetic rotation spectrum of nitric oxide,” J. Chem. Phys. 58, 4968–4970 (1973).
    [CrossRef]
  7. G. Litfin, C. R. Pollock, R. F. Curl, and F. K. Tittel, “Sensitive enhancement of laser absorption spectroscopy by magnetic rotation effect,” J. Chem. Phys. 72, 6602–6605 (1980).
    [CrossRef]
  8. T. A. Blake, C. Chackerian, and J. R. Podolske, “Prognosis for a mid-infrared magnetic rotation spectrometer for the in situ detection of atmospheric free radicals,” Appl. Opt. 35, 973–985 (1996).
    [CrossRef] [PubMed]
  9. A. F. Stalder and W. H. Eberhardt, “Magnetically induced circular dichroism and birefringence in single lines of the electronic spectrum of ICl,” J. Chem. Phys. 47, 1445–1451 (1967).
    [CrossRef]
  10. M. C. McCarthy, J. C. Bloch, and R. W. Field, “Frequency-modulation enhanced magnetic rotation spectroscopy: a sensitive and selective absorption scheme for paramagnetic molecules,” J. Chem. Phys. 100, 6331–6346 (1994).
    [CrossRef]
  11. M. C. McCarthy and R. W. Field, “Frequency-modulation enhanced magnetic rotation spectroscopy of PdH, PdD, NiH and CuH,” J. Chem. Phys. 100, 6347–6358 (1994).
    [CrossRef]
  12. M. C. McCarthy and R. W. Field, “The use of magnetic rotation spectroscopy to simplify and presort spectra: an application to NiH and CeF,” J. Chem. Phys. 96, 7237–7244 (1992).
    [CrossRef]
  13. M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), pp. 372–375.
  14. M. De Angelis, G. Gagliardi, L. Gianfrini, and G. M. Tino, “Test of the symmetrization postulate for spin-0 particles,” Phys. Rev. Lett. 76, 2840–2843 (1996).
    [CrossRef] [PubMed]
  15. R. C. Hilborn and C. L. Yuca, “Spectroscopic test of the symmetrization postulate for spin-0 nuclei,” Phys. Rev. Lett. 76, 2844–2847 (1996).
    [CrossRef] [PubMed]
  16. D. A. Van Baak, “Resonant Faraday rotation as a probe of atomic dispersion,” Am. J. Phys. 64, 724–735 (1996).
    [CrossRef]

1996 (4)

M. De Angelis, G. Gagliardi, L. Gianfrini, and G. M. Tino, “Test of the symmetrization postulate for spin-0 particles,” Phys. Rev. Lett. 76, 2840–2843 (1996).
[CrossRef] [PubMed]

R. C. Hilborn and C. L. Yuca, “Spectroscopic test of the symmetrization postulate for spin-0 nuclei,” Phys. Rev. Lett. 76, 2844–2847 (1996).
[CrossRef] [PubMed]

D. A. Van Baak, “Resonant Faraday rotation as a probe of atomic dispersion,” Am. J. Phys. 64, 724–735 (1996).
[CrossRef]

T. A. Blake, C. Chackerian, and J. R. Podolske, “Prognosis for a mid-infrared magnetic rotation spectrometer for the in situ detection of atmospheric free radicals,” Appl. Opt. 35, 973–985 (1996).
[CrossRef] [PubMed]

1994 (2)

M. C. McCarthy, J. C. Bloch, and R. W. Field, “Frequency-modulation enhanced magnetic rotation spectroscopy: a sensitive and selective absorption scheme for paramagnetic molecules,” J. Chem. Phys. 100, 6331–6346 (1994).
[CrossRef]

M. C. McCarthy and R. W. Field, “Frequency-modulation enhanced magnetic rotation spectroscopy of PdH, PdD, NiH and CuH,” J. Chem. Phys. 100, 6347–6358 (1994).
[CrossRef]

1992 (1)

M. C. McCarthy and R. W. Field, “The use of magnetic rotation spectroscopy to simplify and presort spectra: an application to NiH and CeF,” J. Chem. Phys. 96, 7237–7244 (1992).
[CrossRef]

1980 (1)

G. Litfin, C. R. Pollock, R. F. Curl, and F. K. Tittel, “Sensitive enhancement of laser absorption spectroscopy by magnetic rotation effect,” J. Chem. Phys. 72, 6602–6605 (1980).
[CrossRef]

1973 (1)

F. A. Blum, K. W. Nill, and A. J. Strauss, “Line shape of the Doppler-limited infrared magnetic rotation spectrum of nitric oxide,” J. Chem. Phys. 58, 4968–4970 (1973).
[CrossRef]

1968 (2)

D. B. Keck and C. D. Hause, “Spectra of the 1–0 vibration–rotation band of nitric oxide,” J. Chem. Phys. 49, 3458–3464 (1968).
[CrossRef]

A. D. Buckingham and G. A. Segal, “Calculation of the magnetic rotation spectra of NO in the near infrared,” J. Chem. Phys. 49, 1964–1966 (1968).
[CrossRef]

1967 (1)

A. F. Stalder and W. H. Eberhardt, “Magnetically induced circular dichroism and birefringence in single lines of the electronic spectrum of ICl,” J. Chem. Phys. 47, 1445–1451 (1967).
[CrossRef]

1966 (2)

J. L. Aubel and C. D. Hause, “Magnetic rotation spectra of the 2–0 vibration–rotation band of NO,” J. Chem. Phys. 44, 2659–2664 (1966).
[CrossRef]

A. D. Buckingham and P. J. Stephens, “Magnetic optical activity,” Ann. Rev. Phys. Chem. 17, 399–432 (1966).
[CrossRef]

1960 (1)

G. A. Mann and C. D. Hause, “Magnetic rotation spectra of nitric oxide in the near infrared,” J. Chem. Phys. 33, 1117–1123 (1960).
[CrossRef]

Aubel, J. L.

J. L. Aubel and C. D. Hause, “Magnetic rotation spectra of the 2–0 vibration–rotation band of NO,” J. Chem. Phys. 44, 2659–2664 (1966).
[CrossRef]

Blake, T. A.

Bloch, J. C.

M. C. McCarthy, J. C. Bloch, and R. W. Field, “Frequency-modulation enhanced magnetic rotation spectroscopy: a sensitive and selective absorption scheme for paramagnetic molecules,” J. Chem. Phys. 100, 6331–6346 (1994).
[CrossRef]

Blum, F. A.

F. A. Blum, K. W. Nill, and A. J. Strauss, “Line shape of the Doppler-limited infrared magnetic rotation spectrum of nitric oxide,” J. Chem. Phys. 58, 4968–4970 (1973).
[CrossRef]

Buckingham, A. D.

A. D. Buckingham and G. A. Segal, “Calculation of the magnetic rotation spectra of NO in the near infrared,” J. Chem. Phys. 49, 1964–1966 (1968).
[CrossRef]

A. D. Buckingham and P. J. Stephens, “Magnetic optical activity,” Ann. Rev. Phys. Chem. 17, 399–432 (1966).
[CrossRef]

Chackerian, C.

Curl, R. F.

G. Litfin, C. R. Pollock, R. F. Curl, and F. K. Tittel, “Sensitive enhancement of laser absorption spectroscopy by magnetic rotation effect,” J. Chem. Phys. 72, 6602–6605 (1980).
[CrossRef]

De Angelis, M.

M. De Angelis, G. Gagliardi, L. Gianfrini, and G. M. Tino, “Test of the symmetrization postulate for spin-0 particles,” Phys. Rev. Lett. 76, 2840–2843 (1996).
[CrossRef] [PubMed]

Eberhardt, W. H.

A. F. Stalder and W. H. Eberhardt, “Magnetically induced circular dichroism and birefringence in single lines of the electronic spectrum of ICl,” J. Chem. Phys. 47, 1445–1451 (1967).
[CrossRef]

Field, R. W.

M. C. McCarthy, J. C. Bloch, and R. W. Field, “Frequency-modulation enhanced magnetic rotation spectroscopy: a sensitive and selective absorption scheme for paramagnetic molecules,” J. Chem. Phys. 100, 6331–6346 (1994).
[CrossRef]

M. C. McCarthy and R. W. Field, “Frequency-modulation enhanced magnetic rotation spectroscopy of PdH, PdD, NiH and CuH,” J. Chem. Phys. 100, 6347–6358 (1994).
[CrossRef]

M. C. McCarthy and R. W. Field, “The use of magnetic rotation spectroscopy to simplify and presort spectra: an application to NiH and CeF,” J. Chem. Phys. 96, 7237–7244 (1992).
[CrossRef]

Gagliardi, G.

M. De Angelis, G. Gagliardi, L. Gianfrini, and G. M. Tino, “Test of the symmetrization postulate for spin-0 particles,” Phys. Rev. Lett. 76, 2840–2843 (1996).
[CrossRef] [PubMed]

Gianfrini, L.

M. De Angelis, G. Gagliardi, L. Gianfrini, and G. M. Tino, “Test of the symmetrization postulate for spin-0 particles,” Phys. Rev. Lett. 76, 2840–2843 (1996).
[CrossRef] [PubMed]

Hause, C. D.

D. B. Keck and C. D. Hause, “Spectra of the 1–0 vibration–rotation band of nitric oxide,” J. Chem. Phys. 49, 3458–3464 (1968).
[CrossRef]

J. L. Aubel and C. D. Hause, “Magnetic rotation spectra of the 2–0 vibration–rotation band of NO,” J. Chem. Phys. 44, 2659–2664 (1966).
[CrossRef]

G. A. Mann and C. D. Hause, “Magnetic rotation spectra of nitric oxide in the near infrared,” J. Chem. Phys. 33, 1117–1123 (1960).
[CrossRef]

Hilborn, R. C.

R. C. Hilborn and C. L. Yuca, “Spectroscopic test of the symmetrization postulate for spin-0 nuclei,” Phys. Rev. Lett. 76, 2844–2847 (1996).
[CrossRef] [PubMed]

Keck, D. B.

D. B. Keck and C. D. Hause, “Spectra of the 1–0 vibration–rotation band of nitric oxide,” J. Chem. Phys. 49, 3458–3464 (1968).
[CrossRef]

Lamb, W. E.

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), pp. 372–375.

Litfin, G.

G. Litfin, C. R. Pollock, R. F. Curl, and F. K. Tittel, “Sensitive enhancement of laser absorption spectroscopy by magnetic rotation effect,” J. Chem. Phys. 72, 6602–6605 (1980).
[CrossRef]

Mann, G. A.

G. A. Mann and C. D. Hause, “Magnetic rotation spectra of nitric oxide in the near infrared,” J. Chem. Phys. 33, 1117–1123 (1960).
[CrossRef]

McCarthy, M. C.

M. C. McCarthy and R. W. Field, “Frequency-modulation enhanced magnetic rotation spectroscopy of PdH, PdD, NiH and CuH,” J. Chem. Phys. 100, 6347–6358 (1994).
[CrossRef]

M. C. McCarthy, J. C. Bloch, and R. W. Field, “Frequency-modulation enhanced magnetic rotation spectroscopy: a sensitive and selective absorption scheme for paramagnetic molecules,” J. Chem. Phys. 100, 6331–6346 (1994).
[CrossRef]

M. C. McCarthy and R. W. Field, “The use of magnetic rotation spectroscopy to simplify and presort spectra: an application to NiH and CeF,” J. Chem. Phys. 96, 7237–7244 (1992).
[CrossRef]

Nill, K. W.

F. A. Blum, K. W. Nill, and A. J. Strauss, “Line shape of the Doppler-limited infrared magnetic rotation spectrum of nitric oxide,” J. Chem. Phys. 58, 4968–4970 (1973).
[CrossRef]

Podolske, J. R.

Pollock, C. R.

G. Litfin, C. R. Pollock, R. F. Curl, and F. K. Tittel, “Sensitive enhancement of laser absorption spectroscopy by magnetic rotation effect,” J. Chem. Phys. 72, 6602–6605 (1980).
[CrossRef]

Sargent, M.

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), pp. 372–375.

Scully, M. O.

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), pp. 372–375.

Segal, G. A.

A. D. Buckingham and G. A. Segal, “Calculation of the magnetic rotation spectra of NO in the near infrared,” J. Chem. Phys. 49, 1964–1966 (1968).
[CrossRef]

Stalder, A. F.

A. F. Stalder and W. H. Eberhardt, “Magnetically induced circular dichroism and birefringence in single lines of the electronic spectrum of ICl,” J. Chem. Phys. 47, 1445–1451 (1967).
[CrossRef]

Stephens, P. J.

A. D. Buckingham and P. J. Stephens, “Magnetic optical activity,” Ann. Rev. Phys. Chem. 17, 399–432 (1966).
[CrossRef]

Strauss, A. J.

F. A. Blum, K. W. Nill, and A. J. Strauss, “Line shape of the Doppler-limited infrared magnetic rotation spectrum of nitric oxide,” J. Chem. Phys. 58, 4968–4970 (1973).
[CrossRef]

Tino, G. M.

M. De Angelis, G. Gagliardi, L. Gianfrini, and G. M. Tino, “Test of the symmetrization postulate for spin-0 particles,” Phys. Rev. Lett. 76, 2840–2843 (1996).
[CrossRef] [PubMed]

Tittel, F. K.

G. Litfin, C. R. Pollock, R. F. Curl, and F. K. Tittel, “Sensitive enhancement of laser absorption spectroscopy by magnetic rotation effect,” J. Chem. Phys. 72, 6602–6605 (1980).
[CrossRef]

Van Baak, D. A.

D. A. Van Baak, “Resonant Faraday rotation as a probe of atomic dispersion,” Am. J. Phys. 64, 724–735 (1996).
[CrossRef]

Yuca, C. L.

R. C. Hilborn and C. L. Yuca, “Spectroscopic test of the symmetrization postulate for spin-0 nuclei,” Phys. Rev. Lett. 76, 2844–2847 (1996).
[CrossRef] [PubMed]

Am. J. Phys. (1)

D. A. Van Baak, “Resonant Faraday rotation as a probe of atomic dispersion,” Am. J. Phys. 64, 724–735 (1996).
[CrossRef]

Ann. Rev. Phys. Chem. (1)

A. D. Buckingham and P. J. Stephens, “Magnetic optical activity,” Ann. Rev. Phys. Chem. 17, 399–432 (1966).
[CrossRef]

Appl. Opt. (1)

J. Chem. Phys. (10)

A. F. Stalder and W. H. Eberhardt, “Magnetically induced circular dichroism and birefringence in single lines of the electronic spectrum of ICl,” J. Chem. Phys. 47, 1445–1451 (1967).
[CrossRef]

M. C. McCarthy, J. C. Bloch, and R. W. Field, “Frequency-modulation enhanced magnetic rotation spectroscopy: a sensitive and selective absorption scheme for paramagnetic molecules,” J. Chem. Phys. 100, 6331–6346 (1994).
[CrossRef]

M. C. McCarthy and R. W. Field, “Frequency-modulation enhanced magnetic rotation spectroscopy of PdH, PdD, NiH and CuH,” J. Chem. Phys. 100, 6347–6358 (1994).
[CrossRef]

M. C. McCarthy and R. W. Field, “The use of magnetic rotation spectroscopy to simplify and presort spectra: an application to NiH and CeF,” J. Chem. Phys. 96, 7237–7244 (1992).
[CrossRef]

G. A. Mann and C. D. Hause, “Magnetic rotation spectra of nitric oxide in the near infrared,” J. Chem. Phys. 33, 1117–1123 (1960).
[CrossRef]

J. L. Aubel and C. D. Hause, “Magnetic rotation spectra of the 2–0 vibration–rotation band of NO,” J. Chem. Phys. 44, 2659–2664 (1966).
[CrossRef]

D. B. Keck and C. D. Hause, “Spectra of the 1–0 vibration–rotation band of nitric oxide,” J. Chem. Phys. 49, 3458–3464 (1968).
[CrossRef]

A. D. Buckingham and G. A. Segal, “Calculation of the magnetic rotation spectra of NO in the near infrared,” J. Chem. Phys. 49, 1964–1966 (1968).
[CrossRef]

F. A. Blum, K. W. Nill, and A. J. Strauss, “Line shape of the Doppler-limited infrared magnetic rotation spectrum of nitric oxide,” J. Chem. Phys. 58, 4968–4970 (1973).
[CrossRef]

G. Litfin, C. R. Pollock, R. F. Curl, and F. K. Tittel, “Sensitive enhancement of laser absorption spectroscopy by magnetic rotation effect,” J. Chem. Phys. 72, 6602–6605 (1980).
[CrossRef]

Phys. Rev. Lett. (2)

M. De Angelis, G. Gagliardi, L. Gianfrini, and G. M. Tino, “Test of the symmetrization postulate for spin-0 particles,” Phys. Rev. Lett. 76, 2840–2843 (1996).
[CrossRef] [PubMed]

R. C. Hilborn and C. L. Yuca, “Spectroscopic test of the symmetrization postulate for spin-0 nuclei,” Phys. Rev. Lett. 76, 2844–2847 (1996).
[CrossRef] [PubMed]

Other (1)

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), pp. 372–375.

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Figures (12)

Fig. 1
Fig. 1

Schematic of the experimental setup used for magnetic rotation spectroscopy. The important elements include the laser diode; two polarizers (P’s), the second of which can be rotated an angle of θ measured with respect to the position at which the two polarizers are crossed; a gas cell immersed in the longitudinal magnetic field produced by a long tightly wrapped solenoid; the detector (D); and the data analysis system [lock-in amplifier, oscilloscope, and computer (PC)].

Fig. 2
Fig. 2

Qualitative description of the two effects present in this system, MCB and MCD, and their characteristic signals. In (a) we sketch the energy levels for a simple J=1J=0 transition with no applied field, and in (b) the same levels are shown with a B field applied. The J=1 level splits into three sublevels, which are shifted (Zeeman effect). The resulting difference between nR and nL (MCB signal) is shown without and with the applied field in (c) and (d), respectively. In (e) and (f) we likewise show the difference between κR and κL (MCD signal) without and with the magnetic field, respectively. Finally, we show the first derivatives of the the MCB and the MCD signals in (g) and (h), respectively.

Fig. 3
Fig. 3

Theoretical first-derivative curves for the magnetic rotation signal as a function of laser frequency for several analyzer offset angles. The magnetic field was taken to have a value of 45 G and the pressure to be 75 Torr. Values of θ are -1.0°, -0.5°, 0.0°, 0.5°, and 1.0°, where θ is defined in the text.

Fig. 4
Fig. 4

Theoretical first-derivative curves for the magnetic rotation signal as a function of laser frequency for various magnetic-field strengths. The analyzer offset angle is 1.0°, and the pressure is taken to be 75 Torr. Values of B range from 23 to 138 G in steps of 23 G. The resulting signal amplitude is linear in the applied field for these relatively low fields.

Fig. 5
Fig. 5

Theoretical first-derivative curves for the magnetic rotation signal as a function of laser frequency for three widely differing magnetic-field strengths: (i) B is such that the Zeeman splitting is equal to one-half the Doppler width of the transition; (ii) the Zeeman splitting is equal to the Doppler width; (iii) the Zeeman splitting is equal to twice the Doppler width.

Fig. 6
Fig. 6

For θ=1.0° the theoretical first-derivative curves for the magnetic rotation signal are plotted as a function of laser frequency for oxygen pressures ranging from 21 Torr (Doppler-broadening-dominated regime) to 420 Torr (pressure-broadening-dominated regime). (i) P=21 Torr, (ii) P=105 Torr, (iii) P=210 Torr, (iv) P=420 Torr. The magnetic field has been taken to be 59 G.

Fig. 7
Fig. 7

Experimental results for the magnetic rotation signal as a function of laser frequency for several analyzer settings. Values of θ are -1.0°, -0.5°, 0.0°, and 0.5°, all with an uncertainty of 0°5. The pressure is 88±10 Torr, and the magnetic field is 47±5 G. The signals shown are the output of the lock-in amplifier and represent 1f detection, with a subtraction of background run for zero applied magnetic field.

Fig. 8
Fig. 8

Experimental results for the magnetic rotation signal for various magnetic-field strengths. The pressure is 75 Torr, and the analyzer offset angle is 1.0°. Signals are the 1f lock-in output, with a subtraction of a run with zero applied magnetic field.

Fig. 9
Fig. 9

Plot of the peak-to-peak signal as a function of applied magnetic field with the data in Fig. 8. The solid line is a least-squares straight-line fit to the data.

Fig. 10
Fig. 10

Lock-in output signal 1f as a function of laser frequency for various oxygen pressures. Analyzer offset angle θ=1.0°, magnetic-field strength B=58 G. The pressures are (i) 25 Torr, (ii) 100 Torr, (iii) 337 Torr, (iv) 625 Torr.

Fig. 11
Fig. 11

Plot of the magnetic rotation signal amplitude as a function of pressure for data of Fig. 10. A graph of the theoretical prediction has also been included, with the theory curve scaled to match the experimental one at the maximum signal value.

Fig. 12
Fig. 12

Schematic of the energy levels used in calculating the magnetization. The triplet ground state consists of two levels that interact with the circularly polarized light, |b and |c. The unshifted level |g does not participate in the interaction. The magnetic-field induced frequency shift in the ground-state energy levels is denoted by δ.

Equations (46)

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

2E-1c22Et2=μ0 t(×M).
E=ELeˆL exp[-i(ωt-kz+ϕL)]+EReˆR exp[-i(ωt-kz+ϕR)]+c.c.
eˆL=1/2(xˆ-iyˆ),
eˆR=1/2(xˆ+iyˆ).
I=1/20c{EL2+ER2-2ELER cos[2θ-(ϕR-ϕL)]}.
EL=E0 exp(-κgL)+d/2,
ER=E0 exp(-κgR)-d/2,
ϕL,R=κfL,R.
κ=Nμ0μ2cα2l
fL,R=- 1πΔωexp[-(ω-ωR,L)2/Δω2]×ω(ω-ω)γ2+(ω-ω)2dω,
gL,R=- 1πΔωexp[-(ω-ωR,L)2/Δω2]×ωγγ2+(ω-ω)2dω.
2ln(2)Δω={4π[2 ln(2)kT/m]1/2}/λ,
dIdωddω[I0(d/2E0)κ(gL-gR)],
dIdωddω[2I0κθ(fL-fR)].
δ2πB=1.001J(J+1)[J(J+1)+S(S+1)-N(N+1)]MμB=1.40 MHz/G,
M=Nμ=N Tr(ρμ)=N[ρaba|μ|b+ρbab|μ|a+ρaca|μ|c+ρcac|μ|a],
ρ˙=-i[H, ρ].
H=HA+VI,
HA=Ea|aa|+Eb|bb|+Ec|cc|
VI=-μ·B=-μBR exp(-iωt)-μBL exp(-iωt)+H.c.Vab|ab|+Vac|ac|+Vba|b×a|+Vca|ca|.
ρ˙ab=-i(ω0-δ)ρab-γρab-iVab(ρbb-ρaa),
ρ˙ac=-i(ω0+δ)ρac-γρac-iVac(ρcc-ρaa).
E=exp(ikz)exp(-iωt)[ERR exp(-iϕR)+ELL exp(-iϕL)]+c.c.,
B=exp(ikz)exp(-iωt)[BRR exp(-iϕR)+BLL exp(-iϕL)]+c.c.
BR=-iER/c,BL=iEL/c.
Vab=-iμELcexp(-iϕL)exp(-iωt)exp(ikz),
Vac=iμERcexp(-iϕR)exp(-iωt)exp(ikz).
ρ˙ab=-γρab-i(ω0-δ)ρab-μELcexp(-iϕL)exp(-iωt)exp(ikz)(ρbb-ρaa),
ρ˙ac=-γρac-i(ω0+δ)ρac+μERcexp(-iϕR)exp(-iωt)exp(ikz)(ρcc-ρaa).
ρaa=0,ρbb=ρcc=α=ρgg
ρab=-μαELcexp(-iϕL)exp(-iωt)×exp(ikz) γ+i[ω-(ω0-δ)]γ2+[ω-(ω0-δ)]2,
ρac=+μαERcexp(-iϕR)exp(-iωt)×exp(ikz) γ+i[ω-(ω0+δ)]γ2[ω-(ω0+δ)]2.
M=N μ2αcexp[i(kz-ωt)]×ER γ+i[ω(ω0+δ)]γ2+[ω-(ω0+δ)]2exp(-iϕR)R-EL γ+i[ω-(ω0-δ)]γ2+[ω-(ω0-δ)]2exp(-iϕL)L+c.c.
M=MLL exp[i(kz-ωt)]exp(-iϕL)+MR exp[i(kz-ωt)]exp(-iϕR).
MR=N μ2αERcγ+i(ω-ωR)γ2+(ω-ωR)2,
ML=-N μ2αELcγ+i(ω-ωL)γ2+(ω-ωL)2,
ϕR,Lz=μ0ω2ER,LIm(MR,L),
ER,Lz=-μ0ω2Re(MR,L).
ϕR,L(l)=κFR,L(ω, ωR,L),
ER,L=E0 exp[-κGR,L(ω, ωR,L)].
κ=Nμ0μ2cα2l,
FR,L(ω, ωR,L)=ω(ω-ωR,L)γ2+(ω-ωR,L)2,
GR,L(ω, ωR,L)=ωγγ2+(ω-ωR,L)2.
W(ω, ωR,L)=1πΔωexp[-(ω, ωR,L)2/Δω2].
fR,L-W(ω, ωR,L)FR,L(ω, ω)dω=- 1πΔωexp[-(ω-ωR,L)2/Δω2]×ω(ω-ω)γ2+(ω-ω)2dω,
gR,L-W(ω, ωR,L)GR,L(ω, ω)dω=- 1πΔωexp[-(ω-ωR,L)2/Δω2]×ωγγ2+(ω-ω)2dω.

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