Abstract

The nonlinear spectroscopy of cold atoms in the diffuse laser cooling system is studied in this paper. We present the theoretical models of the recoil-induced resonances (RIR) and the electromagnetically-induced absorption (EIA) of cold atoms in diffuse laser light, and show their signals in an experiment of cooling 87Rb atomic vapor in an integrating sphere. The theoretical results are in good agreement with the experimental ones when the light intensity distribution in the integrating sphere is considered. The differences between nonlinear spectra of cold atoms in the diffuse laser light and in the optical molasses are also discussed.

©2009 Optical Society of America

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  1. S. Tremine, S. Guerandel, D. Holleville, A. Clairon, and N. Dimarcq, “Development of a compact cold atom clock,” 2004 IEEE International Ultrasonics, Ferroelectrics, and Frequency Control Joint 50th Anniversary Conference, 65–70 (2004).
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  6. H. D. Cheng, W. Z. Zhang, H. Y. Ma, L. Liu, and Y. Z. Wang, “Laser cooling of rubidium atoms from vapor backgroud in diffuse light,” Phys. Rev. A, to be published.
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    [Crossref] [PubMed]
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    [Crossref]
  10. J.-Y. Courtois and G. Grynberg, “Probe transmission in a one-dimensional optical molasses Theory for circularly-cross-polarized cooling beams,” Phys. Rev. A 48, 1378–1399 (1993).
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  26. J. Dimitrijevic, Z. Grujic, M. Mijailovic, D Arsenovic, B. Panic, and B. M. Jelenkovic, “Enhancement of electromagnetically induced absorption with elliptically polarized light - laser intensity dependent coherence effect,” Opt. Ex. 16, 1343–1353 (2008).
    [Crossref]
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2008 (1)

J. Dimitrijevic, Z. Grujic, M. Mijailovic, D Arsenovic, B. Panic, and B. M. Jelenkovic, “Enhancement of electromagnetically induced absorption with elliptically polarized light - laser intensity dependent coherence effect,” Opt. Ex. 16, 1343–1353 (2008).
[Crossref]

2007 (1)

J. Dimitrijevic, D. Arsenovic, and B. M. Jelenkovic, “Intensity dependence narrowing of electromagnetically induced absorption in a Doppler-broadened medium,” Phys. Rev. A 76, 013836 (2007).
[Crossref]

2004 (1)

C. Goren, A. D. Wilson-Gordon, M. Rosenbluh, and H. Friedmann, “Electromagnetically induced absorption due to transfer of population in degenerate two-level systems,” Phys. Rev. A 70, 043814 (2004).
[Crossref]

2003 (1)

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]

2002 (1)

C. Affolderbach and S. Knappe, “Electromagnetically induced transparency and absorption in a standing wave,” Phys. Rev. A 65., 043810 (2002).
[Crossref]

2001 (3)

K. Kim, M. Kwon, H. D. Park, H. S. Moon, H. S. Rawat, K. An, and J. B. Kim, “Dependence of electromagnetically induced absorption on two combinations of orthogonal polarized beams,” J. Phys. B 34, 2951–2961 (2001).
[Crossref]

A. Lipsich, S. Barreiro, P. Valente, and A. Lezama, “Inspection of a magneto-optical trap via electromagnetically induced absorption,” Opt. Commun. 190, 185–191 (2001).
[Crossref]

E. Guillot, P.-E. Pottie, and N. Dimarcq, “Three-dimensional cooling of cesium atoms,” Opt. Lett. 26, 1639–1641 (2001).
[Crossref]

1999 (1)

A. Lezama, S. Barreiro, and A. M. Akulshin, “Electromagnetically induced absorption,” Phys. Rev. A 59, 4732–4735 (1999).
[Crossref]

1998 (1)

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]

1997 (1)

T. van der Veldt, J. F. Roth, P. Grelu, and P. Grangier, “Nonlinear absorption and dispersion of cold 87Rb atoms,” Opt. Commun. 137, 420–426 (1997).
[Crossref]

1995 (1)

1994 (4)

H. Batelaan, S. Padua, D. H. Yang, C. Xie, R. Gupta, and H. Metcalf, “Slowing of 85Rb atoms with isotropic light,” Phys. Rev. A 49, 2780–2784 (1994).
[Crossref] [PubMed]

H. X. Chen, W. Q. Cai, L. Liu, W. Shu, F. S. Li, and Y. Z. Wang, “Laser Deceleration of an Atomic Beam by Red Shifted Diffuse Light,” Chin. J. Lasers 21, 280–283 (1994).

D. R. Meacher, D. Boiron, H. Metcalf, C. Salomon, and G. Grynberg, “Method for velocimetry of cold atoms,” Phys. Rev. A 50, R1992–R1994 (1994).
[Crossref] [PubMed]

J. -Y. Courtois, G. Grynberg, B. Lounis, and P. Verkerk, “Recoil-induced resonances in cesium: An atomic analog to the free-electron laser,” Phys. Rev. Lett. 72, 3017–3020 (1994).
[Crossref] [PubMed]

1993 (3)

J. Guo and P. R. Berman, “Recoil-induced resonances in pump-probe spectroscopy including effects of level degeneracy,” Phys. Rev. A 47, 4128–4142 (1993).
[Crossref] [PubMed]

J.-Y. Courtois and G. Grynberg, “Probe transmission in a one-dimensional optical molasses Theory for circularly-cross-polarized cooling beams,” Phys. Rev. A 48, 1378–1399 (1993).
[Crossref] [PubMed]

G. Grynberg, B. Lounis, P. Verkerk, J. Courtois, and C. Salomon, “Quantized motion of cold cesium atoms in two- and three-dimensional optical potentials,” Phys. Rev. Lett. 70, 2249–2252 (1993).
[Crossref] [PubMed]

1992 (3)

J. Guo, P. R. Berman, B. Dubetsky, and G. Grynberg, “Recoil-induced resonances in nonlinear spectroscopy,” Phys. Rev. A 46, 1426–1437 (1992).
[Crossref] [PubMed]

W. Ketterle, A. Martin, M. A. Joffe, and P. E. Pritchard, “Slowing and cooling of atoms in isotropic laser light,” Phys. Rev. Lett. 69, 2483–2486 (1992).
[Crossref] [PubMed]

Weihan Tan, Weiping Lu, and R. G. Harrison, “Approach to the theory of radiation-matter interaction for arbitrary field strength,” Phys. Rev. A 46, 7128–7138 (1992).
[Crossref] [PubMed]

1991 (2)

J. W. R. Tabosa, G. Chen, Z. Hu, R. B. Lee, H. J, and Kimble, “Nonlinear spectroscopy of cold atoms in a spontaneous-force optical trap,” Phys. Rev. Lett. 66, 3245–3247 (1991).
[Crossref] [PubMed]

D. Grison, B. Lounis, C. Salomon, J. Courtois, and G. Grynberg, “Raman spectroscopy of cesium atoms in a laser trap,” Europhys. Lett. 15, 149–154 (1991).
[Crossref]

1990 (1)

1975 (1)

Affolderbach, C.

C. Affolderbach and S. Knappe, “Electromagnetically induced transparency and absorption in a standing wave,” Phys. Rev. A 65., 043810 (2002).
[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]

An, K.

K. Kim, M. Kwon, H. D. Park, H. S. Moon, H. S. Rawat, K. An, and J. B. Kim, “Dependence of electromagnetically induced absorption on two combinations of orthogonal polarized beams,” J. Phys. B 34, 2951–2961 (2001).
[Crossref]

Anderson, R.

Arsenovic, D

J. Dimitrijevic, Z. Grujic, M. Mijailovic, D Arsenovic, B. Panic, and B. M. Jelenkovic, “Enhancement of electromagnetically induced absorption with elliptically polarized light - laser intensity dependent coherence effect,” Opt. Ex. 16, 1343–1353 (2008).
[Crossref]

Arsenovic, D.

J. Dimitrijevic, D. Arsenovic, and B. M. Jelenkovic, “Intensity dependence narrowing of electromagnetically induced absorption in a Doppler-broadened medium,” Phys. Rev. A 76, 013836 (2007).
[Crossref]

Barreiro, S.

A. Lipsich, S. Barreiro, P. Valente, and A. Lezama, “Inspection of a magneto-optical trap via electromagnetically induced absorption,” Opt. Commun. 190, 185–191 (2001).
[Crossref]

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]

Bartlett, C. L.

Batelaan, H.

H. Batelaan, S. Padua, D. H. Yang, C. Xie, R. Gupta, and H. Metcalf, “Slowing of 85Rb atoms with isotropic light,” Phys. Rev. A 49, 2780–2784 (1994).
[Crossref] [PubMed]

Berman, P. R.

J. Guo and P. R. Berman, “Recoil-induced resonances in pump-probe spectroscopy including effects of level degeneracy,” Phys. Rev. A 47, 4128–4142 (1993).
[Crossref] [PubMed]

J. Guo, P. R. Berman, B. Dubetsky, and G. Grynberg, “Recoil-induced resonances in nonlinear spectroscopy,” Phys. Rev. A 46, 1426–1437 (1992).
[Crossref] [PubMed]

Boiron, D.

D. R. Meacher, D. Boiron, H. Metcalf, C. Salomon, and G. Grynberg, “Method for velocimetry of cold atoms,” Phys. Rev. A 50, R1992–R1994 (1994).
[Crossref] [PubMed]

Cai, W. Q.

H. X. Chen, W. Q. Cai, L. Liu, W. Shu, F. S. Li, and Y. Z. Wang, “Laser Deceleration of an Atomic Beam by Red Shifted Diffuse Light,” Chin. J. Lasers 21, 280–283 (1994).

Carter, W. H.

Chen, G.

J. W. R. Tabosa, G. Chen, Z. Hu, R. B. Lee, H. J, and Kimble, “Nonlinear spectroscopy of cold atoms in a spontaneous-force optical trap,” Phys. Rev. Lett. 66, 3245–3247 (1991).
[Crossref] [PubMed]

Chen, H. X.

H. X. Chen, W. Q. Cai, L. Liu, W. Shu, F. S. Li, and Y. Z. Wang, “Laser Deceleration of an Atomic Beam by Red Shifted Diffuse Light,” Chin. J. Lasers 21, 280–283 (1994).

Cheng, H. D.

H. D. Cheng, W. Z. Zhang, H. Y. Ma, L. Liu, and Y. Z. Wang, “Laser cooling of rubidium atoms from vapor backgroud in diffuse light,” Phys. Rev. A, to be published.
[PubMed]

Chipman, R. A.

Clairon, A.

S. Tremine, S. Guerandel, D. Holleville, A. Clairon, and N. Dimarcq, “Development of a compact cold atom clock,” 2004 IEEE International Ultrasonics, Ferroelectrics, and Frequency Control Joint 50th Anniversary Conference, 65–70 (2004).

Courtois, J.

G. Grynberg, B. Lounis, P. Verkerk, J. Courtois, and C. Salomon, “Quantized motion of cold cesium atoms in two- and three-dimensional optical potentials,” Phys. Rev. Lett. 70, 2249–2252 (1993).
[Crossref] [PubMed]

D. Grison, B. Lounis, C. Salomon, J. Courtois, and G. Grynberg, “Raman spectroscopy of cesium atoms in a laser trap,” Europhys. Lett. 15, 149–154 (1991).
[Crossref]

Courtois, J. -Y.

J. -Y. Courtois, G. Grynberg, B. Lounis, and P. Verkerk, “Recoil-induced resonances in cesium: An atomic analog to the free-electron laser,” Phys. Rev. Lett. 72, 3017–3020 (1994).
[Crossref] [PubMed]

Courtois, J.-Y.

J.-Y. Courtois and G. Grynberg, “Probe transmission in a one-dimensional optical molasses Theory for circularly-cross-polarized cooling beams,” Phys. Rev. A 48, 1378–1399 (1993).
[Crossref] [PubMed]

Dimarcq, N.

E. Guillot, P.-E. Pottie, and N. Dimarcq, “Three-dimensional cooling of cesium atoms,” Opt. Lett. 26, 1639–1641 (2001).
[Crossref]

S. Tremine, S. Guerandel, D. Holleville, A. Clairon, and N. Dimarcq, “Development of a compact cold atom clock,” 2004 IEEE International Ultrasonics, Ferroelectrics, and Frequency Control Joint 50th Anniversary Conference, 65–70 (2004).

Dimitrijevic, J.

J. Dimitrijevic, Z. Grujic, M. Mijailovic, D Arsenovic, B. Panic, and B. M. Jelenkovic, “Enhancement of electromagnetically induced absorption with elliptically polarized light - laser intensity dependent coherence effect,” Opt. Ex. 16, 1343–1353 (2008).
[Crossref]

J. Dimitrijevic, D. Arsenovic, and B. M. Jelenkovic, “Intensity dependence narrowing of electromagnetically induced absorption in a Doppler-broadened medium,” Phys. Rev. A 76, 013836 (2007).
[Crossref]

Dubetsky, B.

J. Guo, P. R. Berman, B. Dubetsky, and G. Grynberg, “Recoil-induced resonances in nonlinear spectroscopy,” Phys. Rev. A 46, 1426–1437 (1992).
[Crossref] [PubMed]

Friedmann, H.

C. Goren, A. D. Wilson-Gordon, M. Rosenbluh, and H. Friedmann, “Electromagnetically induced absorption due to transfer of population in degenerate two-level systems,” Phys. Rev. A 70, 043814 (2004).
[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]

Goren, C.

C. Goren, A. D. Wilson-Gordon, M. Rosenbluh, and H. Friedmann, “Electromagnetically induced absorption due to transfer of population in degenerate two-level systems,” Phys. Rev. A 70, 043814 (2004).
[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]

Grangier, P.

T. van der Veldt, J. F. Roth, P. Grelu, and P. Grangier, “Nonlinear absorption and dispersion of cold 87Rb atoms,” Opt. Commun. 137, 420–426 (1997).
[Crossref]

Grelu, P.

T. van der Veldt, J. F. Roth, P. Grelu, and P. Grangier, “Nonlinear absorption and dispersion of cold 87Rb atoms,” Opt. Commun. 137, 420–426 (1997).
[Crossref]

Grison, D.

D. Grison, B. Lounis, C. Salomon, J. Courtois, and G. Grynberg, “Raman spectroscopy of cesium atoms in a laser trap,” Europhys. Lett. 15, 149–154 (1991).
[Crossref]

Grujic, Z.

J. Dimitrijevic, Z. Grujic, M. Mijailovic, D Arsenovic, B. Panic, and B. M. Jelenkovic, “Enhancement of electromagnetically induced absorption with elliptically polarized light - laser intensity dependent coherence effect,” Opt. Ex. 16, 1343–1353 (2008).
[Crossref]

Grynberg, G.

D. R. Meacher, D. Boiron, H. Metcalf, C. Salomon, and G. Grynberg, “Method for velocimetry of cold atoms,” Phys. Rev. A 50, R1992–R1994 (1994).
[Crossref] [PubMed]

J. -Y. Courtois, G. Grynberg, B. Lounis, and P. Verkerk, “Recoil-induced resonances in cesium: An atomic analog to the free-electron laser,” Phys. Rev. Lett. 72, 3017–3020 (1994).
[Crossref] [PubMed]

J.-Y. Courtois and G. Grynberg, “Probe transmission in a one-dimensional optical molasses Theory for circularly-cross-polarized cooling beams,” Phys. Rev. A 48, 1378–1399 (1993).
[Crossref] [PubMed]

G. Grynberg, B. Lounis, P. Verkerk, J. Courtois, and C. Salomon, “Quantized motion of cold cesium atoms in two- and three-dimensional optical potentials,” Phys. Rev. Lett. 70, 2249–2252 (1993).
[Crossref] [PubMed]

J. Guo, P. R. Berman, B. Dubetsky, and G. Grynberg, “Recoil-induced resonances in nonlinear spectroscopy,” Phys. Rev. A 46, 1426–1437 (1992).
[Crossref] [PubMed]

D. Grison, B. Lounis, C. Salomon, J. Courtois, and G. Grynberg, “Raman spectroscopy of cesium atoms in a laser trap,” Europhys. Lett. 15, 149–154 (1991).
[Crossref]

Guerandel, S.

S. Tremine, S. Guerandel, D. Holleville, A. Clairon, and N. Dimarcq, “Development of a compact cold atom clock,” 2004 IEEE International Ultrasonics, Ferroelectrics, and Frequency Control Joint 50th Anniversary Conference, 65–70 (2004).

Guillot, E.

Guo, J.

J. Guo and P. R. Berman, “Recoil-induced resonances in pump-probe spectroscopy including effects of level degeneracy,” Phys. Rev. A 47, 4128–4142 (1993).
[Crossref] [PubMed]

J. Guo, P. R. Berman, B. Dubetsky, and G. Grynberg, “Recoil-induced resonances in nonlinear spectroscopy,” Phys. Rev. A 46, 1426–1437 (1992).
[Crossref] [PubMed]

Gupta, R.

H. Batelaan, S. Padua, D. H. Yang, C. Xie, R. Gupta, and H. Metcalf, “Slowing of 85Rb atoms with isotropic light,” Phys. Rev. A 49, 2780–2784 (1994).
[Crossref] [PubMed]

H. J,

J. W. R. Tabosa, G. Chen, Z. Hu, R. B. Lee, H. J, and Kimble, “Nonlinear spectroscopy of cold atoms in a spontaneous-force optical trap,” Phys. Rev. Lett. 66, 3245–3247 (1991).
[Crossref] [PubMed]

Harrison, R. G.

Weihan Tan, Weiping Lu, and R. G. Harrison, “Approach to the theory of radiation-matter interaction for arbitrary field strength,” Phys. Rev. A 46, 7128–7138 (1992).
[Crossref] [PubMed]

Holleville, D.

S. Tremine, S. Guerandel, D. Holleville, A. Clairon, and N. Dimarcq, “Development of a compact cold atom clock,” 2004 IEEE International Ultrasonics, Ferroelectrics, and Frequency Control Joint 50th Anniversary Conference, 65–70 (2004).

Hu, Z.

J. W. R. Tabosa, G. Chen, Z. Hu, R. B. Lee, H. J, and Kimble, “Nonlinear spectroscopy of cold atoms in a spontaneous-force optical trap,” Phys. Rev. Lett. 66, 3245–3247 (1991).
[Crossref] [PubMed]

Jelenkovic, B. M.

J. Dimitrijevic, Z. Grujic, M. Mijailovic, D Arsenovic, B. Panic, and B. M. Jelenkovic, “Enhancement of electromagnetically induced absorption with elliptically polarized light - laser intensity dependent coherence effect,” Opt. Ex. 16, 1343–1353 (2008).
[Crossref]

J. Dimitrijevic, D. Arsenovic, and B. M. Jelenkovic, “Intensity dependence narrowing of electromagnetically induced absorption in a Doppler-broadened medium,” Phys. Rev. A 76, 013836 (2007).
[Crossref]

Joffe, M. A.

W. Ketterle, A. Martin, M. A. Joffe, and P. E. Pritchard, “Slowing and cooling of atoms in isotropic laser light,” Phys. Rev. Lett. 69, 2483–2486 (1992).
[Crossref] [PubMed]

Ketterle, W.

W. Ketterle, A. Martin, M. A. Joffe, and P. E. Pritchard, “Slowing and cooling of atoms in isotropic laser light,” Phys. Rev. Lett. 69, 2483–2486 (1992).
[Crossref] [PubMed]

Kim, J. B.

K. Kim, M. Kwon, H. D. Park, H. S. Moon, H. S. Rawat, K. An, 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, “Dependence of electromagnetically induced absorption on two combinations of orthogonal polarized beams,” J. Phys. B 34, 2951–2961 (2001).
[Crossref]

Kimble,

J. W. R. Tabosa, G. Chen, Z. Hu, R. B. Lee, H. J, and Kimble, “Nonlinear spectroscopy of cold atoms in a spontaneous-force optical trap,” Phys. Rev. Lett. 66, 3245–3247 (1991).
[Crossref] [PubMed]

Knappe, S.

C. Affolderbach and S. Knappe, “Electromagnetically induced transparency and absorption in a standing wave,” Phys. Rev. A 65., 043810 (2002).
[Crossref]

Kwon, M.

K. Kim, M. Kwon, H. D. Park, H. S. Moon, H. S. Rawat, K. An, 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, R. B.

J. W. R. Tabosa, G. Chen, Z. Hu, R. B. Lee, H. J, and Kimble, “Nonlinear spectroscopy of cold atoms in a spontaneous-force optical trap,” Phys. Rev. Lett. 66, 3245–3247 (1991).
[Crossref] [PubMed]

Lezama, A.

A. Lipsich, S. Barreiro, P. Valente, and A. Lezama, “Inspection of a magneto-optical trap via electromagnetically induced absorption,” Opt. Commun. 190, 185–191 (2001).
[Crossref]

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]

Li, F. S.

H. X. Chen, W. Q. Cai, L. Liu, W. Shu, F. S. Li, and Y. Z. Wang, “Laser Deceleration of an Atomic Beam by Red Shifted Diffuse Light,” Chin. J. Lasers 21, 280–283 (1994).

Lipsich, A.

A. Lipsich, S. Barreiro, P. Valente, and A. Lezama, “Inspection of a magneto-optical trap via electromagnetically induced absorption,” Opt. Commun. 190, 185–191 (2001).
[Crossref]

Liu, L.

H. X. Chen, W. Q. Cai, L. Liu, W. Shu, F. S. Li, and Y. Z. Wang, “Laser Deceleration of an Atomic Beam by Red Shifted Diffuse Light,” Chin. J. Lasers 21, 280–283 (1994).

H. D. Cheng, W. Z. Zhang, H. Y. Ma, L. Liu, and Y. Z. Wang, “Laser cooling of rubidium atoms from vapor backgroud in diffuse light,” Phys. Rev. A, to be published.
[PubMed]

Lounis, B.

J. -Y. Courtois, G. Grynberg, B. Lounis, and P. Verkerk, “Recoil-induced resonances in cesium: An atomic analog to the free-electron laser,” Phys. Rev. Lett. 72, 3017–3020 (1994).
[Crossref] [PubMed]

G. Grynberg, B. Lounis, P. Verkerk, J. Courtois, and C. Salomon, “Quantized motion of cold cesium atoms in two- and three-dimensional optical potentials,” Phys. Rev. Lett. 70, 2249–2252 (1993).
[Crossref] [PubMed]

D. Grison, B. Lounis, C. Salomon, J. Courtois, and G. Grynberg, “Raman spectroscopy of cesium atoms in a laser trap,” Europhys. Lett. 15, 149–154 (1991).
[Crossref]

Lu, Weiping

Weihan Tan, Weiping Lu, and R. G. Harrison, “Approach to the theory of radiation-matter interaction for arbitrary field strength,” Phys. Rev. A 46, 7128–7138 (1992).
[Crossref] [PubMed]

Ma, H. Y.

H. D. Cheng, W. Z. Zhang, H. Y. Ma, L. Liu, and Y. Z. Wang, “Laser cooling of rubidium atoms from vapor backgroud in diffuse light,” Phys. Rev. A, to be published.
[PubMed]

Martin, A.

W. Ketterle, A. Martin, M. A. Joffe, and P. E. Pritchard, “Slowing and cooling of atoms in isotropic laser light,” Phys. Rev. Lett. 69, 2483–2486 (1992).
[Crossref] [PubMed]

McClain, S. C.

Meacher, D. R.

D. R. Meacher, D. Boiron, H. Metcalf, C. Salomon, and G. Grynberg, “Method for velocimetry of cold atoms,” Phys. Rev. A 50, R1992–R1994 (1994).
[Crossref] [PubMed]

Metcalf, H.

D. R. Meacher, D. Boiron, H. Metcalf, C. Salomon, and G. Grynberg, “Method for velocimetry of cold atoms,” Phys. Rev. A 50, R1992–R1994 (1994).
[Crossref] [PubMed]

H. Batelaan, S. Padua, D. H. Yang, C. Xie, R. Gupta, and H. Metcalf, “Slowing of 85Rb atoms with isotropic light,” Phys. Rev. A 49, 2780–2784 (1994).
[Crossref] [PubMed]

Metcalf, H. J.

H. J. Metcalf and P. van der Straten, Laser Cooling and Trapping, Springer-Verlag, New York, (1999).
[Crossref]

Mijailovic, M.

J. Dimitrijevic, Z. Grujic, M. Mijailovic, D Arsenovic, B. Panic, and B. M. Jelenkovic, “Enhancement of electromagnetically induced absorption with elliptically polarized light - laser intensity dependent coherence effect,” Opt. Ex. 16, 1343–1353 (2008).
[Crossref]

Moon, H. S.

K. Kim, M. Kwon, H. D. Park, H. S. Moon, H. S. Rawat, K. An, and J. B. Kim, “Dependence of electromagnetically induced absorption on two combinations of orthogonal polarized beams,” J. Phys. B 34, 2951–2961 (2001).
[Crossref]

Padua, S.

H. Batelaan, S. Padua, D. H. Yang, C. Xie, R. Gupta, and H. Metcalf, “Slowing of 85Rb atoms with isotropic light,” Phys. Rev. A 49, 2780–2784 (1994).
[Crossref] [PubMed]

Panic, B.

J. Dimitrijevic, Z. Grujic, M. Mijailovic, D Arsenovic, B. Panic, and B. M. Jelenkovic, “Enhancement of electromagnetically induced absorption with elliptically polarized light - laser intensity dependent coherence effect,” Opt. Ex. 16, 1343–1353 (2008).
[Crossref]

Park, H. D.

K. Kim, M. Kwon, H. D. Park, H. S. Moon, H. S. Rawat, K. An, and J. B. Kim, “Dependence of electromagnetically induced absorption on two combinations of orthogonal polarized beams,” J. Phys. B 34, 2951–2961 (2001).
[Crossref]

Pezzaniti, J. L.

Pottie, P.-E.

Pritchard, P. E.

W. Ketterle, A. Martin, M. A. Joffe, and P. E. Pritchard, “Slowing and cooling of atoms in isotropic laser light,” Phys. Rev. Lett. 69, 2483–2486 (1992).
[Crossref] [PubMed]

Rawat, H. S.

K. Kim, M. Kwon, H. D. Park, H. S. Moon, H. S. Rawat, K. An, and J. B. Kim, “Dependence of electromagnetically induced absorption on two combinations of orthogonal polarized beams,” J. Phys. B 34, 2951–2961 (2001).
[Crossref]

Rosenbluh, M.

C. Goren, A. D. Wilson-Gordon, M. Rosenbluh, and H. Friedmann, “Electromagnetically induced absorption due to transfer of population in degenerate two-level systems,” Phys. Rev. A 70, 043814 (2004).
[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]

Roth, J. F.

T. van der Veldt, J. F. Roth, P. Grelu, and P. Grangier, “Nonlinear absorption and dispersion of cold 87Rb atoms,” Opt. Commun. 137, 420–426 (1997).
[Crossref]

Salomon, C.

D. R. Meacher, D. Boiron, H. Metcalf, C. Salomon, and G. Grynberg, “Method for velocimetry of cold atoms,” Phys. Rev. A 50, R1992–R1994 (1994).
[Crossref] [PubMed]

G. Grynberg, B. Lounis, P. Verkerk, J. Courtois, and C. Salomon, “Quantized motion of cold cesium atoms in two- and three-dimensional optical potentials,” Phys. Rev. Lett. 70, 2249–2252 (1993).
[Crossref] [PubMed]

D. Grison, B. Lounis, C. Salomon, J. Courtois, and G. Grynberg, “Raman spectroscopy of cesium atoms in a laser trap,” Europhys. Lett. 15, 149–154 (1991).
[Crossref]

Shu, W.

H. X. Chen, W. Q. Cai, L. Liu, W. Shu, F. S. Li, and Y. Z. Wang, “Laser Deceleration of an Atomic Beam by Red Shifted Diffuse Light,” Chin. J. Lasers 21, 280–283 (1994).

Straten, P. van der

H. J. Metcalf and P. van der Straten, Laser Cooling and Trapping, Springer-Verlag, New York, (1999).
[Crossref]

Tabosa, J. W. R.

J. W. R. Tabosa, G. Chen, Z. Hu, R. B. Lee, H. J, and Kimble, “Nonlinear spectroscopy of cold atoms in a spontaneous-force optical trap,” Phys. Rev. Lett. 66, 3245–3247 (1991).
[Crossref] [PubMed]

Tan, Weihan

Weihan Tan, Weiping Lu, and R. G. Harrison, “Approach to the theory of radiation-matter interaction for arbitrary field strength,” Phys. Rev. A 46, 7128–7138 (1992).
[Crossref] [PubMed]

Tremine, S.

S. Tremine, S. Guerandel, D. Holleville, A. Clairon, and N. Dimarcq, “Development of a compact cold atom clock,” 2004 IEEE International Ultrasonics, Ferroelectrics, and Frequency Control Joint 50th Anniversary Conference, 65–70 (2004).

Valente, P.

A. Lipsich, S. Barreiro, P. Valente, and A. Lezama, “Inspection of a magneto-optical trap via electromagnetically induced absorption,” Opt. Commun. 190, 185–191 (2001).
[Crossref]

Veldt, T. van der

T. van der Veldt, J. F. Roth, P. Grelu, and P. Grangier, “Nonlinear absorption and dispersion of cold 87Rb atoms,” Opt. Commun. 137, 420–426 (1997).
[Crossref]

Verkerk, P.

J. -Y. Courtois, G. Grynberg, B. Lounis, and P. Verkerk, “Recoil-induced resonances in cesium: An atomic analog to the free-electron laser,” Phys. Rev. Lett. 72, 3017–3020 (1994).
[Crossref] [PubMed]

G. Grynberg, B. Lounis, P. Verkerk, J. Courtois, and C. Salomon, “Quantized motion of cold cesium atoms in two- and three-dimensional optical potentials,” Phys. Rev. Lett. 70, 2249–2252 (1993).
[Crossref] [PubMed]

Wang, Y. Z.

H. X. Chen, W. Q. Cai, L. Liu, W. Shu, F. S. Li, and Y. Z. Wang, “Laser Deceleration of an Atomic Beam by Red Shifted Diffuse Light,” Chin. J. Lasers 21, 280–283 (1994).

H. D. Cheng, W. Z. Zhang, H. Y. Ma, L. Liu, and Y. Z. Wang, “Laser cooling of rubidium atoms from vapor backgroud in diffuse light,” Phys. Rev. A, to be published.
[PubMed]

Y. Z. Wang,“Atomic beam slowing by diffuse light in an integrating sphere,” in the Proceedings of the National Symposium on Frequency Standards, Chengdu, China, 1979.

Wilson-Gordon, A. D.

C. Goren, A. D. Wilson-Gordon, M. Rosenbluh, and H. Friedmann, “Electromagnetically induced absorption due to transfer of population in degenerate two-level systems,” Phys. Rev. A 70, 043814 (2004).
[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]

Wolf, E.

Xie, C.

H. Batelaan, S. Padua, D. H. Yang, C. Xie, R. Gupta, and H. Metcalf, “Slowing of 85Rb atoms with isotropic light,” Phys. Rev. A 49, 2780–2784 (1994).
[Crossref] [PubMed]

Yang, D. H.

H. Batelaan, S. Padua, D. H. Yang, C. Xie, R. Gupta, and H. Metcalf, “Slowing of 85Rb atoms with isotropic light,” Phys. Rev. A 49, 2780–2784 (1994).
[Crossref] [PubMed]

Zhang, W. Z.

H. D. Cheng, W. Z. Zhang, H. Y. Ma, L. Liu, and Y. Z. Wang, “Laser cooling of rubidium atoms from vapor backgroud in diffuse light,” Phys. Rev. A, to be published.
[PubMed]

Appl. Opt. (2)

Chin. J. Lasers (1)

H. X. Chen, W. Q. Cai, L. Liu, W. Shu, F. S. Li, and Y. Z. Wang, “Laser Deceleration of an Atomic Beam by Red Shifted Diffuse Light,” Chin. J. Lasers 21, 280–283 (1994).

Europhys. Lett. (1)

D. Grison, B. Lounis, C. Salomon, J. Courtois, and G. Grynberg, “Raman spectroscopy of cesium atoms in a laser trap,” Europhys. Lett. 15, 149–154 (1991).
[Crossref]

J. Opt. Soc. Am. (1)

J. Phys. B (1)

K. Kim, M. Kwon, H. D. Park, H. S. Moon, H. S. Rawat, K. An, and J. B. Kim, “Dependence of electromagnetically induced absorption on two combinations of orthogonal polarized beams,” J. Phys. B 34, 2951–2961 (2001).
[Crossref]

Opt. Commun. (2)

A. Lipsich, S. Barreiro, P. Valente, and A. Lezama, “Inspection of a magneto-optical trap via electromagnetically induced absorption,” Opt. Commun. 190, 185–191 (2001).
[Crossref]

T. van der Veldt, J. F. Roth, P. Grelu, and P. Grangier, “Nonlinear absorption and dispersion of cold 87Rb atoms,” Opt. Commun. 137, 420–426 (1997).
[Crossref]

Opt. Ex. (1)

J. Dimitrijevic, Z. Grujic, M. Mijailovic, D Arsenovic, B. Panic, and B. M. Jelenkovic, “Enhancement of electromagnetically induced absorption with elliptically polarized light - laser intensity dependent coherence effect,” Opt. Ex. 16, 1343–1353 (2008).
[Crossref]

Opt. Lett. (1)

Phys. Rev. A (12)

H. Batelaan, S. Padua, D. H. Yang, C. Xie, R. Gupta, and H. Metcalf, “Slowing of 85Rb atoms with isotropic light,” Phys. Rev. A 49, 2780–2784 (1994).
[Crossref] [PubMed]

J.-Y. Courtois and G. Grynberg, “Probe transmission in a one-dimensional optical molasses Theory for circularly-cross-polarized cooling beams,” Phys. Rev. A 48, 1378–1399 (1993).
[Crossref] [PubMed]

J. Guo, P. R. Berman, B. Dubetsky, and G. Grynberg, “Recoil-induced resonances in nonlinear spectroscopy,” Phys. Rev. A 46, 1426–1437 (1992).
[Crossref] [PubMed]

J. Guo and P. R. Berman, “Recoil-induced resonances in pump-probe spectroscopy including effects of level degeneracy,” Phys. Rev. A 47, 4128–4142 (1993).
[Crossref] [PubMed]

D. R. Meacher, D. Boiron, H. Metcalf, C. Salomon, and G. Grynberg, “Method for velocimetry of cold atoms,” Phys. Rev. A 50, R1992–R1994 (1994).
[Crossref] [PubMed]

Weihan Tan, Weiping Lu, and R. G. Harrison, “Approach to the theory of radiation-matter interaction for arbitrary field strength,” Phys. Rev. A 46, 7128–7138 (1992).
[Crossref] [PubMed]

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]

C. Affolderbach and S. Knappe, “Electromagnetically induced transparency and absorption in a standing wave,” Phys. Rev. A 65., 043810 (2002).
[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]

C. Goren, A. D. Wilson-Gordon, M. Rosenbluh, and H. Friedmann, “Electromagnetically induced absorption due to transfer of population in degenerate two-level systems,” Phys. Rev. A 70, 043814 (2004).
[Crossref]

J. Dimitrijevic, D. Arsenovic, and B. M. Jelenkovic, “Intensity dependence narrowing of electromagnetically induced absorption in a Doppler-broadened medium,” Phys. Rev. A 76, 013836 (2007).
[Crossref]

Phys. Rev. Lett. (4)

J. -Y. Courtois, G. Grynberg, B. Lounis, and P. Verkerk, “Recoil-induced resonances in cesium: An atomic analog to the free-electron laser,” Phys. Rev. Lett. 72, 3017–3020 (1994).
[Crossref] [PubMed]

G. Grynberg, B. Lounis, P. Verkerk, J. Courtois, and C. Salomon, “Quantized motion of cold cesium atoms in two- and three-dimensional optical potentials,” Phys. Rev. Lett. 70, 2249–2252 (1993).
[Crossref] [PubMed]

W. Ketterle, A. Martin, M. A. Joffe, and P. E. Pritchard, “Slowing and cooling of atoms in isotropic laser light,” Phys. Rev. Lett. 69, 2483–2486 (1992).
[Crossref] [PubMed]

J. W. R. Tabosa, G. Chen, Z. Hu, R. B. Lee, H. J, and Kimble, “Nonlinear spectroscopy of cold atoms in a spontaneous-force optical trap,” Phys. Rev. Lett. 66, 3245–3247 (1991).
[Crossref] [PubMed]

Other (4)

H. D. Cheng, W. Z. Zhang, H. Y. Ma, L. Liu, and Y. Z. Wang, “Laser cooling of rubidium atoms from vapor backgroud in diffuse light,” Phys. Rev. A, to be published.
[PubMed]

S. Tremine, S. Guerandel, D. Holleville, A. Clairon, and N. Dimarcq, “Development of a compact cold atom clock,” 2004 IEEE International Ultrasonics, Ferroelectrics, and Frequency Control Joint 50th Anniversary Conference, 65–70 (2004).

Y. Z. Wang,“Atomic beam slowing by diffuse light in an integrating sphere,” in the Proceedings of the National Symposium on Frequency Standards, Chengdu, China, 1979.

H. J. Metcalf and P. van der Straten, Laser Cooling and Trapping, Springer-Verlag, New York, (1999).
[Crossref]

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Figures (10)

Fig. 1.
Fig. 1. Pump transition and steady-state population of every ground states of 87Rb in diffuse laser lights
Fig. 2.
Fig. 2. Scheme of recoil-induced resonance. Probe laser (k 1, ω 1) travels along the direction ex , The another beam (k 0, ω 0) is the one of the isotropic laser lights which can cause recoil-induced resonance of the atom with the probe laser.
Fig. 3.
Fig. 3. Calculated probe absorption signal of recoil induced resonance in diffuse pump field. Γ = 6.056MHz (87Rb, Fg = 2 → Fe = 3), γ = 0.05Γ, Δ0 = - 3Γ, T = 200μK, S 0 =2 and S 1 = 0.01.
Fig. 4.
Fig. 4. Clebsch-Gordan coefficients and light shifts of every sub-levels for Fg = 2 - Fe = 3 transition.
Fig. 5.
Fig. 5. Calculation results of probe absorption. Γ = 6.056MHz (87Rb, Fg = 2 → Fe = 3), γ = Γ g = 0.05Γ, Δ0 = -3Γ, S 1 = 0.1, S 0 = 1,5,10.
Fig. 6.
Fig. 6. Experimental setup of the diffuse cooling 87Rb atomic vapor in an integrating sphere.
Fig. 7.
Fig. 7. Experimental signal varying with the detuning of probe laser light at three different diffuse light detunings: (a) Δ0 = -2Γ, (b) Δ0 = -3Γ, (c) Δ0 = -2Γ. Power of injected cooling laser beams are 40 mW/cm2.
Fig. 8.
Fig. 8. Experimental signal varying with power of injected cooling laser: (a) 40 mW/cm2, (b) 32 mW/cm2, (c) 24 mW/cm2, (d) 16 mW/cm2. Δ0 = -3Γ.
Fig. 9.
Fig. 9. (a) The two dot line are the calculated signals of recoil-induced resonances when S 0 = 2.5, S 1 = 0.03 and EIA when S 0 = 3.80, S 1 = 0.03. Their sum is the solid line. (b) Experimentally observed signal under Δ0 = -3Γ when the total power of injected cooling laser beams is 40 mW/cm2.
Fig. 10.
Fig. 10. light-field distribution in the integrating sphere, the injected laser are reflected by the inner surface of the integrating sphere to create diffuse laser light for laser cooling. Before the first-time reflection, the injected laser are two expanded beams due to the fibers have a the numerical aperture. Through the light path of the probe beam, the two expanded beams and the diffuse laser light are all pump light in region (a), while only diffuse laser light are the pump light in region (b).

Equations (52)

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

Δ kv cos θ = 0 ,
H I ( ω 0 , 1 ) = h ̄ Ω 0,1 | e g cos ( k 0 , 1 · X ω 0,1 t ) + h ̄ Ω 0,1 * g e cos ( k 0 , 1 · X ω 0,1 t ) ,
ρ = a , a ρ aa | a , p a , p | .
d dt ρ ˜ ee ( p , p ) = ( Γ + γ ) ρ ˜ ee p p
+ i a = 0 1 Ω a * exp [ i ( Δ a + ω r k a · p m ) t ] ρ eg ( p , p h ̄ k a )
i a = 0 1 Ω a * exp [ i ( Δ a ω r + k a · p m ) t ] ρ eg ( p h ̄ k a , p ) ,
d dt ρ ˜ gg ( p , p ) = Γ N ( q ) dq ρ ˜ ee ( p + h ̄ q , p + h ̄ q ) exp ( i p p m · q t )
γ ρ ˜ gg p p + γW p p
+ i a = 0 1 Ω a * exp [ i ( Δ a + ω r k a · p m ) t ] ρ gg ( p , p h ̄ k a )
i a = 0 1 Ω a * exp [ i ( Δ a ω r k a · p m ) t ] ρ ee ( p h ̄ k a , p ) ,
d dt ρ ˜ gg ( p , p ) = Γ 2 ρ ˜ ge p p
+ i a = 0 1 Ω a * exp [ i ( Δ a + ω r k a · p m ) t ] ρ gg ( p , p h ̄ k a )
i a = 0 1 Ω a * exp [ i ( Δ a ω r k a · p m ) t ] ρ ee ( p h ̄ k a , p ) ,
d dt ρ ˜ eg ( p , p ) = [ d dt ρ ˜ ge p , p ] * .
ρ ˜ aa p p = ρ aa ( p , p ) e [ ( p 2 p 2 ) t / 2 m h ̄ ] e i ω aa t ,
Δ 0,1 = ω 0,1 ω eg ,
ω r = h ̄ k 2 2 m .
N ( q ) = 3 8 π sin 2 θ ,
ρ ˜ ge = ρ ge x t exp ( i k 1 · x i ω 1 t )
= 1 ( 2 π h ̄ ) 3 ∫∫ d p d p [ i ( p p ) · x h ̄ i ( p 2 p 2 ) 2 m h ̄ i + iωt ] ρ ˜ ge p p exp ( i k 1 · x i ω 1 t ) .
Ω 2 γ Δ 1 ω r k 1 v < 1
Δ Γ / 2 ω 1 ω 0
kv γ
h ̄ Δ k b T
θ 0 ,
ρ ˜ ge = 2 i Ω 1 * N 0 Γ + 2 i Δ 1 4 Ω 2 2 Γ 2 + 4 i Δ 0 2 [ 8 ω r Δ 0 ( k 1 2 + k 0 2 + 2 k 1 k 0 cos θ ) v 2 p 2 W ( y ) 2 Γ 4 i Γ ω r ( Γ + 2 i Δ 1 ) γ 2 ] ,
y = 2 p ( ω 1 ω 0 ) k 1 2 + k 0 2 + 2 k 1 k 0 cos θ · v .
Im ( ρ ˜ ge ) = 2 π N 0 Ω 1 * Ω 0 2 ω r m 2 Δ 1 2 k 2 ( 1 + cos θ ) W ( p ω 1 ω 0 2 kv 1 + cos θ ) 2 N 0 Γ 2 Ω 1 * Ω 0 2 ω r Δ 1 5 γ .
4 πm ( ω 1 ω 2 ) = 2 k 1 p x + 2 k 0 ( p x cos θ p y sin θ )
+ h ̄ ( k 1 2 + k 0 2 + 2 k 1 k 0 cos θ ) .
sin θ c 1 + cos θ c = h ̄ k 1 2 2 m k b T
H e i g j I = H e i g j I ( ω 0 ) e i ω 0 t + H e i g j I ( ω 1 ) e i ω 1 t ,
H e i g j I ( ω 0,1 ) = μ e i g j E 0,1 = h ̄ ( 1 ) F e m e F e 1 F g m e q m g Ω 0,1 .
d dt ρ e i e j ( ω 0 ) = ( Γ + i ω e i e j ) ρ e i e j ( ω 0 )
+ i h ̄ k = 2 2 [ ρ e i g k ( ω 0 ) H g k e i I ( ω 0 ) H e i g k I ( ω 0 ) ρ g k e i ( ω 0 ) ] ,
d dt ρ e i g j ( ω 0 ) = [ Γ + Γ g i 2 + i ( ω e i g j ω 0 ) ] ρ e i e j ( ω 0 )
+ i h ̄ k = 3 3 [ ρ e i e k ( ω 0 ) H e k g j I ( ω 0 ) i k = 2 2 H e i g k I ( ω 0 ) ρ g k e i ( ω 0 ) ] ,
d dt ρ g i g j ( ω 0 ) = i ω g i g j ρ g i g j ( ω 0 ) + 7 Γ ρ g i g j s ( ω 0 )
+ i h ̄ k = 3 2 [ H e i g k I ( ω 0 ) ρ g k e i ( ω 0 ) ρ e i g k ( ω 0 ) H g k e i I ( ω 0 ) ] ,
ρ e i e j ( ω 0 ) = ρ e i e j exp ( i ω 0 t ) ,
ρ g i g j s ( ω 0,1 ) = q = 1 1 k , l = 3 3 ( 1 ) m k m l
× F g 1 F e m g i q m e ρ e k e l ( ω 0,1 ) F e 1 F g m e q m g i ,
ρ e i g j = ρ e i g j ( ω 0 ) exp [ i ω 0 t + ] + ρ e i g j ( ω 1 ) exp [ i ω 1 t ] .
d dt ρ e i e j ( ω 1 ω 0 ) = [ Γ + γ δ e i e j i ( ω 1 ω 0 ω e i e j ) ] ρ e i e j ( ω 1 ω 0 )
+ i h ̄ k = 2 2 [ ρ e i g k ( ω 1 ) H g k e i I ( ω 0 ) H e i g k I ( ω 1 ) ρ g k e i ( ω 0 ) ] ,
d dt ρ e i g j ( ω 1 ) = [ Γ + Γ g j 2 + i ( ω e i g j ω 1 ) ] ρ e i e j ( ω 1 )
+ i h ̄ k = 3 3 [ ρ e i e k ( ω 0 ) H e k g j I ( ω 1 ) i k = 2 2 H e i g k I ( ω 1 ) ρ g k e i ( ω 0 ) ]
+ i h ̄ k = 3 3 [ ρ e i g k ( ω 1 ω 0 ) H e k g j I ( ω 0 ) i k = 2 2 H e i g k I ( ω 0 ) ρ g k e i ( ω 1 ω 0 ) ] ,
d dt ρ g i g j ( ω 1 ω 0 ) = [ Γ g i + Γ g j 2 i ( ω 1 ω 0 ω g i g j ) ] ρ g i g j ( ω 1 ω 0 )
+ 7 Γ ρ g i g j s ( ω 1 ω 0 ) + k = 2 2 Γ g i g k δ g i g j ρ g k g k ( ω 1 ω 0 )
+ i h ̄ k = 3 3 [ H e i g k I ( ω 1 ) ρ g k e i ( ω 0 ) ρ e i g k ( ω 1 ) H g k e i I ( ω 0 ) ] .
Im [ i , j H e i g j I ( ω 1 ) ρ e i g j ( ω 1 ) ] ,

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