Abstract

Using a 12 transition as an analytically tractable model, we discuss in detail magneto-optical resonances of both electromagnetically induced absorption (EIA) and electromagnetically induced transparency (EIT) types in the Hanle configuration. The analysis is made for an arbitrary rate of depolarizing collisions in the excited state and an arbitrary elliptical field polarization. The obtained results clearly show that the main reason for the EIA subnatural resonance is the spontaneous transfer of anisotropy from the excited level to the ground level. In the EIA case we predict the negative structures in the absorption resonance at large field detuning. The role of the finite atom-light interaction time is briefly discussed. In addition we study nontrivial peculiarities of the resonance line shape related to the velocity spread in a gas.

© 2005 Optical Society of America

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2003 (8)

C. H. van der Wal, M. D. Eisaman, A. Andre, R. L. Walsworth, D. F. Phillips, A. S. Zibrov, and M. D. Lukin, "Atomic memory for correlated photon states," Science 301, 196-200 (2003).
[CrossRef] [PubMed]

H. Failache, P. Valente, G. Ban, V. Lorent, and A. Lezama, "Inhibition of electromagnetically induced absorption due to excited-state decoherence in Rb vapor," Phys. Rev. A 67, 043810 (2003).
[CrossRef]

C. Goren, A. D. Wilson-Gordon, M. Rosenbluh, and H. Friedmann, "Electromagnetically induced absorption due to transfer of coherence and to transfer of population," Phys. Rev. A 67, 033807 (2003).
[CrossRef]

J. Vanier, M. W. Levine, D. Jansses, and M. Delaney, "On the use of intensity optical pumping and coherent population trapping techniques in the implementation of atomic frequency standards," IEEE Trans. Instrum. Meas. 52, 822-831 (2003).
[CrossRef]

A. B. Matsko, I. Novikova, M. S. Zubairy, and G. R. Welch, "Nonlinear magneto-optical rotation of elliptically polarized light," Phys. Rev. A 67, 043805 (2003).
[CrossRef]

S. Knappe, M. Stahler, C. Affolderbach, A. V. Taichenachev, V. I. Yudin, and R. Wynands, "Simple parameterization of dark-resonance line shapes," Appl. Phys. B: Lasers Opt. 76, 57-63 (2003).
[CrossRef]

A. V. Taichenachev, V. I. Yudin, R. Wynands, M. Stahler, J. Kitching, and L. Hollberg, "Theory of dark resonances for alkali-metal vapors in a buffer-gas cell," Phys. Rev. A 67, 033810 (2003).
[CrossRef]

J. Vanier, M. W. Levine, D. Jansses, and M. Delaney, "Contrast and linewidth of the coherent population trapping transmission hyperfine resonance line in 87Rb: effect of optical pumping," Phys. Rev. A 67, 065801 (2003).
[CrossRef]

2002 (6)

A. Javan, O. Kocharovskaya, H. Lee, and M. O. Scully, "Narrowing of electromagnetically induced transparency resonance in a Doppler-broadened medium," Phys. Rev. A 66, 013805 (2002).
[CrossRef]

C. Y. Ye and A. S. Zibrov, "Width of the electromagnetically induced transparency resonance in atomic vapor," Phys. Rev. A 65, 023806 (2002).
[CrossRef]

M. Stahler, R. Wynands, S. Knappe, J. Kitching, L. Hollberg, A. Taichenachev, and V. Yudin, "Coherent population trapping resonances in thermal 85Rb vapor: D1 versus D2 line excitation," Opt. Lett. 27, 1472-1474 (2002).
[CrossRef]

M. Fleschauer and M. D. Lukin, "Quantum memory for photons: Dark-state polaritons," Phys. Rev. A 65, 022314 (2002).
[CrossRef]

A. S. Zibrov, A. B. Matsko, O. Kocharovskaya, Y. V. Rostovtsev, G. R. Welch, and M. O. Scully, "Transporting and time reversing light via atomic coherence," Phys. Rev. Lett. 88, 103601 (2002).
[CrossRef] [PubMed]

C. Andreeva, S. Cartaleva, Y. Dancheva, V. Biancalana, A. Burchianti, C. Marinelli, E. Marriotti, L. Moi, and K. Nasyrov, "Coherent spectroscopy of degenerate two-level systems in Cs," Phys. Rev. A 66, 012502 (2002).
[CrossRef]

2001 (9)

F. Renzoni, S. Cartaleva, G. Alzetta, and E. Arimondo, "Enhanced absorption Hanle effect in the configuration of crossed laser beam and magnetic field," Phys. Rev. A 63, 065401 (2001).
[CrossRef]

M. Kwon, K. Kim, H. S. Moon, H. D. Park, and J. B. Kim, "Dependence of electromagnetically induced absorption on two combinations of orthogonal polarized beams," J. Phys. B 34, 2951-2961 (2001).
[CrossRef]

K. Kim, M. Kwon, H. D. Park, H. S. Moon, H. S. Rawat, K. An, and J. B. Kim, "Electromagnetically induced absorptionspectra depending on intensities and detunings of the coupling field in Cs vapour," J. Phys. B 34, 4801-4808 (2001).
[CrossRef]

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, "Observation of coherent optical information storage in an atomic medium using halted light pulses," Nature (London) 409, 490-493 (2001).
[CrossRef]

D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, "Storage of light in atomic vapor," Phys. Rev. Lett. 86, 783-786 (2001).
[CrossRef] [PubMed]

M. D. Lukin and A. Imamoglu, "Controlling photons using electromagnetically induced transparency," Nature (London) 413, 273-276 (2001).
[CrossRef]

H. Wang, D. Goorskey, and M. Xiao, "Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system," Phys. Rev. Lett. 87, 073601 (2001).
[CrossRef] [PubMed]

M. Erhard and H. Helm, "Buffer-gas effects on dark resonances: theory and experiment," Phys. Rev. A 63, 043813 (2001).
[CrossRef]

M. Stahler, S. Knappe, C. Affolderbach, W. Kemp, and R. Wynands, "Picotesla magnetometry with coherent dark states," Europhys. Lett. 54, 323-328 (2001).
[CrossRef]

2000 (6)

A. V. Taichenachev, A. M. Tumaikin, and V. I. Yudin, "Influence of atomic motion on the shape of two-photon resonance in gas," JETP Lett. 72, 119-122 (2000).
[CrossRef]

J. Kitching, S. Knappe, N. Vukicevic, L. Hollberg, R. Wynands, and W. Weidemann, "A microwave frequency reference based on VCSEL-driven dark line resonances in Cs vapor," IEEE Trans. Instrum. Meas. 49, 1313-1317 (2000).
[CrossRef]

I. Novikova, A. Matsko, V. A. Sautenkov, G. R. Welch, and M. O. Scully, "Ac-Stark shifts in the nonlinear Faraday effect," Opt. Lett. 25, 1651-1653 (2000).
[CrossRef]

A. V. Taichenachev, A. M. Tumaikin, and V. I. Yudin, "Electromagnetically induced absorption in a four-state system," Phys. Rev. A 61, 011802 (R) (2000).
[CrossRef]

M. Fleschauer and M. D. Lukin, "Dark-state polaritons in electromagnetically induced transparency," Phys. Rev. Lett. 84, 5094-5097 (2000).
[CrossRef]

Y. Dancheva, G. Alzetta, S. Cartaleva, M. Taslakov, and Ch. Andreeva, "Coherent effects on the Zeeman sublevels of hyperfine states in optical pumping of Rb by monomode diode laser," Opt. Commun. 178, 103-110 (2000).
[CrossRef]

1999 (8)

A. Lezama, S. Barreiro, and A. M. Akulshin, "Electromagnetically induced absorption," Phys. Rev. A 59, 4732-4735 (1999).
[CrossRef]

A. V. Taichenachev, A. M. Tumaikin, and V. I. Yudin, "On the spontaneous-coherence-transfer-induced sign change of a sub-natural-width nonlinear resonance," JETP Lett. 69, 819-824 (1999).
[CrossRef]

V. Milner and Y. Prior, "Biaxial spatial orientation of atomic angular momentum," Phys. Rev. A 59, R1738-R1741 (1999).
[CrossRef]

L. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature (London) 397, 594-598 (1999).
[CrossRef]

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]

D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashuk, "Nonlinear magneto-optics and reduced group velocity of light in atomic vapor with slow ground state relaxation," Phys. Rev. Lett. 83, 1767-1770 (1999).
[CrossRef]

M. D. Lukin, A. B. Matsko, M. Fleischhauer, and M. O. Scully, "Quantum noise and correlations in resonantly enhanced wave mixing based on atomic coherence," Phys. Rev. Lett. 82, 1847-1850 (1999).
[CrossRef]

S. E. Harris and L. V. Hau, "Nonlinear optics at low light levels," Phys. Rev. Lett. 82, 4611-4614 (1999).
[CrossRef]

1998 (3)

A. M. Akulshin, S. Barreiro, and A. Lezama, "Electromagnetically induced absorption and transparency due to resonant two-field excitation of quasidegenerate levels in Rb vapor," Phys. Rev. A 57, 2996-3002 (1998).
[CrossRef]

P. D. Featonby, G. S. Summy, C. L. Webb, R. M. Godun, M. K. Oberthaler, A. C. Wilson, C. J. Foot, and K. Burnett, "Separated-path Ramsey atom interferometer," Phys. Rev. Lett. 81, 495-499 (1998).
[CrossRef]

J. Vanier, A. Godone, and F. Levi, "Coherent population trapping in cesium: dark lines and coherent microwave emission," Phys. Rev. A 58, 2345-2358 (1998).
[CrossRef]

1997 (2)

S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50, 36-42 (1997).
[CrossRef]

S. Brandt, A. Nagel, R. Wynands, and D. Meschede, "Buffer-gas-induced linewidth reduction of coherent dark resonances to below 50 Hz," Phys. Rev. A 56, R1063-R1066 (1997).
[CrossRef]

1994 (1)

M. Weitz, B. C. Young, and S. Chu, "Atomic interferometer based on adiabatic population transfer," Phys. Rev. Lett. 73, 2563-2566 (1994).
[CrossRef] [PubMed]

1993 (1)

S. I. Kanorsky, A. Weis, J. Wurster, and T. W. Hansch, "Quantitative investigation of the resonant nonlinear Faraday effect under conditions of optical hyperfine pumping," Phys. Rev. A 47, 1220-1226 (1993).
[CrossRef] [PubMed]

1992 (4)

M. O. Scully and M. Fleschauer, "High-sensitivity magnetometer based on index-enhanced media," Phys. Rev. Lett. 69, 1360-1363 (1992).
[CrossRef] [PubMed]

O. Kocharovskaya, "Amplification and lasing without inversion," Phys. Rep. 219, 175-190 (1992).
[CrossRef]

M. O. Scully, "From lasers and masers to phaseonium and phasers," Phys. Rep. 219, 191-201 (1992).
[CrossRef]

B. Lounis and C. Cohen-Tannoudji, "Coherent population trapping and Fano profiles," J. Phys. II 2, 579-592 (1992).

1991 (2)

A. Akulshin, A. Celikov, and V. Velichansky, "Subnatural absorption resonances on the D1 line of rubidium induced by coherent population trapping," Opt. Commun. 84, 139-143 (1991).
[CrossRef]

P. Marte, P. Zoller, and J. L. Hall, "Coherent atomic mirrors and beam splitters by adiabatic passage in multilevel systems," Phys. Rev. A 44, R4118-R4121 (1991).
[CrossRef]

1990 (1)

X. Chen, V. L. Telegdi, and A. Weis, "Quantitative study of the nonlinear Macaluso-Corbino (resonant Faraday) effect in Cs," Opt. Commun. 74, 301-305 (1990).
[CrossRef]

1989 (1)

1988 (2)

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, "Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping," Phys. Rev. Lett. 61, 826-829 (1988).
[CrossRef] [PubMed]

M. V. Krasheninnikov, V. S. Smirnov, M. B. Sultanov, A. M. Tumaikin, and V. I. Yudin, "Anomalous self-rotation of the polarization plane of a confined light beam in a gas located in a weak magnetic field," Zh. Eksp. Teor. Fiz. 94, 204-209 (1988).

1985 (1)

1984 (1)

A. P. Kasantsev, V. S. Smirnov, A. M. Tumaikin, and I. A. Yagofarov, "Effect of atomic ground state self-polarization in the optical-pumping cycle on the increase of linear light absorption for J-->J+1 transitions," Opt. Spectrosc. 57, 189-191 (1984).

1983 (1)

1976 (1)

G. Alzetta, A. Gozzini, L. Moi, and G. Orriols, "An experimental method for the observation of R.F. transitions and laser beat resonances in oriented Na vapour," Nuovo Cimento Soc. Ital. Fis., B 36, 5-20 (1976).
[CrossRef]

Affolderbach, C.

S. Knappe, M. Stahler, C. Affolderbach, A. V. Taichenachev, V. I. Yudin, and R. Wynands, "Simple parameterization of dark-resonance line shapes," Appl. Phys. B: Lasers Opt. 76, 57-63 (2003).
[CrossRef]

M. Stahler, S. Knappe, C. Affolderbach, W. Kemp, and R. Wynands, "Picotesla magnetometry with coherent dark states," Europhys. Lett. 54, 323-328 (2001).
[CrossRef]

Akulshin, A.

A. Akulshin, A. Celikov, and V. Velichansky, "Subnatural absorption resonances on the D1 line of rubidium induced by coherent population trapping," Opt. Commun. 84, 139-143 (1991).
[CrossRef]

Akulshin, A. M.

A. Lezama, S. Barreiro, and A. M. Akulshin, "Electromagnetically induced absorption," Phys. Rev. A 59, 4732-4735 (1999).
[CrossRef]

A. M. Akulshin, S. Barreiro, and A. Lezama, "Electromagnetically induced absorption and transparency due to resonant two-field excitation of quasidegenerate levels in Rb vapor," Phys. Rev. A 57, 2996-3002 (1998).
[CrossRef]

Alzetta, G.

F. Renzoni, S. Cartaleva, G. Alzetta, and E. Arimondo, "Enhanced absorption Hanle effect in the configuration of crossed laser beam and magnetic field," Phys. Rev. A 63, 065401 (2001).
[CrossRef]

Y. Dancheva, G. Alzetta, S. Cartaleva, M. Taslakov, and Ch. Andreeva, "Coherent effects on the Zeeman sublevels of hyperfine states in optical pumping of Rb by monomode diode laser," Opt. Commun. 178, 103-110 (2000).
[CrossRef]

G. Alzetta, A. Gozzini, L. Moi, and G. Orriols, "An experimental method for the observation of R.F. transitions and laser beat resonances in oriented Na vapour," Nuovo Cimento Soc. Ital. Fis., B 36, 5-20 (1976).
[CrossRef]

An, K.

K. Kim, M. Kwon, H. D. Park, H. S. Moon, H. S. Rawat, K. An, and J. B. Kim, "Electromagnetically induced absorptionspectra depending on intensities and detunings of the coupling field in Cs vapour," J. Phys. B 34, 4801-4808 (2001).
[CrossRef]

Andre, A.

C. H. van der Wal, M. D. Eisaman, A. Andre, R. L. Walsworth, D. F. Phillips, A. S. Zibrov, and M. D. Lukin, "Atomic memory for correlated photon states," Science 301, 196-200 (2003).
[CrossRef] [PubMed]

Andreeva, C.

C. Andreeva, S. Cartaleva, Y. Dancheva, V. Biancalana, A. Burchianti, C. Marinelli, E. Marriotti, L. Moi, and K. Nasyrov, "Coherent spectroscopy of degenerate two-level systems in Cs," Phys. Rev. A 66, 012502 (2002).
[CrossRef]

Andreeva, Ch.

Y. Dancheva, G. Alzetta, S. Cartaleva, M. Taslakov, and Ch. Andreeva, "Coherent effects on the Zeeman sublevels of hyperfine states in optical pumping of Rb by monomode diode laser," Opt. Commun. 178, 103-110 (2000).
[CrossRef]

Arimondo, E.

F. Renzoni, S. Cartaleva, G. Alzetta, and E. Arimondo, "Enhanced absorption Hanle effect in the configuration of crossed laser beam and magnetic field," Phys. Rev. A 63, 065401 (2001).
[CrossRef]

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, "Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping: theoretical analysis," J. Opt. Soc. Am. B 6, 2112-2124 (1989).
[CrossRef]

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, "Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping," Phys. Rev. Lett. 61, 826-829 (1988).
[CrossRef] [PubMed]

Aspect, A.

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, "Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping: theoretical analysis," J. Opt. Soc. Am. B 6, 2112-2124 (1989).
[CrossRef]

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, "Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping," Phys. Rev. Lett. 61, 826-829 (1988).
[CrossRef] [PubMed]

Ban, G.

H. Failache, P. Valente, G. Ban, V. Lorent, and A. Lezama, "Inhibition of electromagnetically induced absorption due to excited-state decoherence in Rb vapor," Phys. Rev. A 67, 043810 (2003).
[CrossRef]

Barreiro, S.

A. Lezama, S. Barreiro, and A. M. Akulshin, "Electromagnetically induced absorption," Phys. Rev. A 59, 4732-4735 (1999).
[CrossRef]

A. M. Akulshin, S. Barreiro, and A. Lezama, "Electromagnetically induced absorption and transparency due to resonant two-field excitation of quasidegenerate levels in Rb vapor," Phys. Rev. A 57, 2996-3002 (1998).
[CrossRef]

Behroozi, C. H.

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, "Observation of coherent optical information storage in an atomic medium using halted light pulses," Nature (London) 409, 490-493 (2001).
[CrossRef]

L. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature (London) 397, 594-598 (1999).
[CrossRef]

Biancalana, V.

C. Andreeva, S. Cartaleva, Y. Dancheva, V. Biancalana, A. Burchianti, C. Marinelli, E. Marriotti, L. Moi, and K. Nasyrov, "Coherent spectroscopy of degenerate two-level systems in Cs," Phys. Rev. A 66, 012502 (2002).
[CrossRef]

Brandt, S.

S. Brandt, A. Nagel, R. Wynands, and D. Meschede, "Buffer-gas-induced linewidth reduction of coherent dark resonances to below 50 Hz," Phys. Rev. A 56, R1063-R1066 (1997).
[CrossRef]

Budker, D.

D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashuk, "Nonlinear magneto-optics and reduced group velocity of light in atomic vapor with slow ground state relaxation," Phys. Rev. Lett. 83, 1767-1770 (1999).
[CrossRef]

Burchianti, A.

C. Andreeva, S. Cartaleva, Y. Dancheva, V. Biancalana, A. Burchianti, C. Marinelli, E. Marriotti, L. Moi, and K. Nasyrov, "Coherent spectroscopy of degenerate two-level systems in Cs," Phys. Rev. A 66, 012502 (2002).
[CrossRef]

Burnett, K.

P. D. Featonby, G. S. Summy, C. L. Webb, R. M. Godun, M. K. Oberthaler, A. C. Wilson, C. J. Foot, and K. Burnett, "Separated-path Ramsey atom interferometer," Phys. Rev. Lett. 81, 495-499 (1998).
[CrossRef]

Cartaleva, S.

C. Andreeva, S. Cartaleva, Y. Dancheva, V. Biancalana, A. Burchianti, C. Marinelli, E. Marriotti, L. Moi, and K. Nasyrov, "Coherent spectroscopy of degenerate two-level systems in Cs," Phys. Rev. A 66, 012502 (2002).
[CrossRef]

F. Renzoni, S. Cartaleva, G. Alzetta, and E. Arimondo, "Enhanced absorption Hanle effect in the configuration of crossed laser beam and magnetic field," Phys. Rev. A 63, 065401 (2001).
[CrossRef]

Y. Dancheva, G. Alzetta, S. Cartaleva, M. Taslakov, and Ch. Andreeva, "Coherent effects on the Zeeman sublevels of hyperfine states in optical pumping of Rb by monomode diode laser," Opt. Commun. 178, 103-110 (2000).
[CrossRef]

Celikov, A.

A. Akulshin, A. Celikov, and V. Velichansky, "Subnatural absorption resonances on the D1 line of rubidium induced by coherent population trapping," Opt. Commun. 84, 139-143 (1991).
[CrossRef]

Chen, X.

X. Chen, V. L. Telegdi, and A. Weis, "Quantitative study of the nonlinear Macaluso-Corbino (resonant Faraday) effect in Cs," Opt. Commun. 74, 301-305 (1990).
[CrossRef]

Chu, S.

M. Weitz, B. C. Young, and S. Chu, "Atomic interferometer based on adiabatic population transfer," Phys. Rev. Lett. 73, 2563-2566 (1994).
[CrossRef] [PubMed]

Cohen-Tannoudji, C.

B. Lounis and C. Cohen-Tannoudji, "Coherent population trapping and Fano profiles," J. Phys. II 2, 579-592 (1992).

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, "Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping: theoretical analysis," J. Opt. Soc. Am. B 6, 2112-2124 (1989).
[CrossRef]

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, "Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping," Phys. Rev. Lett. 61, 826-829 (1988).
[CrossRef] [PubMed]

Dancheva, Y.

C. Andreeva, S. Cartaleva, Y. Dancheva, V. Biancalana, A. Burchianti, C. Marinelli, E. Marriotti, L. Moi, and K. Nasyrov, "Coherent spectroscopy of degenerate two-level systems in Cs," Phys. Rev. A 66, 012502 (2002).
[CrossRef]

Y. Dancheva, G. Alzetta, S. Cartaleva, M. Taslakov, and Ch. Andreeva, "Coherent effects on the Zeeman sublevels of hyperfine states in optical pumping of Rb by monomode diode laser," Opt. Commun. 178, 103-110 (2000).
[CrossRef]

Dehmelt, H.

Delaney, M.

J. Vanier, M. W. Levine, D. Jansses, and M. Delaney, "Contrast and linewidth of the coherent population trapping transmission hyperfine resonance line in 87Rb: effect of optical pumping," Phys. Rev. A 67, 065801 (2003).
[CrossRef]

J. Vanier, M. W. Levine, D. Jansses, and M. Delaney, "On the use of intensity optical pumping and coherent population trapping techniques in the implementation of atomic frequency standards," IEEE Trans. Instrum. Meas. 52, 822-831 (2003).
[CrossRef]

Dutton, Z.

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, "Observation of coherent optical information storage in an atomic medium using halted light pulses," Nature (London) 409, 490-493 (2001).
[CrossRef]

L. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature (London) 397, 594-598 (1999).
[CrossRef]

Eisaman, M. D.

C. H. van der Wal, M. D. Eisaman, A. Andre, R. L. Walsworth, D. F. Phillips, A. S. Zibrov, and M. D. Lukin, "Atomic memory for correlated photon states," Science 301, 196-200 (2003).
[CrossRef] [PubMed]

Erhard , M.

M. Erhard and H. Helm, "Buffer-gas effects on dark resonances: theory and experiment," Phys. Rev. A 63, 043813 (2001).
[CrossRef]

Ezekiel, S.

Failache, H.

H. Failache, P. Valente, G. Ban, V. Lorent, and A. Lezama, "Inhibition of electromagnetically induced absorption due to excited-state decoherence in Rb vapor," Phys. Rev. A 67, 043810 (2003).
[CrossRef]

Featonby, P. D.

P. D. Featonby, G. S. Summy, C. L. Webb, R. M. Godun, M. K. Oberthaler, A. C. Wilson, C. J. Foot, and K. Burnett, "Separated-path Ramsey atom interferometer," Phys. Rev. Lett. 81, 495-499 (1998).
[CrossRef]

Fleischhauer, A.

D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, "Storage of light in atomic vapor," Phys. Rev. Lett. 86, 783-786 (2001).
[CrossRef] [PubMed]

Fleischhauer, M.

M. D. Lukin, A. B. Matsko, M. Fleischhauer, and M. O. Scully, "Quantum noise and correlations in resonantly enhanced wave mixing based on atomic coherence," Phys. Rev. Lett. 82, 1847-1850 (1999).
[CrossRef]

Fleschauer , M.

M. Fleschauer and M. D. Lukin, "Quantum memory for photons: Dark-state polaritons," Phys. Rev. A 65, 022314 (2002).
[CrossRef]

M. Fleschauer and M. D. Lukin, "Dark-state polaritons in electromagnetically induced transparency," Phys. Rev. Lett. 84, 5094-5097 (2000).
[CrossRef]

Fleschauer, M.

M. O. Scully and M. Fleschauer, "High-sensitivity magnetometer based on index-enhanced media," Phys. Rev. Lett. 69, 1360-1363 (1992).
[CrossRef] [PubMed]

Foot, C. J.

P. D. Featonby, G. S. Summy, C. L. Webb, R. M. Godun, M. K. Oberthaler, A. C. Wilson, C. J. Foot, and K. Burnett, "Separated-path Ramsey atom interferometer," Phys. Rev. Lett. 81, 495-499 (1998).
[CrossRef]

Friedmann, H.

C. Goren, A. D. Wilson-Gordon, M. Rosenbluh, and H. Friedmann, "Electromagnetically induced absorption due to transfer of coherence and to transfer of population," Phys. Rev. A 67, 033807 (2003).
[CrossRef]

Fry, E. S.

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]

Godone, A.

J. Vanier, A. Godone, and F. Levi, "Coherent population trapping in cesium: dark lines and coherent microwave emission," Phys. Rev. A 58, 2345-2358 (1998).
[CrossRef]

Godun, R. M.

P. D. Featonby, G. S. Summy, C. L. Webb, R. M. Godun, M. K. Oberthaler, A. C. Wilson, C. J. Foot, and K. Burnett, "Separated-path Ramsey atom interferometer," Phys. Rev. Lett. 81, 495-499 (1998).
[CrossRef]

Goorskey, D.

H. Wang, D. Goorskey, and M. Xiao, "Enhanced Kerr nonlinearity via atomic coherence in a three-level atomic system," Phys. Rev. Lett. 87, 073601 (2001).
[CrossRef] [PubMed]

Goren, C.

C. Goren, A. D. Wilson-Gordon, M. Rosenbluh, and H. Friedmann, "Electromagnetically induced absorption due to transfer of coherence and to transfer of population," Phys. Rev. A 67, 033807 (2003).
[CrossRef]

Gozzini, A.

G. Alzetta, A. Gozzini, L. Moi, and G. Orriols, "An experimental method for the observation of R.F. transitions and laser beat resonances in oriented Na vapour," Nuovo Cimento Soc. Ital. Fis., B 36, 5-20 (1976).
[CrossRef]

Hall, J. L.

P. Marte, P. Zoller, and J. L. Hall, "Coherent atomic mirrors and beam splitters by adiabatic passage in multilevel systems," Phys. Rev. A 44, R4118-R4121 (1991).
[CrossRef]

Hansch, T. W.

S. I. Kanorsky, A. Weis, J. Wurster, and T. W. Hansch, "Quantitative investigation of the resonant nonlinear Faraday effect under conditions of optical hyperfine pumping," Phys. Rev. A 47, 1220-1226 (1993).
[CrossRef] [PubMed]

Harris , S. E.

S. E. Harris and L. V. Hau, "Nonlinear optics at low light levels," Phys. Rev. Lett. 82, 4611-4614 (1999).
[CrossRef]

Harris, S. E.

L. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature (London) 397, 594-598 (1999).
[CrossRef]

S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50, 36-42 (1997).
[CrossRef]

Hau, L.

L. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature (London) 397, 594-598 (1999).
[CrossRef]

Hau, L. V.

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, "Observation of coherent optical information storage in an atomic medium using halted light pulses," Nature (London) 409, 490-493 (2001).
[CrossRef]

S. E. Harris and L. V. Hau, "Nonlinear optics at low light levels," Phys. Rev. Lett. 82, 4611-4614 (1999).
[CrossRef]

Helm, H.

M. Erhard and H. Helm, "Buffer-gas effects on dark resonances: theory and experiment," Phys. Rev. A 63, 043813 (2001).
[CrossRef]

Hemmer, P. R.

Hollberg, L.

A. V. Taichenachev, V. I. Yudin, R. Wynands, M. Stahler, J. Kitching, and L. Hollberg, "Theory of dark resonances for alkali-metal vapors in a buffer-gas cell," Phys. Rev. A 67, 033810 (2003).
[CrossRef]

M. Stahler, R. Wynands, S. Knappe, J. Kitching, L. Hollberg, A. Taichenachev, and V. Yudin, "Coherent population trapping resonances in thermal 85Rb vapor: D1 versus D2 line excitation," Opt. Lett. 27, 1472-1474 (2002).
[CrossRef]

J. Kitching, S. Knappe, N. Vukicevic, L. Hollberg, R. Wynands, and W. Weidemann, "A microwave frequency reference based on VCSEL-driven dark line resonances in Cs vapor," IEEE Trans. Instrum. Meas. 49, 1313-1317 (2000).
[CrossRef]

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]

Imamoglu, A.

M. D. Lukin and A. Imamoglu, "Controlling photons using electromagnetically induced transparency," Nature (London) 413, 273-276 (2001).
[CrossRef]

Janik, G.

Jansses, D.

J. Vanier, M. W. Levine, D. Jansses, and M. Delaney, "Contrast and linewidth of the coherent population trapping transmission hyperfine resonance line in 87Rb: effect of optical pumping," Phys. Rev. A 67, 065801 (2003).
[CrossRef]

J. Vanier, M. W. Levine, D. Jansses, and M. Delaney, "On the use of intensity optical pumping and coherent population trapping techniques in the implementation of atomic frequency standards," IEEE Trans. Instrum. Meas. 52, 822-831 (2003).
[CrossRef]

Javan, A.

A. Javan, O. Kocharovskaya, H. Lee, and M. O. Scully, "Narrowing of electromagnetically induced transparency resonance in a Doppler-broadened medium," Phys. Rev. A 66, 013805 (2002).
[CrossRef]

Kaiser, R.

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, "Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping: theoretical analysis," J. Opt. Soc. Am. B 6, 2112-2124 (1989).
[CrossRef]

A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, "Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping," Phys. Rev. Lett. 61, 826-829 (1988).
[CrossRef] [PubMed]

Kanorsky, S. I.

S. I. Kanorsky, A. Weis, J. Wurster, and T. W. Hansch, "Quantitative investigation of the resonant nonlinear Faraday effect under conditions of optical hyperfine pumping," Phys. Rev. A 47, 1220-1226 (1993).
[CrossRef] [PubMed]

Kasantsev, A. P.

A. P. Kasantsev, V. S. Smirnov, A. M. Tumaikin, and I. A. Yagofarov, "Effect of atomic ground state self-polarization in the optical-pumping cycle on the increase of linear light absorption for J-->J+1 transitions," Opt. Spectrosc. 57, 189-191 (1984).

Kash, M. M.

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]

Kemp, W.

M. Stahler, S. Knappe, C. Affolderbach, W. Kemp, and R. Wynands, "Picotesla magnetometry with coherent dark states," Europhys. Lett. 54, 323-328 (2001).
[CrossRef]

Kim, J. B.

K. Kim, M. Kwon, H. D. Park, H. S. Moon, H. S. Rawat, K. An, and J. B. Kim, "Electromagnetically induced absorptionspectra depending on intensities and detunings of the coupling field in Cs vapour," J. Phys. B 34, 4801-4808 (2001).
[CrossRef]

M. Kwon, K. Kim, H. S. Moon, H. D. Park, and J. B. Kim, "Dependence of electromagnetically induced absorption on two combinations of orthogonal polarized beams," J. Phys. B 34, 2951-2961 (2001).
[CrossRef]

Kim, K.

K. Kim, M. Kwon, H. D. Park, H. S. Moon, H. S. Rawat, K. An, and J. B. Kim, "Electromagnetically induced absorptionspectra depending on intensities and detunings of the coupling field in Cs vapour," J. Phys. B 34, 4801-4808 (2001).
[CrossRef]

M. Kwon, K. Kim, H. S. Moon, H. D. Park, and J. B. Kim, "Dependence of electromagnetically induced absorption on two combinations of orthogonal polarized beams," J. Phys. B 34, 2951-2961 (2001).
[CrossRef]

Kimball, D. F.

D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashuk, "Nonlinear magneto-optics and reduced group velocity of light in atomic vapor with slow ground state relaxation," Phys. Rev. Lett. 83, 1767-1770 (1999).
[CrossRef]

Kitching, J.

A. V. Taichenachev, V. I. Yudin, R. Wynands, M. Stahler, J. Kitching, and L. Hollberg, "Theory of dark resonances for alkali-metal vapors in a buffer-gas cell," Phys. Rev. A 67, 033810 (2003).
[CrossRef]

M. Stahler, R. Wynands, S. Knappe, J. Kitching, L. Hollberg, A. Taichenachev, and V. Yudin, "Coherent population trapping resonances in thermal 85Rb vapor: D1 versus D2 line excitation," Opt. Lett. 27, 1472-1474 (2002).
[CrossRef]

J. Kitching, S. Knappe, N. Vukicevic, L. Hollberg, R. Wynands, and W. Weidemann, "A microwave frequency reference based on VCSEL-driven dark line resonances in Cs vapor," IEEE Trans. Instrum. Meas. 49, 1313-1317 (2000).
[CrossRef]

Knappe, S.

S. Knappe, M. Stahler, C. Affolderbach, A. V. Taichenachev, V. I. Yudin, and R. Wynands, "Simple parameterization of dark-resonance line shapes," Appl. Phys. B: Lasers Opt. 76, 57-63 (2003).
[CrossRef]

M. Stahler, R. Wynands, S. Knappe, J. Kitching, L. Hollberg, A. Taichenachev, and V. Yudin, "Coherent population trapping resonances in thermal 85Rb vapor: D1 versus D2 line excitation," Opt. Lett. 27, 1472-1474 (2002).
[CrossRef]

M. Stahler, S. Knappe, C. Affolderbach, W. Kemp, and R. Wynands, "Picotesla magnetometry with coherent dark states," Europhys. Lett. 54, 323-328 (2001).
[CrossRef]

J. Kitching, S. Knappe, N. Vukicevic, L. Hollberg, R. Wynands, and W. Weidemann, "A microwave frequency reference based on VCSEL-driven dark line resonances in Cs vapor," IEEE Trans. Instrum. Meas. 49, 1313-1317 (2000).
[CrossRef]

Kocharovskaya, O.

A. Javan, O. Kocharovskaya, H. Lee, and M. O. Scully, "Narrowing of electromagnetically induced transparency resonance in a Doppler-broadened medium," Phys. Rev. A 66, 013805 (2002).
[CrossRef]

A. S. Zibrov, A. B. Matsko, O. Kocharovskaya, Y. V. Rostovtsev, G. R. Welch, and M. O. Scully, "Transporting and time reversing light via atomic coherence," Phys. Rev. Lett. 88, 103601 (2002).
[CrossRef] [PubMed]

O. Kocharovskaya, "Amplification and lasing without inversion," Phys. Rep. 219, 175-190 (1992).
[CrossRef]

Krasheninnikov, M. V.

M. V. Krasheninnikov, V. S. Smirnov, M. B. Sultanov, A. M. Tumaikin, and V. I. Yudin, "Anomalous self-rotation of the polarization plane of a confined light beam in a gas located in a weak magnetic field," Zh. Eksp. Teor. Fiz. 94, 204-209 (1988).

Kwon, M.

K. Kim, M. Kwon, H. D. Park, H. S. Moon, H. S. Rawat, K. An, and J. B. Kim, "Electromagnetically induced absorptionspectra depending on intensities and detunings of the coupling field in Cs vapour," J. Phys. B 34, 4801-4808 (2001).
[CrossRef]

M. Kwon, K. Kim, H. S. Moon, H. D. Park, and J. B. Kim, "Dependence of electromagnetically induced absorption on two combinations of orthogonal polarized beams," J. Phys. B 34, 2951-2961 (2001).
[CrossRef]

Lee, H.

A. Javan, O. Kocharovskaya, H. Lee, and M. O. Scully, "Narrowing of electromagnetically induced transparency resonance in a Doppler-broadened medium," Phys. Rev. A 66, 013805 (2002).
[CrossRef]

Leiby, C. C.

Levi, F.

J. Vanier, A. Godone, and F. Levi, "Coherent population trapping in cesium: dark lines and coherent microwave emission," Phys. Rev. A 58, 2345-2358 (1998).
[CrossRef]

Levine, M. W.

J. Vanier, M. W. Levine, D. Jansses, and M. Delaney, "On the use of intensity optical pumping and coherent population trapping techniques in the implementation of atomic frequency standards," IEEE Trans. Instrum. Meas. 52, 822-831 (2003).
[CrossRef]

J. Vanier, M. W. Levine, D. Jansses, and M. Delaney, "Contrast and linewidth of the coherent population trapping transmission hyperfine resonance line in 87Rb: effect of optical pumping," Phys. Rev. A 67, 065801 (2003).
[CrossRef]

Lezama, A.

H. Failache, P. Valente, G. Ban, V. Lorent, and A. Lezama, "Inhibition of electromagnetically induced absorption due to excited-state decoherence in Rb vapor," Phys. Rev. A 67, 043810 (2003).
[CrossRef]

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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).
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C. Y. Ye and A. S. Zibrov, "Width of the electromagnetically induced transparency resonance in atomic vapor," Phys. Rev. A 65, 023806 (2002).
[CrossRef]

A. V. Taichenachev, V. I. Yudin, R. Wynands, M. Stahler, J. Kitching, and L. Hollberg, "Theory of dark resonances for alkali-metal vapors in a buffer-gas cell," Phys. Rev. A 67, 033810 (2003).
[CrossRef]

Phys. Rev. Lett. (12)

M. Weitz, B. C. Young, and S. Chu, "Atomic interferometer based on adiabatic population transfer," Phys. Rev. Lett. 73, 2563-2566 (1994).
[CrossRef] [PubMed]

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A. S. Zibrov, A. B. Matsko, O. Kocharovskaya, Y. V. Rostovtsev, G. R. Welch, and M. O. Scully, "Transporting and time reversing light via atomic coherence," Phys. Rev. Lett. 88, 103601 (2002).
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[CrossRef]

D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashuk, "Nonlinear magneto-optics and reduced group velocity of light in atomic vapor with slow ground state relaxation," Phys. Rev. Lett. 83, 1767-1770 (1999).
[CrossRef]

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Phys. Today (1)

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C. H. van der Wal, M. D. Eisaman, A. Andre, R. L. Walsworth, D. F. Phillips, A. S. Zibrov, and M. D. Lukin, "Atomic memory for correlated photon states," Science 301, 196-200 (2003).
[CrossRef] [PubMed]

Zh. Eksp. Teor. Fiz. (1)

M. V. Krasheninnikov, V. S. Smirnov, M. B. Sultanov, A. M. Tumaikin, and V. I. Yudin, "Anomalous self-rotation of the polarization plane of a confined light beam in a gas located in a weak magnetic field," Zh. Eksp. Teor. Fiz. 94, 204-209 (1988).

Other (5)

D. Budker and S. M. Rochester, "A relation between electromagnetically induced absorption resonances and nonlinear magneto-optics in Lambda-systems," http://arxiv.org/abs/ physics/0310066.

S. G. Rautian and A. M. Shalagin, Kinetic Problems of Nonlinear Spectroscopy (Elsevier, Amsterdam, 1991).

E. E. Mikhailov, I. Novikova, Yu. V. Rostovtsev, and G. R. Welch, "Buffer-gas induced absorption resonances in Rb vapor," http://arxiv.org/abs/quant-ph/0309171.

E. Arimondo, "Coherent population trapping in laser spectroscopy," in Progress in Optics, Vol. XXXV , E. Wolf, ed. (Elsevier North-Holland, Amsterdam, 1996), pp. 257-354.

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge U. Press, Cambridge, UK, 1997), Chap. 7.

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Figures (6)

Fig. 1
Fig. 1

(a) Mutual orientation of the polarization ellipse and the magnetic field; (b) scheme of light-induced transitions.

Fig. 2
Fig. 2

Line shape parameters versus Δ for the case of pure radiative relaxation γ1=0. (a) A (solid curve) and B (dashed curve) are in arbitrary units, and (b) w (solid curve) and Ω0 (dashed curve) are in γeg units. Other parameters are =π/8, Γ=0, and κ=0.1γeg.

Fig. 3
Fig. 3

Line shape parameters versus Δ for the case of total excited-state depolarization γ1=10/γeg. (a) A (solid curve) and B (dashed curve) are in arbitrary units, and (b) w (solid curve) and Ω0 (dashed curve) are in γeg units. Other parameters are =π/8, Γ=0, and κ=0.1γeg.

Fig. 4
Fig. 4

Absorption resonance line shapes in the EIA case γ1=0. (a) Total excited-state population in arbitrary units versus the ground-state Zeeman shift in γeg units at Δ=0; (b) the same but Δ=5γeg. Other parameters are =π/8, Γ=0, and κ=0.1γeg.

Fig. 5
Fig. 5

Line shape parameters versus Δ. (a) A (solid curve) and B (dashed curve) are in arbitrary units, and (b) w (solid curve) and Ω0 (dashed curve) are in γeg units. The case of the pure radiative relaxation γ1=0. Other parameters are Γ=0.005γeg, κ=0.1γeg, and =π/8.

Fig. 6
Fig. 6

Resonance line shapes in a gas. The case of the pure radiative relaxation γ1=0 and γeg=γr/2+Γ. The total excited-state population averaged over Maxwell velocity distribution πev versus the ground-state Zeeman shift Ωg in γr units at different ellipticity parameters (a) =0, (b) =π/10, and (c) =π/5. The Doppler width varies in each panel kv¯=0 (dotted curve), kv¯=γr (dashed curve), and kv¯=20γr (solid curve). Other parameters are δ=0, Γ=0.001γr, and κ=0.2γr. For all curves the background is subtracted, and all curves are normalized to the absorption in the center Ωg=0.

Equations (40)

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E(r, t)=Ee exp[-i(ωt-kr)]+c.c.,
e=ex cos()+iey sin()=e+1 cos(-π/4)+e-1 sin(-π/4),
HˆD-E=-dˆE=κVˆ+H.c.,
Vˆ=Tˆege,
Tˆqab=ma,mbCFbmb;1qFama|Fa, maFb, mb|,
HˆB=-μˆB=a=e,gΩaFˆb(a),
Fˆb(a)=Fa(Fa+1)Tˆaab
ρˆeg=ρ¯ˆeg exp[-i(ωt-kr)],ρˆge=ρ¯ˆge exp[i(ωt-kr)].
(γeg-iδv)ρ¯ˆeg=-iκ(Vˆρˆgg-ρˆeeVˆ)-i[ΩeFˆb(e)ρ¯ˆeg-ρ¯ˆegΩgFˆb(g)],
(γeg+iδv)ρ¯ˆge=-iκ(Vˆρˆee-ρˆggVˆ)-i[ΩgFˆb(g)ρ¯ˆge-ρ¯ˆgeΩeFˆb(e)],
(Γ+γr)ρˆee+γˆcoll{ρˆee}=-iκ(Vˆρ¯ˆge-ρ¯ˆegVˆ)-iΩe[Fˆb(e), ρˆee],
Γ(ρˆgg-ρˆgg(0))=Gˆ{ρˆee}-iκ(Vˆρ¯ˆeg-ρ¯ˆgeVˆ)-iΩg[Fˆb(g), ρˆgg],
Gˆ{ρˆee}=βγrq=0,±1TˆqegρˆeeTˆqeg
Tr{γˆcoll{ρˆee}}=0.
γˆcoll=0,γeg=γr/2+Γ.
γˆcoll{ρˆee}=γ1[ρˆee-ˆe Tr{ρˆee}/(2Fe+1)]
πe=Tr{ρˆee}.
S=|κ|2γeg2+δv21.
(Γ+γr)ρˆee+γ1[ρˆee-ˆe Tr{ρˆee}/(2Fe+1)]=2γegSVˆρˆggVˆ,
Γ(ρˆgg-ρˆgg(0))+iΩg[Fˆb(g), ρˆgg]
=-{(γeg+iδv)SVˆVˆρˆgg+H.c.}+γrq=0,±1TˆqegρˆeeTˆqeg,
πeπe(0)=i=02Ni(Δ, γ˜1, Γ˜, )Ωik=02Dk(Δ, γ˜1, Γ˜, )Ωk,
Ni=j=02Nij(γ˜1, Γ˜, )Δj,
Dk=l=02Dkl(γ˜1, Γ˜, )Δl.
N00=-(5+7γ˜1)[25+115γ˜1+172γ˜12+84γ˜13-c2(15+9γ˜1-76γ˜12-84γ˜13)-4c4γ˜1],
N02=(1+γ˜1)2[-25(5+16γ˜1+12γ˜12)+c2(175+320γ˜1-12γ˜12)-12c4(5-4γ˜1-24γ˜12)],
N11=-40s(1+γ˜1)2[15+48γ˜1+36γ˜12-c2(8-6γ˜1-36γ˜12)],
N20=-12(1+γ˜1)2[12(1+2γ˜1)(5+6γ˜1)-5c2(7-5γ˜1-30γ˜12)],
D00=-(5+7γ˜1)[(5+7γ˜1)(5+32γ˜1+36γ˜12)-4c2(2+5γ˜1-8γ˜12-14γ˜13)],
D02=(1+γ˜1)2[-25(5+32γ˜1+36γ˜12)+4c2(35+194γ˜1+166γ˜12)-32c4(1+γ˜1-6γ˜12)],
D11=-40s(1+γ˜1)2[3(5+32γ˜1+36γ˜12)-c2(4+2γ˜1-24γ˜12)],
D20=-48(1+γ˜1)2[3(5+32γ˜1+36γ˜12)-c2(4-25γ˜12)],
πe=Aw2(Ωg-Ω0)2+w2+Bw(Ωg-Ω0)(Ωg-Ω0)2+w2+C,
Aπe(0)=2(2N0D22+N2D12-N1D1D2-2N2D0D2)D2(4D0D2-D12),
Bπe(0)=2(N1D2-N2D1)D24D0D2-D12,
Cπe(0)=N2D2,
Ω0γegS=-D12D2,
wγegS=4D0D2-D122D2.
Aπe(0)=3(5-2γ˜1)(1+2γ˜1)4(187+565γ˜1+418γ˜12),
πev=(πv¯)-1-+πe exp{-(v/v¯)2}dv,

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