Abstract

We show experimentally that the presence of a buffer gas in a rubidium vapor cell modifies significantly the nonlinear magneto-optical rotation of polarization of near-resonant light propagating through the cell. We observe not only the well-known narrowing of the nonlinear magneto-optical resonances, but also changes in their shape and visibility. We explain these effects in terms of coherence-preserving, velocity-changing collisions between rubidium and buffer gas atoms.

© 2005 Optical Society of America

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  1. D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153-1201 (2002).
    [CrossRef]
  2. E. Arimondo, "Coherent population trapping in laser spectroscopy," in Progress in Optics, Vol. XXXV , E. Wolf, ed. (Elsevier, Amsterdam, 1996).
  3. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge U. Press, Cambridge, UK, 1997).
  4. S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50, 36-42 (1997).
    [CrossRef]
  5. J. P. Marangos, "Topical review: electromagnetically induced transparency," J. Mod. Opt. 45, 471-503 (1998).
    [CrossRef]
  6. C. Cohen-Tannoudji, J. Brossel, and A. Kastler, "La conservation de la phase lors de la collision d'un atome de sodium orienté et d'un atome d'hélium," C. R. Acad. Sci. 245, 1027-1029 (1957).
  7. F. Hartmann, M. Rambosson, J. Brossel, and A. Kastler, "Action des gaz etrangers sur le taux dorientation de la vapeur de sodium, obtenu par pompage optique," C. R. Acad. Sci. 246, 1522-1525 (1958).
  8. W. Happer, "Optical pumping," Rev. Mod. Phys. 44, 169-249 (1972).
    [CrossRef]
  9. C. Ottinger, R. Scheps, G. W. York, and A. Gallagher, "Broadening of Rb resonance lines by noble gases," Phys. Rev. A 11, 1815-1828 (1975).
    [CrossRef]
  10. J. Vanier and C. Audoin, The Quantum Physics of Atomic Frequency Standards , Vol. 1 (Hilger, Bristol, UK, 1989).
  11. M. Graf, E. Arimondo, E. S. Fry, D. E. Nikonov, G. G. Padmabandu, M. O. Scully, and S.-Y. Zhu, "Doppler broadening and collisional relaxation effects in a lasing-without-inversion experiment," Phys. Rev. A 51, 4030-4037 (1995).
    [CrossRef] [PubMed]
  12. E. Arimondo, "Relaxation processes in coherent-population trapping," Phys. Rev. A 54, 2216-2223 (1996).
    [CrossRef] [PubMed]
  13. 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]
  14. R. Wynands and A. Nagel, "Precision spectroscopy with coherent dark states," Appl. Phys. B 68, 1-25 (1998).
    [CrossRef]
  15. M. Erhard, S. Nußmann, and H. Helm, "Power broadening and Doppler effects of coherent dark resonances in Rb," Phys. Rev. A 62, 061802 (R) (2000).
    [CrossRef]
  16. M. Erhard and H. Helm, "Buffer-gas effects on dark resonances: Theory and experiment," Phys. Rev. A 63, 043813 (2001).
    [CrossRef]
  17. J. H. Xu and G. Alzetta, "High buffer-gas pressure perturbation of coherent population trapping in sodium vapors," Phys. Lett. A 248, 80-85 (1998).
    [CrossRef]
  18. S. Gozzini, P. Sartini, C. Gabbanini, A. Lucchesini, C. Marinelli, L. Moi, J. H. Xu, and G. Alzetta, "Experimental study of velocity-changing collisions on coherent population trapping in sodium," Eur. Phys. J. D 6, 127-131 (1999).
    [CrossRef]
  19. J. Vanier, J.-F. Simard, and J.-S. Boulanger, "Relaxation and frequency shifts in the ground state of 85Rb," Phys. Rev. A 9, 1031-1040 (1974).
    [CrossRef]
  20. A. Corney, Atomic and Laser Spectroscopy (Clarendon, Oxford, UK, 1977).
  21. W. Gawlik and J. Zachorowski, "Forward scattering of a strong nonmonochromatic laser light," J. Phys. B 20, 5939-5952 (1987).
    [CrossRef]
  22. W. Gawlik, "Hyperfine interaction versus strong laser field - optical backgoudsmit (Paschen-Back) effect," Am. J. Phys. 59, 706-710 (1991).
    [CrossRef]
  23. G. Wa¸sik, W. Gawlik, J. Zachorowski, and Z. Kowal, "Competition of dark states: Optical resonances with anomalous magnetic field dependence," Phys. Rev. A 64, 051802 (2001).
    [CrossRef]
  24. A. S. Zibrov, I. Novikova, and A. B. Matsko, "Observation of Ramsey fringes in an atomic cell with buffer gas," Opt. Lett. 26, 1311-1313 (2001).
    [CrossRef]
  25. A. S. Zibrov and A. B. Matsko, "Optical Ramsey fringes induced by Zeeman coherence," Phys. Rev. A 65, 013814 (2002).
    [CrossRef]
  26. M. Fleischhauer, A. B. Matsko, and M. O. Scully, "Quantum limit of optical magnetometry in the presence of ac Stark shifts," Phys. Rev. A 62, 013808 (2000).
    [CrossRef]
  27. 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]
  28. I. Novikova, A. B. Matsko, and G. R. Welch, "Detection of non-resonant impurity gases in alkali vapor cells," Appl. Phys. Lett. 81, 193-195 (2002).
    [CrossRef]
  29. D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and R. Zolotorev, "Sensitive magnetometry based on nonlinear magneto-optical rotation," Phys. Rev. A 62, 043403 (2000).
    [CrossRef]
  30. I. Novikova, A. B. Matsko, and G. R. Welch, "Large polarization rotation via atomic coherence," Opt. Lett. 26, 1016-1018 (2001).
    [CrossRef]
  31. E. E. Mikhailov, I. Novikova, Y. V. Rostovtsev, and G. R. Welch, "Buffer-gas-induced absorption resonances in Rb vapor," Phys. Rev. A 70, 033806 (2004).
    [CrossRef]
  32. A. V. Taichenachev, V. I. Yudin, M. Stahler, R. Wynands, J. Kitching, and L. Hollberg, "Nonlinear-resonance line shapes: Dependence on the transverse intensity distribution of a light beam," Phys. Rev. A 69, 024501 (2004).
    [CrossRef]
  33. 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]
  34. S. G. Rautian, G. I. Smirnov, and A. M. Shalagin, "Polarization phenomena and nonlinear interference effects with allowance for collisions," Zh. Eksp. Teor. Fiz. 62, 2097-2110 (1972).
  35. P. R. Berman, T. W. Mossberg, and S. R. Hartmann, "Collision kernels and laser spectroscopy," Phys. Rev. A 25, 2550-2571 (1982).
    [CrossRef]
  36. T. Privalov and A. Shalagin, "Exact solution of the one- and three-dimensional quantum kinetic equations with velocity-dependent collision rates: Comparative analysis," Phys. Rev. A 59, 4331-4339 (1999).
    [CrossRef]

2004 (2)

E. E. Mikhailov, I. Novikova, Y. V. Rostovtsev, and G. R. Welch, "Buffer-gas-induced absorption resonances in Rb vapor," Phys. Rev. A 70, 033806 (2004).
[CrossRef]

A. V. Taichenachev, V. I. Yudin, M. Stahler, R. Wynands, J. Kitching, and L. Hollberg, "Nonlinear-resonance line shapes: Dependence on the transverse intensity distribution of a light beam," Phys. Rev. A 69, 024501 (2004).
[CrossRef]

2003 (1)

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]

2002 (4)

I. Novikova, A. B. Matsko, and G. R. Welch, "Detection of non-resonant impurity gases in alkali vapor cells," Appl. Phys. Lett. 81, 193-195 (2002).
[CrossRef]

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 and A. B. Matsko, "Optical Ramsey fringes induced by Zeeman coherence," Phys. Rev. A 65, 013814 (2002).
[CrossRef]

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153-1201 (2002).
[CrossRef]

2001 (4)

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

G. Wa¸sik, W. Gawlik, J. Zachorowski, and Z. Kowal, "Competition of dark states: Optical resonances with anomalous magnetic field dependence," Phys. Rev. A 64, 051802 (2001).
[CrossRef]

I. Novikova, A. B. Matsko, and G. R. Welch, "Large polarization rotation via atomic coherence," Opt. Lett. 26, 1016-1018 (2001).
[CrossRef]

A. S. Zibrov, I. Novikova, and A. B. Matsko, "Observation of Ramsey fringes in an atomic cell with buffer gas," Opt. Lett. 26, 1311-1313 (2001).
[CrossRef]

2000 (3)

D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and R. Zolotorev, "Sensitive magnetometry based on nonlinear magneto-optical rotation," Phys. Rev. A 62, 043403 (2000).
[CrossRef]

M. Fleischhauer, A. B. Matsko, and M. O. Scully, "Quantum limit of optical magnetometry in the presence of ac Stark shifts," Phys. Rev. A 62, 013808 (2000).
[CrossRef]

M. Erhard, S. Nußmann, and H. Helm, "Power broadening and Doppler effects of coherent dark resonances in Rb," Phys. Rev. A 62, 061802 (R) (2000).
[CrossRef]

1999 (2)

S. Gozzini, P. Sartini, C. Gabbanini, A. Lucchesini, C. Marinelli, L. Moi, J. H. Xu, and G. Alzetta, "Experimental study of velocity-changing collisions on coherent population trapping in sodium," Eur. Phys. J. D 6, 127-131 (1999).
[CrossRef]

T. Privalov and A. Shalagin, "Exact solution of the one- and three-dimensional quantum kinetic equations with velocity-dependent collision rates: Comparative analysis," Phys. Rev. A 59, 4331-4339 (1999).
[CrossRef]

1998 (3)

J. H. Xu and G. Alzetta, "High buffer-gas pressure perturbation of coherent population trapping in sodium vapors," Phys. Lett. A 248, 80-85 (1998).
[CrossRef]

R. Wynands and A. Nagel, "Precision spectroscopy with coherent dark states," Appl. Phys. B 68, 1-25 (1998).
[CrossRef]

J. P. Marangos, "Topical review: electromagnetically induced transparency," J. Mod. Opt. 45, 471-503 (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]

1996 (1)

E. Arimondo, "Relaxation processes in coherent-population trapping," Phys. Rev. A 54, 2216-2223 (1996).
[CrossRef] [PubMed]

1995 (1)

M. Graf, E. Arimondo, E. S. Fry, D. E. Nikonov, G. G. Padmabandu, M. O. Scully, and S.-Y. Zhu, "Doppler broadening and collisional relaxation effects in a lasing-without-inversion experiment," Phys. Rev. A 51, 4030-4037 (1995).
[CrossRef] [PubMed]

1991 (1)

W. Gawlik, "Hyperfine interaction versus strong laser field - optical backgoudsmit (Paschen-Back) effect," Am. J. Phys. 59, 706-710 (1991).
[CrossRef]

1987 (1)

W. Gawlik and J. Zachorowski, "Forward scattering of a strong nonmonochromatic laser light," J. Phys. B 20, 5939-5952 (1987).
[CrossRef]

1982 (1)

P. R. Berman, T. W. Mossberg, and S. R. Hartmann, "Collision kernels and laser spectroscopy," Phys. Rev. A 25, 2550-2571 (1982).
[CrossRef]

1975 (1)

C. Ottinger, R. Scheps, G. W. York, and A. Gallagher, "Broadening of Rb resonance lines by noble gases," Phys. Rev. A 11, 1815-1828 (1975).
[CrossRef]

1974 (1)

J. Vanier, J.-F. Simard, and J.-S. Boulanger, "Relaxation and frequency shifts in the ground state of 85Rb," Phys. Rev. A 9, 1031-1040 (1974).
[CrossRef]

1972 (2)

W. Happer, "Optical pumping," Rev. Mod. Phys. 44, 169-249 (1972).
[CrossRef]

S. G. Rautian, G. I. Smirnov, and A. M. Shalagin, "Polarization phenomena and nonlinear interference effects with allowance for collisions," Zh. Eksp. Teor. Fiz. 62, 2097-2110 (1972).

1958 (1)

F. Hartmann, M. Rambosson, J. Brossel, and A. Kastler, "Action des gaz etrangers sur le taux dorientation de la vapeur de sodium, obtenu par pompage optique," C. R. Acad. Sci. 246, 1522-1525 (1958).

1957 (1)

C. Cohen-Tannoudji, J. Brossel, and A. Kastler, "La conservation de la phase lors de la collision d'un atome de sodium orienté et d'un atome d'hélium," C. R. Acad. Sci. 245, 1027-1029 (1957).

Alzetta, G.

S. Gozzini, P. Sartini, C. Gabbanini, A. Lucchesini, C. Marinelli, L. Moi, J. H. Xu, and G. Alzetta, "Experimental study of velocity-changing collisions on coherent population trapping in sodium," Eur. Phys. J. D 6, 127-131 (1999).
[CrossRef]

J. H. Xu and G. Alzetta, "High buffer-gas pressure perturbation of coherent population trapping in sodium vapors," Phys. Lett. A 248, 80-85 (1998).
[CrossRef]

Arimondo, E.

E. Arimondo, "Relaxation processes in coherent-population trapping," Phys. Rev. A 54, 2216-2223 (1996).
[CrossRef] [PubMed]

M. Graf, E. Arimondo, E. S. Fry, D. E. Nikonov, G. G. Padmabandu, M. O. Scully, and S.-Y. Zhu, "Doppler broadening and collisional relaxation effects in a lasing-without-inversion experiment," Phys. Rev. A 51, 4030-4037 (1995).
[CrossRef] [PubMed]

Berman, P. R.

P. R. Berman, T. W. Mossberg, and S. R. Hartmann, "Collision kernels and laser spectroscopy," Phys. Rev. A 25, 2550-2571 (1982).
[CrossRef]

Boulanger, J.-S.

J. Vanier, J.-F. Simard, and J.-S. Boulanger, "Relaxation and frequency shifts in the ground state of 85Rb," Phys. Rev. A 9, 1031-1040 (1974).
[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]

Brossel, J.

F. Hartmann, M. Rambosson, J. Brossel, and A. Kastler, "Action des gaz etrangers sur le taux dorientation de la vapeur de sodium, obtenu par pompage optique," C. R. Acad. Sci. 246, 1522-1525 (1958).

C. Cohen-Tannoudji, J. Brossel, and A. Kastler, "La conservation de la phase lors de la collision d'un atome de sodium orienté et d'un atome d'hélium," C. R. Acad. Sci. 245, 1027-1029 (1957).

Budker, D.

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153-1201 (2002).
[CrossRef]

D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and R. Zolotorev, "Sensitive magnetometry based on nonlinear magneto-optical rotation," Phys. Rev. A 62, 043403 (2000).
[CrossRef]

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, J. Brossel, and A. Kastler, "La conservation de la phase lors de la collision d'un atome de sodium orienté et d'un atome d'hélium," C. R. Acad. Sci. 245, 1027-1029 (1957).

Erhard , M.

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

Erhard, M.

M. Erhard, S. Nußmann, and H. Helm, "Power broadening and Doppler effects of coherent dark resonances in Rb," Phys. Rev. A 62, 061802 (R) (2000).
[CrossRef]

Fleischhauer, M.

M. Fleischhauer, A. B. Matsko, and M. O. Scully, "Quantum limit of optical magnetometry in the presence of ac Stark shifts," Phys. Rev. A 62, 013808 (2000).
[CrossRef]

Fry, E. S.

M. Graf, E. Arimondo, E. S. Fry, D. E. Nikonov, G. G. Padmabandu, M. O. Scully, and S.-Y. Zhu, "Doppler broadening and collisional relaxation effects in a lasing-without-inversion experiment," Phys. Rev. A 51, 4030-4037 (1995).
[CrossRef] [PubMed]

Gabbanini, C.

S. Gozzini, P. Sartini, C. Gabbanini, A. Lucchesini, C. Marinelli, L. Moi, J. H. Xu, and G. Alzetta, "Experimental study of velocity-changing collisions on coherent population trapping in sodium," Eur. Phys. J. D 6, 127-131 (1999).
[CrossRef]

Gallagher, A.

C. Ottinger, R. Scheps, G. W. York, and A. Gallagher, "Broadening of Rb resonance lines by noble gases," Phys. Rev. A 11, 1815-1828 (1975).
[CrossRef]

Gawlik, W.

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153-1201 (2002).
[CrossRef]

G. Wa¸sik, W. Gawlik, J. Zachorowski, and Z. Kowal, "Competition of dark states: Optical resonances with anomalous magnetic field dependence," Phys. Rev. A 64, 051802 (2001).
[CrossRef]

W. Gawlik, "Hyperfine interaction versus strong laser field - optical backgoudsmit (Paschen-Back) effect," Am. J. Phys. 59, 706-710 (1991).
[CrossRef]

Gawlik , W.

W. Gawlik and J. Zachorowski, "Forward scattering of a strong nonmonochromatic laser light," J. Phys. B 20, 5939-5952 (1987).
[CrossRef]

Gozzini, S.

S. Gozzini, P. Sartini, C. Gabbanini, A. Lucchesini, C. Marinelli, L. Moi, J. H. Xu, and G. Alzetta, "Experimental study of velocity-changing collisions on coherent population trapping in sodium," Eur. Phys. J. D 6, 127-131 (1999).
[CrossRef]

Graf, M.

M. Graf, E. Arimondo, E. S. Fry, D. E. Nikonov, G. G. Padmabandu, M. O. Scully, and S.-Y. Zhu, "Doppler broadening and collisional relaxation effects in a lasing-without-inversion experiment," Phys. Rev. A 51, 4030-4037 (1995).
[CrossRef] [PubMed]

Happer, W.

W. Happer, "Optical pumping," Rev. Mod. Phys. 44, 169-249 (1972).
[CrossRef]

Harris, S. E.

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

Hartmann, F.

F. Hartmann, M. Rambosson, J. Brossel, and A. Kastler, "Action des gaz etrangers sur le taux dorientation de la vapeur de sodium, obtenu par pompage optique," C. R. Acad. Sci. 246, 1522-1525 (1958).

Hartmann, S. R.

P. R. Berman, T. W. Mossberg, and S. R. Hartmann, "Collision kernels and laser spectroscopy," Phys. Rev. A 25, 2550-2571 (1982).
[CrossRef]

Helm, H.

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

M. Erhard, S. Nußmann, and H. Helm, "Power broadening and Doppler effects of coherent dark resonances in Rb," Phys. Rev. A 62, 061802 (R) (2000).
[CrossRef]

Hollberg, L.

A. V. Taichenachev, V. I. Yudin, M. Stahler, R. Wynands, J. Kitching, and L. Hollberg, "Nonlinear-resonance line shapes: Dependence on the transverse intensity distribution of a light beam," Phys. Rev. A 69, 024501 (2004).
[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]

Kastler, A.

F. Hartmann, M. Rambosson, J. Brossel, and A. Kastler, "Action des gaz etrangers sur le taux dorientation de la vapeur de sodium, obtenu par pompage optique," C. R. Acad. Sci. 246, 1522-1525 (1958).

C. Cohen-Tannoudji, J. Brossel, and A. Kastler, "La conservation de la phase lors de la collision d'un atome de sodium orienté et d'un atome d'hélium," C. R. Acad. Sci. 245, 1027-1029 (1957).

Kimball, D. F.

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153-1201 (2002).
[CrossRef]

D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and R. Zolotorev, "Sensitive magnetometry based on nonlinear magneto-optical rotation," Phys. Rev. A 62, 043403 (2000).
[CrossRef]

Kitching, J.

A. V. Taichenachev, V. I. Yudin, M. Stahler, R. Wynands, J. Kitching, and L. Hollberg, "Nonlinear-resonance line shapes: Dependence on the transverse intensity distribution of a light beam," Phys. Rev. A 69, 024501 (2004).
[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]

Kowal, Z.

G. Wa¸sik, W. Gawlik, J. Zachorowski, and Z. Kowal, "Competition of dark states: Optical resonances with anomalous magnetic field dependence," Phys. Rev. A 64, 051802 (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]

Lucchesini, A.

S. Gozzini, P. Sartini, C. Gabbanini, A. Lucchesini, C. Marinelli, L. Moi, J. H. Xu, and G. Alzetta, "Experimental study of velocity-changing collisions on coherent population trapping in sodium," Eur. Phys. J. D 6, 127-131 (1999).
[CrossRef]

Marangos, J. P.

J. P. Marangos, "Topical review: electromagnetically induced transparency," J. Mod. Opt. 45, 471-503 (1998).
[CrossRef]

Marinelli, C.

S. Gozzini, P. Sartini, C. Gabbanini, A. Lucchesini, C. Marinelli, L. Moi, J. H. Xu, and G. Alzetta, "Experimental study of velocity-changing collisions on coherent population trapping in sodium," Eur. Phys. J. D 6, 127-131 (1999).
[CrossRef]

Matsko, A. B.

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]

A. S. Zibrov and A. B. Matsko, "Optical Ramsey fringes induced by Zeeman coherence," Phys. Rev. A 65, 013814 (2002).
[CrossRef]

I. Novikova, A. B. Matsko, and G. R. Welch, "Detection of non-resonant impurity gases in alkali vapor cells," Appl. Phys. Lett. 81, 193-195 (2002).
[CrossRef]

A. S. Zibrov, I. Novikova, and A. B. Matsko, "Observation of Ramsey fringes in an atomic cell with buffer gas," Opt. Lett. 26, 1311-1313 (2001).
[CrossRef]

I. Novikova, A. B. Matsko, and G. R. Welch, "Large polarization rotation via atomic coherence," Opt. Lett. 26, 1016-1018 (2001).
[CrossRef]

M. Fleischhauer, A. B. Matsko, and M. O. Scully, "Quantum limit of optical magnetometry in the presence of ac Stark shifts," Phys. Rev. A 62, 013808 (2000).
[CrossRef]

Meschede, D.

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]

Mikhailov, E. E.

E. E. Mikhailov, I. Novikova, Y. V. Rostovtsev, and G. R. Welch, "Buffer-gas-induced absorption resonances in Rb vapor," Phys. Rev. A 70, 033806 (2004).
[CrossRef]

Moi, L.

S. Gozzini, P. Sartini, C. Gabbanini, A. Lucchesini, C. Marinelli, L. Moi, J. H. Xu, and G. Alzetta, "Experimental study of velocity-changing collisions on coherent population trapping in sodium," Eur. Phys. J. D 6, 127-131 (1999).
[CrossRef]

Mossberg, T. W.

P. R. Berman, T. W. Mossberg, and S. R. Hartmann, "Collision kernels and laser spectroscopy," Phys. Rev. A 25, 2550-2571 (1982).
[CrossRef]

Nagel, A.

R. Wynands and A. Nagel, "Precision spectroscopy with coherent dark states," Appl. Phys. B 68, 1-25 (1998).
[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]

Nikonov, D. E.

M. Graf, E. Arimondo, E. S. Fry, D. E. Nikonov, G. G. Padmabandu, M. O. Scully, and S.-Y. Zhu, "Doppler broadening and collisional relaxation effects in a lasing-without-inversion experiment," Phys. Rev. A 51, 4030-4037 (1995).
[CrossRef] [PubMed]

Novikova, I.

E. E. Mikhailov, I. Novikova, Y. V. Rostovtsev, and G. R. Welch, "Buffer-gas-induced absorption resonances in Rb vapor," Phys. Rev. A 70, 033806 (2004).
[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]

I. Novikova, A. B. Matsko, and G. R. Welch, "Detection of non-resonant impurity gases in alkali vapor cells," Appl. Phys. Lett. 81, 193-195 (2002).
[CrossRef]

A. S. Zibrov, I. Novikova, and A. B. Matsko, "Observation of Ramsey fringes in an atomic cell with buffer gas," Opt. Lett. 26, 1311-1313 (2001).
[CrossRef]

I. Novikova, A. B. Matsko, and G. R. Welch, "Large polarization rotation via atomic coherence," Opt. Lett. 26, 1016-1018 (2001).
[CrossRef]

Nußmann, S.

M. Erhard, S. Nußmann, and H. Helm, "Power broadening and Doppler effects of coherent dark resonances in Rb," Phys. Rev. A 62, 061802 (R) (2000).
[CrossRef]

Ottinger, C.

C. Ottinger, R. Scheps, G. W. York, and A. Gallagher, "Broadening of Rb resonance lines by noble gases," Phys. Rev. A 11, 1815-1828 (1975).
[CrossRef]

Padmabandu, G. G.

M. Graf, E. Arimondo, E. S. Fry, D. E. Nikonov, G. G. Padmabandu, M. O. Scully, and S.-Y. Zhu, "Doppler broadening and collisional relaxation effects in a lasing-without-inversion experiment," Phys. Rev. A 51, 4030-4037 (1995).
[CrossRef] [PubMed]

Privalov , T.

T. Privalov and A. Shalagin, "Exact solution of the one- and three-dimensional quantum kinetic equations with velocity-dependent collision rates: Comparative analysis," Phys. Rev. A 59, 4331-4339 (1999).
[CrossRef]

Rambosson, M.

F. Hartmann, M. Rambosson, J. Brossel, and A. Kastler, "Action des gaz etrangers sur le taux dorientation de la vapeur de sodium, obtenu par pompage optique," C. R. Acad. Sci. 246, 1522-1525 (1958).

Rautian, S. G.

S. G. Rautian, G. I. Smirnov, and A. M. Shalagin, "Polarization phenomena and nonlinear interference effects with allowance for collisions," Zh. Eksp. Teor. Fiz. 62, 2097-2110 (1972).

Rochester, S. M.

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153-1201 (2002).
[CrossRef]

D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and R. Zolotorev, "Sensitive magnetometry based on nonlinear magneto-optical rotation," Phys. Rev. A 62, 043403 (2000).
[CrossRef]

Rostovtsev, Y. V.

E. E. Mikhailov, I. Novikova, Y. V. Rostovtsev, and G. R. Welch, "Buffer-gas-induced absorption resonances in Rb vapor," Phys. Rev. A 70, 033806 (2004).
[CrossRef]

Sartini, P.

S. Gozzini, P. Sartini, C. Gabbanini, A. Lucchesini, C. Marinelli, L. Moi, J. H. Xu, and G. Alzetta, "Experimental study of velocity-changing collisions on coherent population trapping in sodium," Eur. Phys. J. D 6, 127-131 (1999).
[CrossRef]

Scheps, R.

C. Ottinger, R. Scheps, G. W. York, and A. Gallagher, "Broadening of Rb resonance lines by noble gases," Phys. Rev. A 11, 1815-1828 (1975).
[CrossRef]

Scully, M. 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]

M. Fleischhauer, A. B. Matsko, and M. O. Scully, "Quantum limit of optical magnetometry in the presence of ac Stark shifts," Phys. Rev. A 62, 013808 (2000).
[CrossRef]

M. Graf, E. Arimondo, E. S. Fry, D. E. Nikonov, G. G. Padmabandu, M. O. Scully, and S.-Y. Zhu, "Doppler broadening and collisional relaxation effects in a lasing-without-inversion experiment," Phys. Rev. A 51, 4030-4037 (1995).
[CrossRef] [PubMed]

Shalagin, A.

T. Privalov and A. Shalagin, "Exact solution of the one- and three-dimensional quantum kinetic equations with velocity-dependent collision rates: Comparative analysis," Phys. Rev. A 59, 4331-4339 (1999).
[CrossRef]

Shalagin, A. M.

S. G. Rautian, G. I. Smirnov, and A. M. Shalagin, "Polarization phenomena and nonlinear interference effects with allowance for collisions," Zh. Eksp. Teor. Fiz. 62, 2097-2110 (1972).

Simard, J.-F.

J. Vanier, J.-F. Simard, and J.-S. Boulanger, "Relaxation and frequency shifts in the ground state of 85Rb," Phys. Rev. A 9, 1031-1040 (1974).
[CrossRef]

Smirnov, G. I.

S. G. Rautian, G. I. Smirnov, and A. M. Shalagin, "Polarization phenomena and nonlinear interference effects with allowance for collisions," Zh. Eksp. Teor. Fiz. 62, 2097-2110 (1972).

Stahler, M.

A. V. Taichenachev, V. I. Yudin, M. Stahler, R. Wynands, J. Kitching, and L. Hollberg, "Nonlinear-resonance line shapes: Dependence on the transverse intensity distribution of a light beam," Phys. Rev. A 69, 024501 (2004).
[CrossRef]

Taichenachev, A. V.

A. V. Taichenachev, V. I. Yudin, M. Stahler, R. Wynands, J. Kitching, and L. Hollberg, "Nonlinear-resonance line shapes: Dependence on the transverse intensity distribution of a light beam," Phys. Rev. A 69, 024501 (2004).
[CrossRef]

Vanier, J.

J. Vanier, J.-F. Simard, and J.-S. Boulanger, "Relaxation and frequency shifts in the ground state of 85Rb," Phys. Rev. A 9, 1031-1040 (1974).
[CrossRef]

Wa¸sik, G.

G. Wa¸sik, W. Gawlik, J. Zachorowski, and Z. Kowal, "Competition of dark states: Optical resonances with anomalous magnetic field dependence," Phys. Rev. A 64, 051802 (2001).
[CrossRef]

Weis, A.

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153-1201 (2002).
[CrossRef]

Welch, G. R.

E. E. Mikhailov, I. Novikova, Y. V. Rostovtsev, and G. R. Welch, "Buffer-gas-induced absorption resonances in Rb vapor," Phys. Rev. A 70, 033806 (2004).
[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]

I. Novikova, A. B. Matsko, and G. R. Welch, "Detection of non-resonant impurity gases in alkali vapor cells," Appl. Phys. Lett. 81, 193-195 (2002).
[CrossRef]

I. Novikova, A. B. Matsko, and G. R. Welch, "Large polarization rotation via atomic coherence," Opt. Lett. 26, 1016-1018 (2001).
[CrossRef]

Wynands, R.

A. V. Taichenachev, V. I. Yudin, M. Stahler, R. Wynands, J. Kitching, and L. Hollberg, "Nonlinear-resonance line shapes: Dependence on the transverse intensity distribution of a light beam," Phys. Rev. A 69, 024501 (2004).
[CrossRef]

Wynands , R.

R. Wynands and A. Nagel, "Precision spectroscopy with coherent dark states," Appl. Phys. B 68, 1-25 (1998).
[CrossRef]

Wynands, R.

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]

Xu, J. H.

S. Gozzini, P. Sartini, C. Gabbanini, A. Lucchesini, C. Marinelli, L. Moi, J. H. Xu, and G. Alzetta, "Experimental study of velocity-changing collisions on coherent population trapping in sodium," Eur. Phys. J. D 6, 127-131 (1999).
[CrossRef]

Xu , J. H.

J. H. Xu and G. Alzetta, "High buffer-gas pressure perturbation of coherent population trapping in sodium vapors," Phys. Lett. A 248, 80-85 (1998).
[CrossRef]

Yashchuk, V. V.

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153-1201 (2002).
[CrossRef]

D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and R. Zolotorev, "Sensitive magnetometry based on nonlinear magneto-optical rotation," Phys. Rev. A 62, 043403 (2000).
[CrossRef]

York, G. W.

C. Ottinger, R. Scheps, G. W. York, and A. Gallagher, "Broadening of Rb resonance lines by noble gases," Phys. Rev. A 11, 1815-1828 (1975).
[CrossRef]

Yudin, V. I.

A. V. Taichenachev, V. I. Yudin, M. Stahler, R. Wynands, J. Kitching, and L. Hollberg, "Nonlinear-resonance line shapes: Dependence on the transverse intensity distribution of a light beam," Phys. Rev. A 69, 024501 (2004).
[CrossRef]

Zachorowski, J.

G. Wa¸sik, W. Gawlik, J. Zachorowski, and Z. Kowal, "Competition of dark states: Optical resonances with anomalous magnetic field dependence," Phys. Rev. A 64, 051802 (2001).
[CrossRef]

W. Gawlik and J. Zachorowski, "Forward scattering of a strong nonmonochromatic laser light," J. Phys. B 20, 5939-5952 (1987).
[CrossRef]

Zhu, S.-Y.

M. Graf, E. Arimondo, E. S. Fry, D. E. Nikonov, G. G. Padmabandu, M. O. Scully, and S.-Y. Zhu, "Doppler broadening and collisional relaxation effects in a lasing-without-inversion experiment," Phys. Rev. A 51, 4030-4037 (1995).
[CrossRef] [PubMed]

Zibrov , A. S.

A. S. Zibrov and A. B. Matsko, "Optical Ramsey fringes induced by Zeeman coherence," Phys. Rev. A 65, 013814 (2002).
[CrossRef]

Zibrov, A. S.

Zolotorev, R.

D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and R. Zolotorev, "Sensitive magnetometry based on nonlinear magneto-optical rotation," Phys. Rev. A 62, 043403 (2000).
[CrossRef]

Zubairy, M. S.

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]

Am. J. Phys. (1)

W. Gawlik, "Hyperfine interaction versus strong laser field - optical backgoudsmit (Paschen-Back) effect," Am. J. Phys. 59, 706-710 (1991).
[CrossRef]

Appl. Phys. B (1)

R. Wynands and A. Nagel, "Precision spectroscopy with coherent dark states," Appl. Phys. B 68, 1-25 (1998).
[CrossRef]

Appl. Phys. Lett. (1)

I. Novikova, A. B. Matsko, and G. R. Welch, "Detection of non-resonant impurity gases in alkali vapor cells," Appl. Phys. Lett. 81, 193-195 (2002).
[CrossRef]

C. R. Acad. Sci. (2)

C. Cohen-Tannoudji, J. Brossel, and A. Kastler, "La conservation de la phase lors de la collision d'un atome de sodium orienté et d'un atome d'hélium," C. R. Acad. Sci. 245, 1027-1029 (1957).

F. Hartmann, M. Rambosson, J. Brossel, and A. Kastler, "Action des gaz etrangers sur le taux dorientation de la vapeur de sodium, obtenu par pompage optique," C. R. Acad. Sci. 246, 1522-1525 (1958).

Eur. Phys. J. D (1)

S. Gozzini, P. Sartini, C. Gabbanini, A. Lucchesini, C. Marinelli, L. Moi, J. H. Xu, and G. Alzetta, "Experimental study of velocity-changing collisions on coherent population trapping in sodium," Eur. Phys. J. D 6, 127-131 (1999).
[CrossRef]

J. Mod. Opt. (1)

J. P. Marangos, "Topical review: electromagnetically induced transparency," J. Mod. Opt. 45, 471-503 (1998).
[CrossRef]

J. Phys. B (1)

W. Gawlik and J. Zachorowski, "Forward scattering of a strong nonmonochromatic laser light," J. Phys. B 20, 5939-5952 (1987).
[CrossRef]

Opt. Lett. (2)

Phys. Lett. A (1)

J. H. Xu and G. Alzetta, "High buffer-gas pressure perturbation of coherent population trapping in sodium vapors," Phys. Lett. A 248, 80-85 (1998).
[CrossRef]

Phys. Rev. A (17)

J. Vanier, J.-F. Simard, and J.-S. Boulanger, "Relaxation and frequency shifts in the ground state of 85Rb," Phys. Rev. A 9, 1031-1040 (1974).
[CrossRef]

M. Erhard, S. Nußmann, and H. Helm, "Power broadening and Doppler effects of coherent dark resonances in Rb," Phys. Rev. A 62, 061802 (R) (2000).
[CrossRef]

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

M. Graf, E. Arimondo, E. S. Fry, D. E. Nikonov, G. G. Padmabandu, M. O. Scully, and S.-Y. Zhu, "Doppler broadening and collisional relaxation effects in a lasing-without-inversion experiment," Phys. Rev. A 51, 4030-4037 (1995).
[CrossRef] [PubMed]

E. Arimondo, "Relaxation processes in coherent-population trapping," Phys. Rev. A 54, 2216-2223 (1996).
[CrossRef] [PubMed]

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]

C. Ottinger, R. Scheps, G. W. York, and A. Gallagher, "Broadening of Rb resonance lines by noble gases," Phys. Rev. A 11, 1815-1828 (1975).
[CrossRef]

P. R. Berman, T. W. Mossberg, and S. R. Hartmann, "Collision kernels and laser spectroscopy," Phys. Rev. A 25, 2550-2571 (1982).
[CrossRef]

T. Privalov and A. Shalagin, "Exact solution of the one- and three-dimensional quantum kinetic equations with velocity-dependent collision rates: Comparative analysis," Phys. Rev. A 59, 4331-4339 (1999).
[CrossRef]

D. Budker, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and R. Zolotorev, "Sensitive magnetometry based on nonlinear magneto-optical rotation," Phys. Rev. A 62, 043403 (2000).
[CrossRef]

E. E. Mikhailov, I. Novikova, Y. V. Rostovtsev, and G. R. Welch, "Buffer-gas-induced absorption resonances in Rb vapor," Phys. Rev. A 70, 033806 (2004).
[CrossRef]

A. V. Taichenachev, V. I. Yudin, M. Stahler, R. Wynands, J. Kitching, and L. Hollberg, "Nonlinear-resonance line shapes: Dependence on the transverse intensity distribution of a light beam," Phys. Rev. A 69, 024501 (2004).
[CrossRef]

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]

G. Wa¸sik, W. Gawlik, J. Zachorowski, and Z. Kowal, "Competition of dark states: Optical resonances with anomalous magnetic field dependence," Phys. Rev. A 64, 051802 (2001).
[CrossRef]

A. S. Zibrov and A. B. Matsko, "Optical Ramsey fringes induced by Zeeman coherence," Phys. Rev. A 65, 013814 (2002).
[CrossRef]

M. Fleischhauer, A. B. Matsko, and M. O. Scully, "Quantum limit of optical magnetometry in the presence of ac Stark shifts," Phys. Rev. A 62, 013808 (2000).
[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]

Phys. Today (1)

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

Rev. Mod. Phys. (2)

D. Budker, W. Gawlik, D. F. Kimball, S. M. Rochester, V. V. Yashchuk, and A. Weis, "Resonant nonlinear magneto-optical effects in atoms," Rev. Mod. Phys. 74, 1153-1201 (2002).
[CrossRef]

W. Happer, "Optical pumping," Rev. Mod. Phys. 44, 169-249 (1972).
[CrossRef]

Zh. Eksp. Teor. Fiz. (1)

S. G. Rautian, G. I. Smirnov, and A. M. Shalagin, "Polarization phenomena and nonlinear interference effects with allowance for collisions," Zh. Eksp. Teor. Fiz. 62, 2097-2110 (1972).

Other (4)

J. Vanier and C. Audoin, The Quantum Physics of Atomic Frequency Standards , Vol. 1 (Hilger, Bristol, UK, 1989).

E. Arimondo, "Coherent population trapping in laser spectroscopy," in Progress in Optics, Vol. XXXV , E. Wolf, ed. (Elsevier, Amsterdam, 1996).

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

A. Corney, Atomic and Laser Spectroscopy (Clarendon, Oxford, UK, 1977).

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Figures (11)

Fig. 1
Fig. 1

Simplified schematic of the experimental setup.

Fig. 2
Fig. 2

NMOR in  87Rb cell with 0.12 Torr of Kr buffer gas. (a) The transmission resonance contrast C, given by Eq. (4). (b) The rotation rate dϕ/dB(B=0). (c) The polarization-rotation angle ϕ at B=100 mG as functions of the laser frequency; zero detuning corresponds to the F=2F=1 transition; the atomic density is N=5.2×1011 cm-3.

Fig. 3
Fig. 3

Polarization rotation (left column) and transmitted power PEIT/P0 (right column) versus applied magnetic field for different laser detunings observed in the  87Rb cell with 0.12 Torr Kr buffer gas. Zero detuning corresponds to the light resonant with the F=2F=1 transition. Note the scale difference for each graph. The atomic density is N=5.2×1011 cm-3.

Fig. 4
Fig. 4

Dipole matrix elements for the F=1F=0, 1 transitions.

Fig. 5
Fig. 5

Polarization rotation rate dϕ/dB(B=0) for the D2 line of  87Rb in an atomic cell with (a) no buffer gas (the additional rotation peak at detuning near 1 GHz is due to a small admixture of  85Rb), (b) 0.12 Torr of Kr, (c) 1 Torr of Ne. The temperature is adjusted for each cell so that the total transmission on the F=2F transition is ≈80%; zero laser detuning is chosen to coincide with the center of the absorption peak of the F=2F transition.

Fig. 6
Fig. 6

NMOR in  87Rb cell with 10 Torr of Ne buffer gas (cf. Fig. 2). (a) The EIT contrast C, (b) the rotation rate dϕ/dB(B=0), (c) the polarization-rotation angle ϕ at B=100 mG as functions of the laser frequency; zero detuning corresponds to the F=2F=1 transition; the atomic density is N=2.1×1012 cm-3.

Fig. 7
Fig. 7

Polarization rotation (left column) and transmitted power PEIT/P0 (right column) observed in the  87Rb cell with 10 Torr of Ne buffer gas as a function of the applied magnetic field for different laser frequencies (cf. Fig. 3); zero detuning corresponds to resonance with the F=2F=1 transition; the atomic density is N=2.1×1012 cm-3.

Fig. 8
Fig. 8

(a) Polarization-rotation angle as a function of magnetic field. Inset, central narrow resonance; the dotted curve is a dispersive Lorentzian fit. (b) The transmission of the laser field through the cell. Inset, central narrow absorption resonance, corresponding to the narrow rotation feature.

Fig. 9
Fig. 9

(a) Width of the rotation resonance and (b) the maximum polarization rotation angle for both narrow and wide structures. The dashed and dotted curves are to guide the eyes, except for the linear fit for the wide resonance’s width.

Fig. 10
Fig. 10

(a) Simplified double-Λ level scheme that was the basis for the analytical calculations. (b) Level scheme as basis for the numerical simulations.

Fig. 11
Fig. 11

Absorption and polarization rotation of linearly polarized light interacting with F=2F=1, 2 transition of D1 line of  87Rb in a vacuum atomic cell for different values of coherence decay rate γ0. The decay rates are (from top to bottom) γ0=4×10-4 γr, γ0=10-4 γr and γ0=10-5 γr; Rabi frequency is equal to γr; Doppler broadening is equal to 100 γr.

Equations (52)

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

γ0=2.4052Dr211+6.8l/r+2σdphv¯rnb.
ϕ=1/2 arcsinS1-S2S1+S2.
ϕ=-2μBBγ0 lnPoutPin
C=PEIT-P0P0,
|d10=Ω+|++Ω-|-|Ω+|2+|Ω-|2=|++|-2,
|d11=Ω+|+-Ω-|-|Ω+|2+|Ω-|2=|+-|-2,
ϕ(B)=ϕmaxwBw2+B2,
Ω1b=Ω exp(iϕ),Ω1c=Ω exp(-iϕ),
Ω2b=-Ωγr2γr11/2 exp(iϕ),
Ω2c=Ωγr2γr11/2 exp(-iϕ).
H0=Δ|a1a1|+(Δa+Δ)|a2a2|+δ(|cc|-|bb|),
Vint=Ω1b|a1b|+Ω1c|a1c|+Ω2b|a2b|+Ω2c|a2c|+H.c.,
ρˆ˙=-i[H, ρˆ]-12{Γ, ρˆ}.
Ecz=i2πνcN1ρ1c+2ρ2cv,
Ebz=i2πνcN1ρ1b-2ρ2bv,
-i(Ec*Eb-Eb*Ec)2ϕ(|Ec|2+|Eb|2)forϕ1:
Iz=iνN×Ω1c*ρ1c+Ω2c*ρ2c+Ω1b*ρ1b+Ω2b*ρ2bv+c.c.,
(Iϕ)z=1/2νN×Ω1c*ρ1b-Ω2c*ρ2b-Ω1b*ρ1c+Ω2b*ρ2cv+c.c.
|d1=(1/2)(|b+|c),
|d2=(1/2)(|c-|b).
H˜0=Δ|a1a1|+(Δa+Δ)|a2a2|,
-V˜int=Ω11|a1d1|+Ω22|a2d2|+Ω12|a1d2|+Ω21|a2d1|-δ|d1d2|+H.c.,
Ω112Ω,Ω222γr2γr11/2Ω,
Ω12-iϕ2Ω,Ω21-iϕ2γr2γr11/2Ω.
Iz=iνNΩ11*ρa1,d1+Ω22*ρa2,d2+Ω12*ρa1,d2+Ω21*ρa2,d1v+c.c.,
(Iϕ)z=-1/2 νN×Ω11*ρa1,d2+Ω22*ρa2,d1-Ω12*ρa1,d1-Ω21*ρa2,d2v+c.c.
IziνNΩ11*ρa1,d1+Ω22*ρa2,d2v+c.c.
Iϕz=-1/2 νN×Ω11*ρa1,d2+Ω22*ρa2,d1+Ω12*ρa1,d1+Ω21*ρa2,d2v+c.c.
ρak,dkiΩkkρdk,dk-ρak,akγrk+i[Δ+(k-1)Δa],
ρak,ak=Sk1+Skρdk,dk,
Sk=|Ωkk|2γrk2+(Δ+(k-1)Δa)2.
Iz-38πNλ2Iγr12γr12+Δ2ρd1,d11+S1+γr22γr22+(Δ+Δa)2ρd2,d21+S2v.
ρd1,d1=12γ0+2γr2S2/(S2+1)γ0+γr1S1S1+1+γr2S2S2+1+(γr1+γr2)S1S2(S1+1)(S2+1) ,
ρd2,d2=12γ0+2γr1S1/(S1+1)γ0+γr1S1S1+1+γr2S2S2+1+(γr1+γr2)S1S2(S1+1)(S2+1) .
Iz
-38πNλ2I
×8π3νγ0Iλ2γr12(Δ+Δa)2-γr22Δ22γr12γr22+γr12(Δ+Δa)2+γr22Δ22+2γr12γr222γr12γr22+γr12(Δ+Δa)2+γr22Δ2v.
Iϕnlz=-νNδ(ρd2,d2-ρd1,d1)×γr12(Δ+Δa)2-γr22Δ24γr12γr22+[γr1(Δ+Δa)-γr2Δ]2v.
Iϕnlz-νNδ[γr12(Δ+Δa)2-γr22Δ2]2{4γr12γr22+[γr1(Δ+Δa)-γr2Δ]2}[2γr12γr22+γr12(Δ+Δa)2+γr22Δ2]v.
ρcb-12γr12(Δ+Δa)2-γr22Δ2γr12(Δ+Δa)2+γr22Δ2.
Δ0=-Δaγr1γr1+γr2.
Iz-νγ0Nρbc2v.
ρcbc=-Wc2π-γr12(Δ+Δa+Δκ)2-γr22(Δ+Δκ)2γr12(Δ+Δa+Δκ)2+γr22(Δ+Δκ)2×dΔκWc2+Δκ2=-12γr12(Δ+Δa)2-γr22Δ2+Wc2(γr12-γr22)[γr1(Δ+Δa)+Wcγr2]2+(γr2Δ-Wcγr1)2.
Δ˜0=-Δaγr1γr1+γr2-Wc2(γr12-γr22)2Δaγr1γr2.
Iϕnlz
-4νNδ×ρcbρcb2γr12γr22+γr12(Δ+Δa)2+γr22Δ24γr12γr22+[γr1(Δ+Δa)-γr2Δ]2v,
ρcbρcb1-2γr2γr1WcΔa<1.
Iz-ζ(Δ)νγ0Nρcbρcbv,
ρd2,d2-ρd1,d1=[ρd2,d2(0)-ρd1,d1(0)]cos 2δt-i[ρd1,d2(0)-ρd2,d1(0)]sin 2δt,
ρd1,d2-ρd2,d1=[ρd1,d2(0)-ρd1,d2(0)]cos 2δt-i[ρd2,d2(0)-ρd1,d1(0)]sin 2δt.
ρd1,d2-i2(ρd2,d2-ρd1,d1)sin 2δτR.
Iϕnlz-4νNδρcbρcb+ξsin 2δτR2δτRρcb1×2γr12γr22+γr12(Δ+Δa)2+γr22Δ24γr12γr22+[γr1(Δ+Δa)-γr2Δ]2v,

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